[petalas]

Summary

Scoped proof with exact finite corroboration. The scalar Neumann locality tail admits the smaller prescription

 g = 1/(16p), target tail < 2^(-16p)
-J = 512p²
+J = 226p² for every integer p >= 1

The coefficient 226 is optimal among nonnegative integer coefficients in J=c p² required to work for every positive precision. Coefficient 225 fails at p=1. The uniform proof uses a four-term binomial bound and geometric induction; sixteen direct cases corroborate its constants. This extends Jose Manuel's PR #3851 and James Reed's PR #3864 without replaying their results as the contribution. No new κ or selected-frontier change.

Evidence

Before: PR3851 uses 512p² and PR3864 checks that coefficient for precisions 101–200 and at precision 1. After: a written all-positive-precision proof for 226p², exact integer exclusion of 225, sixteen full-tail comparisons using both integers and fractions, and four calibration-domain rejection controls.

From the repository root, with CPython 3.12.15 and the standard library:

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

All passed locally. Direct normal and optimized checks took 0.08 and 0.09 seconds; registered replay took 0.41 seconds on the first measured run. Output includes PASS_OPTIMAL_NEUMANN_QUADRATIC_TAIL coefficient=226 excluded=225 direct_precisions=16 domain_controls=4 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES for all five files and the declared source-history commit. Optimized replay also passed in a standalone copy outside Git. Generated hash-bound receipt creation and refusal to overwrite an occupied receipt passed.

Limitations

The same contraction and prefactor are hypotheses. The proof does not establish rounded evaluation, physical costs, inverse-norm assumptions, caller integration, complete outer tensor or integer recovery. The p=1 obstruction does not establish optimality on the production domain p>=101. No Gaussian/native replay, Lean, Docker or independent human/formal-kernel review is claimed.

Open next questions

• Determine the smallest integer coefficient restricted to p>=101, beginning with monotonicity of the normalized logarithmic tail.
• Integrate the smaller J and path margins into the exact PR3851 caller while retaining every setup, rounding, routing and ownership obligation.

Related open work

The refreshed open-work list contains no PR studying this same scalar tail or quadratic coefficient. Adjacent contributions have distinct hypotheses and quantities:

• Prove sharp theta order for broad Gaussian edge inverses (no κ) #5150 proves the sharp inverse-norm order at broad Gaussian phase-cell edges. This grants a fixed contraction and prefactor; it does not derive that inverse estimate or compare its constants.
• Prove sharp quadratic rounding guards for closed words (no new κ) #5153 proves an endpoint-conditioned coefficient-rounding guard for finite words. This bounds an exact scalar truncation tail and leaves rounded evaluation open; no numerical comparison is claimed.

PR #3851 and PR #3864 are merged predecessors. The ongoing Gaussian caller work in Discussion #4202 retains separate operator, rounding and global-reference obligations; this scalar theorem does not discharge them.

CI notes

Both hosted checks passed for head 3133085844aabba9b4aa703a4799eb1f81d896f2 in Registered research packages run 39942989456: plan and replay (research/optimal-neumann-quadratic-tail) / replay (research/optimal-neumann-quadratic-tail). The registered replay runs both normal and optimized exact verifiers. This certifies the recorded automated checks, not independent acceptance of the written proof or inherited Gaussian premises. All local checks listed above also passed.

Merge Danger

Door: two-way. Blast radius: isolated. Only the new research package is added; no caller, shared code or selected result is changed.