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The Spin-7 Lattice Model: Phase Transitions and Statistical Mechanics of the GTE Polynomial

Nova Spivack

Published Jun 9, 2026• 0 citations
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Abstract

Programme status: This record belongs to an active research programme. It is not a finished or final theory. The framework is currently undergoing a systematic audit, together with corrections and improvements prompted by reader feedback. Several significant issues have been identified, and some claims, derivations and interpretations in the current version may be revised, qualified or withdrawn in future versions. Revised versions will be released when this work is complete. See PROGRAM_STATUS.md. Status as of October 2026. The polynomial p(L,C,R) = C + R - CR - LCR over F_7, derived as the unique MDL-minimal substrate encoding the Standard Model from first principles, defines a natural statistical mechanics system: a 7-state nearest-neighbor spin model with three-body interactions. We analyze this spin-7 model in one and two spatial dimensions. In 1D, the exact transfer matrix acts on pair states (s_i-1, s_i); its Perron eigenvalue at = 1 is _1 = 1.4846, the correlation length grows from 2.9 sites at = 1 to 15.0 at = 2 and diverges as , and the local zero-energy windows number 43 = _6(7) (consistent with the Cyclotomic Point Count Theorem). The global zero-energy rings are exactly the three uniform configurations \0^n, 1^n, 5^n\ at every ring length n 3 - an exact threefold ground-space degeneracy, machine-certified in Lean 4 - so the ground space realizes the ternary repetition code of length n and distance n. In 2D, Metropolis Monte Carlo simulations on L L tori (L = 4-16) reveal a phase transition at _c 0.35 (T_c 2.86\,J) with Z_3-like symmetry breaking among three degenerate global ground states: the uniform configurations \0, 1, 5\, algebraically characterized as the roots of p(x,x,x) = -x(x-1)(x-5) 7. A secondary feature at _c,2 1.7 exhibits strong hysteresis consistent with domain-wall locking. The zero-temperature support of the 1D transfer matrix is the deterministic pair-state digraph of the CMCA vacuum sector, and the thermodynamic entropy rate S_CMCA() = _1() interpolates from 7 = 1.946 nats at infinite temperature through 0.395 nats per site per update at = 1 to 0 at zero temperature, where only the three rigid ground states survive. The three ground-state sectors \0, 1, 5\ map to the color charge sectors of the Standard Model, and the NL identity NL(p) = 1 - c_H/7^2 connects the polynomial's cryptographic nonlinearity to the Higgs branch capacity c_H = 13. A minimal experimental realization using 7-level Rydberg atoms in optical tweezer arrays is proposed.

