[petalas]

Summary

A scoped written proof with exact finite corroboration establishes that the absolute rounded-E oracle contract cannot guarantee inverse error o(p*2^-p), even with unlimited adaptive queries. For every integer p>100, two entrywise nonnegative operators share one odd, zero-preserving rounded oracle but have separated inverse answers:

h=2^-p, delta=(p/3-1)h, S=coordinate swap
E_+/-=(2/5 +/- delta)S
same T = nearest-grid rounding of (2/5)S
u=(1,-1), k=p
half the inverse-answer separation = delta/[2(9/25-delta^2)] > (4/9)ph

Every deterministic algorithm has error greater than (4/9)ph on one instance. The same lower bound holds for worst-case randomized expected error. At any radius below the exact two-instance radius, one instance has success probability at most one half. Together with Hanson Sun / bbMisk's cited PR1430 Horner upper bound <ph, the optimal error order is Theta(ph). The exact minimax constant for the whole class is unclaimed.

This contribution was produced by Nick Petalas. Credit Hanson Sun for the numerical contract, upper bound and stronger-structure question; eumemic, icekylinx and the retained Gaussian/resampling lineage; and classical rounding and oracle indistinguishability. Predecessor scientific suites are cited, not audited or replayed. The target is PR1430's single fixed-precision oracle, distinct from PR6490's changed access to oracles at every requested precision. No new kappa or selected-frontier change.

Written proof. Advance CLAIM. Canonical base 7c49fe3, priority revision 8, selected PR6030. The original uniform-transfer-audit card explicitly changes to this numerical-interface side theorem.

Evidence

Before: PR1430 supplies an O(ph) rounded inverse evaluator and a quadratic-accumulation example for the older evaluator. After: a new identical-oracle pair proves that no algorithm using only that fixed oracle improves the error order.

• Integer and separate Fraction rounding agree on all 68,556 enumerated complex disk-grid points at precisions 0..7; all 25 rounding residue pairs are covered.
• All 416 precision cases, 101..512 plus 768,1024,2048,4096, pass inverse equations, the exact separation and lower inequalities. There are 13,312 local complex-component error comparisons and six literal tail-power checks.
• All 3,104 scalar two-answer decisions, 136 randomized mixtures, six invalid-precision rejections, four false-inverse rejections and four semantic discriminators pass. The all-p and all-algorithm claims rest on the written proof, not finite extrapolation.
• Warning-strict direct normal/optimized commands and both registered commands pass with generated six-file hash reports. The direct registered command forms were timed locally at 0.237 and 0.260 seconds, below their 120-second limits. These are deadline observations, not performance claims.
• Package metadata, exact committed bytes, modes and source ancestry pass at head 19947aa. Official committed export integrity/input coverage passes. Optimized read-only, Git-free, empty-PATH export replay passes, with all seven exported files unchanged.

Local environment: CPython 3.12.15, macOS 27.0.1 arm64, standard library only. From the repository root:

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

Another CPython 3.12 executable can replace the absolute path. Expected markers are PASS_NEUMANN_ORACLE_ERROR_FLOOR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Default replay writes nothing and imports no predecessor implementation.

Limitations and unrun obligations

E is hidden and only one fixed-precision T is available. Exact E-dependent data, a Gaussian formula, precision refinement or a stronger error law can evade this bound. The two operators are order-preserving swap multiples; they are not asserted to be actual Gaussian operators. This is a numerical-interface obstruction, not a physical time or native-query tariff.

Actual Gaussian structure, coefficient preparation, physical implementation, dirty restoration, precision/tape/stock costs, complete supplier/outer admission and all five selected framework assumptions remain open. No selected cold replay, Lean or Docker build was run. No required local package check was deferred to CI. Independent human and formal proof-kernel review are unclaimed. Python allocation and the two fixed rejecting-control callback targets have bounded deviations documented in the README.

Open next questions

1. Derive an actual Gaussian local error law that excludes this pair, then price its implementation and caller.
2. Determine the exact minimax constant for the fixed absolute oracle class; the two-instance radius here is a lower bound only.

Related open work

• PR6490 reduces Horner oracle precision volume under access to a family at every requested precision. This theorem permits only one fixed T and does not bound that changed-access model or its precision volume.
• PR371 accounts for final-grid error in an actual Gaussian block routine. This abstract shared-oracle pair neither audits that repair nor instantiates its Gaussian operator.
• PR6413 removes setup exponential calls in a dense Gaussian recovery backend; this proof supplies no backend implementation or setup saving.
• PR6487 varies factor-row precision in a cyclic-band solve. Its source-specific factor-error contract differs from this hidden rounded-E oracle.

Live open work, recent account work, handoffs, local reservations and main/merged object and general-form searches were checked before committing. The overlap snapshot covered 546 open PR bodies and 6,571 Discussion records with complete reported remote pagination; sparse local coverage was 124 of 8,917 notes. Git-object searches supplemented it. Unpublished work and arbitrary historical diffs remain limits. No same-instance numerical disagreement was identified or investigated.

Completion refresh inspected all six other account open PRs created since the CLAIM at 2026-11-03T11:13:41Z, namely #6596, #6597, #6598, #6599, #6600 and #6601. None studies this fixed-precision rounded-E oracle error bound or computes a competing same-instance value. No additional mathematical computation checked those contributions. PR6490 remains the related changed-access model described above.

CI notes

Both hosted checks passed at exact head 19947aa37abbc9cfd1fe3f2b2b9c5449f19a1652 in run 41626862965: plan and replay (research/neumann-oracle-error-floor). The completed checks were read once with gh pr checks 6602 --repo CrocSwap/integer-mult-bounds; separate run metadata confirms completed success and the matching head.

Both registered normal/optimized commands and all required local checks also passed. Hosted replay certifies execution of the declared finite checks and source validation, not independent acceptance of the universal written proof. No CI failure remains and no required local check was deferred to CI. Independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: package.

Six new files in one isolated research package. Shared implementation, frozen sources, predecessor credit and the selected record are unchanged.