[cbethin]

PR985 gives a conservative O(r³ W_total) allowance for sequential factor lookup. This research audit proves that its unchanged balanced word admits quadratic factor-tape travel under an explicit contiguous, fixed-width layout.

For every n ≥ 1 and both call directions, the closed index travel is less than 2N, where N = 3qn − 2q² and q = 2^floor(log2 n). A reader that copies every B-bit descriptor and restores its head uses fewer than 4NB descriptor-head moves and exactly NB payload reads. The complete lookup bound is P_layout + P_parse + O(G + N(B + log²(nB+2))); setup, parsing, common parameters, preflight, copies, and amortized control are explicit. The proof also gives the exact quadratic variation on powers of two.

This refines a safe access allowance, not a false predecessor theorem. Preparation, numerical precision, arithmetic, mutable-array routing, recursive children, and physical/compiler obligations remain paid separately. It does not certify PR1054's complete invoice, improve the selected κ, or claim a measured runtime gain.

Verification

python3 -I -O -B research/balanced-descriptor-travel/test_package.py

Expected: PASS_DESCRIPTOR_PACKAGE with a nested PASS_BALANCED_DESCRIPTOR_TRAVEL receipt.

• 150,126 actual lookups from the byte-exact PR985 dispatcher across 78 signed cases; two further cases vary array geometry.
• A separate compressed word algebra checks 4,241 counts. Literal bit-head and countdown models check 102 tape cases and 1,025 counter cases.
• Nine mathematical/tape controls and five package faults reject. The full audit passes from a read-only relocated copy.
• All five vendor files match their pinned Git blobs. Selected-record integrity and all 25 selected-result tests pass. CPython 3.15.0, standard library only.

The all-size claim rests on the written induction and paid reader construction. The main dispatcher fixtures use length-one arrays and zero coefficients to exercise the fixed dispatch structure; these are not numerical precision tests. Independent checking means separate implementations/models, not independent human review. Full repository/cold supplier replay, hosted CI, and formal verification of the multiplication theorem are not claimed.

Scope and attribution

Only research/balanced-descriptor-travel/ is added. The publication base is 1b59b624d21924f8e3537cdc1ac52d3be6eda2fe; the experiment source is PR985 da1cdf8ed4ac9b1b8378dfec73f77ff4e81a4f70. The local seed-65 claim records continuation, finite domains, inspected heads, current discussion review and overlap checks through PR1206. PR1054, PR1085, PR1135, and discussion1202 concern different precision, magnitude, or representation questions.

Credit Ali Romman for PR985, Chafik Boukhalfa for PR628's balanced commutator and preparation lineage, and 0xkilgore for PR663's leaf count. Retained licenses and predecessor notices are unchanged. Prepared by Charles Bethin with substantial OpenAI Codex assistance in derivation, implementation, verification, and writing. No literature-priority claim or external human review.

[Mister-Meeseeks]

100-PR batch review at 80cb75bc135c9efc112e91801e3fa50ac300a8d9.

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

Reviewed scope: All-length balanced-word descriptor travel: variation at most 2(N−n), paid sequential lookup O(G+N(B+log²(nB+2))) plus explicit layout/parsing. Inductive trace and bit-tape proof; no full precision/compiler theorem.

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.