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Pitkänen, M., Combinatorial Hierarchy: Two Decades Later

Exploration

Combinatorial Hierarchy: Two Decades Later

Matti Pitkänen 1

Abstract

Combinatorial Hierarchy (CH) is a hierarchy consisting of Mersenne integers M(n) = MM(n−1) =

2M(n−1) − 1 and starting from M1 = 2. The first members of the hierarchy are given by 2, 3, 7, 127,
M127 = 2127 − 1 and are primes. The conjecture of Catalan is that the hierarchy continues to some
finite prime. It was proposed by Peter Noyes and Ted Bastin that the first levels of hierarchy up to
M127 are important physically and correspond to various interactions. I have proposed the levels of
CH define a hierarchy of codes containing genetic code corresponding to M7 and also memetic code
assignable to M127. In this article I consider the argument that the hierarchy ends at M127 and find
that it should end already at M7 for which the condition used saturates and which corresponds to
genetic code in TGD interpretation. The failure of condition at M127 level has interesting ””Gödelian
interpretation”. I find also that in TGD Universe genetic code and its memetic counterpart are realized
at the level of fundamental particles. Already earlier I have ended up with alternative realizations at
the level of dark nucleons and sequences of 3 dark nucleons.

1 Introduction

Combinatorial Hierarchy (CH) [1, 2] is a hierarchy consisting of Mersenne integers M(n) = MM(n−1) =

2M(n−1)−1 and starting from M1 = 2. The first members of the hierarchy are given by 2, 3, 7, 127,M127 =
2127−1 and are primes. The conjecture of Catalan is that the hierarchy continues to some finite prime. It
was proposed by Peter Noyes and Ted Bastin that the first levels of hierarchy up to M127 are important
physically and correspond to various interactions (see http://tinyurl.com/hszo9wb). I have proposed
the levels of CH define a hierarchy of codes containing genetic code corresponding to M7 and also memetic
code assignable to M127 [3].

Pierre Noyes and Ted Bastin proposed also an argument why CH contains only the levels mentioned
above. This has not been part of TGD view about CH: instead of this argument I have considered the
possibility that CH does not extend beyond M127. With the inspiration coming from email discussion I
tried to understand the argument stating that CH contains M127 as the highest level and ended up with a
possible interpretation of the condition. Zero energy ontology (ZEO) and the representation of quantum
Boolean statements A→ B as fermionic parts of positive and negative energy parts of zero energy states
is essential. This led to several interesting new results.

1. To my best understanding the original argument of Noyes does not allow M127 level whereas prime
property allows. States at M127 level cannot be mapped to zero energy states at M7 level. Allowing
a wild association with Gödel’s theorem, one could say that that there is hube number of truths at
M127 level not realizable as theorems at M7 level.

A possible interpretation is that M127 level corresponds to next level in the abstraction hierarchy
defined by CH and to the transition from imbedding space level to the level of ””world of classical
worlds” (WCW) in TGD. The possible non-existence of higher levels (perhaps implied if MM127

is
not prime) could be perhaps interpreted by saying that there is no ””world of WCWs”!

2. Rather remarkably, for M7, which corresponds to genetic code [3], the inequality serving as consis-
tency condition is saturated. One can say that any set of 64 mutually consistent statements at M7

1Correspondence: Matti Pitkänen http://tgdtheory.com/. Address: Köydenpunojankatu 2 D 11 10940, Hanko, Finland.
Email: matpitka@luukku.com.

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http://tinyurl.com/hszo9wb
http://tgdtheory.com/
mailto:matpitka@luukku.com


DNA Decipher Journal | December 2016 | Volume 6 | Issue 3 | pp. 170-175 171

Pitkänen, M., Combinatorial Hierarchy: Two Decades Later

level can be represented in terms of 64 Boolean maps at M3 level representable in terms of zero
energy states. One obtains an explicit identification for the Boolean algebras involved in terms of
spin and isospin states of fermions in TGD framework at level M7 so that genetic code seems to be
realized at the fundamental elementary particle level thanks to the dimension D = 8 of imbedding
space. Even more, the level M127 corresponding to memetic code emerges in the second quantization
of fermions at M7 level. Here color triplet property of quarks and color singletness of leptons and
the identification of elementary particles as pairs of wormhole contacts are in essential role.

