Exact contextuality gaps for graphs up to eleven vertices: certified maximizers and the algebraic degrees of the landscape

Abstract

New in v2.5. At eight vertices three independent enumerations — the per-graph file published with arXiv:2605.12828, the 2012 Amselem–Danielsen–López-Tarrida–Portillo database, and this sweep — give the same 498 graphs, identical as graph6 strings, with zero disagreement on the independence number across all 11 117 graphs. It is a check on their enumeration as much as on ours; what it does not establish is stated beside it, since all three enumerate with McKay's software. Two printed bounds are corrected, neither ever wrong in the data: a dual certificate had been printed against Δ without subtracting α = 4, and an upper bound on T(13,5) had been truncated downwards. A pre-flight now checks every printed fraction against the decimal beside it, reading the built PDF rather than the source. Two integrity fixes found only by running the checks on a clean clone and on the archive itself. core.autocrlf had been stripping CR from four of the authors' CSVs before commit, so the copy here was not the file the source published; sources/ is now stored byte-for-byte. And the seal check, given no git history, had failed every comparison against the hash of an empty file — which reads as tampering; an unpacked archive now gets a clear NOT APPLICABLE instead. Extremal graphs are algebraically harder than typical ones. The fraction of graphs whose Lovász theta value has no closed form runs 1% to 61% across n = 8..11 in the top 100 by contextuality gap (0.01, 0.10, 0.28, 0.61), against 0% to 14% in a uniform random sample of the same size among graphs with gap greater than zero (0.00, 0.00, 0.07, 0.14). The sign is the same on all four sizes. The sealed threshold was not met and it is not being lowered: the preregistered criterion required the gap to reach 0.25 on at least three of the four sizes, and it does so on one, so the recorded verdict is not confirmed. The bar was fixed by SHA-256 before any graph was measured, and a direction consistent on 4 of 4 sizes is a description, not a verdict. The third reversal of how this project reads its own series. The surprise was never that the series of closed forms stops at eleven vertices, which is the normal behaviour of an 11x11 semidefinite program. The maximizers look like the algebraically hardest part of the landscape rather than the easiest, and the reading this invites is that the six closed forms up to n = 10 were not found in easy country and lost when the ground got hard, but six times running in the hardest country there is here. That is an interpretation of a measured direction, not an established fact: the threshold sealed before the measurement was met on one of four sizes, not three, and all three sealed hypotheses of the stage failed their criteria, as stated above. The measurement. Algebraic degrees for 1286 graphs at n = 5..11, three samples per size: the top 100 by gap, a uniform random sample of equal size among graphs with a positive gap, and a zero-gap control. Found 1137, not found 122, declined by the instrument 27. 97.4% of the relations found are degree 1 to 4. Degree 5 was searched for and never occurred once. Degrees 11, 13, 15 and the other odd values above 10 were never searched, so nothing about the tail's parity follows. The sharpest cut is by layer. At n = 11 the top-100 graphs with independence number 4 are out of reach 45 of 47 (96%), against 8 of 41 (20%) in the random sample at the same size and the same layer. The best predictor is edge count, and only at the top: Spearman rho = -0.30, -0.44, -0.42, -0.76 inside the top sample at n = 8..11, all significant; inside the random sample +0.09, 0.00, +0.08, -0.10, none significant. With the limit in the same breath: edge count, independence number and layer are nearly one quantity (rho = -0.87 at n = 11), and they separate on 5 cells of 13. Symmetry explains nothing anywhere. The hypothesis that degree tracks the automorphism group is refuted: absolute rho below 0.20 on three sizes of four, signs inconsistent. All three sealed hypotheses failed their criteria. The stage's preregistration said in advance that this closes the stage rather than voiding it: the measurement is the deliverable. Literature: no measurement of the algebraic degree of theta across a class of graphs was found, and no work relating it to the automorphism group. Recorded as the result of a search on 2026-08-26, not as a claim about the literature. Two precisions agreeing is not evidence of accuracy. This may matter more than the degrees. The rule for counting trustworthy digits used since Stage 2, the matching prefix of a run at precision p and one at 2p minus five, measures the stability of an iteration, not its correctness. On a degenerate optimum Gauss-Newton settles on a fixed point that is not the optimum and every level reproduces it: for one eight-vertex graph the values at 960, 1920 and 3840 digits agree with each other to 465 and 945 digits while agreeing with the truth theta = alpha = 3 to 359. The failure is not a blank, it is a plausible wrong answer. Feed an integer-relation search those 940 claimed digits and at degree 1 it finds nothing, while at degree 3 it returns x^3 - 9x^2 + 27x - 27, that is (x - 3)^3, because the cube of the 1e-359 error falls below the claimed tolerance. An integer polynomial, an unremarkable degree, everything looking right. At an honest 350 digits the same value gives x = 3 at once. Both are checks in the test suite. This is the same mechanism as the false positive the original paper's authors withdrew, said without any accusatory edge, because we walked into it in our own instrument and it took three restarts to see. It also corrects our own Stage 2 method, where the agreement-between-levels criterion was used as a test of correctness. The residual settles it by mechanism rather than by a tuned threshold: a converged run has residual about 1e-dps and loses hundreds of orders per doubling, a stalled one returns the identical residual at every precision. Convergence is now verified before any digit count is believed; 27 of 1286 graphs were declined on that test, 2.1% against a preregistered kill threshold of 40%. Honest lines that are not being removed. All three sealed hypotheses failed their criteria. The direction was measured, the threshold was not met, and the threshold is not being lowered. Section 1 of the Stage 9 preregistration contains a factual error, marking a source UNVERIFIED when it had already been verified two days earlier; the sealed file is not edited and the correction is a dated amendment in the report. The three instrument defects were ours, not anyone else's. What was already here, and remains. Every connected graph on up to eleven vertices enumerated, 1006700565 of them at the last size, the contextuality gap computed for each, and the maximum for every size determined. For each maximizer the Lovasz theta value is proved rather than fitted: an exact primal-dual certificate where a closed form exists, a certified rational enclosure where none does. The eight-vertex value is the root of x^4 - x^3 + 23x^2 - 155x + 158, the value the original paper left open after withdrawing its own integer-relation candidate. The inheritance conjecture this project raised in v2.0 was refuted in v2.1 by an exact certified counterexample on thirteen vertices. What this work is not about. It concerns the number theta and not physical realisability. Where the minimal dimension d* of an orthogonal representation is known it is given, but d* is a lower bound from a non-convex heuristic rather than a proved dimension, we have no upper bounds on eta_3 and neither does the work this continues, and nothing here is a claim about experimental accessibility. Twelve vertices will not be enumerated: 164 059 830 476 connected graphs is about 490 days at our measured rate, and the next step is a proof about the layers or a construction that reaches further, not a bigger sweep. Reproduction. Every preregistration is SHA-256 sealed and committed before its stage ran, and scripts/verify_seals.sh checks byte-identity and commit order. scripts/verify_from_scratch.sh rebuilds every published number from a clean checkout, including the Stage 9 degree measurement, and exits non-zero if anything differs. A standalone verification pack with no dependencies beyond the Python standard library is attached as a separate file: 57 checks, offline, about five minutes. This work continues U. Tamer, O. E. Mustecaplioglu, A. Dizdar and Z. Gedik, The Quad-C5 Graph: Maximum Contextuality Gap on Eight Vertices, arXiv:2605.12828, whose eight-vertex result is reproduced here independently. Carried out with an AI assistant: specification, kill criteria and preregistered predictions formulated in dialogue before each stage; computation and checking performed by the assistant. The certificates are machine-checkable independently of who produced them.