Gamma-theta conjecture: certified order-12 frontier v1.0.0

Gamma-theta conjecture: certified order-12 frontier v1.0.0

This release publishes one current paper and its exact reproducibility
archive:

Alec Kriebel, A Certified Order-Twelve Extension of the
gamma-theta Frontier in One-Guard Eternal Domination (17 pages,
26 July 2026).

Result and exact boundary

Assume MacGillivray, Mynhardt, and Virgile's published exhaustive result that
there is no counterexample through order 11. Then no finite simple graph of
order at most 12 satisfies

gamma(G) = gamma-infinity(G) < theta(G).

Equivalently, every counterexample, if one exists, has order at least 13.
Here theta is the clique-cover number, equivalently the chromatic number of
the complement, and gamma-infinity is the standard one-guard-moves eternal
domination number.

This is a finite frontier result. It is not a universal proof, a
counterexample, a lower bound of 14, or an independent re-enumeration of all
graphs at orders 10 and 11. The universal conjecture remains open.

Proof architecture

• Common parameter 3 is excluded by a complete odd-hole template split and
  independently replayed proof certificates.
• Common parameter 4 is excluded for connected order-12 graphs by an exact
  18,381-variable graph-to-CNF theorem and a checked 228,381,671-byte LRAT
  refutation.
• Common parameter 5 is excluded analytically by a simplicial
  closed-neighborhood reduction and the McCuaig-Shepherd domination bound.
• Classical parameter and component reductions prove that these are the only
  possible order-12 cases.

Reproduce

From gamma_theta_eternal_domination/, verify all theorem bindings and replay
the decisive LRAT proof without invoking a SAT solver:

python3 -I -B repro/c050/replay.py --full

Expected verdict:

VERIFIED_ORDER12_FRONTIER_BINDINGS_AND_EXACT_LRAT

The accepted order-13 complement-C9 template exclusion is separately
replayable with:

python3 -I -B -W error repro/c057/replay.py

That second claim is included as current campaign context; it is not part of
the order-12 theorem and does not complete order 13.

Release hashes

b35d4bd795ddfbfa61be18bdd60ddb6d23492b0a63a7449e2ec0190170e6e9d2  paper/order12_frontier/main.pdf
e3b093085bafd124c228a29ef98c86341a45316dc02e11b565a138afe983d57a  results/order12_frontier_acceptance.json
14284db1f0b9cfb37b91d834fbabac1d0ca06d36e0d2782683e35cbd04a976e7  instances/order12_k4_connected_doublelex/instance.cnf
edc0f6b76bc96b2b26677f399566ae437cc47a6fc0cc921eaff81d77b72a50da  certificates/order12_k4_doublelex_seed0_lrat_publication/proof.converted.lrat.zst

Authorship and verification status

Alec Kriebel is the author. The research, programs, certificate design,
audits, manuscript, and publication materials were developed with heavy
assistance from ChatGPT 5.6 Sol under his direction. No outside individual
was contacted, and the result has not been peer reviewed. Exact certificate
checking is evidence about the encoded finite statements, not independent
expert validation or proof of worldwide priority.

Public pages:

• https://aleckriebel.github.io/Math/research/gamma-theta-conjecture/
• https://aleckriebel.github.io/Math/papers/gamma-theta-order-12-frontier/

H(668) research checkpoint v1.0.0 — exact obstructions, no matrix

H(668) research checkpoint v1.0.0

No Hadamard matrix of order 668 was found. This release pauses the
headline search and publishes three scoped, provisional, computer-assisted
papers together with their exact verification artifacts.

