Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Exact Rigidity Certificates for Kochen-Specker Sets, the Graded Flex Dictionary, and the Peres-Penrose Circle

Manuel Flores Gordillo

Published Jul 22, 2026• 0 citations
Publisher

Abstract

Within the program of identifying rigid Kochen-Specker (KS) sets (Xu-Saha-Bharti-Cabello, PRL 132, 140201; Trandafir-Cabello, PRA 111, 052204; Kernaghan, arXiv:2603.16988), this paper contributes exact-arithmetic certificates, one unification, and one correction. The two flexible islands of Kernaghan's landscape (Peres-33 and Z[sqrt(-2)]-33) are the two antipodes of a single exact circle of complex 33-ray KS sets, the closed form of the Gould-Aravind phase family. We certify flex=1 exactly at both islands, prove local completeness, and correct Kernaghan's Observation 28: the flex is finite, not merely infinitesimal. The circle contains no degenerations (proved exactly) and its Sum P = 11 I mechanism is resolved via hyperoctahedral orbit tight frames. Exact rigidity certificates across the full known census make Peres-Penrose-33 the unique flexible KS set. The Hermitian rigidity matrix block-diagonalizes at real points (flex_C = flex_R + flex_skew) with the graded cycle split proved, and a torsion layer is proved flex-invariant. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / LITERATURE / OPEN). The attached verification suite (35+ self-checking scripts) reproduces every certificate; companion hardware notebooks (IBM Quantum, two devices) confirm the witness-invariance prediction Sum P = 11 to 0.45% and measure the loop holonomy eigenphase to ~2%, consistent with the exact value 2*pi*sqrt(1867)/33. Note on methods: large-language-model assistants (Anthropic Claude) were used substantially (drafting, verification-script development, literature search, proof proposals); all mathematical claims are machine-verified by the attached suite, and the author takes full responsibility. Code and data: https://github.com/manuflog/contextuality-obstructions

DOI
10.5281/zenodo.21488473

Related papers

Geometry and arithmetic of the Kochen-Specker circle

Manuel Flores Gordillo

CORRECTION IN v3 (2026-07-28). Versions 1 and 2 cited the companion flagship paper as "Preprints.org 224406 (2026)". That citation was premature: the companion was submitted to Preprints.org in July 2026 but was not posted, so no public record under that identifier exists. The reference is corrected to the companion's Zenodo concept DOI, 10.5281/zenodo.21488473. Every self-citation across all papers in the program was audited at the same time, and stale or missing identifiers elsewhere were corrected. No mathematical claim in this paper depended on the erroneous reference. 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). The Gould-Aravind circle - the unique positive-dimensional flexible family of 33-ray Kochen-Specker sets, with Peres-33 and Penrose-33 as antipodes - is studied as a geometric object in its own right. Results: (1) an arithmetic classification of the circle's exact ring points (ring points exist iff 2 ramifies; exactly 8 special phases: Z[sqrt2] x2, Z[i] x4, Z[sqrt-2] x2; dimension 7 excluded by an exact denominator-4 obstruction); (2) the Berry/Wilczek-Zee holonomy of the KS loop: exact rational per-ray Berry phases summing to zero, trivial abelian holonomy, and the non-abelian eigenphase identified and proved in closed form, phi = 2*pi*sqrt(1867)/33 (rotating-frame reduction to a constant connection; 1867 = 43^2+3^2+3^2), with its section dependence characterized; (3) exact imaginarity profiles along the circle (Im v(theta) = sin(theta) y identically; total l1 imaginarity (8+20*sqrt2/3)|sin theta|); (4) hardware experiments on IBM Quantum devices: witness invariance confirmed on ibm_marrakesh (grand mean 11.049, +0.45% of the exact prediction 11, 297 circuits), and the holonomy eigenphase measured on two devices (Hadamard test and a constant-connection shortcut circuit), consistent with sqrt(1867)/33 to about 2%. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / LITERATURE / OPEN); each label is backed by a script in the attached verification suite (also public: https://github.com/manuflog/contextuality-obstructions/tree/main/rigidity, with the executed hardware notebooks in rigidity/hardware_results/). 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.

