The alphabet theorems: cancellation mechanisms, mechanism moduli, and the unique flexible family of two-symbol Kochen-Specker sets

Abstract

Companion paper to "Exact rigidity certificates for Kochen-Specker sets, the graded flex dictionary, and the Peres-Penrose circle" (https://doi.org/10.5281/zenodo.21488473). A complete algebraic theory of Kernaghan's two-symbol alphabet landscape (arXiv:2603.16988). Results: (1) the exhaustive 80-case d=3 classification: exactly FOUR independent cancellation mechanisms (rational, modulus-2, phase, golden), not two - a fifth candidate is proved identical to the phase mechanism under the alphabet's own x<->1/x gauge duality - and any generator avoiding all four provably yields a 3-colorable pool (structure theorem via Godsil-Zaks); (2) the d=4 extension: a term-alphabet lemma, 16 new mechanisms from the 280-case 4-term classification, and no colorable regime (Peres-24 anchors every pool); (3) the mechanism-moduli audit across the census, proving the forward flex bound in general; (4) the uniqueness theorem: a machine-certified KS-coloring of the full 145-ray M3-line-stable graph proves the Gould-Aravind circle is the unique positive-dimensional flexible family of two-symbol d=3 KS sets - including the honest retraction of a tempting arithmetic explanation that failed direct verification; (5) the M2 mirror: the maximal circle-stable graph is KS-uncolorable, compactness pins the M2 circle to exactly five integer-trace exact points, and the Gaussian points x=+-(1+i) carry a NEW 49-ray KS core (SAT-certified with a three-anchor-validated encoder, critical, and exactly rigid) distinct from all six census islands and from the survey's own Gaussian island (shown here to reduce to Peres-33). Flexibility tracks the circle's four conjugation-fixed points exactly (Hypothesis L), and a J-splitting construction makes the real/skew flex decomposition well-posed at conjugation-symmetric non-real points. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / CONJECTURE / OPEN), each backed by a script in the attached verification suite (also public: https://github.com/manuflog/contextuality-obstructions/tree/main/rigidity). Methods note: large-language-model assistants (Anthropic Claude) were used substantially in drafting and in developing the verification suite; every mathematical claim is machine-verified independently of that assistance, and the author takes full responsibility for the results. Version 4 (July 2026) — status of the Observation 28/29 point. This paper extends and in one place corrects Part II of Kernaghan's survey (arXiv:2603.16988). The specific disputed point, the reported non-persistence of the Peres/Penrose flex, was conceded by its author in v8 of that preprint, in a labelled correction note that also publishes the cause. References are updated to v8 and the passage now describes the matter as settled. As elsewhere in this series, v8 is not treated as independent confirmation of finiteness: its continuation is double precision and its modular rank computation bounds flex only from above, as its author states. Version 5 (July 2026) — a proof step is withdrawn and replaced. The theorem is unchanged. Versions up to v4 justified the last step of the structure theorem (necessity of a mechanism for the raw pool) by invoking Godsil–Zaks, paraphrased as “rational-coordinate projective ray sets in R^n are always 3-colorable”, and concluded that the x=3 graph is 3-colorable. Both halves are false and are withdrawn. The paraphrase is false: the x=2 raw pool is a rational-coordinate ray set (49 rays, 138 orthogonal pairs) with chromatic number 4. The application is false at the pool it was applied to: the x=3 graph is not 3-colorable either, also chromatic number 4. Godsil–Zaks concerns rational points of the unit sphere, and a rational ray such as (1,1,1) does not normalize to one, so the theorem never applied to these pools at all. The theorem itself is unaffected. Graph 3-colorability was only ever a sufficient route to KS-colorability, and KS-colorability at x=3 is established directly by exhaustive certificate — 31488 distinct valid KS colorings, recomputed independently for this version. Both chromatic numbers were verified twice, by DSATUR backtracking and by SAT (Glucose3). An earlier section of the paper had already retracted this same invocation for the Re(x) = -1/2 line; that retraction should have been applied to the structure theorem at the same time and was not. It is now. Also corrected: earlier versions described cross-product completion as “the closure step that rescues the Golden and CK-31 islands from initial colorability”. That is wrong for CK-31, whose raw pool at x=2 is already KS-uncolorable (49 rays, 138 pairs, 26 triads, no valid coloring). Completion enlarges it but is not needed for uncolorability. Only the Golden island is genuinely rescued by completion. Version 6 (July 