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

notes/task-card-agent40-rank2-touch.md

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+# Task card — agent-40 — same-column rank-2 retiming fusion at touch depth ≥ 2
+
+Task ID / proposed scope key: `complex-rank-shape-decoder` / drawn scope key
+`cad3ba853d7f324f16fa`, refined to "same-column rank-1 climb-pair retiming fusion
+at touch depth >= 2 on the pinned gauge44 local program" (scope status: refined;
+the card's generic rank-shape question is unchanged, the object is this concrete
+fusion family).
+
+Base commit: `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2` (origin/main at entry).
+Chooser OS nonce: `06773f60d3e6d07377dbdf84678e2c84` (unseeded draw, track
+`complex-mechanisms`, priority revision 4, catalogue SHA-256
+`d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7`, selected-record
+SHA-256 `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`).
+
+Exact predecessors and dependency heads (local paths at the base commit):
+
+- `research/complex-rank2-retiming-seed73506/` (Cinna Davis, PR #1703): the
+  one-touch class (exactly one intervening scalar event) has 690 candidates per
+  direction and maximum 674 disjoint fusions, below the 720 needed for the 5%
+  gate. Its open question 1 asks for the larger chronological class and names
+  pairs with two intervening scalar operations as the first step.
+- Discussion #1659 comment 18916048 (Navinor, PR #1671 lineage): for the 1,441
+  consecutive same-column rank-(1,1) climb pairs, the intervening referencing
+  event counts are `{1: 690, 2: 473, 3: 136, 4: 66, 5: 72, >=6: 4}`; every one of
+  the 2,782 consecutive same-column climb pairs has at least one referencing
+  event between the climbs.
+- `research/complex-local-word-ceiling/ceiling.py` (pinned engine, PR #1240 /
+  PR #1012 coverage identity).
+- Adjacent, different families: `research/complex-gather-child-seed7a6d`
+  (PR #1814, consecutive 3-source gather windows), `research/complex-source-frame-fanout-seedc309`
+  (consecutive shared-source fanout), `research/complex-staggered-scratch-screen-seed4c027`
+  (scratch-capped rank-3 packets), PR #2014 / `research/chronological-rank23-majorization`
+  (fixed-order contiguous rank-2/3 packing within source runs).
+
+Concrete object and question: on the pinned gauge44 local program
+(`research/community-frontier1094-audit/selected-replay.tar.gz` member
+`candidate/strict/global/base/providers/gauge44/LOCAL-PROGRAMS.json.gz`), take
+consecutive same-column rank-1 climb pairs (climb-path adjacent). A pair's touch
+depth is the number of intervening events referencing that column (scalar,
+adapter, residual_adapter). Question: does the maximum set of disjoint fusions
+of such pairs, over the touch depth <= k class, reach the 720 fusions whose
+ideal zero-overhead profile beats the selected word's ceiling by 5%? Where does
+the line fall in k, and what does the exact ceiling say at each depth?
+
+Scope status: refined. Reason: the generic card is saturated; this is the exact
+first step named by the predecessor's open question, extended in one parameter
+(touch depth) on the same pinned word and the same profile shape
+`{1: 2412 - 2n, 2: 418 + n, 3: 432, 4: 142, 5: 82, 8: 8}`.
+
+Existing restriction being changed: the predecessor's one-touch restriction
+(exactly one intervening scalar event); retiming of the intervening referencing
+events is granted for free in the same favorable screen (no operation, copy,
+frame, hosting, precision, interface or restoration cost charged).
+
+Method; explicit parameter ranges: exact per-column climb-path reconstruction
+from the pinned literal event lists (both orientations); intervening referencing
+event counts recomputed independently (anchors: 2,782 adjacent pairs, 1,441
+(1,1) pairs, distribution above, one-touch 690/674); maximum disjoint matchings
+for classes k = 1, 2, 3, 4, 5, all, cross-checked by greedy path matching and an
+independent DP; exact ceilings via the pinned `ceiling.py` at width m = 73, 896
+families, 9 stages, centre 64; required target = 1.05 x the certified selected
+upper endpoint (recomputed, expected a_target = 0.00177455681454345); exact
+least fusion count needed in this profile shape (expected 720). Falsification
+test: an exact rational value-map check on one concrete two-touch window (the
+smallest instance) to see whether a rank-2 child plus free scalar work can carry
+the two dirty dependencies at all.