•Jul 22, 2026•0 citations

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

Manuel Flores Gordillo

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.

•Jul 22, 2026•0 citations

A second Kochen-Specker circle, in dimension four

Manuel Flores Gordillo

Fourth paper of the rigidity program (companions: https://doi.org/10.5281/zenodo.21488473, https://doi.org/10.5281/zenodo.21498342, https://doi.org/10.5281/zenodo.21498385). In 2010, Blanchfield asked how common parameterized generalisations are among Kochen-Specker sets, and found every d=4 set she examined fully determined by its orthogonalities; the only known continuous family lived in dimension 3 (the Gould-Aravind circle, whose uniqueness among two-symbol d=3 alphabets the companion papers prove). This paper answers her question in dimension four. Results: (1) an anchor theorem: every two-symbol d=4 mechanism-stable graph contains Peres-24 and is therefore KS-uncolorable, so the d=3 uniqueness argument structurally cannot run in d=4; (2) the M9 family: for the unimodular mechanism |x|^2=1, an 89-ray, 35-basis KS core with entries {0, +-1, +-e^(i theta)} whose 433 orthogonalities hold identically in theta, KS-uncolorable at every point of the circle, with exact flex = 1 - to our knowledge the first continuous fixed-graph KS family in dimension 4; (3) a complete triage of all 16 two-symbol d=4 mechanisms: exactly two carry families - the M9 circle (flex 1) and the M10 line Re(x)=0 (exact flex 2, 21 rays) - while the off-center-circle mechanisms are proved rigid by a Moebius-stability method; (4) the geometry of the M9 circle: its Wilczek-Zee connection is exactly constant, with eigenphase quartic irreducible over Q with Galois group S4 (so no closed form of the d=3 type 2*pi*sqrt(1867)/33 exists - an honest structural contrast), and det W = exp(2*pi*i*23/35), nontrivial where d=3's is trivial; (5) a structure card: all four conjugation-fixed points of the circle degenerate and are exactly rigid while generic points flex - the reverse of the d=3 pattern. 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). A final adversarial literature guard (including the Pavicic-Megill coordinatization program, the Waegell-Aravind catalogs, and the 2022 RMP review) found no prior continuous fixed-graph d=4 KS family; the claim is stated as first known. 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. v2 (2026-07-23): adds the refined reading of M10's flex (Remark 7.x): the second deformation is an exactly-integrable core-local twin-ray rotation, not portable to the full stable pool, so the mechanism-level flex of M10 is 1 = dim V; adds the conditional d=4 uniqueness results (decoupling certificates for all 8 rigid mechanisms; integer-pool sweep: 2082 exact flex computations, all critical {0,+-1}^4 cores rigid) and the guard-mandated citations (RMP 2022 review; Trandafir-Cabello rigid-sense disambiguation; Pavicic-Megill coordinatization distinguished; Waegell-Aravind catalogs). Verification suite updated with branch_d4unique.py, branch_intflex.py, branch_m10x.py. v3 (2026-07-25): structural pass on Sections 5-7. v4 (2026-07-27): the stale "SCAFFOLD DRAFT" banner and a stale Discussion paragraph removed; and, importantly, an attribution remark added -- the anchor property used here is NOT ours, it is due to Pavicic and co-authors (Pavicic/Waegell, Pavicic/Megill), and earlier versions did not say so. This deposit is document version v4, and the Zenodo version counter now reads v4 to match; do not read the two as independent.