2026) — a theorem is scoped, its closed-pool reading withdrawn, and the resolution identified. The structure theorem (“no mechanism, no KS set”) was stated through v5 without an alphabet hypothesis. For Hermitian-closed complex pools it is FALSE, and the cause is a term the classification missed: with x* as an entry, (x*)*·x = x² appears, so the closed term set is {±1, ±x, ±x*, ±N, ±x², ±(x*)²}. Counterexample, verified twice in independent exact implementations: x = √3 + i avoids every locus M1–M5, yet its closed pool (145 rays / 390 pairs / 30 triads) is KS-uncolorable; unclosed it is colorable. The identity x² + (x*)² = N holds along the whole lines arg x ≡ ±30° (mod 180°), at every modulus — a positive-dimensional uncolorable locus outside the classification. The resolution, also in this version: the new lines carry no new mechanism at core level and no new flexible family. Along them the ratio x/x* is a primitive sixth root of unity at every modulus, and projective rescaling identifies the lines’ uncolorable content with the known Eisenstein island: the line-stable graph is constant in the modulus away from a finite exceptional set, all 23 of its minimal uncolorable cores (complete enumeration) consist of modulus-independent rays, every core has mechanism flex exactly 0, and all 43 rays of the certified core embed projectively into the closed ω-pool. The pool’s one genuine deformation (flex exactly 1) lives entirely on colorable decoration rays. The uniqueness theorem — the Gould–Aravind circle as the unique positive-dimensional flexible family, scoped to M1–M4 — is unaffected. A complete x²-extended classification of closed-pool identities is recorded as open. The theorem is restated with the unclosed-alphabet hypothesis (under which it is true, and true for all real x); the term-alphabet lemma, correct as literally stated, now carries its scope explicitly. Found by the program’s own adversarial verification gate; every number above is from exact arithmetic with SAT and backtracking engines agreeing. Version 7 (July 2026) — the open problem left by v6 is closed, and the two-mechanism dichotomy is restored. v6 corrected a scope error (with x* as an entry the product x² appears, so the published 80-case classification is incomplete for Hermitian-closed pools) and recorded the complete x²-extended classification as open. This version carries it out. The classification. The closed term set has six types, giving 224 three-term cases; 126 are impossible for every nonzero x, and the rest reduce to the published loci M1–M5 together with exactly four new positive-dimensional loci — x²+(x*)² = ±N (the lines arg x = ±30° and ±60°) and x²+(x*)² = ±1 (two rectangular hyperbolae) — plus eight isolated points, all of which lie on the 60° line. Restricted to the 80 published cases the same computation returns exactly the 54 impossible cases and exactly M1–M5, reproducing the original theorem. The corrected dichotomy, now a theorem rather than an open problem. The closed raw pool is KS-uncolorable if and only if (A) |x|² ∈ {2, 1/2} or x ∈ {±2, ±1/2}, or (B′) x/x* is a primitive third or sixth root of unity. Established by an audit of all 58 realizable locus-membership classes, using the fact that the index-labelled closed graph depends only on which loci x belongs to. Why this is deflationary, and the point of the version. A ratio lemma explains the new uncolorable loci rather than adding a mechanism: projective rescaling by 1/x* carries the degree-one sub-pool isomorphically onto the pool over {0, ±1, ±r} with r = x/x*, multiplying every inner product by 1/N and so preserving orthogonality exactly. Since |r| = 1, the uncolorable cases are precisely those where r is a primitive third or sixth root of unity, and there the sub-pool is the unclosed Eisenstein pool at every modulus. The two hyperbola loci are genuinely new identities but carry no uncolorable content at any point tested; the arg = 45° locus, where r = i has order four and the pool is colorable, shows that being a root of unity is necessary but not sufficient. Consequence for the two-mechanism reading. Hermitian closure does not break the dichotomy reported in arXiv:2603.16988; it relocates one half of it. The literal rule “|x|² ≥ 3 and x not a root of unity implies colorable” fails — x = √3 + i is a counterexample — but that x has x/x* = e^{iπ/3}, and the corrected statement is again exactly two mechanisms: modulus, acting on x; phase, acting on x/x*. We read this as identifying the correct invariant for the phase mechanism rather than as a counterexample to the pattern. Verification scripts x2class_derive.py, x2class_pools.py, x2class_scan.py, x2class_deflate.py and x2class_closure.py accompany this version in the public repository; every colorability verdict was decided by SAT and by an independent backtracker in agreement, in exact arithmetic in the appropriate number field.