+
+Nearest overlapping PRs and why this adds something: PR #1703 stops at
+one-touch (674 < 720) and explicitly leaves two-touch unsearched; Discussion
+#1659 publishes only the touch-count distribution, not the per-column
+matchings; PR #2014/PR1960 keep scalar order (contiguous packing inside source
+runs) and do not retime through referencing events; #2058 is cross-destination
+centre-source fanout. This package computes the exact capacity line of the
+retiming family itself.
+
+Optimistic full cost budget: the favorable zero-extra-charge relaxation of the
+predecessor (all retiming, copies, frames, hosting, precision, interface and
+restoration free). Required observable improvement: ideal certified ceiling at
+least 5% above the selected word's certified upper endpoint; a pass is only a
+capacity/shape survival result, never a physical bill; a miss with a strict
+moment margin closes this fusion family for the pinned word.
+
+Success evidence; falsification test; time/compute cap: `PASS_*` markers from a
+stdlib-only verifier with negative controls (a mutated touch bin, a reused
+climb, a wrong target threshold must all fail); seconds of compute; one session.
+
+Predecessor authors; intended artifact and reproduction command: Cinna Davis
+(complex-rank2-retiming-seed73506, gather-child), Navinor (Discussion #1659
+touch distribution), Douglas Colkitt / Sagar Thakrar (ceiling model), selected
+PR1094 lineage. Artifact: `research/complex-rank2-touch-depth-seed06773f60/`
+with `verify.py`; reproduction
+`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`.
+
+Pair-collision note: card `p104-side-correction` (drawn card 2) was passed over
+because its exact object has a fresh claim and PR from 19:08-19:09 UTC today
+(Discussion #2092 / PR #2089, Cycle 16) plus adamgall PR #2083/#2096;
+`complex-new-helper-inventory` (card 3) has active fresh claims (#2058 at
+18:26, #2057 at 18:24, #2077 at 18:47). No fresh claim matches this object at
+entry (checked 2026-10-14 19:35-19:40 UTC).
+
+Aaron Wang, with help from opencode.

research/complex-rank2-touch-depth-seed06773f60/README.md

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+# Same-column rank-2 retiming at touch depth >= 2: capacity line on the pinned gauge44 word
+
+**Result type:** source-pinned capacity/shape screen with exact rational ceilings. Scoped
+negative result for the two-touch class, scoped positive capacity for deeper classes. No
+retiming legality, physical bill, supplier, selected-record, kappa or frontier change is
+claimed. Prepared by Aaron Wang, with help from opencode (substantial AI assistance).
+
+## Question and scope
+
+`research/complex-rank2-retiming-seed73506` (Cinna Davis, PR #1703) screens fusions of
+consecutive same-column rank-(1,1) climb pairs on the pinned selected gauge44 local
+program, with exactly one intervening scalar event ("one-touch"): 690 candidates per
+direction, maximum 674 disjoint fusions, below the 720 fusions its zero-overhead profile
+needs to beat the selected word's ceiling by 5%. Its open question 1 asks whether a larger
+chronological class reaches 720, naming pairs with two intervening scalar operations as
+the first step.
+
+This package answers that step and draws the full capacity line. A pair's **touch depth**
+is the number of intervening events referencing its column (scalar operand `["L", col]`,
+adapter, or residual_adapter). For each class (touch <= 1, 2, 3, 4, 5, and unrestricted)
+we compute the maximum set of vertex-disjoint fusions and the exact ceiling of the
+resulting optimistic profile, in the profile shape
+
+```text
+{1: 2412 - 2n, 2: 418 + n, 3: 432, 4: 142, 5: 82, 8: 8}
+```
+
+at width m = 73, 896 families, 9 stages, centre charge 64. The favorable model is exactly
+the predecessor's: retiming the intervening events, copies, frames, hosting, precision,
+interface and restoration are all free; every incumbent charge stays.
+
+## 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 them rank (1,1),
+touch distribution `{1: 690, 2: 473, 3: 136, 4: 66, 5: 72, >=6: 4}` (Discussion #1659
+comment 18916048), one-touch 690 candidates with 674 disjoint fusions (PR #1703), and the
+720-fusion threshold (719 certified fail, 720 certified pass).