•Jul 23, 2026•0 citations

The canonical class of the MASA context stack and the Peres-Mermin obstruction

Manuel Flores Gordillo

Quantum measurements need not reveal preexisting values: the outcome assigned to an observable can depend on which compatible observables are measured with it. This contextuality underlies the Peres-Mermin proof of the Kochen-Specker theorem and, in quantum computation with magic states, tracks the even/odd parity of the local dimension separating the classically simulable from the genuinely quantum. ATTRIBUTION: the even/odd value classification of state-independent qudit contextuality (the parity dichotomy and the fixed obstruction value d/2) is due to Abramsky, Cercelescu and Constantin, FSCD 2024; this paper supplies its geometric and cohomological ACCOUNT, not a rediscovery of it. Organizing all measurement contexts of M_n(C) into the stack [X/PU(n)] of maximal abelian subalgebras, we show the lifting class of U(1) -> U(n) -> PU(n) is vacuous on the action groupoid U(n) x X but, on PU(n) x X, is a canonical degree-2 class of order exactly n. Its real shadow, through the signed-permutation group B_n, is a Z_2 extension splitting if and only if n is odd, with "-I is a rotation square iff n is even" as the mechanism; at n=4 it restricts to the real two-qubit Pauli class, whose pairing with the Peres-Mermin word is -1, exactly the obstruction to a noncontextual value assignment. We compute H^2 = Z/n (+) Z_2 for n >= 4 and identify the canonical class as a generator of the Z/n summand. STATUS AND SCOPE. Every step is tagged Complete, Sketch (reduced to a named standard theorem whose hypotheses are verified), or Gap, and a status table closes the paper. The finite and algebraic core is machine-checked by scripts in the public suite. The geometric layer rests on standard but non-elementary facts about differentiable-stack cohomology and the Lyndon-Hochschild-Serre spectral sequence; that bookkeeping is reduced to named standard theorems and AWAITS SPECIALIST REVIEW. This is a preprint, not a peer-reviewed article, and should be read at that strength. The finite Weyl value spectrum is developed in a companion paper. Code, data and verification suite: https://github.com/manuflog/contextuality-obstructions Note on methods: large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance, and the author takes full responsibility for the results, including any errors.

•Jul 27, 2026•0 citations

The Algebraic Landscape of Kochen-Specker Sets in Dimension Three

Michael Kernaghan

We present a computational survey of Kochen-Specker (KS) uncolorability in three-dimensional Hilbert space across coordinate alphabets drawn from quadratic, cyclotomic, golden-ratio, and cubic Pisot number fields, principally the two-symbol family A = {0, +-1, +-x}. In every tested alphabet, KS sets arise only when x supports a cancellation identity: an algebraic relation making dot products vanish exactly. Three mechanisms occur: modulus-2 cancellation (|x|^2 = 2, with the integer case 1+1=2 as the degenerate additive instance), phase cancellation (a vanishing sum of unit-modulus terms, as in 1 + omega + omega^2 = 0), and minimal-polynomial cancellation, in which the defining relation of a cubic algebraic unit supplies the vanishing sum directly, as in the supergolden ratio psi^3 = psi^2 + 1. Generators with |x|^2 >= 3 that are not roots of unity produce orthogonal triples but not KS-uncolorability in our survey. This pattern explains why constructions cluster into at least eight discrete algebraic islands among the tested fields, two of them cubic. Two yield potentially new KS graph types: the Heegner-7 ring Z[(1+sqrt(-7))/2] (43 vectors) and the golden ratio field Q(phi) (52 vectors, revealed only by cross-product completion); Z[sqrt(-2)] gives a new realization of a known Peres-type graph. Using SAT-based bipartite KS-uncolorability we verify and extend the input counts of Trandafir and Cabello across all six non-cubic islands. The two-mechanism pattern reported in versions 1-8 of this paper is refuted by the supergolden island, which motivates the third mechanism; whether the three-mechanism classification is complete remains open.

quant-pharXiv:2603.16988•Mar 18, 2026•0 citations

A tower of Kochen-Specker circles: even-dimensional unimodular families and the maximal-Galois holonomy law

