[petalas]

Summary

For every odd h >= 3 and every field F of characteristic three, the disjointness matrix A_h on the nonempty proper subsets of [h] satisfies rank_F(I-A_h)=2^h-2. Thus its additive identity repair needs exactly 2^h-2 scalar rank-one terms. This is a scoped algebraic proof with an explicit inverse and bounded exact corroboration; no new kappa or physical supplier is claimed.

K^4 = -I                    (characteristic three)
T = K tensor ... tensor K   (odd h gives T^4 = -I)
R = (I-T)^-1 = 2(I+T+T^2+T^3)
R[empty,full] = [[2,2],[2,0]] is invertible
=> deleting those endpoints leaves I-A_h invertible

Six new files in research/ternary-disjointness-repair/ contain the proof, standalone checker and source-bound metadata. Scientific base: a5df48c55ea806d7952b65acc9c76d878ff09b13; contribution head: 50ee4f7069d7aabdcdfc44ebadf0456d8cf82e0c.

Evidence

• Before: Ali Romman's merged Prove the exact rank of additive Boolean disjointness repair #4962 proves the binary repair rank; characteristic three is outside its theorem. Its tensor/deletion approach is credited and its proof/README are pinned as provenance. No predecessor verifier is imported or rerun.
• After: the written proof covers every odd h >= 3 and every characteristic-three field. Exact elimination gives ranks 6,30,126,510 at h=3,5,7,9. Literal full and corrected interior inverses pass at h=3,5,7. An independent census checks all 729 kernel vectors at h=3. Six controls reject invalid domains and omitted/wrong-sign endpoint corrections. The h=2 kernel vector (1,1) demonstrates the oddness boundary.

CPython 3.12.15, standard library only. Run from the repository root:

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

The two registered commands each passed locally in 0.73–0.74 seconds; the complete launcher passed in 1.67 seconds. These timings establish bounded replay, not a performance improvement. Source-history/committed-byte preflight passed for all six files. Official committed export integrity and a read-only, Git-free isolated optimized replay with PATH=/nonexistent passed; its generated source-hash report equals the direct replay's report. git diff --check passes.

Credit Ali Romman #4962, mikey1201 #172 and the icekylinx/#144 compiler/bookkeeping lineage. The new theorem and verifier are produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.

Limitations and unrun obligations

This counts additive scalar rank-one terms. A ternary physical caller, coefficient preparation, paid reads/copies, dirty restoration, frames, compiler/stock/precision/tape costs and complete recursive/multiplication bounds are not supplied. No binary ledger is transferred. No selected cold replay, predecessor scientific suite, Lean or Docker run was performed.

The advance claim sought a first original rank statement; the first proved result settles odd h, and even dimensions remain open under the stop rule.

Related open work

• Boolean-lattice (zeta) finite networks: compiler-level construction, in-tree assembly projection 15x-48x, obligations named (research record, draft) #172 studies the same Boolean-lattice disjointness endpoint in a binary construction. This contribution changes the coefficient field and proves an additive repair obstruction; it does not evaluate that PR's projected invoice or resolve its physical semantic obligations.

Live targeted searches for boolean-disjointness-repair and zeta-family, main-tree/merged searches and recent claims found no other same-question open work. Sparse local notes, arbitrary diffs and unpublished work remain coverage limits. Completion refresh examined this account's open PRs created since the advance CLAIM: nine including this PR. None of the other eight studies this Boolean-lattice disjointness repair or supplies a directly comparable rank value, so no additional related-work line or numerical comparison is required.

Open next questions

• Determine rank_F3(I-A_h) for even h >= 4. Begin with the full tensor eigenspaces and the two-endpoint restriction when T^4=I; the odd-dimensional inverse is unavailable.
• Decide whether a nonadditive physical repair can evade this obstruction. First supply an exact all-input ternary identity and complete paid bill; neither is assumed here.

CI notes

Both hosted checks passed for exact head 50ee4f7069d7aabdcdfc44ebadf0456d8cf82e0c in Registered research packages run 39948488925: plan in 4m21s and the focused package replay in 1m12s. The replay runs both registered normal and optimized exact checks. Run metadata confirms that head and success. This certifies the automated replay, not independent human or proof-kernel acceptance of the written theorem. All local checks recorded above also passed.

Merge Danger

Door: two-way. Blast Radius: package.

This adds one isolated research package. The selected frontier, shared code and frozen evidence are unchanged.