Rank-2 retiming capacity line at touch depth >= 2 on the pinned gauge44 word (no new kappa)

[navinor]

Rank-2 retiming capacity line at touch depth >= 2 on the pinned gauge44 word

Aaron Wang, with help from opencode (substantial AI assistance; no independent human review claimed). No new kappa, supplier, selected-record or frontier change.

Scope

One bounded object: on the pinned selected gauge44 local program (research/community-frontier1094-audit/selected-replay.tar.gz, member candidate/strict/global/base/providers/gauge44/LOCAL-PROGRAMS.json.gz), screen the complete family of disjoint fusions of consecutive same-column rank-(1,1) climb pairs into rank-2 children, with the intervening referencing events retimed for free, parameterized by the number of intervening referencing events (touch depth). Package: research/complex-rank2-touch-depth-seed06773f60/ on branch research/agent-40-rank2-touch-06773f60, head commit on top of base main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2.

This is the exact first step named by research/complex-rank2-retiming-seed73506 open question 1 (Cinna Davis, #1703): that screen covers only pairs with exactly one intervening scalar event (690 candidates, maximum 674 disjoint fusions) and stops below the 720 fusions its zero-overhead profile needs to beat the selected word's ceiling by 5 percent. Drawn card complex-rank-shape-decoder (scope key cad3ba853d7f324f16fa, chooser OS nonce 06773f60d3e6d07377dbdf84678e2c84, priority revision 4); scope status refined, recorded in the package TASK.json and the local task card.

Exact result

Both orientations agree on every number. The reconstruction reproduces the published anchors exactly: 2,782 consecutive same-column climb pairs, 1,441 of rank (1,1), touch distribution {1: 690, 2: 473, 3: 136, 4: 66, 5: 72, >=6: 4} (Discussion #1659 (comment)), one-touch 690 candidates with 674 disjoint fusions, and the 720-fusion threshold (719 certified fail, 720 certified pass). Gains are certified ceiling ratios against the selected bracket upper endpoint 0.001690054109089; the gate is the recomputed target a = 0.00177455681454345, equivalently 720 disjoint fusions in the profile shape {1: 2412 - 2n, 2: 418 + n, 3: 432, 4: 142, 5: 82, 8: 8}.

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)
at most 4	1365	854	5.9869 percent	pass (capacity only)
at most 5	1437	901	6.3374 percent	pass (capacity only)
unrestricted	1441	905	6.3673 percent	pass (capacity only)

1. Two-touch is falsified at the capacity level: 716 disjoint fusions is four short of 720, and the exact relaxation at the target is certified above one even with every implementation charge waived. Per autoresearch section 4 this kills the two-touch candidate before any synthesis or value-map check, and answers the predecessor's open question 1 in the negative for its named first step.
2. Touch depth 3 is the first class whose capacity can pass (790 at least 720). Whether a legal retiming through three referencing events exists is not established here and is the binding open obligation.
3. The whole family's optimistic cap is 6.37 percent (905 disjoint fusions, touch unrestricted), in a model that charges nothing for realizing any fusion. A physical bill must come out of that margin.

Cost boundary and limits

Every incumbent source, helper, recursive-child, centre and coverage charge stays in the profile (local mass 5,586, idle mass 1,646 at idle rank cap 51, halving guard checked). The screen grants retiming, copies, hosting, precision, interface transitions and dirty-state restoration for free, exactly as in the predecessor's favorable model. A pass is a capacity/shape survival of this profile family only. Nothing here proves that any pair can be retimed; mixed-rank pairs, non-adjacent climbs, other words, changed coverage laws and every physical supplier obligation are out of scope. Not checked: the selected cold replay, any formal kernel, precision or all-size admission.

Reproduce

Stdlib-only, Python 3.12, from the repository root:

python3.12 -I -B research/complex-rank2-touch-depth-seed06773f60/verify.py
python3.12 scripts/research_package.py --package research/complex-rank2-touch-depth-seed06773f60 --check-only
python3.12 scripts/research_package.py --package research/complex-rank2-touch-depth-seed06773f60

Expected markers include PASS_RANK2_TOUCH_DEPTH adjacent=2782 pairs11=1441 dist={'1': 690, '2': 473, '3': 136, '4': 66, '5': 72, 'ge6': 4} matched k1=674 k2=716 k3=790 k4=854 k5=901 all=905 min_fusions=720 gate_all=True, PASS_CONTROLS {...}, and PASS_REGISTERED_RESEARCH_PACKAGE research/complex-rank2-touch-depth-seed06773f60 outcome=completed. All three commands pass locally on CPython 3.12. Controls: forward/backward agreement, greedy/DP matching equality with vertex-disjointness, 719/720 threshold, touch-bin mutation, inflated target, and exact RESULT.json equality on replay. Source pins (archive, member, ceiling engine, selected record) are in SOURCES.json and checked by the verifier.

Coordination and overlap

Claim (expires 2026-10-14 22:15 UTC): #1693 (comment) with a short pointer in #1166 (comment). Pre-submission overlap refresh at 2026-10-14 20:13-20:15 UTC covered 386 open PRs and 792 discussion records with complete pagination and found no competing touch-depth retiming claim beyond the predecessor. Pair-collision note: drawn card 2 p104-side-correction was passed over (fresh Cycle-16 claim and PR at 19:08-19:09 UTC today, #2092 and #2089, plus #2083 and #2096); card 3 complex-new-helper-inventory has active fresh claims.

Credit

The one-touch class, its 690/674 census, the 720 threshold and the favorable screen model are Cinna Davis's (research/complex-rank2-retiming-seed73506, #1703). The touch distribution of the 1,441 (1,1) pairs is Navinor's (#1659 (comment)). The ceiling engine and coverage identity are Douglas Colkitt's and Sagar Thakrar's (PR 1240 / PR 1012). The selected gauge44 word, compiler and owner lineage is credited in the selected record's review. Adjacent but distinct families: research/complex-gather-child-seed7a6d (#1814), research/complex-source-frame-fanout-seedc309, research/complex-staggered-scratch-screen-seed4c027, and the fixed-order contiguous rank-2/3 packing of PR 1960 / #2014 (research/chronological-rank23-majorization), which does not retime through referencing events. New package material is Apache-2.0; source licenses and notices are unchanged.

Remaining obligations

A dirty-state/frame legality lemma for three-touch pairs, a mixed-rank weighted matching screen, and an allowed-overhead budget for the surviving classes (see the package README's open next questions). No supplier admission, kappa change or frontier change is requested; merging remains a maintainer action.