Manuel Flores Gordillo

What this paper does. It builds a tower of Kochen–Specker circles: continuous one-parameter families of KS sets in even dimensions 4, 6, 8, 10 and 12, all carried by the unimodular cancellation mechanism |x|2 = 1 over two-symbol coordinate alphabets. For each rung it computes the Wilczek–Zee holonomy of the KS loop and determines the Galois group of the holonomy's characteristic polynomial, which is the full symmetric group Sd at all five even rungs for cores with irreducible coupling. It also proves a trace law — |Tr S| = d/2 identically on the working stratum, hence det W(2π) = 1 — and assembles a p-cycle certificate route whose required prime exists at every even dimension. Attribution. The three-dimensional Peres/Penrose family originates with Gould and Aravind; this program's contribution there is the gauge reduction showing the moduli space is a circle rather than a 3-torus, which corrects their pairwise-inequivalence claim. The Kochen–Specker uncolorability of the integer {0,±1} ray pools is due to Pavičić and collaborators, not to this program; the paper records only an independent re-derivation and a mechanism-independence observation downstream of their result. The dimension-three two-symbol classification is Kernaghan's (arXiv:2603.16988). The Galois machinery is classical and is not claimed as new: Jordan's 1873 theorem, the block argument of Wielandt (Finite Permutation Groups, Thm. 13.9), the standard practice recorded by Isaacs and Zieschang, and Nagura's 1952 prime-interval theorem. Only the application of that machinery to these holonomy polynomials is this program's. Status and scope. This is a preprint, not a peer-reviewed article. Every claim in the paper carries an explicit evidence label (proved, exact, numerical, conjecture, literature, open), and the scope limits are stated rather than implied. In particular: the Sd law is established only at the rungs actually built, and the general even-dimensional statement is a conjecture; the hypothesis of irreducible coupling is necessary, and the paper exhibits the counterexamples that show it; existence of cores beyond d = 12 is not claimed constructively; and flex certificates have been run at d = 10 only, not at d = 8 or d = 12. All arithmetic is exact; no claim rests on floating-point evidence. Relation to arXiv:2603.16988. An earlier version of that survey reported the Peres/Penrose flex as infinitesimal only. In v8 its author retracts that statement in a labelled correction note and publishes the cause, a constraint-ordering error. The present paper describes the matter as settled, and does not treat v8 as independent confirmation of finiteness: v8's own continuation is double precision and its modular rank computation bounds flex only from above, as its author states. The finite-flexibility verdict rests on the exact certificates of the companion paper. Use of AI assistance. This work was produced with substantial assistance from AI language models, used for exact symbolic and combinatorial computation, for drafting, for literature search, and for adversarial review of the arguments. Every mathematical claim was independently recomputed, and an adversarial pre-publication audit was run against the manuscript. The author is responsible for the content. This disclosure is made unconditionally. Verification code. https://github.com/manuflog/contextuality-obstructions Version 2 (July 2026) — a theorem is corrected and a claim is withdrawn. Version 1 stated a “reachability law” asserting that the unimodular cancellation mechanism |x|²=1 is algebraically realizable in dimension d if and only if d is even and d ≥ 4, and concluded that a flexible KS circle of this kind can exist in every even dimension and in no odd dimension. The odd-exclusion is false and is withdrawn. The proof partitioned the terms of a vanishing sum by symbol type into four groups, {1}, {x}, {x*}, {N}. The alphabet also contains 0, and the zero terms were omitted from that partition; padding with zeros defeats the parity argument. Explicitly, in C&sup5; the rays u = (1, x, 0, 0, 0) and v = (1, −x, 0, 0, 0), both over {0, ±1, ±x}, have inner product 1 − |x|², which vanishes exactly when |x|² = 1 — the unimodular mechanism, in an odd dimension. What survives, and is now stated as the theorem, is the full-support form: a vanishing sum of m nonzero alphabet terms forcing |x|² = p/q independently of arg x exists iff m ≥ p+q and m ≡ p+q (mod 2), so the unimodular mechanism needs an even m ≥ 4. Since an overlap in C^d contributes one term per coordinate and zeros are permitted, the mechanism is available in every dimension d ≥ 4, of either parity. This is not a technicality about phrasing, and the correction is stated at length in the paper rather than absorbed. Every core built here from d = 6 upward is drawn from the support-four stratum, so its overlaps carry at most four nonzero terms whatever d is, and the identity actually in use is the four-term one available at all d ≥ 4. The parity law governs the full-support case only, which is not the case the tower is built from. Consequently the evenness of the rungs constructed here is no longer explained by this paper, and no obstruction to an odd rung is claimed. Why the constructed rungs are all even is now recorded as open. Everything else is unchanged: the cores at d = 4, 6, 8, 10, 12, the S_d Galois certificates, the trace law, and the anchor obstruction are unaffected, since none of them uses the withdrawn direction.

