IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS THE POLYNOMIALS [ n.. Knots–FIGURES ] EDGE - POINTS VIBRATION & M-GEOMETRY BY Markos Georgallides 1Larnaca–Cyprus (Expelled from Varosha-Famagusta town occupied by the Barbaric Turks , in Aug - 1974) Cyprus , Civil - Structural O Engineer (NATUA) , Athens . Abstract The Interactions : One of the most important concept in Geometry is , distance , which is the Quanta in E- geometry , while in Material-Geometry the composition of Opposite , where the Material - Point is the Quanta in Chemistry and Physics . As in Algebra Zero ,0, is the Master-key number for all Positive and Negative numbers and this because their sum and multiplication becomes zero, and the same on any coordinate-System where ± axes pass from zero . In PNS Space , The Rolling of the Positive ⊕ constituent on the Negative ⊝ constituent , creates the Neutral Material Point which Equilibrium . Angular-Momentum is identical with Spin and consists the First-Discrete-Energy-Monad which occupies , Discrete Value and Direction , in contradiction to the Point which is Nothing , Dimensionless and without any Direction . Quaternions [(+)↻↺(-)] ≡ Box 𝐁 𝐑 =𝐀𝐁 carries the Principal stress 𝛔 𝐀 , 𝛔 𝐁 between Points A(+) , B(-) which σ , as Centripetal-acceleration is the minimum Energy becoming from the in-storage AB acceleration and is equal to the Gravity Force g . [108 – 110] . Because of the Revolving and Periodic acceleration of Gravity g ≡  σ exists as the First Energy-Box-𝐁 𝐑 , while in the Second Box 𝐁𝐏 is followed the Local-Extreme-case where Gravity g ≡  σ , and is altered Locally by changing the Principal-stress σ with an Local-uniform-Pressure → 𝐠𝐋 ≡ g k = g . [ Force/Area ] = G ← i.e. The minimum Local - Energy acceleration is the known , Universal Gravitational-constant G = g k = 𝐤𝐄 g = 𝐤𝐋 σ , such for Macrocosm and for Microcosm , Obeying the Newton`s Laws of motion . G ⏊ σ This Energy in Hydrogen-Cave as E-M , Conductor ≡ Edge Points Vibration ≡ The Pin of Atom → Plug Into their Sockets , which are the Orbit – Bracket–Hooks ≡ The Hands of Atoms ← i.e. The Atoms Plug with their Pins into the other Atoms-Drains = Holes , and so Bond and carry Informations . 1 https://doi.org/10.5281/zenodo.16737369 mailto:georgallides.marcos@cytanet.com.cy 2 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . This Resonance frequency of Hydrogen is Common to all Atoms and to all Compounds in this Cosmos . The Energy-Quaternion w ̅̅ ̅, B̅, Monad-Magnitudes exist as DUAL- Nature for Any { ⊕ , ⊝ }, { Position , Motion },{ Universe , Black-Holes } , { Gravity ,Antigravity} , {Action → . ← Reaction} , Edge Points Vibration creating Electric = [⊕] and Magnetic = [⊝] Forces as [⊕↔⊝] ,The Light and others .The STPL- Line Conductors on the [STPL]- Mechanism , are the Physical-Rotors for the Origination of the Cosmic –Particles which transfer Informations as the Signals - Spectrum .[110] From Mechanics-Physics , all Systems Possessing Elasticity ≡ motion and Reaction to the motion , the called mass , are capable of free vibration or vibration ≡ Periodic - motion taking Place in the Absence of External Excitation .This Principle issues for Both Systems Closed {The Atoms Nucleus} or Open Systems {The Orbitals}. For instance , In order that shifting of an u-d-Quarks from an Anti-Proton into a Proton , the Spin-Pair requires extra input of Energy in (MeV) , so that would the Proton Paired with a Neutron and be Stable. transfer Informations as the Signals . Vibration in a cave ∆ , means the Wave Pattern . Cave–Spin-�̅� of PNS Space is �̅� = r m v and is the first Monad occupying 4-Spaces , i.e. From Number n , of the Equilibrium number of masses m 𝐧 in a System. a.. The n - Spaces of Monad �̅� are the Polygons 𝐒 𝐧 with n = 1 ≈ ∞ Knots , b.. The n Anti-Spaces of Monad - �̅� are the Polygons - 𝐒 𝐧 with n =1≈ ∞ Knots , c.. Sub-Spaces of Monad ≡ ± √ �̅� 𝒏=𝟏−∞ are the Polygons with n = 1 ≈ ∞ Knots All n-Regular Polygons End to equations of n-degree Segment , by finding a suitable value of the Segment , x , That is we have in the general case to solve one or two equations of the form : A .R0. xn - B .R2. xn−2 + C .Rn−6. x³ – D .Rn−4. x² + E .Rn−2.x1– F. Rn. x0 = 0 for The Even Polygons , and A .R 2. xn−2 - B.Rn−2. xn−3 + C.R2(n−4). x³ -D.R2(n−3). x² +E.R2(n−2). x1– F.R2(n−1).x0 = 0 for The Odd Polygons , where A , B , C , D are constants . The Presented Geometrical method is the solution of the above equation in the general case . Because , the nth - degree - equations are → the Vertices (n) and the Sides (𝐚𝐧= 𝛌𝐧) of the n-Polygon in circle ← number , π , is their common Mould . [ 62 ] . The Natural Mechanism Continuously Originates the Elementary Particles and Compounds , Atoms and Molecules with the One Action from Opposite . Article [111] encloses many Paragraphs of [110] , in order to Distinguish and Prove the Way of Energy and the Stresses- Paths in the Spaces . For the Regular Polygons is given such the Geometrical Solution as well the Algebraic . [106] = Programming the Atoms and Compounds , is an Program which solves the Problem of the n-Knots Figures and gives the Energy Spectrum of any Complex-Forced-Vector . Article [114] an Way of Deceptioning the Cells is prepared . Keywords : The Figure n-Knots , The Polynomials n-Knots , Spaces and Figure n-Knots , The Waves of Spaces . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 3 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . CONTENT ABSTRACT Page ….................1 , A… THE STRUCTURE OF THE EUCLIDEAN GEOMETRY 1a.. : The Points , Vectors , Lines , Planes , Volumes , n-Spces Page ….................4 , 2a.. : The Fundamental Principles of E-Geometry Page ……….4.Fig-1 , 3a.. : The Quantization of the Energy – Space – Universe Page …………....7 , 4a.. : The Correlation of Spaces and The Primary motion Page ………11.Fig-2 , 5a.. : The Figure-eight Knot & The Figure n = 1 ≈ ∞ Knots Page ………13.Fig-3 , 6a.. : The Origination of The Primary motion in Spaces Page …… ..21.Fig-4 , 7a.. : The Mechanism and Charges for the DUAL Originations Page …...............23 , 8a.. : The Material Geometry E-Geometry Physicks and Chemistry Page ……....24.Fig-5 , [a].. The { S TP L } Natural Mechanisms Page …..26.Fig-6,7.7a,7b,7c , [b].. The { TVPM } Natural Mechanisms Page ... 32.Fig-8a , 8b , [c].. The { TC I C } Natural Mechanisms Page ..35.Fig-9,9a,b,9c,9d,9e , [d].. The { TOBM } Natural Mechanisms Page ......... 43.Fig-10 , 9a.. : The Flow Plan of the Space - Energy Universe Page ……..45.Fig-11 , 10a..:The E-Spaces and the Energy Regular-Polygons Page.49.Fig-12a,12b,12c,12d , B… : GENERAL : 1b.. : The Spaces and the Energy States Page …... 54.Fig-12- , 2b.. : The Complex Numbers and Quaternion. Page …............. 56 , 3b.. : The Stability of Space - Anti-Space , IN the Neutral-Space Page …….59.Fig-13 , 4b.. : The Quantum Interference of the Duality-Photon Page …….59.Fig-14 , C.. : THE ORIGINATION OF THE PHYSICAL LOOPS ≡ CAVES 1c.. : The Gravitation Force G , and the Kick-Start Page ….............. 62 , 2c.. : The United-Coulomb-Newton Law for Interactions Page ….............. 69 , 3c... : The Origination of Photon`s Light velocity c Page ……..........69 , 4c... : The Origination of Electron-charge e̅ ≡ q̅ Page ……..........72 , 5c... : The Origination of the Hydrogen Cave H & Explanations Page ……..........73 , 6c.: The Epilogues of Prior Page…78.Fig-17a,17b,17c,17d,17e , D... THE ORIGINATION OF THE FUNDAMENTAL PARTICLES 1d… : The Balancing of the Space and Anti-Space Page ……..99.Fig-18- 2d… : The Vibration of Particles in all Levels Page ……100.Fig-19- 5d… : The Natural Electromagnetic E-Tetrahedron , Cube , Spheres Page …..102.Fig20.b- 8d.. : The Stability of Fundamental Particles Conductors & Atoms Page ……108.Fig26 - 9d.. : The Origination of Proton-Neutron Funta-Particles , Objectivity Page ……110.Fig28 - E.. : REFERENCES Page…….……113 , IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 4 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . A… THE STRUCTURE OF THE EUCLIDEAN GEOMETRY . [6 -12 ] a1.. Point is nothing , has not any Position and Dimension , and may be anywhere in the known Space therefore , the Primary point A , being nothing also in no Space , it is the Only - Point and no-where ( i.e. Primary Point is the Only Space and from this all the others ) . [ 6 ] a2.. Straight line is 0 , Positive , Negative , consisted of ± ∞ Points which are nothing , and since it is composed of infinite Points which are filling the line , then nature of line is that of Point i.e. ( the all is one for Lines and for Points ). a3.. Plane is Positive , Negative , ± Neutral lines and since Points are Imaginary lines it is composed of infinite Straight lines which are filling Plane , so then , nature of Plane is that of Line and that of Points ( i.e. the all is one for Planes , Lines and Points ) . a4.. Space is + Positive , - Negative , ± Neutral Planes and since Points are Imaginary Planes it is composed of infinite Planes which are filling Space , so then , nature of Space is that of Plane and that of Points (i.e. the all is one for Spaces , Planes , Lines and Points) . 2a… THE FUNDAMENTAL PRINCIPLES OF E-GEOMETRY . 2a1.. The first dimensional Unit AB is of two Points A , B not coinciding which is the geometrical Shape that has as Position the Point of direction AB⃗⃗⃗⃗ ⃗ , BA⃖⃗⃗⃗⃗⃗ , and as Magnitude (the length |AB| = 0 → N → ∞ ) . |AB| is a straight line through Points A , B . ( F.1b) , [ 6 ] ↓ A C B A C B CA + CB = AB , where , ▫ ▫ ▫ —▫—— ▫ ————▫— F.1(a) F.1(b) ds = An Infinitely small increment of length AB in the direction AB⃗⃗⃗⃗ ⃗ or BA⃖⃗⃗⃗⃗⃗ . ∞ = An Infinitely great magnitude |AB| in the directions AB⃗⃗⃗⃗ ⃗ , BA⃖⃗⃗⃗⃗⃗ ) . Any Point C is on Straight line AB , only when exists the equation CA + CB = AB , i.e. The Whole AB is equal to the Parts CA and CB . ( as equation F 1.b ) . In case that CA + CB > AB then Point C is not on line AB , and this is the main difference between Euclidean and Non-Euclidean geometries . In Definition 2 ( a line AB is breathless length) is altered as for any Point C on line AB exists CA + CB =AB. Edge Points A , B not coinciding on Monad AB , keep the Properties of Complex numbers with Imaginary Part which differs between the infinite Point [11] – [13] Unit AB creates Spaces with two basic Elements , The Position ( which are the three directions AB⃗⃗⃗⃗ ⃗ , BA⃖⃗⃗⃗⃗⃗ , AB⃖⃗⃗⃗ ⃗ ) and the Dimension ( which is the Magnitude N such that N = 0 → |AB| → ∞) , and when exists , A = B = The Principle of the Equality . A ≠ B = The Principle of the Inequality . A→↔← B = ∞ The Principle of Virtual Displacements . W = ∫ P.ds = 0 , ds = ∂W / ∂P IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 5 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . PA⃗⃗⃗⃗ ⃗ + PB⃖⃗⃗⃗⃗⃗ = 0 The Principle of Stability PA⃗⃗⃗⃗ ⃗ + PB⃖⃗⃗⃗⃗⃗ > < 0 The Principle of Motion . 2a2.. The Monads = Quantum = 𝐝𝐬⃗⃗⃗⃗ = AB / ( n = ∞ → 0 ) = [ a ± b i ] = 0 → ∞ , are simultaneously ( Actual Infinity ) and also ( Potential Infinity ) in Complex number form and this defines , infinity which exists between all Points which are not coinciding and where ( ds > 0 ) , and because ds comprises any two edge Points with Imaginary Part then this Property differs between the infinite Points. The Plank length is a Monad ds = 1,62 x 𝟏𝟎−𝟑𝟓 m , for the two Points A , B , and for the moment is accepted as the smallest possible size . Monad is also infinitely divided because edge Points A , B are not coinciding i.e. ... ds = 1x𝟏𝟎−𝑵 < ∞ , where N is any number . 1.. Spaces of Unit-AB are (in Plane) the Infinite (+) Circumscribed-Regular-Polygons on the circles with AB as Side (repetition of Unit AB) the Nth Space , the Nth Unit Tensor of the N equal finite Elements ds , and for the ∞ Spaces , the line AB ↔ . ( Fig-1 ) The diameter of this circles extends to infinity ( it is of Potential nature ) . [11] 2.. Anti - Spaces of Unit AB are ( in the three dimensional Space ) the Symmetrically Infinite (-) Circumscribed- Regular-Polygons on the circles or ( a Solid in Cube in Sphere with AB as side of the Solid ) . The Harmonic Repetition of Unit BA , symmetrical to AB ) is the Nth Anti-Space , the Nth Unit Tensor of the N equal finite Anti - Elements , and for the ∞ Spaces , the Sphere through line BA .The diameter of this Spheres extends to the infinity ( it is of Potential nature ) . [11] 3.. Sub- Spaces of Unit AB are ( in Plane) the Infinite Inscribed-Regular-Polygons in the circle with AB as diameter ( are the Harmonic Repetition of the Roots in Unit AB ) and in Nth Sub-Space , the Nth Unit Tensor of the N finite Roots and in case of the ∞ Elements are the Points on the circle , and for 3D-Space , the Points on Sphere N.AB ) . The Superposition of Spaces , Anti-Spaces and Sub-Space Layers of Unit AB is shown in Figure . Remark : (+) Spaces , (-) Anti -Spaces , ( ± ) Sub-Spaces , of a Unit AB are between magnitude ( Point = 0 = Nothing ) and the Infinite magnitude ( ↔ = AB = ± ∞ = Infinite ) which means that All Spaces exist in One Space only . Because in Spaces and Anti-Spaces , the ∞ Spaces of Unit AB is Sector AB ↔ , and in Sub-Spaces the ∞ Sub-Spaces of Unit AB are the Points on the circle with AB as diameter , then this ordered continuum for Points on the circle of Unit AB and on line AB shows the correlation of Spaces in Unit AB . i.e. Monads ds = 0 → ∞ are Simultaneously, Actual Infinity (because for n = ∞ then ds = [ AB / n = ∞ ] = 0 , or A Point ) and , Potential Infinity , ( because for n = 0 then ds = [ AB / n = 0 ] = ∞ is the straight line through AB . Infinity exists between all Points IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 6 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . which are not coinciding , and because Monads ds comprises any two Edge-Points with Imaginary Part , then this Property is the difference between the infinite Points . [11] 4.. Neutral-Spaces of Unit-AB are ( in Plane) the Infinite Circles with AB as diameter ( the Harmonic Repetition of the Roots in Unit - AB ) and in Nth Sub-Space , the Nth Unit Tensor of the N finite Roots , and in case of the ∞ Elements are the Points on the circle , and for the 3D-Space , the Points on the Unit - Sphere AB ) . Since Spaces , Anti – Spaces and Sub-Spaces are created from Unit AB , and are Property of this Unit only , therefore these are a Restrained System ( S ) Presupposition for Unit |AB| = ds (displacement ds) is Point A to move at the New Position B (A ≡ B) which means an IMPULSE ( P ) transfers Point A to B . Since in each Restrained System ( S ) the Work done ( W) by Impulse ( P ) on a Virtual displacement ( ds > 0 ) is zero , or W = ∫ 𝑃𝑑𝑠 𝐵 𝐴 = 0 Therefore , Each Unit |AB| = |ds| > 0 exists , by this Inner Impulse ( P ) which means that , The Position and Dimension of all Points which are connected across the Universe and that of Spaces , exist because of this Static Inner Impulse P between A and B , on the contrary , should be one Point only ( Primary-Point = Black-Hole → ds = 0 → P = ∞) Impulse P = ∞ , and may be Vacuum , Momentum or Potential or Induced Potential and it is a Type of Effect of Push nature . 2a3..The Superposition of Plane Space , Anti-Space Layers and Sub-Space Layers The simultaneously co-existence of Spaces , Anti-Spaces and Sub-Spaces of any Unit |AB| , Unit AB = 0 → ∞ , ( A ≡ B ) i.e. , Euclidean , Elliptic , Spherical , Parabolic , Hyperbolic, Geodesics , Metric and Non-metric geometries , exist in Euclidean Model as an Sub-case within . The Interconnection of Homogeneous and Heterogeneous bounded Spaces Anti-Spaces and Subspaces of the Universe . The Unity of Opposites is also the Quantized Property of Euclidean geometry < all is one > as it is Discrete ( for Monads AB ) and Continuous (for Points A , B ) . For Primary Point A it is the only Space [17] The Unique Case where P = 0 and Δs > 0 = AB → ∞ is the Primary Planck`s Space [ PS ] The Spherical connection of the same Points in PS , is related on the influence ( P = 0 → AB⃗⃗⃗⃗ ⃗ → ∞) of the infinite ( ∞ ) Equilibrium Primary-Spaces and Anti-Spaces . For every Point in [ PS ] exist the three Spatial dimensions ( x , y , z ) and the infinite dimensions of the ( i ) Layers at this Point existing from the other Layers ( at small scale using coordinate system ) of Primary Anti-Space and Sub-Space where i = 1 → ∞ . It has been also shown that , Every Point M of Primary Planck`s Space [ PS ] exists in this Space with a deficit of Impulse P = 0 → ∞ , due to the influence of the other Equilibrium Primary Spaces . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 7 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Unique Case where P = 0 and Δs = 0 → 0 is the Primary Neutral Space [ PNS ] The ⊕ Material Point ≡ The Space , continually Rotates around The ⊝ Material Point ≡ The Anti-Space , on 2R circle . [PNS] Space is a This Property of Points in Neutral Space is the cause of their Continuously Survive . The Quantization of Points becomes through Vector Unit d�̆� = |AB| , and for Energy is done in bound States ( loops ) which withhold the diffusion ( flow ) .The two fundamental dimensions (Quanta of Points (ds) and Quanta of Energy (d P) , are connected on| AB|, So at any Point M exist the Centripetal force of two Opposite [⊕↻↺⊝] .All Points of [PNS] are Homogenous ( i.e. all Points of PNS Space are Equivalent ≡ ) and Isotropic ( i.e. all directions , ⇉ , of AB are Equivalent ) → [ A ≡ A ] , For the Primary dipole and for Any dipole Ai-Bi issues , The Position of Dipole in the equilibrium Space Anti- Space creates charge = momentum and Angular momentum ( Spin = the intrinsic twist of Space , Anti-Space ) which inextricably unify Geometry of Space and motion . [10 ] Figure – 1 . The 4-Primary-Spaces of Euclidean-Geometry and the Primary Rotational Quantized motion ≡ The Unit-Angular-Momentum-Energy in PNS Spaces . 3a… THE QUANTIZATION OF THE → { ENERGY – SPACE UNIVERSE } ← In a Conservative System the Total Energy ≡ Motion is constant and the Differential equation of motion can be established by the Principle of Conservation of Energy . For the free vibration of an Undamped System , the Energy is Party Kinetic = 𝐄 𝐊 which is stored in IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 8 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . the mass by virtue of its velocity and Party Potential = 𝐄 𝐏 which is stored in the form of Strain energy in elastic deformation , or the Work done in a Forced-Field such as gravity . Newton`s second law is the first basis for examining the motion of a Free – Vibration in a System . When the Deformation of a Spring in the Static - Equilibrium Position is Δ and the Spring force k.Δ equal to the gravitation al force mg , then issues k.Δ = mg …. (a) Applying Newton`s second law to the mass m , or mass can be replaced by a mass Moment of Inertia denoting the Resistance to motion Δ , and then m �̈� = - k x for an x – x axis . The Natural – Period of the oscillation is established from relation wn T = 2 π where issues [7] T = 2 π √ m 𝐤 , 𝐟 𝐧 = 1 / T = 1 2π √ k 𝐦 , 𝐟 𝐧 = 1 / T = 1 2π √ g Δ ….…(b) i.e. The Kinetic Energy 𝐄 𝐊 is applicable in a Single - DOF System including Rotation . Since Issues E K = E P so and E K + E P = constant . From the theory of Vibrations , { All Systems possessing mass= m and Elasticity = E = 1 / k are capable of free vibration in the absence of external excitation which is its Natural frequency of vibration = 𝐰 𝐧 and the equation of motion is → �̈� + 𝐰 ²𝐧 x = 0 ←where w ²n= k m , Gravitational force G = m g } i.e. Since the Hydrogen-cave System H , as E-M Conductor , Possesses mass 𝐦 𝐇 , and light-velocity c , then vibrates IN – OUT the Space-Positions (x , y , z) of all Universe with its Natural frequency 𝐰𝐇 =√ k 𝐦 𝐇 . or and frequency 𝐟 𝐇 = 𝐰 𝐇 𝟐𝛑 , and is a Wave – function providing the complete Description of the Physical Reality . From , Work = W = Force - 𝐅 𝐢,𝐣,𝐤 x Displacement 𝐝𝐬 𝐢,𝐣,𝐤 = ( 𝐅 𝐢,𝐣,𝐤 x 𝐝𝐬 𝐢,𝐣,𝐤 ) = ( F x ds ) , and for the Unit of time dt , → W / dt = F x ( ds / dt ) = ( F x �⃡� i , j , k ) = Power = P . i.e. Power ≡ Force x Velocity or ( P = F x �⃡� ) Remarks , from Mechanics - Physics : 1…Energy E = 1 2 m H . c ² , or → 2E = 𝐦 𝐇 . c ² ← and Velocity = The light-velocity c , 2…Wave length λ H = 𝟐𝛑 𝐜 𝐰 𝐇 , and Wave Amplitude 𝐀 𝐇 = λ H 𝟐𝛑 , 3…Hydrogen Power 𝐏 𝐇 = 0,5. 𝐦 𝐇 . [ 𝐀 𝐇 ] ² . [ c ] ² 4…The Hydrogen Stress-Vector 𝛔 𝐇 is related to c as , c = 𝚽 . 𝛔 𝐇 5…Since Hydrogen cave is an E-M Conductor then it is a Wave with Natural frequency , and since occupies the Property of →Atoms-Plug ≡ Pin and Atoms-Drain = Hole so are Bonded ← in a multi-DOF System at the Stationary Conditions . 6…To find the Eigenvalues and Eigenvectors of a multi-DOF System , Iteration method of motion is formulated by either the Flexibility Matrix [a] or the Stiffness Matrix. 7…Momentum ≡ ± SPIN ≡ [Gravity and Antigravity] because the light velocity �̅� = [ 𝐆 𝚽 𝐀 ] , IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 9 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . and Precedes that of A-Momentum  �̅� = r m �̅� . Exists Vector B ⊥ r-c Plane . 8…Since Angular-velocity |w̅| , a vector , Rotates in one axis k with respect to one variable only and this happens IN {PNS-Space ,Atoms-Space ,Objective-Reality and in Universe} then Quaternion - Momentum , B̅ & w̅ , and W ≡ Motion are Stabilized in the DUAL – TYPE , such in Microcosm , as in Macrocosm , or Both as Omnipresent . 1…The DUAL-Momentum B̅ & w̅, as Real ≡ Existing-Universe , Imaginary≡ Black-Holes 2…The DUAL - Photon , as Real ≡ Space ≡ Particle and, Imaginary ≡{ E M - Wave }. 3…The DUAL - Spin B̅ , as Real ≡ Space ≡ Gravity and, Imaginary ≡ { Anti - Gravity }. 4…The DUAL – Matter B̅ , as Real ≡ Space ≡ Atoms and, Imaginary ≡{ E M -Wave }. 5…The DUAL-Refraction of Energy as Real ≡ The Space angle φ , Imaginary ≡ �̅� 1, �̅� 2 { The Energy ≡ W from the different velocities , �̅� 1, �̅� 2 }. The Results in Hydrogen - cave : 1…Hydrogen-Cave , IN – OUT Universe occupies mass 𝐦 𝐇 , velocity �⃡� , and Power 𝐏 𝐇 2…Electron in Hydrogen-cave Precesses and Nutates due to the Gravitational constant G. The produced Work is stored in form of→ Stress Energy as Hydrogen-Bracket-Hook← The Electron Precesses from the continuous and immense-communication to gravity, g . Electron-Spin is the Angular-momentum-vector �̅� and rotates according to equation dB dt 3…The Stationary → Tetrahedron , In-Sphere , Cube , Ex-Sphere ←construction of Atoms Permits the Space – coordinate - Structure of Atoms , The Wave-eigenfunctions of many non-commuting Physical operators as Momentum Power, from the Quantum-Mechanical description of the Physical-Reality is Complete i.e. IF-known the Physical Operators , their coordinates are simultaneous Physical reality. 4…The Interactions of Two or more Systems with known Status can be calculated any time by the Bioelectronic-Spectrum of →{ Carrier-Modulating-Modulated , Demodulation Process Mechanism } ← using Markos Program < Programming Atoms-Bonding and Their Compounds > SO , Energy ≡ motion is of Wave nature which enters the Energy caves and becomes a Particle or Wave or Both . In case of Photons exists this DUAL- Property , Wave – Particle . Thus The Historical Doubt of Einstein - Podolsky - Rosen for the Q-Mechanics Completion VANISHES . [100,104] . 5… The Programming of Atoms Bonding is the Quantization of Atoms -Wave -Energy to all Possible Equilibrium Positions of the [⊕↔⊝] constitutes Reactions . The Wave-Energy as Vibration travels at 75-90 % of the light speed c , while the Wave-Energy in Black Holes is n π c times of the light speed . [100-101] 6… From all the Possible Reactions in Compounds , the Bonding or the Releasing of energy is the Vital rule of the Theory of Vibrations .The Program Programming the Atoms and their Compounds , analyses the Interactions of two or more Energy Systems with known IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 10 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Status using the → Carrier–Modulating–Modulated–Demodulation Waves Process ← Some following Wave Properties that are Defined : 6a... The Equilibrium Mode – Shapes , Φ , Diagrams . 6b1..The Circular – Frequencies , 𝐖 𝐑 , in 1015 Hz , 6b2..The Natural – Frequencies , 𝐟 𝐑 , in 1015 Hz , 6c.. The Resultant Energy – State , 𝐄 𝐑 , in e-Volt , 6d...The Electric–Magnetic Field , 𝐄 𝐑 - 𝐌 𝐑 , in 10−12 Ampere -10−6 Tesla , 6e.. The Intensity of the λ-Electric–Field , 𝐈 𝝀 , in 10−12 Ampere , 6f.. The Intensity of Magnetic–Field , 𝐌 𝝀 , in 10−6 Tesla , 6g.. The Wavelength , λ , in .10−10 meters , 6h.. The Wave Velocity , �̅� , in 105 meters / s , 6i1..The Resultant Voltage at Wavelength-sides , 𝐕 𝝀 , in Volt , 6i2..The S-Bands Voltages at Wavelength-sides , 𝐕 𝝀 , in Volt , 6j.. The Radius of the Helical motion , r = 𝐀 𝐑 in .10−10 meter , 6k.. The Total Carrier Power , 𝐏 𝐂𝐓 , in 10−20 Watt , 6l.. The Total Side-Bands Power , 𝐏 𝐓 , in 10−20 Watt , 6m.. The Side-Bands Amplitude , 𝐀 𝐁 , in 10−10 meter , 6n.. The Side-Bands Coefficients , a - m , in n-n , 6o.. The Modulating Phase Shift , 𝛗 , in Rad /2π , 6p.. The Modulating Factor , 𝒎 , in n-n , 6q.. The Total Energy-Status , a - m , in H-W , Figure-2.. The ± Energy in Hydrogen and Atoms exists from Hydrogen Common-Atom . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 11 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The two Hydrogen Bonding → H + H = H1 1 + H1 1 = H1 1−1 H1−1 1 = H1 0 + H0 1 = H2 Water H2O = O1 H2= O2 2 H2 2 = T 4 4 = O2 2−2H2−2 2 = O2 0 H0 2 = ES 2=T/2 2=T/2 , Still. [102] 4a.. THE CORELATIONS OF SPACES AND THE PRIMARY MOTION . Euclidean Geometry Spaces occupy Position (x , y , z) and Dimension |⇅| at any Point P, Cauchy Stress Tensor defines the State of Stress (σ , τ) to the Traction-Vector (Te= e. σ) The Position of Spaces is defined from any 3-dimension Coordinate (x , y , z) or from Angular-System ( φ , θ , ω ) or any other System with axis from any fixed Point O . The Dimension of Spaces is defined as the distance between any two Points O , P , and from any fixed Point O the magnitude |OP| = 𝐧 |OP| , where n⃡ is The Unit Line - length direction and is identified by vector n⃡ ≡ |AB| = The Unity Monad . In Fig-1-1 , On AB Diameter = 2R = The Monad of Neutral-circle exist the Spaces with their Sides 𝛌𝐧= 2,√R2 − a²n , and their Critical-sides 𝐚𝐧 = 1 2 √4R2 − λ²n , as below 1… For Spaces , the Polygon Side is λ n(S) = 2 .√R2 − a²n(S) , 2… For Anti-Spaces , the Polygon Side is λ n(AS) = 2 .√R2 − a²n(AS) , 3… For Sub-Spaces , the Polygon Side is λ n(SS) = 2 .√R2 − a²n(SS) , 4… For Neutral - Spaces , the Polygon Side is λ n(NS) = 0 , while in Material-Geometry becomes from the Centripetal Force FC = σ = m.v³ r = J w² r , where the Unity Monad r ≡ monad R = σ J.w² = 𝛔 �̅� = Stress Angular−momentum̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ , and Polygon Side 𝛌 𝐧(𝐍𝐒) = 2 √R2 − a²n , All becoming from Pythagorean relation [ R ] ² = [ an] ² + [ λn 2 ] ² . In Fig-3 , is shown the Geometrical construction of the Inscribed and the Circumscribed n-Regular Polygons . With 4 , 5 , 6 , different Vibrating masses for each Space and by using the Argand Diagram for a Vector of Amplitude AB = 2R = The Monad , then the 4 - Spaces are measured using the Euler equation for exponential functions as follows , On the SPACES , the Vector of Amplitude can be represented as the Complex quantity on → S  Z + n = + 2R.𝑒𝑖.𝑛𝜔𝑡 = + 2Rn.[ cosw. t + i. sinw. t ] = x + i .y on → AS  Z − n = - 2R.𝑒𝑖.𝑛𝜔𝑡 = - 2Rn.[ cosw. t + i. sinw. t ] = x + i .y on → SS  Z ± 1/𝑛 = ±2R 1/n. ei.θ/n = ± √2R n . √[ cosw. t + 𝐢. sinw. t ] n = x + i .y where The Space Vector–Amplitude 2R = √ λ n(S) 2 − a²n(S) , for the n-masses . The Anti-Space Vector–Amplitude 2R = √ λ n(AS) 2 − a²n(AS) , for the n-masses . The Sub - Space Vector–Amplitude 2R = √ λ n(SS) 2 − a²n(SS) , for the n-masses . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 12 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Remarks : a…From Argand Diagram the Vector Z + n =2R.𝑒𝑖.𝑛𝜔𝑡=+2Rn.[cosw. t + i. sinw. t] = x + i .y meaning the Wave Pattern of Spaces in Planck`s cave 10−35 m .The Vector Amplitude 𝐀𝐁̅̅ ̅̅ = 2R = The Monad ,is the Complex quantity 𝐀𝐁̅̅ ̅̅ ≡ �̅� ≡ x + i .y , where x = 2R → the Geometry of PNS - Space and y = 𝑒𝑖.𝑛𝜔𝑡→ the motion in Planck`s Space . b…Since Anti-Space Z − n ≡ - Z + n , means the Conjugate Space to the Space Z + n . c…Since from Wave-Action of Spaces issues Z − n x Z + n = [2R] 2n . e i.2nθ , this means that ,The conservation of Energy = Work , between the Spaces and the Anti-Spaces happens by doubling the number , n , of the Vertices of the Regular Polygon . d... The Natural frequency of an x-x linear vibration , [⊕↔⊝], becomes from 𝐰𝐧= 2π.f n meaning that the infinite frequencies f n are stored in Subspaces by doubling the number , n , of the Vertices with 2n-Sides to a Regular Polygon of initial n.Sides . e... The Natural frequency of an Rotational ↻ r ↺ DOF vibration System becomes from �̅� = r.m.v n= r.m.v n= r.m.w n.r = m.w n.r ² = m.r².w n , meaning the Natural Identities of the ∞ Monads in our Universe . The Equations of motion in a Material Point related to the Inner motion was referred before where the In-between-magnitude Circumscribed – Inscribed Polygon , The Part , becomes the Outer AB as the ,Whole or the Self-Growth .This Augmentation-Property , of these two Golden-Ratios , exists in the Material-Point and on Frequency fn , which motion ≡ Growth , and Spread in Two-Closed-Transverse Planes as is in the Propagating Electromagnetic Wave where E ⊥ H . From Planck E = h.fn ≡ [ 1+√5 2 ] hσ 2πr = [ nσ 8 r² ].B̅ , 𝐟𝐑 ≡ [ f1=N ,f2 ,f3 , f R = w²N ] , Euler`s 𝐞−𝐢.𝐀( 𝛑 𝟐 +𝟐𝐤𝛑).𝐛 = e−i.A( π+4kπ 2 ).b = A.cos[ π+4kπ 2 ]. b – i . Asin[ π+4kπ 2 ].b , L = [ B̅/2].w , so then w = √2π. fR = 2L / B̅ , and the Wave Equation is y = 2A.sin( 2π.x λ ).cos . wt ....(a) f... The Natural frequency of the Unit-Energy-Vectors , �̅�.�̅� = m.r².w ²n = m.v²n= KE . From Momentum �̅� = r m v = m �̆� r² = ± [ 𝛑.𝐫𝟐 𝟐 ].�̅� x �̅� , �̅�.�̆� = ± [ π.r2 2 ].c̅ ² , c̅ = σ Φ , i.e. The Primary Space is Originated from the Stress  σ , and from the Angular Momentum  �̅� , �̅� , Produced from the Two - Opposite [⊕↔⊝] components. g...Since Monads are Complex quantities as AB ≡ �̅� ≡ x + i. y , then are Quaternions where Material-Points ≡ Quaternions , which Carry the → Motion ≡ Energy ← from the two Edge-Poinds [ �̅�↔�̅� ] , as this is from Point A , to Point B circularly . Simultaneousley On AB Conductor is created an Electric Field 𝐄 𝐅 between A–B Points and ⊥ on AB axis directed to the motion , and an Transverse (⊥AB) Magnetic Field as , 𝐌 𝐅 ⊥ 𝐀𝐁̅̅ ̅̅ ⊥ 𝐄 𝐅 . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 13 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 5a.. The Figure-eight Knot & The Figure n = 1 ≈ ∞ Knots : Figure -3- : The Geometrical construction of the 4-Spaces and the , Spaces Waves of PNS . In Argand Diagram AB = 2R → x = -1 , O , +1 , y = i , exist the Vector of Amplitude �̅� ≡ x + i . y , with angle θ ᶱ = [ Z – OC] ᶱ , [i = √−1] .x . The Wave's " strength " and " direction ", where x is a number that represents the magnitude or length of the vector z , and from Definition of the complex Vector Space , is the Set of Vectors of length (the n Vertices which form the n Polygon Sides , or the Norm of the Complex-Vectors ) In (1) , (2) , are the 4-Geometrical Spaces consisting the Geometrical - Mould . In (2) , are the Vectors of Amplitude ON the 4 -Spaces consisting the E-Form-Mould . It is constructed in Space → The Inscribed and the Circumscribed to the circle of diameter AB , the → Regular Tetra Pleuron ← It is constructed in Anti-Space → The Inscribed and the Circumscribed to the circle of diameter AB , the → Regular Penta Pleuron ← It is constructed in Sub-Space → The Inscribed and the Circumscribed to the circle of diameter AB , the → Regular Hexa Pleuron ← In (3) , are the Norm (magnitudes) of the 4 -Vectors consisting the Strength -Mould . In Fig-3 The time-harmonic function �̅� (t)≡Wave , x is the amplitude, w the angular frequency , and Phase of �̅� (t) to be determined by the number n , n+1 of Vertices . The complex – Vector ± �̅� (t) uses the Norm Iz̅I = 2.R , and the number n , of the Space which Determines the Number of Knots and Sides of Regular Polygons as , IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 14 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 1…The Space is consisted of 4 masses [99] which Produce the Space - Wave = 2R.𝑒𝑖.𝑛𝜔𝑡 , 2…The Anti-Space is consisted of 5 masses Producing the Anti-Space–Wave= - 2R.𝑒𝑖.𝑛𝜔𝑡 3…Sub-Space is consisted of 6 masses Producing the Sub-Space–Wave = ± 2R 1/n. ei.θ/n 5a1.. The Physical notion of the Regular Polygones : [ 62 ] According to Archimedes , Geometric means , speaking of numbers , whether solid or square , observes that , Between Plane One - mean suffices , but to connect two solids two – means are necessary .This denotes that between two square numbers there is one mean proportional number and between two cubes there are two means proportional numbers . It was Proved that Odd numbers become from any two consequent Even numbers , so the sum of two irrationals may be either rational or irrational . The Cattle – Problem of Archimedes may be further analyzed reaching to equations of any degree. It was shown before that , all n-Regular Polygons End to equations of n-degree Segment , by finding a suitable value of the Segment , x , That is we have in the general case to solve one or two equations of the form : A . R 0. x n - B . R 2. x n−2 + C . R n−6. x ³ – D . R n−4. x ² + E . R n−2. x1 – F . R n. x0 = 0 for The Even Polygons , and A .R 2. x n−2 - B .R n−2. x n−3 + C .R2(n−4). x ³ - D .R 2(n−3). x ² + E .R 2(n−2). x1 – F .R 2(n−1). x0 = 0 for The Odd Polygons , where A , B , C , D are constants . The Presented Geometrical method is the solution of the above equation in the general case . Because , the nth - degree - equations are → the Vertices (n) and the Sides (𝐚𝐧) of the n- Polygon in circle ← so number , π , is their Mould . The Knot is a Closed – Curve in Space ( x , y , z ) , that does not intersect itself . A simple Parametric representation of the Figure-eight Knot is as the Set of all ( x , y , z ) Points where , x = ( 2 + cos 2𝑡 ). cos(3𝑡) , y = ( 2 + cos 2𝑡 ). sin(3𝑡) , z = sin(3𝑡) , For the Cave-Spin , �̅� = r m v , which is a Monad and is consisted of 4-Spaces exists → (( The Spaces n.�̅� Polygons , The Anti-Spaces - n.�̅� Polygons , The Sub-Spaces ± √𝐧. �̅� 𝒏=𝟏−∞ Polygons )) ← , with n = 1 ≈ ∞ Knots It has been Proved [62] that , Projecting the vertices of the Regular n-Polygon on any tangent of the circle , then the Sum of the heights 𝐲 𝐧 is equal to , n  R . This is a linear relation between Heights , h , and the radius of the circle , the monad . This Property on the circle yields to the Geometrical construction ( As Resemblance Ratio of Areas which is controlled ) and the Algebraic measuring of the Regular Polygons as follows, when : R = The radius of the circle , with a random diameter IAB I . 𝐚𝐧 = The side of the Regular n -Polygon inscribed in the circle n = Number of Sides , a , and Knots of the n -Polygon , then exists : IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 15 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . n . R = 2 . R + 2 . y1 + 2 . y2 + 2 . y3 +……… 2 . 𝐲𝐧 …………. (n) The heights yn are as follows yB = [ 2 . R ] , y1 = [ 4.R ² - a ² ] /. ( 2 R) …(y) Equation (y) is a Space-Wave depended on 𝐚𝐧 , meaning n Informations on 𝐲𝐁 Height . y2 = [ 4.R4 – 4. R2. a2 + a4 ]/(2. R3) y3 = [ 8. R6 – 10. R4. a2 + 6 . R2. a4 – a6 ] –a2 . 64. R 8- 96. R6.a²+52. R4. a4 –12.R². a6 +a8 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 2. R5 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ yn = [ …………….. ] / 2. Rn Figure – 3A- In Fig-1-1 , On AB Diameter = 2R = The Monad of Neutral-circle exist the Spaces with their Sides 𝛌𝐧 = 2,√R2 − a²n , and their Critical-sides 𝐚𝐧 = 1 2 √4R2 − λ²n , as below 1… For Spaces , the Polygon Side is λ n(S) = 2 .√R2 − a²n(S) , 2… For Anti-Spaces , the Polygon Side is λ n(AS) = 2 .√R2 − a²n(AS) , 3… For Sub-Spaces , the Polygon Side is λ n(SS) = 2 .√R2 − a²n(SS) , 4… For Neutral - Spaces , the Polygon Side is λ n(NS) = 0 , while in Material-Geometry becomes from the Centripetal Force FC = σ = m.v³ r = J w² r , where the Unity Monad r ≡ monad R = σ J.w² = 𝛔 �̅� = Stress Angular−momentum̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ , and Polygon Side 𝛌 𝐧(𝐍𝐒) = 2 √R2 − a²n , All becoming from Pythagorean relation [ R ] ² = [ an] ² + [ λn 2 ] ² . 5a2 .. The General Equations of motion in a Polygon-Vector–Conductor d = Side 𝛌 𝐧(𝐍𝐒) The Displacement at Point x of a Conductor d , is x + dx and Strain will be u + ( ∂u ∂t ).dx , and the Unit Strain is ( ∂u ∂x ) . From Hook`s law , σ = E u , The ratio of Unit-Stress to Unit-Strain is equal to the `modulus of Elasticity E and A = the cross section area of the Conductor , then issues ∂u ∂t = Force A .E = F A .E and by differentiation becomes → A .E. 𝛛²𝐮 𝛛𝐱² = 𝛛𝐄 𝛛𝐱 …….(1) Applying Newton`s law of motion for the element dx and equate the Unbalance-force ∂P ∂x , to the Product of the mass and acceleration of the elements , and then ∂E ∂x dx = ρ A dx ∂²u ∂t² ….(2) Eliminating ∂u ∂x then we have The Partial differential equation ∂²u ∂t² = [ E ρ ] ∂²u ∂ x² .…(3) and 𝛛²𝐮 𝛛 𝐱² = [ 𝛒 𝐄 ] . 𝛛²𝐮 𝛛𝐭² ≡ [ 𝟏 𝐜² ] . 𝛛²𝐮 𝛛 𝐱² …(4) where �̅� =√ E ρ , and it IS The velocity of Propagation of the Displacement dx , along the Conductor or → is The Stress–Wave ← Applying equation (4) in an Conductor [ d ≡ 𝛌 𝐧(𝐍𝐒) ] of Space Anti-Space Equilibrium formation of Polygon STPL line with both ends Free , then the Natural - frequencies must IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 16 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . be zero because charges ⊕ , ⊝ interchange at both ends , A , KA , and issues ∂u ∂t = 0 such at x = 0 , and as at , x = d = λ n(NS) = AKA . The Two equations of the below (5) correspond to these , Stretched ≡ linear , boundary conditions which are as follows , { ∂U ∂x } 𝑥 =0 = A 𝟏 𝐜 [C sin wt + D cos wt ] = 0 { ∂U ∂x } 𝑥 =0 = 𝐰 𝐜 .