[petalas]

Summary

PR #801: odd stages 3..13 only
New tail: every odd s >= 15 has X(s,4,25/40) < 9/1000
Together: every allowed stage has X(s,4,25/40) < 1/99

Scoped written mathematical proof with exact finite corroboration. This answers the cost-25 part of Rohan Guliani / Rohansguliani's #801 Open next question 1. The new infinite-tail inequality follows from a weighted Jensen bound and an exact degree-five polynomial whose coefficients in K-7 are all positive. The earlier finite enclosures are cited, not replayed or audited. Increasing the copy cost preserves the exclusion.

The result concerns the historical B_K forced-slot model with normalized window h/k=4 and copy ratio x/k>=25/40. No decoder, physical supplier, new kappa or selected-frontier change is claimed. This contribution was produced by Nick Petalas. Credit Rohansguliani #801 for the question and finite census, DreamingOfClouds for the format/slot framework and retained #446/#386 lineage, Jonathan Lam #798 and GamingPuzzled / Joel Pulikkan #620 for the retained Eisenstein setting, and the predecessors credited in #801. The logarithm, exponential and Jensen inequalities are classical.

Evidence

• Before: Research: certified format ceilings for Eisenstein/Gaussian/Hurwitz E8 copy floors (no κ claim) #801 explicitly leaves s>=15 uncovered. After: a single positive-coefficient polynomial proves the bound for every K>=7, where s=2K+1.
• The standalone verifier checks the exact symbolic identity, all 128 independently reconstructed literal schedules through K=128, all 122 tail cases with a separate rational midpoint-integral lower bound, and 22 rejection/scope controls. Three disposable source mutations reject a changed coefficient, underpriced idle work and a shifted bridge boundary.
• Local CPython 3.12.15 on macOS arm64: normal and optimized registered commands pass in 0.259 seconds each against 120-second limits. Metadata/source pins, registered replay, exact committed bytes/source history, official export integrity, and both read-only Git-free exported replays pass. Unexpected CLI arguments reject. Generated stdout receipts bind all seven package files by SHA256.

From the repository root:

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

Any CPython 3.12 installation can replace that local interpreter path. The direct marker is PASS_EISENSTEIN_STAGE_TAIL s>=15 X<9/1000 no_new_kappa.

Limitations

The universal statement rests on the written proof; finite execution corroborates its stated domain. The all-stage 1/99 corollary additionally cites #801's earlier finite enclosures. Its field-to-rational and residue-field realization, native operators, physical delivery, dirty restoration, ownership, compiler, precision/tape, stock and outer admission remain outside this result. Changed coverage laws and helper-target contracts need a different model. The historical a/(1+a) conversion is not applied to the current assembly. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected-operator replay were not run; an external Python linter was unavailable.

Open next questions

The other cost/window rows of #801 still need their own all-stage bounds. Start with a complete slot formula and a parameter-specific tail certificate. A physical extension-field supplier still needs its native and caller interfaces before such a ceiling has an algorithmic consequence. No further construction expansion is claimed here.

Related open work

• Speed complete Eisenstein replay by 35% with exact rational rank #5954 accelerates rational-rank calls in an Eisenstein verifier. It changes neither this stage-count model nor its ceiling, and computes no competing tail value.

The merged #1794 instead gives an all-stage complex-side cost screen for windows h>=7 at a historical selected target. Its scope differs from this normalized h=4, 1/99 question. #801, #620, #798, #814, #1977 and #2607 supply distinct census, geometry or observation-model predecessors. No additional computations were run to check their results.

The completion refresh inspected all ten other open account PRs created since the advance CLAIM: #5961, #5962, #5963, #5964, #5966, #5967, #5968, #5969, #5970 and #5971. They study different quantities and supply no competing value for this stage-count tail. No extra computations were run to check those results.

CI notes

Both hosted checks passed at head 1714abe1bdae078eb781482efa64221247c418b6 in run 40539826649: plan in 4m38s and replay (research/eisenstein-stage-tail) in 4m47s. The completed checks were read once, and the run head matches the PR head. The replay certifies the bounded automated checks, not independent human or formal proof-kernel acceptance of the universal proof. All listed local checks passed; no CI failure remains.

Merge Danger

Door: two-way. Blast radius: isolated. Seven new files under research/eisenstein-stage-tail; shared code, frozen evidence and the selected result are unchanged.

Advance CLAIM.