[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. For the standard prefix-product encoder on every finite group G of order q:

complete endpoint operator norm                 = 1
best fixed positive transfer-metric norm bound  = sqrt(q)
feasible contractive metrics                   = t I, t > 0

The loss persists under every fixed positive blocking length and every invertible complex virtual basis change. This is a new sharp limitation of the sufficient certificate in Ali Romman's merged PR5451, using zxv's merged PR5441 physical-transfer contract and prefix-XOR example. The predecessor theorems are cited, not audited or replayed. Trace invariance, congruence and weighted operator norms are classical. No new kappa or selected-frontier change.

Only research/prefix-encoder-metric-loss/ is added. The proof handles all finite groups and all positive block lengths. The finite checker is corroboration, not the universal proof or formal-kernel acceptance. Advance CLAIM.

Evidence

Before, the cited criterion supplied a sufficient error estimate without a sharp optimum for this encoder. The new proof characterizes every feasible positive metric and gives matching boundary lower and upper bounds.

Local CPython 3.12.15 checks pass in isolated warning-strict normal and optimized modes, with identical generated receipts. They enumerate six groups including noncommutative S3, 21 blocked tables, 787 literal blocked paths, 85,873 metric-commutator entries, 17,792 complete endpoint entries, 45 explicit negative metric witnesses and 12 rational gauge certificates. The 63 controls comprise seven invalid group/block cases, one changed terminal boundary, one imaginary off-diagonal metric and 54 strict boundary/rate underclaim witnesses.

Registered metadata and both scientific commands pass. Each scientific command runs in under one second locally. Exact committed bytes and declared source-history ancestry pass at 3443b645a127f2bd30f08d43b850d56941f6ccb9. The official nine-file export passes integrity checking; both modes pass from its read-only Git-free tree with an empty PATH, preserving all ten extracted files' bytes and modes. AST compilation, function-length inspection and whitespace checks pass. Receipts are generated by execution and bind the seven package files and two source inputs.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/prefix-encoder-metric-loss/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/prefix-encoder-metric-loss/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/prefix-encoder-metric-loss
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/prefix-encoder-metric-loss --check-source-history

Other machines may substitute their CPython 3.12 path. The scientific command prints PASS_PREFIX_ENCODER_METRIC_LOSS; the registered runner prints PASS_REGISTERED_RESEARCH_PACKAGE. Standard library only.

Limitations and unrun obligations

This fixes the standard representation and the single fixed positive-metric criterion. Larger or different virtual representations, boundary-aware methods and position-dependent metrics are outside the theorem. The example is a complete algebraic endpoint, not a physical native/tape implementation or a priced supplier. Physical synthesis, dirty restoration, paid costs, compiler closure, precision, stock and exact recovery remain open. No selected cold replay, native suite, Lean or Docker build, independent human review or proof-kernel acceptance is claimed.

Open next questions

• Construct a sound boundary-aware certificate attaining norm one for this encoder while retaining coherent physical inputs and complete residuals.
• Determine whether an explicitly different virtual representation can remove the loss. Invertible basis changes of this representation are already covered.

Related open work

A fresh scan of this account's open PRs created since the CLAIM found 14 other submissions and no additional work on this prefix-encoder norm or transfer-metric optimum. The complete cached open-PR search and current live PR/claim and main/merged searches likewise located no competing result. PR5451 and PR5441 are merged predecessors. Sparse local notes, arbitrary historical diffs and unpublished work remain search limits.

• PR6050 targets the shared onboarding-documentation blocker described below. It has no mathematical overlap with this result.

CI notes

CI uncertified. At head 3443b645a127f2bd30f08d43b850d56941f6ccb9, run 40726682190 failed plan and skipped replay (research/prefix-encoder-metric-loss). The onboarding suite ended with exactly failures=2, skipped=23:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects literal **0.003125804** in docs/research/current-status.md. The document instead puts the decimal and fraction inside one bold span.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in the current section of docs/ci-verification.md. That section still describes the PR5599 replay.

These are inherited documentation mismatches from main commit 721c03822c421b7f7fcbb2598993b947c4d88419, the PR6030 kappa admission pushed with [skip ci]. It changed the selected record without matching the required navigation and decimal formatting. This PR changes neither documentation file. The same two failures are confirmed on PR6120, run 40691776027. Open PR6050 targets the fix.

The package replay did not run in CI. This PR is therefore uncertified by CI and needs a maintainer re-run after main is fixed. The passing local registered normal/optimized replays, committed-byte/source-history checks and read-only exported replays remain the available execution evidence. No branch update or documentation fix was made for this CI failure.

Merge danger

Door: two-way. Blast radius: package. The change adds one isolated research package with two declared source-document inputs.