[ A cos | 𝐰 .𝐀𝐊𝐀 𝐜 | - B sin| 𝐰 .𝐀𝐊𝐀 𝐜 | ].[ C sin wt + D cos wt ] …..(5) Equation (5) must be true for any time t , Therefore from the First equation A = 0 . From the Second equation B must be finite in order to have vibration and is satisfied when , sin | w . AKA c | = 0 , or , | w.n . AKA c | = | 𝐰 𝐧 𝐝 𝐜 | = 𝟐.𝐧 �̅� 𝐜 = π , 2π , 3π , ... nπ , .…(6) where , n = The order of the mode and the equation of motion In Conductor becomes d = λ n(NS) =AKA which is an longitudinal-vibration u = 𝐮𝟎 cos [ 𝐧 𝛑 𝐝 ] x sin [ 𝐧 𝛑 𝐝 ].c ..(7) a Cosine-Wave having n nodes in Pipe d . The equation of motion for an Longitudinal wave in a Conductor d = λ n(NS) =AKA is → y(x ,t) = N. sin [ kx - wt + φ ] ← …(8) where , c = the light velocity-vector , N = the amplitude , w = the angular velocity-vector , φ = The Phase angle , and k = 2π λ = π rL = Propagating-Energy Constant =Wave-number , λ =The Wave length , �̅� = w k = λ T =√ T m =√ E ρ = The light-velocity in Conductors . The frequency of the Longitudinal wave is → f L = [ n ] �̅� 2λ = 0 , 1.c 2λ , 𝟐.c 2λ , 𝟑.c 2λ …. . . 𝐧.c 2λ , The equation of motion of a Standing-Wave in Conductor d = λ n(NS) =AKA ≡ 2 |rA−K𝐴| ≡ r L ≡ λ is as , y(x ,t) = 2N sin[k x].cos [(w =2πc/λ) t] ...(9) where Longitudinal - frequency f L = ( n + 1 2 ) �̅� 2λ . The Amplitudes become zero at y = 0 when k x = 0 , which implies k x = n π , n = 0 , 1 , 2 , 3 , …n . and Since the sine-function repeats itself after every 2π change in angle which is The Wavelength λ , of the Wave , and from equation → sin[k x] ≡ sin[k x +2nπ] ≡ sink.[x+ 2n.π k ] ← then for n =1 , The wavelength λ = [ 2.1.π k ] = 2π k , and is The-Unit-Energy , The Quantum , k = 𝟐𝛑 𝛌 .....(10) and Wave-equation → sin[k x] ≡ sin[k x +2nπ] ≡ sink.[x+ 2n.π k ] ≡ sin[ 𝟐𝛑 𝛌 ].[ x + λπ ] ....(10a) Since , k λ = 2π constant and λ , is the displacement for n =1 , then implies that k , is Force and because Force x Displacement = Work ≡ Energy ≡ motion , therefore k λ ≡ E ≡ Constant ≡ The Unit-Energy ≡ Energy- Quantum . In case of Energy Equilibrium, then Frequency in Conductor d = AKA= λ / 2 and , f L = (n + 1 2 ) �̅� 2λ ≡ 0,5 c 2λ , 1,5.c 2λ , 2,5.c 2λ , 3,5.c 2λ ,,,,, (2n+1) .c 2 2λ = 1 2 [ c 2λ ] , 3 2 [ c 2λ ] , 5 2 | c 2λ | ,,,,,, 2n+1 2 | c 2λ | The Wave is a Force 𝐐 AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = F = |⊕ => AKA| , an Energy-Vector | ⊕ 𝐐 ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ |≡| ↔ | that Propagates from the Place where it was created , The Point A , so when is found an Obstacle at End-Point 𝐊𝐀 then the Particles of the Conductor medium oscillate Perpendicular |↕|| , because IT IS THE - ONLY - EXIT , to the direction of the Wave - Propagation , and by this Way the motion continues transverse as Longitudinal Wave in Conductor d = AKA as , |⇉ ⇈| ≡ ® , or in Conductor d = λ n(NS) =AKA , The two Perpendicular Energy-Vectors , |k 1 x k 2| = k 1┴2 = k = 𝟐𝛑 𝛌 = 𝟐𝛑𝐟=𝐰 𝐜 and Longitudinal Wave -Energy → c { k 1 x k 2 } = 2π.f , or | 𝐤 𝟏 𝐱 𝐤 𝟐 𝟐𝝅 |.c = 𝐜 𝛌 = f . In Figure -3 - and -9- IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 17 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Obstacle at Edge 𝐊𝐀 Point , is The Presentation of the , ⊕ constituent and The Lack of ⊝ constituent . The Transverse - Propagating-Wave , or the moving Energy -Vector |⊕ 𝐐 ⇈ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ | = |↕| at node KA , and with the Energy-Vector | ⊕ 𝐐 ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ | ≡ |↔| at node A , consist The Electromagnetic - Wave in d = AKA= 2r Conductor . The Energy in this Stationary Wave of wavelength λ = 2d is , E = h f L = h 2n+1 2 | c 2λ | = 2n+1 2 | hc 2λ | = | 3.h.c 4λ | ……… (11) and from relation λ = c T = c f = 2π k , Then k = 2π λ = 2πf c = w c = | v̅ c |.| 1 r | = | v̅ c |.| 2 λ | , or the Propagating-Transverse-Wave carries the Standing-Wave-Energy-Constant-magnitude Constant-motion k = 2π λ = 2πf c = w c = | v̅ c |.[ 2 r[A−KA] ] .…...(12) The two Perpendicular Energy-Vectors , |𝐐 ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↔ | and |𝐐 ⇈ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↕ | ,in Conductor d ,of this moving Energy FORM The Energy-Square , {| ↔ |.| ↕ |≡|𝐐 ⇉ |}.{| 𝐐 ⇈|} ≡ u0 , on Conductors AKA . with Conductors - Pivot-axis Propelled on their Merging -Resultant -axis of direction |⇉ ⇈ ≡ ⤱| . In case of an closed Conductor above motion with Energy = k = 2π λ = 2πf c = w c , and with velocity �̅� = w k = λ T =√ T m =√ E ρ , consists the Stationary–Energy-Storage with fundamental frequency f 1 = (n + 1 2 ) �̅� 2λ ≡ 3.�̅� 4 λ ≡ 3.𝛔 𝚽 4 λ ….…(13) . The Obstacle at KA Point is the ⊕ constituent which executes an Coulomb force F [+ ↔ +] = [⊕↔⊕] r² = [σ.σ] r² = | σ. r | 2 =| e. r | 2 = | 2πf Φ | 2 = | w Φ | 2 . The Energy k at Point KA is the Total mass MT of Conductor d = AKA = r = λ n(NS) equal to the Moment of Inertia JT = m.r² 3 ,where m is the stiffness per meter of Conductor r . The Impact of this mass at THE end-Point is the Energy = J w and Impact → m v = J w r = [ m.r² 3 ] . wr , or velocity after impact velocity → v = [ r³ 3 ].2πf = [ 2πf.r³ 3 ] .......(14) Coulomb force with fundamental frequency becomes F [+↔+] = | 2πf. Φ | 2 = [ 2π Φ | 3.�̅� 4 λ |] ² = | 3πv 2λΦ | 2 …….(15) Placing velocity after Impact (14) in Equation (15) then the concentrated Energy at Point KA is F [+↔+] = { 3π 2λΦ [ 2πf.r³ 3 ]}² = | π2r4f λr.Φ | 2 =| B=(π2r4f) λr.Φ | 2 =| �̅� λr.Φ | 2 ≡ Angular-momentum ≡ SPIN …...(16) In Short-distance Conductor d = 6,195543.10−7m , the Magnetic-field B̅ = m.v q̅ r = 9.10−31Kg.2,9979.108 𝑚/𝑠 1,6.10−19C.[6,195543.10−7] = 2,718088.1013 Telsa , which is Perpendicular , normal , to the Conductor`s axis . ⇉ i.e. In a Closed Energy-Conductor 𝐫 [+↔+] , the fundamental frequency 𝐟 𝟏 becomes the Spin. The Energy- Square - Prism , |AKA| .{𝐐 ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ }.{𝐐 ⇈ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } = |Qx.Qy|.|AKA|=kx.ky|.|AKA| = |↔|.|↕|.|AKA| …..(17) , where |↔| = |kx| the Conductor`s Axial-energy , |↕| =| ↺ | the Conductor`s Rotation-Energy = SPIN , and |↔|.|↕| the Conductor`s Resultant-energy-Vector is an Cruise-missile . When a Conductor d = AKA = r = λ n(NS) has a Hole at an Point 𝐀𝐄 such that AAE= 2 r ,{ A -2r- 𝐀E -r- 𝐊A}= d then The Energy k at Point A E is the Total mass MT of Conductor AAE = 2 r equal to the Moment of Inertia JT = | m.4r² 3 | , where m is IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 18 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . the stiffness per meter of Conductor 2r . From equation (15) Coulomb force F [+↔+] =| 2πf. Φ | 2 = [ 2π Φ | 3.�̅� 4 λ |]² =| 3πv 2λΦ | 2 becomes as F[+⇅ ⇵+] ≡ F[+↔+] , equal and Perpendicular and {𝐐 ⇉ AAE⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } = {𝐐 ⇈ AAE⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } , |Qx| = |Qy| , |kx| = |ky| , |↔| = |↕| …....(17a) Transverse motion |↕| , Presupposes an |←| Impact of the mass at end-Point A E and is the Energy = - J w and because lack of Obstacle Impact → mv = - Jw² = [ m.r² 3 ].w² , or velocity after impact → v = - [ 2πf.r³ 3 ] = - [ v.r² 3 ] ..(14a). Placing velocity after Impact (14a) in Equation (15a) then the concentrated Energy at Edge Point KA is F[+↔+] = { 3πw 2λΦ [ −2πf.r² 3 ]}² = | 𝑤[π2r4f ] rλΦ | 2 = | w.B=(π2r4f) r.λΦ | 2 = w².| �̅� rλΦ | 2 ≡ | 2L= w.�̅� rλΦ | 2 ≡ The Total Energy 2L to the Twin circles squared , or to Angular- momentum ≡ SPIN squared …….(16) ⇉ i.e. In an Energy – Slit - Conductor 𝐫 +{ 𝐀 −2r− 𝐀E −r− 𝐊A}= 𝐝 + , the fundamental frequency 𝐟 𝟏 becomes the Spin , Or , The carrier of the Total-Energy 2L of the moving - System . At AE Point Energy - Square-Prism is → |AAE|.{𝐐 ⇉ AAE⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ }.{𝐐 ⇈ AAE⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } = |Qx.Qy|.|AE KA | = |kx.ky|.|AE KA | = ≡ |↔|.|↕|.|AE KA | .........(18) , where |↔| = |kx| the Conductor`s Axial-energy , |↕| = |ky| the Conductor`s . Transverse - Energy , |↺| the Conductor`s Rotational-Energy = SPIN , where |kx.ky| = |↔|.|↕| ≡ ⤱ = The Dot-Product = Area of Vectors , which is a Scalar magnitude 𝐀 𝐱.𝐲 = |𝐤𝐱|.|𝐤𝐲|.𝐜𝐨𝐬 𝛉 , where θ , is the angle between the two vectors , kx , ky and , The Cross-Product =The Resultant-Vector =®= �̅� = 𝐤𝐱 x 𝐤𝐲 which is Perpendicular to kx,kyvectors where , r =√k²x + k²y , x = r.cos θ , y = r.sin θ , θ = tan−1( y x ) , and �̅� = 𝐤𝐱 x 𝐤𝐲 = 𝐰 𝐜 = 𝟐 𝛑 𝛌 The Conductor`s Resultant-Energy-Vector , ® = �̅� , is as an Non-guided Cruise – missile . From Total Energy relation 2L = B w ≡ E K + E U ≡ |kx|+ |ky| is seen that , kx , ky , consist the 2-Line , Plane-Launcher of { Energy ≡ motion- missile } into → Two Perpendicular and Equal Conductors , with Conductors-Pivot-axis Propelled ON their Merging-Resultant axis of the two directions ⇉ ⇈ ≡ ⤱ . In One-Conductor , kx= ZA̅̅̅̅ , k y= 1,MM̅̅ ̅̅ ̅̅ ̅̅ , Resultant R̅ = Z1MO̅̅ ̅̅ ̅̅ ̅̅ , while for (2) → In Two-Conductors , kx= ZB̅̅̅̅ , k y= ZA̅̅̅̅ , R̅ = Z ,1_2_3 , MO̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ ≡ ⤱ The Total Energy 2L = B w ≡ E K + E U ≡ |kx|x|ky|x|AKA| ≡ |kx|x|ky|x|λ| ≡ An Propagating- Wave in λ , i.e. Energy-Square-Prism is a Moving-Energy-Volume and Because from Cauchy equations of Stresses in three dimensions , The Energy Stress remains Flat ONLY WHEN the Plane - Section becomes a circle , And Because from Mechanics the Sphere occupies the Least - Resistance to the motion , then the Square - Prism is altered to the Equivalent , Energy-Sphere - Cone → ⸿ = k πr ³ ← Fig-9a ,9b , 9c. where the Cube [| ↔ |.| ↕ |.|𝐀𝐊𝐀|] = x ³ becomes an Sphere–Cone ≡ 𝐐.𝐀𝐊𝐀 ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ | ⇉ | ↑↓ 𝛌 | → k π r³ = x ³ . From the max-sin | w .AKA c | = 1 , of equation (6) then u = [ C sin wt + D cos wt ].sin|. 𝐰 .𝐀𝐊𝐀 𝐜 | ……..…(14a) The condition u = y (AKA , t ) = 0 , leads to the equation → sin | 𝐰 .𝐀𝐊𝐀 𝐜 | = 0 , or to …(14) For the Duality-Photons issues , v̅ . [ fn̅ + fn ] ≡ | 𝐯 𝛑². 𝐫⁴𝐧 |. B̅n + | c̅ | fn ..……...(15a) Or v̅. [ fn̅ ≡ (n + 1 2 ) �̅� 2.|AKA| ] + [f n ≡ n �̅� 2λ ] ≡ → �̅� { | �̅� 𝟐.𝛌 | (n + 𝟏 𝟐 ) + 𝐧 𝟐𝛌 } ..........(15) IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 19 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . E-Vectors |𝐐 ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↔| x |𝐐 ⇈ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↕ |x|AKA| ≡ |Qx.Qy|.|AKA| ≡ | ↔ |x| ↕ |y| λ | ≡ k π r³. From Wave-Energy-constant k = 2π λ = 2πf c = w c = | v̅ c |.{ 1 r = 2 r|A−K𝐴| }so c= w k = λ f , or | �̅� 𝟐.|𝐀𝐊𝐀| |≡ f i.e. Frequency f AKA of Pipe |AKA| is the Stationary-Photon Energy-Storage fn̅ ≡ | �̅� 𝟐.|𝐀𝐊𝐀| | ≡ | �̅� 𝛌 | ...(16) and Photon v̅. [ fn̅ + fn ] ≡ �̅�. { | �̅� 𝛌 | + 𝐟𝐧 } i.e. , as Particle ≡ v̅. | �̅� 𝛌 | ≡ E U ≡ Potential Energy , and as Wave ≡ v̅. fn ≡ E K ≡ Kinetic Energy . Since Energy- Constant-magnitude , k , denotes the Total motion then needs a Scalar -Magnitude for measurements and a Vector- Magnitude for Directions . In Conductor D = |AKA| ,| D | ≡ The Scalar Magnitude between Points , which gives also the direction of motion with Points , A , KA , and AKA ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ vector the Direction which defines the Vector AKA ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ . The Total Energy k = | v̅ c |.[ 2 r[A−KA] ] of Photon-wave is Split as, E U+ E K= k ≡ Potential + Kinetic Energy where E U ≡ v̅. | �̅� 𝛌 | ≡ Particle ≡ �̅� { | �̅� 𝟐.𝛌 | (n + 𝟏 𝟐 ) } ≡ | 𝐯 𝛑². 𝐫⁴𝐧 |. B̅n = | 𝐯 𝛑². 𝐫⁴𝐧 | r².q.B̅M ≡ | 𝐪.𝐯 𝛑². 𝐫⁴𝐧 | x B̅M is the Stationary-Storage ≡ Particle ≡ Magnetic-field with velocity , v , in cave λ = 2r = 2π / k and E K ≡ v̅.fn ≡ Wave ≡ Kinetic Energy = 2π λ = q k.A = q k.πr ² = σ.Φ k.3rB̅M = { σ.Φ k.3r } 𝟏 B̅M = 𝐜/𝐄 B̅M , and �̅�𝐄 ≡ c�̅�𝐌 , and the moving Electromagnetic-wave is , E̅E = |E̅0|.sine [kz - wt] , B̅M = |B̅0|.sine [kz - wt] , k = 2π λ , Motion is kept in Caves as Electric , Magnetic fields and as frequencies and conserved in them when the Space-Points become Energy-Conductors . The in caves Energy-Import is shown in Figures . Examples : 1.. In an Conductor d =AKA= 2r , with ⊕ obstacle at KA Point occupies zero velocity and ⊕ Charge is accelerated . The Initial light-velocity is accelerated as v² = v̅o²+2a.r , or a = −v̅o² 2.r = [2,9979.108]² 2.r = 8,9874044.1016 2.r m/s² . For Conductors , r =10−16,100,1016 then , a1 = −v̅o² 2.r = 8,9874.1016 2.10−16 = 4,4937.1032 m s² , a2 = −v̅o² 2.r = 8,9874044.1016 2.100 = 4,4937022.1016 m/s², a3 = −v̅o² 2.r = 8,9874044.1016 2.1016 = 4,4937022.100 m/s². Force on Charge becomes from the Electric field as F̅K = m.a̅x= q.E̅F =+e. E̅F and E̅F= m.a q = e , or E̅F = m.a e = m e [a] = 9.10−31 1,6022.10−19 [4,4937022.1032] = 25,242366. 1020 N C E̅F = m.a e = m e [a] = 9.10−31 1,6022.10−19 [4,4937022.1016] = 25,242366. 10 4 N C E̅F = m.a e = m e [a] = 9.10−31 1,6022.10−19 [4,4937022.100] = 25,242366. 10−12 N C i.e. The ⊕ Charge is accelerated Opposite to initial direction of ⊝ Charge in an Electric-field E̅F . This Property of Acceleration , creates the Magnetic-Fields in Electric Conductors as Atoms are . 2.. Uniform circular motion of electron in Caves constitutes a current I = e T = e.f , and T= 2πr c , I = e.c 2πr = 1,6022.10−19.2,9979108. 2π.r = 7 ,644811.10−12. [ 1 r ] , and for r =𝟏𝟎−𝟏𝟔, 𝟏𝟎𝟎, 𝟏𝟎𝟏𝟔 m then I = e.c 2πr =7 ,644811.10−12. [ 1 r =10−16, ] = 7 ,644811.104 - 7 ,644811.10−12 – and 7 ,644811.10−28 Ampere. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 20 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 3.. The Electric-Potential V = 𝐪 𝐫 = 8,99.109 Nm² C² [ 1,602.10−19. r = 10−16 ]=14,4038.106 =1,44038.107Volt V = 𝐪 𝐫 = 8,99.109 Nm² C² [ 1,6022.10−19. r = 100,10 16 ] = 14 ,403778.10−10 V , 14,403778.10−26 V , 4.. Electric-Field �̅�𝐅 = 𝚫𝐕 𝐝 = [ 14,403778.106. r =10−16,100,,10 16 ] =14,403778.1022, 14,403778.10−10 V 5.. Force on electron is F = q .�̅�𝐅 =1,6022.10−19.{14 ,403778.1022}= 23,077733.103 N and F = q .�̅�𝐅 =1,6022.10−19.{14 ,403778.10 6−16}=23,07.10−13 N,23 ,077733.10−35 N , and 6.. The Produced Work is W = F.ds = F r = 23 ,077733.103.{10−16 } = 23 ,077733.10−13 J . W = F.ds = F r = 23 ,077733.103.{100,1016 } = 23 ,077733.103, 23 ,077733.1019 J . 𝑒 0 7.. Intensity I E−r is , 𝐈 𝐄−𝐫= c.E² 2 = [ 2,998.108.1,602210−19(14 ,4038.1022)² 2 ]=498,26077.1033W/m2 𝐈 𝐄−𝟐= c.E² 2 = 498 ,261.101W= 4,9826077.103W and 𝐈 𝐄−𝟑= c.E² 2 = 4,9826077.10−29 W/m² 8.. The Power 𝐏 𝐄−𝐫= E² Z=πr² = (14,403778.1022)² π.(10−16)² = 6,60394.1077W , 6,60394.1057, 6,604.1013W 9.. Two-Capacitors Conductor of area A and Charge q Charge density is on Plates σ = q A and Electric field E = σ 𝑒 0 = q A 𝑒 0 and from Potential difference V= E d then E = V 𝑒 0 and V d = q A 𝑒 0 or V = q.[ d A .e 0. ] = q A = E d . From above → A = q.d V = [ 1,602210−19(10−16) 14,403778.(r=106) ] = m2 1,11235.10−42 ,1,11235.10−26,1,11235.10−10 m2 ,Stress σ = q A = 1,44038.1023 ,7 ,−9 T m² Remarks : 1.. The Stress-Strain Relationship , from Static theory of Elasticity solves the Problem of static Indeterminacy. That is , the Stress Parameters σ x , σ y , are bound in , (1) By the Stereo static conditions (2) and by the Constraint that the Strain accompanying the Stress s x , s y , remains Geometrically Continuous . This Property of the Stresses allows .the Transportation of the Informations on Knots to the Polygon-Sides. 2.. When a Body contains Potential Energy , During any Unloading it gives off Mechanical - Work , since the Points of application of the GRADUALLY - withdrawn Loads are Displaced. If the Unloading is ABRUPT , then Kinetic Energy is converted into Potential and after is Kinetic , converted back into Kinetic and so on . That is, the body Performs an Oscillating motion until the Mechanical Energy is extinguished due to a State of Equilibrium or Alters . 3.. The Action of the Complex–Vectors create the Waves of Motion ≡ Energy Waves , which are closed curves and Transported . The Pattern of Disturbances with the Informations , Propagates from the One-Edge-Point to the other Edge-Point of Conductors , OR is on the [Sub-Units] . Action of Vectors exist on Polygon–Conductor`s d = Side λ n(NS) of Polygons. 4.. Action of a SIGNAL which Strikes the Tetrahedron–Cube-Spheres Mechanism . Unloading happens on the three Tetrahedron - Sides which withdrawn Loads are Displaced On the Anti-Parallel-Strand . If the Unloading is ABRUPT, then Kinetic Energy of 1-Strand is converted in the 2-Strand as Potential and so on . That is , CUBES Perform an Oscillating motion until the Mechanical Energy is extinguished to their Vertices , which are the stores of Information . 5.. The Harmonic displacement of Support-Points on a Regular Polygon vertices . The Projections of Heights y n of a Regular Polygon are as 2.y n= n R , which allows to the Vector- Force Polygon to follow the Space-Vector-Wave = 2R.𝑒𝑖.𝑛𝑤𝑡 = 2R.[cos w t + i .sin wt] . i.e. Complex frequency Response , Happens on the Regular Polygon Vertices only . 6a.. THE ORIGINATION OF PRIMARY MOTION IN E - SPACES . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 21 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Stress - State (σ , τ) in Spaces is defined from the Cauchy Stress Tensor where , Exists at any Point inside a material in the deformed State Placement or Configuration. The second order Tensor consists of nine components and relates a unit length direction vector 𝐧 to the Traction-Vector 𝐓𝐧 across an Imaginary Surface (in M-Geometry is Φ) Perpendicular to n⃡ , Tn⃖ = n⃡ σ , or Tj n⃖ = ∑σ i j n i , obeying to the Tensor transformation law for Stress Analysis . A contact Force ∆F exerted at Point P of element area ∆A , the ratio ∆F/∆A becomes dF/dA and the couple stress vector ∆M vanishes and prior tensor becomes Tj n⃖ = lim ∆S=0 ∆Fi/∆S = dFi 𝑑𝑆 . Equation means that the Stress-Tensor depends on the Position (location) in Body and the orientation of the Plane on which it is acting . From Spaces (4) in Material-Geometry of PNS -Space (Fig-1-3) the eternal Rotation of the two Opposite [⊕↻↺⊝] constitutes , and because of their in between Stress σ which is equal to the Centripetal-Force then issues FC = σ = 𝐦.𝐯³ 𝐫 = J w² r = �̅� r ,where the monad r ≡ monad R = σ J.w² = 𝛔 �̅� i.e. The Stress σ , which is created between the Opposite [⊕↔⊝] , and through the only Possible Rotational-motion , Originates the Angular-velocity �̆� = 𝛔.𝚽 𝐫 = 2πr.f and the Angular Momentum �̅� = 𝛔. 𝐫 = 𝛔. 𝐑 , which when Placed on AB Diameter = 2R = The Monad consist the Neutral – Spaces - Energy. The Planck`s cave , 10−35 m , is The Safety-Valve ≡ The Critical State for the Stress Ellipsoids , B̅ w̅, Vectors in order that this Total-Energy to Superflow and be Still into Vacuum ≡ [10−62-10−35m] . This Vacuum between the |Gravity cave - Planck`s cave| CONSISTS The BLACK - HOLES - CHAOS , as the Recycling–Energy into which → The { Energy – Space - Monad |�̅�| ≡ |AB| } , Decomposes ← {Fig-14} The Rolling of Polhode Unit-Space into the Herpolhode Space-monad Cone. Figure -4- The Central Axial-Ellipsoid of ⊕ constituent Rotating through constant Point O . In (1-2) The ⊕ Material Point OK , {⊕ ≡ O,[OK = R] ≡ The Space}, continually Rotates around The ⊝ Material Point OK, { ⊝ ≡ O , [OK] ≡ The Anti-Space }, on 2R circle . Their Common circle center , forms the material angle φ = φP.t = ( vP √c²−r² ).t , which then IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 22 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . CREATES at any contact Point P the Centripetal Force 𝐅𝐂 = mv² r = 𝜋r² 2r [w². r²] = 𝝅.𝐫³ 𝟐 [w ²]. The vector �̆� of Angular velocity w , formulates the Polhode Cone of Base-radius a , and of OP = S side forming the 𝐰-angle with z axis , while the vector �̆� of Angular Momentum B , formulates the Herpolhode Cone of Base-radius r , of OP = S side and 𝛉-angle with z axis In (3) , is the MPMP Space where Time is not existing t = 0 , and the same for motion also . In PNS-Space the 3-dimension motion of the Primary constituents [⊕↻↺⊝] the time is as Period T , and Momentum is the Angular-velocity �̆� and Angular Momentum �̅� , where the Momentum �̅� = r m v = m �̆� r² = ± [ 𝛑.𝐫𝟐 𝟐 ].�̅� x �̅� and ± [ 𝛑.𝐫𝟐 𝟐 ] ≡ the Hemi - circle area . The Polhode is of radius a , and rotates with an angular velocity w̆ = 2π.fP in the Cone πa².s . The Herpolhode is of radius r , and is the Stationary Cone πr².s . with an angular Momentum B̆ = r m v = r m (w r) = m �̆� r² . The Polhode Cone is rolling on OP and contact common line , into the Herpolhode Cone without friction . i.e. The motion in PNS-Space exists as the Rotation of Space Point P (+) on [ O ,OK ] circle Anti-Point P ` (-) on [ O ,OK ] of Anti-circle and defines The Angular-momentum �̅� , so The Position of Spaces Points P (+) and Points P (-) and Momentum are simultaneously Determined and measured , consisting a direct violation of the Uncertainty Principle . The Primary - Rigid -Body Analysis : a.. The Position of the Herpolhode Cone is , Fig-4. x = a.sin θ. coswt , y = a.sin θ. sinwt , z = a - a.cos θ = a.[ 1 - cos θ ] where exists , a = The Polhode Cone Base-radius , r = The Herpolhode Cone Base-radius [ x , y . z ] = Coordinate System such that z-axis ⊥ [KOK] line , and angle - 𝛉 between OP , Oz Axis . b.. The Energies of the Polhode - Herpolhode Cones are from Lagrange`s Principle where for any Particle , The Kinetic 𝐄 𝐊 and Potential 𝐄 𝐔 Energy are measured as below , 𝐄 𝐊 = 1 2 m.[ ẋ ² + ẏ ² + ż ² ] = 1 2 m a².[ θ̇ ² + w².sin ² θ ] , and 𝐄 𝐔 = m g z = - m g a.cos θ The Lagrangian 𝐋 𝐊𝐔 = m.a².[ θ̇ ² 2 - V EFFECTIVE ] , where the Effective Potential V EFFE is V EFFE = - 1 𝑚,𝑎² [ m g a.cos θ + 1 2 m a².w².sin ² θ ] = - [ g .cosθ 𝑎 + w².sin ²θ 𝑎 ] …..(e) The equation of motion for the Bead becomes from Lagrangian which is �̈� = [ ∂V EFFE ∂θ ] ..(b) Since θ is constant then issues θ̇ = θ ̈ = 0 , and for the Bead-Herpolhode motion ∂V EFF ∂θ = 0 ..(c) Equation (c) is Stationary at Positions which satisfy relation g.𝐬𝐢𝐧𝛉 = a.w².𝐬𝐢𝐧𝛉.𝐜𝐨𝐬 𝛉..(c1) 1…The Solutions of (c1) are at Position sin θ = 0 where issues 𝛉 = 0 2…The Solutions of (c1) are at Position sin θ.cos θ = sin 2θ = 0 and is 𝛉 = π & 2𝛉 = 2π 3…The Solutions of (c1) are at Position cos θ = g a,w ² and for θ = 0 then w ² = 𝐠 𝐚 Remarks : 1… Solution (1) for θ = 0 exists always and from Figure-4-(2) the velocity vector �̆� is applied at the center of mass 𝐊 𝟎 , Anti-clockwise Spinning . Fig-4-(3) 2… Solution (2) for θ = π exists always and from Figure-4-(2) the velocity vector �̆� is IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 23 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . applied at the center of mass 𝐊 𝟎 , Clockwise Spinning . Fig-4-(3) 3… Solution (3) where since θ = 0 exist the Prior and from Figure-4-(2) velocity vector �̆� is applied at the center of mass 𝐊 𝟎 , and the ± Spinning is very fast from the center and is such that w ² ≥ 0 , and happening when { from relation g a } the radius a  0 . 4…The Solution w ² ≥ 𝐠 𝐚 , Determines the Stability of the Polhode System . The Solution w ² < 𝐠 𝐚 and θ = 0 , Determines Stability at Point O Position and−V EFFE. The Solution w ² < 𝐠 𝐚 and θ = π , Determines Stability at Point O Position and+V EFFE. For Solution w ² > 𝐠 𝐚 and θ = 0 , The Position is Unstable at Point O with −V EFFE. For Solution w ² > 𝐠 𝐚 and θ = π , The Position is Unstable at Point O with +V EFFE. Meaning the Black – Holes Origination . From Solutions w ² = 𝐠 𝐚 , the θ = 0 = π =2π Positions are Unstable and below Point O with+V EFFE. meaning an evolving and Operational Black-Hole Stable Origination. 5…All above Stable Solutions result to the determination of the two Vector - magnitudes that of Angular-velocity �̆� and that of Angular Momentum �̅� , differently Black-Hole. ----------------------------------------------------------------------------------------------------------- Both magnitudes define the Unit-Energy Vectors → { Unit-E Vector R = σ J.w² = 𝛔 �̅� } ← When the Unit-E.Vector R is Placed ON [AB≡2R S.Diameter of Spaces-Neutral-circle] Then the Energy-Monad |�̅�| exists on Space-Monad |AB| ≡ |�̅�| as {E+S -Monad |�̅�|}, i.e. IS THE { Energy - Space - Monad |�̅�| } , with the magnitudes , Spaces Sides 𝛌𝐧= 2,√R2 − a²n , & , Quantum Critical-Side 𝐚𝐧 = 1 2 √4R2 − λ²n , ----------------------------------------------------------------------------------------------------------- The Quantization of Energy in Spaces , happens when follow the Archimedes formula for Doubling the Vertices n , 𝛌𝟐𝐧=√2R2 − R√4R2 − λ²n & Critical-Sides 𝐚𝟐𝐧= 1 2 √4R2 − λ²2n where then Energy is divided and spread on the double number of the Polygon Sides . Monad R = σ J.w² = 𝛔 �̅� remains the same , while Spaces Sides change as above 𝛌𝟐𝐧 , 𝐚𝟐𝐧 . 7a.. THE MECHANISMS & CHARGES , FOR DUAL - ORIGINATIONS , THE NATURAL - Spaces of AB Diameter is the { AB̅̅ ̅̅ Unit Complex Vector } . 1… Spaces ≡ The Infinite (+) Circumscribed-Regular-Polygons in AB̅̅ ̅̅ . [Fig-1,3]. 2…Anti - Spaces ≡ The Infinite (-) Circumscribed-Regular-Polygons in AB̅̅ ̅̅ .[Fig-1,3]. 3…Sub. - Spaces ≡ The Infinite (+) Inscribed-Regular-Polygons on Diameter AB̅̅ ̅̅ .[F-1,3]. 4…Neutral-Spaces ≡ The Infinite Circles on Diameter AB̅̅ ̅̅ .(The Sphere on AB), [Fig-1,3]. 5…{MFMF} ≡ The Gravity cave = 10−62 m . 6…{ PNS } ≡ The Planck`s cave Length = 10−35 m . 7…Vacuum ≡ [ 10−62 -- 10−35m ] = The in-between distance  THE Origination-Mechanisms in Spaces & the Origin of Charges. 8…Action-Spaces ≡ The Infinite Quaternion - monads → �̅� ≡ ( s + v̅.i ) ≡ 𝐀𝐁̅̅ ̅̅ ≡ ≡ √s2 + [ v̅. i ]² ] , or �̅� ≡ x + i . y ≡ 𝐀𝐁̅̅ ̅̅ ≡ moduli , r , is their magnitude [ r = | r | = √ x2 + y2 ] , on Unit-Diameter AB̅̅ ̅̅ . The [QMAS] . For Quaternion v = s 𝐳 * 𝐳 = (s + v̅.i )² = s² +[v̅]² +2.sv̅.i = s² – v̅ ² ± 2.s v̅ = |s|² -|s̅|² ± 2.|s|.|s̅| , for |v̅| = s . 8a.. THE Material - Geometry , E-Geometry , Physics AND Chemistry . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 24 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 8..[a].. The { STPL } Mechanisms in Spaces , from Charges in Under-Planck`s-Length ≡ The Six Triple Points Line Mechanism ≡ The Complex Vector-Spaces Rotor , It is a Mechanism on where the Charges of Opposites , [⊕↔⊝] , React as Centrifugal force FC which in turn changes → (Action-Reaction ≡ + Spin – Anti Spin) ←of [MFMF] Space To Rotational motion Angular-velocity = �̅� & Angular-Momentum = �̅� . [Fig-6]. 8..[b].. The {TVPM} Mechanisms , from Charges on Two-Perpendicular-Vectors-Edges ≡ Markos Two - Vectors Three - Poles Mechanism . [Fig-8a , 8b]. In Fig-8|4| , Vector CP̅̅̅̅ ⏊ CA̅̅̅̅ Vector , which changes the Energy ≡ The Stress σ of The | CP̅̅̅̅ x CA̅̅̅̅ | Square , to 2 - Halve ± Squares on Vectors-axis for Equilibrium . 8..[c].. The { TCIC } Mechanisms , from Charges In Three-Vectors-Conductors-Edges ≡ Markos Tetrahedron Cube Inscribed-Circumscribed Circles Mechanism . [Fig-27-29]. In Fig-6 , The Tetrahedron Conductors -Vectors DA̅̅ ̅̅ ⏊ DB̅̅ ̅̅ ⏊ DC̅̅ ̅̅ , CARRY Energy as Electromagnetic-Fields (Forces-Frequencies-Stresses-any motion) TO Inscribed Sphere (Nucleus of Atoms), TO Cube Vertices (Electrons-Position) ,TO Circumscribed Sphere (The Electrons Orbits) for Equilibrium and for transferring Information and Signals . 8..[d].. The {TOBH} Mechanisms , from Charges In Two-Orbits-Pins & Drain. Markos Atoms Bonding Bracket - Hook Mechanism . [Fig-2]. In Fig-9 , The ⊝ - Electron Precesses due to the Gravity and Produces Work which is stored in a second Orbit . This 2-Orbit equilibrium exists from the Existing ⊝ -Electron ≡ DRAIN and from the New ⊕ -Nucleus ≡ The PIN . The Second -Orbit called as the Bracket – Hook -Mechanism , with which Atoms and Compounds are Bonded .[99-104] The { LNCS } Mechanisms , from Spaces - Action ON Quaternion - monads 𝐀𝐁̅̅ ̅̅ is the Lagrange`s Non Conservative Systems for generalized coordinates , where the System is subjected to forces that do not have Potential or , d (T + U) ≡ δW𝐍𝐏𝐅 = 0 , as exists in the Gravitational Force G and from which are created → The Elementary Particles using the STPL – Mechanism , and all Universe using the ISTCCS – Mechanism ← and The DUAL → The Black - Holes ← as equation δW𝐍𝐏𝐅 = ∑ Qi. δqiN i=1 , [Fig-3, 21 , 22]. 8..[e].. MATERIAL-GEOMETRY & E-GEOMETRY IN PHYSICS and CHEMISTRY Figure – 5 - : The Relation between The [+] Spaces , [-] Anti-Spaces , [+ -] Sub-Spaces IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 25 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . of Euclidean & Material-Geometry in a circle (R,OA) of Three-Dimensions. In (1) , The Euclidean Geometry , E-G , is defined upon the Number of Points which can define the Specific Space i.e. The → One-Point in E-G is defining One-Material-Point-Space , the M-P-S . The → Two-Points are defining the Line-Segment-Space , the Line-M-P-S . The → Three-Points are defining the Plane-Triangle-Space , the Plane-M-P-S .. The → Four-Points are defining the Volume Tetrahedron-Space , the Volume-M-P-S . The → Five-Points are defining the Volume Regular Pentahedron-Space the SV-M-P-S. and so on , representing the Steady-Stable-Regular Geometrical-Formations. The corresponding to the Material-Geometry-Units are for : 1.. One-Point → Two-Units , [⊕],[⊝] with Dimension as , the Material-Vector [⊕→ ⊝] 2.. Two-Points → Four-Units , [⊕],[⊝] with One Dimension as , the Material-Line 3.. Three-Points → Six-Units , [⊕],[⊝] with Two Dimension as , the Material-Plane 4.. Four-Points → Eight-Units , [⊕],[⊝] with Three Dimension as the Material-Volume Keeping the Property of Edge-Points to be the Vector ⊕ → ⊝ then Bond is the Potential to the Unique-Steady-Stable-Regular Material-Geometry-Formations. In (2-3-4-5). All Units in Vectors follow Quaternion q = [s+v̅.i] = AB̅̅ ̅̅ Properties i.e. [⊕] → the Positive constituent of Quaternion at , Point A . [⊝] → the Negative constituent of Quaternion at , Point B . Pythagorean relation [ R ] ² = [ an] ² + [ λn 2 ] ² . In Euclidean-Geometry E-vector AB̅̅ ̅̅ carries Point A to Point B as , Vector A → B In Material-Geometry M-vector AB̅̅ ̅̅ carries Energy , ⊕ , from Point A to ⊝ , Energy of Point B as , ⊕ moves to ⊝ , which is as The Periodic - Pattern . Velocity �̅� = The rate of change ( m / s ) in AB , Therefore Units are Formatted according to the Steady-Material-Geometry-Formations which are : The Line-Vector in cave , r ,which is the Simplest with Double number of 4-Units The Plane-Regular-Triangle in Orbits , is the most Stable Shape of 6-Plane-Units The Volume-Regular-Tetrahedron in Space is the most Stable Shape of 8-Volume Units , the Cube , which are the Crystals . The Regular n-Hedron are for all others . This is the Why The Glue-Bond , Bond is the Potential , between the 6-Units formulates the Regular-Hexagon as the First Steady - Plane - Formulation in nature as in Fig-18- Remark : The Constraint , that Strain accompanying the Stresses s x , s y remains Geometrically Continuous is , because the Material Point is consisted of the two Units ⊕ , ⊝ .only IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 26 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 8..[a].. The { STPL } Physical Mechanisms : The Six Triple Points Line Mechanism . Figure -6- : The Physical Three – Phase - Mechanism , Producing The Fundamental Particles and the Primary Forces IN Absolute System A ≡ R . γ = R. 𝐬𝐞𝐜𝛗. [97] In –Fig. ABC is any Right-angled Triangle at , A = (The-Space) , KAKBKC = Triangle is the (Anti-Space) ,, AEBECE = Triangle is the (Sub-Space) respectively.The Instaneous Pole P of Rotation is off the Circle of diameter |BC| . The Poles of Rotation lie on{DA- PA} Reference System .The Reference System {𝐃𝐀- 𝐏𝐀}≡ [R] (x' , y', z', t') moves with velocity �̅�, Parallel to , x - x', Axis with respect to the Fixed and Absolute System {𝐃𝐀- O} ≡ [S]-(x , y , z , t ) . The Geometrical expression of Lorentz factor γ = 𝒄 𝒗 , is the 𝐬𝐞𝐜𝛗 = γ = O𝐃𝐀 : A𝐃𝐀 = ± 1 / [ √1– (v/c) ² ] = c / [√c² – v²] , which is the Conchoidal of Nicomedes {s = a + b. secφ} and which acquires the material Angle , φ = 𝐯 √c²−r² , i.e. issues |A| ≡ |R| . γ [a]-1…The [STPL] line is an Extreme - Spaces–Physical -Originating-Wave Pattern . Extreme-Spaces (are the Extreme Space-Triangles ABC≡ ⊕) meet Anti-Spaces (Extreme Anti-Space-Triangles , KAKB KC≡ ⊝ ) through The Only-Gateway which is the Circularly Charged + , 0 , - , or [ ⊕  ⊝ ],[ ⊝  ⊕ ],[ ⊕ ⊝  ] Formulation which is a Physical Geometrical formation - Mechanism called The [STPL] line . Figures-[ 6 ,7 ] → The [STPL] - line is a Physical – Mechanism as , In-circle  Triangle  Ex-circle  Ex - Triangle consisted from The Three Spaces , On which the Three Breakages [ s² = ⊕ , 2s ² =  , - s ² = ⊝ ] are Circularly- Charged On the Three – Extreme - Triangles {A B C} , {KA KB KC} , {AE BE CE} ← Producing The Energy-Quantities ± 𝐐𝐩−𝐩 . This Circular-Charge from Breakages IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 27 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . On The-Three -Triangles is the THRUST upon The Energy-Quantity Produced , for Each needed Circular Charge , To Shake Off Quantity 𝐐𝐩−𝐩 . The 3-Circular Charges correspond to the 3-Phase Rotors Produce the 3-Phase Physical-Current , Since Common Circle exists on their Common - Sphere , THEN the Infinite Straight- lines , Launch The E - Quantity 𝐐 𝐩−𝐩 OFF the Fundamental Particles to all Universe and INTO The Planck`s Confinement Only . i.e. The Physical-Three-Phase- STPL Mechanism is the only Absolute-System [A] On which circularly exist the Relative-System [R] ≡ [ ⊕  ⊝ ] ↻ [ ⊝  ⊕ ] ↻ [ ⊕ ⊝  ] of The three constituents [ ⊕  ⊝ ] , which Produces The Fundamental Particles and The Existing Objectivity as [A] ≡ [R].γ , where γ = sec φ = [ 𝐎DA 𝐀DA ] = ± [ 𝟏 √1−(v/c)² ] = [ 𝐜 √c²−v² ] The material Angle φ = 𝐯 √c²−r² , and {A ≡ DA- O}≡{R≡ DA-PA}. γ , or |A| ≡ |R| . γ The Euclidean Geometry is the SPRING , for the Geometrical Relative -Velocities . When the Points [ A , B , C ] , [𝐀𝐄, 𝐁𝐄, 𝐂𝐄] , [ 𝐊𝐀, 𝐊𝐁 , 𝐊𝐂 ] are Applied on the three lines (𝐊𝐀𝐊𝐁 , 𝐊𝐁 𝐊𝐂 ,𝐊𝐂𝐊𝐀) , then the SIX Pairs of the corresponding lines , extended are concurrent at Points PA , PB , PC for the Triple Pairs of lines ( The Pascal`s Perspectivity of Points in Euclidean Geometry ) , [ A𝐀𝐄, 𝐁𝐂𝐄, 𝐂𝐁𝐄 ] , [B𝐁𝐄, 𝐂𝐀𝐄, 𝐀𝐂𝐄] , [C𝐂𝐄, 𝐀𝐁𝐄, 𝐁𝐀𝐄] and at Points DA , DB , DC for the Triple Pairs of Lines [𝐊𝐁 𝐊𝐂 , BC , 𝐁𝐄𝐂𝐄] , [𝐊𝐀𝐊𝐂 ,AC ,𝐀𝐄𝐂𝐄] and [𝐊𝐀𝐊𝐁 , AB , 𝐀𝐄𝐁𝐄] , ( Desargues`s Perspectivity of Points in Euclidean Geometry ) and all the 18 common Points lie on a Straight line The  STPL Mechanism This Mechanism Stabilizes the Nucleus Forces for the{ ⊕ - Space [ A B C] } and for the {⊝ - Anti-Space [ KA KB KC ]} and this succeeded by The Pascal`s –launched caves r through the STPL Circuit of the Ptolemy`s and the Ceba`s Geometry theorem for The Complex Vector - Forces which are , Circularly and Diagonally Equilibrium .i.e. → STPL is The In-Sphere , Tetrahedron-Cube , Out-Sphere Physical Mechanism ← For its 27-Conductors ≡ Electric - Magnetic Fields , Space-Energy CPT is invariant . 8..[a]-2…The Explanation of The Absolute and Relative Motion. . Fig – 6 Because Properties In and On [STPL] line , are relative to the only one Equilibrium and also Absolute System ± Λ = r.mv̅ = r.m.w̅.r = mr².w̅ , so exists that what is called Relativity. Absolute System let it be the [A] ≡ {DA- O} ≡ The STPL mechanism , and as Relative System ( Reference , Affine ) [R] ≡ {DA- O} The Absolute motion happens on [A] ≡ {DA- O} and on [R] ≡{DA-PA} of the two Systems . It was shown , that in {DA- O}, (x ,y ,z , t ) , System �̅� , �̅� , vectors are isochrones i.e. Period T = L / V = 2πR/V = 2π / [c/𝑟𝑐] = 2π/[v/𝑟𝑐] → c/rc = v/rv → c.rv= v. rc , where rv , rc are the radius of their intrinsic rolling circles . This relation is Geometrically expressed as , 𝐬𝐞𝐜𝛗 = O DA : A DA = γ = 𝒄 𝒗 = ± 1 / [ √1–(v/c) ²] = c / [√ c² – v²] , and it is a IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 28 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Geometrical Cycloid Property equal to Lorentz`s , γ , factor . The Newton`s laws are true into the Reference System [R] ≡ {DA- PA} as follows Proof , Considering { DA- O},(x ,y ,z , t ) , as the Fixed frame [A] of the coordinate System in the Gravity cave (d = 2r) , and Point A= (x, y , z) is fixed on circle (O,OA) which is Rotating with a velocity v̅ = w ̅̅ ̅r , of angular velocity w̅ = 2π /T = 2πf , in where Period of Rotation T , is then also constant . Since the acceleration ,a, for a Quaternion z = (s + v̅. i) is a = [d²z/dt²] = (ds/dt.v̅. i) + s. d(v̅. i)/dt = 0 + s. d(w r)/dt = 0 + 0 , and this because w̅ is constant for both therefore velocity �̅� = constant also , i.e. → The Centrifugal velocity of the Absolute System [ A ] is any constant , �̅� , and this because Angular velocity , �̅� , is constant also and thus , is not needed to accept A priori this constancy of velocity �̅� = 0 → �̅� → ∞ on circle (O, OA) and to exist In Frame , ( It has been Proved that this velocity �̅� ≡  σ Φ ) so , Automatically is defined the conversion factor t = time , between the conventional time units (second) and the length units (meter = A.DA) or as c̅. rv = v̅. rc , → c̅ (v)(T/2π) = v̅(c).