•Jul 29, 2026•0 citations
Within the program of identifying rigid Kochen-Specker (KS) sets (Xu-Saha-Bharti-Cabello, PRL 132, 140201; Trandafir-Cabello, PRA 111, 052204; Kernaghan, arXiv:2603.16988), this paper contributes exact-arithmetic certificates, one unification, and one correction. The two flexible islands of Kernaghan's landscape (Peres-33 and Z[sqrt(-2)]-33) are the two antipodes of a single exact circle of complex 33-ray KS sets, the closed form of the Gould-Aravind phase family. We certify flex=1 exactly at both islands, prove local completeness, and correct Kernaghan's Observation 28: the flex is finite, not merely infinitesimal. The circle contains no degenerations (proved exactly) and its Sum P = 11 I mechanism is resolved via hyperoctahedral orbit tight frames. Exact rigidity certificates across the full known census make Peres-Penrose-33 the unique flexible KS set. The Hermitian rigidity matrix block-diagonalizes at real points (flex_C = flex_R + flex_skew) with the graded cycle split proved, and a torsion layer is proved flex-invariant. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / LITERATURE / OPEN). The attached verification suite (35+ self-checking scripts) reproduces every certificate; companion hardware notebooks (IBM Quantum, two devices) confirm the witness-invariance prediction Sum P = 11 to 0.45% and measure the loop holonomy eigenphase to ~2%, consistent with the exact value 2pisqrt(1867)/33. Note on methods: large-language-model assistants (Anthropic Claude) were used substantially (drafting, verification-script development, literature search, proof proposals); all mathematical claims are machine-verified by the attached suite, and the author takes full responsibility. Code and data: https://github.com/manuflog/contextuality-obstructions
CORRECTION IN v3 (2026-07-28). Versions 1 and 2 cited the companion flagship paper as "Preprints.org 224406 (2026)". That citation was premature: the companion was submitted to Preprints.org in July 2026 but was not posted, so no public record under that identifier exists. The reference is corrected to the companion's Zenodo concept DOI, 10.5281/zenodo.21488473. Every self-citation across all papers in the program was audited at the same time, and stale or missing identifiers elsewhere were corrected. No mathematical claim in this paper depended on the erroneous reference. 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). The Gould-Aravind circle - the unique positive-dimensional flexible family of 33-ray Kochen-Specker sets, with Peres-33 and Penrose-33 as antipodes - is studied as a geometric object in its own right. Results: (1) an arithmetic classification of the circle's exact ring points (ring points exist iff 2 ramifies; exactly 8 special phases: Z[sqrt2] x2, Z[i] x4, Z[sqrt-2] x2; dimension 7 excluded by an exact denominator-4 obstruction); (2) the Berry/Wilczek-Zee holonomy of the KS loop: exact rational per-ray Berry phases summing to zero, trivial abelian holonomy, and the non-abelian eigenphase identified and proved in closed form, phi = 2*pi*sqrt(1867)/33 (rotating-frame reduction to a constant connection; 1867 = 43^2+3^2+3^2), with its section dependence characterized; (3) exact imaginarity profiles along the circle (Im v(theta) = sin(theta) y identically; total l1 imaginarity (8+20*sqrt2/3)|sin theta|); (4) hardware experiments on IBM Quantum devices: witness invariance confirmed on ibm_marrakesh (grand mean 11.049, +0.45% of the exact prediction 11, 297 circuits), and the holonomy eigenphase measured on two devices (Hadamard test and a constant-connection shortcut circuit), consistent with sqrt(1867)/33 to about 2%. Every claim carries an explicit evidence label (PROVED / EXACT / NUMERICAL / LITERATURE / OPEN); each label is backed by a script in the attached verification suite (also public: https://github.com/manuflog/contextuality-obstructions/tree/main/rigidity, with the executed hardware notebooks in rigidity/hardware_results/). 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.
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.