The conclusion would be that in TGD Universe genetic code and its memetic counterpart are realized
at the level of fundamental particles. Already earlier I have ended up with alternative realizations at the
level of dark nucleons and sequences of 3 dark nucleons [L1].

2 Summary of Combinatorial Hierarchy

I summarize first the basics of CH.

1. One considers the space algebra of Boolean statements of n bits which can be also extended to
complex linear space -quantum Boolean algebra. One can give it linear structure as Z2 algebra
for binary coefficients with Z2 sum having set theoretic interpretation. This linear space has some
basis. That the coefficient field for linear structure is Z2 does not seem to be absolutely essential.
In TGD framework one considers the linear space defined by quantum Boolean algebra with qubit
interpretation generated by fermionic oscillator operators: one operator for every bit.

2. One assigns to the linear n-D space the n2-D space of linear maps of it to itself. One can also
consider the space of maps of quantum Boolean algebra to itself and also require that this defines a
Boolean homomorphism. Dimensions would be the same: only coefficient field would be different.

3. To CH level, which corresponds to Mersenne prime M(n) = MM(n−1) (n = 2, 3, 7, 127, 2127 − 1, ...)
one assigns vector space with dimension

D(n− 1) = [(M(n− 1) + 1)]2 ,

and requires that the space formed by

D1(n) =
(M(n) + 1))

2

bit sequences, which represent a subset of mutually consistent Boolean statements as subset of
M(n) + 1 bit sequences are representable as a subset of bit sequences with D(n − 1) bits. This
demands

D1(n) ≤ D(n− 1)

giving

M(n) + 1)

2
≤ [M(n− 1) + 1]2 .

4. This criterion is satisfied for the primes of CH up to M7 but not for M127: 2127 − 1 > 1282 so that
M127 should not included if I have understood the criterion correctly.

For M7 = 27−1 = 127 one obtains the condition 26 = 64 ≤ 8×8 = 64 so that condition is saturated.
Remarkably, 64 is the number of DNA codons!

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DNA Decipher Journal | December 2016 | Volume 6 | Issue 3 | pp. 170-175 172

Pitkänen, M., Combinatorial Hierarchy: Two Decades Later

5. The numbers of CH are also known as Catalan Mersenne numbers. Catalan Mersenne primes
are special case of double Mersenne primes MMn

(see http://tinyurl.com/j4tqwch). Catalan
conjecture that Catalan Mersennes are primes up to some limit. After the first non-prime the
remaining Catalan Mersenne numbers are necessarily composite. The known double Mersennes
are given by MMp

: p = 2, 3, 5, 7. No other cases are known. These primes are good candidates
for labelling scaled up variants of say hadron physics. To my opinion Catalan criterion is more
plausible.

6. Classical number fields are in key role in TGD [9, 10, 11] and have dimensions D = 1, 2, 4, 8. Also
CH involves these dimensions. D(n− 1) = M(n− 1) + 1 giving dimensions 2, 4, 8 for M2,M3,M7.
For M127 one would obtain D = 128, which does not correspond to any division algebra. This might
relate to the above observation.

3 CH as a prediction of quantum TGD

In the following the interpretation of Boolean map in ZEO is proposed. Also it is shown that M7 level
allows a natural realization in terms of spin-isospin states of fermions and that M127 level is obtained in
second quantization meaning going from the level of imbedding space to the level of WCW.

3.1 Interpretation of the lower level Boolean map in terms of ZEO

One can ask, why one should have this kind of map? One interpretation is that the space of Boolean
statements at given level is imbeddable to the space of quantum Boolean maps at previous level. Quantum
Boolean maps would represent Boolean rules A→ B, ””theorems” or ””laws of physics”.

1. In TGD framework the interpretation of CH would be as a hierarchy of statements about statements
about... The number of statements about N statements is indeed 2N . One statement corresponding
to all bits equal to 0 (in set theoretic realization empty set) is thrown away so that one has 2N − 1
statements instead of 2N .

2. ZEO means that physical states are pairs of states with opposite conserved quantum numbers: they
correspond to physical events, which replace states as fundamental entities in ZEO. The fermionic
parts of positive and negative energy parts of states would be pairs of many-fermion states allowing
interpretation as elements of quantum Boolean algebra. Zero energy states themselves would corre-
spond to pairs of these fermionic states and thus to ”theorems” A→ B or maps from Boolean algebra
to itself. The allowed statement pairs would satisfy fermion number conservation and conservation
of various quantum numbers and would indeed represent laws of physics.

