Research: higher-rank paired boundary at a single ADD gate

[1337isnot1337]

Scope

Cycle seed 3444f4b8606069e8d454447c1dc60c12; track complex-mechanisms; task complex-rank-shape-decoder; priority revision 3; exact base 0b1cccf86e2f5132ed286a26ef0a681a495740ea. Selected-record SHA256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

This is a finite conditional interface screen for one binary ADD gate with two positive frame climbs where at least one quotient rank exceeds one. It extends the rank-one J2 proposal in draft PR #1764 by treating the higher-rank pair (r,s) as one hypothetical J_{r,s} child of rank r+s. It does not admit that supplier or claim a new κ.

Exact mechanism and census

On the pinned selected local word (compressed FULL-WORD.json.gz SHA256 49596aa4ed4fd36cd4582fd0e8ddeb09fd80370a7d8aea1ede52a135b536d289; decompressed SHA256 c2b294300760e1fd6eeab8c22c4d13e4633dca40e52cac8da0d68163a76e3ca8), each qualifying gate is the exact shear T_c(x,y)=(x+c y,y), determinant one with inverse T_-c. If the old dirty frames are U,V and the common target is F, then U×V has codimension r+s in F×F; the invertible shear preserves its dimension, so the joint boundary has codimension r+s.

The full scan checks A, B and scheduled cleanup-tail operations in both orientations, exact rational coefficients, monotone frame inclusion and old/new endpoints. It finds 359 higher-rank gate pairs:

• (1,2): 72; (1,3): 52; (1,4): 35; (1,5): 26
• (2,2): 14; (2,3): 5; (2,4): 19
• (3,3): 122; (3,4): 6; (4,4): 8

CENSUS.tsv records the exact source gate and frame identifiers for each row; RESULT.json is reconstructed by the verifier.

Conditional cost result

The comparison starts from the selected profile {1:2412,2:418,3:432,4:142,5:82,8:8}. For a like-for-like conditional extension it reconstructs PR #1764's 753 rank-one pairs (590 same-gate pairs plus 163 terminal pairs), then applies only these 359 higher-rank gate pairings:

• PR Research: rank-two interface candidate for paired frame transitions #1764 candidate profile: {1:906,2:1171,3:432,4:142,5:82,8:8}
• Extended profile: {1:721,2:1047,3:197,4:132,5:96,6:167,7:6,8:16}
• Rank mass remains 5,586; calls per local word fall from 2,741 to 2,382.

Using the selected fixed-word ceiling routine on a 10^-12 grid, the conditional ceiling improves by at least 5.956346700% over the PR #1764 candidate and 11.510179694% over the selected baseline profile. This assumes zero extra joint-interface cost.

At 1.05 times the upper endpoint of the PR #1764 ceiling, the exact rank-one-equivalent overhead allowance is 43 additional child costs per local call; 44 fail. A hypothetical one-rank-one adapter per each of the 359 boundaries gives a relaxation moment above one and misses the 5% incremental target. The complete source, copy, helper, hosting, restoration, phase and precision ledger is in the README.

Reproduce and limits

python3 -B research/complex-joint-higher-rank-seed3444f4/verify.py
python3 -I -O -B research/complex-joint-higher-rank-seed3444f4/verify.py
python3 scripts/research_package.py --package research/complex-joint-higher-rank-seed3444f4 --check-only
make selected-record-check

Both direct replays pass under local Python 3.14.7; package source-pin validation and make selected-record-check pass. Python 3.12 is unavailable in this environment, so the registered Python 3.12 replay was not run locally. Four mutation controls reject a changed coefficient, rank sum, inverse matrix and old-frame endpoint.

Predecessors are PR #1764 / Discussion #1765 (rank-one joint pairs), PR #1814 / Discussion #1167 (three-source consecutive gather), and PR #1817 / Discussion #1596 (diagonal rank-one batching). This package does not compose their profiles. The proposed J_{r,s} supplier, physical adapters and complete-call hosting remain open; no global assembly, selected-result change or new κ is claimed.

Substantial OpenAI Codex assistance is disclosed in the package. The research claim and final overlap snapshots are recorded there; the claim is Discussion #1167, comment #18922740.