(T/2π) → c̅(v)/w = v̅(c) / w , which is happening with the same w , without setting any restrictions , in contradiction to the General Relativity which is an axiom a priori . Additional , This is the why that the conversion factor , t = time , has not any essence in all universe but it is only a meter of changes in all Relative Systems . STPL Mechanism is the only Absolute System On which circularly exist [ ⊕  ⊝ ],[ ⊝  ⊕ ],[ ⊕ ⊝  ] , the three constituents , [ ⊕  ⊝ ] , and Originates this Existing - Objective - Cosmos . This circularity of the constitutes exist because of their Natural frequency of vibrations . Because [STPL] line of the fixed frame becomes from this System [A] , then this relative Frame [R} is common to the fixed one (common DA) and let it be [R] ≡ (x' , y' , z' , t') . From figure , sin φ = (�̅� / �̅�) , meaning that the Relative System , [R] ≡ (x' , y' , z', t') , ( the Affine Frame) is the Projection of Absolute Frame [A] ≡ {DA-O} - (x , y , z , t) where exists as the Simultaneity for all motions where issues , [R] ≡ {DA- A} ≡ [ (x' , y' , z' , t') ] , [A] ≡ {DA- O} ≡ (x , y , z , t) = [R] .γ ≡ ( x' , y' , z', t' ) . γ Considering Point DA as the common center and [STPL] as the x-x axis of the two Systems , then becomes DA ( x , y=y ', z = z', t) and for all linear Systems DA ( x', y'= y , z '= z , t' ) This Specific state of constancy , i.e., the Centrifugal velocity of Absolute System [A] to be a constant c̅, and the rectilinear motion with respect to one another defines the natural Inertial frames , and because of uniformity of Space and motion , therefore occupies the same meter for their changes , (i.e. the Time) . Since also Points O , A remove to Point DA isochrones by their intrinsic Property motion which are → their wavelengths are a Stationary wave in cycloid ← following Lorentz`s factor γ , then this following happens also to all Frames which make this motion , and so issues for all the System relations { DA – O } = γ.{ DA – A } . Fig-6 On this Relative System DA( x', y' = y, z' = z , t`) are conveyed , the Breakages [ ± (wr)² , 2(wr)² ] of (O,OA) circle after the colliding with the rotating velocity v̅ = w̅.r of the [A] System , and are the Fundamental Particles , Fermions and Bosons, or by escaping consisting the Rest Field and Gravity , or Dark matter and Dark Energy as analytically was shown .[39] IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 29 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Angular-Momentum Ellipsoid Nibs on →{ Herpolhode and Polhode Ellipsoids}. In any Rotational motion , B̅xw̅ are Constants and The Angular-velocity-vector �̅� , describes the Ellipsoid of Angular Velocity and its nib describes a Cone of which Plane-Base is Fixed . Simultaneously .The Angular-Momentum �̅� describes the Ellipsoid of Angular Momentum and its nib describes a Cone also of which Plane-Base is also Fixed . The Nib of Angular - velocity-vector , �̅� , describes on the Tangential-Plane of the Angular-Momentum Ellipsoid , the Polhode , while The Nib of Angular-Momentum �̅� , describes on the Tangential-Plane , of the Angular – Velocity - Ellipsoid , the Herpolhode . or The Fixed Tangential-Planes on �̅� , �̅� nibs which are alternately Perpendicular to B̅ , and w̅ , common Central Axes of Rotation , or The Angular-Momentum Ellipsoid Nibs , in Herpolhode and on Polhode Ellipsoids as → Electro magnetic Fields ← and which are Space-Energy CPT invariants . The moving Energy-Vector is |⊕ 𝐐 ⇈ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ | = |↕| →{ The Electron-Unit-Charge }, and since Charges correspond to the 3-Phase Rotors is ,The Unit-Charge Complex-Vector. Remarks : Material Point A ≡ ± |(w̅.r)²| of the Fixed System {DA- O} travels with velocity v̅ at Point DA , so geometrical distance A.DA in the Relative System [R] ≡ {DA- PA} is A.DA = x'+ v̅ t', and because of the Isochrones motion in the Fixed System [A] ≡ {DA- O} , then holds , x = (x'+ v̅.t') .γ or x = (x'+ v̅.t').γ ≡ [ x'+ v̅.t'] / [√1-(v/c) ²] ……. (a) Inversely , by using (a) , where [A] ≡ { DA-A} ≡ {DA- O}/ γ , then if Material Point A of the Fixed - System { DA- O} travels with velocity v̅ at Point DA , where then the geometrical distance ADA in the Fixed System is , [A] ≡ {DA - O} is → A.DA = x - v̅.t and in the Relative System , [R] ≡ { DA- PA} is → x' = (x-v t).γ = [x-v t] : [√1 - (v/c)²] …...(b) Fig-6 An application to Gravity [72] and Photon [95] , Because Gravitational Force is equal to → the Geometric-Resultant of the light-velocity c , acting on Electron-Unit-Charge �̅� ← or is , G = c √𝟐 �̅� , then Electron - Charge is , q̅ = G c √2 = 6,680561 .10−11 1,41429.[2,9979346.10 8] = 1,58.10−19 Coulomb , which is the known Coulomb`s charge Quantity . From gL ,kL Universal constants the Newton`s Gravitational constant Force G ≡ g.kE ≡ g .[gL kL ] ≡ [ T²P 𝐚³ ].[gL kL ] ≡ 9,808238* 6,8116.10−12 ≡ 6,68056.10−11 𝐦³ 𝐍𝐬² The in depth Position of electrons allows to the Work Produced in Prior Phase - Areas as Golden Ratio Pattern , to be Stored in the Next . Energy = motion / T ≡ ( v 2πr ) . [σ + σ Φ] = �̅� . [ 𝛔 𝟐𝛑𝐫 + 𝛔𝚽 𝟐𝛑𝐫 ] ≡ �̅� . [. fn̅ + 𝐟𝐧 ] ≡ Moving - Storage →[ �̅� . fn̅ ] ← + Moving-Frequency → [ �̅�.𝐟𝐧 ] ← ≡ The Material-Point . i.e. The Energy Produced in Photon-Cave is consisted of Two-moving-Storages , that travel as Wave [�̅�.𝐟𝐧] →[ �̅� . fn̅ ] → [ �̅� = �̅� = λ f Φ ] → [ S ≡ EM-R ≡ f1=N , f2 , f3, fD ,,f n= w² ] , and as Particle →[ f1= (E²+H²) = n (1+√5)σ 2πr = nB̅ π² r⁴ ] →{W≡EM-R≡ [εE² + μB²] = 2.λc.sin.2φ} and is The Duality of the Energy-Storage S ≡ { [⊕← 𝐫 →⊝] + Motion M . [87-89] .The three elements [Σ = n . h]  [s² = ⊕ , 2s²= , - s² = ⊝] of This {STPL}-Mechanism formulate the 3-Knots Figures ( n = 3 ) and determinates the Elementary-Particles Vector Mould . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 30 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 8.[a]-3..The circularly existing Unit-Charge Complex-Vector [⊕  ⊝],[⊝  ⊕],[⊕ ⊝ ], Figure - 7 - : The , STPL , Mechanism is the Generator of the Cosmic-Particles . The Unit-Charge is as Complex-Vector Velocity Thrust ( v̅ = w̅.r ) , with and from the Breakages → 𝐳 * 𝐳 = (s + v̅.i )² = s² + [v̅]² +2.sv̅.i . 7-a = The Positive Breakage Quantity Q ≡ [ ⊕  ⊝ ] , for Leptons and Quarks . Figure –7- a : The , STPL , Mechanism is the Generator of the Cosmic-Particles . The Unit-Charge is as Complex-Vector Velocity Thrust ( v̅ = w̅.r ) , with and from the Breakages → 𝐳 * 𝐳 = (s + v̅.i )² = s² + [v̅]² +2.sv̅.i . V-Trust ≡ [ ⊕  ⊝ ] IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 31 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 7-b = The Positive Breakage Quantity Q ≡ [⊝  ⊕] , for Anti-Leptons , Anti-Quarks . Figure -7- b : The , STPL , Mechanism is the Generator of the Cosmic-Particles . The Unit-Charge is as Complex-Vector Velocity Thrust ( v̅ = w̅.r ) , with and from the Breakages → 𝐳 * 𝐳 = (s + v̅.i )² = s² + [v̅]² +2.sv̅.i . V - Trust ≡ [⊝  ⊕] 7-c = The Positive Breakage Quantity Q ≡ [⊕ ⊝ ] , for Bosons Origination . Figure -7- c : The , STPL , Mechanism is the Generator of the Cosmic-Particles . The Unit-Charge is as Complex-Vector Velocity Thrust ( v̅ = w̅.r ) , with and from the Breakages → 𝐳 * 𝐳 = (s + v̅.i )² = s² + [v̅]² +2.sv̅.i . V - Trust ≡ [⊕ ⊝ ] , Since the Primary motion ≡ [PM] exist from the three Breakages only , and STPL is the n =3 Mechanism creating Elementary Particles  [n-Knots=3] in 𝐕 GRA = 7,593.1035 V. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 32 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 8..[b].. The { TVPM } Physical Mechanisms : Two - Vectors Three - Poles Mechanism . Figure -8- a : The Two-Perpendicular Complex-Vectors , CA ⊥ CP , Slide on (O,OA) circle , and through the Three Poles , A , C , P , carry the Stress σ , of Square , ACPC’, to the Two Equal and Opposite Squares |COAB| , |COPB’| for Halve Area - Stress - Conservation b-1... The Conservation of Energy IN Geometry - Mechanism : [62] In(1)…On AA` diameter circle (O,OA) exists the Inscribed -Square OA x OA = OA², and the Circumscribed Square CP x CP = CP². Since Vector CP = [OC= OA].√𝟐 , therefore the Inscribed Square is Halve the Circumscribed . The same to the Circles π [ 𝐂𝐏 √𝟐 ] ² = π [CP] ² = π [OA.√𝟐] ² = 2.π [OA]² . This Linear Expansion of Archimedes-Circles and Squares Passes from a Point M , such that π r ² = CMNH Square . [ 47 - 51 ] In(2)... On Two-Perpendicular Vectors , CA ⊥ CP , Point C` Slides on (O,OA = OP) circle forming the CAC`P to CMNH to CBAO Squares with Prior of (1) Properties .This Sliding-Rotating Method Is the Markos { 2 -Vectors 3 -Poles Mechanism } which Circles and Squares as Areas Pass from the Point M determined by the Position 𝐀 𝐞 , such that π r ² = CMNH Square . The Squaring of the circle π [ 𝐂𝐀 √𝟐 ]² = CM² . [ 47 - 51 – 98A ] In(3).. If the Two-Perpendicular Vectors Are Conductors with Prior Properties then Square CAC`P to CMNH to CBAO Square , are a Rotating and Changing-Squares-System , CP ² ↑ = |∷| ,↻, → To 2-Half and Equal Area Squares CB ² = ⤭ =[⊕ ∷ 2 +⊝ ∷ 2 ] CB`² , by turns in OABC ⊥ OPB`C Planes , which Equilibrium and Consist , The Neutral-Space . Placing Stress in Squares becomes the Physical World as , Physics Mechanics and Chemistry , by Conservation of Energy as Stresses in Areas . As in Electricity , The flow of a current in an Conductor , CA = d , Represents the created around the Conductor by the Circular-Lines of Forces , Magnetic field , The flow of ⊕→⊝-Charge in d IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 33 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Conductor , is the Torsional Force �̅�𝐅 = v⃗ .r = w⃗⃗⃗ .r ² = 2πf. r ² = σ Φ r …....(1) . Since edge-velocities v̅C = v̅A= 0 , therefore CA =d= ½ (aT²) and Acceleration , a = 2d /T² = 2d.f², and then the velocities diagram is Energy-Triangle → v.[Δ-CAP] = v.[½ (2d. |v⃗ |)] = v.[d w r ] = v d.2πf.r = v.[σ Φ d] ..(2) The Current flow in Electric-Field �̅�𝐂𝐀𝐏 of CAP Plane is transported into the Perpendicular to CA axis Opposite Magnetic-Field �̅�𝐏𝐀 in PA-Plane due to the Force �̅�𝐅= �⃗� .r = I = 𝟐𝛑𝐫 𝛍 �̅�𝐏𝐀= σ Φ r .(3) Stress σ = 𝐅𝐨𝐫𝐜𝐞 𝐀𝐫𝐞𝐚 = 𝐆 𝐯.𝐯 = G √2 .c = The �̅� Electron Charge = G √2 .c = 6,6736923 .10−11 1,41429.2,9979346.108 =1,574.10−19 C The Spin of cave , r, is Equal to the Angular-momentum-Vector → Spin ≡ |�̅�| ≡ r σ Φ ← which contains and it is the Golden-Radio-frequency Φ as Pressure , σ , in cave r . i.e. [a] The Forces are Spread on Surfaces as Stresses σ , and in case of Area = 0 then are Spread on the Velocity – square – Surfaces , and for the light c , on Resultant √𝟐.c . b-2... For Conductor-Case which is an Screw-motion where a Torque is created in Perpendicular Plane of Con-axis . Markos { 2-Vector 3-Poles Rotating Mechanism },{ −A ∷ ↻ +𝐂 ↑ [⊕ ∷ 2 +⊝ ∷ 2 ]𝐏 } carries motion (+) → (-) from Point A through an Perpendicular-Conductor CA , to Point B , into an Changeable and Rotating-Squares-System , PA ↑ = |∷| ,↻, → To 2-Half and equal Area Squares CB = ⤭ = [⊕ ∷ 2 +⊝ ∷ 2 ] CB` , by turns in OABC ⊥ OPB`C Planes which Equilibrium [⊕ ∷ 2 +⊝ ∷ 2 ]=0 (The Neutral-Space) . b-3..For the motion of the ⊕→⊝-Charge of any Conductor d = CA becomes Stress - σ in length d Motion is created in Conductors d ≡ [⊕← 2𝐫 →⊝] following the [⊕↔⊝] Interactions .This Motion is Conserved and kept in Caves as Electric and Magnetic fields and as frequencies which are conserved in them when the Space-Points become Energy-Conductors and Energy k = 2π λ = 2π d . b-4...The in caves Energy Imports are in Squares which equilibrium through MPM ≡ Markos of ⊝ Charge in an Electric–field E̅F . For Velocity-Vectors at Point C issues , CP⃖⃗⃗⃗ = CA⃖⃗⃗⃗ and CP⃖⃗⃗⃗ ⊥ CA⃖⃗⃗⃗ and the Resultant is the Velocity-Vector C⃡C`AP , Electric-Vector = πσΦ 4 [d]² , Magnetic-Vector = πσΦ 4 [d]² . Similarly on Material Vector CP⃖⃗⃗⃗ carries the Maximum Conjugate-Force CC`= C⃡KA from E-Square CP² to two Equal and Opposite squares – CB`PO ,+ CBAO from CAC`P ≡ +𝐂𝐎𝐀𝐁 −𝐂𝐎𝐏𝐁 ` [⤭] which Rest in the Circumscribed circle π.OA² = 2π( 𝐎𝐀 √𝟐 )², to the Screwdriver Inscribed to , CA⃖⃗⃗⃗ , CD⃖⃗ ⃗⃗ circles , π[ 𝐂𝐀² 𝟒 ] , π [ 𝐂𝐏² 𝟒 ] . Motion to Rest , Before the Halve areas , circles Pass through the equal to CMNH Square where exist Quadrature at Position M and to Half-Area circle π [ 𝐎𝐁 √𝟐 ]² = π[ 𝐎𝐁 ² 𝟐 ] at Position M b-5..This Property of Any Two Equal and Perpendicular Vectors at Point C , issues for any Point N , which has a Screwdriver –Material - Vector- CN . The Quantization of E-Geometry to its Dimensional-moulds : The- One , Two , Three Vectors and Three - Poles Mechanism . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 34 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Figure -8 - b : The Quantization of E-Geometry to its moulds , The Point , the Linear , the Plane , the Space (volume) Mould for E-geometry Quantization , to Opposite. b-6…For (1) monad EA , to Antimonad EC–For (2) AB line to Parallel line MM` - For (3) CE Radius to the CM Square Segment - For (4) KA Segment to the KD Cube Segment . From (3) the Two-Perpendicular Vectors , CA ⊥ CP , Slide on (O,OA) circle and through the Three Poles , A , C , P , carry the Stress σ , of Square ACPC , To the Two Equal and Opposite Squares |COAB| , |COPB’| for still to the Halve Area - Stress - Conservation . For the Rotating Square CMNH exist as Upper-Still-State the Square CAC`P and as Lower - Still-State the Square CBAO , related [ CAC`P ] = 2.[ CBAO ] . In(3)..On Two-Perpendicular Vectors , CA ⊥ CP , Point C` Slides on (O,OA = OP) circle forming the CAC`P to CMNH to CBAO Squares with Prior of (1) Properties .This Sliding Rotating Method is Markos { 2-Vectors 3-Poles Mechanism } which Circles and Squares as Areas Pass from the Point M determined by Position 𝐀 𝐞 , such that issues π r ² = CMNH Square .The Squaring of the circle is π [ 𝐂𝐀 √𝟐 ]² = CM² = |CM`|² , Fig-15, [ 47-51 ] , [62] Remarks : 1… The Two - Vectors Three - Poles Mechanism is a Mechanism from Two ⊥ Vectors in Geometry , where the Action of Energy on their Vectors Square and is Conservated . 2… The Mechanism is a Rotating and Changing-Squares System , on a Circle which changes The Inscribed to the circle Square , changes to 2 –Halve Opposite and equal Squares .The same Procedure issues for the Inscribed and Circumscribed circles to the Circle . 3… Since from Wave-Action of Spaces issues Z − n x Z + n = [2R] 2n . e i.2nθ , this means that ,The conservation of Energy = Work , between the Spaces and the Anti-Spaces happens by Doubling the number n , of the Vertices of the Regular Polygon . i.e. The Surplus energy is stored in 2.n vertices , and on 𝛌𝟐𝐧=√2R2 − R√4R2 − λ²n Sides , where then Energy is divided and spread on the double number of the Polygon Sides. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 35 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 8..[c].. The { TCIC } Mechanism : The Tetrahedron , the Cube , and the Inscribed – Circumscribed to the Circle Mechanism . Figure - 9- : The Active-Conductors of Elements and The Space , Anti-Space formation . In Figure ZA , ZB , ZC , are the three Conductors on the Three Diagonals of Cube with 1 , 2 , 3 , 4 , 5 , 6 , 7, 8 , Vertices . At Point Z is Acting a Complex-Vector Z̅ and which transform the Vibration to the three conductors in Space Pattern . It was shown [61] that The Stresses of Complex-Vectors are Spread in Surfaces only for Still states. The number of Moulds of Stationary-States are → 𝟐n=1−4 Points ≡ 2 , 4 ,8 ,16 ≡ 4–Types← (2) = For the Proton = Nucleus , (4) = Tetrahedron , (8) = Cube = The Neutral-caves , (16) = The Double Number of Vertices for the 8-Stationary Neutral-caves and called TCIC-Pattern →{ The Sphere – IN TETRAHEDRON – IN Cube , IN External Sphere }← The Active-Conductors of Elements and The Space , Anti-Space formation . Point , in E-geometry is considered nothing , without Position and direction while in , MG ≡ Material - Geometry [1 Positive ⊕ and 1 Negative ⊝] . From Nothing (i.e. the Point Z ) to Existence ( i.e. to be another Spherical Point P ) issues the Zero Virtual work law ,where zero Work is the equilibrium of two Equal and Opposite forces on Points and is [⊕ - d - ⊝] . Thus Space [S] is Z-Point ⊕ , Anti-space P-Point ⊝ , and d = ZP = The Conductor of Points Since ,The Position and the Number of Points, create the Spaces so the Only Three elements are → {⊕ , [⊕↔⊝] , ⊝} ≡ [+, 0 , -] ← for Material Geometry exist linearly in moulds . The Stresses of Complex-Vectors Act on a Surface , A circle of Sphere , between the Sub Units Axially and Rotationally following Ptolemy`s and Cebas Theorem of Equilibrium. The Space Anti-Space and , Neutral-Space Conductors are self-Sustaining . Stability : IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 36 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The number of Moulds of Stationary-States are as→ 2n=1−4 Points ≡ 2 ,4 ,8 ,16 ≡ 4–Types← i.e. Monads are formed from the first 4- Moulds as → The 1-Point which is everywhere, the 2- Points which form Vectors , the 3- Points which form the Planes , the 4 - Points which form the Volumes . For n > 4 as this is the Atoms Structure, follows the Sustainability of the Spaces and Anti-spaces. Since Space-Energy Structures follow the M-Geometry Stability these follow the Mould of Spaces 2 n = 3 = 8 Elements which is The-Cube as shown before . Instances , 1.. In the constant , Hydrogen Conservative-Cave-System , which creates only closed Orbit shapes as Circles , Ellipses and Eight-shape , ∞ , are Placed the above elements , π g ≡ Energy ≡ [meter of area * meter of force] ≡ Electrons on Orbits , and also the Unit-Space ≡ Massive-Complex-Unit-Space ≡ →[+�̅�.s²]← jointed through the Neutral Material-Points [ (+) [↔] (-)] with the Strong-force → S F = h .fn ≡ h .{ [S ≡ BP ≡ EM-R ≡ f1=N , f2 , f3, fD ,,f n ] ≡ h.n (1+√5) σ 2πr ≡ h [ n.B̅ 4𝜋2. r4 ] ….(mg) Equations (m) , (mg) are the Energy-Space-constituents in Hydrogen-System . c-1.. The Protons and the Nucleus Structure : From above MP ≡ [(+)[↔](-)] ≡ Neutral = n , then a.. For 1 Proton [(+)[↔](-)]↔(+) ≡ Neutral → n = 1 p = 1 b.. For 2 Proton (+)↔[(-)↔(+)][(+)↔(-)]↔(+) ≡ Neutral → p n n p c.. For 3 Proton (+)↔[(-)(+)](+)[(-)(+)](+)[(-)(+)] ≡ Neutral → [ 0 0 0 0 2⃡ 0 1e 0 0 ] d.. For 4 Proton (+)↔[(-)(+)](+)[(-)(+)](+)[(-)(+)](+)[(-)(+)] → [ 0 0 0 0 2⃡ 0 1e 1e 0 ] e.. For k Proton (k1)↔[(-)(+)](k2)[(-)(+)](+)[(-)(+)](+)[(-)(𝐡𝐤)] → [ ke ke ke ke 𝑘 ∶ 8 − 18⃖⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ke ke ke ke ] f.. The Inscribed to the Tetrahedron-Sphere consists the Nucleus ≡ The ⊕ Space , and in Circumscribed to the Tetrahedron-Cube Sphere consists the Orbitals ≡ The ⊝ Space . c-2.. The Annihilation of the Fundamental Particles . Considering the , Matter Antimatter , with the classical known theory of Annihilation as the common Constant Energy then issue for a Force ↔Anti Force , Action ↔Reaction Electron = e−= ⊝→ mass 𝐦𝐞 = 9,11.10−31 Kg → Charge 𝐂𝐞 = 1,602.10−19 eV → Cave a = 5,0.10−17 m 1.. Electron - Positron  e−→ e+ ≡ 2 γ ≠ 0 The Common Me→p is from relation 1 MT = 1 me + 1 mp = 1 me + 1 me = 2 me = 2 9,11.10−31 = 1 4,555.10−31 From energy equation E = 1 2 Me→p c² = 1 2 [4,555. 10−31].[2,998. 108]² = 6,827945. 10−15 J = J / (1,6022.10−19 eV) = 4,2616059. 10 4 eV = 42,616059 .KeV The common frequency is f e→p = E h = 6,827945.10−15 J 6.62607.10−34 Js. = 1,0304667. 10 19 H  i.e. 𝐟 𝛄 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 37 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . c-3.. The Equilibrium of the Three-Spaces , The Space Anti-Space , Neutral-Space The Stability of Elementary Particles , Conductors and the Atoms -Nucleus : [60] Figure– 9a- : The Electric and Magnetic Field of 3 Conductors On 2 Anti-Parallel-Strands The Waves - Motion ≡ Energy , is Transported and , The Pattern of Disturbances with Informations , Propagates from the One-Edge-Point to the other Edge-Point of Conductors , OR is on [ Sub-Units ] . To Prove the Stability of the System , The Equations of motion become everywhere from any Opposite Space ≡ ⊕ , Anti- space ≡ ⊝ . During these motions as , ds . is Done a Constant-Work , k , or and from their velocities �̅� and this Work may be Positive or Negative for Equilibrium . In F-23 Cycloid Anti-cycloid motion is as , D. The Origination of Gravity g , Antigravity [- g ] In Fig-9a , Conductor D = d = AKA where ⊕ constituent attacks ⊝ by a Hook`s law force as Stress σ = E u , and an Coulomb force F [+ ↔ −] = [⊕↔⊝ ] r² = [2 σ] r² , creating Work ≡ motion IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 38 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . as was shown It was shown that , The Stability of Nucleus Forces , is succeeded through The Pascal`s - Ceba`s triangle launched caves r , while Leptons from their Anti - Leptons existence in the same launched Pascal`s - caves r . [58] The Equilibrium of the Three-Spaces is Done on the three Cyclic-Quadrilaterals (ABEAECE) , (BCEBEAE) , (CAECEBE) , of the 6 Conductors each for , [a , b , c , d , p , q ] x [3-ABC] = 18 Conductors . The Stability is 1..Linearly through the Products of the Diagonal - connectors as , → → → [AAE].[BECE] = [ACE].[AEBE]+[ABE].[AECE] , [BBE].[CEAE] = [BAE].[BECE]+[BCE].[BEAE] , [CCE].[AEBE] = [CAE].[BECE]+[CBE].[CEAE] and , 2..Rotationally through the Ratio of the Diagonals ≡ The Conductors | AKB AKC |x| BKC BKA |x| CKA CKB | = 1 or  Rotational-Equilibrium as , [AKB].[CKA].[BKC]=[AKC].[BKA].[CKB], of all Opposites  Tangential-Electromotive Forces . 3.. From Mechanics-Physics , all Systems Possessing Elasticity ≡ motion and Reaction to motion , the called mass , are capable of free vibration or vibration taking Place in the Absence of External Excitation [7] .This Principle issues for Both Closed or Open Systems. i.e. Either for the Nucleus where issues The Pascal`s - Ceba`s triangle Stability , or for the Orbitals where issues the Tetrahedron-Cube -Sphere-Mould .A Signal as the Electric and Magnetic Field IN 3-Conductors Is transformed On the 2 Anti-Parallel – Strands . c-4... The Natural Electromagnetic Energy-Tetrahedron and , The 3- Conductors Into The Energy - Cube and Spheres : [102] In Fig- 9c - The Electric - Magnetic Field of 3-Conductors which lie on the DABC occurs INTO the {⊝ 8-Anti-Space Cube , [DA1A D1 , B B1CC1]}, and The Positive → { ⊕ 8 - Space – Cube , [DA1A D1 , B B1CC1]} , INTO The {⊝ 16-Anti-Space [ 1 ,2 ,3 ,4 5 , 6, 7, 8 - 9 ,10 ,11, 12 , 13 ,14 , 15, 16] } Sphere Cube on Vertices.= Point [The Point 16 ≡ |⇉|Z|] consists the Navel-String Gate into the Triangle-Circle-System {Z-(∆D,ABC), ∆[T1T2 T3]} of the Stationary Bases which Results on the 2 -Anti-Parallel-Strands . The Dual Photon v̅ [ σΦ 2πr + σ 2πr ] ≡ v̅.[ fn̅ + fn ] , occupies Stresses = σ and velocities v̅ , in The Tiny-caves r . The Colours in light , are the Still-Sub-Units in Storage →[v̅ . fn̅ ]← and exist as Frequencies of → Violet , Blue , Green , with their Complementary Colours Yellow , Orange , Red ← Every 8-electrons are vibrations on Atoms-Cube-Structure {A Tetrahedron in Cube in a Sphere} whether it be Sound or Light . Above Structure is followed by all Compounds. The Molecules are Systems consisted of The PERIFERAL ≡ Skeleton 𝐌 𝟏𝟔 and The CENTRAL ≡ Fittings 𝐌 𝟖 , and the ACCESSORIES ≡ 𝐌 𝟒 or 𝐌 𝟐 , 𝐌 𝟏 . This Property of Atoms and of the Compounds is The Critical - Valve of Switching the motion as this happens in the Electro-magnetic Solenoid Valves with High or low Pressure and flow rates directly or Not . Remarks : Work of Elastic Deformation . Elastic Deformation requires (1) That every New Stress that is applied to a New infinite Element is always added to the existing one .(2) Be always Unity Work of Deformation. When a Body contains Potential Energy , During any Unloading it gives off Mechanical - Work , since the Points of application of the GRADUALLY - withdrawn Loads are Displaced. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 39 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . If the Unloading is ABRUPT , then Kinetic Energy is converted into Potential and after is Kinetic , converted back into Kinetic and so on . That is, the body Performs an Oscillating motion until the Mechanical Energy is extinguished due to a State of Equilibrium or Alters . Figure – 9b- : The Electric - Magnetic Field of 3 - Conductors Results on 2 -Anti-Parallel - Strands .The Stability of →{ ⊕ 4-Space [D A B C]} Regular - Tetrahedron , IN {⊝ 8- Anti-Space Cube , [ DA1B D1 , C C1AB1]}, and The →{ ⊕ 8- Space [ DA1BD1 , CC1AB1]} Cube, INTO The {⊝ 16-Anti-Space [ 1 ,2 ,3 ,4 ,5 , 6, 7, 8 - 9 ,10 ,11, 12 , 13 ,14 , 15, 16]} Sphere Cube Vertices . The Point [16 ≡ | ⇉ |Z|] consists the Navel - String Gate in System . and are { Z - (∆D , ABC) , ∆ [ T1 T2 T3 ] } Stationary Bases . The Stability Analysis in [88] c-5.. : The Mater Antimatter 3-D Origination Mechanism [⊕↔⊝] Figure – 9c- : The 4-Dimentional Spinning , Tetrahedron - Cube – Spheres - Mould . The Stationary-States are → 2n=1−4 Points ≡ 2 , 4 , 8 , 16 ≡ 4–Types ← IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 40 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Figure – 9d.. :The Hydrogen , Carbon , Oxygen , Atoms-Conductors Into the Sphere-Cube- Mould . Deka hexahedron , Octahedron ≡ Cube , Tetrahedron ≡ 4-Points , Triangle . The Nucleus ≡ ( In-Sphere ) , Tetrahedron ≡ ( D - A.B.C ) , Cube ≡ ( O,2,A,4,B,6,C,8 ) , Orbitals ≡ ( Ex-Sphere ) , Mould . When Vector ≡ 2-Points ≡ The Closed-Conductor , 1-Point ≡ The Opened -Conductor ≡ The Bracket-Hook . Oxygen is the Element which fills 8 - 2 = 6 vertices of Cube as EL → { 1𝑒 0𝑒 1𝑒 1 𝑒 2⃡ 0 𝑒 1𝑒 1𝑒 1𝑒 } ≡ (± 2 2 ) , where ⓪ is the first , Zero-Acting Cube filled with 1e , therefore is the only Steady element which occupies the first Energy-Level ① with 2 Electron-Positions as ① = 2e = 2⃡ Positions at 1⃗ , 1⃗⃖ The Electron being in the Hydrogen Precesses because of the different axis of Rotation and Nutation`s , from the continuous and immense-communication to the effect of gravity g . Since Electron is continually oscillating , with the Nutation-frequency 𝐟 𝐍 , so Produces a continuous and oscillating Magnetic-field –M- which in turn is the source of an oscillating Electric-field – E - which implies the Regeneration of each other , i.e. it is a Propagating Electromagnetic-Wave where E = B c , and with a quantum-Energy E = h fN or E = 2μ.B , This Energy E consists the Hydrogen Bracket Hook [HBH] , or [BH⃗⃗⃗⃗ ⃗] and it is the only free monad which can Bond to all Energy Structures . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 41 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Spheres-Cube-Structure of Atoms In Sphere Center –O- allows The Stability of the System, Per–Two electrons in Dipole Positions , 1 ↔ 6 , 2 ↔ 5 , 3 ↔ 8 , 4 ↔7 , consisting The-Positive Dipole-[BH⃗⃗⃗⃗ ⃗] , to be able to move → Into-In- Outside it ← and allow the Atoms- Bonding as this in Energy levels . The Cube-Structure of Atoms and Orbits in the Out- Sphere formulates +[BH⃗⃗⃗⃗ ⃗] to those Dipole Positions which lack of one electron and so can PLUG-In the LOOP of the others . The Atoms with Zero Stable State are , The Hydrogen Bracket-Hook Into the Carbon encloses the Inscribed Sphere for the Nucleus , Into the Oxygen for the Orbitals , Into the Sulfur enclosed into the Circumscribed Sphere for the Atoms-Conductors Sustains . The First 4-Dimentional Spinning ,Tetrahedron-Cube–Sphere-Mould are , 9c--2 1… O-1-3-5-7 - → The First Solid Tetrahedron Structure in Nature , 2… O-1-2-3-4-5-6-7-8 → The First Cube with max. Positions e = 2.n² = 8 , 3… R - [ O , A , B , C ] → NUCLEUS IS IN the First Inscribed Sphere into Tetrahedron. 4… R-O-1, 2, 3, 4, 5, 6, 7, 8 → ORBITS ARE IN Enveloped Cube-Sphere of Tetrahedron. Deka Hexahedron , Octahedron ≡ Cubes , Tetrahedron ≡ 4-Points , Triangle ≡ 3-Points ,Vector ≡ 2-Points ≡ The Closed-Conductor , 1-Point ≡ The Opened -Conductor ≡The Bracket-Hook . The Hydrogen Nucleus is kept in the Inscribed to the Tetrahedron-Cube-System where exist the Protons and Neutrons , while The Electrons of Hydrogen Orbits are kept into the Circumscribed , to the Tetrahedron-Cube-System . Both Equilibrium by following the Menelaus - Ceba`s Principles . [70] Any Two Perpendicular Energy-Vectors , |𝐐 ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↔ | , |𝐐 ⇈ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↕ | of this Propagating Energy , FORM The Energy-Square , |↔|.|↕|≡|𝐐 ⇉ |.| 𝐐 ⇈| ≡ q0 , on Conductor AKA and Then Energy-Circular-Prism = |AKA| .{𝐐 ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ }.{ 𝐐 ⇈ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } ≡ [�̅� 𝐱�̅� 𝐲].|AKA| = | ↔ |.|↕|.|AKA| , i.e. An Energy-Circular-Cone which is a Moving Energy -Volume , and Because from Cauchy equations of Stresses are of three Dimensions , then The Energy Stresses remains Flat ONLY- WHEN the Plane-Section is a circle [𝐄𝟐 ] and The Square becomes circle, Anti-Quadrature.. Because from Mechanics , the Hydrogen Nucleus is kept in the Inscribed to the Tetrahedron-Cube-System where exist the Protons and Neutrons , while The Electrons of Hydrogen Orbits are kept into the Circumscribed , to the Tetrahedron–Cube-System . Both Equilibrium by following the Menelaus-Cebas Principles . Oxygen is the Element which fills 8 - 2 = 6 vertices of Cube and for Neon 10 -2 = 8 , as Ox → { 1𝑒 0| 0/ 1 𝑒 2|⃖⃗ ⃗/ 1 𝑒 1𝑒/ 1𝑒| 1𝑒 }≡ ( −2 +2 ) , ② ≡ 𝐍𝐞 1x8 2⃡ = 2⃡ + 8 = 10 → { 1 1 1 1 2⃡ 1 1 1 1 }, The Possible Orbits of Oxygen are two as , → 1e /-- 2⃡/-- / 0 ← and → 1e |-- 2|⃖⃗ ⃗-- 0 | ← for all the 8-Cube Positions . ⓪ is the first , Zero-Acting Cube and is filled with 1e , therefore is the only Steady element which occupies the first Energy-Level ① with 2 Electron-Positions as ① = 2e Positions at 1⃗ , 1⃗⃖ . In the same way ① is the first One-Acting Cube and is filled IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 42 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . with 2e, and so on , The Electron being in the Hydrogen -cave Precesses because of the different axis of Rotation and Nutation`s , from the continuous and immense-communication to the gravity g . Since Electron is continually oscillating with the Nutation-frequency 𝐟 𝐍 so Produces a continuous and oscillating Magnetic-field –M- which in turn is the source of an oscillating Electric-field – E - which implies the Regeneration of each other , i.e. it is a Propagating Electromagnetic-Wave where E = B c and with a quantum-Energy E = h fN or E = 2μ.B , This Energy E consists the Hydrogen Bracket Hook HBH , or [BH⃗⃗⃗⃗ ⃗] and it is the only free monad which can Bond to all Energy Structures . The Sphere-Cube-Tetrahedron Structure of Atoms In Sphere Center –O- allows to issue The Ceba`s Theorem for 6-Stresses on the Inscribed Tetrahedron which Secure The Stability for the Nucleus-System , and for the Outer Electron-System the Stability is Obtained Per–Two electrons the Dipole Positions 1 ↔ 6 , 2 ↔ 5 , 3 ↔ 8 , 4 ↔7 , consisting The- Positive Dipole-[BH⃗⃗⃗⃗ ⃗] , to be able to move → Into-In- Outside it ← and allow the Atoms- Bonding as this in Energy levels . The Cube-Structure of Atoms and Orbits in the Out-Sphere formulates the +[BH⃗⃗⃗⃗ ⃗] to those Dipole Positions which lack of one electron and so can PLUG-In the LOOP of the others. Conclusion : 1… The Forces of Atom`s NUCLEUS , are Holded from → The Inscribed Sphere In the Tetrahedron , which follows the Ptolemy`s and Ceba`s Vectors Structure . 2… The Forces of Atom`s ORBITALS , are Holded from → The Into the Cube , Circumscribed–Sphere Complex Vectors Structure . 3… The Forces of Atom`s POSITIONS , are Holded from → The on Cube Inscribed Tetrahedron Complex Vectors Structure . 4… The Stability of the {TCIC} Mechanism follows the Ptolemy`s & Ceba`s Complex Vectors Structure [62] The Action of the Complex–Vectors create the Waves of Motion ≡ Energy Waves which are closed curves and Transported . The Pattern of Disturbances with the Informations Propagates from the One-Edge-Point to the other Edge-Point of Conductors , OR is on the [ Sub-Units ] . Coulomb force [⊕→⊝] = F [+ ↔ −] = [⊕↔⊝ ] r² = [ 𝟐 𝛔 ] 𝐫² = F = σ .A = [ 2πrf Φ ].A = w r.[ 𝐀 𝚽 ] = v .[ 𝐀 𝚽 ] , and Acts on a Surface , A , between the Sub-Units Axially and Rotationally following Ptolemy`s and Cebas Theorem of Equilibrium . Figure –9e- Remarks : The Work of Elastic Deformation On the { TCIC } Mechanism : Every New Stress that is Applied to D - Element is always added to the existing And the Unity Work of Deformation is transformed to three Tetrahedron Sides. When a SIGNAL Strikes the TCIC – Cube`s Mechanism . The Signal is Energy-Wave carrying motion . Unloading happens on the three Still-Spaces in Tetrahedron - Sides which withdrawn Loads are Displaced On the Anti-Parallel-Strand . If the Unloading is ABRUPT, then Kinetic Energy of 1-Strand is converted in the 2-Strand as Potential and so on . That is , CUBES Perform an Oscillating motion until the Mechanical Energy is extinguished to their Vertices , which are the stores of Information . Anti-Parallel-Strand is the Body containing the Potential Energy , During any Unloading on CUBES , it gives off Mechanical-Work , since the Points of application of the Gradually - withdrawn Loads are Displaced . The Stability of Kinetic – Potential Energy is continues . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 43 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Figure –10- The Hydrogen BRACKET – HOOK of 2 = Orbitals  [ (+) [↔] (-)] Stationary-State happens as → [⊝] ≡ ORBIT-1 ↔ [⊕] ≡ ORBIT-2 ← 8..[d].. The { TOBH } Mechanisms Atoms Bonding Bracket - Hook Mechanism . d-1.. : The Precession and Nutation of Electron : Gravitational constant G , is The Pulling and the Cohesive Force on all the Quantized Energy Structures which communicates with everything due to the Periodic excitation on all Spaces.[84] The Newton`s laws issues for the masses and also the same to Electrons in caves as below , G ≡ g.kE ≡ g .[gL kL ] ≡ [ T²P 𝐚³ ].[gL kL ] ≡ 9,8078925* 6,8116.10−12 ≡ 6,68056.10−11 𝐦³ 𝐍𝐬² The Electron being in Hydrogen-cave Precesses because of the different axis of Rotation and Nutation`s , from the continuous and immense-communication to the gravity , g . Electron-Spin is the Angular-momentum-vector B̅ and rotates according to equation dB dt = [u̅B̅] = u B.[k̅k̅`] in Gravitational Potential Ug = [ mg] .s.cos θ = - s Q . [k̅k̅`] so the change of angular momentum B̅ is → dB dt = u = s.Q B = s.Q J3.w3. ← and from the 1-degree equation of motion u , become �̈� + w ² u = 0 , with solution a Period of Nutation T = 2π u = 2π.J3w sQ , and its natural Frequency f N = sQ 2π.J3w ….(1) with Total Energy of Nutation E N = J1 2 [w1² + w2²] + J3 2 w3² ….(2) [70] . For Material-Points , the chains of Spins due to Periodic excitation [↔] is created in Orbit a Magnetic field due to LRC-circuit and which is tuning to the critical Quantum critical-State 𝐠𝐆 . The Light velocity vector �̅� = �̅� , by Acting on cave , r = 𝐋 𝐏 , finds the Impedance m g , and becomes the Centrifugal-Force 𝐅𝐠 of Cave equal to Gravity g . The chains of Spins for the , ONE-WAY Pointy vibrating , is the Resonance – Frequency fR = (1+ √5 ] ) .σ 4πr = σ 𝛷 2πr of vectors v̅ and B̅ , of the-Stationary-Photon-cave , where B̅ ≡ S̅ ≡ Spin , i.e. The fR is common to both . The Moving Electron in Orbit of charge �̅� ≡ ⊝ , energy E with the Orbit -Velocity - Vector �̅� =√ 2 𝑚 [E − { 𝐤 𝐫 + 𝐋 𝟐 𝟐𝒎 𝒓𝟐 }] which creates IN Orbit , r , the Varying and Perpendicular Magnetic IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 44 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Field �̅� , which in time-turn Creates an Electric-field �̅� ⊥ �̅� , with the Resultant force F acting on Electron Velocity �̅� and which is composed of the 𝐕𝐩 , Perpendicular to the Magnetic-circles O ⊥ B , and 𝐕 𝐯 , Parallel to the Magnetic-field-Vector |�̅�| , is tending in such way that L ≡ S ≡ Spin . The resulting motion of Electron in trajectory is the Helical motion and Work Produced during this motion . The Conservation of Energy exist in Orbit between Protons and Electrons [p↔ e] therefore Energy as frequency exists in Orbit only , and differs of those of Energy- levels. Since frequency 𝐟 𝐍 = 𝐟𝐑 = 2, 839845.𝟏𝟎𝟏𝟎 s−1 and exists in all Atoms , due to the Hydrogen first cave ,then this 𝐟𝐍 is the Resonance - frequency 𝐟𝐑 between all Atoms and Molecules. It was proved that → Since fR = σ 𝛷 2πr and related to Hydrogen cave r and Stress σ only ← It is the Unique Common and Neutral-Monad , in all Chemistry structures and Composites . d-2..Energy from equation E= h.𝐟 𝐍 = 6.62607.10−34 . 2,8398447.1010 = 18,817009.10−24 J / (1,6022. 10−19 ) = 1,174463 .10−4 eV , and conserved as Thermal-Energy 𝐄 𝐓 in kilo Cal and which is continually increasing as , E T = 18,817009. 10−24 J / [(4,19. 103 ) = kcal] = 4,49093. 10−27 kcal . Since 1eV = 96,363636 kJ/mol thermal energy E T = 4,49093. 10−27 kcal / [96,363636.kcal/mol=1eV] = E T = 4,660399.10−29 eV. This happens because of the closed Energy - Orbit-Rims r , constant light velocity c , and from Spin equation S = r m c . Taking into consideration the Thermal - Energy of a Photon when it is Pressing the surface of 1 m2 for 60s then , 𝐄 𝐏 = 20 kcal = 20.(4,19. 103 ) = 8,3777.104 J , and the ratio , [ E T / E P] = 4,49093. 10−27/20 = 2,24546.10−28 , which is a Quantity Not - detected . The Hydrogen caves created in the Sun 1 Million Years-ago =106.365.24.3600 =3,1536.1013 sec ,is accumulated a Thermal-Energy of a magnitude E Therm= 3,1536.1013.4,49093.10−27 = 1,41626.𝟏𝟎−𝟏𝟑 kcal , i.e. The Stationary Hydrogen-wave–cave needs 1- Quadrillion years to conserve 1-kcal Thermal-Energy . For Half-frequency fR / 2 =1,4199223.1010 s−1, the Kinetic Energy is Zero and the Potential-Energy is U = E = h f = 6,62606957.10−34 . 1,4199223.1010 = 9,4085039.10−24 J / (1,6022.10−19 eV) = 5,8722405.10−6 eV , which agree with the Bohr-Magneton . Remarks : 1.. The Real Shear - Stress that exists in the Force – Carriers . From the Static Theory of Equivalent Tension (Moore's cycle) is (1) = σ = √(𝛔𝟏 − 𝛔𝟐)² + 𝟒 𝛕𝟏𝟐 , and σ1,2 = (σ1+σ2 )/2 ± (½)√(σ1 − σ2)² + 4 τyz² , instead of the classical case which is σ = 2.τ max = √σ1 2 + 4. τ ² , . (2) . Because nature always follows the uniform tension so it follows the (1) case. where exist the equation σ = √(𝛔𝟏 − 𝛔𝟐)² + 𝟒𝛕𝟏𝟐 , and σ 1,2 = (σ1+σ2) /2 ± (½)√(σ1 − σ2)² + 4τyz² = σ 2 [√5+1 ] = σ.[ √𝟓+𝟏 𝟐 ] , which is the Golden ratio for stresses . 