Exact finite-field rank and kernel audit of W_{3,5}(15), followed by Wilson's integral signed-design solvability criterion.

modular block-incidence obstruction

Compute the GF(2) and GF(3) column images, identify their complete left kernels, test all four fixed-family residual pools, and evaluate the full integer signed-block lattice divisibility conditions.

Hypothesis: GF(2), GF(3), or the full integer block-incidence lattice supplies a compatibility obstruction beyond the established total, point, and pair margins.

Test: Determine the exact ranks and left kernels of the 455 by 3003 triple-block incidence matrix, repeat the ranks after each normalized five-column deletion, and check Wilson's four integral divisibility families.

The finite-field nullities exactly equal independently reconstructed star-space dimensions, so there are no hidden linear conditions. Established degree and pair equations make all star residues zero. Wilson's necessary-and-sufficient integer criterion then makes every global signed-lattice divisibility condition automatic. These facts exclude only the stated linear and signed-lattice routes.

• rank_GF2(W_{3,5}(15))=440, and its left kernel is exactly the 15-dimensional point-star span.
• rank_GF3(W_{3,5}(15))=351, and its left kernel is exactly the 104-dimensional pair-star span.
• Deleting the five fixed columns of any of the four complete pair-normalized types leaves ranks 440 over GF(2) and 351 over GF(3).
• Every integer triple-multiplicity vector satisfying total 540, point sums 108, pair sums 15+3s_xy, and integral triple entries lies in the full integer column lattice of W_{3,5}(15).

• python3 scripts/modular_block_incidence_obstruction_v1.py produced rank_GF2=440, rank_GF3=351, and unchanged ranks for types 1 through 4.
• python3 checkers/check_modular_block_incidence_obstruction_v1.py reproduced the ranks using Wilson's formula and independent incremental column bases.
• The independent checker evaluated 360,360 star-block dot products.
• python3 checkers/check_integral_incidence_lattice_corollary_v1.py accepted all four Wilson divisibility families.
• Seven material mutations were rejected and regenerated-result.json is byte-identical to result.json.
• sha256sum -c artifacts/modular-block-incidence-obstruction-20260811/manifest.sha256 passed.

• .proof-experiments/20260811-163437-ee9b26: producer completed in 0.176 seconds and closed the GF(2)/GF(3) route.
• .proof-experiments/20260811-163444-6cfb70: independent formula and column-basis check passed.
• .proof-experiments/20260811-163453-1f75fa: four modular mutations rejected.
• .proof-experiments/20260811-163505-36e724: byte-identical producer regeneration.
• .proof-experiments/20260811-163653-25b7eb: integral-lattice corollary independently checked.
• .proof-experiments/20260811-163719-8fa349: three integral-certificate mutations rejected.

check_modular_block_incidence_obstruction_v1.py does not import the producer: it uses Wilson's formula for full ranks, direct combinatorial annihilation checks, and incremental column bases for branch deletions. check_integral_incidence_lattice_corollary_v1.py separately reconstructs all four divisibility families.

not_requested

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome

no_progress

Public classification

no_progress

• Finite-field inclusion-matrix theory -> predict that kernel dimensions can replace residue-vector enumeration -> observed exact dimensions 15 and 104 with complete star bases.
• Signed-design solvability theory -> predict that all integral congruences reduce to lower-subset divisibility -> observed automatic divisibility under the C(15,5,3) margins.

• W_{3,5}(15) has GF(2) rank 440 and GF(3) rank 351.
  Packed row elimination and independent Wilson-formula computation agree. · The full 455 by 3003 inclusion matrix. · computed
• The GF(2) and GF(3) left kernels are exactly the point-star and pair-star spans.
  Explicit star ranks equal the corresponding nullities, and all star-block dot products vanish. · The full inclusion matrix. · proved
• All four normalized 2998-column pools retain ranks 440 and 351.
  Producer row elimination and independent incremental column bases agree for every type. · The four complete multiplicity-five-pair normalized branches. · computed
• All full signed-block lattice divisibility conditions are automatic under the established margins.
  Wilson criterion specialized to divisors 10, 6, 3, 1 and numerators 540, 108, 15+3s_xy, m_T. · The unrestricted integer column lattice; not nonnegative or binary realization. · proved

