[petalas]

Summary

Scoped written proof and deterministic exact constructor for the seven-vertex theta(2,3,3) query mask, produced by Nick Petalas.

Before (#5218): residual-state construction, up to product_i(d_i+1) states
After:          fixed 1,583 incidence templates and 267 half-covers
                O(B) bit operations and working bits for B-bit widths
Output:         eight shared counts, seven private counts, a half-cover and rank

The constructor rounds at most one fractional odd cycle and returns an integer matching/cover certificate proving optimality. It also gives the exact fixed-cover formula nu(d)=floor(min_y d dot y / 2), with y in {0,1,2}^7 covering every edge to weight at least two. The source fitting-rank theorem rank=sum(d)-nu(d) is cited, not replayed or audited. This is a theoretical encoded-width improvement, not a measured runtime speedup or new kappa.

Credit JoyboyBrian #5218 for the exact theta contract, matching reduction and earlier constructor; JoyboyBrian #5023, Tekalign #3362, Nick Petalas #4909, Sagar Thakrar #2484 and the retained query-mask lineage. LP duality, half-integrality and incidence basis enumeration are classical. Advance CLAIM.

Evidence

Local CPython 3.12.15 on macOS arm64, standard library only:

• All 2,187 width vectors in {0,1,2}^7 agree with an independently implemented one-copy matching recurrence; 76 have half-integer fractional optima.
• 130 expanded attaining matrices pass 390 rank calculations over F2, F3 and F101 and all 2,109 fitting-entry checks.
• Nine compressed fixtures reach 4,099-bit widths. Fourteen malformed-input/certificate controls reject.
• Warning-strict normal and optimized direct checks pass in 8.840 s and 9.159 s. Both registered checks pass in 19.121 s total; each is below its 120-second timeout. These are verification durations, not comparative performance claims.
• Metadata/source pins, source compilation, staged whitespace and committed bytes/modes/source history pass. Official export integrity passes. Read-only Git-free normal/optimized exported replays and actual constructor-corruption controls are recorded before publication.
• Generated receipts bind every package file and both cited source documents. No receipt is handwritten. No predecessor executable or census is imported.

Exact reproduction, from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/theta-compressed-constructor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/theta-compressed-constructor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/theta-compressed-constructor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/theta-compressed-constructor --check-source-history

Expected tokens: PASS_THETA_COMPRESSED_CONSTRUCTOR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Either direct command accepts --receipt /absolute/path/outside/checkout.json.

Limitations

The all-width claim rests on the written proof, including classical LP strong duality and the cited #5218 all-field fitting theorem. Finite enumeration corroborates the new constructor. Independent human and formal proof-kernel review are unclaimed.

The graph is fixed. Writing a full fitting matrix still costs its output size. No physical copying, coefficient delivery, frames, owner lifetimes, restoration, precision, fixed-tape lowering, compiler/supplier admission, new kappa or complete multiplication theorem is established. The selected frontier and its five framework assumptions remain unchanged. Lean, Docker and the selected scientific cold replay were not run because this is a scoped constructor contribution.

Open next questions

Can an actual recursive caller supply this exact ordered read mask with a favorable complete invoice? Start with a literal source incidence map and charge copies, coefficient delivery, owner lifetimes and restoration. A source-bound producer/consumer and physical tariff remain missing. No wider search is included.

Related open work

• Construct rank-135 fitting matrix on 336 alternating-plane ports (no κ) #5950 constructs a fitting matrix on 336 alternating-plane ports. Its graph and rank quantity differ from this fixed theta mask; no shared numerical value is compared.
• Bound E6 and E7 F2 shadow minranks by 8 and 11 (no new κ) #5959 gives binary shadow-minrank upper bounds on E6/E7 root-line patterns. This construction retains the all-field theta contract and does not compute either root-line minrank.
• Checkpoint symmetric two-star P112 supports and inconclusive COPY63 search #5088 explores symmetric two-star P112 supports and COPY63. This abstract theta constructor supplies no P112 geometry or COPY tariff.
• Checkpoint eight masked-decoder basis searches (exploratory; no new K) #5064 explores masked-decoder basis searches. This package supplies one exact fixed-mask matching constructor, without a physical decoder or basis-search claim.

No open PR studying this same theta constructor was found in the refreshed live list and complete cached open-PR corpus. Unpublished work, arbitrary diffs and unindexed prose remain coverage limits. No extra computations were run to validate another contribution.

Completion refresh: all ten other open account PRs created after the CLAIM at 2026-10-27T16:08:05Z were inspected: #5961, #5962, #5963, #5964, #5965, #5967, #5968, #5969, #5970 and #5971. They concern minor probabilities, chronological widths, cyclic-call tariffs, rounded P104 errors, Eisenstein stage tails, Clifford tensor thresholds, observer-safe adapters, scalar rewrites, borrowed-shear endpoints and invariant lattices. None studies this theta matching number or constructor. No common numerical value or disagreement was identified, and no extra computations were run to validate those contributions.

• Save one boundary XOR at four calls with both output updates nonzero #5968 optimizes observer-safe adapters at a fixed four-call boundary. It changes the observation contract; this theta package optimizes a different abstract ordered read mask and supplies no physical adapter tariff.

CI notes

Both hosted checks passed at the exact contribution head aab0f750bfa6c34b48cf5400e536e898a063cf0b in run 40539824972: plan in 6m46s and replay (research/theta-compressed-constructor) in 2m24s. The completed check list was read once. The run head and PR head match. Hosted replay certifies the registered package checks at this commit; it does not certify the universal mathematical proof or a multiplication supplier.

All declared scientific commands also passed locally, as recorded above. The read-only Git-free normal and optimized exported replays passed in 9.911s and 10.171s and preserved every exported byte. Three actual optimized CLI corruptions were rejected: altered cycle rounding (nonoptimal matching certificate), erased matching values (primal/dual gap), and weakened optimality validation (negative control accepted). The final source-hash receipt agrees between both exported replay modes. No package or mathematical changes were needed after publication.

Merge danger

Door: two-way. Blast radius: isolated.

Seven new files in one opt-in package. Shared code, frozen predecessor packages, review ledgers and the selected frontier are unchanged.