3. A possible interpretation of the map would be that the statements at given level M(n+ 1) must be
representable as theorems at previous level M(n). For M(n) > M7 = 127 this would not hold true
anymore. Could this have some deep mathematical meaning as the wild association with Goedel’s
theorem suggests?

In the model of genetic code and its generalizations [3] I have proposed that each level of CH defines
a maximal number of mutually consistent statements identifiable as ””axioms”: the number is 2n−1

for 2n n-bit statements. For M7 = 127 the number is 64, the number of DNA codons, which would
thus have interpretation as axioms or ””fundamental truths”. In this case the representability would
still hold and map would be bijection. At the next level one would have ””memetic code” with 2126

codons representable as sequences of 21 DNA codons with stop codon included (126 = 21×6). By the
proposed criterion, at memetic level only vanishingly small subset of truths would be representable
as theorems at genetic level.

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DNA Decipher Journal | December 2016 | Volume 6 | Issue 3 | pp. 170-175 173

Pitkänen, M., Combinatorial Hierarchy: Two Decades Later

3.2 Representation of M7 level in TGD framework

Could the saturation for M7 have some physical meaning? The maps would be from 8-D space to itself.

1. Bits can be represented in terms of spin and electroweak spin giving 2×2 = 4 states and imbedding
space-spinors (H = M4 × CP2) of given H-chirality (quark or lepton like), given fermion number
(fermion or antifermion) and physical helicity. If also unphysical helicities with fixed fermion number
are allowed one would have 4 + 4 = 8 states. The condition that helicity is physical would reduce
the number of states by one half. This applies to both quarks and leptons since color is not spin
like quantum number in TGD (colored states correspond to partial waves in CP2).

2. What could be the interpretation for 27 − 1 = 127 states containing as subset n = 26 states. Could
n = 26 correspond to the number of states in the tensor product formed by pairs of 8 leptons and 8
antileptons allowed to have also unphysical polarizations? Same would apply to quarks. Allowing
both quark-antiquark and lepton-antilepton type states one would have 128 states. The physicality
condition for boson polarizations could drop the number of states to 64. What the dropping of one
state would correspond: to the dropping of νR− νR pair having no electroweak and color couplings
perhaps?

One can imagine two alternative identifications for the two tensor factors.

(a) In TGD framework fundamental bosons correspond to fermion antifermion pairs with members
at opposite throats of wormhole contact connecting two space-time sheets. Could the genetic
code correspond to 64 elementary bosons with physical polarizations and the maps to those
assigning to 8 fermions 8 antifermions?

(b) An alternative identification is suggested by ZEO. The tensor product of fermionic Boolean al-
gebras at opposite boundaries of causal diamond (CD) would replace that at opposite wormhole
throats. This would in accordance with the interpretation of zero energy states as statements
A→ B represented as Boolean maps.

3.3 Representation of M127 level in TGD framework

What about the physical interpretation of M127 level in TGD framework?

1. The first thing to observe is that physically p = M127 corresponds in TGD to the p-adic prime p
characterizing electron in p-adic mass calculations: Compton length is proportional to the p-adic
length scale and thus proportional to

√
p. The remaining Mersenne primes correspond to completely

super-astrophysical Compton lengths. Hence M127 has a very special role. The Mersenne primes
3, 7, 31, 127 giving rise to double Mersenne primes correspond to extremely short p-adic length scales.

Recall that the ratio of mCP2
/me is approximately mCP2

/me = 2127/2/
√

5 + x, where x ∈ [0, 1]
characterizes the second order contribution to electron mass from p-adic mass calculations [5]. The
ratio of Planck mass to proton mass equals to mPl/mp = 1.307 × 1019. For x = 0 this gives
mPl/mCP2 = (mp/me)× 3.96 = 7.271× 103, which is not far from 213 ' 8.912× 103. The value of
213 is very attractive number theoretically and would be obtained for x = .5, again power of 2.

2. The states at this level should correspond to statements about statements at the lower level repre-
sented in terms of quark lepton state space as many-fermion states assignable to wormhole throat
or several wormhole throats (elementary corresponds to two wormhole contants and 4 wormhole
throats). The construction of infinite primes can be interpreted as a process of forming repeatedly
statements about statements and the physical analog is repeated second quantization [8].