2.. The Bernoulli Astute about Stresses-Linear-Distribution is equallyTrue for Bending too. 3.. In the case of concentrated Stresses (such as Closed or Open Conductors) ,the Doubling of the main Stresses Observation is True , and this because of the Golden ratio Pattern. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 45 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 1d.. The Spaces and the Energy States : In Articles [87-89 ] is Elucidated the Duality-Photon as Particle and as Wave . The Photon`s Charge �̅� Photon = v.σΦ 2πr , Transfers Information`s and Energy-Storage in Nature . In Articles [87-88-89-90 ] are Elucidated The Gravitational constant G , The Stresses σ , The Light-velocity �̅� , The Gravity �̅� , The Hydrogen cave H , The electron - �̅� , The electron-charge �̅� 𝐞 , The Weak-Forces �̅� 𝐅 , The Strong – Forces 𝐐𝐓 , and The Uniform-Resonance - Frequencies 𝐟 𝐑 , for Atoms Structure . In Articles [91] is Shown How ,The Three Circularly-Placed Physical-Charged- Breakages on The-Three Spaces [⊕ -  - ⊝] , Or on the 3-Extreme -Triangles of STPL- Mechanism which is The -Thrust for The-Origination of all the Cosmic- Particles in Linear & Rotational Conductors Frequencies , either into the Atoms or to their Compounds [91] . 9a.. THE FLOW - PLAN OF THE → SPACE - ENERGY ← UNIVERSE . 1.. It was shown in [9-18] what is Primary-Neutral-Space [PNS] as well as what is Infinity [15] , and Rotational Energy ≡ Λ ≡ Torque ≡ momentum [22] so [PNS] → [A𝐏𝐀 ↔ 𝐏𝐁 B] ≡ Work W =|EP |= [ |Λ|. + Λ x ] → W = ∫ P.ds = 0 → and so exists Time T = ds / v = 0 . The Cause is , because Primary Point , A , is nothing and then is Quantized as → an Point B (which is following the Principle of Virtual Displacements W = ∫ P.ds = 0 = Force x Displacement = Energy x Space , and according to the ancient Greek Philosopher Anaximander [ The non-existent ( i.e. the Point A ) , Exist when is Done , it occurs as ( Point B ) i.e. ≡ [ Το μη Ον , Ον γίγνεσθαι ] = [ Το Τίποτα υπάρχει Όταν αυτό Γίνεται ] Time is Not existing because ds = ∞ . 0 = constant [21-22] . The relative Range is the Displacement AB → which is The Monad-Space as AB = 0 → k → The Infinity Anti-space. 2.. [PNS] The Energy-Quaternion [ |Λ|. + Λ x ] ≡ �̅� ≡ [ λ ± Λ x ] ≡ |�⃡�𝐨| 𝐰. 𝐞^{ [ �⃗⃖� i / √ Λ' �⃗⃖� ] . [Arc Cos (w|λ|/2 . |√ �⃡� `𝐨. �⃡�𝐨|] } which is the Beyond-gravity Forced - field . [25] . As in case (1) → there is No change of ds → so Time T = ds / v = 0 . The Cause is the Lever arm moment of the Primary Forces and is Quantised as the Torque , < The Torque modelling of microscopic description > , and later the Momentum [19] . The relative Range of the Infinite Points is the Displacement AB = �⃡� to Infinity ∞ . 3.. [PNS] → The Temperature-Quaternion[ λ , ± Λ x ] = z o = Λ = n.R.T / V( λ = C) → The Gas equation → No change of ds → so Time T = 0 . The Cause is the Heat causing vibration on molecules and is Quantised as → Intensity ( Pressure σ = 𝟐𝛑𝐫.𝐟 𝚽 = 𝟐𝛑𝐫.𝐜 𝚽.𝛌 = IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 46 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 𝟐𝛑𝐫.𝛔 𝛌 ) between them . The relative Range of the Infinite Points in Displacement is AB = λ = 𝟐𝛑𝐫 which is the wavelength λ. 3.a → k1 ≡ z = |𝑧 0| . e - i. ( 9.π /2 ).10 = The Energy Under Planck length , The Tank Cavity of Gravity where �̅� 𝐄 = 0 and ET = Λ.v B + Λ x v.�̅� and is the accelerating removing , Rotating energy Λ to v�̅� , m = 0 and ET = Λ.v.B + Λ x v.�̅� and it is the linearly removing , Energy Λ towards v B , so No change of ds → therefore Time T = 0 . The Cause is the High Heat Conservational Balanced Tank of gravity and is Quantised as The Fundamental Particles ( Bosons and Fermions) . [30] The relative Range is of the Infinite Points in Displacement |AB| to Infinity . 3.b → k2 ≡ z = |𝑧 0|.𝐞− 𝐢( 𝟓𝛑 𝟐 )𝟏𝟎 = The Energy In Planck length → ∞ changes of ds → so Time T = t . Cause are the Infinite changes of Space and is Quantised as → Matter , Energy and the Existent as caves . The Material-Points occupy a relative Range in Planck`s length as In Fig-1- 3.c. → k3 ≡ z = |𝑧 0 =Λ|. e - i. ( π / 2 ) . 10 = The Black Hole Temperature , Balanced Tank Energy length , where PV = n.R.T and ( PA = Wd = σ.T 4 ) → No change of ds → so Time is T = Constant . Cause is the Very high Heat causing vibration on molecules and is Quantized as → Intensity = ( Pressure = Fd = C. �̅� = ± Co .w.[√ A² - x²] ) i.e. Cause → (Constant Co) → Quantized as New monad and relative Range is Infinity , or The meter of Space-Energy- changes is time T , which exists in k2 Quantized-Region only. From Quaternion equation �̅� = [s  �̅�. i] = [s² ±�̅�.s²] = [ �̅�↔�̅� ] = [ +|�̅�|² ↔ - |�̅�|² ] is seen that motion , +|�̅�|² at Point A , is transferred as motion - |�̅�|² at Point B . This issues from Quaternion equation �̅� = [s  �̅�. i] = [s² ±�̅�.s²] = [�̅�↔�̅�] = [ +|�̅�|² ↔ - |�̅�|² ]. Fig -1- i.e. The Vectors-Sum of , n , Vectors , either On The Same Linear-Direction Or , On Non Linear-Directions Results on The – Resultant - Vector �⃗⃖� of the Linear-Vibration and between the Quaternion Edge-Poinds [ �̅�↔�̅� ] . Conclusions –1- : Energy In Primary-Neutral-Space [PNS] and , In Under Planck1s length ≡ The Tank Cavity of Gravity is an Standing Wave with NO-Time existing as T = ds / v = 0 . Conclusions –2- : Since In Primary-Neutral-Space [PNS] and , In Under Planck1s length ≡ The Tank Cavity of Gravity existing as T = ds / v = 0 . the only meter of changes is the Period of Rotation of the [⊝ ↻ ⊕] as in the next Figure-11 . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 47 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Figure –11-The Rotation of Opposites [⊝ ↻ ⊕] and the How and Why Spin is equal to 1 , 𝟏 𝟐 In (1). The Glue-Bond-Pair of Opposites [⊝ σ ⊕] of the Rectilinear motion for both → Great circles and the Small circles , creates on the Stress-common-curve Rotation of circle of radius r , the velocity v̅ = w.r = 2π T .r = 2πr. f = [ σ 2 ].(1+√5) = ± σ.Φ , and the frequency f = (1+ √5 ] ) . σ 4πr = σ Φ 2πr , of a Period T = 4πr σ(1+√5) = 2πr σ.Φ and , ± σ , where then exist two equal and Opposite Forces , → the Centripetal , Fp , and the Centrifugal , Ff force . The Produced work as Energy is → E = h f = �̅� = Momentum = r m v = r [ 𝜋 𝑟² 2 ] w r = [ 𝜋 r4 2 ]w = [ 𝜋 r4 2 ] 2πf = π². r4.f = �̅� , where f = (1+ √5 ).σ 4πr = σ.Φ 2πr , meaning that the Stress - Pointy Rotation creates the Angular – Momentum and momentum as , �̅� , �̅� , in the Zero Wave-note . [70] The Glue-Bond Pair of Opposites [⊝ - ⊕] in the → Left Direction of Small circles creates a Rotation on circle of radius , R , with velocity v = w.2r = 2π T .2r = 4πr. f = [ σ 2 ].(1+√5) , where frequency f = (1+ √5 ] ).σ 8πr = σ Φ 2πr , Period T = 8πr σ(1+√5) = 2π(2r) σ.Φ , where ± σ are the two equal and opposite Centripetal ,Fp , Centrifugal ,Ff force , with Energy → E = h.f = h.σΦ 2πr , in One Wave- note. The Glue-Bond Pair of Opposites [⊝ - ⊕] in the → Right Direction of Small circles creates a Rotation on circle of radius , R = 2 r , with velocity v = w.2r = 2π T .2r = 4πr .f = [ σ 2 ].(1+√5) , where frequency f = (1+ √5 ] ) . σ 8πr , Period T = 8πr σ(1+√5) = 4πr σ.Φ and the ± σ are the two equal and Opposite Stresses , Or Centripetal , Fp, Centrifugal , Ff force . Energy is → E = h f = (1+ √5 ] ) .σh 8πr = σΦh 4πr , in the One Wave - note . In (2) The hollow-Cone of 90ᵒ , between Angular-momentum-vector �̅� and Angular-velocity- vector �̅� when illuminated by a circularly Polarized light beam , then any changes in Spin IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 48 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . ≡ �̅� of the Polhode Cone POT and the exchange of Linear and Angular Momentum between Electromagnetic fields and the material Media are shown as the Profiles of the Phase and the Angular-velocity-vector in the POT- cone cross-sectional Plane , i.e. The Space can be eternally twisted but cannot disappear . [70] Measurements of Physical-Properties such as Position , Momentum , Spin and Polarization , performed on Entanglement Particles gives rise to seemingly Paradoxical effects considering Systems as a whole and the ERP Paradox . The answer is given in Figure 1-(the right-2) where is shown each Property of the Material-Point . The Spins [⊝ ↻ ⊕] Presuppose [⊝ , ⊕] Material-Points which are The caves in PNS Space . Conclusions - The Energy in Spins , Presuppose that the Caves to be in [PNS] Space and in the Under Planck1s length . The ± Spin is created from the Opposite Centripetal , 𝐅𝐩 , Centrifugal , 𝐅𝐟 , forces . The Energy E is initially absorbed in [PNS] and in all quantized regions , either as momentum (mv) or as Rotational Momentum (Λ = r. mv) . From momentum relation �̅� = m r v = m r² w = m r² (2πf) = J.w 2 = [ πr⁴ 4 ].[2πf] = π ²𝐫⁴f 2 , then the Mass of the elementary Particles is → m = 2π³𝐫⁴,f ² r²2πf = 𝛑²𝐫² 𝟏 f = [ 𝛑𝐫.𝐯 𝟐 ]v , Or 𝐦 𝐄𝐏 = 𝛑.𝐫² 𝟒 = π.(1,07.10−7)² 4 = 8,992023.10−15 Kg , in the min Planck`s cave 𝐝 𝐏𝐜 = 1,07. 10−7 m → which agree with the Diffraction Energy mechanism for all Space Levels of quantization and which are the Energy Particles only ,where for base e = 2,71828 and k = 0 → 𝐋 𝐯= e^i.( ± π/2).b the Space is e^ (-15,7079) = 1,78118 .10 ̄ 7 m Since velocity v = s / t = distance / time then for constant s , then v is also constant and t = T , the Period of vibration and for any constant momentum Λ = λ2 m2 v²2 / 2 = λ2 m2 [wλ2 ] ² = 4π². m2 . λ³2 .f ² = [4π². m2 . λ³2 ] f ² = Co. f ² = Λ , i.e. In all Rotational Systems time T is constant and equal to the Period of Rotation which is the meter of changes . For monads in k2 ≡ PNS region ET = [Λ +Λ x ] = [Λ.M+ΛxM ] = √ m. v²E + [Λ. vB + Λ 𝐱 vB ]² = = √ m. v²E + [Λ. vB + S ² ] ² = h . f = h / T = h .v / λ , i.e. In every monad issues λ.mv/2 = h f = λ Λ and Kinetic-energy Λ = h f / λ = Co.f where the constant 𝐂𝐨 = E/ f ² a gauge magnitude depended on the angular velocity �̅� , the velocity �̅� , and then wavelength λ is conserved as momentum (mv) or angular momentum ( Λ = r. mv̅ = m. wλ²/4) or both . Considering oscillatory motion as the simplest case of Energy dissipation , and this because any other motion is not differently conserved , then The Work embodied in Dipole is an < Spring-mass System with viscous dumping > with co variants , energy E , mass m , velocity �̅� = �̅�E = ẋ wavelength λ , and then Energy dissipation is the damping force equal F d = C ẋ where C is a constant and ẋ the velocity. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 49 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Following the steady-state of displacement and velocity then → x = A. sin(wt - θ) ← and ẋ = w.A .cos(wt-θ) = w x , where the Energy dissipated per cycle is the executed work which is , W d = F d .dx = ∲ C x̅ dx = ∲ C x̅ ²dt = π C w A² = (π/4) C wλ² = [π²/2] C. f . λ² → The Energy dissipated per Cycle Wd by the damping Force 𝐅 𝐝 , is mapped by writing velocity ẋ in the form ẋ = wA.cos(wt - θ) = ±wA.[√1 − sin²(wt − θ) ] =  w . [ √A² − x² ] where then the Dumping force F d = C. ẋ = ± Co .w. [ √A² − x²] and by rearranging [ cos²(wt-θ) +sin²(wt-θ) =1] then is Fd ² / [ Co .w. A ] ² + x² / A² = 1 , i.e. Energy = motion is mapped as an Ellipse with 𝐅 𝐝 ,x mapped along the vertical and horizontal axes respectively and the Energy Dissipated Per - Cycle is given by the area enclosed by the above ellipse . The Total Energy E = λ2 . Λ which is embodied in monad AB̅̅ ̅̅ is moving as an Ellipsoid in the Configuration of co variants λ , m , v = �̅�E , as Kinetic ( Energy of motion Ω̅ ) and Potential Energy (The Stored Energy in λ , m , v) by Rotation and Displacement , on the Principal axis (through the centre of monad ) , which is mapped out , as in Solid material configuration by the nib of the Rotational-Vector (Ω̅ ) = δ r̅ c) = [ v̅ c + w̅ . r̅ n] . δt , as the Inertia-ellipsoid [ The Poinsot's - Ellipsoid of construction ] in an Absolute Frame which instantaneously Rotates around vector axis w̅ , θ , with the constant Polar distance w̅.Fd/|Fd| and the constant angles ϑs , ϑb , traced on Reference - Cone [Body Frame] and on [ Absolute Frame ] cone , which are rolling around the common axis of , w̅ vector , without slipping , and if (Ω̅ ) = Fe , is the Diagonal of the Energy Cuboid with the three dimensions a̅ , b̅ , c̅ then follow Pythagoras conservation law , with the three magnitudes [J,E,B ] of Energy- State following → Cuboidal , Plane , or the Linear Diagonal direction . [ 25 - 26 ]. The Transportation of Energy on Edge - Points happens in Material Geometry , while the Transportation of the Edge-Points as , Positions , exists in Vector Geometry only. Since in Vibrations the Deformation of the Spring is an equilibrium Position in ∆ , and the Spring-Forse k ∆ = w = m g = Gravitational force w acting on mass m , then exists → m �̈� = ΣF = w = k [ ∆ + x] ….(1) , and because k ∆ = w we obtain w ²n = k/m.. (2) and (1) becomes → �̈� + 𝐰 ²𝐧 = 0 ,…(3) ← i.e. The Transportation of Energy is a Harmonic - Vibration in cave ∆ , means the Wave Pattern . It is shown [92] that in PNS-Space the two equal and Opposite Stresses ± σ , are the Two Centripetal , 𝐅𝐩 , Centrifugal , 𝐅𝐟 force where Energy is → E = h f = (1+ √5 ] ) .σh 8πr = σΦh 4πr , in the One Wave - note .and the frequency f = (1+ √5 ] ) . σ 8πr , Period T = 8πr σ(1+√5) = 4πr σ.Φ . 10.a.. The E-Spaces and the Energy Regular - Polygons . a. THE GEOMETRICAL SOLUTION : a.. The Even and Odd n-Polygons [ 80 ] : IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 50 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Figure –12a- An Even and an Odd n-Polygon in circle O ,OA with diameters , Ak A2k , Passing from Point A2k , as Vertex (apex) of the Polygone , and diameters , Ak+2 M1 To be Perpendicular to Side A1A2 . The Projections of the Heights 𝒉 𝒏 . Let be the n-Polygon A1 , A2 , A3 , Ak , Ak+1 , Ak+2 , A2k , in circle (O , OA1) , (e) a straight line not intersecting the circle d1 , d2 , d2k , The heights of the vertices to (e) line , h1 , h2 , h2k+1 , The heights of the midpoints Mk Mk+1 of the sides to (e) line φ h−λ , , The Angle of the vertices and the ( ⏊ ) Semi -Sides , and OK = h , The height from the center O to (e) line . To Proof : In any n - Polygon , The Sum , Σ = Σ (h) , of the Heights , d1 , d2 , d2k , of the Vertices A1 , A2 , A3 , Ak , Ak+1 , Ak+2 , A2k , where n = 2k , from any straight line (e) is equal to Σ = Σ (h) = n . OK = n . h . The Geometrical Proof in [ 80] Remarks : 1.. In the System of Regular - Polygons the , Interior angles ( w ) and Gradient ( φ ) , Heights ( h ) and their differences , Δh , – Summation and Subtraction of Heights are Interconnected and Intertwined at the Common Circle [ A , Δh = 𝐡 𝐀- 𝐡 𝐁 ] Producing the Common ( n+1 ) , Odd – Regular - Polygon . This Property allows to refer the Regular - Polygons to the Tangents of any Triangle A,B,C, as it is the 8..[a]  { STPL } Mechanisms : The Six Triple Points Line Mechanism . i.e. The Fundamental Particles Origination happens from The Electric and Magnetic Field of the 3 Conductors On 2 Anti-Parallel-Strands , Fig– 9a- 2.. 𝐬𝐢𝐧 𝛗 = ( 𝐡 𝛗 𝛌 𝛗 ) , is the Harmonic - Mean between [ 𝐡 𝐀 - 𝐡 𝐁 ] , [ 𝟏 𝟐 (𝐫 𝐚 − 𝐫 𝐛) ] IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 51 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Force Harmonic Vibration & The Unbalance Vibration System : Figure –12b- The Complex Frequency Response from the Vector-Force-Polygon is seen that the terms of the Forced Harmonic Vibration , ( m ẍ + cẋ + k x ) = 𝐅 𝟎 .sin(w t) , are Perpendicular to the y-y axis .When the force changes to = 𝐅 𝟎 .cos(w t) , then the terms are Perpendicular to the x-x axis , i.e. Vector-Force is 𝐅 𝟎 .[ cos(w t) + sin(w t) ] = 𝐅 𝟎 .e i.[w t] The Rotation Unbalance Vibration System : For an Unbalance-Rotational System , the Source of vibration excitation IS considered an Spring mass System constrained to move in the vertical dirtection and is excited by an Unbalanced rotation. The Unbalance is represented by an eccentric mass m ,with eccentricity e , and Rotates with angular velocity w , on the Nonrotating mass M - m , with displacement x + | 𝐞 √𝟐(𝐧) | , The equation of motion is , [ M-m] ẍ + m d² dt² [x +| e √2(n) |] + c ẋ + kx = 0 , which can be rearranged to → M ẍ + c ẋ + k x = m | 𝐞 √𝟐(𝐧) | ← …... (c-2) . as in Fig-12b-2. By tuning the System to the frequency of the exciting force Fo= 1 such that w ² = k 2 m 2 , then System acts as an Vibrator-Absorber and reduces the motion of the main mass M ≡ m16 ≡ m1 and m 8 ,4 ,2 ≡ m2 ≡ m , to Zero . Making the Substitution , w11² = k 1 m1 , w22² = k 2 m 2 , and assuming the motion to be Harmonic , then the Steady-State solution is , XS = m | e √2(n) | w ² √(k−Mw2)2+ [cw]2 , and tanφ = c w k−Mw ² and so the reduced to Non-dimensional form equation for amplitudes is , X S = | m e M√2(n) |.| [ 𝐰 w n. ] ² √(1−[ 𝐰 w n. ] 2) 2 + [2𝜁 𝐰 w n. ] 2| , and tanφ = 2 ζ .[ 𝑤 w n ] 1−( 𝐰 wn )² …... (c-3) which is identical with , m ẍ + c ẋ + k x = Fo.sin wt ….. (b-4) for Steady- State solution . The Resonance frequency 𝐟 𝟏 = w1 2π. = 1 2π. √ 𝑘 𝑚 − 𝑥²1 2m². , From (c-3) , a). Choosing the Damping factor-values ζ and the eccentric weight exciter e , is Possible to define the Resonant amplitude X = 1 2 ζ. [ m e M ] , and by altering the eccentric mass-m the Prober X = m e M . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 52 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . b). Above Un balance happens in Rotating machines or wheels , which can be repaired by Adding or Detaching ± masses , or the opposite ∓ Detaching – Adding masses. 1). For small values of , 𝐰 wn << 1 , both the Inertia m.w².X , and Damping forces c w X , are small which results in a small Phase angle φ . The magnitude of the impressed force F0 , is then nearly equal to the Spring-Force , k X , as in Fig.12.b-2 2). For value , 𝐰 wn = 1 , the Phase angle φ = 90 °, as in diagram .The Inertia Force mw²X , is larger and balanced by the Spring force k X . whereas the impressed force F0 is >> the Damping force . The Amplitude at Resonance is X = F 0 𝒄.wn = F 0 𝟐𝛇 𝐤 , or from diagram. 3). For large values , 𝐰 wn >> 1 , the Phase angle approaches φ = 180 ° , and the impressed force F0 , is expended almost in overcoming the large Inertia force , mw²X . a. THE NATURAL ANALOGOUS OF THE ENERGY MOTION THROUGH THE PATHS . The Propagation Path of Vibrations & The Vibration-Absorber . Figure –12c- The Electromagnetic-Wave {�̅�. fn̅ + �̅�.𝐟𝐧 } into the Atoms-Bracket-axes-Hook. The Process of the Photo elastic Fringe – Pattern [96A] In-1- The Unloaded Beam ( F = 0 ) is the Length AB̅̅ ̅̅ . In-2- The Loaded Beam ( F = F ) is the Deflection-Curve AB⏟ dh , the Strain , where exist , a.. The Shape of the Beam is the Space , the Strain , the Figure of the System , b.. The Shape of the Line-Stresses is the Figure-Fringe the Path of Energy . c.. The Colure of the Figure-Fringe . is the motion = the Energy the Forced Vector. In-3- The Loaded and Deflected-Beam are the Spaces = the Figures of System. The Loaded and Deflected-Beam Strain-Lines-Fringes , are the Path of motion , The Loaded and Deflected-Beam Colure - Fringes , are the kind of Forced Vector, IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 53 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Complex Force – Vector Harmonic - Vibration & The Vibration-Absorber . Figure –12d- The Complex -Force -Vector is 𝐅 𝟎 .[ cos(w t) + sin(w t) ] = 𝐅 𝟎 .e i.[w t] . In-1- The Terms = ( m ẍ , cẋ , k x ) , of Vector Force Polygon are Perpendicular to y-y axis. In Case of Complex -Force -Vector the Polygon becomes an Absorber which Reduces the motion from Kinetic Energy to Potential Energy (a Magnetic Field) . In-2- The 2-DOF System with Vector Force such that , w ²11 = k 1 m1 , w ²22 = k 2 m2 , acts as an Vibrator Absorber , and reduces the motion of the mass m 1 to zero .The two Natural Frequencies , f 1 , f 2 , are Perpendicular [ ⏊ ] each other as the Diagrams. This happens for every action on the 8.[ a , b , c , d ] Physical - Mechanisms . In-3- The{ STPL } - Six Triple Points Line Mechanism is formed from Three Tangents from A ,B ,C Spaces on 2R - Circle . Since Pascal-Desargues lines meet on Markos STPL Mechanism lines ,Then transfer the Properties & Information ( f 1 , f 2 , ,, f n ) From The Complex-Vector-Absorber = 𝟐𝐑̅̅ ̅̅ , to The Vector-Force-Polygons Remarks : 1.. The n-Knots Figures is any N-DOF System , which Characteristic equation results to an Algebraic equation of 3-n Degree , with main equation of motion [ λ.M + K ] X = 0 2.. The n-Knots Figures exist in Figures where the Vector -Force - Polygons have the same number n , as the in Figures . In (3) The Circularly- Charged Breakages are the three- 3 elements [Σ = n . h]  [s² = ⊕ , 2s²= , - s² = ⊝] which formulate the 3-Knots Figures. This {STPL}-Mechanism of 3 elements ( n=3 ) determinates the Mould which denotes the number n , of the Spaces for Elementary-Particles Vector. 3.. The case of {TCIC}≡ [ Σ = n . h ]  8 x ① which is the Physical Mechanism , gives the n-masses of Atom`s &Compounds determining the Atoms Complex-Vector-Response. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 54 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . B… GENERAL : 1.b. The Spaces and the Energy States . 1…The Objective Reality is composed of two elements , that of Distance . The Space ds , and that of Motion , or else the called Energy , and it is the content of all sciences [32] 2…The Euclidean-Geometry describes the Space only but Not the Energy as is motion [48] 3...The Solution of all the Unsolved-Ancient-Greek-Problems [50] opened the way to the Material-Geometry [54] which Incorporates the Motion → In-Space as (⊕↔⊝) ≡ 0,[61] The Space exists in Energy - Caves as Energy-Quantum-Quantities [39] , while Motion or Energy exist in Caves as an Confined - Stationary-wave which is either Static or an moving Energy-Storage or an Energy-Box [68] . The Any-Two Opposites (+) , (-) exist in Nature and are found everywhere from → Zero-Point (0) ≡ [+ = -] or as (+ A) + (- A) = 0 in E-Geometry and → [0] = [⊕↔⊝] ≡ f n , [⊕←ds→⊝] ≡ ds in Material-Geometry , to the aperon , ± ∞ where → The Space ≡ ds , is the nodes distance , and motion exist as the Vibration Type ← [52] 4..The Electron-Nutation in Hydrogen cave Produces Energy due to g effect , as an minimum frequency fn ≡ 2,8398447. 1010 H and thus exists in all Atoms . This Energy in Hydrogen- Cave is an Electrical -Magnetic Conductor, which is The Pin of Atom-Plug and enters Into the other Atom-Sockets , which are the Orbit-Bracket–Hooks . These are the Hands of Atoms , i.e. → The Atoms-Plug , Placing their Pins into the other Atoms Drains ≡ Holes , and so is done that what we say the Bonding of Atoms . 5.. Because of the Resonance frequency between all Atoms , and because of the Common Hydrogen , so Atoms Bond by Pinning and form Molecules and all Compounds in this Cosmos . This is Possible because the Periodic-System of Atoms is NOW modulated into 4 New - Categories with Only-One New- Criterion which is The Number of (±) Energy Conductors in Atoms , related to their Stability . 6..The first Solid (4 Points not coinciding ) is The Tetrahedron , and the first Material Space for the Two elements (+) , (-) , are (4 x 2 = 8 Positions for Electrons ) The-Cube , i.e. the Tetrahedron is Into the Cube and Into its Ex - Sphere , and this because all the Circles on Tetrahedron exist on Sphere . 7.. When 1-8 Electrons ⊝ are Placed on the Cube's Vertices and into this Onion - System , then Automatically are formed the 1- 4 , Energy-Conductors , where issues [⊕ ↔ ⊝] = [ p ↔ e ] = [Atom-Plug = + Electric Vector ↔ - Magnetic Vector ≡ Drain ≡ Hole ] , and are the Actions between Opposite . With this New modulation [ The ,1 , 2 ,3 , 4 , Energy- Conductors ≡ Atoms-Four Fundamental frequencies] , Conductors exist on System and ARE ACTIVE + as → The-Electric-Vectors ← and simultaneously PASSIVE ≡ Drains (-) as → The-Magnetic Vectors ≡ Holes ← . The Zero-State Conductors are those whose 8 Cube- Positions are all filled with electrons . The Multiple – Plug-Pins Atoms can Plug all the less number Electrons - Atoms and can Be-Plugged from all the greater number Electrons- Atoms .With these Four modulations are formed all Atoms and their Compounds these are the Mechanisms of , Water , the Cells , RNA , DNA , Cancer ,and everything in this Energy - Space Universe . 8..In the same way , is that where an 110 Volt Refrigerator Can’t-Plug into 220 Volt Socket and this because are different in construction , the same to Atoms-Plug which differ to the IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 55 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Number and the Length of their Pins . Hydrogen is the Multiple-Plug-Pins Atom which can Plug all Atoms and itself. 9..The New-Atom-Structure is consisted of → a) The Nucleus IN-Sphere b) The Dynamic Electro-Magnetic -Tetrahedron , c) The N-Outer- Cube , d) The Orbitals in EX-Sphere. 10..In the New-Atom-Structure , the Master-Keys - Resonances are the Communications- 11.. Article [110] is the completion of [72-82] and of Physical interpretation of the three constants of nature , that of Planck constant 𝐋 𝐏 , Newton`s Gravitational constant G , and the Gravity Constant g ,with a rigorous Geometrical and Mechanical logic . It was shown [33-36] that Un-clashed Fragments through the center , O , consist the Medium- Field Material-Fragment → [ ± s² ] = [MFMF] ≡ The Chaos , the Base for all motions ← and Gravity which is a force [ i] while the clashed with the constant velocity , c̅ , consist the Dark matter [ ± .c̅.s ] and the Dark Energy [c̅. i] , so declaring that → Antimatter-Galaxies and Antimatter-Asteroids can exist only as Dark-matter or and Dark-Energy and Not as Antimatter light , - c . From the velocity Breakages [ ± s² = ± (w r)²] , [ i = 2(w r)² ] and the between instress σ , become the Waves {The distance ds = r =|AAE| which is the Work embedded in monads and it is what is vibrated as frequency f } of the Material-Point with their Vibrating equations of motion and after to become the Waves . A → The Particles with Inherent Vibration occupying the Distances r = ds =Positions |AAE| the {Monad = 𝐎𝐊𝟎 = R} ≡ Energy In Vacuum Spaces are the Fundamental Particles , as [ ± v̅.s²] → Fermions , Quarks and Leptons , and → [  v̅. i] → the Bosons , B → The Gravity-Antigravity Field-Energy g without Vibration being a Stationary-Rotating Spinning Material Point M-P. [ ± s² ] → The [MFMF] Neutral Field ≡ The Energy Chaos , and their Negative Energy Binder - Field [ i ] → as ± Positions , For �̅� = ± �̅� , then + �̅� = + Gravity – Spin = + 𝛑.𝐫𝟑𝛔.𝚽 𝟐  The Gravity Spin - �̅� = - Gravity – Spin = - 𝛑𝐫𝟑𝛔.𝚽 𝟐  The Antigravity Spin C → The Dark-matter and the Dark-Energy constituents are as below , [ ±c̅.s² ] → The Dark-matter , and the Binder Gravity-Force g̅ ≡ [ i] , [ c̅. i ] → The Expanding Dark -Energy , Positive-Energy , which both are moving with light - velocity , c̅ , causing the Universe to Grow. From above in , A , and ,C, case → Energy as Velocity ,v̅, and , c̅ , exists in the Discrete monads ,± v̅.s² , and , ± c̅.s² only . 2 , The Case of Transportation of Energy , from Chaos to Stationary-Material-Points . Dark Energy DE ≡ [ �̅� . i] (©) , Acting  Positive-Energy , on the Five Constituents { [ ( i) , (+s²) , (-s²) , (+cs²) , (-cs²) ] } then Produces , [ ± s² ] → The [MFMF] Field [ ± c̅.s² ] → DM-DE Field of , DM = Dark matter , and DF = Anti matter = Dark Energy . [ ±v̅.s² ] → The PNS Space , Particles Fermions , and [ i ] → Gf ≡ Gravity-Force in DM-DE Stationary Rotating = Field . [ v̅. i ] → The Bosons , [ c̅. i ] ≡ The DE → Dark Energy → The Travelling-Energy with , �̅� , velocity. 𝐜 x (©) [ i ] ≡ Gf → The Gravity Force . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 56 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . In all above issue Kepler - laws , denoting that Macrocosm and Microcosm Obey the Newton`s Laws of motion , in all Scales , as was Proofed . Figure - 13 - Vector-Geometry follows the Geometry rules as the left figure. EuclideanVector Carry Point A to Point B as , AB , following the Geometry rules , while The Material – Points ≡ Quaternions , Carry the Motion ≡ Energy , from Point A , to the Point , B , linearly . In both cases the Points are nothing Possessing Zero-magnitude and Infinite- directions . The Two Points in E-Geometry consist the Straight-line-Segment Possessing |AB| - magnitude and the |A→B| Directions while the Two-Points in Material-Geometry denote Two-Elements as one (+) and one (-) Performing the Space ≡ |AB| of motion = Energy ≡ [A ↔ B] in |AB| Space .The above comparison can bee seen in Figure-1- also . Conclusions – 3 - : From above Analysis is Seen the Path which Energy as motion follows to Propagate . This Path is the Resultant of the Vector-Sum , or the Quaternion as Material–Point . The Vector-Sum-length as a Geometrical-Resultant-Vector is the Quaternion`s Edge Points Energy-vibration which transfers the motion from the vibrating-Edge-Points . Thus Stress → σ = [ 2πr Φ ² ] . f n ← is the Way of Information and Storage in Nature . Extending above to all Planetary Systems and to Universe is seen that BIG - BANG . has Never been existed , Inversely Energy ≡ Motion ≡ Matter followed the (+) Stress-Storage , ± σ = [ 2πr Φ ² ] . f n , and Anti-Matter followed the (-) Stress-Storage and Material-Geometry. 2.b. The Complex Numbers & Quaternion . 1… De Moivre´s formula for complex numbers states that the multiplication of any two Complex numbers say z1 , z2 , or [z1 = x1 + i. y1 , z2 = x2 + i. y2] where x = Real [z] IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 57 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . the Real-Part and y = Imaginary[z] . the Imaginary Part of , z , is the multiplication of their moduli r1 , r2 , where the moduli , r , is the magnitude [ r = | r | = √x² + y² ] and the addition of their angles , φ1 , φ2 , where φ = arg.(z) = atan2.(y , x) and so , z1 * z2 = ( x1 + i. y1) . ( x2 + i. y2) = r1. r2 [cos(φ1+ φ2) + i . sin(φ1+ φ2) and when z1 = z2 = z , and φ1 = φ2 = φ , then z = x + i y and z . z = z² = r².[cos2φ+i.sin2φ) and for , w , complex numbers becomes the Euler`s , zw = rw. [ cos(wφ) + i. sin(wφ)] and so for r = 1 then → zw = 1w. [ cos φ+i. sin φ ] w = [ cos.(wφ) + i sin.(wφ) ] . The n.th root of any number z , is a number b [ √z n = b] such that bn = z and when z is a Point on the Unit circle r =1 , the first vertex of the Polygon where φ = 0 is [ b = (cosφ + i sinφ)] ⁿ = b ⁿ = z = cos(nφ) + i .sin(nφ)= [cos(360/n) + i. sin(360/n)] ⁿ = cos 360 + i.sin360 = 1+ 0.i = 1 i.e. The w-Spaces which are the repetition of any Unit complex number z (multiplication by itself ) is equivalent to the addition of their angle and the mapping of the Regular Polygons on circles with unit sides , while the n Spaces which are the different roots of unit 1 and are represented by the Unit circle and have Points z = 1 as one of their vertices , are mapped as these Regular Polygons inscribed the unit circle , Figure-1- (1) Since also 𝐳𝐰 = 𝐳− 𝐧 and 𝐳− 𝐧 = 𝐳−𝟏/ 𝐧 = 𝐳+ 𝐧 therefore complex numbers are even and odd functions , i.e. are Symmetrical about y , axis ( mirror ) and also about the origin . Euler`s rotation in 3D Space is represented by an axis (vector) and an angle of Rotation which is a Property of complex number and defined as �̅� = [ s ± �̅�.i ] where s , �̅� are real numbers and i the imaginary Part such that i ² = - 1 . Extending imaginary Part to three dimensions , 𝐯𝟏 , i , 𝐯𝟐 , j , 𝐯𝟑 , k → �̅�. i becomes Quaternion which has 1 + 3 = 4 degrees of freedom i.e. [ dx – dy – dz – df ] ≡ The Space - Energy- Mode . 2… De Moivre´s formula for the n-th roots of quaternion ,where q = k.[ cos.φ+ [ i].sin.φ] is for w = 1/n , 𝐪𝐰 = 𝐤𝐰. cos.wφ + ε . sin.wφ] where , q = z = ± ( x+y.i ) decomposed into its scalar (x) and vector Part [y.i] and this because all the inscribed regular Polygons in the Unit circle have this first vertex at points , 1 , or at ,- 1 , (for real part φ = 0 , φ = 2π ) and all others at imaginary Part where , k = 𝐓 𝐳 = Tensor (the length) of vector , z , in Euclidean coordinates which is k = 𝐓 𝐳=√𝐱 ² + 𝐲²𝟏 + 𝐲²𝟐 + 𝐲²𝐧 and for imaginary Unit vector �̅� (𝐚𝟏, 𝐚𝟐, 𝐚𝟑, 𝐚𝐧,w) the Unit vector ,ε, of imaginary Part is → ε = (y.i / 𝐓𝐲) = [y. i ] / [𝐓𝐲] = ± ( 𝐲𝟏 . 𝐚𝟏+ 𝐲𝟐 . 𝐚𝟐) / √ 𝐲²𝟏 + 𝐲²𝟐 + 𝐲²𝐧 , the rotation angle 0 < = φ < 2π , φ = ± sin - 1 ( 𝐓𝐲 / 𝐓𝐳 ) , cosφ = x / 𝐓 𝐳 , which follow Pythagoras theorem for them and for their reciprocal quaternions �̅�` [�̅� . �̅� `= 1] . Since also the directional derivative of the scalar field relation y.( 𝐲𝟏 , 𝐲𝟐 , …. 𝐲𝐧 ,,,) in the direction , i , is → i ( 𝐲𝟏 , 𝐲𝟐 …. 𝐲𝐧 ) = 𝐢 𝟏. 𝐲𝟏 + 𝐢 𝟐. 𝐲𝟐 + i.n.𝐲𝐧 and is defined as → i. grad y = 𝐢 𝟏.( 𝝏𝒚 𝝏𝟏 ) + 𝐢 𝟐.( 𝝏𝒚 𝝏𝟐 ) +…= [ i. ].y , which gives the change of field y , in the direction → i , IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 58 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . and [i. ] is the single coherent Unit , Therefore coexistance between Spaces ,Antispaces and Sub-Spaces of Any monad → 𝐳 = x + y .i = 𝐀𝐁̅̅ ̅̅ , is happening through the general equation which follows 𝐦 [𝐬+𝐯 𝐢] = 𝐪𝐰 = (𝐓𝐪) . [cos.wφ + ε .sin.wφ] , where , m = 𝐥𝐢𝐦(𝟏 + 𝟏/𝐰)𝐰 for w = 1 → ∞ , q = z = ± ( x+y.i ) sinφ = y / √𝐱² + 𝐲² 𝟐 , cos.φ = x / √𝐱 ² + 𝐲 ² 𝟐 , |z| = √𝐱² + 𝐲² 𝟐 𝐓𝐪 = √𝐱 ² + 𝐲²𝟏 + 𝐲²𝟐 + ⋯ . . 𝐲²𝐧 , Ty = √ 𝐲²𝟏 + 𝐲²𝟐 +. . … 𝐲²𝐧 , ε = (y.i / 𝐓𝐲) = [y. i ] / [𝐓𝐲] = ± ( 𝐲𝟏.𝐚𝟏+ 𝐲𝟐.𝐚𝟐) / √ 𝐲²𝟏 + 𝐲²𝟐 + ⋯ 𝐲²𝐧 i.e. The Spaces , Antispaces and Sub-Spaces of Any monad → 𝐳 = x + y.i = 𝐀𝐁̅̅ ̅̅ follows the Euclidean Geometry (2a) and the Energy flow (2b) , as in Figure -1-3-21. The Vectors-Sum of Vectors , Either on Points on The Same - Linear-Direction of Vectors , Or on Non-Linear-Directions , Results onto The-Resultant-Vector �⃗⃖� , of all System-vectors and becomes A Linear-Vibration between Quaternion-Edge-Points. This Remark denotes that Quaternions which are composed of Real (+charge) and of Imaginary Part (- charge) act only on the Resultant-Direction End-Points carry Energy from One - Edge to another- Edge as in Fig – 6 - The Above Property of Imaginary-Part ≡ Energy shows The How Elementary Particles are Originated from the three Charges (of The Spaces [⊕ -  - ⊝]) , Circularly-Placed on The-Three-Triangles of STPL- Mechanism . [93] From Planck`s Energy E = h.fP = [ 6,6262.10−34 J.s ] x [ [ 3,28393.1015 / s] = 2 ,176.10−18 J , and in eV becomes → [ 2 ,176.10−18] / [1,6.10−19] = 13,6 eV or , Energy in Gravity-Layer g  13,6 eV = h f = h / T = h /√𝐠. 𝐚³. \Figure - 14 - : The Action of quaternion �̅� . �̅� = [s + �̅�. i] ² ≡ s² - |�̅�| ² + [2.�̅�].|s r|. i ≡ ≡ s² - s² + 2.𝐬² and follows Pythagoras theorem of the Euclidean Geometry. 3b…: The Stability of Space - Anti-Space , IN the Neutral-Space The number of Moulds of Stationary-States are as , 2n=1−4 Points ≡ 2 ,4 ,8 ,16 ≡ 4–Types i.e. Monads are formed from the first 4- Moulds as → The 1-Point which is everywhere, IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 59 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . the 2- Points which form Vectors , the 3- Points which form the Planes , the 4- Points which form the Volumes . For n > 4 as this is the Atom Structure, follows the Sustainability of the Spaces and Anti-Spaces . Since Space-Energy Structures follow the M-Geometry Stability these follow the Mould of Spaces 2 n = 3 = 8 Elements which is The-Cube as was shown before for the Atoms structure Fig-5-28 . The Atoms-Bonding follows → Space ≡ Electric-field & Anti-Space ≡ Magnetic-field ← 4b… The Quantum Interference of the Duality-Photon : Quantum Interference is the Phenomenon when , a Photon Interferes with itself .[92E]. This Proposition is Right when Photon is considered having a twin-existance , [101]-3c.P-21. The Problem is to determine the expression of the Power developed by Gravitational Force. From Dual-Photon-equation , v̅.[ fn̅ + fn ] ≡ | 𝐯 𝛑².𝐫⁴𝐧 |. B̅n + | c ̅| fn ,when c = 0 or c = c⃖ , then Photon motion becomes → | 𝐯 𝛑².𝐫⁴𝐧 | . B̅n ← ≡ Spin . i.e. The Stationary Photons exist as their Spin only. This Rest-Space is the Part | 𝐯 𝛑².𝐫⁴𝐧 |. B̅n and its Communication Code to be the term | 𝐯 𝛑².𝐫⁴𝐧 |=| 𝐰 𝐫 𝛑𝟐.𝐫⁴ | = | 𝟐 𝐯 𝟐𝛑 𝟑𝐫 ⁴ | ,The Rest Space is a Stationary Wave with fundamental frequency f 1= 1 T 1 = λ̅ c , with an Electric Wave , that is acting on vector c ̅ = ( i. cx + j. cy ) . It was Proved [78] that the Newton`s Gravitational constant G does NOT effect on Plane Surfaces , But on Volume Surfaces only . This Property of , G , means that the only one Force which effects on the Rest-Space in Torsional motion , and Onto the light velocity vector c ̅, is the gravity force g only. Figure - 15-The Stability of {⊕- Space [Electric Wave},{⊝-Anti-Space [Magnetic Wave] of Sub-Space { [⊕↔⊝] ≡ ds ≡ [⊕←ds→⊝] – or –The Neutral-Space [A E , B E , C E]} : IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 60 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Since c is constant , it is analyzed to , ( cx , cy ) in any two directions x , y where x ⏊ y and y to be the gravity direction on where gravity is acting . The Produced Work is in y - Range , S y = cy t - 1 2 g 𝑡2 = c sin 𝑎𝑡- 1 2 g 𝑡2 , where maximum Range is S max−y = cy 2/ 2g = (c sin𝑎𝑡) ² 2 g = [ c ² 2g ] sin² 𝑎 , happening at angle a = 90° and where sin² 𝑎 = 1 . Since at Point (1) the velocity v y = 0 , the motion in 0 -1 , may be considered as falling in linear motion and so issues v y = g t or time t = v y g = (c sin𝑎𝑡) g , as halve Period with total Period → T = 2 (c sin𝑎𝑡) g . In figure are seen the 4-Point Positions , at where change the Phase and the Work done in each Phase of Photon`s inner motion . The 4- Phases correspond to the Wavelength . λ , as λ / 4 . distance with 90° = 𝜋 2 to be the change of Phase . This means that the Produced Work in Prior Phase - Area as the Golden – Ratio Pattern is Stored in the Next and Perpendicular Phase-Area . The Total distance for doing Work is the Wavelength λ , in total Period T as λ = cx T = c .cos 𝑎 .T or λ = c . cos 𝑎 . 