Papers

1. Exact repair obstructions around a 64-modular Hadamard matrix of
   order 668 — fixed-(q), the distance-(80/41) frontier, one certified
   long support exclusion, all nine short support exclusions, and the
   orientation cascade for twenty open long cases.
   • PDF SHA-256:
     0f203088cb77ab68b92424f0f59f4fa1dcc8d921fff4743d06a382aa010b1d39
2. Exact shell descent and finite-field norm gates in a
   prescribed-compression chart for Legendre pairs of length 333 — five
   shell-two orbits, eighteen dense orbits in the stated census scope,
   physical placement geometry, primitive units, and three pair-resultant
   norm keys.
   • PDF SHA-256:
     98df2842ae7ab6e3a505f6cb237e847b7ae837b0b662bde986c73125c8ca5da2
3. Semiregular (C_{37}) conference lifts at order 334: quotient
   classification, modular realizations, and low-rank obstructions —
   (625) quotient classes, the universal diagonal law, exact
   characteristic-two supports, and specified formal obstructions.
   • PDF SHA-256:
     c8193ce8485a2642044d11ead0e10bea15589e76bff20021218401a85153012e

Production ledgers

• h668-short-case-production-manifests-v1.0.0.tar.gz contains all 2,304
  retained production ranges behind the complete nine-short-case Eliahou
  certificate.
  • SHA-256:
    6e38f08e9c3c9798bc5ca87a46ae9ccffdb54c6f1ed3901d6ac9ce3fd5d69084
• h668-dense-shell-production-v2-v1.0.0.tar.gz contains all 729
  dense-shell production shards, the manifest, frozen aggregate, compiled
  helper, enumerator, and detached replay code.
  • SHA-256:
    493f73884ff5b5454f179b7754c0207178eeb70c70c27750daa610f3bda6c2df
  • A fresh extraction strictly reaggregates to
    3bccde87f456bfcd2f0c3da6ac8cf9cb3635538e831a95951003068ae87cae86.

Exact endpoint

The last continuation gate produced three exact pair-resultant norm
equalities in (\mathbb F_{167^3}^{*}), but every audited marginal image
is full and every triple image has affine rank three. A complete
margin-conditioned join still requires
5,091,993,547,996 invariant evaluations and approximately 81.5 TB under
the stated naive exact-key layout. The search is paused until a new
algebraic compression, proof-producing decomposition, broader construction
theorem, or materially different exact-computation architecture becomes
available.

No solver timeout is promoted as evidence of nonexistence. The release does
not construct or exclude an (H(668)), a Legendre pair of length 333, a
base sequence in BS(84,83), or a conference graph on 333 vertices.

Verification and responsibility

The paper-level SHA256SUMS, package READMEs, claim-to-artifact tables, and
the public research page give the shortest replay paths. Large historical
computations are published as inspectable ledgers but remain trusted
computer-assisted claims pending independent reimplementation.

The work was developed with heavy assistance from ChatGPT 5.6 Sol under
Alec Kriebel's direction. It has not been peer reviewed. Alec Kriebel is a
non-specialist and cannot independently certify every derivation,
implementation, citation, or novelty assessment.

Public checkpoint:
https://aleckriebel.github.io/Math/research/hadamard-matrix-order-668/

Regular degree-18 endpoint capacity theorem and verifier v1.0.1

Regular degree-18 endpoint capacity theorem and verifier v1.0.1

This corrected packaging release preserves the catalog-conditional exclusion
of the regular degree-18 endpoint
[
(e(A),e(H))=(85,128)
]
for a hypothetical ((5,5;43))-graph.

The mathematical result and all frozen classifications are unchanged from
v1.0. The packaging is hardened so that the one-command verifier works both
from a Git checkout and after repeated runs: it ignores only Git's private
.git metadata and interpreter-generated __pycache__/*.pyc or .pyo files,
while continuing to reject every other unlisted file. The deterministic
archive builder applies the same exclusions.

The reusable ingredient is a minimum-miss transversal-capacity inequality.
Applied to the two byte-pinned published catalogs, it excludes 61,939 of
62,382 fixed-side pairs strictly. Equality rigidity eliminates the remaining
443 pairs. Conditional on the published catalog completeness and extremal-edge
statements, every vertex of an 18-regular ((5,5;43))-graph therefore lies in
at most 84 triangles.