Fourth paper of the rigidity program (companions: https://doi.org/10.5281/zenodo.21488473, https://doi.org/10.5281/zenodo.21498342, https://doi.org/10.5281/zenodo.21498385). In 2010, Blanchfield asked how common parameterized generalisations are among Kochen-Specker sets, and found every d=4 set she examined fully determined by its orthogonalities; the only known continuous family lived in dimension 3 (the Gould-Aravind circle, whose uniqueness among two-symbol d=3 alphabets the companion papers prove). This paper answers her question in dimension four. Results: (1) an anchor theorem: every two-symbol d=4 mechanism-stable graph contains Peres-24 and is therefore KS-uncolorable, so the d=3 uniqueness argument structurally cannot run in d=4; (2) the M9 family: for the unimodular mechanism |x|^2=1, an 89-ray, 35-basis KS core with entries {0, +-1, +-e^(i theta)} whose 433 orthogonalities hold identically in theta, KS-uncolorable at every point of the circle, with exact flex = 1 - to our knowledge the first continuous fixed-graph KS family in dimension 4; (3) a complete triage of all 16 two-symbol d=4 mechanisms: exactly two carry families - the M9 circle (flex 1) and the M10 line Re(x)=0 (exact flex 2, 21 rays) - while the off-center-circle mechanisms are proved rigid by a Moebius-stability method; (4) the geometry of the M9 circle: its Wilczek-Zee connection is exactly constant, with eigenphase quartic irreducible over Q with Galois group S4 (so no closed form of the d=3 type 2*pi*sqrt(1867)/33 exists - an honest structural contrast), and det W = exp(2*pi*i*23/35), nontrivial where d=3's is trivial; (5) a structure card: all four conjugation-fixed points of the circle degenerate and are exactly rigid while generic points flex - the reverse of the d=3 pattern. 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). A final adversarial literature guard (including the Pavicic-Megill coordinatization program, the Waegell-Aravind catalogs, and the 2022 RMP review) found no prior continuous fixed-graph d=4 KS family; the claim is stated as first known. 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. v2 (2026-07-23): adds the refined reading of M10's flex (Remark 7.x): the second deformation is an exactly-integrable core-local twin-ray rotation, not portable to the full stable pool, so the mechanism-level flex of M10 is 1 = dim V; adds the conditional d=4 uniqueness results (decoupling certificates for all 8 rigid mechanisms; integer-pool sweep: 2082 exact flex computations, all critical {0,+-1}^4 cores rigid) and the guard-mandated citations (RMP 2022 review; Trandafir-Cabello rigid-sense disambiguation; Pavicic-Megill coordinatization distinguished; Waegell-Aravind catalogs). Verification suite updated with branch_d4unique.py, branch_intflex.py, branch_m10x.py. v3 (2026-07-25): structural pass on Sections 5-7. v4 (2026-07-27): the stale "SCAFFOLD DRAFT" banner and a stale Discussion paragraph removed; and, importantly, an attribution remark added -- the anchor property used here is NOT ours, it is due to Pavicic and co-authors (Pavicic/Waegell, Pavicic/Megill), and earlier versions did not say so. This deposit is document version v4, and the Zenodo version counter now reads v4 to match; do not read the two as independent.
Quantum measurements need not reveal preexisting values: the outcome assigned to an observable can depend on which compatible observables are measured with it. This contextuality underlies the Peres-Mermin proof of the Kochen-Specker theorem and, in quantum computation with magic states, tracks the even/odd parity of the local dimension separating the classically simulable from the genuinely quantum. ATTRIBUTION: the even/odd value classification of state-independent qudit contextuality (the parity dichotomy and the fixed obstruction value d/2) is due to Abramsky, Cercelescu and Constantin, FSCD 2024; this paper supplies its geometric and cohomological ACCOUNT, not a rediscovery of it. Organizing all measurement contexts of M_n(C) into the stack [X/PU(n)] of maximal abelian subalgebras, we show the lifting class of U(1) -> U(n) -> PU(n) is vacuous on the action groupoid U(n) x X but, on PU(n) x X, is a canonical degree-2 class of order exactly n. Its real shadow, through the signed-permutation group B_n, is a Z_2 extension