2 (c sin𝑎𝑡) g = [ 2 c ² g ] sin 𝑎 , cos 𝑎 = [ 𝐜 ² 𝐠 ] 𝐬𝐢𝐧 𝟐𝒂 , ……(1) since from trigonometry exists , sin 2𝑎 = 2 sin 𝑎 . cos 𝑎 . The maximum wavelength λ happens when 𝐬𝐢𝐧𝟐𝒂 =1 or 2a = 90 ° and a = 45 ° , i.e. →The Total motion in the Wavelength , λ , corresponds to the angle a = 45 °, meaning that the motion of Space , The Electric-field , and Anti-Space , The Magnetic-field , equilibrium as an Bellow . The two Wings in 90° Type simultaneously oscillate to its 45 ° Bisector -axis , and the Produced Work from The Electric-field-area is Stored into the Magnetic-field - area . The common axis of the two Perpendicular fields , carry the G-Ratio Surfaces , Runs with the same velocity c ̅ to the direction of the inner motion or, From 90°→ 45° , the created Work W = c y g , is Symmetrically Stored from 0°→ 45°. The created Work in [ 0-1 ] is in → λ / 4 of transverse surface and is stored in the horizontal and perpendicular surface [ 1`-2 ] = λ / 4 . Surface [ 0 -1 -1` ] = ⏊ Surface [ 1` -2` -2 ] . The created Work in [ 1-2 ] is in → λ / 4 of transverse surface and is stored in the horizontal and perpendicular surface [ 2-3 ` ] = λ / 4 . Surface [ 1 -2 -1` ] = ⏊ Surface [ 2` -3` -2 ] . The created Work in [ 2-3 ] is in → λ / 4 of transverse surface and is stored in the horizontal and perpendicular surface [ 3 `-4 ] = λ / 4 . Surface [ 2 -3 -3` ] = ⏊ Surface [ 3` -4` -4` ] . The created Work in [ 3-4 ] is in → λ / 4 of transverse surface and is stored in the horizontal and perpendicular surface [ 3 `-4 ] = λ / 4 . Surface [ 3 -4 -2 ] = ⏊ Surface [ 4` -5` -4 ] . Since from Trigonometry issues sin 2𝜋= sin(180 − 2𝑎) = sιn 2(90 − 𝑎) then (1) becomes . λ = c cos 𝑎 2 (c sin𝑎𝑡) g = [ 2 c ² g ] sin 𝑎 , cos 𝑎 = [ 𝐜 ² 𝐠 ] 𝐬𝐢𝐧 𝟐𝒂 = [ 𝐜 ² 𝐠 ].sin(𝜋 − 2𝑎) = [ 𝐜 ² 𝐠 ].sin( 𝜋 2 − 𝑎) …(2) since 𝐬𝐢𝐧 𝟐𝒂 = sin(180 − 2𝑎) = sin(𝜋 − 2𝑎) , and also 𝐬𝐢𝐧 𝟐𝒂 = sin(180 − 2𝑎) = sin( 𝜋 2 − 𝑎) ……(1a) IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 61 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . From Dual-Photon-equation , v̅.[ fn̅ + fn ] ≡ | 𝐯 𝛑².𝐫⁴𝐧 |. B̅n + | c ̅| fn ,when c = 0 or c = c⃖ , then Photon motion becomes → | 𝐯 𝛑².𝐫⁴𝐧 | . B̅n ← ≡ Spin . i.e. The Stationary Photons exist as their Spin only . The Rest-Space is the Part | 𝐯 𝛑².𝐫⁴𝐧 |. B̅n where v = n.π.c , and Communication Code to be the term | 𝐯 𝛑².𝐫⁴𝐧 |= | n.𝜋𝑐 𝛑².𝐫⁴𝐧 | = | 𝑛.𝑐 𝝅.𝒓⁴ |=| 𝐰 𝐫 𝛑𝟐.𝐫⁴ | = | 𝟐 𝛑.𝐟 𝛑 𝟐𝐫³ | = | 𝟐 .𝐟 (𝛑 𝐫 ³) | with Electromagnetic Radiation the in monads frequency-f , which is satisfied by the relation | 2rw 2 v |= π /2 , π , 3π /2 .,,, n.[π/2] , and w = n.[ �̅� 2r ]x[ π 2 ] , where n = 1 → ∞ . When two Photons strike each other c = c⃖ , or reflect on a wall , then c = 0 , and from Spin �̅� = B̅ = π ². r ⁴. f1 its angular velocity w = n.[ �̅� 2r ] x [ π 2 ] = 2π.𝐟𝟏 , where then its fundamental frequency 𝐟𝟏 = n.[ �̅� 𝟐𝐫 ]x[ 𝟏 𝟒 ] = n.[ �̅� 𝟖.𝐫 ] which for n = [ 16.𝑟² v.c ] increases to λ c . r3 and Photon-motion becomes → | 𝐯 𝛑².𝐫⁴𝐧 | . B̅n ← ≡ Spin , i.e. The Stationary Photons are Spin only and consist an Stationary- Harmonic - Vibration , (1) – (1a) , with the same Period T of the two Waves [92-E}. The effective-Range of the Wavelength λ , for the Dual-Photon is related to the consistence v x , v y , velocities to any two Perpendicular directions of Electromagnetic fields E x , E y . The maximum displacement in y , direction is the centrifugal λ x = v² y 2 E y and sisin(φ) λ y = = c2. sin ²(φ) 2 E y , where φ = The Phase difference between the two vibrations . Since time t , is the same between the two vibrations where then t = v y E y = c.sin(φ). E x and since the Period = 2.t then T = 2c.sin(φ). E x …... (2) From the in x → direction velocity v x = c . cos(φ) , and displacements , λ x , λ y , become , λ x = v x T = c. cos(φ){ 2c.sin(φ). E x } = λ x = { 2.𝑐² 𝐸 𝑥 }. sin(φ).cos(φ) = { c ² E x }. sin(2. φ) (3) Since in trigonometry issues , sin(2φ) = sin(𝜋 − 2φ) = sin( 𝜋 2 − φ) and in Physics E x = g , IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 62 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . then → 𝛌 𝐱 = { 𝐜 ² 𝐠 } . 𝐬𝐢𝐧(𝝅 − 𝟐𝛗) , 𝛌 𝐲 = { 𝐜 ² 𝐠 } . 𝐬𝐢𝐧( 𝝅 𝟐 − 𝛗) ← …….(4) Since Monads are Complex quantities as AB ≡ �̅� ≡ x + i. y , then are Quaternions where Material-Points ≡ Quaternions , which Carry the → Motion ≡ Energy ← from the two Edge-Poinds [ �̅�↔�̅� ] , as this is from Point A , to Point B circularly . Since λ x ⏊ λ y , then issues λ y = [ i = √−1 ] * λ x , or equivalent to multiplying the quantities along the x – x axis , by i = √−1 . i.e. Photons occupy 1.. The Dual-Property of Particles and Waves , 2.. The Dual-Property of Stationary and Spin , 3.. The Dual-Property of Simultaneous existence , 4.. As Electric and Magnetic Fields in , 𝛌 𝐱 , 𝛌 𝐲 , 5.. The Bellow motion in transverse E-M- Waves as is the way to fly , 6.. The Transverse E-M - Waves are ⏊ to the Propagation-Direction , 7.. The Electric Effective Range is thrusted through G only. [ Fig-5 ] . 8.. With the same initial velocity is succeeded the same Length-Range with Two complementary angles . since v y = c. , C.. : THE ORIGINATION OF THE PHYSICAL LOOPS ≡ CAVES 1c.. The Gravitation Force G , and the Kick - Start : [78-79] Figure – 16. The Newton`s Universal Laws in Primary-Material-Point , Above , is consisted of the Two-Primary-Opposite-Spaces , {+} {-} , Poles , with infinite Points and Parallel-lines such that G is a Uniform-Pointy–Force . From where G , is Produced ? → It was Prior referred that , Periodic excitation between Primary Space (+) and Anti-Space (-) may exist only as collision of Opposite , so Energy captured in Box-𝐁𝐏 , Planck`s Loop , contains these IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 63 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . three constitute elements [(+) ,[↔], (-)] without the inner acceleration but the Material– extreme case of Periodic acceleration [→ ←] = 0 and which is the Reciprocating motion So , above Property of Primary Space exists also and for all Primary Complex- Spaces. Complex Spaces either at , [(+)] or [(-)] , constitutes may be realized , exist , separately because of the → Primary-Opposites which Pull each other ← therefore the Initial Property of Attraction , G , between Primary-Opposites continue issues and for All the other infinite Composite and Complex Structures as well as in all Groups of them . Since Force G is a Uniform-Pointy–Force , therefore needs a Conductor , a Layer , a kind of mass , to be spread and act on it as Surface-force , a Layer which is the Base or the Conductor Spin S , or the Stress g , as this is the Growing-Golden-ratio-Pattern Φ. Because of this Stress-Layer g , all the Energy-Structures Present → Reaction to any motion ← i.e. mass . The Gravity acceleration g , produced from the in Material-Points acceleration is equal to the Pressure σ , and is the Conductor to all Waves as , Force G - Over-Layer - A → [A ≡ mass * g ] → Gσ ≡ G , and Stress = g = σ² r².Φ² 8 From Figure -15(3) is seen that Gravitational Force is Attractive following Newton inverse square law, and it is an Attractive-force [↔] , On any two Opposite-Elements [ (+) , (-)] realized through a Layer of Unit-Stresses [σ = g ] , and becomes from an HUGE Magnet in Planck`s - Loop containing the whole Universe [89] . G - Kick-Start , is Planck`s frequency → fP = 1 / TP = w 2π = √ 1 g.a ³ 2 and in this world the How this frequency can → Enter , Format and cohesive , to the first or any other Energy-Cave in Planck`s length , E-Unit Monad , 1 = g .𝐟 ²𝐧 . 𝐚 𝟑 = [ 𝟒𝛑² 𝐆𝐌 ] . 𝐟 ²𝐧 . 𝐚𝟑 , where g = [ 𝟒𝛑² 𝐆𝐌 ] as this was shown in [79] for Unit-Monad-relation . An Energy-Rim = Cave is a Plane - Surface , an orbit , representing an Constant Energy becoming from the squared Frequency 𝐟𝐧², Representing the Imaginary Part , and r P³ , Representing the Real – Space - Part of monad as equation 1 = k.𝐟𝐧².𝐫𝐏³ . The Stationary Energy is spread in one Plane as this happens for the Stationary-waves in Caves , while in Propagating - Energy in Two , as are the Electromagnetic Transverse waves . All these Energy-Rims consist the Quantized-Plane-curves as Fig-15 . The two different motions of ⊕ Space , ⊝ Anti-space , in any cave r and in a Finite Space , The Revolving and Periodic Excitation Create an Eternal frequency which influence all other Spaces , the caves . The minimum quantized-energy is stored in the Gravity-Layer , g , becoming from the Unit Energy of the sinus orbit . From equation , g = T ² / a³ , and minimum - a then Period is T = √g. a³ = √9,8076925. (2,1145016. 10−11)³ = 3,04513.10−16 s , Since the Unit-Work = Sine Integral =∫ sin t t 𝑡 0 dt =1 , the semi-major axis , a , is a = 2,1145016.10−16 m , where frequency T −1 = fP = 3,28393..1015 /s , corresponds to a Sphere-loop in Planck`s scale as (4 / 3).π.r³ = 3,96 . 10−32 m³ , and then the Energy in this Planck`s loop is the minimum quantized . From Planck`s Energy E = h.fP = [ 6,6262.10−34 J.s ] x [ [ 3,28393.1015 / s] = IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 64 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 2 ,176.10−18 J , and in eV becomes → [ 2 ,176.10−18] / [1,6.10−19] = 13,6 eV or , Energy in Gravity-Layer g  13,6 eV = h f = h / T = h /√𝐠. 𝐚³. The above quantity of Energy consist the Hydrogen minimum Energy-Rim , which becomes from Loop-equation a = √𝐓𝟐/𝐠 𝟑 = √ [3,04513.10−16]² 9,80769251 3 = 2,1145016.10−16 m , and it is the Energy-cave for the Unit Energy quantity . From Newtonian Constant of Gravitation relation issues , G = E = h . fn = [ 𝐜.𝐫³ 𝐚³ ].[ gL kL ] = g .k E = g . [ gL kL ] , and since for The First Chemical-Neutral-material-cave , r , constants gL ,kL are equal to unity i.e. 𝐠𝐋 = 𝐤𝐋 = 1 , then above Energy of E = 13,6 eV , in Hydrogen-Plane-orbit corresponds to the minimum-Energy-Cave , or → The Phys-Quantized-Energy-Structure ← Since G Pushes → g , on the Earth-Unit-coefficient , k E , and because is the Starting for the first time beginning , of this Mechanism then from G = g.[gL kL] ≡ g.[1*1] ≡ → g ← or G = g , meaning that In Earth System of Gravity , the Newton`s Gravitational constant G , and Gravity g are equal , while in all other relative Systems are equal to the proportionality of their Local-constant 𝐤 𝐋 . Gravity , g , is the minimum Energy becoming from the in-storages angular Velocity acceleration a = v²/ r ≡ 9, 8076941 . in the Material-Point . [72] Photon is a Material-Point , Box BR , with fix-ends Inward-cave r , and which is the Energy Storage 𝐁𝐑, Outward-cave-r is an Electromagnetic-Radiation on wavelength λ = c T = c / fP which EM-Radiation , carries the Box BR . Universal Gravitational constant G = g k related to g , kR , is a Force and through the Static g becomes the Principal stress  σ , or frequency f R which exists in nature as motion in the minimum Resonance Golden-ratio-frequencies fR = fn = 1 , and this because of the Periodic motion , in Excitation , where issues the Coulomb-Dipole law and which Coulomb inverse law is as , F = kc [q1.q2 / r²] = kc [⊕→←⊝] /r² = 8 πr(1+√5) [ B r² ] , and Coulomb constant kc = 9.109 Nm2/C² , i.e. Because of the Periodic excitation between the , Space (+) and Anti-Space (-) , [⊕→←⊝] exists only collision of opposite where the inner acceleration is equal to zero Gravity g , and this because of the Material extreme-case of the Periodic acceleration {[→←]=0} which is zero , the Net force vanishes , issues the Coulomb Dipole-law where Static - Stress g , becomes from the Stationary constant Dipole moment �̅� ≡ �̅� ≡ as momentum which is the analogous in the Revolving motion where then issues → G = g k = 𝐤𝐄 g = g.𝐤𝐑 𝐠𝐑 = �̅� , 𝐠𝐜 = σ = 𝐅𝐨𝐫𝐜𝐞 𝐀𝐫𝐞𝐚 = 𝐠.𝐌𝐚𝐬𝐬 𝐀𝐫𝐞𝐚 For Earth-System mass ME = 5,9723.1024 Kg and Radius RE = 6378,137 Km = 6,378.106 m , then Earth-constant 𝐤𝐄 = R²E ME = [6,378.106]² 5,98.1024 = 6,811551810−12 and G = g 𝐤𝐄 = [ 9,8076941] . 6,8116.10−12 = 6 , 68056.𝟏𝟎−𝟏𝟏 m3/ N.s² becoming from g , 𝐤𝐄 only , i.e. G as force Pushes → g as Stress energy in → 𝐤𝐄 constant . Now is Proved that , the Gravitational Constant G is the mechanism or the mould , for the First-kick-Start upon this Unit-Granular-Energy-Stress-Layer , g , to IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 65 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . formulate in that orbit a , into Planck`s cave the lightest and the less-energy mass Particle of this Universe , which is the Hydrogen with the minimum Quantized-energy of 13,6 eV. The Centripetal acceleration in Photon cave due to Pressure σ ,where issues only Rotational Energy E R = B̅ . w̅ = 2L= J.w² , and 2E = B f , or → 2.(0,5.m.c²) = m.c² = 0,5.π.𝑟4.𝑐2/4 , then 4. �̅� . f = 𝐜𝟐. 𝐫𝟒 , where the Golden ratio frequencies → fn = [ n.σ.Φ 2.π.r ] ≡ n.(1+√5 ) .σ 2.𝜋.𝑟 = = 𝐜𝟐. 𝐫𝟒 / 4.�̅� = E h ← i.e. The box 𝐁 𝐏 , or the Planck`s Cave , carried with Light velocity c and Pressed into the Planck`s Loop , as n 𝐜𝟐 = n. c c , becomes the Gravity g , which is a stress by which Gravitational-force G is communicating as , g G = [ π r v4 2 ].𝑒3 = 9,808238 m/s . Gravity is the minimum Unit-Stress-monad in Planck`s scale becoming from a M-Point and which is Under -Planck`s - length region . Since also acceleration in a Material-Point ( Centrifugal-Centripetal ) becomes from the Principal Stresses  σ , therefore constant g G ≅ ± σ = Force Area = g.Mass Area = G k , and for Unit-G constant k becomes , k = System Area System Mass G where → 1 𝐠 = k G = System Area System Mass and G = k g G , and for the Earth G ≡ g.kE ≡ g .[gL kL ] ≡ [ T²P 𝐚³ ].[ gL kL ] ≡ 9,808238* 6,8116.10−12 ≡ 6,68056.10−11 𝐦³ 𝐍𝐬² So G , is The Pulling and Cohesive Force on all the Quantized-Energy-Structures which communicates with everything due to Periodic excitation on all Spaces . From equation g = G / [gL kL ] , is seen that g becomes from G under Local-laws , And also , Gravity g , is the Unit-Energy-Path , Quanta of Gravitation , due to M-Point Frequency fg ≅ ± 𝜎 from the Revolving motion , and also the Unit-Energy-Stress per Surface and for the Force - G , is the Permitted Path , or the communication Conductor on all Energy Wave Structures , In and Out Planck`s length . Remarks : Gravitational Force ,G, is the Pressure which every Object in the Universe Exerts on every other , whether Small or Big and is equal to Fgrav = G.M.m d² = g.𝐑²𝐄 M.m Md² = g.𝐑²𝐄 .m d² = mg.𝐑²𝐄 d² , and G = g.𝐑²𝐄 M.m.d² Md² = g.[ 𝐑²𝐄 ] M = g . kE , where kE = [ 𝐑²𝐄 ] M , and g = The minimum Quantized work as acceleration ≡ 9, 8076941 m/s² kE = The Earth Local-coefficient between the two constants g , G . For Earth (E) → k E = r ²E / m E , For Bodies (B) → k B = r ²B / m B , For any Body → Local Gravity g L = kE / kL From relation G = g 𝐤𝐄 = 9,8076941.6,8116.10−12 = 6,68056.𝟏𝟎−𝟏𝟏 m3/ N.s² and from relation 1 = c. r³. 𝐟𝐏² → fP = √1/cr³ and from f = E/h then 𝐄 𝐡 = √1/cr³ or → E = h . √1/c. r³ , E² h² = 1 cr³ → and , E ² = 𝐡² 𝐜 𝐫³ , is an Energy relation between c , r . Instances : [77] 1.. For Earth-System mass ME = 5,9723.1024 Kg and for Area → Radius 6378,137 Km = 6,378.106 m then the Earth-constant 𝐤𝐄 = [6,378.106]² 5,9723.1024 = 6,811551810−12 and G ≡ σ.𝚽³ = g 𝐤𝐄 = 9,8076941* 6,8115518.10−12 = 6,6805616*𝟏𝟎−𝟏𝟏 m³/Kg .s² i.e. Gravitational-constant G becomes from g , 𝐤𝐄 , and issues G = 6,6805616.10−11 m3 / N.s² ……..….(7) For Black-Holes-System mass M BH = 4,0.10 52 Kg and for the Area IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 66 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Radius 3,08.1025 m , then Black-Hole-constant kBH = [3,08.1025]² 4,0.1052 = 2,3716.10−2 𝐠𝐁𝐇 = k E k BH = 6,81155.10−12 / 2,3716.10−2 = 2, 872.10−10. gE → i.e. For Earth-Unit-coefficient kE = 6,81155.10−12 → Black-Hole –Unit-coefficient kBH = 2,3716.10−2 m2/Kg , and Gravity acceleration of a Black-Hole is , 𝐠𝐁𝐇 = 2,872.10−10. gE , an expected explanation . For all Planes issues G = 𝐤𝐄 g = 𝐠 . 𝐤𝐋 𝐠𝐋 , and for Black-holes also where , G = g . 𝐤𝐁𝐇 . 𝐠𝐁𝐇 = 9,8076941* 2,3716.10−2 * 2,872.10−10 = 6,6805616*𝟏𝟎−𝟏𝟏 Meaning that Gravitational constant is the Same for all Systems . The Gravitational Force , G , is the Pressure which every Object in the Universe exerts on every other , whether Small or Big and from Newton are equal to → 𝐅𝐠𝐫𝐚𝐯 = 𝐆.𝐌.𝐦 𝐝² = 𝐠.𝐑²𝐄 𝐌.𝐦 𝐌𝐝² = 𝐠.𝐑²𝐄 .𝐦 𝐝² = 𝐦𝐠.𝐑²𝐄 𝐝² , G = 𝐠.𝐑²𝐄 𝐌.𝐦.𝐝² 𝐌𝐝² = 𝐠.[ 𝐑²𝐄 ] 𝐌 = g . 𝐤𝐄 where 𝐤𝐄 = 𝐠.[ 𝐑²𝐄 ] 𝐌 = 𝐠.𝐌 𝐆 g =The minimum Quantized Work as change of velocity → acceleration ≡ 9,807694 m/s² 𝐤𝐄 = The Earth Local-coefficient between the two Constants , g , G , and 𝐤𝐄.G = g .M For Earth (E) → 𝐤 𝐄 = 𝐫 ²𝐄 / 𝐦 𝐄 , For Bodies (B) → 𝐤 𝐁 = 𝐫 ²𝐁 / 𝐦 𝐁 , For any Body → Local Gravity 𝐠 𝐋 = 𝐤𝐄 / 𝐤𝐋 2.. Coulomb Electrical Force , 𝐅𝐞𝐥𝐞𝐜𝐭𝐫𝐨𝐧 = 𝐤𝐜 𝐐𝟏.𝐐𝟐 𝐝² = [⊕→←⊝] 𝐝² = 𝟖 𝛑𝐫(𝟏+√𝟓) [ 𝐁 𝐫² ] , where Coulomb constant 𝐤𝐂 = 9.𝟏𝟎𝟗 Nm²/C² 3.. The Work done by the Electric field to Rotate the Dipole is W = 𝐅𝐞𝐥𝐞𝐜𝐭𝐫𝐨𝐧 . 𝐄 𝐅𝐢𝐞𝐥𝐝 . 4.. The Work done in Material Point needs a Path to exit from the Box 𝐟𝐑 = [ 𝐁𝐏𝐇 ] ≡ [ 𝐟𝟏=𝐍 , 𝐟𝟐 , 𝐟𝟑 , 𝐟 𝐑 = 𝐰²𝐍 ] ≡ [E² + H²] , and from where is Propagated . Resonance-Path happens as the Force , EM-Radiation in Two directions , which can travel in any closed System , and for solids through Cauchy-Stress-Tensor where the two Conveyers E ⊥ B ⊥ r ≡ 𝛔𝟏 ⊥ 𝛔𝟐 ⊥ 𝛔𝟑 , can carry the Energy Storage r in System [ + - 0 ] and Change the Inner-Structure of this System to another or Destroy it. In [70] , the Work Produced In Material-Point 𝐀𝐁⃖⃗⃗⃗ ⃗ is equal to → W = 2L = �̅� . �̅� = J.w² ← consisting the First-Energy-Store which is a Stationary Wave with , n lobes as , 𝐖𝐧(𝐧+𝟏) = [ 𝟒𝝅𝐫²𝐟𝟏 𝟑 ].n.(n+1) and wavelength 𝛌𝐍 = 𝛔.(𝟏+√𝟓) 𝟒𝛑𝐫 = 𝒏 .�̅� 𝟒𝛑𝐫² = 𝟐𝒏 .�̅� 𝐰 of r = 𝒎 .�̅� 𝐪.𝐁 related to M-Field B In [75-B] ,The Permeable-Resonance-Path for M-Point is → The Resonance-frequencies 𝐟𝐑 [S≡ 𝐟𝟏=𝐧 ,𝐟𝟐, 𝐟𝐑 =𝐰²] with 𝐟𝐧= [n (𝟏+√𝟓)𝛔 𝟒𝛑𝐫 = 𝐧𝛔.�̅� 𝟖 𝐫² ] related to Spin �̅� , Stress 𝛔 and cave r . Since the frequency in Material-Point is → IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 67 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 𝐟𝐧 = ( 𝐧𝛔 𝟖 𝐫 ² ) .�̅� ≡ [ 1+√5 2 ] 𝛔 𝟐𝛑𝐫 ≡ [ nσ 8 r ² ].B̅ ≡ [ 1+√5 2 ] Stress Perimeter ≡ [ 1+√5 2 ] Force [N] Area∗L[m3] ≡ 𝐄 𝐡 , Then it occupies the Property of the Golden-ratio Pattern Φ , and equation defines that the Material Point of frequency 𝐟𝐧 , when collide with another Material Point , or with any other Particle or Particles then Produces another monad as → 1 ≡ A New Quaternion and the first continuous to be of the same Identity , frequency 𝐟𝐧 , as before and from Euler`s , Rigid Body Dynamics work → W = 2L = B̅ . w̅ = J . w² ≡ h. 𝐟𝐧 ← i.e. The Frequency of Photon , is embodied with the → Growing-Golden-ratio-Pattern Φ and It Uses the Vibrating Physical Structures , or the same the Granular Material-Instruments , to Kick - Start all of them and everything in this world . The How In [75-B] Page 34 . Because of the Periodic excitation between , Space (+) and Anti-Space (-) , [⊕→←⊝] exists only Collision of Opposite , where the inner acceleration is equal to zero Gravity g and this because of Material extreme-case of the Periodic acceleration {[→←] = 0} , and which is zero and the Net force vanishes , and in where issues The Coulomb Dipole-law and where acceleration g becomes from the Stationary constant Dipole moment �̅� ≡ �̅� ≡ momentum and which is the analogous in the Revolving motion in where then issues → Gravitational force  G = g k = 𝐤𝐄 g = g . 𝐤𝐑 𝐠𝐑 = �̅� . 𝐠𝐜 = σ = 𝐅𝐨𝐫𝐜𝐞 𝐀𝐫𝐞𝐚 = 𝐌𝐚𝐬𝐬 𝐀𝐫𝐞𝐚 g In Page-33 [77] , the Earth-constant 𝐤𝐄 = R²E ME = [6,378.106]² 5,98.1024 = 6,811551810−12 and the Gravitational Force G = g 𝐤𝐄 = [ 9,8076941] . 6,8116.10−12 = 6,68056.𝟏𝟎−𝟏𝟏 m3/N.s² becoming from g , 𝐤𝐄 only, i.e. G is a force which Pushes → g as energy in → 𝐤𝐄 From Gravity relation G = g 𝐤𝐄 =9,8076941.6,8116.10−12= 6,68056.𝟏𝟎−𝟏𝟏 m3/ N.s² and from Energy-loop relation 1= c. r³. 𝐟𝐏² → fP=√1/cr³ , from f = E / h then f = 𝐄 𝐡 = √1/cr³ or → E = h.√1/c. r³ , and E² h² = 1 cr³ → E ² = h² c r³ , an Energy relation between the , c , r . This relation is used for Black- Holes Energy where r , result to zero , [71] The Permissible - Permeable - Resonance - Paths for the means , is as below . 1) Solids → The Normal-mode-Vibration - System { - w² [M] + [K] (X) } = 0 2) Liquids → The Cauchy Stress-Tensor as Momentum equation { ∇.σ = -∇p +∇.τ } 3) Gazes → The combined Avogadro`s Pressure - law { PV = n RT = n.mv²/3 } 4) Crystals → The Cauchy Ellipsoid-Stress-tensor where { E⊥B⊥r ≡ σ1⊥ σ2⊥σ3 } 5) Molecules → The Lattice-Crystal-Arrangement with their Chemical Bonds relation 6) Atoms → The Energy-Rims with Chemical Bonds , Ionic , Covalent , relation 7) Particles → The Resultance of the One-Dimensional-Collision v̅ij= v̅j - v̅i = w̅ij. r̅ij IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 68 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 8) M-Points → The Resonance-frequencies 𝐟𝐑 [S ≡ f1= n ,f2, f3,fR=w² ] = 𝐟𝐧 = [⊕ ↔ ⊝] = n (1+√5)σ 4πr = n.σ.B̅ 8 r² → i.e. the Golden-ratio-Pattern , which is related to Spin. 9) Cave-Orbit→ The Relation c.L s = Lv , Light-velocity 3.108 m/s*1.10−42 m cave =3.10−34 m²/s are the Cave-Energy-Plane-Rims in Atom`s , Planet orbits. In Vibrations of Systems , for Orbits issues → W = 4π² . 𝐫³ 𝐓² . �̅� = 4π².r ³. 𝐟²𝐩 . �̅� ← which is the Energy in the Conic-Sections of Kepler`s Unit - Orbits . The Energy-Storages r = n.[ λ 2 ] ≡ Wn(n+1)= [ 4𝜋r²f1 3 ].n.(n+1) , are travelling through Bodies and follow, Lame Stress Ellipsoid n1² + n2² + n3² = T1² σ1² + T2² σ2² + T3² σ3² =1 on Principal Stresses σ1, σ2, σ3 , which is the Passage through which Forces (The EM-Radiation) travel in any Solid either it is in Motion or is at Rest . Laplace`s Orbital Angular-momentum ei.2πn = 1 and for n = 0 ,1,2 ,3,. n , consist the Eigenvalues Operator Lz which agree with prior Resonance-frequencies 𝐟𝐑 [S≡ f1=N , f2, f3,f R = w² ] as wavelengths λ ≡ [f1 , f2…fn= w²] ≡ the n lobes , or → fN = n (1+√5)σ 4πr = nσ.B̅ 8 r² , as the Principal-Stresses σ , and Resonance- frequencies 𝐟𝐑 relation , which is Energy stored in the MP-lobes. [70] Type - Particles at sites - Type of Bounding-Force - Properties - EM-Radiation Ionic : ⊕ , ⊝ Ions - Electrostatic ⊕ ↔⊝ - non-conductors - Infrared Molecular : Atoms or Molecules - Dipole Attraction - Repulsion - non-conductors - Chemical-Bonds . Covalent : Atoms - Network- Bonds between Atoms -non-conductors - EM-Spectrum Metallic : Atoms - Ions and Electrons Attraction - Conductors - E-conduction . The Kinetic-Energy E K of a moving Material-Point , as this is the Photon , is stored as motion in its Storage , r = [ n. λ/2] , with the n frequencies f n = n.f 1 , with the n lobes , and with an fundamental frequency f 1 . From above is seen the Passage and The-How EM- Radiation can travel in Crystals and which are the Cauchy-stress-Tensor , where exists E ⊥ B ⊥ r ≡ σ1 ⊥ σ2 ⊥ σ3 , in-where Energy Propagates along Directions without any Birefringence , and carries the Energy-Storage r , which radiation is The Conveyer. Above procedure can be used in Cells , where cells are cases of an Birefringence material and the Resonance-Passage happens as the Force , an EM-Radiation in Two directions , can travel in Cell through Cauchy-stress-tensor where the two Conveyers E ⊥ B ⊥ r ≡ σ1 ⊥ σ2 ⊥ σ3 , and can carry the Energy-Storage , r , in Cell , and alter the Inner-Structure of Cell to another desirable Property. From Inner-velocity equation v = w r = (2π/T).r =2π.f1 r , wavelength λ = IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 69 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . cT=c /f 1, cave r = n.[λ/2], then r = n.(c/2f1) and v =2π.f 1[n.c/2f1] = n.π.c or v= n.π.c ....(4) showing that velocities in lobes are , n.π ,times that of light and for n = 1 then v = π.c , more than three times faster of light velocity. Because of the above velocity v , an E field is produced , which produces the 𝛛𝐃/𝛛𝐭 field , which in turn produces the H field which produces the 𝛛𝐁/𝛛𝐭 field and which again produces the E field , i.e. the Total EM-field regenerates itself as it Rotates , and is a Phenomenon happening in a Propagating Plane-Wave . Its Permeable - Resonance - Path is impossible in an Three-times Stronger EM-field because issues … (4) 2c.. The United-Coulomb-Newton Law for Interactions : For a Stationary interaction the Coulomb-Law-Force is 𝐅𝐂 = 𝐤[𝐪𝟏𝐪𝟐] 𝐫² , and for Charges 𝐪𝟏, 𝐪𝟐 where Constant k = 9,0.109 N.m²/C² , and r , is the distance between the Charges. For a Stationary interaction Newton Law for masses m1, m2 is 𝐅𝐍 = 𝐆[𝐦𝟏𝐦𝟐] 𝐫² , and for v̅=r̅ Conjugating Interaction , Energy E = 𝐯² 𝟐 [m1 + m2] = 𝐫² 𝟐 [ F1 g + F2 g ] = 𝐅.𝐫² 𝟐 [ 1 g + 1 g ] = 𝐅.𝐫² 𝟐 = = 𝐅.𝐫² 𝟐 [ 2.g g ² ] = 𝐅𝐍 .𝐫² 𝐠 , where 𝐅𝐍 is Newton Force, and E is the Energy of the Field-System.[ g+g g∗g } Equating the two forces then k [q1q2] r² = gE r ² , or  k [𝐪𝟏, 𝐪𝟐] = g . E ……..(1) Equation (1) relates any two Charges q1, q2, with Field-Energy E , under vector �̅� = �̅� = �̅� 3c.. The Origination of Light velocity c : Gravity g ≡ 𝐓² 𝐚³ ≡ 𝟏 f ²n.𝐚³ ≡ 9, 8076941 → The minimum Granular Quantized Energy , The Forces Beyond-Planck-Length are → F = Φ³. [⊕ 𝛔 ⊝] = σ x 𝚽 ³ ← (1)-[87] Force G acts → on Spinning M-Points on �̅� through g̅ → on Planck`s - B̅ → on 𝐠𝐆 , through the Resonance frequency 𝐟 𝐑. It was Proved that G = σ Φ³, and σ = 𝐆 Φ³ = 𝐆.𝛔³ c ³ , or → σ ² G = c ³…(2) , where σ is a Stress between the frequencies . From Force-relation F = σ A = (2πf r) A Φ = w r A Φ = �̅� A Φ , is seen that force G becomes a velocity �̅� . In the case of Planck`s length this velocity is �̅� = w r = 𝛔.𝚽 𝐫 𝐫𝐩 = σ x Φ = | 𝐆 𝐋𝐏 . r Φ³ |Φ = [ 𝐆.𝐋𝐏 𝐫.𝚽² ] and from Gravity relation G ≡ σ x 𝚽³ , then issues → [ �̅� = 𝐆 𝐋 𝐏 𝐫.𝚽² ] = [ �̅� = 𝟐.𝐆 𝐋 𝐏 𝝀.𝚽² ] which is the Light-velocity vector . Since r , becomes from the Beyond-Planck`s-Region then  �̅� = F.Φ A = [ G Φ A ] …...(f) Equation (f) may be written as Force G = �̅� 𝐀 𝚽 , which is a Viscus- Damping- Force where A Φ is a constant of Proportionality and the equation of motion is 𝐅𝐆 = [ 𝐀 𝚽 ] . �̇� Force G is applied in all Quantized-Universe A as Velocity �̅� and as Stress σ on Spaces Anti-spaces as Spaces-relation 𝐋𝐩 = ei.( π 2 +2kπ).b ,where from Material-Geometry [24-58] , Space ≡ 1 and Anti-Space ≡ √1 = + 1 , - 1 , and √−1 = i = The Imaginary Part into the → Anti-Space + Space ≡ motion ≡ i ≡ √−1 2 ≡ √−1 4 ≡ e−i.( π 4 ).b = 0,707106781 . b . The Base b , which for Natural logarithms issues << The Natural logarithm ln(x) of a Magnitude x , is the Power to which , e , would have to be raised to equal x >> and is Defining that → ln(x) is the Period-needed to Grow x , as this is in Integration ∫ x , so ex IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 70 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . is the Amount of Growth after Period x , and the Possible-Repetitive-Permutations for moulds and elements in which issues 𝐌𝐨𝐮𝐥𝐝 𝐄𝐥𝐞𝐦𝐞𝐧𝐭𝐬 = The - Growth-Periods i.e. when Mould = Elements then is succeeded maximum and for e is ee, and any logx x . For logx x and Base x = 10 then log10 10 =1010 and for the two elements [⊕ , ⊝] is 10[10]² = 1020 Positions ≡ Distances ≡ r and since 10−x = 1 10x then b = 10−20 , and The Anti-Space + Space-Positions are , 0,707106781.10−20 = 1, 0707106781.10−19. In this Way The Non-Dimensional-number [ 0,707106781*10−1 = 0, 0707106781] , Is Quantized in cave , r , as distance and Becomes The-Dimensional-Space in the Decimal - System b =10 as cave r , and which cave is monad r = 1, 0707106781 or , r = 1, 0707106781 m , having Unit-Area π.r² =3.601588 m2 , and the QUA-Universe is A = 3.601588 . 𝟏𝟎−𝟏𝟗 m2/s ,which are The Space +Anti-Space Positions in Universe ...(i) From (h) , light-velocity �̅� = [ 𝐆 𝚽 𝐀 ] = 𝟔,𝟔𝟕𝟑𝟔𝟗𝟐.𝟏𝟎−𝟏𝟏𝟏,𝟔𝟏𝟖𝟎𝟑𝟑𝟗𝟖𝟖𝟕 𝟑,𝟔𝟎𝟏𝟓𝟖𝟖 .𝟏𝟎−𝟏𝟗 . = 2.99819938 .𝟏𝟎𝟖 m/s i.e. Gravity Force G ≡ motion , exists without mass and is Quantized → Spread in {Space and Anti-Space} or ≡ QUA ≡ {is The Quantized-Units In-Area of Space and Anti-Space }, and as a Mould 𝚽 Golden-Ratio , exists in the Impedance b . Impedance in Mechanics is the Friction Coefficient , where for Force G to be Proportional to the Light-velocity c = �̅� , then the Harmonic – Oscillation is an Dumped-Oscillator , and Can - Oscillate due to the excitation as , a.. with a frequency lower than in the Undamped case and an Amplitude Decreasing with time , the Under-Damped-Oscillator which Originates the Quantum of motion which is a limiting case between the oscillatory and Non-Oscillatory motion the Critical Damping , and which Originates the Light-velocity �̅� , [49] b.. with Undamped case frequency and Decay to the Equilibrium-Position without oscillation and The Over-Damped-Oscillator Originating the n-times velocity c . What is Quantization of motion and Impedance is analytically referred in [83] A Parallel solution becomes also from the attendant Geometry logic , The Three Elements ≡ Digits of Material-Geometry are →{⊕ , [⊕↔⊝] , ⊝} ≡ [+ ,0, -]← The Permutation , arrangement of the Two-Elements P1 2 = 2 , i.e. are→ [⊕,⊝] - [⊝,⊕]← The Three-Elements in Space need P 1 3 = 3.(3-1).(3-2) = 6 Positions and the same for Three-Elements in Anti-Space need P 1 3 = 3.(3-1).(3-2) = 6 Positions , and Total Places → P1 3 . P1 3 = 6 x 6 = 36 Positions = A , for Spaces and Anti-Spaces as Impedance and as before for logx x and Base x = 10 then log10 10 =1010 and for the two elements [⊕ , ⊝] the Growth is 10 [10] ² = 1020 Positions ≡ Distances ≡ r , and since issues 10−x = 1 10x then b = 36.10−20 , and  �̅� = F Φ A = [ G Φ A ] = [ 6,673692 .10−11.1,6180339887 36 .10 −20 . ] = 2.9795163. 108 m/s i.e. The Ubiquity of Material - Geometry in Electromagnetism is Everywhere . The Gravity constant g , is the Permeable Path for inner stress σ , to pass the Material`s Point a surface 4πr³/3 and to expenditure its Energy .The same exists also to the Electromagnetic force which is associated with a fundamental Property of matter which is the Electric - Charge and which is a clue to the Ubiquity of Electromagnetism . From equation of Gravitation G = kE g = g. [ kR gR] seems that the two constants are related i.e. act each other through Local-coefficients or through Field-lines , called the Medium or Permissible Path which are as σ = 𝐅 𝐀 = 2πrf Φ = w r Φ = 𝐯 𝚽 , velocity vector in a Unit-Space 𝚽. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 71 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . It was shown that the first Path is Gravity , g , and Original Field-lines of Force , G , are distorted by these Charges , Local-coefficients , the Layers following Newton`s laws . The Original Field-lines terminate at the surface on one side of the Medium , and new field lines originate from the other side of it . It was shown that the Momentum vector ,�̅� , is equal to spin S , because it is following the Stationary - Wave - Nodes - Principle in the Material-Points creating the minimum quantized Energy which is conserved in lobes . This Property is extended also to the Number of lobes as well as to , π , number as velocity { v = n.π.c } which is the minimum Number relating Lines and Surfaces . Analogous happens in equation when v = c , and → r = c . From Inner-velocity equation v = w.r = (2π / T).r = 2π.f1. r , of fundamental frequency f1 , of wavelength λ = c.T = c / f1 , and cave r = n.[λ/2] , then r = n.(c /2f1) and v = 2π.f1 . [n. c / 2f1] = n.π.c or v = n . π . c ……..(π) Equation (π) shows that velocities in lobes are , n.π times that of light , following , π , number in circle , i.e. In Material-Points exist velocities multi-times that of light and the minimum Surface-constant, Unit π , or the Growth of the velocity-vectors occurs in lobes by following the logarithm laws of Energy-constant c which acts on Space constant π. From velocity , v = n.π.c, is seen that light-velocity is the Quantum of Unit-velocity in Planck`s length . The Why velocity c and π , is such in [42-51-63] Kepler`s Laws → Explain How the Planets move around the Sun But NOT the WHY. Newton`s Laws → Explain the WHY by filling this Gap by a Force F = G.m 1.m 2/ r². acting instantly between the bodies that are moving around each other But NOT their Nature and NOT the HOW Force is Acting . Markos-Spaces → Explain the WHY by filling the Gap with a Double - Ocean of the Pointy-Spinning- Material-Points , becoming from the Stationary- Material-Points , Photons or Electrons , and from the moving-Energy- Storages , the Photons , which Orientate and Re-orientate the Stationary-Spins , and explain the HOW and the WHY all motions follow the ubiquity of the Electromagnetism , starting from the Primary- Material-Points, Photons ,which through the Golden-Ratio-frequency-Growth effect and conserve the Whole Natural-World from microcosm to macrocosm . a) In Vacuum ≡ [10−62-10−35m] , The {Energy-Space-Monad |�̅�| ≡ |AB|},Vanishes. b) In Vacuum exists only Pointy vibrating ( The circular motion only ) between the Opposite Dipole [⊕↻↺⊝] generating the , w , B , momentum. c) As Unit-Energy Vectors �̅� = σ / B̅ → exists ON {Unit-Space-Monad |AB| ≡ |R̅|← And Then → {The Energy–Space - Monad |�̅�| ≡ |AB| ≡ The Existing Universe} d) STPL-Line-Machine is the Physical - Rotor for the Cosmic-Particles origination . The Gravitational-Force G , Acting in the Beyond-Planck- Cave , and on the light Velocity Vector �̅� , creates Electron-Charge �̅� Electron = G c √2 = 1,574.10−19 C . which agrees with that of Coulomb`s charge q̅ , and the Material-Points �̅� Photon= G √2.f , while by effecting on the Whole-Planck-Cave 𝐋 𝐏 ≡ e i . (- 5π/2) .10 , creates the Pointy-Gravity Force as this is the Ocean of Spins g̅ , and which Oriented-Spins , Originate Gravity �̅� = [↓= ↻] , and Antigravity - g = [↑= ↺] as Electron e̅ [72] . The Light-Velocity c̅ of Photons becoming from the Action of G on Planck`s Length , when act on a-cave , r = 𝐡 𝐜.𝐙𝐜 ≠ 𝐋 𝐏 then originates the Hydrogen-cave E = h fm , because Angular-Momentum �̅� = 2L w̅ = 2L 2πf = [ 2L 2π ].[ 1 f ] = Constant 2π [ 1 fm ] = SPIN [70]. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 72 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Four Energy-Space Constants in Nature are , [99] 𝐋 𝐏 ≡ 𝐞−i.( π 2 +2kπ).b ≡ ei .(- 5π/2) .10 ≡ ei .(-5π/2) .10 ≡ { √3.π. 1,616199.10−35 m } → and is The Planck`s length cave , g ≡ 𝐓² 𝐚³ ≡ 𝟏 f ²n.𝐚³ ≡ 9, 8076941 → The minimum Granular Quantized Energy , from the gravity frequency fg , representing the Bedding-Quantum-Raw-material of force G , on all Energy structures . G ≡ σ.𝚽³ ≡ g. kE ≡ g .[gL kL ] ≡ [ T²P 𝐚³ ].[gL kL ] ≡ 9,8076941* 6,8116.10−12 ≡ 6,68056.𝟏𝟎−𝟏𝟏 m3/ N.s² The Pulling and Cohesive Bond , of all the Quantized-Energy-Structures in all Spaces . 𝐟𝐧 ≡ { [S ≡ BP ≡ EM-R ≡ f1=N , f2 , f3, fD ,,f n] ≡ n (1+√5)σ 4πr = nσ.B̅ 2 r² and from cave 𝛌𝐍 = 4𝜋.r c nσ.(1+√5) = 𝜋2.𝑟 4 c n.B̅ ] } , The Amount of Energy , the unit meter of motion , in the Torsional and Resonance-Augmented-Golden-Ratio-Pattern Φ , and �̅� = �̅� = σ . Φ = F A Φ = [ G Φ A ] = [ 6,673692 .10−11.1,6180339887 36 .10 −20 . ] = 2.9985163. 108 m . 4c.. The Origination of Electron-charge �̅� ≡ �̅� : From relation G ≡ g.kE ≡ g .[gL kL ] ≡ [ T²P 𝐚³ ].[gL kL ] = g k , is seen that Gravitational-Force G , is The Spring-like central-force from a fix Point , the Source, on an attached , Probe, mass as → F = - k r = - k r.r̅ , as the 1-DOF equation , r̈ + w ² r = 0 , where k = The Unit-Spring-Force ≡ [meter of area].[meter of force ≡ stress] ≡ π g ≡ with a general solution r = A sinwnt + B coswnt , where A , B are constants and evaluated from the initial velocity-conditions which become r = [x́(0) / wn] . sinwnt + x(0).coswnt . Above equation represents the Natural-frequency of the Celestial-Structures , and for Planck`s length the only one Primary-Particle occupying the less frequency and Negative-charge , which is the electron equation and with the solution as , 𝐰𝐧 𝟐𝛑 = 𝐟𝐞 = 1 2𝜋 √ k m , or 4 π² f ²e . me = k = π g and or → 𝐦𝐞 = 𝐠 𝟒 𝛑 𝐟 ²𝐞 From Planck`s min-equation f e = E / h = [-13,6 x 1,6.10−19 = 2,176.10−16 Joule] / [ 6,6262.10−34 J.s ] = 3, 283998.𝟏𝟎𝟏𝟔 /s ,where in Hydrogen atom energy E = h f = -13,6 eV Substituting all the minimum-meters of Planck`s scale then , Electron mass is , 𝐦𝐞 = g 4 π f ²e = 9,808238 4.π.[3,28399.1016 ]² = - 7 , 2373149. 𝟏𝟎−𝟑𝟎 kg ……… (4b) 𝐟 𝐞 = 3, 283998.𝟏𝟎𝟏𝟔 /s , and Loop 𝐋 𝐞 = 2 , 3762992.𝟏𝟎−𝟏𝟔 m …..(4c) Relation 4π²f²e .me = k = π g , denotes that Electric-field generated by an electron has both , a component oscillating at the Electron`s - Compton - frequency 𝐟 𝐞 , which is IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 73 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Energy-Part , and the Non-oscillating component , the Electric-field-lines as the Energy-Conductor , The Space-Part , which is connecting Gravity with Electric-field. So , Electron , is the minimum-Energy massive Particle in Hydrogen-cave {-13,6 eV} , becoming from Material-Point In Planck`s-length , minimum-cave, occupying the less Unit frequency in Planck`s – Length ,with Negative-charge { 𝐟𝐞= 3,283998.1016 /s }. Because of the Negative-Energy in Hydrogen-cave , and the less-Negative Energy of Rims which consist the Plane-traces-Conductors or the Paths in-where electrons roll so Energy Rims consist the Energy-minimum-Paths in Hydrogen-cave-Potentials , as En - En−1 = [me.v²] .[2n-1] , where the electrons , the Electric–Charges , roll . Electron Charge �̅� Becomes from Magnetic Field M which creates the Electric-Field E , which is acting on Charge q , and the acting Force per second creates Work which is conserved and coincide with the Planck`s constant h . This is because h → J s = N m s = Power ,where from, Energy = Power x Time , issues the Beyond Planck`s length 𝐋𝐏 , and The Voltage V is → q̅ ≡ KE VP =1 = me 𝑐 ² 2 = g c ² 8πf ².1 , and V̅ ≡ VP ≡ c .Charge Total−Energy= h = c .q̅ h ...(1) Using the two Energy-equations for Linear-motion → fn = 1 2π √ k m 2 and Orbital-motion a =√ 1 k .f ² 3 , for Unit-Energy-Space-frequency k = g , a = π , then → g .f ². π ³ = 1 ...