The attached archive is self-contained, standard-library-only, and independent
of the multi-gigabyte research history. It includes:

• the standalone theorem and proof;
• a focused primary-source novelty review;
• the two byte-pinned source catalogs and their provenance;
• the deterministic producer and a separately implemented checker;
• the complete 62,382-line classification stream;
• 17 focused semantic tests; and
• a one-command clean-room verifier.

Run python3 verify_bundle.py from the extracted directory. The final replay
returned "valid": true, reproduced every frozen artifact byte-for-byte, and
passed all 17 tests.

Integrity identifiers:

• source research checkpoint:
  b14f50dd3b048b2d0e51e6aabd63bb608662f053;
• compact release commit:
  b6d7f1a87382d63debeb8c96c76caa230e44331d;
• bundle manifest SHA-256:
  7aac9cf56dad8787bef798dc7507ca437ec2e662645d77f7ddaf565932e671fe;
• normalized archive SHA-256:
  de541d6c7ed8be496784397ea0ee3f1b12c2b93cdbc42ba908160095c1d79cc4.

The archive was rebuilt twice and was byte-identical. It contains no .git
metadata or generated Python cache files.

Scope: this closes only the ((85,128)) endpoint layer. It does not close the
remaining regular degree-18 layers, determine (R(5,5)), or improve the
current bound (43\le R(5,5)\le46). The focused novelty review found a
positive but moderate novelty signal, not proof of global priority.

Version v1.0 remains mathematically valid when used as a clean extracted
archive, but v1.0.1 supersedes it for source-checkout and repeat-run
reproducibility.

Regular degree-18 endpoint capacity verifier v1

Regular degree-18 endpoint capacity theorem and verifier

Superseded for packaging by
v1.0.1.
The mathematical result and frozen classifications are unchanged. Use
v1.0.1 for repeat-run and source-checkout reproducibility.

This compact release preserves the catalog-conditional exclusion of the
regular degree-18 endpoint
[
(e(A),e(H))=(85,128)
]
for a hypothetical ((5,5;43))-graph.

The reusable mathematical ingredient is a minimum-miss transversal-capacity
inequality. Applied to the two pinned published catalogs, it excludes 61,939
of 62,382 fixed-side pairs strictly. The remaining 443 equality cases are
eliminated by uniqueness of one size-six minimizer and a forced
high--high-edge contradiction. Conditional on the published catalog
completeness statements, every vertex of an 18-regular ((5,5;43))-graph
therefore lies in at most 84 triangles.

The attached archive is self-contained, standard-library-only, and independent
of the multi-gigabyte research history. It includes:

• the standalone theorem and proof;
• a focused primary-source novelty review;
• the two byte-pinned source catalogs and their provenance;
• the deterministic producer, independently written checker, and 17 tests;
• the complete 62,382-line classification stream; and
• a one-command verifier.

Run python3 verify_bundle.py from the extracted directory. The clean replay
completed in about 67 seconds with peak memory below 150 MB and returned
"valid": true.

Integrity identifiers:

• source research checkpoint:
  b14f50dd3b048b2d0e51e6aabd63bb608662f053;
• compact release commit:
  26d4fbbb2a5f0c3829507d9a38d80ec1fdf4e726;
• bundle manifest SHA-256:
  17a02f86268dd7682813b263120f6f0dccca4bb97701c87248203aeb567af2be;
• archive SHA-256:
  f73b9b3d61e4b831623c36a0bc99ee769cd6bb11a9699704e00ac6403853624d.

Scope: this closes only the ((85,128)) endpoint layer. It does not close the
remaining regular degree-18 layers, determine (R(5,5)), or improve the
current bound (43\le R(5,5)\le46). The focused novelty review found a
positive but moderate novelty signal, not proof of global priority.