[petalas]

Result

Scoped written proof with exact finite corroboration. No new κ, no word, no selected-frontier change.

unpaidmoon's open #7050 studies paired-star words with one internal thin gap in every stage. It proves that the own-leaf degree-1 core-to-leaf block has rank at most 2 rank(Z), deduces total thin rank at least q g / 2, and asks in its open question 1 whether the factor is 1 under the square condition P_s Z P_f = 0 for every half width q. This package answers that question.

1. Rank lemma. For complementary idempotents P + Q = I over any field and any Z with Q Z P = 0: rank(P Z P + Q Z Q) = rank(P Z P) + rank(Q Z Q) <= rank(Z). In a basis adapted to im P ⊕ im Q, Z is block triangular and the pinching is its block diagonal.
2. Factor 1, sharp. For every q, every B12, B21 and either half order, the own-leaf degree-1 block has rank at most rank(Z). With B = 0 and Z the identity from the leaf into the core, the block is I_{2q} and rank(Z) = 2q, so 1 is the best constant at every q.
3. Bound, conditional on Prove every-stage internal gaps need thin rank at least q g / 2 #7050's schedule analysis. A successful every-stage word has total thin rank at least 2 q ceil(g/3). At g = 31, q = 26 this is 572. Prove every-stage internal gaps need thin rank at least q g / 2 #7050 proves 403, and its predicted factor-1 weight count would give ceil(2 q g / 3) = 538. The proof adds the per-leaf inequalities on leaves 0, 3, 6, ..., whose contributing gaps are pairwise disjoint.

The same inequalities admit a covering point of value 2 q ceil(g/3), so rank counting on the core-to-leaf blocks gives nothing higher. That is a limit of the method. It is not a word, and 572 is not shown to be attained.

Package: research/every-stage-gap-rank-factor (PROOF.md, README.md, verify.py). Card joint-query-controller, priority revision 8, scope changed: the card asks for a changed query demand or caller, and this package answers a rank question on one fixed schedule family.

Evidence and reproduction

From the repository root, CPython 3.12.15, standard library only:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/every-stage-gap-rank-factor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/every-stage-gap-rank-factor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/every-stage-gap-rank-factor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/every-stage-gap-rank-factor --check-source-history

Local results at head 26f8bbfb23feb1f24ba0eb3fb6c4187b331fa057:

• both verifier modes end with PASS_EVERY_STAGE_GAP_RANK_FACTOR factor=1 bound_g31_q26=572 certificate=e42b24ab6676fb20 no_new_kappa, about 2.7 s each;
• PASS_REGISTERED_RESEARCH_PACKAGE research/every-stage-gap-rank-factor outcome=completed, 5.9 s for both registered commands;
• PASS_RESEARCH_COMMIT_BYTES with 6 files and source history commit e54523517d0f03e58352e647c403a5af3e525a57.

What the verifier checks, all exact:

• every 4 by 4 matrix over F_2, every B, both half orders: 524,288 cases, of which 32,768 satisfy the square condition; the lemma and the block bound hold on each;
• control: 168 of 1,800 rank-one cases have block rank 2, and each violates the square condition;
• 4,500 seeded random cases on the actual stage halves at q = 2, 3, 4;
• 3,200 seeded random cases of the lemma over F_2, F_3, F_5, F_7, each with a random submatrix;
• the sharpness witness at q = 1, 2, 3, 5, 26;
• 1,170 of 30,600 rank-one maps at q = 1 satisfy the square condition and meet leaves t-1, t, t+1 at rank one each;
• contributor sets recomputed from the monomials, disjointness and the covering point for every g <= 64, with two rejection controls.

The random cases corroborate the proof. They are not a census.