(2) Frequency fn =√ 1 g.π ³ 2 =√ 1 9,808238.π ³ 2 =1,8133418.10−3 , i.e. The Unit-Charge-Cave , q̅ into Hydrogen cave [a =1,82043047.10−12m].[1,813342.10−3/s] = 3,3010625.𝟏𝟎−𝟏𝟓 C From equations Charge and Voltage is the Self-Growing Property of frequency 𝐟𝐧 in Material-Point , therefore and for Hydrogen-cave is equal to → q̅.Φ , Because Gravitational Force is equal to → the Geometric-Resultant of light-velocity c , acting on Electron-Unit-Charge �̅� ← or , G = c √𝟐 �̅� , then Electron-Charge is , q̅ = G c √2 = 6,680561 .10−11 1,41429.[2,9979346.10 8] = 1,58.10−19 Coulomb , the known Coulomb`s charge quantity, From gL ,kL Universal constants the Newton`s Gravitational constant Force is , G ≡ g.kE ≡ g .[gL kL ] ≡ [ T²P 𝐚³ ].[gL kL ] ≡ 9,808238* 6,8116.10−12 ≡ 6,68056.10−11 𝐦³ 𝐍𝐬² The in depth Position of electrons allows to the Work Produced in Prior Phase-Areas as Golden Ratio Pattern , to be Stored in the Next . Energy = motion / T ≡ ( v 2πr ) . [σ + σ Φ] = �̅� . [ 𝛔 𝟐𝛑𝐫 + 𝛔𝚽 𝟐𝛑𝐫 ] ≡ �̅� . [. fn̅ + 𝐟𝐧 ] ≡ Moving - Storage →[ �̅� . fn̅ ] ← + Moving-Frequency → [ �̅�.𝐟𝐧 ] ← ≡ Material-Point i.e. The Energy produced in Photon-Cave is consisted of Two-moving-Storages , that travel as Wave [�̅�.𝐟𝐧] →[ �̅� . fn̅ ] → [ �̅� = �̅� = λ f Φ ] → [ S ≡ EM-R ≡ f1=N , f2 , f3, fD ,,f n= w² ] , and as Particle →[ f1= (E²+H²) = n (1+√5)σ 2πr = nB̅ π² r⁴ ] →{W≡EM-R≡ [εE² + μB²] = 2.λc.sin.2φ} and is The Duality of an Energy-Storage S ≡ { [⊕← 𝐫 →⊝] + Motion M . [87-89] 5c.. The Origination of Hydrogen Cave H : [99] From Kepler third law , the Energy-Closed-Space equation and Newton`s Laws of motion Constant k = v². r = ( w r ) ².r = [ 𝟐𝛑 𝐓 𝐫 ] ². r = 𝟒𝛑² 𝐫² 𝐓² . r = 𝟒𝛑² 𝐫 ³ 𝐓² = 4π². 𝐫³ 𝐓² = 4π².r³.𝐟²𝐩 ……..(k) IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 74 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Because (k) is constant , r ³.f ²p , is also a Constant multiplication of cave , r , and the frequency f , and the Work ≡ motion are conserved in cave , r , as the , n , frequencies fN = n (1+√5)σ 2πr = nB 𝜋2r4 , and for a Damping-cave → r(t) = r(t + w) ← as Planck`s scale is with min-Damping = 1 and Unit-Energy-Quantity 𝐖𝐮 , ( the critical-energy-unit in min , r ) , is this Unit-Stress-Gravity g , as k = E = T² a³ = g = 1 f ².a³ , i.e. → Stress �̅� , when is entering into the minimum cave , a , of a minimum Surface , then from the Period of Rotation T on the Perimeter , is created in Surface the minimum Quantity of Energy-cave and is the Hydrogen-Atom , where gf ² ≡ the Energy-Part embodied with stress �̅� , and the cave a³ , is the Space-Part , in 3-DOF Space as → T ² = g a³ = 9,808238.[ 2 ,1145016.10−11 ] ³ = 9 , 2728158.10−32 s , and Period T = 3,04513.10−16 s , or frequency f = 3,2839982.1015 / s . From equation E = h f = 6.62607.10−34 . 3,2839982.1015 = 2,175999. 10−18 J / (1,6022. 10−19 ) = 13,59999 eV Above Quantized Energy of 13,6 eV , corresponds to the Hydrogen-Atom-Cave . It was shown that in Conservative Systems of a Central-Force , the Total energy E is conserved and at Periapsis , energy E = 𝐆.𝐌𝐦 𝟐 𝐚 and e = √1 + 2EL2/G²M²m³ and for e = 0 then → E = - G²M² m³ 2 L² , i.e. Energy is always Negative . Hydrogen cave constant is k = E = T² a³ = g = [ 4π² GM ] therefore GM = 4π²a g and from Newton Total-energy E = - GM m 2 a , Rotational-Momentum L = √(1 − e2). GMm². a Eccentricity e = 0 , G Mm = - 2a.E , and then → L² = GMm².a = - 2aE[a] = - 2a²E . i.e. Equation L² = - 2a².E , denotes that Angular -Momentum L , in Orbit - Rims is always Negative and equal to , L = - a √𝟐. 𝐚𝐄 . So , Hydrogen-Atom , is the minimum-Energy-cave In Planck`s-Length , occupying the minimum Negative-Energy in Rims ≡ Caves , the Space , where Quantum-Energy-Rims are the Plane Traces-Conductors in where can roll the ⊝ charged Electrons . In Hydrogen-cave , the Resonance Frequency of an Electron ⊝ , and a Proton ⊕ , happens from the Stationary Unit-Cave of a System of Kepler second Planetary-law as equation , 4 π² m f ²o = k , and from constant law of Areas 1 = k .𝐟 ²𝐨 𝐚 𝟑 .Their common Constant-Energy k , is → k = 4 π² m f ²p = 1 𝐟 ²𝐩 𝐚𝟑 or , f ⁴p = 1 4π²ma3 and fp = √ 1 4π²m.a3 4 . A loop in Planck`s length 2a = L P = π.1,616199.10−35 m with frequency f = c 𝜆 , where then issues f ⁴p = 1 4π²ma3 = [ c 𝜆 ] ⁴ = [ 1 𝜆 ] ⁴.{𝑐4 = 1 [4πa3]mπa } , and for the Unit Intensity I = Energy Area = E = hf A = 4πa3 = 1KW A = 1 [4πa3] = 1 𝜆⁴ {mc4. πa}= 1 𝜆⁴ {1}.80,760865.1032. (3,14).0,8081.10−35 =0,5865.{ 1 𝜆⁴ }, and E-Intension is → I = 0,5865 . { 𝟏 𝝀⁴ } ← or , is the Rayleigh Scattering , i.e. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 75 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Quanta of Energy , Passing through a Unit-Surface or Volume , which becomes from the Kepler`s Second and third law Mechanism , is Inversely Proportional to the fourth Power of the wavelength 𝝀 . In Outer Electromagnetic fields , frequency , w , is that of inner motion → w = 2π T = 2π.fN ≡ [ 1+√5 2 ] σ 2πr = 𝛔.𝚽 𝐫 , which continuous to follow the Growth-Golden-Ratio-Pattern . Photon occupies such the Property of Material Point as the Growth-Pattern .With this way Photon`s frequency → fP = 1 / TP , Kicks-Start everything found on its-way . From the Resonance constant k = 1 f ²n.a ³ → or 1 = k .f ²n. a3 ← and from Kepler`s laws and Newton`s Lagrange laws for Orbits , the cave-Semi-major-axis is as , a = √T2/ k 3 = √ 1 g.f ² 3 =√ 4.π².r² g.σ².Φ² 3 = √ [4π2].x r² [g.Φ2].x σ² 3 , and 𝐟 𝐑 = wR 2π = √ k=1 g.a ³ 2 = √ 𝛔 g .Φ2.a³ 2 ≡ √ k md eV …(a) i.e. It is the Resonance-frequency fR , in Photons loop , and which is the Energy-Space Quanta , following the Growth-Golden Ratio-Pattern , which travels in all Cosmos. a)... It was shown that when two Photons strike each other , c = c⃖ , or reflect on a wall , then c = 0 and so their fundamental frequency f1 = n.[ �̅� 2r ]x[ 1 4 ] = n.[ �̅� 8.r ] which for n = [ 8.𝜋𝑟𝑐 v.λ ] increases to λ c , therefore for n = N.[ 8.𝜋𝑟𝑐 v.λ ] = N.| c 𝜆 | and for N → ∞ then f1 = E / h =∞ ...(b) In Dual Photon v̅ [ σΦ 2πr + σ 2πr ] ≡ v̅ . [ fn̅ + fn ] , Colours are the Still-Sub-Units of Storage [ v̅ . fn̅ ] , and exist as Frequencies , Violet , Blue , Green , which Stabilize with their Complementary colors Yellow , Orange , Red , of wavelength d = λ → 𝝀 𝐦𝐢𝐧  𝝀 𝐦𝐚𝐱 = 3.10−7m  3.√3.10−7m . b).. The Quanta of Energy , Passing through a Unit-Surface or Volume is Inversely Proportional to the fourth Power of the wavelength 𝜆 as I = 0,5865 . { 1 𝜆⁴ } . When fBE = E / h = ∞ then the Energy-Intension Increases and the wavelength 𝝀 Decreases and for fBE = ∞ then 𝝀 = 0 . This is what happens in Black-Holes where d = 𝜆 = 0 , and in where E = fBE = ∞ . Hydrogen-Energy E = h.fP = [ 6,6262.10−34 J.s ] . [ 3,28393.1015 / s] = 2,176.10−18 J and in eV → [2 ,176.10−18] / [1,6.10−19] = 13,6 eV , Above Quantity of Energy consist the Hydrogen minimum Energy-Rim , becoming from equation a = √𝐓𝟐/𝐠 𝟑 = √ [3,04513.10−16]² 9,80769411 3 = 2,1145016.10−11 m , for Unit energy cave . From Angular-Momentum B = r.mv = r [ πrv 2 ] v = 𝛑 𝐫² 𝟐 v² = 𝛑 𝐫² 𝟐 v² = πr³ 2 [n.π.c]² = 𝛑³ 𝐫³ 𝟐 c² , or Black-Hole-Energy → 𝐁𝐄 = 2.π⁵.r³.f ² = (𝛑𝐫)³.w² ← i.e. Velocity in Black-Holes is Related to Cave , r ³, and Energy w² times of light velocity. c).. Since the Primary motion ≡ [PM] exist from the three Breakages only , and STPL is the n =3 Mechanism creating Elementary Particles  [n-Knots=3] in 𝐕 GRA = 7,593.1035 V. Explanations : IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 76 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Action-Spaces ≡ The Infinite Quaternion - monads → �̅� ≡ ( s + v̅.i ) ≡ 𝐀𝐁̅̅ ̅̅ ≡ ≡ √s2 + [ v̅. i ]² ] , or �̅� ≡ x + i . y ≡ 𝐀𝐁̅̅ ̅̅ ≡ moduli , r , is their magnitude [ r = | r | = √ x2 + y2 ] , on Unit-Diameter AB̅̅ ̅̅ . The [QMAS] . For Quaternion |v̅| = s Breakages 𝐳 * 𝐳 = (s + v̅.i )² = s² +[v̅]² +2.sv̅.i = s² – v̅ ² ± 2.s v̅ = |s|² - |�̅�|² ± 2.|s|.|s̅| , i.e. In Figure -3- the Vector of Amplitude is Quaternion �̅� ≡ x + i . y , with vector �̅� , to be from Definition of the Complex Vector Space , the Set of Vectors of length (the n Vertices which form the n - Polygon Sides , or the Norm of the Complex - Vectors ) a...The 4-Geometrical Spaces , [ Space , Anti-Space , Neutral-Space , Sub-Space ] consist → The Geometrical - Mould of 𝐀𝐁̅̅ ̅̅ - Monad ≡ [PM-2R-M] ← b...The Vectors of Amplitude z = 2R , ON 𝐀𝐁̅̅ ̅̅ Monad , consist The Energy-Mould . of 2R-Form .The Physical -Mechanisms → [ Physical Mould of 2R- Monad ] ≡ [GM-AB-M] ← are such and dependent on the n - Number of Accessories they use . A… The Gravity Force G ≡ motion ≡ σ.𝚽³ , exists without mass and is Quantized Spread in{Space and Anti-Space Or ≡ QUA ≡ {is The Quantized-Units In-Area of Space and Anti-Space }, and as a Mould 𝚽 Golden-Ratio , exists as Impedance b. Placing [ Physical Mould of 2R- Monad ] ON [ Geometrical - Mould of 𝐀𝐁̅̅ ̅̅ - Monad ] [ PM-2R-M ] → ⇅ ← [ GM-AB-M ] THEN → Gravitational-Force G ≡ [ Σ = n . h ] , and n-NA = 1  Creates & Constructs, For n =1 ABM = [Σ = n h]  G / c = Charge �̅� = 𝐆 𝐜 √𝟐 = 1,58.10−19 Coulomb . 1)..→ g ← G ≡ σ.𝚽³ ≡ g. kE ≡ g .[gL kL ] ≡ [ T²P 𝐚³ ].[gL kL ] , and g = 𝐆 kE = [ 6,673692 .10−11 6,8116 .10 −12 . ] ≡ 9,8076941 . Gravity-Acceleration in Black-holes g G = [ π r v4 2 ].𝑒3 = 9,808238 m/s . For b = 36.10−20 , �̅� = F Φ A = [ G Φ A ] = [ 6,673692 .10−11.1,6180339887 36 .10 −20 . ] = 2.9895163.108 m/s 2).. → c̅ ← �̅� = �̅� = σ . Φ = F A Φ = [ G Φ A ] = [ 6,673692 .10−11.1,6180339887 36 .10 −20 . ] = 2.9985163.108 m . 3).. → q̅ ← from G = c √𝟐 �̅� , Electron-Charge is , q̅ = G c √2 = 1,58.𝟏𝟎−𝟏𝟗 Coulomb , B… The{TVPM} Mechanisms , from Charges on Two-Perpendicular-Vectors-Edges ≡ Markos Two-Vectors Three-Poles Mechanism is case {QUA} ≡ [ Σ = n . h ]  ∞ x 𝚽 which is the Physical Mechanism , and gives the ∞ - Impedance b , determining the Objective Reality which is the existing Universe . The two ⏊ Vectors Produce the ∞ Squares ∞ , 𝐀𝐁̅̅ ̅̅ Monad ≡ ∞ - [ GM-AB-M ] with ∞ , 2R-Form [ PM-2R-M ] and n ≡ ∞ Number of Accessories , as from monad R = σ J.w² = 𝛔 �̅� and velocity �̅� ≡  σ Φ . C…Primary motion ≡ [PM] exist from the three Breakages only [s² = ⊕ ,2s² = , - s² = ⊝] , on Triangle 3-Knots with n ≡ 3 Number of Accessories , therefore [STPL] Mechanism is the n = 3 Accessories-Mechanism creating the Elementary Particles , [n-Knots=3] in the Gravity Voltage  𝐕 GRA = 7,593.1035 V. This Physical Mechanism creates the Elementary Particles their n = 1 - ∞ , masses of Atom`s & Compounds determining their Atoms Complex-Vector-Response .These sparse Particles exist in all of the IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 77 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . universe in the sense that they are uniformly distributed . D…The { TCIC } Mechanisms ≡ [ Σ = n . h ]  8 x ① is the Physical Mechanism , from Charges In Three-Vectors-Conductors-Edges ≡ Markos Tetrahedron Cube Inscribed – Circumscribed Circles .The Tetrahedron Conductors -Vectors DA̅̅ ̅̅ ⏊ DB̅̅ ̅̅ ⏊ DC̅̅ ̅̅ Curry Energy as Electromagnetic-Fields (Forces-Frequencies-Stresses-any motion) TO the Inscribed Sphere (Nucleus of Atoms), TO Cube Vertices (Electrons-Position) ,TO the Circumscribed Sphere (The Electrons Orbits) for Equilibrium and for transferring Information and Signals . a.. The Protons and the Nucleus Structure : From above [GM-AB-M] ≡ 8 x [(+)[↔](-)] ≡ Neutral = n , then a... For 1 Proton [(+)[↔](-)]↔(+) ≡ Neutral → n = 1 p = 1 b... For 2 Proton (+)↔[(-)↔(+)][(+)↔(-)]↔(+) ≡ Neutral → p n n p c... For 3 Proton (+)↔[(-)(+)](+)[(-)(+)](+)[(-)(+)] ≡ Neutral → n p n p n p d... For 4 Proton (+)↔[(-)(+)](+)[(-)(+)](+)[(-)(+)](+)[(-)(+)] → p n p n 0 n p n p h... For k Proton (k1)↔[(-)(+)](k2)[(-)(+)](+)[(-)(+)](+)[(-)(𝐡𝐤)] → kp kn kp kn 0 kn kp kn kp where n-NA ≡ k ≡ The Number of Accessories of {TCIC} Mechanisms ≡ [ Σ = n . h ] On → kp kn kp kn 0 kn kp kn kp k - Space [ GM-AB-M ] . From above is seen the Way of Atoms Construction. (Geometrical-Mould of 𝐀𝐁̅̅ ̅̅ -Monad). E…From Energy = motion / T ≡ ( v 2πr ).[σ+σ Φ] = �̅�.[ 𝛔 𝟐𝛑𝐫 + 𝛔𝚽 𝟐𝛑𝐫 ] ≡ �̅� . [. fn̅ + 𝐟𝐧 ] ≡ Moving Storage →[ �̅� . fn̅ ] ≡ [GM-AB-M] ← + Moving-Frequency → [�̅�.𝐟𝐧] ≡ [PM-2R-M] ← ≡ [ Σ = n . h ] , and n-NA = 1  Material-Point i.e. The Energy produced in Photon-Cave is consisted of Two-moving-Storages that travel as Wave [�̅�.𝐟𝐧] →[ �̅� . fn̅ ] → [ �̅� = �̅� = λ f Φ ] → [S≡ EM-R ≡ f1=N] & as Particle→[ f1= (E²+H²) = n (1+√5)σ 2πr = nB̅ π² r⁴ ]→{W≡EM-R≡ [εE²+μB²] = 2.λc.sin.2φ} and The Duality of an Energy-Storage ≡{[GM-AB-M] ≡ S = [⊕← 𝐫 →⊝] + Motion ≡ [PM-2R-M] ≡M≡ �̅� − 𝐕𝐞𝐜𝐭𝐨𝐫 } , issuing [PM-2R-M] = √−1 .[GM-AB-M] , Therefore → Photon-Mechanism is for Both travelling as Particle and as Wave , IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 78 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 6c.. THE EPILOGUES OF PRIOR . Figure – 17 : The Periodic motion in Caves follows Material - Geometry rules . In (1). Is shown the Geometrical Expose of , Dynamic-Space-Energy relation g .r ³.𝐟²𝐩= 1 In (2). Is shown the Mechanical Impress of the , Orbit-Space-Energy relation g .r ³.𝐟²𝐩= 1 In (3). Is shown the Extreme Design of the , Dynamic-Space-Energy relation g .r ³.𝐟²𝐩= 1 In (4). Is shown the Geometrical Creation of The Energy-monads-motion & The Physical Monads - motion from their Unit - relation g .r ³.𝐟²𝐩= 1 THE ORIGINATION OF ANTI – GRAVITY - STRESS - g : 6c1.. The Origination & The Nature of Gravity g . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 79 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Figure -17a-:The Stability of ±SPIN as an Electromagnetic Wave of Two frequencies ,0, c From relation Angular Velocity c = w r = 2πfn.r = 2πr .fn = σ Φ . Since light-velocity is created from G beyond the Planck`s length r P =1,61626.10−35 m , therefore issues c = w r = w r P = 2π rP .fn = σ Φ , and from the light-velocity Properties becomes , a).. The light velocity c is constant . Since c is constant then is analyzed to , ( cx , cy ) in any two directions x , y where x ⏊ y and y to be the gravity direction on where gravity is acting . In figure (8) are seen the 4-Point Positions , at where Velocity Phase changes and the Work done in each Phase of Photon`s inner motion . The 4-Phases correspond to the Wavelength . λ , as λ / 4 . in distance with 90° = 𝜋 2 to be the change of Phase . This means that the Produced Work in the Prior Phase-Area is as the Golden-Ratio Pattern which is Stored in the Next and Perpendicular Phase-Area . The Light velocity �̅� , changes Phase every 90° and the created Work { W = c y x [Quarter circle 1-O-4] } is Symmetrically Stored from 0° → 90° alternately + , - , - , + , in [Quarter circle 1-O-4] in the [Quarter circle 4-O-2] into [Quarter circle 2-O-3] into [Quarter circle 3-O-1] The Momentum ≡ Spin ≡�̅�≡ r m v = [ 𝛑.𝐫𝟐 𝟒 ].�̅� x �̅� where ± [ 𝛑.𝐫𝟐 𝟒 ] = The Quarter circle. The Quarter circle , + + , consist the + Spin ≡ �̅� which is the Gravity Force , and the Quarter circle , - - , consist the -Spin ≡ �̅� which is the Gravity Force from Total ,+, -,-,+, When r → 0 Or when �̅� doesn`t act on the Quarter circle then happens Black-Hole. In [108] is determined the when Newton’s Gravitational Force 𝐆 , Stops Acting thrusting , on The Light velocity Vector �̅� , and happen the Black-Holes . b).. The Dual Property of the Simultaneously existing Spin as Electric and as Magnetic IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 80 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Field in wavelengths , λ x , λ y , as Effective Range , is thrusted through the Light velocity �̅� , and finally is from the Newton’s Gravitational Force 𝐆 . In Planck`s length r P =1,616255.10−35 m the created Work from �̅�, on 1-4 Position is stored in the [1-O-4] Quarter-circle surface and at Phase-4 where c = 0 is then stored in the [4-O-2 ] Quarter circle surface . At Phase-2 where �̅� = - �̅� motion is stored in the [2-O-3 ] Quarter circle surface , and At Phase-3 where �̅� = 0 motion is stored in the [3-O-1] Quarter circle surface , i.e. is alternately repeated the + [ 𝛑.𝐫𝟐 𝟒 ] =The +Quarter circle [1-O-4] , and the - [ 𝛑.𝐫𝟐 𝟒 ] =The - Quarter circle [4-O-2] both in the same Plane-surface [1-4-2-3] of motion . The Surface [ 1 – 4 – 2 – 3 ] is ⏊ on the SPIN Axis → U – D . From fn = σ 2πr Φ and c̅ = w̅.r = 2πfn.r = 2πr .fn = σ Φ , are frequencies , 0 , c , relation. At c = 0 exist minima or maxima where is the change of two Energy-Semicircles. c).. The origination of Spin in cave H . From Orbit vibration fn = (1+√5)σ 4πr = σ 2πr Φ and from relation g.a³.f ²n = 1 , f ²n = σ² 4π²r² Φ² = = 1 g .a³ then g.a³.Φ² = 4𝜋²𝑎² 𝜎² , or 1). a = 1 g [ 2𝜋 σ Φ ] ² , and from Work W = 2E = B w = J.w² , or 2E = 2π.f and �̅�f = E π then 2). The Energy in Planck-scale-cave 2E = B̅fn = [ (1+√5)σ 4πr ].B̅ = [ σ 2πr ].B̅ Φ , or E = [𝚽 𝛔 𝟒𝛑𝐫 ].�̅� i.e. The Energy in Planck-cave-Particles is Dependent on their Spin only , and for Electron r = a e → E = [Φ σ 4π.r ].B̅ =1,6180339[ 1 4𝜋.1,6840307.10−17 ]x x 2,845976. 10−17= 21,76.10−19 1,6.10 19 . = 13,6 eV , which is the min-Voltage in Hydrogen cave . d).. The Angular Momentum in cave c̅ - r , In cave , r , The Energy-Vector �̅� is Positive for O3⃗⃗⃗⃗ ⃗ , O1⃗⃗⃗⃗ ⃗ , O4⃗⃗⃗⃗ ⃗ , Directions while , The Energy-Vector �̅� is Negative for O3⃐⃗⃗⃗⃗⃗ , O2⃐⃗⃗⃗⃗⃗ , O4⃐⃗⃗⃗⃗⃗ , Directions . Since Total-Energy L = B w = J.w 2 w = J.w² 2 then 2L = J.w² , then Angular Momentum relation B̅ = r.mv = r [ π.r⁴ 2 ]. 2πf r = π ².𝐫𝟔.f . and B = J w i.e. Angular Momentum �̅� = r m v = J.w 2 = [ π.𝐫𝟒 4 ] c r = [ 𝛑.𝐫𝟑 𝟒 ].�̅� = [ 𝛑.𝐫 𝟐 𝟒 ].�̅� x �̅� ......(a) IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 81 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . For �̅� = ± �̅� , then + �̅� = + Gravity – Spin = + [ 𝛑.𝐫𝟐 𝟒 ].�̅� x �̅� , The + Gravity Force - �̅� = - Gravity – Spin = - [ 𝛑.𝐫𝟐 𝟒 ].�̅� x �̅� , The -Antigravity Force From Momentum relation B̅ = m r v = m r² w = m r² (2πf) = J.w 2 = [ πr⁴ 2 ].[2πf] then Spin S ≡ Angular-Momentum �̅� ≡ �̅� = J.w 2 = πr⁴ 4 [2πf]= π².r⁴.[f] = 𝛑.𝐫³𝚽.𝛔 𝟐 ≡ [𝟏+√𝟓 ].𝛔.𝛑.𝐫³ 𝟒 = 𝛑𝐫𝟑𝛔.𝚽 𝟐 ..(b) For �̅� = ± �̅� , then + �̅� = + Gravity – Spin = + 𝛑.𝐫𝟑𝛔.𝚽 𝟐 , The Gravity Spin - �̅� = - Gravity – Spin = - 𝛑𝐫𝟑𝛔.𝚽 𝟐 , The Antigravity Spin From Spin-Momentum relation S = �̅� = 𝐄 𝐰 = 𝐄 𝟐𝛑𝐟 = 𝐄𝐫 𝛔𝚽 = 𝐫𝐆 𝛔𝚽 , and �̅�² = 𝐉²𝛔² 𝐫² Φ², so → [�̅�]² = [ 𝚽𝐉𝛔 𝐫 ]²← while from Gravity-force 𝐅𝐆 ≡ 𝐦𝐆g= g. [σ c].r = 𝐦𝐆 𝐜² 𝐫 = Jw².𝐠𝐆 = 𝐯² 𝐫 . [ 𝛑𝐫𝟒 𝟐 ] 𝐯² 𝐫² = 𝛑𝐫𝐯𝟒 𝟐 , and Gravity-Acceleration ± 𝐠𝐆 = [ 𝛑𝐫𝐯𝟒 𝟐 ]. 𝒆𝟑 , Geometry Growth [ 𝛑𝐫𝐯𝟒 𝟐 ]=[ 𝟑,𝟏𝟒𝟏𝟓𝟗𝟐𝟔([√𝟓+𝟏]. √𝟐 𝟒 .𝟏𝟎−𝟑𝟓).(𝟐,𝟗𝟗𝟕𝟗𝟑𝟒𝟓𝟖.𝟏𝟎𝟖 )𝟒 𝟐 ].𝒆𝟑 = = 6,044981.𝟏𝟎−𝟑𝟓. 80,773378.𝟏𝟎𝟑𝟐. 20,085536 = 𝐠𝐆 = ± 9,8076925 . i.e. Gravity + 𝐠𝐆 = + 9,8076925 = 𝛝𝐫c̅ 𝛝𝐭 = + �̅� , and Antigravity= - 𝐠𝐆 = - 9,8076925 = 𝛝𝐫c̅ 𝛝𝐭 = - �̅� are the Two Opposite Forces as Spin and for the Stability of the Spin - �̅� , Equilibrium. For c = 0 exist minima or maxima and thus SPIN is equal to the Negative Gradient of the Potential Energy in the two Interchange Energy Semi circles . Centripetal force Originates the Two-Momentum ± �̅� , �̅� = r m v = [ 𝛑𝐫𝟑𝛔.𝚽 𝟐 ] ≡ Gravity – Antigravity – Spin. 6c2.. The Stability of Equilibrium : In the Conservative System , of cave �̅� - r , the Total Energy 𝐄 𝐓 remains constant and the Total-Energy is 𝐄 𝐓 = 𝐄 𝐊 + 𝐄 𝐔 = 𝟏 𝟐 [�̇�] ² + U(x) = constant …..(1) where 𝐄 𝐊= The Kinetic Energy and 𝐄 𝐔 = The Potential Energy . For y = �̇� then , y = �̇� =± √𝟐. [𝐄 − 𝐔(𝐱)] …(2) The differential equation of motion in cave �̅� - r , for Conservative Systems is ẍ = f(x) and Because ẍ = ẋ (d�̇� /dẋ ) then ẋ. dẋ = f(x)dx =0 ..(3) By Integrating then ẋ² 2 - ∫ 𝑓(𝑥)𝑑𝑥 𝑥 0 = IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 82 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . E , and by comparison to Priors then → U(x) = - ∫ 𝑓(𝑥)𝑑𝑥 𝑥 0 or f(x) = - 𝐝𝐔 𝐝𝐱 = �̈� …(4) i.e. For the Conservative System , of cave �̅� - r , the force = �̅� = r m �̅� = J.w² 2 = [ 𝛑.𝐫𝟐 𝟐 ].�̅� ² is Equal to the Negative Gradient of the Potential Energy Into 2-Energy Semi-circles. For y = �̇� then (3) becomes dy dx = f (x) y = Φ (x , y ) , and the Singular Points correspond to f (x) = 0 , and y = ẋ = 0 and hence are the Equilibrium Points . For every Point x , y in the Phase-Plane for which Φ (x y) is not indeterminate exists a unique slope of the trajectory. When the velocity vector is at Points 1,2 then y = 0 and f(x , y) = C ≠ 0 then the Slope of the trajectory is infinite and angle φ = 90ᶱ . Since motion in cave �̅� - r , is a single-DOF oscillator , then motion equation is ẍ + w² x = 0 .Equation indicates that at the Equilibrium Singular Points , 3 , 4 , the Slope of the Potential Energy curve U(x) must be zero, meaning minima or maxima , and which is the ± change of direction for the Four Energy- Quarter circles [1-O-4] , [4-O-2] , [2-O-3] , [3-O-1] in ( cx , cy ) where 𝐜𝐲 ̅̅̅̅ = �̅� = Spin Direction , Conclusions : 1… From A-Momentum �̅� = r m �̅� = J.w² 2 = 𝟏 𝟐 [�̇�]² +U(x) = 𝟏 𝟐 [ y]² + 𝟏 𝟐 k x² = constant , then Spin S ≡ �̅� = 1 2 [y]² + 1 2 k x² = [y] ² + w ² x² = C = Constant , or → y² + w² x² = C ←….(a) Equation (a) is a series of Ellipses determined by C , and for w = 1 y/w = y then reduce to circles y² + x² = C , in �̅� - r Plane and Perpendicular to S since , [ �̅� - r ] ⊥ �̅� . 2… In the Conservative System of cave �̅� - r , where �̅� = Light-Velocity = 2.9985163. 108 m/s , r = Planck`s -Length = √2 4 . 10−35 m , motion in cave is an single-DOF oscillator as equation ẍ + w² x = 0 . Equation of motion is a circle on where exist the 2-Singular Points 3 , 4 , meaning minima or maxima , where issues dy dx = f (x) y = Φ (x , y ) , and are in the Mode-Shapes at the Points of Equilibrium . At Singular Points 3 , 4 , Velocity 𝐜𝐲 ̅̅̅̅ = �̅� = 0 and Spin �̅� = Gravity = 0 , and since dy dx < 0 , then Spin �̅� changes Direction becoming - �̅� = Antigravity = 0 , meaning that at Singular Points Gravity & Antigravity are Zero . 3…At Point 1 , Velocity 𝐜𝐲 ̅̅̅̅ = + �̅� because of 𝐜𝐲 ̅̅̅̅ , +↓ direction and corresponds the Spin �̅� = Gravity = �̅� = r m �̅� which is Constant and equal to Work = 𝐜𝐲 ̅̅̅̅ x [Quarter circle 1-O-4] At Point 2 , Velocity 𝐜𝐲 ̅̅̅̅ = - �̅� because of 𝐜𝐲 ̅̅̅̅ , -↑ direction and corresponds the Spin �̅� = Gravity = �̅� = r m �̅� which is Constant and equal to Work = 𝐜𝐲 ̅̅̅̅ x [Quarter circle 2-O-3] where the [Semicircle 4-1-3] = + �̅� ≡The Gravity , and [Semicircle 4-2-3] ≡ - �̅� = The IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 83 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Antigravity .The produced Work from the eternal Rotation is store in the Semicircles as , (+) Total Work = [Semicircle 4-1-3] = + �̅� and (-) Total Work =[Semicircle 4-2-3] = - �̅� 4…From Newton`s law of Universal Gravitation m r. �̅� ² = G 𝐦.𝐌 𝐫 ² , and when m = 1 , issues �̅� ² = 𝐆.𝐌 𝐫 ³ , and B = J w , therefore angular velocity �̅� = ± √ GM r³ 2 = √ Gπ.r4 2.r³ 2 = √ π.Gr 2 2 . This Angular-momentum is identical with Spin which is trapped in caves`s loops which are in Phase with each other. The amplitude of Oscillation varies from Zero at Nodes to maxima at Antinodes .Since Frequency ≡ 𝐟𝐧 = ( nσ 8 r ² ).�̅� , so Energy-caves are Stationary Wave-Fringes 5… The Total-Energy , E T , in Caves , [70] a… → In any Moving System E T = E R + E K = πc² 8 r³ + 1 2 m.v² = 3,535.1016. r ³ + 1 2 m.v² , b… → Light-Velocity c̅ of Photons becomes from the Action of G on Plancks Length LP = r P =1,6162.10−35 m cave r, and when cave r = 𝐡 𝐜.𝐙𝐜 ≠ 𝐋 𝐏 originates Hydrogen-cave. It is shown later that when the G action Area - Resistance ≡ Gravitational-Impedance Z G−I = 6,344907.10−35 Kg differs for any reason then Black-Holes happen. c… → Euler`s Rotating Rigid Body .The D-Work is W = 2L = B̅ . w̅ = J .w² ≡ h. 𝐟𝐧 , and for m mass W = h fm ,Angular-Momentum �̅� = 2L w̅ = 2L 2πf = [ 2L 2π ].[ 1 f ] = Constant 2π [ 1 fm ]= SPIN d… → In Central-motion ,Kepler constant k = 4π². r³. fP², or 1 = [ 4𝜋² 𝑘 ] r³. fP², 1 = c . r³. 𝐟𝐏² where for → r → 0 then 𝐟 𝐏 → ∞ ← , a case of a Black-Hole in any Central motion . e… →The Vibrations in Orbit-Systems are as → W = 4π² .[ 𝐫³ 𝐓² ] . �̅� = 4π². r ³. 𝐟²𝐩 . �̅� ← f… → In [PNS] Neutral Space , Dumping force is Fd = C . ṙ = ± Co .w. [√A2 − r ²] and Fd ² / [ Co .w. A ] ² + r ² / A² = 1 , i.e. Energy ≡ motion is mapped as an Ellipse with axis , 𝐅𝐝 , x , and Energy Dissipated Per - Cycle , the Area enclosed by ellipse. g… → The Energy in [PNS] , is either as momentum (mv) or as Rotational Momentum (Λ = r.mv) . Velocity �̇� , in form ẋ = w A cos.(wt - θ) = ± wA.[ √1 − sin²(wt − θ) ] =  w . [ √A² − x² ] where momentum relation �̅� = m r v = m r² w = m r² (2πf) = J.w 2 = [ πr⁴ 4 ].[2πf] = π2r4f 2 ,and Mass of Elementary-Particles is m = π2r4f 2w.r² = π2r3 2v f = π1r2 4 , or The repeated Quarter circle [ 𝛑.𝐫𝟐 𝟒 ] Surface of the ± Spin Dual Property In Planck`s length and area , 𝐀 𝐐𝐂 = [ 𝛑.𝐫² 𝟒 ] = 𝟑,𝟏𝟒𝟏𝟓𝟗𝟐𝟔.( √𝟐 𝟒 .𝟏𝟎−𝟑𝟓)² 𝟒 = 1,1107206.10−70 m² . Since issues the monad of mass to be [ eV c² ] = 1,782662.10−36 Kg , then Energy Dissipated Per - 𝐴 𝑄𝐶 = 𝐄 𝐐𝐂 = ( 1,1107206.10−70 m²/ 1,782662. 10−36 = 6,230715. 10−35 Kg , Or from the G action Area - Resistance ≡ Gravitational-Impedance Z IQC , exists ZIQC =√2𝑐3A QC / G , and 𝐦 𝐏𝐍𝐒 ≡ ZIQC = √2.𝑐3A QC / G = 1,4143.(27.10"24)1,1107206.10−70 𝟔,680056.10−11 = 6,344907.10−35 Kg . The result agree with Prior because in [PNS] Dipole of Space exists the Impedance ZIQC , which Doesn`t exist in [MFMF] Space . [22-26] IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 84 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . h… → In [MFMF] Space , Work = motion = Energy is zero as W = ∫ P. ds A B = 0 , and this because ds = v⃗ = 0 and Time is not existing and v⃗ = 0 as relation , s = �⃗� .T . Each Unit AB⃗⃗⃗⃗ ⃗ = ds > 0 exists by this Inner Impulse ( P ) and so P A+P B = 0 . i.e. The Position and Dimension of all Points which are connected across Universe and that of Spaces exists because of this Static Inner Impulse , on the contrary should be one Point only Or , Primary Point A = Black Hole → ds = 0 and P = ∞ . Since Impulse is ∞ then may be → Vacuum , Momentum or Potential or Induced Potential and it is a Type of Effect , to Push the nature → (Impulse ≡ Motion ≡ Energy) ← 6c3.. The Nature of Antigravity (- g ) . From Mechanics in case that , in an Axisymmetric Rotating Body , with constant Angular-Velocity , w , the moment of Inertia , 𝐉𝐱 = 𝐉𝐲 , is equal about the two of the three Principal axis , and then , 1.. The Angular – velocity - vector , �̅� , describes the Ellipsoid of Angular Velocity and its nib describes a Cone of which Plane-Base is Fixed Simultaneously , 2.. The Angular-Momentum , �̅� , describes the Ellipsoid of Angular-Momentum and its nib describes a Cone also of which Plane-Base is also Fixed . 3.. The Nib of Angular - velocity - vector , �̅� , describes on the Tangential-Plane of the Angular-Momentum-Ellipsoid , the Herpolhode , while , 4.. The Nib of Angular-Momentum , �̅� , describes on the Tangential-Plane , of the Angular-Velocity-Ellipsoid , the Polhode . 5.. The Fixed-Tangential-Planes on , �̅� , and , �̅� , nib are alternately Perpendicular to , B̅ , and , w̅ , common Central Axes of Rotation . 6.. The Kinetic Rotational – Energy of monad , which is the Work in monad , is the Scalar quantity , L , The Vector Angular Momentum quantity is , �̅� , and the Vector of Angular-Velocity quantity is , �̅� , which Three monads are related as �̅�.�̅� = 2L = J.w² where J = The moment of inertia around the axis of rotation = ⇅ 7..All above happen in , Material-Point , where the Positive ⊕ constituent The + �̅� , is Eternally self - rolling on the Negative ⊝ constituent ,The - �̅� , with Angular-Velocity w , in Infinite Spherical traces , either at Great-circles [+ or - ] , or Small-circles , [+] where is the Clockwise Left direction and [-] where is Anti-Clockwise Right direction or any other close Spherical-curve , and Applying all laws of Mechanics into this Energy Chaos , is thus created the → First–Discrete-Energy–monad �̅� , the Material Point ← i.e. The Angular Momentum �̅� , is the Quantum of Physics and is , for all the Energy – Space-Universe .The Energy = E dissipated per cycle is equal to the Work Wd = E = h f = h(1+ √5 ] ) 4π .[ 𝜎 𝑟 ] = 8.kζ(πr²) and is Stored in monad , which is a Stationary Wave with and - in , n , loops as 𝐖𝐧 = [ 𝟒𝝅𝒓² 𝟑 ].𝐟𝐧 = [ 𝟒𝝅𝒓² 𝟑 ].𝐧𝐟𝟏 = n (𝟏+√𝟓)𝛔𝐫 𝟑 = n [𝛔𝚽.𝐫] 𝟏,𝟓 . 8.. Geometry has as Monad ,The discrete continuity 𝐀𝐁⃖⃗⃗⃗ ⃗ -length becoming from the Zero IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 85 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Point ≡ 0 , and Mechanics-Physics the Recent-Acquisition of the Material - Geometry, where Zero-Point , 0 =  = {⊕+⊝} = The Material-Point = The Quantum of Space , as , The Positive Space and The Negative Anti-Space . [58] In Fig-8 , The Light velocity vector �̅� = 2,9979. 10 8 m/s Rotates In the Planck`s length Perimeter 𝐫 𝐏 = 1,616255.10−35 m , and is thrusted through The Newton’s – Gravitational Force 𝐆 , During Rotation is created Work which is stored in the circle Surface as Angular - Momentum �̅� = r m v = J.w 2 = [ π.𝐫𝟒 4 ] c r = [ 𝛑.𝐫𝟑 𝟒 ].�̅� = [ 𝛑.𝐫 𝟐 𝟒 ].�̅� x �̅� , in Semi circle {+[ 4 -1-3]} , and Semi circle { - [ 4-2-3] }, for the Stability of Equilibrium . i.e. Gravity + g ≡ The Vector of Angular-Momentum �̅� ≡ + Spin ≡ ↓↻ ≡ The Work stored in the [4-1-3] Semi circle of the , �̅� - �̅� Vectors circle - Surface Produced from the Gravitational-Force , G , as ACTING into the – Planck- Cave , AND on the light Velocity Vector in , + ↓ �̅� Direction . Antigravity - g ≡ The Vector of Angular -Momentum �̅� ≡ - Spin ≡ ↑↺ ≡ The Work stored in the complementary [4-2-3] Semi circle of the , �̅� - �̅� Vectors circle-Surface which is Produced from the Gravitational-Force , G , as ACTING into the - Planck- Cave , AND on the light Velocity Vector in The , - ↑ �̅� Direction . Figure – 17b - The Material , LRC Circuit on Orbit , on Focus-Planet-Sector |F↔P| In (1). Force g , as Wave , is directed to the center of rotation F , and is proportional to the distance PF ≡ Focus - Planet .The Gravitational Potential-Energy 𝐠𝐆 = 9, 8076925 is stored in → Focus-Planet-Sector ≡ FP ← which is the Material Capacitor Stores charge as that of Material-LRC-circuit , and Inductors . Because of the chains of Spins and of their Periodic excitation [↔] , is thus created a Magnetic field due to the LRC-circuit and which is tuning to the critical Quantum and critical-State 𝐠𝐆 . The chains of Spins are Pointy - vibrating with their characteristic IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 86 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . frequencies f = (1+ √5 ] ) .σ 4πr B̅ , and filling up the entire universe . Since Inner-stresses 𝛔𝟏,𝟐 = σ1/2 ± (½)√σ1² + 4. σ1² = σ1/2 [1±√5] follow the Golden ratio on stresses then this Quantum-energy gG Produced , is the State causing them to Magnetically- Resonate . In Material-Point-chain of caves r = λ/2 , and since ẋ = w.r → w = ẋ/r then Golden ratio Damping-Force is the Work produced with Unit Damping-ratio ζ=1 .Since also ẋ = v then exists , Fd = c. ẋ = 1m.wn .ẋ = 1.[ (π r²/2) v2]. ẋ ² 𝑟 = [ 𝛑𝐫𝐯𝟒 𝟐 ] = g = 𝟐 𝒓 [ σ (1+√5) ] ² = 𝟒𝝈² 𝒓 [ 3+√5 ] . This dissipation of Energy is determined under conditions of cyclic oscillations , and is dependent on Glue-bond σ , and r , cave . Since r is in denominator then for the very small caves , the under Planck`s caves, Damping-Force becomes infinite independently of Glue-bond . This may be considered as a Type which happens in Black holes as this happens in algebra inverse fractions also . In (2) is Presented the Back-Up Electromagnetic current flowing in Opposite direction FP , by changing the Spin Direction of the Sector-Material-Points such that work W = g . From Kepler`s 2nd law the , Area S , swept by any Focus-Planet-Sector ≡ FP is the constant k , and equal to, S² = L² T² 4m² = π²a².[ b = πa ( L² 2mE ) ] , or T ² a ² = 4π²m 2E = 4π² 2E/m = 4π² a GM and → 𝐓 ² 𝐚 ³ = 𝟒𝛑² 𝐆𝐌 = k and k = 𝟏 𝐟 ²𝐧 .𝐚 ³ , from which issues → 1 = k . 𝐟²𝐧 . 𝐚 𝟑 and , k = 4π² GM where Is seen that , Resonance between Focus-Planet exists through Space a and Energy f . From the Second-order differential equation excited by a Harmonic external force , Ft .sin wt , and is as , m d²x dt² + c dx dt + k . x = Ft . sin w t , corresponds Physically to the free damped vibration , where x = the displacement , dx / dt = the velocity and , d²x / dt² = the acceleration of monad , m ,c ,k constants , with the general solution given by the equation x = A . 𝑒𝑠1.𝑡+ B.𝑒𝑠2.𝑡 + X sin(wt-φ) ……(1) The Equation → L d²q dt² + R dq dt + 1 𝐶 q = E t . sinwt .Corresponds Physically to the free damped vibration , where Charge q = is the Physical Property of matter that causes it to experience a force which can be Positive or Negative , dq / dt = the least quantized amount of charge , and d²q / dt² = the Space distribution of charge , and L , R , C Inductance , Resistance , Elasticity constants with general solution given by the equation q = A . 𝑒𝑠1.𝑡+ B. 𝑒𝑠2.𝑡 + X sin(wt-φ) …...(2) Equations (1) , (2) give the analogic relation of the Classical mechanics [Space Position x] and the Electromagnetism [Quanta of Energy , q ] of Storing and Removing of energy in Energy-Space cosmos .Distributed forces L1- L2 = L (di/dt) , R1- R2 = R . i , C1- C2 = q / C respectively , showing the Identification of the Mechanical and Physical laws . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 87 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The way that Potential-Energy is stored , is that of Material - LRC- Circuit , which is for the Gravitational-Potential-Energy the Material-Capacitor or the → Focus-Planet–Sector – Stores-Change ← which develop a Voltage in response to that charge .The coil of wire is the infinite Stationary-Dipole-Spinning Material-Points of this → Focus – Planet - Sector ← which develops the Back-Electromagnetic-force ,when current through them changes . In-Cave-Box exist two motions ,Revolving and Periodic, the acceleration of Gravity g ≡  σ exists for the First Box-𝐁𝐑 , while Local-Extreme-case for the Second Box-𝐁𝐏 . In this Gravity g ≡  σ , is Locally altering S , by changing the Principal-stress σ with Local uniform Pressure 𝐠𝐋 ≡ g kE = g * [Force/Area] = G , i.e. is Proved that the minimum Local-Energy acceleration is the known Universal Gravitational-constant G = g k = 𝐤𝐄 g = 𝐤𝐋 σ = g gL kL such for Macrocosm as for Microcosm , and both cases Obeying Newton`s Laws of motion . 4.. Since Electron is rotating in Orbits so Electromagnetic Wave is also rotating , therefore is needed a Proper-Stationary-Magnet on which Rotation becomes Oscillation in order to succeed a clear 3-dimensioned , Φ ³, Magnetic-Resonance-Imaging [The MRI ] and the other Systems use this Property of Bottom-Information . This Property of Spin in all depths of Energy - caves allows the Granularity of Energy in Energy- loops , and after to be Bonded .This Property of Electron may be used for Cancer Chromosomes . 6c4.. The Conservation of Energy in → velocity �̅� and in gravity �̅� Caves ← Since Gravity- �̅� is the Energy ≡The Angular-Momentum {�̅� =r m v}in Planck`s length r P , and it is a Stationary-Wave or an Electromagnetic Wave-Signal which transmits Informations Then follows the Waves-Frequency-Signal-Spectrum .The Gravitational Constant G is the Active force on light velocity �̅� and consequently on g̅ also . From Waves theory → Position x = A.sinwt and Acceleration �̈� = A w ² ← where the Signal-Spectrum is consisted of the Amplitude 𝐀 𝐂 and of the Angular frequency �̅� , from where all other Properties follow. Since �̈� ts the acceleration Force of Position x , then for Planck`s length issues ẍ = G = A²G.w ², or �̅̅̅� = √G/A² = √6,68. 10−11/[2,6262. 10−35]² so Gravity-frequency→ �̅̅̅� = 0,5057.1030 H and Gravity-Wave-frequency 𝐟 𝑮𝑾=8,05.1028 H ← THE GRAVITATIONAL FORCE ACTION : [[G]] → {[�̅� ] ± 𝐄𝒏}≡ �̅� ≡ 𝐄𝒏+{[G] ± [�̅� ]} and → 𝐄 𝐑 = 𝐄 𝐆 + 𝐄 �̅� ← where , [[G]] = [Gravitational Constant G] ≡ The Signalling.(W c) → The Carrier - Wave [[�̅�]] = Any Action of [G - ON- �̅�] ≡ Any Signalling.(�̅�) Modulating Velocity-Wave [[�̅�]] = The New Modulated [G ↔ �̅�] Signal ≡ f �̅� m+wc 2 + f �̅� m−wc 2 ≡ [MAS] COMPOUND = The Demodulated [MAS] ≡ f w+2wc 2 + f w−2wc 2 ≡ [DMAS] . �̅� , since [ G = The CARRIER ] →[ �̅� = The MODULATOR ]→[ G+ �̅� = The MODULATED ] and [ G+ �̅� ] + �̅� ↔ [ DM = ANTIDOTES ] = The DEMODULATION ] i.e. The Communications–Frequencies In Planck`s Cave , are the Wave Resonances which consist the Master-Keys of these wave - caves . The Conservation of Energy in Caves and in Atoms Compounds . [106] IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 88 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . THE ATOMS – ACTIONS : [[A]] → {[B]±𝐄𝒏} ≡ C ≡ 𝐄𝒏+ {[A] ± [B]},→ 𝐄𝒏= 𝐄𝐂 + 𝐄𝐀 ← and where [[A]] = [Atom OR Compound] ≡ The Signalling.(Wc) → The Carrier - Wave [[B]] = Any[A-OR-C] ≡ Any Signalling.(w) Wave = The Modulating - Wave [[C]] = The New Modulated Wave [A] Signal ≡ f w+wc 2 + f w−wc 2 ≡ [MAS] ANTIDOTE = The Demodulated [MAS] ≡ f w+2wc 2 + f w−2wc 2 ≡ [DMAS] , �̅� , as The Atoms Action (1) ... [A] → {[B]±𝐄𝐍} ≡ [C] ≡ ± 𝐄𝐍 + {[B] + [A]} , , [A] ≡ [C] +{ 𝐗 𝐍} [A] = The Initial , [C] = The Final , [ X N ] = {[B] ± [EN]} ≡ The Antidote . The Atoms Action (2) …. [A] → [ ± E N ] ≡ [C] ≡≡≡ ± 𝐄𝒏 + [A] , , , , [A] ≡ [C ] + [ E N ] [A] = The Initial , [C} = The Final , [ E N ] ≡ The Antidote , The Circular-Signal-Action is as , I → F = [ I + F ] and [ I + F ] ↔ [ DM ] ≡ Antidote , or Still-Actions [ I = CARRIER ] →[ F = MODULATOR ]→[ I+F = MODULATED ] → → [ I+F ] + M ↔ [ DM = ANTIDOTES] = THE DEMODULATION ] The Communications –Frequencies In Atoms - Caves , are their Wave Resonances which consist the Master-Keys of all the wave - caves . 6c5.. The Energy - Superflow in MFMF-PNS Space-Levels for Black-Holes Origination. 