[petalas]

Summary

Scoped mathematical proof with exact finite corroboration. The exact initially clean capacity for destructive reversible overwrite of the raw Gaussian tensor C^tensor n x mod pi^P is

K(n,P) = sum_j binom(n,j) min(2j,P) bits

T_n -> rotate (live,lost,clean) to (clean,live,lost) -> S_n

Arbitrary finite dirty workspace cannot reduce this minimum, even if its final value may change. The explicit permutation attains it while returning dirty workspace unchanged. For normalized precision p>=n, the minimum is n*2^n bits. These bits retain kernel information and cannot all be reset from the modular response alone.

This answers the destructive-overwrite portion of Open next question 4 in Ali Romman's #5821. Its image-information theorem and Dugongue's #5815 Smith factors are cited, not audited or replayed as this contribution. The source-preserving array route retains different information. No new kappa or selected-frontier change is claimed.

Evidence

• Before: Execute Gaussian Smith circuits and prove sharp precision requirements #5821 explicitly leaves reversible-host truncation and cleanup open. Its owned array reset is not a destructive reversible interface.
• After: PROOF.md gives the exact lower bound, an attaining full-state permutation and inverse, and the cleanup obstruction. Local CPython 3.12.15 warning-strict normal and optimized checks pass.
• New finite evidence covers all 17,898 states of 14 complete augmented permutations, 646 clean endpoints, 432 larger interfaces with 272 clean endpoints, 1,492 individual retained-bit witnesses, 216 dirty-capacity comparisons, ten invalid-input controls and four semantic controls.
• Both registered commands pass and each direct check takes about 0.81 seconds locally, below its 30-second timeout. Metadata, staged whitespace, committed-byte/source-history preflight and the official seven-file export pass.
• Read-only Git-free normal and optimized exported replays pass with empty PATH. All eight exported files, including the export manifest, retain their bytes and modes. The verifier generates a source-hash-bound stdout receipt and leaves no generated files in the checkout.

Exact reproduction from the repository root:

PY=/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12
"$PY" -W error -B research/gaussian-smith-reversible-capacity/verify.py
"$PY" -O -W error -B research/gaussian-smith-reversible-capacity/verify.py
"$PY" -B scripts/research_package.py --package research/gaussian-smith-reversible-capacity
"$PY" -B scripts/check_research_commit.py --package research/gaussian-smith-reversible-capacity --check-source-history

The standard-library replay needs no predecessor code, pending Git objects or network. The written all-size argument uses the cited predecessor theorems. Finite executable bounds are n<=8 and P<=16; the complete state enumeration is capped at 14 total bits. These are corroboration bounds, not bounds on the theorem.

Limitations

The auxiliary capacity starts clean and finishes with retained information. Reusable clean restoration, physical reversible primitives, operation counts, tape routing and a complete caller remain open. The original source-preserving contract is outside the overwrite lower bound. All five selected framework assumptions remain. No Lean, complete rounded multiplier, independent human review or formal proof-kernel acceptance is claimed. Hosted replay has not yet certified this package.

Open next questions

• Implement and price the one-axis attaining permutation on one permitted reversible native/tape host, including representation changes and every bit move.
• Compare source-preserving compute-copy-uncompute with overwrite plus retained kernel information at an actual caller cut, using its lifetimes and complete paid bill.

Related open work

• Execute Gaussian Smith circuits and prove sharp precision requirements #5821 supplies the image-information theorem and explicit reversible-host question; this adds the clean-capacity optimum and attaining destructive interface without rechecking its census.

• Construct exact Gaussian corrections with pruned and mixed precision circuits #5815 supplies the Gaussian Smith factors and complete array schedule; this uses those factors and prices a different information-retention contract, without changing its arithmetic invoice.

• Execute pruned Gaussian corrections and bound the standalone Newton-W route #5825 executes spectral corrections and leaves reversible cleanup open; it does not compute the same overwrite clean-capacity quantity.

• Construct Gaussian spectral and precision compilers from the ETH preprint #5813 proposes Gaussian spectral/precision routes; this addresses the modular Smith overwrite boundary rather than spectral delivery or runtime.

• Construct optimal reversible scratch schedules with replay certificates #5780 plans reversible scratch schedules under permitted compute/erase primitives; this instead states the exact information capacity needed for one noninjective Gaussian overwrite.

• Prove sharp tensor composition of dyadic helper capacity (no new κ) #5882 computes dyadic unit-block capacity under tensor composition over Z[1/2]; this computes initially clean bit capacity for a modular Gaussian overwrite, a different ring and quantity. No common numerical value is compared.

• Prove the binary-affine call floor with clean erasure (no new κ) #5879 permits irreversible scalar erasure when bounding binary-affine paid calls; this requires a reversible Gaussian endpoint and bounds clean storage. The operation and cost contracts differ, and no common numerical value is compared.

Completion refresh inspected the nine other account PRs opened since the CLAIM: #5876, #5877, #5878, #5879, #5881, #5882, #5883, #5884 and #5885. None computes the same Gaussian overwrite clean-capacity quantity; no conflicting numerical value was found in their stated scopes. No extra computations were run to validate these contributions.

CI notes

At head 20165927fce9064a148774bcea412adf789a44f4, run 40491196305 finished with plan failed; replay skipped. The only failed step is Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22), exit code 1. The errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight assertion failures concern frozen-card transformation, missing current/historical references and sparse omission lists.

All ten failure headers and exception/assertion messages match PR #5879's plan log. This confirms the reported shared onboarding failure on main de5ec2e7a. The package is unchanged, and no shared source or tests were modified or rerun. Maintainers will handle the shared repair. Hosted replay has not certified this package; only the local scientific checks listed above have run.

Merge Danger

Door: two-way. Blast radius: package. Only the new research/gaussian-smith-reversible-capacity directory is added; no shared implementation, predecessor artifact or selected record changes.

This contribution was produced by Nick Petalas. Credit Ali Romman #5821, Dugongue #5815, classical finite-ring and reversible-embedding arguments, and the unchanged Gaussian/native/controller/compiler predecessor lineage. Apache-2.0.