• Use standalone GF(2) or GF(3) block-incidence compatibility to prune a global pair-skeleton or triple-excess target.
  All established-margin targets for W_{3,5}(15). · The complete kernels consist only of already enforced point-star and pair-star equations. · result.json and independent-check.json · A genuinely nonlinear modular condition or additional fixed literals that produce a newly proved image restriction.
• Use GF(2) or GF(3) image rank loss caused solely by the five fixed blocks of a normalized type.
  All four complete pair-normalized fixed families. · Every 2998-column residual matrix has the same rank as the full matrix over both fields. · Independent branch_column_basis_ranks for types 1 through 4. · A deeper fixed family or different modulus with an independently demonstrated residual image change.
• Use unrestricted signed integer-block congruences to prune a margin-feasible triple target.
  The full 3003-column integer lattice. · Wilson's necessary-and-sufficient divisibility conditions are identities under the established margins. · integral-lattice-corollary.json and integral-lattice-independent-check.json · A nonnegative, binary, fixed-column, or bounded-coefficient theorem not implied by signed lattice membership.
• Repeat the Terra-recommended explicit joint endpoint-header/internal-skeleton census.
  The 91 surviving type-4 headers under the current fixed-family action. · Epoch 77 already proves any explicit quotient has at least 12,247,501 cells, exceeding the gate. · artifacts/type4-weighted-skeleton-frontier-gate-20260811/result.json and independent-check.json · A proved complete representation below the one-million-cell gate.

• Loss-aware incoming-family generation.
  The prior random all-hitter generator produced no favorable exchange; using unique-triple losses during construction directly addresses that failure signature. · Generate a small deterministic tranche, join by exact point-incidence vector, and independently check defect and healed-versus-new-loss counts. · high · open
• Residual integer-lattice equality after normalized fixed-column deletion.
  Finite-field ranks agree, but equality of the full residual integer lattices was not proved. · Seek compact signed trades expressing each omitted fixed column using only remaining columns, with an independent integer check. · low · open
• Materially new proof-capable literal-block branching.
  A complete SAT exclusion remains decisive, but existing aggregate splits and encodings regress. · Require a new complete branch transformation with independently checked equivalence and a matched propagation gain before any proof-scale run. · normal · open

Objective: Test whether loss-aware construction escapes the verified defect-ten basin.

First action: Freeze a protocol whose generator scores incoming blocks by healed missing triples minus newly exposed unique triples before exact-incidence joining.

Stop condition: Close the generator if a small deterministic pilot yields no exact-degree move with defect below ten or strictly positive loss budget; redirect immediately on checker disagreement.

• Implement a loss-aware incoming generator around the locked defect-ten seed that uses unique-triple losses during candidate construction.
• Run a small deterministic exact-degree pilot and independently reconstruct every accepted exchange.
• Advance only on a defect below ten or a strict healed-minus-new-loss improvement.
• Do not repeat the fixed-pair link, explicit weighted-skeleton quotient, lambda_45 count split, or global modular/signed-lattice filters.

• https://www.coveringrepository.com/systems.aspx?k=5&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?k=5&t=3&v=15 ↗
• https://arxiv.org/abs/2009.05202 ↗
• https://arxiv.org/abs/0709.3144 ↗
• https://arxiv.org/abs/2607.23766 ↗

Codex GPT-5 Sol principal audited the workspace and primary sources, rejected the stale joint-skeleton recommendation, designed the discriminator, implemented the artifacts, and interpreted the result. Two advisory GPT-5.6 Terra delegates supplied reconnaissance only; model agreement was not validation. Python 3.12.3, exact packed GF(2)/GF(3) arithmetic, Wilson rank and integral-solvability formulas, SHA-256, independent checkers, mutation testing, and the computational-researcher experiment harness were used. No SAT solver, CAS, proof assistant, lab job, package installation, system change, external write, publication, or Git operation was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1187.6s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260811-164238-7498de

No human review recorded.