1)..It was Prior Proved that Angular-Momentum of the Primary-motion in PNS Space is a Quaternion of dimension Power ( s + �̅� i) ¹/ d = √( s + �̅� i) d , which follows that of Spaces Origination . 2)- It was Prior Proved [70] that in PNS - Space the inner Forces of a System , are the two equilibrium Centripetal and Centrifugal Forces due to the Eternal , ± σ , Stresses of Opposites [⊕↔⊝] Originate motion u , as 𝐉𝟏. 𝐮² 𝐜𝐨𝐬𝛝 - 𝐉𝟑 𝐰𝟑 .u + s . Q = 0 . This Quadratic equation has 4 - Solutions where the Angular velocity Complex number u = |�̅�| = s Q B = s Q J3w = |�̅�| = dφ dt = [ J3. w3 2 J1.cos ϑ ] ± [ k PNS 2 J1.cos ϑ ].i = s ± v i , or → Angular-velocity |�̅�| = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ± √− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ) ² 𝟐 = s ± {√�̅� }.i ← where s = [ J3. w3 2 J1.cos ϑ ] , �̅� = [ k PNS 2 J1.cosϑ ] , 𝐤 𝐏𝐍𝐒 = √ [ J3. w3]² − 4. sQ. J1 cos ϑ 2 The constant Work exists on the common Central Axes of Rotation which is Fixed to the Tangential-Planes of �̅� & �̅� nib , being alternately Perpendicular to , B̅ , and , w̅ , and found to be as �̅� . �̅� = 2L = J.w² . The frequency monad of rotation in the PNS Space is as → 𝐟 𝐏𝐍𝐒 = 𝐬 𝐐 𝟐𝛑.𝐉𝟐.𝐰 ← with the following 4 - Solutions for |�̅�| , Solution-1  | 𝑤 1 | = [ s ] + [ √�̅� ] → Real Root ≡ Huge Minerals at Edge of Universe. Solution-2  | 𝑤 2 | = [ s ] - [ √�̅� ] → Real Root ≡ Atoms & Compounds into Structures. Solution-3  | 𝑤 3 | = [ s ] + [ √�̅� ].i → Imaginary Root ≡ Huge BH at centres of Galaxies. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 89 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Solution-4  | 𝑤 4 | = [ s ] - [ √�̅� ].i → Imaginary Root ≡ Micro BH at centre of Molecules. where BH = Black Hole . From 4-above issues , A-Velocity |𝑤1 ̅̅ ̅̅ | = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ±√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 ≡ 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 +√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 A-Velocity |𝑤2 ̅̅ ̅̅ | = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ±√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 ≡ 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 -√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 A-Velocity |𝑤3 ̅̅ ̅̅ | = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ±√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 ≡ 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 +√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬 𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 . i A-Velocity |𝑤4 ̅̅ ̅̅ | = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ±√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 ≡ 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 -√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 . i Angular-velocity �̅� is a vector , and Rotates in one axis k with respect to variable �̅� Momentum and , the Ellipsoid of |�̅�| ≡ Quaternion of dimension Power (s ± �̅� i)¹/d The Inertia J-Ellipsoid of |�̅�| gives two Anti-Symmetrical Inertial Ellipsoids , a Real and an Imaginary in their Polhode and Herpolhode Cones as below, The-Real �̅� ≡ { Polhode ≡ Cone OPS , Herpolhode ≡ Cone OPT , [ ↻ Rolling ] on the the Ellipsoid axis , ↓ Normal ON the Unmovable & Attitude-Plane of the [ PSPT ] Truncated-Polar-Cone }. The-Imaginary �̅� ≡ {Polhode = Cone OPS , Herpolhode = Cone OPT [↺ Rolling ] on the Anti-Symmetrically-Ellipsoid-axis, ↑ Normal ON the Unmovable & Attitude-Plane of the [ PSPT ] Truncated-Polar-Cone } . It has been Proved that �̅� , �̅� Vectors exist as , E = Energy =motion , S = Space =Length ---------------------------------------------------------------------------------------------------------------- Both magnitudes define the Unit-Energy Vectors → { Unit-E Vector R = σ J.w² = 𝛔 �̅� } ← When the Unit-E.Vector R is Placed ON [AB≡2R S.Diameter of Spaces-Neutral-circle] Then Energy-Monad |�̅�| , exists on , Space-Monad |AB| ≡ |�̅�| and is {E+S -Monad |�̅�|} Or is The { Energy – Space - Monad |�̅�| ≡ |AB| } , where the magnitudes, Spaces Sides 𝛌𝐧 = 2,√R2 − a²n , & the Quantum Critical-Side 𝐚𝐧 = 1 2 √4R2 − λ²n , Every=Monad occupies AB = 2|R| Diameter = Space & Unit-E-Vector R= σ J.w² = σ B̅ = motion The Quantization of Energy in Spaces happens by following the Archimedes formula for Duplication of Sides as 𝛌𝟐𝐧=√2R2 − R√4R2 − λ²n & Critical-Sides 𝐚𝟐𝐧= 1 2 √4R2 − λ²2n The Energy-Monad |�̅�| = σ J.w² = 𝛔 �̅� remains the same in PNS - Space while The Spaces Sides change as 𝛌𝟐𝐧 , 𝐚𝟐𝐧 , when the Monad |�̅�| enters the 2n . -------------------------------------------------------------------------------------------------------------------------- The 4 - Solution for Real and Imaginary Root is as below , The |w1 ̅̅ ̅̅ | Solution is the Real -Root giving the Strongest in Amplitude Energy-Structions. The |w2 ̅̅ ̅̅ | Solution is the Real -Root giving the Weakest in Amplitude Energy-Structions. The |𝑤3 ̅̅ ̅̅ | Solution is the Imaginary-Root giving the Mega Tapered-Amplitude E-Structions. The |𝑤4 ̅̅ ̅̅ | Solution is the Imaginary-Root giving the Micro Black-Holes Structions . From Gravity relation G = g 𝐤𝐄 =9,8076941.6,8116.10−12= 6,68056.𝟏𝟎−𝟏𝟏 m3/ N.s² from IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 90 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Energy-loop relation 1 = c. r³. 𝐟𝐏² → fP=√1/cr³ , and from f = E / h then f = 𝐄 𝐡 = √1/cr³ or → E = h .√1/c. r³ , and E² h² = 1 c r³ or , E ² = h² c r³ , an Energy relation between the , c , r . This relation is used for Black- Holes Energy where r , result to zero , The Permissible-Permeable-Resonance-Paths for their means are for , 7) Particles → The Resultance of the One-Dimensional-Collision v̅ij= v̅j - v̅i = w̅ij. r̅ij 8) Particles → The Cauchy Ellipsoid-Stress-tensor where { w̅ ⊥ B̅ ⊥ r̅ ≡ σ1⊥ σ2⊥σ3 } 9) M-Points → The Resonance-frequencies 𝐟𝐑 [S ≡ f1= n ,f2, f3,fR=w²]= 𝐟𝐧 = [⊕↔⊝] = n (1+√5)σ 4πr = n.σ.B̅ 8 r² → i.e. The Golden-ratio-Pattern , which is related to Spin. -------------------------------------------------------------------------------------------------------------- From Solution-3 the magnitude D becomes maximum when 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 ≡ maximum or ,when |θr̅| axis is ⊥ B̅ and Cone-angle 𝛉  90 . From Energy relation E = W = 2L = B̅ .w̅ = J.w² Angular-Momentum �̅� = [3 ,3770539.10−96J].[5,613308.1045Hz] = 1,895644.𝟏𝟎−𝟓𝟎 J/s, i.e. The Planck`s cave ,10−35 m , is The Safety-Valve ≡ The Critical State for the Stress Ellipsoids ,B̅ w̅, Vectors in order that this Total-Energy to Superflow and be Still into Vacuum ≡ [10−62-10−35m]. This Vacuum between the |Gravity cave - Planck`s cave| CONSISTS The BLACK - HOLES - CHAOS , as the Recycling–Energy into which → The { Energy – Space - Monad |�̅�| ≡ |AB| } , Vanishes ≡ Decomposes ← -------------------------------------------------------------------------------------------------------------- In Physics exist ,Two different Potential-Energies between Planck Spaces and Gravity Spaces as UPLANC and UGRAVITY .The difference ∆U = UGRA- UPLA is the Flow-Energy between the Two Energy Levels L PLA = 6,6262.10−34 m , and L GRA= 1,777.10−62 m , From cave L PLA = 6,6262.10−34m and fP = √ g a³ 2 = √ 9,8078 (6,6262.10−34)³ 2 = 1,83585.10 50 H then → E PLA = h.fP = 6.62607.10−34 . 1,83585.10 50 = 1,216447. 1017 J The minimum Charge in Planck`s cave → q PLA = 1Cb = 1,6 .10−19 eV , From Voltage relation ∆U = Energy Charge issues , 1Volt = 1Joule 1Culomb or 1V = 𝟏𝐉 𝟏𝐂 ….(v) In order that 1Cb may overflow and Flow from Planck`s cave To the Gravity cave , then must exist a Voltage ∆U > = 7,593.10 35 eV , and be as ....(v) 𝐕 𝐆𝐑𝐀 = 𝐡.𝐟𝐏 . 𝟏𝐂𝐛 = 1 ,216447.1017 J 1,6022.10−19eV = 7,593.𝟏𝟎 𝟑𝟓 V , i.e. → When the Energy in Planck`s cave LPLA , Increases to an Voltage >= 7,593.10 35 V then the motion≡ Energy between the Two Opposite elements of MP-Dipole [⊕↔⊝] Becomes a Stationary Flow [⊕,⊝] ≡ Charges [⊕--⊝ ] in Gravity Cave L GRA. So Energy ≡ motion , exists in L GRA as an → Pointy Rotational Centripetal Force ← IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 91 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . → When the Energy in Planck`s cave LPLA , Decreases to an Voltage =< 7,593.10 35 V then motion ≡ Energy between the Two Opposite elements of M-P-Dipole [⊕↻↺⊝] Creates the A-velocity-vector �̅� which Rotates in the axis k with respect to variable �̅�. So when Unit-Energy Vectors �̅� = σ / �̅� exists on the Unit-Space-Monad |AB| ≡ |�̅�| , Then is → {The Energy –Space - Monad |�̅�| ≡ |AB| ≡ The Existing Universe} ← For The Chaos - Motion → Angular – Momentum ≡ The Rotational Energy �̅� ← For The PNS – Spaces → Angular – Momentum ≡ The Rotational Energy �̅� ← From Fig-13 and Solution-3 , magnitude D becomes maximum when 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 ≡ maximum or , when |θr̅| axis is ⊥ B̅ and Cone-angle θ  90 , where Energy E =W= 2L= B̅ .w̅ = J.w² Angular-Momentum �̅� = [3 ,3770539.10−96J].[1,15346.1051Hz] = 3,895312.𝟏𝟎−𝟒𝟓 J/s, i.e. The Planck`s cave , 10−35 m , is The Safety-Valve ≡ The Critical State for the Stress Ellipsoids ,B̅ w̅, Vectors in order that this Total-Energy to Superflow and be Still into Vacuum ≡ [10−62-10−35m] . This Vacuum between the |Gravity cave - Planck`s cave| CONSISTS The BLACK - HOLES - CHAOS , as the Recycling–Energy into which → The { Energy – Space - Monad |�̅�| ≡ |AB| } , Decomposes ← {Fig-13} IN FIGURE , THE HERPOLHODE- CONE PLANE IS A CIRCLE WHERE VECTOR B̅ ROTATES , AND WHEN 𝛉  90 . THEN CONE BECOMES A CIRCLE , where Energy follows [ ⊕  ⊝ ] mode as → + when 𝛉 = 0 = < 90 , and - when 𝛉 = 90 – 180 ← It was shown that , the Energy from Chaos [⊕ -  - ⊝ becomes Momentum �̅� , �̅� , and is Composed ≡ Quantized through the Physical-[STPL]-line-Mechanism , In-circle  Triangles  Ex-circle  Ex-Triangles consisted with the three Spaces . On this PM exist the Three Breakages [s² = ⊕ , 2s ² =  , - s ² = ⊝] which Circularly-Charge ON the Three – Extreme - Triangles {A B C} , {KA KB KC}, {AE BE CE} and thus launch The Energy-Quantity ± 𝐐𝐩−𝐩= [ 𝒄.𝑺 𝒓 ] ≡ Centripetal-Force from Chaos to PNS Space , On the Pascal`s -Desargues Points-Line through the STPL Circuit caves r , and from the Ptolemy`s & Ceba`s Still-mechanism . {Fig-24} Remark . 1.. Electron , e , me = 0 , 511MeV = 0, 511.10−6 eV . [1,80. 10−27] = 9,198.10−34 Kg 𝐚𝐞 = 5,0.10−18 m ,Charge 𝐂 𝐞 =1,602.10−19 C , Spin 𝐒𝒆 = 𝐒 2 =2,845976.10−34{Kg/m/s}, Using 𝐄 𝐞𝐊 = 𝐤 𝐫 + 𝒄𝐒 𝟐 𝟐𝐦.𝐫𝟐 = 36.10−20 7,10−18 + 3.108(5,691952.10−34)² 2.9,198.10−34[5.10−18]² = 51,428 eV + 2,111843.1010J = 51,43 eV+ 1,32 .1029 eV , i.e. The Energy Increases inverse analogous to the cave . 2…Since Primary motion ≡ [PM] exist from the Three Breakages only STPL is the n =3 Mechanism creating Elementary Particles as [n-Knots=3] in 𝐕 GRA = 7,593.1035 V . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 92 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Unit-Space -Vectors , �̅�.�̅� , & The Unit-Energy-Vectors , �̅�.�̅� , Ellipsoids . Figure.17c .. Shown the Geometrical-meaning of ao̅̅ ̅ ri̅ , √ri² − (ao̅̅ ̅ ri̅)² terms of radius OA i In (1) is shown the Inertial - Ellipsoid ( O, a̅) of Radii , a̅, and the Interchangable Momentum–Ellipsoid (O , ρ̅) of Radii , ρ̅. (O , ρ̅) . The same for {B̅ , (O, B̅)} In (2) is shown the Momentum - Energy - Ellipsoid ( O, B̅) of Radii , B̅, and the Interchangable Angular Velocity – Ellipsoid (O , w̅) of Radii , w̅ . S = Space In (3) is shown in Mohr-method , the Geometrical construction from the two Interchangable Ellipsoids→The Energy-Rotational Momentum–Ellipsoid ( O , ρ̅) , where { Work } = The Angular-Velocity-vector -Unit-sphere , S =The Inertial- Ellipsoid (O , a̅) , { Force }= The reaction to the velocity-change-motion , S = The Mass - Ellipsoid { Mass } = ( GD = Ja ≡ M̅ ) , and Between the Types of Ellipsoids issues ρ̅ . a̅ = J a ..(12) The above Property of the two Interchangable-Ellipsoids defines the deep relation between , The Angular-velocity-Ellipsoid → J1w1² + J2w2² + J3w3² = 2L = C = Ja w̅ ² ….(13) and The Momentum-Energy-Ellipsoid → 1 J1 B1² + 1 J2 B2² + 1 J3 B3² = 2L = C = Ja w̅ ² .....(13a) Conclusions : 1..From the generated Radius �̅� =The INERTIAL-Ellipsoid of a Solid Fig-17-1 corresponds another Radius B̅ , Fig-17-2 Momentum –ENERGY– Ellipsoid , to the Common-Point O of Rotation .The Radius �̅� is Perpendicular to the Tangential-Plane at the Nib of the �̅� Angular Velocity-Ellipsoid . Similarly the Radius �̅� is Perpendicular to the Tangential-Plane at the Nib of the �̅� of the Angular-Momentum-Ellipsoid . Relation ρ̅ . a̅ = J a= a constant , means ρ̅ ⊥ a̅ ≡ Space + Anti-Space ≡ motion  �̅� ≡ [ i ≡ √−1 2 ] * ρ̅ ≡ The Argand-Diagram. From relation (12) , the vertical-Projection of the radius , ρ̅ , at the corresponding conjugate radius a̅ , of the Unit Sphere , denotes the meter to the Solid last Angular momentum . Because the Unit-Energy-Vectors , �̅�.�̅� , are related as �̅� = AB = σ / B̅ = σ / J.w² , i.e. the common velocity vector c = w r = 2.B / π r ³ , therefore issues and for all the 5-Spaces . 2..Inversely The Unit-Energy-Vectors , �̅�.�̅� , Decompose when Bcrit=J w = 𝜋r3 2 [c/r] = [ 𝜋r2 2 ]c Or when �̅� becomes maximum , i.e. for Gravity and Antigravity happens when |θr̅| axis is Perpendicular (⊥) to B̅ and the Cone-angle θ  90 , and Energy E =W= 2L= B̅ .w̅ = J.w². From the �̅� Physical -Magnet , �̅� is Quantized to the 2-Constitutes ⊕ ⇉ ⊝ as the Prior . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 93 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 6c6.. The De-Quantization of Energy from , PNS  MFMF , ≡ Empty Space . Figure.17d.. The Angular-velocity |�̅�| = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ±√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ) ² 𝟐 = s ± {√�̅� }.i  |𝑤3 |= [s]+[√�̅� ].i , s = [ J3. w3 2 J1.cos ϑ ] , �̅� = [ k PNS 2 J1.cosϑ ] , 𝐤 𝐏𝐍𝐒 = √ [ J3. w3]² − 4. sQ. J1 cos ϑ 2 From 10.c, the Unit-Energy-Vectors , �̅�.�̅� , fP = 1,836.1045Hz in PNS–Space become 1).. From relation w̅ =2π.f PNS , then 𝐰 PNS = 56,13308 .1044Hz = 5,613308 .1045Hz , 2).. From Centripetal Force in MFMF Space FC = σ = m.v³ r =J w² r =B̅ w r , then B̅ =J w ,or B̅ = J w = [ π.r4 2 ].w = π[1,6162.10−35]4 (5,613308 .1045) 2 = 3 ,3770539.10−96 J and for eV , B̅ = 3 ,3770539.10−96 J / 1,6022 .10−19 C = 3,754935.10−76 eV. 3).. From Sol -3 the magnitude D becomes maximum when 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 ≡ maximum or , when |θr̅| axis ⊥ B̅ and Cone-angle 𝛉  90 . From Energy relation E = W = 2L = B̅ .w̅ = J.w² Angular-Momentum �̅� = [3 ,3770539.10−96J].[5,613308.1045Hz] =1,895644.𝟏𝟎−𝟓𝟎 J/s, i.e. The Planck`s cave ,10−35 m , is The Safety-Valve ≡ The Critical State for the Stress Ellipsoids ,B̅ w̅, Vectors in order that this Total-Energy to Superflow and be Still into Vacuum ≡ [10−62-10−35m]. This Vacuum between the |Gravity cave - Planck`s cave| IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 94 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . CONSISTS the BLACK - HOLES - CHAOS , as the Recycling–Energy , into which → The { Energy – Space - Monad |�̅�| ≡ |AB| } , Vanishes ≡ Decomposes ← → In Physics exist , Two different Potential-Energies between Planck Spaces and Gravity Spaces as UPLANC and UGRAVITY .The difference ∆U = UGRA- UPLA is the Flow-Energy between the Two Energy Levels L PLA = 6,6262.10−34 m , and L GRA= 1,777.10−62 m , The Total Energy in Planck`s cave is → E PLA = 1,216447. 1017 J , and The minimum Charge in Planck`s cave → q PLA = 1Cb = 1,6 .10−19 eV , From Voltage relation ∆U = Energy Charge issues , 1Volt = 1−Joule 1−Culomb or 1V = 𝟏𝐉 𝟏𝐂 ….(v) In order that 1Cb may Overflow and Flow from Planck`s cave To the Gravity cave , then must exist a Voltage ∆U > = 7,592354 .10 35 eV , and be as ..….(v) 𝐕 𝐆𝐑𝐀 = 𝐡.𝐟𝐏 . 𝟏𝐂𝐛 = 1 ,216447.1017 J 1,6022.10−19eV = 7,593.𝟏𝟎 𝟑𝟓 V , i.e.  When the Energy in Planck`s cave = LPLA , Increases to an Voltage = 7,593.10 35 V then the motion≡ Energy between the Two Opposite elements of MP-Dipole [⊕↔⊝] Becomes a Stationary Flow [⊕,⊝] ≡ Charges [⊕ -  - ⊝] in Gravity Cave L GRA . So Energy ≡ motion , exists in L GRA as an → Pointy Rotational Centripetal Force ←  When the Energy in Planck`s cave LPLA , Increases to an Voltage 𝐕𝐂𝐑 > 7,593.10 35 V then motion ≡ ⇄ ≡ Energy between the Two Opposite elements of M-P-Dipole [⊕↻↺⊝] Creates the Negative-Angular -velocity-vector �̅� ↓ , which Rotates in the Symmetrical to Neutral circle axis k = ↓ with respect to the Interchangable variable �̅� ↓ . This is the Case of The Black-Holes - Origination .  When the Energy in Planck`s cave LPLA , Decreases to an Voltage = < 3,695.10 30 V then motion ≡ ⇄ ≡ Energy between the Two Opposite elements of M-P-Dipole [⊕↻↺⊝] Creates the Positive-Angular -velocity-vector �̅� ↑ , which Rotates in the Symmetrical to Neutral circle axis k = ↑ with respect to the Interchangable variable �̅� ↑ . This is the Case of The Universe - Origination . So thus , are defined the Unit-Energy - Vectors , �̅� = σ / �̅� = σ / J.w² , which Exist ON the Unit – Space – Monad |AB| ≡ |�̅�| , and which is { The Energy – Space – Monad |�̅�| ≡ |AB| ≡ The Existing Universe }. In Fig-14 are shown the 5-States for angle → θ  0ᶱ , 45ᶱ , 90ᶱ , 135ᶱ , 180ᶱ . 1.. For angle → θ  00ᶱ , �̅� ≡ �̅� , Voltage = 7,593.10 35 V , ↑ On Neutral – Circle 2.. For angle → θ  45ᶱ , �̅� / �̅� , Voltage < 7,593.10 35 V , < ≠ Neutral – Circle 3.. For angle → θ  90ᶱ , �̅� ⏊ �̅� , Voltage ≡ 7,593.10 35 V , ⇉ In Neutral - Circle 4.. For angle → θ  135ᶱ , �̅� \ �̅� , Voltage > 7,593.10 35 V , ≠ > Neutral – Circle 5.. For angle → θ  180ᶱ , �̅� ⏊ �̅� , Voltage >=7,593.10 35 V , ⇅ Off Neutral-Circle 4).. From Solution-4 , The {Energy-Space-Monad |R̅| ≡ |AB|} Vanishes in BH-Chaos , where Energy Increases to an Voltage >> 7,593.10 35 V and then motion ≡ Energy between the Two Opposite elements collides them to the MP-Dipole [⊕↔⊝]. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 95 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 5).. From Solution-5 , The {Energy-Space-Monad |R̅| ≡ |AB|} Vanishes in BH-Chaos , & Energy is Quantized to the 2-Constitutes from the �̅� , Physical-Magnet ⊕ ⇉ ⊝. Figure.17e..The Stability of Photon as Electromagnetic Wave with the Two frequencies . The Dual Property of The Simultaneous existence as Electric and as Magnetic Fields in wavelengths , λ x , λ y , as Effective Range .which is thrusted through the Newton’s Gravitational Force G only , becoming from the Trigonometry relations [The Space ] , sin(2φ) = sin(𝜋 − 2φ) = sin( 𝜋 2 − φ ) , and from Physics [Energy] E x = g , Maxwel Electric Displacement current 𝝑D / 𝝑t is created from the ( 𝐜𝐱 , 𝐜𝐲 ) Vectors motion happening from gravity continuous action , and Producing the Work W = 𝐜 𝐲 g For Photon { Markos Rotating Method of{ 2-Vectors 3-Poles Mechanism }which Circles and Squares as Areas Pass from the Point M determined by the Position 𝐀 𝐞 , and where is such that issues π r ² = CMNH Square then .The Squaring of the circle becomes at the equilibrium - Positions which are from relation → π [ 𝐂𝐀 √𝟐 ]² = CM² = |CM`|² ← namely The square | CMNH | = square | CM`N`H`| . Because of the Bellow motion then , The Total motion in the Wavelength , λ , corresponds to the angle a = 45 °, meaning that the motion of Space ≡ The Electric-field , and Anti-Space ≡ The Magnetic-field equilibrium as an Bellow . The two Wings in 90° Type simultaneously oscillate to its 45 ° Bisector -axis , and the Produced Work from The Electric-field-area is Stored into the Conjugate Magnetic-field-area . The common axis of the two Perpendicular fields , carry the Golden-Ratio Energy-Surfaces , and Runs with the same velocity c ̅ to the direction of the inner motion . The EPILOGUS in [89 - 96] IT WAS PROVED and ELUCIDATED THAT , a.. Monad is Quaternion [x + i y] = λ̅ type , and its Energy is Vector as Q⃗⃗ = {λ}.X , where X = The Displacement . Since 1 = c. r³. fP² = constant , so → r³. fP² = constant . Since also IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 96 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . λ = 2r then r³. fP² = λ ³. fP² = c , and Energy becomes →Vector Q⃗⃗ = {λ}.√c/ fP² 3 ← [𝐄 𝟏] b.. In Mechanics an Energy-Vector is an Conductor AKA ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ = |Q⃗⃗ |  , and is The Quaternion in Material-Geometry where Zero-Point 0 ≡  ≡ {⊕+⊝}on where motion |Q⃗⃗ | ≡ �̅� 𝐱 as an Electric field on AKA axis and as Q⃗⃗ |  ≡ �̅� 𝐲 ≡ Magnetic field ⊥ AKA , is Transferred from One Edge Point A , to the Other Edge Point K A as → �̅� , �̅� 𝐱 , �̅� 𝐲 ← [𝐄 𝟐] c.. Energy is the Work , The motion , Produced in Energy-monads and for Photon is Equal to W = 2L = B̅ . w̅ = r m v = [r. p] = J w² ≡ [ �̅� 𝐱 ⊥ �̅� 𝐲 ] where �̅� 𝐱 = �̅� 𝐲 , in the Conductor AKA ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ = 2.r ≡ λ where , λ Is -The -Wavelength and p = m c = The Momentum . [𝐄 𝟑] d.. It was Prooved That → Any Diameter D of a circle in the STPL – Circuit , the Three – Poles –Squares-Rotational - System is an Conductor where D = AKA ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ = |Q⃗⃗ |  = | 𝐜.�̅� 𝐫 ² | = 𝐈𝐄 The - Electric-Wave-Intensity , And A ↑ KA ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ ≡ 𝐈𝐌 is The - Magnetic–Wave Intensity. Their Square -Vectors – Area = [ AKA ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ]x[A ↑ KA ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ ] = [�̅� 𝐱�̅� 𝐲] = The Energy - Square ≡ CM x CM = CMNH square = on circle ( E , EA ) , and from relation �̅� = σ . Φ = F A Φ area A = F / σ = |CM|² / σ , i.e. Plane-force F = σ . |CM|² , and in Space Volume-Force F = σ . |CM| ³ ≡ [𝐄 𝟒] or [ AKA ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ]x[A ↑ KA ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ ]x[D] = CMNH x 𝐐 ⤱ ⇉⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = CMNH x �̅� ⤱ ⇉ , where |𝐐 ⤱⇉⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ | ≡ �⃗⃗� , which is the Energy-Quantum In The Propagating Electromagnetic Wave where | 𝐈𝐄 ⊥ 𝐈𝐌 | . Since motions B̅xw̅ , are Constants from (b) and from Poinsot`s solution w1² + J2 w2² = a² , then Angular velocity-Ellipsoid Polar-Paths are the Parallel circles and Perpendicular to the Angular-Momentum Ellipsoid Nibs.. [𝐄 𝟓] , [70] e.. When is found an Obstacle at Point 𝐊𝐀 , Then The Particles of Conductor medium Oscillate Perpendicular . This Obstacle at KA Point , is The Presentation of the , ⊕ constituent and the Absent of ⊝ constituent and so  , Transverse- Propagating-Wave , Or moving Energy-Vector |⊕ 𝐐 ⇈ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ |=|↕| at Node KA Is Equal to E-Vector| ⊕ 𝐐 ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ |≡ |↔| at Node A , and Energy Volume is Doubled as , 2L = B̅ .w̅ , in KAA⃐⃗⃗⃗ ⃗⃗ ⃗⃗⃗ Conductor meaning that the meter of Volume is the Duplication of Total Energy L .The Reaction to the motion of Squares {|QX =CO ⇉| x|Qy= CB ⇈|} ≡ |𝐐 ⤱⇉⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ | ≡ [ 𝐜𝐒 𝐫² ].|𝐐 ⇉⃗⃗⃗⃗⃗⃗ ⃗⃗ | is equal to Quantum �⃗⃗� . [𝐄 𝟔] f.. Any Two Perpendicular Energy-Vectors , |𝐐 ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↔ | , |𝐐 ⇈ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↕ | of this Propagating Energy , FORM The Energy-Square , |↔|.|↕|≡|𝐐 ⇉ |.| 𝐐 ⇈| ≡ q0 , on Conductor AKA and Then Energy-Circular-Prism = |AKA| .{𝐐 ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ }.{ 𝐐 ⇈ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } ≡ ≡ [�̅� 𝐱�̅� 𝐲].|AKA| = | ↔ |.|↕|.|AKA| , i.e. An Energy-Circular-Cone which is a Moving Energy -Volume and Because from Cauchy equations of Stresses in three Dimensions , The Energy Stress remains Flat ONLY- WHEN the Plane-Section is a circle [𝐄𝟐 ] and The Square becomes circle ,Anti-Quadrature. Because from Mechanics , The Sphere occupies The Least-Resistance to motion where the Geometrical Square–Cone [CM x CH x[CO⃗⃗⃗⃗ ⃗ ↑] IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 97 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . is altered to Equivalent Energy- Circular - Cone [π.[ 𝐂𝐌 𝟐 ]² (x) [CO⃗⃗⃗⃗ ⃗ ↑] and To the Energy Unit -Volume CO³.[𝐐 ⇉⃗⃗⃗⃗⃗⃗ ⃗⃗ ] ≡ 2 .Sphere 𝟒𝛑 𝟑 r ³ And To Energy-Sphere Cone ⸿ = k π r³ , where The Geometry- Cube [|↔|.|↕|.|𝐀𝐊𝐀|] = x ³ so becomes The Energy -Sphere – Cone as → k π r³ = x ³ ← [𝐄 𝟕] g.. Since Frequencies fn are The Quantum of Energy , and Electromagnetic Vectors are always Perpendicular between them as | 𝐅 𝐄 ⊥ 𝐅 𝐌 | , so are everywhere in nature from microcosm to macrocosm . Because Gravitational Force exists Beyond Planck`s length therefore the Propagating-Plane-force 𝐅𝐏 = σ . |CM|² , is continually Perpendicular to the two Perpendicular constituents of Electromagnetic Vectors as |𝐅𝐏 ⊥ 𝐅 𝐄 ⊥ 𝐅 𝐌 |. Distance is the Quantum of E-Geometry , while Material-Point is the Quantum in Physics and in Material – Geometry which is the composition of Opposites and are The Elements in Chemistry and Physics . As in Algebra Zero ,0, is the Master-key number for all Positive and Negative numbers because their Sum and multiplication is Zero , and the same on coordinate - System , ± , axes pass from zero . [𝐄 𝟖] The Rolling of Positive ⊕ , constituent on the Negative ⊝ , constituent , creates the Neutral Material Point which Equilibrium . Spin �̅� , is Its Angular momentum and also the First -Discrete-Energy-monad which occupies Discrete-Value and Direction , in contradiction to the Point which is nothing , Dimensionless and without any Direction . Point-caves Are the Energy-Magnets from which the Quantum of Energy are Generated . h.. The Space is Quantized as Energy-Caves under the effect of Gravitational-force G , as well in Conductors and this Energy or motion is Quantized as Frequency in Energy- Caves , Or in Conductors , following the Kepler`s First-law of Equal Areas where in equal intervals and in Conductors , is the Electricity or Electromagnetic - Waves . [𝐄 𝟗] Quaternion [(+)↻↺(-)] is a Quantum-Mould for Space [(+)(-)] and Energy ≡ motion ≡ Force x Displacement as [↻↺ or ↑↔↓] ≡ Standing Box 𝐁𝐐 ≡ An Material Point which carries the Principal Stress σ between Positions A (+) , B (-) , and σ , is the Centripetal - acceleration of minimum Energy becoming from the in-Storage AB acceleration and which is equal to the Gravity g . From Quaternion Quantum Mould [r + v̅ i ]𝟏/𝐰 = 𝐞− 𝐢.(𝛑/𝟐+𝟐𝐤𝛑).𝐰 is created the min-Space ≡ cave r = 1, 07.𝟏𝟎−𝟕 m and the min-Energy ≡ v̅ = w r = 2πr f , as frequency 𝐟𝐦= 2,839844 . 𝟏𝟎𝟏𝟎 H . G Pushes the → min-Quantum-Energy ≡ motion 𝛔𝐄 , into the Inscribed-Cave π.[ 𝐂𝐌 𝟐 ] ² ≡ π.EA² . Gravitational-force G effecting on light velocity �̅� creates the Electron-charge �̅� and Electron 𝐞 ̅, while acting on Planck`s-cave on the Gravity g ≡  σ , as G = g k = 𝐤𝐄 g = 𝐤𝐋 σ = g.gL kL , and �̅� effecting on the min-Planck-cave in LP formulates Hydrogen-cave H with its electron as He . [𝐄 𝟏𝟎] , [109] i.. The Physical-Notion of The Duplication of the Cube , in The Quadrature Physical Energy Mould Shows that Nature follows the Geometry-Rules in Order to Exist as Space and as Energy , which is motion of The ,+, Constituent to The , - , Constituent . This motion is conserved by Propagating in caves either for Standing ≡ | 𝐯 𝛑².𝐫⁴𝐧 |. B̅n or for moving caves as this is the Duality-Photon → v̅ . [ fn̅ + fn ] ≡ | 𝐯 𝛑².𝐫⁴𝐧 |. B̅n + | c̅ | fn .Material-Geometry meters for Quantum-length is the Quaternion ≡ Dipole ≡ The Two-Points joint with motion between them ,while for The Quantum-Surface is The Unit circle π.r ² = π , and is r =Unit 1. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 98 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . For the Quantum-Volumes , is the. Unit Volume Sphere 4 3 π.r ³ = [ 4.r 3 ] (π.r ²) = 4 π 3 , for r = Unit 1 . The Unit-Volume is formulated from The Inscribed-Square of Unit Circle-Side CO = CB = 1 /√2 , and when Placed The-Unit-Energy-magnitude OC on Square CO x CO then is Unit-Volume [CO x CO x CO⃗⃗⃗⃗ ⃗ ↑] . This Unit-Volume when Placed in , The Three -Poles - Squares-Rotating-System ,Then at Point KN , Unit Volume [CO x CO x [CO⃗⃗⃗⃗ ⃗ ↑] becomes [CM x CH ] x CH⃗⃗⃗⃗ ⃗ ↑] = Circular-Cone [π.( 𝐂𝐁 √𝟐 )²] . [CO⃗⃗⃗⃗ ⃗ ↑] = Unit - Sphere 4π 3 r ³ , Equal to Quantum of Energy . Because Physical-Magnet ⊕ ⇉ ⊝ compels Force �̅� ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ , Between A and KA Points and Then Unit-Sphere 4π 3 r ³ becomes The Double - Unit-Sphere 2.{ 4π 3 r³} and because the Energy Square [CAKAP] is Double that the Unit Square |CB x CO| , then Force �̅� ⇉ AKA⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ is The Pivot of Spinning indicating ± �̅� ≡ �̅� , in all Energy monads .[𝐄 𝟏𝟏] j.. Gravity and Antigravity are the two Halve - Spins from the Angular Momentum �̅� ≡ ± Work created from the velocity vector �̅� by Rotating ↻ on Planck`s Cave Diameter.[𝐄𝟏𝟐] This Property , ± �̅� ≡ �̅� , is in all Energy monads and when Applied on Initial , ⊕ ⇉ ⊝ , Answers to the 3D-Stability of All Elementary-Particles and Cosmology Equilibrium . Gravity + g ≡ The Vector of Angular-Momentum �̅� ≡ + Spin ≡ ↓↻ ≡ The Work stored in the [4-1-3] Semi circle of the , �̅� - �̅� Vectors circle-Surface ,Produced from the Gravitational-Force , G , as ACTING into the -Planck- Cave , AND on the light Velocity Vector of , + ↓ �̅� Direction . [107] Antigravity - g ≡ The Vector of Angular -Momentum �̅� ≡ - Spin ≡ ↑↺ ≡ The Work stored in the complementary [4-2-3] Semi circle of the , �̅� - �̅� Vectors circle Surface which is Produced from the Gravitational-Force G , as ACTING into the-Planck-Cave , AND on the light Velocity Vector of The , - ↑ �̅� Direction . k.. The Black-Holes exist as Parallel origination either as  Planar Black-Holes or  Atoms - Black – Holes , when G - Force is ACTING On the -Planck – Cave . [𝐄 𝟏𝟑] Momentum B̅ & w̅ , as Real ≡ The EXISTING - UNIVERSE as is the Real Objective Reality |�̅�| ≡ The Numeric Vector-value of The Material-Point [⊕, r ,⊝] in [PNS] ≡ The Work stored in POLHODE - CONE of the |𝐎𝐒̅̅ ̅̅ | Rolling Nib -Vector Produced from the Eternal CENTRIPETAL - FORCE , 𝐅 𝐂 AS THE Newton`s ACTION in the Planck`s - Cave, AND from the �̅� Angular Momentum in the , + ↓ B̅ , Direction . The Real Type is applied on all different Monads and their Compounds Monads . [109] Momentum B̅ & w̅ , as Imaginary ≡ The BLACK - HOLES as is their Real Objective Reality ± �̅�.i ≡ s  The Numeric Amplitude-Vector-value of Material-Point [⊕,r,⊝] in [PNS] and The Imaginary - �̅�.i ≡ CENTRIPETAL - FORCE 𝐅 𝐂 ≡ The Wave Amplitude -value ≡ The Work stored in the HERPOLHODE CONE of the |𝐎𝐁̅̅ ̅̅ | Rotating �̅� -Vector and also Produced from the Eternal CENTRIPETAL - FORCE 𝐅 𝐂 , AS THE Newton`s REACTION into the Planck`s - Cave AND from the �̅� Angular Momentum in , - ↑ �̅� Direction. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 99 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . l.. In case of Points A , B , C being on a Sphere then [STPL] becomes a Cylinder as this is the Angular-Momentum �̅� = 2L w̅ = 2L 2πf = [ 2L 2π ].[ 1 f ] = Constant 2π [ 1 fm ] = SPIN [89] D... THE ORIGINATION OF THE FUNDAMENTAL PARTICLES 1d… The Balancing of the Space and Anti-Space . [64] ↺ ↻ ↻↺ → ↻↺ + ↺ + ↻ Figure -18- : The work W, of Two-Quaternion-monads 𝐳 = (s + v̅. i) ≡ 𝐀𝐁̅̅ ̅̅ ≡ �⃗⃗� is the Action Of any two Tangential & Opposite Equilibrium-Dipole [⊕↔⊝] with v = w⃗⃗⃗ .r. For Quaternion Actions exists the square law for both Real and Imaginary Part , as 𝐳 * 𝐳 = (s + v̅. i )² = s² +[v̅. ]² +2.sv̅. i = s² – v̅ ² ± 2.s v̅ = |s|² -|s̅|² ± 2.|s|.|s̅| , for |v̅| = s Vorticities ± Λ (Rotating Energy) , Becomes from the Collision on Common-circle and from the Thrust on the Velocity-Breakages .The work is , W = [ n . P ] = [ λ.Λ] , where λ = Displacement of A to B and it is a scalar magnitude called wavelength of dipole AB. Λ = the amount of Rotation on Dipole AB ( this is Angular Momentum and it is a vector ) . Momentum ± Λ̅ = r.m.v = r m wr = mr².w̅ , where w̅ is the angular velocity (Spin) which maps velocity vector v̅ on the Perpendicular to ± Λ̅ Plane with two components v̅ E ⊥ v̅ B . The Tangential velocity v̅ E = wr , is the quaternion v̅ E = w r = z̅ = [ s + v̅. i ] where s = v̅ E = |r.w̅| and v̅. i = |w̅ x r̅ |. In a Spherical cave the Biaxial Ellipsoid (σx = σy) exists as momentum +Λ on caves of diameter 2r with parallel circles → 0 .The Biaxial Anti-Ellipsoid (- σx = - σy) exists as equal and Opposite momentum - Λ on the same diameter 2r with Anti Parallel circles → 0 . Equilibrium of the Two Ellipsoids ± Λ , Presupposes a Stabilizer- System attached to Ellipsoids such that Opposite Momentum is distributed to the Center of Mass of the Total - System , recover equilibrium , which is the center of the Spherical cave. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 100 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . The Biaxial - Ellipsoid and Anti-Ellipsoid are inversely directed and Rotated in the same circle , so the two velocity vectors , v̅ ,− v̅ , collide .This collision of the two Opposite Velocity-Vectors is the Physical-Action ( Thrust ) of the Two-Quaternion and their Action is , ( s + v̅. i )(©)( s + v̅. i ) = [s + v̅ . i] ² = s² + | v̅| ². i ² + 2|s|.|v̅ |. i = s² - |v̅| ² + 2|s|.| w̅ r|. i = s² - |v̅| ² + [2w̅].|s| |r|. i , where , s² = (r w) ² → Is the Real-Part of the New-Quaternion, - |v̅|² = |w̅ x r̅|² = - (w.r) ² → The always Negative-Anti-Space ( a vector ⊥ to w r Plane ) , [2 �̅�].|s|.|�̅�. i = 2w.(sr). i → The double Angular-velocity-Term , and for the PNS Space where |s| = |v̅|² then s² = (w.r) ² ≡ ⊕ ≡ s ² = + Charge - |v̅|² = |w̅xr̅|² = - (w.r) ² ≡ ⊝ ≡ - s ² = - Charge [2w̅].|v̅||.r̅. i = 2 |v̅|² = 2 (w̅.r) ≡ 2[⊕↔⊝] ≡ 2s ² = 0 Charge i.e. For the Equilibrium-Recovery (maybe a surface cylinder with 2r diameter) , Fig-16-17 2d.. The Vibration of Particles in all Levels [54] . Figure - 19- : The Glue-Bond Electrons of Nucleus Rotor , form Rhodonea-like - Curves Curves created by Rotor`s electron are such that they keep a continuous Pressure on Protons. 1.. The Continuous action of Negative ⊝ , on One Positive ⊕ , is equivalent to a Circular motion of the ⊝ , ⊕ , with velocity v = √F. r/m , and this because exists l = r = constant . 2.. The Continuous action of Negative ⊝ , on Two Positive ⊕⊕ of glued Bond , is equal to a Circular - and Other of one-step longer motion of ⊝ to ⊕⊕ , or Circular motion on IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 101 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . the Common–length , l , in Polar equations is , l = a . cos(c.θ) or l = a . sin(c.θ) where a = the length of the base =2r+2r = 4r , c = Rim`s number which Produce a c-Petaled rose , If c is odd or 2c-Petaled Rose and if c is even number. The Period of both cos(cθ) , sin(cθ) is 2π/c , and the number of petals for the period is [ 0 ,2π/c] will be c or 2c , according to as c is odd or c is even , so the degree of the Cartesian equation of the curve is , c +1 for odd-c , and 2(c+1) for even-c . 3.. The Continuous action of Negative ⊝ , on Three Positive ⊕⊕⊕ of glued Bond , is equivalent to a Circular-and Other of two-step longer motion of ⊝ to ⊕⊕⊕ , or Circular motion on the Common–length . Length , l , in Polar equations is , l = a . cos(c.θ) or l = a . sin(c.θ) , where a is the length of the base = 2r + 2r + 2r = 6r , c = Rim`s number which produce a c-petaled rose , where c is odd or 2c - petaled rose number . This is seen and from equations issuing in Figure.21-2 → r = rD.sin φ and R = RD.sinφ where a , are the equal radius of Point A of Polar path . Rhodonea curves follow the Odd or and Even number of Protons in Nucleus by changing their Rolling- length a = (2r) , and the Polar – length l = a . cos(c.θ) depending on ,θ, as above . The common bonding Point of the Negative circle and the Positive line of the Rolling circle obliges that Point to follow the Cycloid motion in maximum degree and thus executing Hypocycloid motion . Particles Vibration in all Levels of Material-Geometry . 4.. The n-Knots Figures exist in Figures where the Vector -Force - Polygons have the same number n , as the in Figures . For The Circularly- Charged Breakages the 3-elements [Σ = n . h]  [s² = ⊕ , 2s²= , - s² = ⊝] formulate the Primary-Particles 3-K,Figures. Since n =3 , the only Shape in Spaces is the Regular Triangle , and from this n < 4→ n there are vast regions of space that are very nearly empty of them , 5.. The case of {TCIC}≡ [ Σ = n . h ]  8 x ① which is the Physical Mechanism , gives the n-masses of Atom`s &Compounds determining the Atoms Complex-Vector-Response. These Particles exist in all of the universe in the sense that they are uniformly distributed throughout space and consist the Objective Reality which is the existing Universe IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 102 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Figure – 20- : The Carbon , Nitrogen , Oxygen , Sulfur , Atoms-Conductors Sustains Into the First 4-Dimentional Spinning , Tetrahedron - Cube – Sphere - Mould . Figure– 20a- :The Hydrogen , Carbon , Oxygen , Atoms-Conductors Into the Sphere-Cube- Mould . Deka hexahedron , Octahedron ≡ Cube , Tetrahedron ≡ 4-Points , Triangle ≡ 3-Points , Vector ≡ 2-Points ≡ The Closed-Conductor , 1-Point ≡ The Opened -Conductor ≡ The Bracket-Hook . Oxygen is the Element which fills 8 - 2 = 6 vertices of Cube as EL → { 1𝑒 0𝑒 1𝑒 1 𝑒 2⃡ 0 𝑒 1𝑒 1𝑒 1𝑒 } ≡ (± 2 2 ) , where ⓪ is the first , Zero-Acting Cube filled with 1e , therefore is the only Steady element which occupies the first Energy-Level ① with 2 Electron-Positions as ① = 2e = 2⃡ Positions at 1⃗ , 1⃗⃖ The Electron being in the Hydrogen-cave Precesses because of the different axis of Rotation And Nutation`s , from the continuous and immense-communication to the effect of gravity g . Since Electron is continually oscillating , with the Nutation-frequency 𝐟 𝐍 , so Produces a continuous and oscillating Magnetic-field –M- which in turn is the source of an oscillating Electric-field – E - which implies the Regeneration of each other , i.e. it is a Propagating Electromagnetic-Wave where E = B c , and with a quantum-Energy E = h fN or E = 2μ.B , This Energy E consists the Hydrogen Bracket Hook HBH , or [BH⃗⃗⃗⃗ ⃗] and it is the only free monad which can Bond to all Energy structures . 