[petalas]

Summary

Prove that the fixed PR5756 two-tail/capacity inequality family has minimum integer block length 112, using explicitly cited scalar bounds from Ali Romman's PR5775. The unique cutoff/floor pair at the minimum is (16,17), with rational offset 1/7 and decay 1951/2000.

g(c,u)<rho, d(c,b)<rho, rho^L<1/16, c^L<b^(L-2)
L=3,...,111: impossible for every permitted c,b,u,rho
L>=112: the displayed rational witness works

Claim type: scoped written proof using cited predecessor mathematical premises. This changes the fixed exponent 128 to a variable exponent and settles the sharp threshold of that algebraic gate family. It does not establish a complete compiler at 112 or a new kappa. The fractional shortening within this family is exactly 1/8.

New proof and checker produced by Nick Petalas. Credit Ali Romman #5775 for the envelope and scalar premises, Joel Pulikkan / GamingPuzzled #5756 for the complete invoice, and the retained #5751/#5718 P104/native/controller and compiler lineage. Three predecessor files are retained verbatim with exact origin paths, commit, Git blobs and SHA256 hashes. No predecessor scientific replay or audit is the deliverable.

Advance CLAIM. The result matches the declared gate-family scope.

Evidence

Before, PR5775 fixed the block length at 128. After, the new proof excludes all smaller lengths through 111 and supplies a strict witness at 112. The decisive new comparisons are 16*a_num^111 > a_den^111, 16*1951^112 < 2000^112, and 16^112 < 17^110, where a=487740894096357/500000000000000 is a cited global tail lower bound.

Local CPython 3.12.15 warning-strict normal and isolated optimized checks both pass:

PASS_MINIMUM_TAIL_BLOCK minimum=112 cutoff=16 floor=17 excluded_lengths=109 controls=10
PASS_REGISTERED_RESEARCH_PACKAGE research/p104-minimum-tail-block outcome=completed

The checker enumerates the 109 lengths 3 through 111 as finite corroboration of the written universal argument, checks both charged tails and the unique minimal integer pair, and rejects ten semantic or source-corruption controls. Each registered command was timed locally once, at 0.023 seconds normal and 0.027 seconds optimized, well below its 120-second timeout. These are execution durations, not a performance comparison.

Metadata/pins, committed-byte/source-history checks and AST/compilation diagnostics pass. The official committed export passes integrity checking and direct isolated optimized replay with empty executable PATH, no Git directory and read-only files. Every exported byte and mode remains unchanged. Generated receipts bind all ten package files. External premise bytes are retained inside the package; their unmerged Git objects are not runtime dependencies. The source-history preflight checks the declared main-ancestor provenance separately.

Reproduction

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/p104-minimum-tail-block/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/p104-minimum-tail-block/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-minimum-tail-block
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-minimum-tail-block --check-source-history

The direct checker is standalone and uses only the standard library. An optional --receipt /absolute/output/path generates a hash-bound receipt outside the package.

Limitations and unrun obligations

The predecessor's scalar inequalities are explicitly cited premises, not independently re-proved here. The new proof covers the displayed two-tail/capacity subsystem only. It does not supply a changed block decomposition or its normalization, exit, rounding and nonrecursive costs. Other compiler gates, carrier height, native realization, physical delivery and all-size supplier admission remain open. No full source replay, Lean/kernel proof or complete multiplication theorem is claimed. The selected frontier is unchanged. No external linter was available; warning-strict execution and AST/compilation diagnostics passed.

Open next questions

Can length 112 be realized by a complete charged compiler for the fixed invoice? First specify the changed block decomposition and rederive its normalization, exit, rounding and nonrecursive bill, then test the remaining gates. The missing block rule and complete ledger prevent a carrier-size conclusion.

Related open work

• #5775 proves the fixed-length 128 envelope and supplies the cited scalar bounds; this contribution proves the new minimum over variable lengths.

• #5756 supplies the complete fixed R22 invoice. This contribution changes no word, paid/target histogram or kappa.

• #5751 supplies the paid-meeting construction inherited by Conditional κ .00311313795 (+1.219e-7 over #5751): the #5739 block move on the paid-meeting word #5756; no scientific replay or admission claim is added here.

• #5787 changes finite carrier-size parameters for a different frozen Complete coupled-output caller: conditional κ = 0.003027288235 (+5e-12 over #5400) #5413 caller and a complete charge. This gate-family threshold does not change or certify that caller.

• #5901 proves a worst-case concave-invoice pruning obstruction for synthetic complete bills. This contribution instead fixes one source invoice and determines its tail-gate block threshold; no common numerical instance is computed.

Completion refresh inspected this account's eight other open PRs created since the CLAIM: #5897, #5898 and #5900 through #5905. None computes the fixed PR5756 tail-gate minimum block length. No extra computations were run to validate their results or compare different quantities.

CI notes

At head d73fa51b4d06e42c73e78af89754603e3ab257e8, run 40500871500 finished with plan failed; replay skipped.

The only failed step is Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern frozen-card/current-command metadata, missing-reference rejection and sparse omitted-path expectations: 0 != 16, ValueError not raised, and empty omitted-path lists where helper/historical-reference paths are expected.

The ten failure/error headers and final summary match the inspected PR #5898 plan log, based on the same canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a. This confirms the reported shared onboarding failure. Maintainers will handle it; this contribution changes no shared tests or helpers.

Hosted replay has not certified this package. The package's scientific checks have run only locally. The local normal, optimized, registered, metadata, publication and read-only export checks above passed.

Merge Danger

Door: two-way. An isolated research package can be reverted independently.

Blast Radius: package. No shared implementation, selected record or frozen predecessor is changed.