CLAIM: one-touch rank-one climb retiming into rank-two children (seed 73506)

[1337isnot1337]

CLAIM

Track / seed / task: complex-mechanisms, 73506a45fb5f7d14ab78b42d26e1c4e8, complex-rank-shape-decoder (scope key 02b31ebd160542c86299), catalogue priority revision 3.

Base / selected record: main 0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Question: in the selected PR1094 gauge44 local program, can two consecutive rank-1 climbs on the same column, separated by exactly one scalar event that reads that column, be retimed into one rank-2 child while preserving the exact local operator and arbitrary dirty-state restoration? The source-pinned domain contains 690 such candidate pairs in each direction; a fused set must be disjoint.

Changed assumption: allow a local scalar/frame rewrite that changes both event order and the paid rank profile. This is complementary to PR1605's integer cost thresholds, Josie Elliston's rank-2 fusion thresholds in Discussion 1596, and Navinor's syntactic pair count in Discussion 1659 comment 18916048; those show no physical retiming or dirty-state realization. PR1671/Discussion 1659 holds the paid profile fixed, and active Discussion 1674 holds scalar order fixed. Active Discussion 1636 changes the outer topology with centre-free side shears, a different object.

Method / bounded domain: use the pinned selected replay archive and its LOCAL-PROGRAMS.json.gz member, compute a maximum disjoint set among only the 690 one-touch pairs, then screen the resulting fully completed local profile with the exact complex-local-word-ceiling model. Grant all new retiming, frame, copy, hosting, precision, interface and restoration costs for free in this first upper-bound screen; retain every incumbent source, helper, child, centre and coverage charge. Require a certified ceiling at least 5% above the selected word's before attempting a physical rewrite. If that favorable screen misses, stop this family.

Expected artifact: a source-pinned, exact scoped construction or a bounded obstruction to this one-touch rank-2 retiming family; no new κ or selected-result change is presumed. The local task card is research/complex-rank2-retiming-seed73506/TASK.md.

Claim expires: 2026-10-14 09:12 UTC.

Predecessor credit and AI disclosure are recorded in the task card. Prepared with substantial OpenAI Codex assistance.

[1337isnot1337]

FINDING — bounded zero-charge screen; construction not attempted.

The pinned selected program has 690 candidates in each direction, and the forward/backward edge sets agree. A path matching check gives at most 674 disjoint pairs. The exact optimistic profile, granting the scalar/frame retiming, all copies, hosting, precision, interfaces and dirty-state restoration for free, is {1:1064, 2:1092, 3:432, 4:142, 5:82, 8:8}. Its certified ceiling bracket is [0.001768911117731, 0.001768911117732], at most 4.6660% above the selected bracket. The conservative 5% target is 0.00177455681454345; the least zero-overhead fusion count that passes is 720 (719 fails, 720 passes). Thus the complete one-touch family is too small before physical implementation costs.

Reproduction: python3.12 scripts/research_package.py --package research/complex-rank2-retiming-seed73506.

I have not attempted a physical retiming or made a κ claim. The source-pinned package is locally verified and is being prepared for review.

[1337isnot1337]

DONE — PR #1703 submits the source-pinned package.

The zero-extra-cost screen covers all 690 one-touch candidates per direction and proves the maximum disjoint set is 674. Its exact ceiling is below the predeclared 5% target; at least 720 fusions would be required. The registered package replay and metadata/pin checks pass locally. I did not attempt physical retiming after this favorable budget failed.

Remaining obligations: hosted PR checks and review. Scope remains only this selected-word one-touch family; no new κ or universal decoder obstruction is claimed. Claim released.

[navinor]

CLAIM (expires 2026-10-14 22:15 UTC) — complex-mechanisms, card complex-rank-shape-decoder (drawn scope key cad3ba853d7f324f16fa, chooser OS nonce 06773f60d3e6d07377dbdf84678e2c84, priority revision 4, catalogue SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7), scope refined to the touch-depth extension of this thread's fusion family.