5d.. The Natural Electromagnetic Energy-Tetrahedron and , The 3- Conductors Into The Energy - Cube and Spheres : [102] IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 103 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Figure – 20.a- The Dynamic Loading of → Tetrahedron – Cube - Sphere ← Atom –Structure. Material Geometry uses the Only-Three Elements for the Origination of Primary - Particles , as Weak , W−Zo and Strong W+Z+0,− Forces , Spin B̅ = S̅ , Hydrogen cave , r H , and Electron e−. where [0] ≡  ≡ The Rest-Energy ≡ [⊕ ↔ ⊝] → [ ⊕ ∷ 2 +⊝ ∷ 2 ] , Split into Two . and , [0] ≡ ZD̅̅̅̅ ≡ The Rest Space ≡ Conductor − DM⃗⃗⃗⃖ ← + Z E⃗⃗ ⃗ → , from relation [⊕←ds→⊝] , where their Properties and Corelations are all motions into their Spaces , from Energy. In Fig-20b The Electric - Magnetic Field of 3-Conductors which lie on the DABC Tetrahedron .The Stability of the → { ⊕ 4-Space [D A B C]} Regular – Tetrahedron , occurs INTO the {⊝ 8-Anti-Space Cube , [DA1A D1 , B B1CC1]}, and The Positive → { ⊕ 8 - Space – Cube , [DA1A D1 , B B1CC1]} , INTO The {⊝ 16-Anti-Space [ 1 ,2 ,3 ,4 ,5 , 6, 7, 8 - 9 ,10 ,11, 12 , 13 ,14 , 15, 16] } Sphere Cube on Vertices.= Point [The Point 16 ≡ |⇉|Z|] consists the Navel-String Gate into the Triangle-Circle-System {Z-(∆D,ABC) , ∆[T1T2 T3]} of the Stationary Bases which Results on the 2 -Anti-Parallel-Strands . The Dual Photon v̅ [ σΦ 2πr + σ 2πr ] ≡ v̅.[ fn̅ + fn ] , occupies Stresses = σ and velocities v̅ , in The Tiny-caves r . The Colours in light , are the Still-Sub-Units in Storage →[v̅ . fn̅ ]← and exist as Frequencies of → Violet , Blue , Green , with their Complementary Yellow , Orange , Red ← Every 8-electrons are vibrations on Atoms-Cube-Structure {A Tetrahedron in Cube in a Sphere } whether it be Sound or Light . Above Structure is followed by all Compounds . Molecules are Systems consisted of The PERIFERAL ≡ Skeleton 𝐌 𝟏𝟔 and The CENTRAL ≡ Fittings 𝐌 𝟖 , and the ACCESSORIES ≡ 𝐌 𝟒 or 𝐌 𝟐 , 𝐌 𝟏 . This Property of Atoms and of the Compounds is The Critical-Valve of Switching the motion as this happens in the Electro-magnetic Solenoid Valves with High or low Pressure and flow rates directly or Not . Figure – 20.b- :The Electric - Magnetic Field of 3, Conductors Results on 2 -Anti-Parallel- Strands .The Stability of →{ ⊕ 4-Space [D A B C]} Regular - Tetrahedron , IN {⊝ 8-Anti- Space Cube , [ DA1B D1 , C C1AB1]}, and The →{ ⊕ 8- Space [ DA1BD1 , CC1AB1]} Cube, INTO The {⊝ 16-Anti-Space [ 1 ,2 ,3 ,4 ,5 , 6, 7, 8 - 9 ,10 ,11, 12 , 13 ,14 , 15, 16]} Sphere IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 104 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Cube Vertices . Point [16 ≡ |⇉|Z|] consists the Navel-String Gate in {Z-(∆D,ABC),∆[T1 T2 T3]} Stationary Bases . Stability Analysis in [88] DATA : A.. The Atoms Cube Coordinate-System : The relation of a Cube of Side a ,inscribed in a Sphere of Diameter d =2r , is in their Volumes as V Sphere = 4.𝜋 r ³. 3 = 𝜋 d ³. 6 , V Cube = a³ and since d² = 2a² + a² = 3,a² , then a = d. √3 = d.,√3 3 = √3,d 3 = 2√3,r 3 ,and the Coordinates of the 8-Cube-Vertices are for Center O → 𝐗 𝐎 = 0 , 𝐘 𝟎 = 0 , 𝐙 𝟎 = 0 ← 𝐗 𝟏 = − a. 2 , 𝐘 𝟏 = − a. 2 , 𝐙 𝟏 = − a. 2 , 𝐗 𝟐 = + a. 2 , 𝐘 𝟐 = − a. 2 , 𝐙 𝟐 = − a. 2 , 𝐗 𝟑 = + a. 2 , 𝐘 𝟑 = + a. 2 , 𝐙 𝟑 = − a. 2 , 𝐗 𝟒 = − a. 2 , 𝐘 𝟒 = + a. 2 , 𝐙 𝟒 = − a. 2 , 𝐗 𝟓 = − a. 2 , 𝐘 𝟓 = + a. 2 , 𝐙 𝟓 = + a. 2 , 𝐗 𝟔 = − a. 2 , 𝐘 𝟔 = − a. 2 , 𝐙 𝟔 = + a. 2 , 𝐗 𝟕 = + a. 2 , 𝐘 𝟕 = − a. 2 , 𝐙 𝟕 = + a. 2 , 𝐗 𝟖 = + a. 2 , 𝐘 𝟖 = + a. 2 , 𝐙 𝟖 = + a. 2 , Since Tetrahedron in Cube is formed in three Equal-Diagonals 𝐝 𝟏−𝟑 = a.√2 , 𝐝 𝟏−𝟓 = a.√2 , 𝐝 𝟏−𝟕 = a.√2 of the Perpendicular 3-Planes , P X−Y = P Y−Z = P Z−X = a² , and since issues 𝐝 𝟏−𝟑 ⏊ 𝐝 𝟏−𝟓 ⏊ 𝐝 𝟏−𝟕 , these consist The-Scanning Vectors-Plotting for all Systems . Since the Cube’s Outer-Sphere diameter d , is equal to the Cube-Diagonals [1-8] , [2-5] ,[3-6] , [4-7] , therefore consist 4-Big circles on Sphere and Perpendicular between them , in Contradiction to the 3 circles in the 3-Planes which consist the Small circles . The Geometrical Matric of Tetrahedron-Cube-Sphere Mould = ( STCS ) where , 𝐏𝟏 𝐏𝟐 𝐏𝟑 𝐏𝟓 𝐏𝐨 𝐏𝟒 𝐏𝟔 𝐏𝟕 𝐏𝟖 P O = The Coordinates X O , Y O , Z O of the Common Center of 2-Spheres . P 1 = The Coordinates X 1 , Y 1 , Z 1 of the Tetrahedron- Cube Point -1 P 2 = The Coordinates X 2 , Y 2 , Z 2 of the Sphere - Cube Point - 2 P 3 = The Coordinates X 3 , Y 3 , Z 3 of the Tetrahedron- Cube Point -3 P 4 = The Coordinates X 4 , Y 4 , Z 4 of the Sphere - Cube Point - 4 P 5 = The Coordinates X 5 , Y 5 , Z 5 of the Tetrahedron- Cube Point -5 P 6 = The Coordinates X 6 , Y 6 , Z 6 of the Sphere - Cube Point - 6 P 7 = The Coordinates X 7 , Y 7 , Z 7 of the Tetrahedron- Cube Point -7 P 8 = The Coordinates X 8 , Y 8 , Z 8 of the Sphere - Cube Point - 8 On the above Geometrical Coordinates SYSTEM are Placed the 4-Spaces from the OZ = OD - EXTENDED CONDUCTOR and equal to Monads . For the Cave-Spin , �̅� = r m v which is a Monad , exist the 4-Spaces as Fig-1 → (( The n - Spaces of Spin �̅� are the Polygons 𝐒𝐧 , The n Anti-Spaces of Spin - �̅� are the Polygons - 𝐒𝐧 )) ← The Sub-Spaces ± √ �̅� 𝒏=𝟏−∞ are the Polygons with n = 1 ≈ ∞ Knots Since Knots are the Degrees of the Polynomials of each Monad then , The OZ Monad Determines the Number of Knots beginning from Number 3 , IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 105 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 3  is the Figure of Regular Triangle in Poly-3rd Degree . The - 3 and ± √ OZ̅̅̅̅ 𝑛=3 4  is the Figure of Regular Square in Poly-4rd Degree . The - 4 and ± √ OZ̅̅̅̅ 𝑛=4 5  is the Figure of Regular Pentagon in Poly-5rd Degree . The - 5 and ± √ OZ̅̅̅̅ 𝑛=5 6  is the Figure of Regular Hexagon in Poly-6rd Degree . The - 6 and ± √ OZ̅̅̅̅ 𝑛=6 7  is the Figure of Regular Heptagon in Poly-7rd Degree . The - 7 and ± √ OZ̅̅̅̅ 𝑛=7 8  is the Figure of Regular Octagon in Poly-8rd Degree . The - 8 and ± √ OZ̅̅̅̅ 𝑛=8 9  is the Figure of Regular Eniagon in Poly-9rd Degree . The - 9 and ± √ OZ̅̅̅̅ 𝑛=9 N  is the Figure of Regular N-gone in Poly-Nrd Degree . The - N and ± √ OZ̅̅̅̅ 𝑛=𝑁 Remarks : 1… Gravity Force G ≡ motion , exists without mass and is Quantized → Spread in {Space and Anti-Space} or ≡ QUA ≡ {is The Quantized-Units In-Area of Space and Anti-Space }, and as a Mould 𝚽 Golden-Ratio , exists in the Impedance b . 2…The case of {QUA} ≡ [ Σ = n . h ]  ∞ x 𝚽 which is the Physical Mechanism , gives the ∞ - Impedance b determining the Objective Reality which is the existing Universe. For n =2 then [Σ = n h]  2 x G /Φ , IS the light-velocity �̅� = [ 𝐆 𝚽 𝐀 ] =2.9982.108 m/s. For n =1 then [Σ = n h]  G / c = Charge q̅ = 𝐆 𝐜 √𝟐 = 1,58.10−19 Coulomb . 3… For n = 3 is STPL Mechanism which creates the Elementary Particles , [n-Knots = 3] in Gravity-Voltage 𝐕 GRA = 7,593.1035 V. This is because the Primary motion ≡ [PM] exist from the three - 3 Breakages elements [Σ = n . h]  [s² = ⊕ , 2s²= , - s² = ⊝] B.. The Atoms-Elements Mass , Stiffness - Flexibility [101] 6d..The Dimensioning of the [STPL] – Mechanism Defining the lengths , a = KB KC , b = KC KA , c = KA KB , d = [ a+b+c 2 ] =The semi-perimeter then the Inscribe radius r = √d(d−a)(d−b)(d−c) d = |OA| , The Coordinates for the Point K=DA are [ bc b+c−a ] : [ ca c+a−b ] : [ ab a+b−c ] , while for Point O are [ b+c−a a ] : [ c+a−b b ] : [ a+b−c c ] . Fig-17 The STPL Mechanism is the Mould consisted from any Common circle O,OA=[OA`≡ OAE], O,OB = [OB`≡ OBE] , O,OC = [OC`≡ OCE] , and the Common lines DA-PA , DB-PB , DC-PC all on a line of STPL . On the infinite Sectors ADA-APA , BDB-BPB , CDC-CPC vibrate , the IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 106 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Breakages [± s² = ± (w r)²] and [ i = 2(w r)²] forming all families of curves and the Euler - Savary Coupler - curves of the Cubic-Of-Stationary - Curvature Mechanism of Spaces , Anti-spaces Vibrating the End-curves . The Dimensioning of the Mechanism is Possible by using the analytical geometry. [91] The Dimensioning of Proton – Neutron In Fig-26 , and others with Compounds In Fig-27 .The Dimensioning of Hydrogen and Atoms In Fig-28-30, The Linear-Vibrations [ ẍ + w ² x = 0 ] of Three-masses , Occur on Two-Line-Vectors Perpendicular each other , vibrating on Straight-line , ÿ + w ² y = 0 , of x ⊥ y Plane and follow the Lissajous Shapes , [83] , where for , a.. Difference of Phase dφ = 90⁰ emission is → The Eight-Shapes Ꝏ . b.. Difference of Phase dφ = 0⁰ emission is → The Ellipse-Shapes ∝ . c.. Difference of Phase dφ = 45⁰ emission is → The Double-Saddle-Shapes . Ց , Ѡ . For Planck length [73] P-49, was shown that the Rotated Energy case , when s = 0 and cosφ = 0 , exists for angle φ = π /2 and Quaternion ( s + v̅ i )𝟏/𝐰 = 𝐞− 𝐢.(𝛑/𝟐+𝟐𝐤𝛑).𝐰...(1) the dimension power → w = b ← and for k = 1 then (1) becomes , [84] -P.74 𝐞− 𝐢.(𝛑/𝟐+𝟐𝐤𝛑).𝐰 = 𝐞− 𝐢.(𝛑/𝟐+𝟐𝐤𝛑).𝐛 = 𝐞− 𝐢.(𝟓𝛑/𝟐).𝐛 = 𝐞− 𝐢.(𝟓𝛑/𝟐).𝟏𝟎 [86] …... (2) Equation (2) fits , as minimum cave , in the Planck length and is 𝐋 𝐩 = 𝐞− 𝐢.(𝟓𝛑/𝟐).𝟏𝟎 .....(3) Equation (3) is the smallest Energy-Unit of Space , and this because is for s = 0 and k = 1. It was shown [31] that Space and Energy is quantized and measured on the two Constant and Natural numbers e , π , where for base the natural logarithm , e , and exponent the decimal base , b = 10 . From → z¹/w = ( s + �̅� i ) ¹/ w = |zo|−w.[cos.(φ + kπ)/w + i.sin.(φ + kπ)/w] =|zo|−w . e−i.(φ+kπ).w for cos.(φ+kπ)/w = 0 then exists only the Imaginary part of monad (�̅� i ) ≠ 0 , where φ = π /2 and then , z¹/w = |zo| ̄ w . e i.(φ+kπ)/w = e−i.( π 2 +kπ).10 which is the Diffraction Energy mechanism for all Space Levels of quantization which are the Energy Particles only i.e. The Energy particles in Stationary caves are z¹/w = |zo| ̄ w. Lv = E-Monad. Extending quantization of Energy according to exponential formula→ 𝐋 𝐯 = 𝐞− 𝐢.(𝟓𝛑/𝟐).𝟏𝟎 then 𝐋 𝐯 on the decimal base b = 10 and for k = ± 0 → ± ∞ , are the Energy caves as , For base e = 2,71828 and base b = 10 then e^ - (13,8155) = 1 .10 ̄ 6 m For base e = 2,71828 and k = 0 Lv = e^i.( ± π/2)b then e^ (-15,7079) = 1,78118 .10 ̄ 7 m For base e = 2,71828 and base b = 10 then e^ - (16,1181) = 1 .10 ̄ 7 or r = 1,07.𝟏𝟎−𝟕 m. Placing r , in the Dynamic-Space-Energy relation when g =1 then r ³.𝐟²𝐩= 1 and 𝐟²𝐩 = 1 r ³ = 8,0647139.1020 m and occurs the , minimum frequency 𝐟𝐦𝐢𝐧 = 2,839844. 𝟏𝟎𝟏𝟎 H …(4) 7d.. The Cosmic Particles in The Cycloid , Anti-cycloid Evolute : The moving monads in Cycloid Equilibrium in the Anti - Cycloid ( Evolute ). The equilibrium of the Resultant forces is obtained by the above Cycloid , Anti-cycloid Evolute Forces , or by the Rotor and Spin in one Rim , as was shown. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 107 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . In Rims is possible the forming velocities greater than that of light , and this because of different , r , and the Tangential velocities are kept with the same angular velocity , the Spinning , unless Spinning changes in such way to keep light-velocity constant . Figure 25 The Equilibrium of the Resultant Forces , in the Cycloid and for motion Wavelength , and of any other Removal Monad . The Intrinsic Space is (Wavelength = [λ] ) , as the Energy Quanta . The Equilibrium of the Three-Spaces , The Space Anti-Space , Neutral-Space : [60] The Stability of Elementary Particles , Conductors and the Atoms -Nucleus Fig-24 The Equations of motion become everywhere from any Opposite Space ≡ ⊕ , Anti- space ≡ ⊝ . During these motions as , ds . is Done a Constant-Work , k , or and from their velocities �̅� and this Work may be Positive or Negative for Equilibrium . In F-23 Cycloid Anti-cycloid motion is as , D. The Origination of Gravity g , Antigravity [- g ] In Fig-20.b , Conductor D = d = AKA where ⊕ constituent attacks ⊝ by a Hook`s law force as Stress σ = E u , and an Coulomb force F [+ ↔ −] = [⊕↔⊝ ] r² = [2 σ] r² , creating Work ≡ motion as was shown It was shown that , The Stability of Nucleus Forces , is succeeded through The Pascal`s - Ceba`s triangle launched caves r , while Leptons from their Anti - Leptons existence in the same launched Pascal`s - caves r . [58] The Equilibrium of the Three-Spaces is Done on the three Cyclic-Quadrilaterals (ABEAECE) (BCEBEAE) , (CAECEBE) of the 6 Conductors each for , [a , b , c , d , p , q ] x [3-ABC] = 18 Conductors . Linearly is through the Products of the Diagonal - connectors as , → → → [AAE].[BECE] = [ACE].[AEBE]+[ABE].[AECE] , [BBE].[CEAE] = [BAE].[BECE]+[BCE].[BEAE] , [CCE].[AEBE] = [CAE].[BECE]+[CBE].[CEAE] and , Rotationally through the Ratio of the Diagonals ≡ The Conductors | AKB AKC |x| BKC BKA |x| CKA CKB | = 1 or  Rotational-Equilibrium as , [AKB].[CKA].[BKC] = [AKC].[BKA].[CKB] , of all Opposites  Tangential-Electromotive Forces . From Mechanics-Physics , all Systems Possessing Elasticity ≡ motion and Reaction to motion , the called mass , are capable of free vibration or vibration taking Place in the Absence of External Excitation [7] .This Principle issues for Both Closed or Open Systems. i.e. either for the Nucleus where issues The Pascal`s - Ceba`s triangle Stability , or for the Orbitals where issues the Tetrahedron - Cube – Sphere – Mould . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 108 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 8d.. The Stability of the Cosmic Particles Conductors and Atoms – Nucleus . Figure -26- : The Stability of → { ⊕ - Space [A B C] },{⊝ - Anti-Space [KA KB KC]} , of The Sub-Space [A E , B E , C E]}, and Neutral-Space {[⊕↔⊝] ≡ ds ≡ [⊕←ds→⊝]}. Since , Energy of the first Wheel-Rim or Orbital is distributed to the couple of (2e) Electrons of the Orbital , then this is a Cycloid - Anti-cycloid - Stability System . This Still Systems of Space –Anti space DOES NOT mean Matter–Antimatter but The State of Equilibrium in Caves ≡ The Ceba`s Energy triangle . a…The Space Anti-Space and , Neutral-Space Conductors : [60] The Equations of motion become everywhere from any Opposite , Space ≡ ⊕ , and Anti-space ≡ ⊝ . During these motions as , ds . is done a Constant-Work , k , or and from their velocities �̅� . In Figure Conductor D = d = AKA where ⊕ constituent attacks ⊝ by a Hook`s law force as Stress σ = E u , and an Coulomb force F [+ ↔ −] = [⊕↔⊝ ] r² = [2 σ] r² , creating Work ≡ motion as was shown . [58] It was shown that , The Stability of Forces is succeeded in The Pascal`s - launched caves r while Leptons , from their Anti - Leptons existence in the same launched Pascal`s-caves r . The Equilibrium of the Three-Spaces is succeeded on the three Cyclic - Quadrilaterals → (ABEAECE) , (BCEBEAE) , (CAECEBE) of 6 Conductors each [a , b , c , d , p , q ] x [3-ABC] = 18 Conductors . Linearly is through the Products of the Diagonal - connectors as , → → → [AAE].[BECE] = [ACE].[AEBE]+[ABE].[AECE] , [BBE].[CEAE] = [BAE].[BECE]+[BCE].[BEAE] ,[CCE].[AEBE] = [CAE].[BECE]+[CBE].[CEAE] and , Rotationally through the Ratio of Diagonals ≡ The Conductors | AKB AKC |x| BKC BKA |x| CKA CKB | = 1 or the  Rotational-Equilibrium [AKB].[BKC].[CKA] = [AKC].[BKA].[CKB] ,of all above Opposites → Tangential-Electromotive-Forces ← From Mechanics ,all Systems Possessing Elasticity ≡ motion and Reaction to motion , called mass , are capable of free vibration or vibration taking place in the Absence of External Excitation . [7] From Lorentz-Force , F = �̅� �̅� x �̅� 𝐅 , where , q̅ ≡ Charge , v̅ ≡ The velocity of Charge , B̅ F ≡ The Strength of Magnetic field and Centrifugal Force , 𝐅 𝐂𝐅 = m.𝐰²𝐑.r . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 109 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . From Cave-Spin , �̅� = r m v then mass m = ( S r.v ) .Equating equations then , F ≡ FC F ≡ q̅ v̅ x B̅ F , and 𝐐  = m.w²R.r = ( S r.v ).w².r = S.w² v = S.v ² r².v ≡ | 𝐯.�̅� 𝐫² | ….(EMF) where exists rmin= √πb−n 2 , [95] , [96BB] . Electromotive-Forces , F = �̅� �̅� x �̅� 𝐅 = 𝐐  = 𝐯.�̅� 𝐫² ≡ | 𝐜.�̅� 𝐫 ² |, are The Elementary-Particles {F, L ,Q }. From Electromotive-Force in min-cave r = 1,7724538.10−3 m. exists Electron- Force-Charge q̅ e as 𝐐  = | 𝐜.�̅� 𝐫 ² | = [2,998.108].[1,0546.10−34]J (1,7724538.10−3) ² = 1,6022.10−19 eV , which is the Electron-Force-Charge and also from the Gravitational-Stress-Charge �̅� Electron = G c √2 = 6,6736923 .10−11 1,41429.2,9979346.108 = 1,574.10−19 C . Electromotive-Force in the min-cave 𝑒−15,572707= r = 1,7252787.10−7 m , and the Gravitational-Force G becomes also as , 𝐐  = | 𝐜.�̅� 𝐫 ² | = [2,998.108].[6,62606957.10−34] (1,7252787.10−7) ² = 6,673692. 10−11 , which is the G - Physical-Force- Charge. When the Heap of the Elementary Particles on DA-PA line , is compressed and gets out the line then is a reason for Black Hole creation . b…The relation of the n , Degree of Polynomials and Electromotive-Forces Vibration . The Transportation of Energy on Edge-Points happens in Material Geometry , while the Transportation of the Edge-Points as , Positions , exists in Vector Geometry only. Since in Vibrations the Deformation of the Spring is an equilibrium Position in ∆ , and the Spring-Forse k ∆ = w = m g = Gravitational force w acting on mass m , then exists → m �̈� = ΣF = w = k [ ∆ + x] ….(1) , and because k ∆ = w we obtain w ²n = k/m.. (2) and (1) becomes → �̈� + 𝐰 ²𝐧 = 0 ,… (3) ← i.e. The Transportation of Energy is a Harmonic - Vibration in cave ∆ , meaning the Wave Pattern . From before the Cave-Spin , �̅� = r m v is a Monad , and occupies the 4-Spaces as the Fig–1 . i.e. → (( The n - Spaces of Spin �̅� are the Polygons 𝐒𝐧 , The n Anti-Spaces of Spin - �̅� are the Polygons - 𝐒𝐧 )) ← The Sub-Spaces ± √ �̅� 𝒏=𝟏−∞ are the Polygons with n = 1 ≈ ∞ Knots Since the number of Knots n , are the Degrees N of the Polynomials of each Monad and Since the n - masses - Spring 𝐦 𝐧 is an equilibrium Position in ∆ , of the equation of motion → �̈� + 𝐰 ²𝐧 = 0 ← then the 4-Spaces of Monad are the Polygons of n-Knots and equal to the N Degrees Roots of the Polynomials . THE PROCESS OF ACTIONS IN THE 4=SPACES 1…A Heap of n-masses Vibrates and Equilibrium . 2…A Heap of n-masses Equilibrium at the n-Mode Shapes Waveform . 3…A Heap of n-masses of n-Mode Shapes occupies the Resultant E-Spectrum 4…Each-One of the n-masses occupies n-Mode Shapes , n-Element .Spectrum , 4-P.Spaces , 4.n.. Regular-Polygons of n-Degree , with 4.n..Knots and Roots . 5…Each-One of the n-masses is an E -Monad occupying 4-Primary-Spaces , and IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 110 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Forms 4.n.. Regular-Polygons of the n-Degree , and with 4.n..Knots as Roots .. 6…Each-One of the n-masses is an E-MONAD occupying all above Properties (Signals). 9d.. The Origination of Proton and Neutron Cosmic Particles . The Elements in the Proton & Neutral-caves → { Masses – Charges - Forces } [89] Figure-27 : The Sphere-Structure of the Hydrogen-Nucleus , ⊕ Proton ,[⊕↔⊝]=Neutron Proton is consisted of Three-Primary-Opposite-Spaces ⊕ , ⊝ , [⊕↔⊝] , having masses m p , Charges q̅p , and caves a p = r . The Nucleus`s Cave r , exists in the Inscribed to the Tetrahedron Sphere . The elements in Proton are the Two u-Quarks and One d-Quark . The Hydrogen-Nucleus is in the Inscribed Sphere from the { Inscribed Sphere -Tetrahedron – Cube - Circumscribed Sphere - System }. In order to shift the u-d-Quarks of an Anti-Proton into a Proton Spin-Pair require an extra input of Energy-(MeV) , so would Proton Paired with a Neutron be Stable . The Total mass MT in Proton follows , The Parallel Connections Resistors inverse law , where The Proton Total-Harmonic mass ≡ MT is → 1 MT = 1 mu + 1 mu + 1 md , The System Total- Harmonic-Charge ≡ QT ≡ 2.qu + qd = 2.(2/3).e – (1/3) e = + 3 3 e = + 1,6022.10−19 C , and the System-Resonance-Charge QT = + 1, 6022.10−19 C ...(2) The Proton Total- Harmonic- Charge → QT ≡ 2.qu + qd ,and for the Resonance Frequency of the Unit-Cave is the Stationary-System becoming from Kepler second Planetary-law as equation , 4 π² m f ²o = k , and constant law of Areas 1 = k .𝐟 ²𝐨 𝐚 𝟑 . Their common k , is the IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 111 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Constant-Energy → k = 4 π² m f ²p = 1 𝐟 ²𝐩 𝐚𝟑 or , f ⁴p = 1 4π²ma3 and f p = √ 1 4π²m.a3 4 . The measured magnitudes are as follows , a-Proton⊕→ mass 𝐦𝐩 = 1,672.10−27 Kg →Charge 𝐂 𝐩 =1,602.10−19 C → a = 8,4.10−16 m b-Electron⊝→mass 𝐦𝐞 = 9,11.10−31 Kg →Charge 𝐂 𝐞 =1,602.10−19 C → a = 5,0.10−17 m c-Neutron[⊕↔⊝]→mass 𝐦𝐧 = 1,672.10−27 Kg →Charge 𝐂 𝐧 = 0 ,0 C → a = 1,7.10−15 m d-u-Quark →from equal masses mp =2.mu + md =3.mu =1,672.10−27 Kg and Quark masses 𝐦𝐮 = 𝐦𝐝 = mp / 3 = 5,573.10−28 Kg , and 1 MT = 1 mu + 1 mu + 1 md = 3 mp + 2.3 mp = 9 mp and Proton Resonance mass MT = m p 9 = 1,857777.10−28 Kg ≈ 16,7267 m Ev/c² …….(m) For Proton issues qp = 2.qu + qd = 2. 2 3 e - 1 3 e = e , and the Stability of forces is axial . Electron-Charge 𝐂𝐞 =1,602. 10−19 C , while 𝐂𝐪𝐮 = 2 3 e = + 𝟐 𝟑 1,602.10−19 C 𝐂𝐪𝐝 = - 1 3 e = - 𝟏 𝟑 1,602.10−19 C , and from QT ≡ 2.qu + qd , then Proton Resonance charge QT = 3.q p 3 = 1,602.10−19 C ………..(c) Proton-Resonance frequency f p =√ 1 4π²m.a3 4 = √ 1 4π²5,573.10−28(8,4.10−16)³ 4 = 5,26241.1017 H Using the United Newton-Coulomb Electro-Mechanical Equation, q B̅L=2π.m f , the Proton Magnetic-field B̅F = |2π.m T| QT f = 2𝜋.1,85777.10−285,262409.1017 1,6022.10−19 (Kg/Cs) = 3,83389.109 Tesla which is the Strength of a Magnetar , i.e. A type of Neutron-Star having an extremely Powerful Magnetic-field , and Electric-Forces to be over Ten-Thousands-Newton . The Electric Force between the u-Quarks and d-Quarks in Proton is from Coulomb law 𝐅𝐮𝐝−𝐩 = C q1.q 2 r ² = 8,9875.109(Nm2/c²). 2 9 [1,602.10−19 C]² 1 (10−16)² = 1,997222.10 6 N ...(F p) and The Electric Force between the u-Quarks and d-Quarks in Neutron is 𝐅𝐮𝐝−𝐧 = C q1.q 2 r ² = 8,9875.109(Nm2/c²). 2 9 [1,602.10−19 C]² 1 (10−16)² = -1,997222.10 6 N ..(F n) i.e. Forces between the Opposites Equilibrium-Linearly ←[d-u-d]→ or →[u-d-u]← For the Neutral-cave issues qn = 2.qd + qu = - 2. 1 3 e + 2 3 e = 0.e , and the Stability of forces is axial as in Proton and this because the Dynamic-Strip-Polygon doesn`t close . For , IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 112 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . 1.. Neutrinos , v , mass m = (3,11 .10−6) MeV/c² x 1,8.10−28 = 5 , 598 .𝟏𝟎−𝟑𝟒 Kg The Spin is , 𝐒𝒗 = 5,691952.10−34{Kg/m/s}, 𝐚𝐯 = 7,0.10−21 m , 𝐟𝐯 = [𝐄 𝐯𝐊 = mc²] / h = 5,6.10−34 [1017].1,6022.10−19 = 8,96912.10−36 J/[6,626.10−34 Js = 13,536244. 𝟏𝟎𝟏/s From cave r = 7.10−21 m then f = √ 1 g.a³ 2 = √ 1 9,81(7.10−21)³ 2 = 5,347345.𝟏𝟎𝟑𝟎 H and exists , 𝐄 𝐯𝐊 = h 𝐟𝐯 = 6,62607.10−34J s.[ 5,347.1030] /(1,6022.10−19 eV) = 2,2114518.𝟏𝟎 𝟏𝟒eV Using 𝐄 𝐯𝐊 = 𝐤 𝐫 + 𝒄𝐒 𝟐 𝟐𝐦.𝐫𝟐 = 36.10−20 7,10−21 + 3.108(5,691952.10−34)² 2.3,922.10−36[7,0.10−21]² =51,428 eV+ 2,528774.1017eV = 252,8774, [1015] TeV → The Total Energy of the Sun striking Earth-face per second. 2…Electron , e , me = 0 , 511MeV = 0, 511.10−5 eV . [1,80. 10−27] = 9,198.10−31 Kg 𝐚𝐞 = 5,0.10−18 m ,Charge 𝐂 𝐞 =1,602.10−19 C , Spin 𝐒𝒆 = 𝐒 2 =2,845976.10−34{Kg/m/s}, From cave r = 5.10−18 m then f = √ g a³ 2 = √ 9,8078 (5.10−18)³ 2 = 2,801114.10 26 H , and by Using 𝐄 𝐞𝐊 = 𝐤 𝐫 + 𝒄𝐒 𝟐 𝟐𝐦.𝐫𝟐 = 36.10−20 7,10−18 + 3.108(5,691952.10−34)² 2.9,198.10−34[5.10−18]² = 51,428 eV + 2,111843.1010J = 51,43 eV+ 1,32 .1029 eV → The Energy of 133 gr to fall 1 meter against gravity. Remarks : 1.. Gravitational force → G ≡ σ A ≡ [ 2πrf Φ ] A ≡ v̅ [ 𝐀 𝚽 ] ≡ σ.Φ³ ≡ Φ ² . [{σ Φ}] ≡ G ≡ Φ². [{σ Φ}≡ 2πfP r ≡ w r ≡ v̅ ≡ m g = c̅ = 2.B πr³ ] → i.e. G is related To → motion ≡ work W , Spaces r , Anti-Spaces 1/r , Stresses σ , Areas A , Caves a , → Periods T , Frequencies f , Angular-waves w , Angular-Momentum B , → Spin B ≡ S , velocities v , Light-velocity c , Impedances Zn, Masses m , → Gravity g̅ , Charges , q̅ , Electromagnetic – Fields , E̅ , M̅ , Hydrogen H , → Atoms , Molecules , Golden-Ratio Φ , i.e. All-Universe . Markos 9/4/2020. By Sweeping the Φ Surface through Stresses σ ≡ [ 2πrf Φ ] , and scanning frequencies f , Explanation : The Action-Spaces ≡ The Infinite Quaternion - monads → �̅� ≡ ( s + v̅.i ) ≡ 𝐀𝐁̅̅ ̅̅ ≡ ≡ √s2 + [ v̅. i ]² ] , or �̅� ≡ x + i . y ≡ 𝐀𝐁̅̅ ̅̅ ≡ moduli , r , is their magnitude [ r = | r | = √ x2 + y2 ] , on Unit-Diameter AB̅̅ ̅̅ . The [QMAS] . For Quaternion |v̅| = s Breakages 𝐳 * 𝐳 = (s + v̅.i )² = s² +[v̅]² +2.sv̅.i = s² – v̅ ² ± 2.s v̅ = |s|² - |�̅�|² ± 2.|s|.|s̅| , i.e. In Figure -3- the Vector of Amplitude is Quaternion �̅� ≡ x + i . y , with vector �̅� , to be from Definition of the Complex Vector Space , the Set of Vectors of length (the n Vertices which form the n - Polygon Sides , or the Norm of the Complex - Vectors ) a...The 4-Geometrical Spaces , [ Space , Anti-Space , Neutral-Space , Sub-Space ] consist → The Geometrical - Mould of 𝐀𝐁̅̅ ̅̅ - Monad ≡ [PM-2R-M] ← b...The Vectors of Amplitude z = 2R , ON 𝐀𝐁̅̅ ̅̅ Monad , consist The Energy-Mould . of 2R-Form .The Physical -Mechanisms → [ Physical Mould of 2R- Monad ] ≡ [GM-AB-M] ← are such and dependent on the n - Number of Accessories they use . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS 113 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . A… The Gravity Force G ≡ motion ≡ σ.𝚽³ , exists without mass and is Quantized Spread in{Space and Anti-Space Or ≡ QUA ≡ {is The Quantized-Units In-Area of Space and Anti-Space }, and as a Mould 𝚽 Golden-Ratio , exists as Impedance b. Placing [ Physical Mould of 2R- Monad ] ON [ Geometrical - Mould of 𝐀𝐁̅̅ ̅̅ - Monad ] [ PM-2R-M ] → ⇅ ← [ GM-AB-M ] THEN → Gravitational-Force G ≡ [ Σ = n . h ] , and n-NA = 1  Creates & Constructs, Figure -28- : On any Energy - Monad A ↔ B , exist the 4-Primary Spaces which are the Roots of the Nth Degree Polynomial , Becoming from the equation of motion . F... REFERENCES : [ 1] Matrix Structure of Analysis by J.L.MEEK library of Congress Catalog 1971. [ 2] Der Zweck im Rect by Rudolf V. Jhering 1935. [ 3] The great text of J. 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[58] Big-Bang or the Rolling Glue-Bond of Space ,Anti-space , book@scirp.org ,http://www.scirp.org/ [59] STPL Mechanism is the Energy – Space Generator . http://viXra.org/abs/1612.0299 [60] The Chaos becomes Discrete through the STPL mechanism which is Energy-Space Generator (http://www.ijrdo.org/) [61] The How Energy from Chaos becomes Discrete Monads http://www.ephjournal.net/ [61-A] The How Energy from Chaos , becomes Discrete Monads . http://www.ijrdo.org/ [62-B] The Geometrical solution of All Regular n-Polygons . http://www.irjaes.com/ [62] The Geometrical Solution of All Odd – Regular – Polygons , and the Special Greek problems http://www.irjaes.com/ [63] The Geometrical Solution of All Odd – Regular – Polygons , the Special Greek Problems and their Nature . http://www.ijerd.com/ [63] [A] The Geometrical Solution of The- Regular – Polygons , the Special Greek Problems and Their Nature . http://vixra.org/ [63] [B] The Geometrical Solution of The- Regular – Polygons , the Special IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS http://www.sciencepublishinggroup.com/j https://www.lap-publishing.com/ http://www.ijirset.com/ http://vixra.org/abs/1510.0328 http://www.submission.arpweb.com(id-44031-sr-015.0/ http://www.ijera.com(id-512080.0/ http://www.oalib.com(id-1102480.0/ http://gpcpublishing.com/index.php?journal=gjp&page=index http://www.ijerm.com(id-0306031.0/ http://ijemcs.in/ http://www.iosrjournals.org/ http://ephjournal.com/ http://www.technicaldean.org/ http://www.globaljoyrnals.org/ http://www.scirp.org/ http://vixra.org/abs/1612.0299 http://www.ijrdo.org/ http://www.ephjournal.net/ http://www.ijrdo.org/ http://www.irjaes.com/ http://www.irjaes.com/ http://www.ijerd.com/ http://vixra.org/ 116 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Greek Problems and Their Nature .( http://iosrmail.org/) [64] [A] The How energy from chaos becomes the → Spin , of the Discrete Elementary monads . http://www.i-b-r.org. / .?? [64] The How energy from chaos becomes the → Spin , of the Discrete Elementary monads : (http://www.ijrdo.org/) [65] The Spin of monads and their Energy-Stores . www.ajer.org . [66] The Energy-Stores in Photon . http://www.i-b-r.org. / .??? [67] The Energy Structure of Atoms and Photon . http://viXra.org/. [68] The Moving Energy - Storages and Photon . www.sfjqp.com [69] The Moving and the Stationary Particles . http://science MPG [70] The How Energy from Chaos becomes the Spin of Monads and Photon http://www.ijrdo.org/ [70] Energy from Chaos becomes the Spin of Monads & Photon ,http://science MPG [70] The How Energy from Chaos becomes the Spin of Monads and Photon . www.ijera.com . [71] The Gravity and Photons . http://asir@sholink.org [72] The origin of Gravity and universe . [mailto:editorusa@globaljournals.org] [72A] [M] The Origin of Gravity Gravitational Constant and Universe . http://saiconference.com/FTC [73] [M] Planck`s Constant , The Gravitational and Gravity Constant . : https://ijrdo.org/index.php/mce/issue/current [74] [M] The Newtonian Constant of Gravitation and Gravity Constant . www.iosr.Org [75] [M] The Newtonian Constant of Gravitation and Gravity Constant . The Physical Interpretation http://science MPG [76] [M] The Newtonian Constant of Gravitation Gravity Constant , and The Galileo Principia http://www.i-b-r.org. / . [76A] [M] Origination of The Nutation-motion , and Atom-model .http://www.i-b-r.org. /.? [77] [M] The Newtonian Constant of Gravitation and Gravity Constant . Their Physical Interpretation .. [78] [M] The Physical Interpretation of Gravity-Constants , Electron and Photon https://saiconference.com/FICC2020/Submit [79] [M] The Planck`s Constant and Speed of light . http://science MPG [80] [M] The Origination of the Nutation-motion , the New-Energy-Atom-Model and the United-Coulomb-Newton-Laws for Interaction . https://www.akinik.com/publishbookchapter/-research [81] [M] The Physical Interpretation of Gravity – Constant , Electron and Photon http://vixra.org/abs/1906.o468, https://www.scribd.com/doc/ [82] [M] The Physical Interpretation of Gravity – Constant , Electron and Chemistry http://science MPG [83] [M] The EPR Argument and The Quantization of Energy in Spaces http://www.ajer.org/volume9 issue 1.html [83A] [M] The EPR Argument under the Critic of Material-Geometry and Elementary Particles. , http://www.ajer.org/volume9 issue 1.html [84] [M] The unification of Energy-monads , Black Holes ,with Geometry Monads , Black Matter, through Automobile Forces in monads . [85] [M] Quantization of Points and Potential and the Unification of Energy-Space . [86] [M] [M] The Physical Interpretation of Gravity – Constant , and IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS http://vixra.org/ http://vixra.org/ http://www.ijrdo.org/ http://www.ajer.org/ http://vixra.org/ http://science/ http://www.ijrdo.org/ http://science/ mailto:editorusa@globaljournals.org http://saiconference.com/FTC https://ijrdo.org/index.php/mce/issue/current http://www.iosr.org/ http://science/ http://vixra.org/ http://vixra.org/ https://saiconference.com/FICC2020/Submit http://science/ http://vixra.org/abs/ https://www.scribd.com/doc/ http://science/ http://www.ajer.org/volume9 http://www.ajer.org/volume9 117 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . Applications in Chemistry http://science MPG [87] [M] Photon Particle , Photon Wave Or Duality Photon ? and Applications http://vixra.org/abs/2003.0601 [87] [M] Photon Particle , Photon Wave Or Duality Photon ? and Future Technologies https://www.scribd.com/doc/1/ [87B] [R] Photon Particle , Photon Wave Or Duality Photon ? An Answer to WHY and HOW is the Objective-Reality www.IJRDO-Journals.org) [88] [M] [M] The Origination of Nutation motion and a New Electromagnetic Structure of Atom http://science MPG [89] [M] The Wave & Particle Duality Photon and Elementary Particles Origination. http://science MPG ≡ Markos Georgallides [89A] [M] The Duality Photon and The Physical Interpretation of Photon Spectrum http://science MPG ≡ Markos Georgallides [90] [M] The New-Structure of Atom , the Nutation-motion and their Application . http://science MPG ≡ Markos Georgallides [91] [M] The Origin of The Fundamental-Particles in Planck`s Confinement . http://science MPG ≡ Markos Georgallides [92] [M] The Wave & Particle Duality-Photon and Elementary Particles-Origination. www.IJRDO-Journals.org) , http://science MPG ≡ Markos [93] [M] The Fundamental-Particles and the Fundamental-Forces of Nature . http://science MPG ≡ Markos Georgallides [94] [M] The Cosmic-Particles Origination and their Bonding , [95] [M] The Wave and Particle Duality Photon , Cosmic-Particles-Origination and their Bonding STAIR AWARDS 2021 [294] [M] The Wave & Particle Duality - Photon , Cosmic-Particles Origination and their Bonding http://science MPG ≡ Markos [95] [M] The How Intensity-Squares of Electromagnetic-Cosmic-Particles follow Quadrature of Square-Prism to Equivalent-Energy-Sphere-Cone , http://vixra.org/3 / 2021 [96] AA The Wave and Particle Duality Photon , Cosmic-Particles-Origination and their Stability In Cosmology < ejas@scholarpublishing.org > [97A] [M] Duality Photon and the Mechanical and Chemical , DNA-Helix ,Construction Functions< ejas@scholarpublishing.org > , http://science MPG ≡ Markos Georgallides [97B] [M] The Duality Photon , DNA , and the Cruise-Missile Elementary-Particles , Conductors , http://science MPG ≡ Markos Georgallides [98A] [M] The Nature of Greek-Special-Problems & their functioning in Electromagnetic Forces In DNA - Conductors , https://www.lap-publishing.com/. E-book. LAP LAMBERT Academic Publication . [98B] [M] The Why , the How and When , the Atoms – Bond . http://science MPG ≡ Markos Georgallides [99A] [M] The How , Why and When , the Atoms – Bond . https://mts.Intechopen.com.book.process/ ≡ Dragan Miljak [99AAA] [M] The How , Why and When , the Atoms – Bond . http://science MPG ≡ Markos Georgallides [99CCC] [M] The 4-Frequencies of Atoms – Bonding related to Octave Periodic-Table . http://science MPG ≡ Markos Georgallides IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS http://science/ http://vixra.org/abs/2003.0601 https://www.scribd.com/doc/1/ http://www.ijrdo-journals.org/ http://science/ http://science/ http://science/ http://science/ http://science/ http://www.ijrdo-journals.org/ http://science/ http://science/ http://science/ http://vixra.org/3%20/%202021 mailto:ejas@scholarpublishing.org mailto:ejas@scholarpublishing.org http://science/ http://science/ https://www.lap-publishing.com/ http://science/ http://science/ http://science/ 118 The Fundamental Particles Origination - Mechanism in MFMF - PNS Caves . [99] [M] The Bonding of Cosmic-Particles and Resonance Photoelasticity , http://science MPG ≡ Markos Georgallides [100] [M] Programming , The Atoms and their Compounds Energy , http://science MPG ≡ Markos Georgallides [101] [M] The Why and How Atoms and Compounds Bond ,and Chemistry-Programming [102] [M] The Markos-Method of Conservation of Motion = Energy , by Division . [103] [M] The Markos-Method of Conservation of Motion in Nature and Medicine . [104] [M] Programming of Atoms and their Compounds and their Modulated-Energy, http://science MPG ≡ Markos Georgallides [105] [M] The ERP Argument , Under the Critic of the Material—Geometry AND , The Space Energy Universe . [106] [M] Programming the Atoms and Compounds and the Unification of Physics AND Chemistry —- ["JMSEAT - 2024"] atul@scientificadvances.co.in [106] [M] Programming the Atoms and Compounds and the Unification of Physics AND Chemistry —- ASRP.hspublishing@gmail.con [107-A] [M] The Planck`s << Duality Angular - Momentum >> as GRAVITY and . ANTIGRAVITY Research International Journal [107] [M] The Planck`s << Duality Angular - Momentum >> as GRAVITY and ANTIGRAVITY http://science MPG ≡ Markos Georgallides [108] [M] The Planar and Atoms Black Holes , Under the Critic of Material-Geometry & Planck–Dual Spin as Gravity-Antigravity http://science MPG ≡ Markos Georgallides [109] [M] The DUAL Quaternion Momentum -B as { The Existing Universe } & The Black -Holes , http://science MPG ≡ Markos Georgallides [110] [M] The Origination-Mechanism of Fundamental Particles in the Energy-Planck`s Confinement and their existence, http://science MPG ≡ Markos Georgallides [111] [M] The Polynomials , n-Knots Figures , Edge Points Vibration & M-Geometry . by http://science MPG ≡ Markos Georgallides [112] [M] An Way for Deceptioning the Normal OR the Dangerous Cells by http://science MPG ≡ Markos Georgallides [113] [M] The Origination of Universe from Energy ≡ Motion into the E-Geometry . Markos Georgallides comes from Cyprus and currently resides in the City Larnaca , Cyprus after being expelled from his Home Town Famagusta-Varosha , by The Barbaric Turks in August 1974. He works as a consultant Civil and Architect Engineer having his own Business . He is also The author of numerous scholarly Articles focusing on Euclidean and Material Geometry , and mathematical To Physics and Mechanics related Subjects . He obtained his degree from the Athens , National Technical , Polytechnic University [NATUA] Athens- Greece , and subsequently studied in Germany , Math Theory of Photoelasticity . IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Volume 08 | Issue 08 | August 2025 | https://ijojournals.com/index.php/m/index IJO JOURNALS http://science/ http://science/ http://science/ mailto:atul@scientificadvances.co.in mailto:ASRP.hspublishing@gmail.con mailto:editor@g http://science/ http://science/ http://science/ http://science/ http://science/ http://science/