splitting if and only if n is odd, with "-I is a rotation square iff n is even" as the mechanism; at n=4 it restricts to the real two-qubit Pauli class, whose pairing with the Peres-Mermin word is -1, exactly the obstruction to a noncontextual value assignment. We compute H^2 = Z/n (+) Z_2 for n >= 4 and identify the canonical class as a generator of the Z/n summand. STATUS AND SCOPE. Every step is tagged Complete, Sketch (reduced to a named standard theorem whose hypotheses are verified), or Gap, and a status table closes the paper. The finite and algebraic core is machine-checked by scripts in the public suite. The geometric layer rests on standard but non-elementary facts about differentiable-stack cohomology and the Lyndon-Hochschild-Serre spectral sequence; that bookkeeping is reduced to named standard theorems and AWAITS SPECIALIST REVIEW. This is a preprint, not a peer-reviewed article, and should be read at that strength. The finite Weyl value spectrum is developed in a companion paper. Code, data and verification suite: https://github.com/manuflog/contextuality-obstructions Note on methods: large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance, and the author takes full responsibility for the results, including any errors.
We present a computational survey of Kochen-Specker (KS) uncolorability in three-dimensional Hilbert space across coordinate alphabets drawn from quadratic, cyclotomic, golden-ratio, and cubic Pisot number fields, principally the two-symbol family A = {0, +-1, +-x}. In every tested alphabet, KS sets arise only when x supports a cancellation identity: an algebraic relation making dot products vanish exactly. Three mechanisms occur: modulus-2 cancellation (|x|^2 = 2, with the integer case 1+1=2 as the degenerate additive instance), phase cancellation (a vanishing sum of unit-modulus terms, as in 1 + omega + omega^2 = 0), and minimal-polynomial cancellation, in which the defining relation of a cubic algebraic unit supplies the vanishing sum directly, as in the supergolden ratio psi^3 = psi^2 + 1. Generators with |x|^2 >= 3 that are not roots of unity produce orthogonal triples but not KS-uncolorability in our survey. This pattern explains why constructions cluster into at least eight discrete algebraic islands among the tested fields, two of them cubic. Two yield potentially new KS graph types: the Heegner-7 ring Z[(1+sqrt(-7))/2] (43 vectors) and the golden ratio field Q(phi) (52 vectors, revealed only by cross-product completion); Z[sqrt(-2)] gives a new realization of a known Peres-type graph. Using SAT-based bipartite KS-uncolorability we verify and extend the input counts of Trandafir and Cabello across all six non-cubic islands. The two-mechanism pattern reported in versions 1-8 of this paper is refuted by the supergolden island, which motivates the third mechanism; whether the three-mechanism classification is complete remains open.
What this paper does. It builds a tower of Kochen–Specker circles: continuous one-parameter families of KS sets in even dimensions 4, 6, 8, 10 and 12, all carried by the unimodular cancellation mechanism |x|2 = 1 over two-symbol coordinate alphabets. For each rung it computes the Wilczek–Zee holonomy of the KS loop and determines the Galois group of the holonomy's characteristic polynomial, which is the full symmetric group Sd at all five even rungs for cores with irreducible coupling. It also proves a trace law — |Tr S| = d/2 identically on the working stratum, hence det W(2π) = 1 — and assembles a p-cycle certificate route whose required prime exists at every even dimension. Attribution. The three-dimensional Peres/Penrose family originates with Gould and Aravind; this program's contribution there is the gauge reduction showing the moduli space is a circle rather than a 3-torus, which corrects their pairwise-inequivalence claim. The Kochen–Specker uncolorability of the integer {0,±1} ray pools is due to Pavičić and collaborators, not to this program; the paper records only an independent re-derivation and a mechanism-independence observation downstream of their result. The dimension-three two-symbol classification is Kernaghan's (arXiv:2603.16988). The Galois machinery