DOI
10.5281/zenodo.20603066

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Programme status: This record belongs to an active research programme. It is not a finished or final theory. The framework is currently undergoing a systematic audit, together with corrections and improvements prompted by reader feedback. Several significant issues have been identified, and some claims, derivations and interpretations in the current version may be revised, qualified or withdrawn in future versions. Revised versions will be released when this work is complete. See PROGRAM_STATUS.md. Status as of October 2026. We apply 't Hooft's cellular automaton interpretation of quantum mechanics to the GTE cellular automaton f_MDL on Z_7^5, the 16,807-state visible-sector ring encoding the Standard Model generation structure. The complete orbit decomposition of f_MDL () reveals a single cycle: the vacuum fixed point (0,0,0,0,0) with E=0. The physical Hilbert space is one-dimensional; f_MDL belongs to 't Hooft's information-loss regime. SM generation states are transients with tail lengths (gen_1)=3 > (gen_2)=2 > (gen_3)=1, matching the empirical stability hierarchy (). A direct eigenvalue-to-mass correspondence is falsified on every formulation tested (). The Two-Role Structural Principle () separates roles: the cogwheel construction provides Hilbert space structure, unitarity, and the Born rule (, Lean-certified); the GTE N_eff cascade provides the mass spectrum. The seven winding-equivalence classes of Z_7^5 (, gauge\_arithmetic\_identification) are identified as 't Hooft's information-equivalence gauge classes (): the five SM winding sectors \0,2,3,4,6\ yield U(1)_EM, SU(2)_L, and SU(3)_c; sector W=1 predicts the SU(5) Y-leptoquark; the X-leptoquark (Q=+4/3) cohabits W=4. The AFCA causal graph is update-schedule-independent (Lean-certified); discrete Minkowski causal cones are embedded with coordinate surjection certified. Full Minkowski bijection remains open ().
Programme status: This record belongs to an active research programme. It is not a finished or final theory. The framework is currently undergoing a systematic audit, together with corrections and improvements prompted by reader feedback. Several significant issues have been identified, and some claims, derivations and interpretations in the current version may be revised, qualified or withdrawn in future versions. Revised versions will be released when this work is complete. See PROGRAM_STATUS.md. Status as of October 2026. The quantum measurement problem - why a superposition produces a single definite outcome, and why with probability |c_k|^2 - remains unresolved across standard interpretations of quantum mechanics, which either postulate collapse, multiply unobservable branches, or explain apparent classicality without resolving outcomes. This paper provides a formal derivation of the measurement mechanism from first principles within the GTE (Generative Triple Evolution) framework. The GTE polynomial p(L,C,R) = C+R-CR-LCR over 7, the unique Minimum Description Length (MDL)-minimal encoding of the Standard Model generation structure, carries a complete formal certificate of the quantum measurement mechanism. Its Perfect Self-Containment (PSC) projection is , the discrete adjudicator that defines the measurement orbit GEN_1 GEN_2 GEN_3 VAC - the only PSC-consistent trajectory in the cellular automaton dynamics - and this projection is characterized in three parts: agrees with p on the binary sector \0,1\^3 (Rule 110), overrides p on the ten SM-orbit neighborhoods to enforce generation-orbit consistency, and projects to zero on all remaining 325 neighborhoods. The polynomial's real diagonal encodes the Self-Referential Renormalization Group (SRRG) adjudication fixed point phi = (5-1)/2 0.618 (machine-certified, ). A Three-Level MDL Unification Theorem () shows that (X) = _x X K(x PSC) operates at three nested, non-circular domains: theory selection (p from rule space), field dynamics ( via the PMDL variational principle), and quantum event adjudication (Born rule via transputation). The polynomial p is the connecting object: it is the output of Level 1, enters the PMDL action at Level 2, and its PSC-projection defines the orbit used at Level 3. A two-sided computability classification (; hardness core ) locates transputation exactly: record-truth on the diagonal fragment is ^0_1-complete, the adjudicator is limit-computable with boundedly many mind changes under a named premise bundle, and hence _T(diag) = 0' exactly - the Gold-Putnam trial-and-error stratum, with no computable convergence modulus. Quantum measurement is trial-and-error computation at the halting degree, and no stronger: physics requires exactly one Turing jump. The same degree-0' structure reappears at the field level: the irreducible residual of the PMDL action is a left-c.e.\ real of degree exactly 0', so the cosmological constant inherits the transputation classification. Differential Self-Adjudicative Computation (DSAC) is positioned as the continuum member of the [D] adjudicator class, bridging the discrete projection and the transputation selector _D: its relaxation-to-coherence dynamics is the physical realization of the computable stage sequence whose limit is the adjudication outcome.
Programme status: This record belongs to an active research programme. It is not a finished or final theory. The framework is currently undergoing a systematic audit, together with corrections and improvements prompted by reader feedback. Several significant issues have been identified, and some claims, derivations and interpretations in the current version may be revised, qualified or withdrawn in future versions. Revised versions will be released when this work is complete. See PROGRAM_STATUS.md. Status as of October 2026. This tutorial explains one of the most surprising results in the GTE framework: quantum measurement has a precise location in the mathematical hierarchy of computational difficulty. It sits at exactly one level above an ordinary Turing machine — no more, no less. That level is Turing degree 0' (zero-prime), the halting problem. This is a theorem machine-certified in Lean 4 with zero sorry, not a metaphor. The result, called Transputation, shows that the Born-rule probability update during quantum measurement is exactly as hard to compute as the halting problem: it is decidable given an oracle for the halting problem, but not by any ordinary Turing machine. This tutorial explains the computability classification, the GTE derivation that places measurement at degree 0', and the physical interpretation of why quantum indeterminacy sits at precisely this level of the arithmetical hierarchy.