[petalas]

Summary

Ring.inverse, above all borrowed/caller-resident words
  four Euclidean words
  + quotient and remainder
  + four words inside modular multiplication
  = ten additional words, sharp

This closes one arithmetic-interior component of the first BIT node reserve handoff. The written lifetime proof covers arbitrary legal word widths in the pinned implementation. If L words are live on entry and words have b bits, inversion occupies at most (L+11)*(b+1) non-LEFT symbols, including its permanent template. The package also proves sharp bounds for division, addition, subtraction, negation and multiplication.

Only research/selected-word-scratch/ is added. It contains the proof in README.md, a fixed receipt, explicit replay hook, literal allocation/cell checks, failure controls and two unchanged source/notice exports. Scientific source: Jose Manuel/jmbarrancoml's PR2697 at a53fdf4bcbe33c6732f0bec1badc8c023380f9d1. SNAPSHOTS.json retains original paths, Git blobs and SHA256 hashes; the complete predecessor notice is preserved. The arithmetic algorithms and implementation are not claimed as new.

Task selection drew bit-helper-contract, nonce 1875c0a59c67cabd5d78220a2c84dcc9, priority revision 5. TASK.json explicitly changes scope to this live numerical-storage handoff because the helper-contract leads have newer completions. This complements PR2708/2731/2743 payload bounds, PR2749's count-storage component and the adjacent serialized-continuation and larger-child claims. It does not repeat them.

Evidence

• Before: total numerical reserve remains open in PR2697; a nine-word scratch budget or twelve-slot arena rejects the actual inverse witness.
  After: ten additional words suffice, and the unmodified M=5,a=2 routine attains ten and returns 3. Three borrowed words plus the template give exactly 70 occupied non-LEFT symbols. Seven unrelated borrowed words raise the measured total to 105. Thirteen arena slots suffice.
• Registered CPython 3.12.15 replay passes 12,345 finite arithmetic calls and 56 alias/width cases. Results agree with independent integer arithmetic; literal cell scans, operand/head preservation, result ownership and full erasure pass. Nonunit inversion also reaches ten and releases all temporaries.
• Thirteen focused tests pass under -O, including one-word-underbudget rejection for all six primitives, exact capacity, constant aliasing, nonunit cleanup, explicit caller rollback, source rejection before import, dependency/path/blob/pin corruption and an understated receipt. A Git-free export remains byte-unchanged.
• All 70 repository research-workflow tests and scoped Ruff diagnostics pass. Optimized direct replay also passes on CPython 3.13.5. Platform: Linux x86_64 under WSL2.

uv run --python 3.12 --no-project python -I -Werror -B \
  scripts/research_package.py --package research/selected-word-scratch
uv run --python 3.12 --no-project python -B -m unittest discover \
  -s tests -p 'test_research_*.py'

The finite domain checks the actual code; the uniform bound rests on the written lifetime invariant. Evidence is author-checked, without a separate reviewer process or formal kernel. Caller-resident matrices, callback frames/index words, other storage categories, physical placement and total reserve admission remain open. Allocation exceptions require caller rollback. No selected GL64 execution, full caller/cold replay, supplier adoption or new kappa is claimed.

Started from canonical main e93f2ebd9c60b79e564b07acffdbbb99b477c28c. Submission refresh checked 936fac15b1b1b59dc5285c713a77f9f3c07b2906, unchanged guidance/selection/runner and unchanged PR2697 head. Complete overlap covered 705 open PRs and 1393 Discussion records without warnings; the scoped thread was also read directly. Metadata coverage is not a novelty guarantee.

Merge Danger

Door: two-way

Additive research package with immutable predecessor exports. No selected frontier, production implementation, guidance or frozen evidence is changed.

Blast Radius: research