+
+| Touch class | Candidates | Max disjoint fusions | Ceiling gain over selected | 5% gate |
+|---|---:|---:|---:|---|
+| <= 1 (predecessor) | 690 | 674 | 4.6659% | certified fail |
+| <= 2 | 1163 | 716 | 4.9712% | certified fail |
+| <= 3 | 1299 | 790 | 5.5134% | pass (capacity only) |
+| <= 4 | 1365 | 854 | 5.9869% | pass (capacity only) |
+| <= 5 | 1437 | 901 | 6.3374% | pass (capacity only) |
+| unrestricted | 1441 | 905 | 6.3673% | pass (capacity only) |
+
+Gains are the certified upper-endpoint ratio of the optimistic profile's ceiling to the
+selected word's certified ceiling `[0.001690054109088, 0.001690054109089]`; the gate is
+the recomputed target `a = 1.05 x 0.001690054109089 = 0.00177455681454345`, equivalently
+720 disjoint fusions in this profile shape.
+
+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.
+2. **Touch depth 3 is the first class whose capacity can pass** (790 >= 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%** (905 disjoint fusions, touch
+   unrestricted). That is the family's ceiling headroom against a 5% gate in a model that
+   charges nothing for realizing any fusion, so a physical bill has little room.
+
+## Cost boundary
+
+Every incumbent source, helper, recursive-child, centre and coverage charge remains in the
+profile (local mass 5,586, idle mass 1,646 at rank cap 51). The screen grants retiming,
+copies, hosting, precision, interface transitions and dirty-state restoration for free. A
+pass is a capacity/shape survival of this profile family only. The result does not prove
+that any pair can be retimed, does not cover mixed-rank pairs, non-adjacent climbs, other
+words, changed coverage laws or any physical supplier, and is not a lower bound on complex
+decoders.
+
+## Reproduce
+
+From the repository root, Python 3.12 (stdlib only):
+
+```sh
+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
+```
+
+`verify.py --write` regenerates `RESULT.json`; the default run requires exact JSON
+equality. Expected markers:
+
+```text
+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 {...}
+```
+
+The verifier pins the selected record, the compressed source archive and its
+`LOCAL-PROGRAMS.json.gz` member (SHA-256 `241fde27d4e43f5a9cb6b128f9f76c5131a53d09f0371676163f914d0b0fef3a`),
+and the ceiling engine (SHA-256 `86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc`);
+checks forward/backward agreement, greedy/DP matching equality with vertex-disjointness,
+the 719/720 threshold controls, a touch-bin mutation control and an inflated-target
+control.
+
+## Open next questions
+
+- **Does a legal retiming exist for three-touch (1,1) pairs?** This matters because touch
+  depth 3 is the first class whose capacity (790 >= 720) can pass the ideal gate. A first
+  step is one concrete three-touch window with its exact frame/value bookkeeping for a
+  rank-2 child and free scalar work. The blocker is that the pinned event schema carries
+  frame transitions and scalar references but no documented dirty-state map contract for
+  fusing across referencing events.
+- **Do mixed-rank adjacent pairs help?** 1,341 of the 2,782 adjacent pairs are not (1,1);
+  fusing a (1,2) pair yields a rank-3 child. A first step is a weighted matching priced by
+  each fusion's exact ceiling delta. Not attempted here; nothing blocks the screen.
+- **How much added paid rank can the surviving classes absorb?** The unrestricted family's
+  ideal margin is 6.37% against a 5% gate. A first step is a uniform per-fusion rank
+  charge solved against the target at n = 790 and n = 905.
+
+## 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`, PR #1703); the touch
+distribution of the 1,441 (1,1) pairs is Navinor's (Discussion #1659 comment 18916048,
+PR #1671 lineage); the ceiling engine and coverage identity are Douglas Colkitt's and
+Sagar Thakrar's (PR1240 / PR1012); the selected gauge44 word, compiler and owner lineage
+(including Jeremy Kothe and Rohan Arun) is credited in the selected record's review.
+Adjacent but distinct families: `research/complex-gather-child-seed7a6d` (PR #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 / PR #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.