Research audit: sharp physical counter costs with direction reversals (no new kappa)

[Kairos-Ess]

The existing physical ripple-counter theorem correctly bounds calls in one fixed direction. This research note / adversarial mathematical audit supplies an exact signed-potential identity for arbitrary directions and a sharp run-based invoice: for width w >= 1, N calls in K maximal constant-direction runs cost at most 4N + 2(w-1)K literal transitions. If H runs contain wraps, the sharper bound is 4N + 2(w-2)K + 2H.

Alternating wraparound calls attain the uniform bound; alternating nonwrapping calls across the middle edge attain the no-wrap bound. At width 4 and seven calls, these cost 70 and 56 transitions respectively, whereas an invalid extension of the old fixed-direction bound gives 36. This establishes that excluding wraps alone does not justify arbitrary-direction O(N+w) for this primitive. It identifies no error in the existing fixed-direction theorem or selected multiplication construction, and claims no new kappa.

Eight isolated files under research/counter-reversal-audit/ contain the handwritten uniform proof, an independent single-cell tape interpreter, exact deterministic receipt, source pins, fault-injection tests, scope/attribution, validation and local sprint claim. The primitive invoice excludes controller dispatch, preparation, payload, reset/restoration and recursive scheduling; those remain explicit obligations. Finite replay is not a formal proof of the universal statement or full multiplication theorem.

Reproduce with Python 3.10+ (standard library):

python3 -I -B -O research/counter-reversal-audit/verify.py
# PASS_COUNTER_REVERSAL_AUDIT schedules=130305 literal_executions=16376 controls=10
python3 -I -B -O research/counter-reversal-audit/test_audit.py
# Six tests pass.
make selected-record-check machine-transfer-check

All commands pass on Linux / Python 3.14.7; the audit and tests also pass without -O. Six source SHA256/Git-blob pins are checked. Exhaustive coverage is all words of widths 0..10 in both directions with four exterior patterns, plus all words of widths 0..7 and all direction strings of lengths 0..8. Another 256 seeded length-257 schedules are sampled. The fixed domain and stopping rule are recorded. Fault injection checks omitted head returns, suppressed wraps, incorrect endpoints and source drift under optimized Python. git diff --cached --check passed. No full make verify, selected supplier cold replay, Lean rebuild or remote CI pass is claimed for this isolated addition.

Base: f567884aeb8851b6e794af1184119a3bfbe1e631 (still upstream main at submission). Assignment seed 67, drawn with OS-backed secrets.randbelow(100). The accepted result was followed through the #799 certificate, construction and review; selected κ remains 1661634653669567/10^18. No existing files or upstream/ changed. No coordination issue exists, so the claim and its result are local. A final scan of 656 open PRs through #859 found no matching reversal audit; #565 is distinct counter-copying work and is not a dependency.

Credit rfu08 for the physical counter, fixed-direction formalization and already documented direction limitation, published at 2d970ab3ca0271cfb2b609dc9aae1b7073b7774d, and the repository manuscript authors for the scan argument. Contribution by Kairos Ess (@Kairos-Ess) with substantial OpenAI Codex assistance in derivation, implementation, verification and writing. No separate human or delegated-agent review is claimed. Apache-2.0; predecessor notices retained.

[Mister-Meeseeks]

100-PR batch review at 7113da8d458f5c73fe70b1bd5371ea1e44bc75d4.

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

Reviewed scope: Counter-reversal traffic bound with explicit direction-run charges; fixed primitive model, not a full tape machine.

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.