[petalas]

Summary

Proves the least universal coefficient for the exact integer split R=qJ+r>J, 0<=r<J, at every real 0<t<1:

q J^t + r^t <= K_t R J^(t-1)
K_t = (1+s_t^t)/(1+s_t) = t s_t^(t-1)
t s_t^(t-1) - (1-t)s_t^t = 1,  0<s_t<1

higher-moment sufficient charge:
alpha^v + 2 gamma_u^v + 2 delta^v
             becomes
alpha^v + K_t gamma_u^v + 2 delta^v

The coefficient is optimal over integer jobs and satisfies 1<K_t<2. At t=1/2 it is (1+sqrt(2))/2; at t=1/3 it is 4/3, attained by J=8,R=9.

Claim type: scoped analytic proof with exact finite corroboration. Only research/sharp-integer-split-charge/ is added. This tightens one estimate of zxv's #4433 while retaining the tensor, native and completion premises. It does not assert a new kappa or a smaller tensor power. The calculus is elementary; no literature-priority claim is made. The merged #4483 direct compiler removes these steps for certain profiles under its own gates; this does not improve its sufficient ranks.

Produced by Nick Petalas. Credit zxv/#4433, Cinna Davis/#4412/#4414/#4416 and Jacob Alan Sussman for the retained strictification, chronological product and integer compiler, with their existing predecessor attribution. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.

Evidence

• Before: the retained split-tail estimate uses coefficient 2. After: the written maximization and integer approximation prove the optimal K_t; no predecessor executable is used as the deliverable.
• Both registered warning-strict normal and optimized checks pass all 56,448 enumerated split cases, seven integer sharpness sequences, the half/third-exponent closed forms and eleven rejection controls. Rational polynomial and integer-root enclosures use no floating-point acceptance.
• Locally timed registered commands took 0.25 and 0.26 seconds on CPython 3.12.15, macOS arm64. Package metadata/replay, diff checks and six-file committed-byte/source-history preflight pass at 3f60fe0c90573a60e52d016b0001f4d24f6db2ae.
• A committed-byte export passes optimized isolated replay with no Git and PATH=/nonexistent; all six source/export files remain unchanged. Separate altered-source and altered-receipt copies reject with receipt or source-hash mismatch.

From the worktree root:

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

Another CPython 3.12 installation can replace that interpreter path. Expected marker: PASS_SHARP_INTEGER_SPLIT_CHARGE cases=56448 sharpness_sequences=7 rejected_controls=11.

Advance claim. Canonical main remained 624de9dc66858af0945e5f8a0942488d62280d43 at the pre-commit refresh. Remote overlap coverage was complete for 348 open PR bodies and 3,786 Discussion records; sparse local helper coverage was incomplete. Distinctive all-main README and merged-PR searches were checked separately.

Limitations and unrun obligations

The all-real-exponent statement is a written analytic proof. Finite checks cover six exponents on J<=48,q<=8 plus seven sharpness sequences through J=10^6; they are not a formal proof. Native correctness/admission, a new supplier inventory, a reduced tensor power, physical hosting/precision, the selected cold replay and the complete multiplication theorem were not checked or claimed. The sharp constant here is for the linear-rank comparison; the subsequent higher-moment transfer need not be sharp.

Open next questions

• Apply the coefficient to one pinned, fully charged inventory to decide whether it reduces a sufficient tensor power; preserve the certified moment intervals, completion and outer bill.
• Determine the least direct coefficient for fixed sigma>1 in split_t(R,J)<=K_(t,sigma) R^sigma J^(t-sigma), starting with one normalized quotient.

Related open work

No open PR found in the inspected live bodies targets this sharp integer split coefficient. #4814 studies ordinary BIT finite leaf backends, a separate supplier/leaf comparison; this contribution computes none of its saving numbers and does not assess its admission claims.

This account's open PRs created since the CLAIM at 2026-10-22 21:13:18 UTC were refreshed before DONE. The nine other new open submissions concern different objects or quantities; none computes this split coefficient or the same oversized-paid tail term. No additional same-object related-work entry or numerical disagreement was found.

CI notes

Both hosted checks passed on 3f60fe0c90573a60e52d016b0001f4d24f6db2ae in run 39856299519: plan in 2m58s and the registered package replay in 4m25s. The replay executes both warning-strict ordinary and optimized commands. Run metadata confirms the exact head and successful conclusion. Hosted CI certifies execution of these bounded checks; independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way. One isolated opt-in research package.

Blast Radius: research. No shared code, frozen predecessor, selected record or frontier changes.