Research: exact restoration contracts and resource bounds for rounded borrowing

[aliromman]

PR #1057 proves exact restoration with three dirty arrays when every child is additive. The rounded leaves from PR #985 and the precision proof in PR #1054 do not supply that contract. This package identifies the actual borrowed boundary and proves its necessary and sufficient translation condition.

Review files: Proof and scope · Evidence · Source pins and credit.

The result is a scoped obstruction and resource tradeoff. It makes no new kappa claim.

• An exact additive wrapper on the same arbitrary fine grid has integer slope. It cannot approximate the valid PR #985 map x -> 4x/3 with error below 1/3 on the physical interval [-1,1], even with more guard bits or unlimited restored workspace. Restricted source grids and different output grids are explicit escapes.
• For componentwise nearest rounding of Pz/q, the safe translations are exactly Ph=0 mod q. A correction that cannot reread the old helper needs one saved state per quotient class. For a coprime scalar on L entries this gives q^L states when the admitted source and displacement ranges contain a complete residue domain. Smaller finite domains can need fewer states. Readable current data and displacement can recover the old helper without a saved summary; the proof accounts for that escape at the actual borrowing boundary.
• A nonzero additive result from fixed signed offset averaging needs at least q variable calls. The matching q-call construction returns the q-scaled integer map. A separate exact integer lift pays for denominator growth, initialized low bits, all local operations, source access, and inverse calls. It is a comparison construction, not a fast rounded supplier.

The written proof is supported by deterministic exact checks: 716 reduced scalar ratios, 3,495,180 residue responses, all 700 normalized functions on cyclic groups of orders 2 through 5 at the emitted child boundary, six matrix quotients, 380 pinned rounded leaf executions, 22 finite exact lifts, and nine mathematical controls. A read-only relocated replay under optimized Python passes. Three real subprocess controls reject a changed pinned source, a discarded residue, and an occupied output file.

Reproduce from the repository root with the Python standard library:

OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 VECLIB_MAXIMUM_THREADS=1 python3 -I -B research/rounded-borrowing-contract/test_package.py

Expected final markers:

PASS_ROUNDED_BORROWING_CONTRACT ratios=716 responses=3495180 rounded_leaves=380 exact_lifts=22 controls=9
PASS_ROUNDED_BORROWING_PACKAGE controls=3 relocation=1

The selected-record check and all 25 selected unit tests passed on the exact experiment base. Full make verify, the selected cold replay, remote CI, and a physical tape compiler were not run. This change only adds the isolated research package. Finite replay is not a formal proof of the full multiplication theorem.

Experiment base: abdc6a7002abdbcd9e21bfc5e1a795784deacfdf. Source pins: PR #1057 78a062df1dcab2bffea5ff4d8243f3da6e465414; PR #985 da1cdf8ed4ac9b1b8378dfec73f77ff4e81a4f70; PR #1054 6bc675be1554b3a3be771147f21f9dc2c6dae50a. The package freezes exact source and context bytes. During the work, main ba82555c64e1879e4b7decce28186f208fcb8262 selected PR #1042 at conditional kappa=1672188119205421/10^18; this package does not alter that selection or any supplier call moment.

The proof, failed approaches, finite domains, overlap checks, and validation receipt are in research/rounded-borrowing-contract/. It credits Gareth Davidson (@bitplane)'s PR #1057 and the retained predecessor lineage. Prepared for Ali Romman (@aliromman) with substantial OpenAI Codex assistance in the research, proof, implementation, verification, and text. New material is Apache-2.0. No independent human or proof-assistant review is claimed.

[Mister-Meeseeks]

100-PR batch review at 4c35687d7c0da8e6cf33f17e418ecf31c95ff3ad.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Same-grid rounding fails the exact additive borrowed-child contract; safe-translation lattice, restricted summary-state bound and fully charged modular/integer escapes.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.