In the recent situation second quantization would correspond to the formation of many-fermion
states at partonic 2-surfaces defined by the throats of wormhole contacts. This would automatically
give rise to M127 states if one has 127 single fermion states to begin with.

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DNA Decipher Journal | December 2016 | Volume 6 | Issue 3 | pp. 170-175 174

Pitkänen, M., Combinatorial Hierarchy: Two Decades Later

Physically this step would correspond to a step from the spinor modes of imbedding space to the
spinor modes of WCW identifiable as fermionic Fock states assignable to partonic 2-surfaces so that
indeed a huge abstraction is in question. I have proposed that anyonic states could be this kind of
states for large value of heff = n×h implying that the size of wormhole throat becomes nano-scopic
[7].

3. One has 127 boson states but how to obtain 127 (or 128 = 27) single fermion states? Counting only
spin and weak isospin gives n = 8 + 8 = 24 (n = 4 + 4 = 23) single fermion states if one allows
(does not allow) also unphysical polarizations. The simplest option is that each single fermion state
has 23 (24) additional states. The location of fermion at one of the 4 wormhole throat could give
4 additional degrees of freedom. This would leave 2 (4) additional states per fermion state still
missing.

4. A good guess is that quark color realized as color partial waves comes in rescue and gives the needed
states. Light quarks must move in color triplet states and leptons in singlet states. Thefore quarks
have 3 × 8 = 24 modes and leptons 8 modes giving altogether 32 modes altogether. There are 4
wormhole throats so that 4× 32 = 128 modes are obtained and if right-handed neutrino is thrown
out one has 127 states as required if no constraints on polarizations are posed. It therefore seems
that TGD physics codes CH naturally at elementary particle level!

There is indeed a rich set of ””vibrational” degrees of freedom giving also rise to color degrees of
freedom. The symplectic group of ∆M4

± assignable to either boundary of causal diamond (CD)
defined as the intersection of future and past directed light-cones of M4 with points replaced with
CP2 gives rise to products of S2 and CP2 partial waves. Besides this there is a conformal weight
labelling the states correlating with S2×CP2 partial wave Light quarks massless before massivation
by p-adic thermodynamics move in color partial waves and color triplets are obtained as the color
excitations for them corresponding to higher conformal weights and having CP2 mass as mass scale.

I have already earlier ended up with the proposal that genetic code is realized at the level of dark
nuclear physics. Either the states of dark proton or sequence of 3 protons could be organized naturally
states corresponding to 64 DNAs, 64 RNAs, 20 aminoacids, and 40 tRNAs and vertebrate genetic code
follows from very simple assumption that opposite spins are paired [6, 4] [L1] (see http://tinyurl.com/

jgfjlbe). These findings suggest that genetic code and memetic code are also realized at the elementary
particle level.

Acknowledgements: I am gratetul for James Bowery for raising the question about the possible
relevance of CH for TGD.

References

[1] Bastin T et al. 7:445, 1979.

[2] Noyes P. The combinatorial hierarchy - an approach to open evolution. Available at: http://

tinyurl.com/hszo9wb, 1980.

[3] Pitkänen M. Genes and Memes. In Genes and Memes. Onlinebook. Available at: http:

//tgdtheory.fi/public_html/genememe/genememe.html#genememec, 2006.

[4] Pitkänen M. Homeopathy in Many-Sheeted Space-Time. In Bio-Systems as Conscious Holo-
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[5] Pitkänen M. Massless states and particle massivation. In p-Adic Physics. Onlinebook. Available at:
http://tgdtheory.fi/public_html/padphys/padphys.html#mless, 2006.

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Published by QuantumDream, Inc.

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[11] Pitkänen M. From Principles to Diagrams. Onlinebook.Available at: http://tgdtheory.fi/

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[L1] Pitkänen M. About Physical Representations of Genetic Code in Terms of Dark Nuclear Strings.
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ISSN: 2159-046X DNA Decipher Journal www.www.dnadecipher.com

Published by QuantumDream, Inc.

http://tgdtheory.fi/public_html/neuplanck/neuplanck.html#nuclstring
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	Introduction
	Summary of Combinatorial Hierarchy
	CH as a prediction of quantum TGD
	Interpretation of the lower level Boolean map in terms of ZEO
	Representation of M7 level in TGD framework
	Representation of M127 level in TGD framework