is classical and is not claimed as new: Jordan's 1873 theorem, the block argument of Wielandt (Finite Permutation Groups, Thm. 13.9), the standard practice recorded by Isaacs and Zieschang, and Nagura's 1952 prime-interval theorem. Only the application of that machinery to these holonomy polynomials is this program's. Status and scope. This is a preprint, not a peer-reviewed article. Every claim in the paper carries an explicit evidence label (proved, exact, numerical, conjecture, literature, open), and the scope limits are stated rather than implied. In particular: the Sd law is established only at the rungs actually built, and the general even-dimensional statement is a conjecture; the hypothesis of irreducible coupling is necessary, and the paper exhibits the counterexamples that show it; existence of cores beyond d = 12 is not claimed constructively; and flex certificates have been run at d = 10 only, not at d = 8 or d = 12. All arithmetic is exact; no claim rests on floating-point evidence. Relation to arXiv:2603.16988. An earlier version of that survey reported the Peres/Penrose flex as infinitesimal only. In v8 its author retracts that statement in a labelled correction note and publishes the cause, a constraint-ordering error. The present paper describes the matter as settled, and does not treat v8 as independent confirmation of finiteness: v8's own continuation is double precision and its modular rank computation bounds flex only from above, as its author states. The finite-flexibility verdict rests on the exact certificates of the companion paper. Use of AI assistance. This work was produced with substantial assistance from AI language models, used for exact symbolic and combinatorial computation, for drafting, for literature search, and for adversarial review of the arguments. Every mathematical claim was independently recomputed, and an adversarial pre-publication audit was run against the manuscript. The author is responsible for the content. This disclosure is made unconditionally. Verification code. https://github.com/manuflog/contextuality-obstructions Version 2 (July 2026) — a theorem is corrected and a claim is withdrawn. Version 1 stated a “reachability law” asserting that the unimodular cancellation mechanism |x|²=1 is algebraically realizable in dimension d if and only if d is even and d ≥ 4, and concluded that a flexible KS circle of this kind can exist in every even dimension and in no odd dimension. The odd-exclusion is false and is withdrawn. The proof partitioned the terms of a vanishing sum by symbol type into four groups, {1}, {x}, {x*}, {N}. The alphabet also contains 0, and the zero terms were omitted from that partition; padding with zeros defeats the parity argument. Explicitly, in C&sup5; the rays u = (1, x, 0, 0, 0) and v = (1, −x, 0, 0, 0), both over {0, ±1, ±x}, have inner product 1 − |x|², which vanishes exactly when |x|² = 1 — the unimodular mechanism, in an odd dimension. What survives, and is now stated as the theorem, is the full-support form: a vanishing sum of m nonzero alphabet terms forcing |x|² = p/q independently of arg x exists iff m ≥ p+q and m ≡ p+q (mod 2), so the unimodular mechanism needs an even m ≥ 4. Since an overlap in C^d contributes one term per coordinate and zeros are permitted, the mechanism is available in every dimension d ≥ 4, of either parity. This is not a technicality about phrasing, and the correction is stated at length in the paper rather than absorbed. Every core built here from d = 6 upward is drawn from the support-four stratum, so its overlaps carry at most four nonzero terms whatever d is, and the identity actually in use is the four-term one available at all d ≥ 4. The parity law governs the full-support case only, which is not the case the tower is built from. Consequently the evenness of the rungs constructed here is no longer explained by this paper, and no obstruction to an odd rung is claimed. Why the constructed rungs are all even is now recorded as open. Everything else is unchanged: the cores at d = 4, 6, 8, 10, 12, the S_d Galois certificates, the trace law, and the anchor obstruction are unaffected, since none of them uses the withdrawn direction.