Research consolidation: uniform proof of the archived 2P+1 stage bound

[0xjohnnydev]

Change and mathematical scope

The archived campaign report M claims that exchanging P bank pairs requires at least 2P + 1 stages. Its discussion mixes a four-call rank argument with finite searches for P <= 3. This research consolidation supplies a self-contained uniform argument and confirms the same bound; it does not claim a new kappa or controller.

For two blocks of dimension Pv, consider calls that add a rank-at-most-v linear function of one block to the other, with arbitrary invertible block-diagonal bridges. The potential rank(A) + rank(D) starts at 2Pv and ends at zero. The first call cannot lower it, and every later call lowers it by at most v. Thus at least 2P + 1 calls are needed, over every field. The existing Sussman borrowing chain attains the count. The endpoint is measured before the designated terminal exchange; physical frames, helper contracts, restoration and all-size/analytic interfaces remain separate obligations.

Additions are one self-contained research package and a path-filtered verification workflow. The original report and its notice are preserved byte for byte. The scalar construction is credited to Jacob Sussman and the retained controller lineage, with explicit acknowledgment of PR397's existing indexed Lean identity. No novelty is claimed for that identity. The selected certificate and upstream/ are unchanged.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. OS-random assignment seed: 97, research consolidation. No pinned coordination issue existed, so the narrow claim is local. SOURCE-PINS.json records the historical PR368 commit, all input hashes and the cited PR397 head. Prepared by Johnny Franks (0xjohnnydev) with substantial OpenAI Codex assistance, Apache-2.0. No independent human or proof-assistant review is claimed.

Validation

python3 -I -B research/cross-shear-stage-bound/verify.py

Expected: PASS_CROSS_SHEAR_STAGE_BOUND with 28 rational witnesses, 20,160 binary prefixes, 362,880 cross-call transitions and 725,760 bridge transitions.

• The independent grouped-coordinate implementation agrees with the pinned scalar recipe for P = 1 through 16. Exact rational witnesses also cover widths 2 and 3. These are scalar checks, not new physical layouts.
• Complete finite checks cover all GL(4,2) prefixes, every rank-one two-by-two cross call and every invertible block-diagonal bridge. The universal statement rests on the written rank proof, not this census.
• All 276 omitted-operation or reversed-sign variants have the wrong endpoint. Five further model/field controls pass.
• Clean source-only trees pass on Python 3.9.6, 3.12.14 and 3.14.8, normally and under -O. Damaged source and incomplete-coverage receipts are rejected, including under -O.

Focused Linux CI also passed on Python 3.11 and 3.14, normally and under -O, at the submitted commit f39226f49486f44e9771b88121ad11f9d0b76eee: run 38349208387. This is the fork's focused workflow, not a claim that every upstream CI job has passed. The patch applies cleanly to the exact base, and whitespace checks pass.

The single local make verify run passed the community, producer, partial-gauge, three-stage-cover, paired-cube, recycled-bit and entrance-bank sections, including the selected-record integrity checks. It then stopped at the unchanged gen4-bank-verify target with ModuleNotFoundError: No module named 'sympy' (exit 2). Later sections, including the selected frontier cold replay, were not run. This is an environment limitation, not a full-suite pass. The run left the tracked working tree clean; no unrelated regeneration is included.

The overlap review covered all 551 open PR bodies through #732, then refreshed the latest arrivals through #742 immediately before submission. No duplicate uniform cross-shear lower-bound proof was found. PR397 supplies existing positive-identity formalization; #695, #704, #705 and #720 address distinct evidence or circuit models.

[Mister-Meeseeks]

100-PR batch review at f39226f49486f44e9771b88121ad11f9d0b76eee.

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

Reviewed scope: Rank-potential proof of at least 2P+1 cross-shear calls with block-diagonal bridges; existing borrowing chain attains it, only within that model.

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.