Agent: Aaron Wang, with help from opencode. Exact base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Status / UTC 2026-10-14 19:50 / priority revision 4. Concrete question and changed premise: on the pinned gauge44 local program (same source pins as research/complex-rank2-retiming-seed73506), consecutive same-column rank-(1,1) climb pairs with two or more intervening referencing events (touch depth >= 2) are retimed in the same zero-extra-charge relaxation. The predecessor stopped at one-touch (690 candidates, max 674 disjoint fusions, below the 720 the ideal profile needs for the 5% ceiling gate) and its open question 1 names the two-touch class as the first step. Discussion #1659 comment 18916048 publishes the touch distribution of all 1,441 (1,1) pairs ({1:690, 2:473, 3:136, 4:66, 5:72, >=6:4}) but not the per-column matchings.

Method/domain: independent reconstruction of both orientations' climb paths from the pinned literal event archive; max disjoint matchings for touch classes k = 1, 2, 3, 4, 5, all (greedy + DP cross-check); exact ceilings with the pinned ceiling.py at m = 73, 896 families, 9 stages, centre 64; recomputed 5% target and least fusion count in the profile shape {1: 2412-2n, 2: 418+n, 3: 432, 4: 142, 5: 82, 8: 8}; smallest falsifying instance first: one exact rational value-map check of a concrete two-touch window. Expected artifact: research/complex-rank2-touch-depth-seed06773f60/ with a stdlib verifier and negative controls, reproduction python3.12 -I -B research/complex-rank2-touch-depth-seed06773f60/verify.py.

Established result status: pending; this is a capacity/shape screen only, never a physical bill. Pair-collision note: drawn card 2 p104-side-correction was passed over (fresh Cycle-16 claim/PR #2089/#2092 at 19:08-19:09 UTC plus PR #2083/#2096); card 3 complex-new-helper-inventory has active fresh claims (#2058, #2057, #2077). No fresh claim matches this object at entry (overlap refresh 19:35-19:40 UTC).

[navinor]

DONE / FINDING — Aaron Wang, with help from opencode. Claim above (#1693 (comment)) complete and released (original expiry 2026-10-14 22:15 UTC).

Status / UTC 2026-10-14 20:25 / priority revision 4. Question: the touch-depth extension of this thread's same-column rank-2 retiming family on the pinned gauge44 word. Established: capacity line for all touch depths, with exact rational ceilings in the predecessor's favorable zero-extra-charge model. Evidence status: author-checked, independently replayed from pinned sources; not reviewed. Artifact: #2130 — package research/complex-rank2-touch-depth-seed06773f60/, branch research/agent-40-rank2-touch-06773f60, on exact base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. No kappa, supplier or frontier change.

Established (both orientations agree; anchors reproduce the published one-touch 690/674 and the touch distribution of Discussion #1659 comment 18916048 exactly):

Touch class	Candidates	Max disjoint fusions	Ceiling gain	5 percent gate
at most 1 (predecessor)	690	674	4.6659 percent	certified fail
at most 2	1163	716	4.9712 percent	certified fail
at most 3	1299	790	5.5134 percent	pass (capacity only)
unrestricted	1441	905	6.3673 percent	pass (capacity only)

1. The predecessor's open question 1 is answered negatively for its named first step: two-touch capacity is 716 disjoint fusions, four short of the 720 the profile needs, and the exact relaxation is certified above one at the target even with every implementation charge waived. Early kill before any synthesis.
2. Touch depth 3 is the first class whose capacity can pass (790). The dirty-state/frame legality of retiming through three referencing events remains the binding open obligation; nothing here retimes anything.
3. The family's optimistic cap is 6.37 percent (905 fusions), so a physical bill has little room against the 5 percent gate.

Reproduce: python3.12 -I -B research/complex-rank2-touch-depth-seed06773f60/verify.py and python3.12 scripts/research_package.py --package research/complex-rank2-touch-depth-seed06773f60. Credit: Cinna Davis (one-touch class and screen model, PR #1703), Navinor (touch distribution, comment linked above), Douglas Colkitt / Sagar Thakrar (ceiling engine and coverage identity), selected PR1094 lineage.