Limitations and unrun obligations

• The bound in item 3 is conditional on Prove every-stage internal gaps need thin rank at least q g / 2 #7050's schedule hypothesis, square condition and monomial comparison. Those are cited and were not audited or replayed. Only the exponent comparison for the contributor list is recomputed, from Prove every-stage internal gaps need thin rank at least q g / 2 #7050's stated formulas.
• Nested and crossed stage intervals are outside the schedule.
• The leaf-to-core blocks are not used.
• No g = 31, q = 26 word is constructed or excluded outside this schedule.
• No independent human review and no formal proof-kernel review. No Lean, Docker or selected-supplier replay was run.

Open next questions

1. Do the leaf-to-core blocks raise the every-stage bound above 2 q ceil(g/3)? First step: write the leaf-to-core degree-1 block of one internal gap under P_s Z P_f = 0 and test whether a map that meets leaves t-1, t, t+1 at full rank can also cancel its own leaf-to-core monomials.
2. Is total thin rank 2 q ceil(g/3) compatible with the other monomials of the endpoint at g = 3, q = 1, where the bound is 2? First step: an exhaustive census of one internal map of rank at most 2 in stage 1 against every core-to-leaf monomial.
3. Does the block-triangular argument extend to nested or crossed intervals, where the square condition changes? First step: derive the squared-monomial condition for one nested pair.

Related open work

• Prove every-stage internal gaps need thin rank at least q g / 2 #7050 (unpaidmoon): source of the question, the schedule and the factor-2 bound 403. This package builds on it. On the shared q = 1 domain the census here agrees with its two recorded observations: no excess under the square condition, and rank-one matrices with block rank 2 when the square is allowed.
• Prove one internal paired-star gap still needs thin rank at least q(g… #7025 (unpaidmoon): one internal gap in at most one stage, bound q(g-3) = 728. A smaller schedule family; not replaced and not recomputed.
• Prove consecutive paired-star words need thin rank at least g q. #7005 (unpaidmoon): consecutive halves with no internal map, bound q g = 806. A smaller schedule family; not replaced and not recomputed.
• Prove one rank-q internal map raises the paired-star external bound to q(g-1) #7053 (unpaidmoon): one rank-q internal map and the external bound q(g-1). A different schedule; no shared number.
• Prove one interleaved g=31, q=26 word misses the endpoint. #7085 and Record nested and crossed pure products for the g=31, q=26 paired star. #7084 (unpaidmoon): one interleaved g = 31, q = 26 word, and nested and crossed pure products. Outside the every-stage hypothesis; no shared number.
• Prove that adjacent paired-star halves need ceil(g/2) thin maps. #7010 (unpaidmoon): adjacent halves need ceil(g/2) thin maps. A map-count bound on a different schedule; no shared number.

Credit

unpaidmoon's #7050 supplies the schedule, the square condition, the degree-1 block, the monomial comparison, the factor-2 bound, the q = 1 census observation and the question. unpaidmoon's #7025 and #7005 supply the neighbouring bounds. The paired-star contract and the commuting map-count are from Ali Romman's #5747 with its predecessors #5673, #5631 and #5617, and Cinna Davis supplied the controller and simplex lineage those packages retain. The rank inequality for block triangular matrices is classical. This contribution was produced by Nick Petalas.

CI notes

CI did not certify this package. At head 26f8bbfb23feb1f24ba0eb3fb6c4187b331fa057, run 42905330981 failed in the plan job (job 130141720290), and the package replay job was skipped because plan failed.

The failing step is "Onboarding helper tests": 760 tests ran, 1 failed, 24 were skipped. The one failure is

FAIL: test_current_ci_section_matches_selected_record (test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record)
AssertionError: '`make frontier7000-verify`' not found in '# CI verification\n\n## Current selected-family replay: #6030 ...'

The cause is on main, in files this PR does not touch. At base e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json has "reproduce": "make frontier7000-verify", while docs/ci-verification.md still opens with "Current selected-family replay: #6030". This PR changes only research/every-stage-gap-rank-factor/.

The same single test is the only failure, with the same 760-test count, in three other runs read from their logs: 42905094314 (#7124), 42903313790 (#7123) and 42905202459 (#7125).

No fix or retrigger was pushed. The evidence for this package is the local run recorded above: both registered commands pass in under 3 s each.