Research: flag-43 head-step repair co-hosts at update 3364 with no added channel mass (no new kappa)

research/p104-flag43-head-step-repair/MASTER-HANDOFF.md

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+# Master handoff — `research/p104-flag43-head-step-repair/`
+
+## Candidate
+
+- Branch: `cursor/dirty-restoration-repair-bill-eed4` (VictorLiwentao fork). The
+  `-eed4` suffix is this worker's environment-required branch suffix, not `-60b9`.
+- Content tip: `cc77db2b62d1eaa6a1f7b9ffec7e694a910bba54`. This is the scientific
+  package commit, and its parent is the pinned base.
+- Packaging tip: the child commit that adds only this handoff file. Read its SHA
+  with `git rev-parse origin/cursor/dirty-restoration-repair-bill-eed4`.
+- Package: `research/p104-flag43-head-step-repair/`
+- Base: CrocSwap main `6f100d4bb5083f8c35d96fd95a539811c62f245c`
+- Card: `p104-side-correction` (refined), priority revision 6
+- κ: unchanged at `0.001672660197076918`
+- Author: Wentao Li `<wentaol4@illinois.edu>`, with no co-author trailer.
+
+## Claim
+
+On present-head flag 43 (`X21` → `Y59`, head `15`), a changed schedule R0 makes
+`Y59`'s last update co-host the basis head. That update is event 3364,
+`Y59 += D17/4`, the ADD that settles `Y59`'s final `X21` coefficient `1/2`. It now
+runs in the chart `[95,447,6,32,20,15]` while `X21` is in the full chart and still
+has four later reads.
+
+R0 keeps every published event in order, with the same ADDs and coefficients. It
+retargets nine transports of `Y59`, `D17`, `Y103` and `D28`: each changed frame is
+the published frame plus `15`. It also relabels three ADD charts.
+
+The published `check_native.validate()` passes on the repaired word. Eight
+published native edges are swapped for eight newly lowered ones. All 280 framed
+rows, the 72 dirty-helper restorations and the 208 endpoint adapters hold.
+
+**Bill.** Events (4,058), ADDs (2,624), transport occurrences (1,410), rank mass
+(2,368) and distinct native edges (740) are all unchanged. The paid profile delta
+is `-3H1 + 5H2 - H3 - H4`. The published ranks of the changed transports
+majorize the repaired ranks, so the delta is at most zero under any convex
+per-rank price and exactly zero under any affine one.
+
+**Comparison R1.** R1 moves `Y103`'s 3365–3366 ahead of `Y59`'s update and adds
+one rank-1 lift to `D17`. It changes only `Y59` and `D17`, at a cost of +1 event,
++1 transport occurrence and a `+2H2 - H4` profile delta. R1 is also natively
+validated.
+
+**Obstruction.** Consider repairs that keep the published event order and ADDs,
+keep endpoint frames and add no rank mass; inserted refinement transports are
+allowed. All 280 non-private registers sit at their rank floor, so every path in
+this class stays monotone.
+
+- 14 of `Y59`'s 15 updates force the head into a register whose final frame
+  excludes it.
+- 3364 is the only admissible update. It forces exactly `D17` and `Y59` (from
+  3364), `Y103` (from 3366) and `D28` (from 3462).
+
+Without inserted events, all nine retargeted transports are forced.
+
+## Why this is not a chronology reprint
+
+The headline is a constructed repaired word, its native validation and its bill.
+The package cites, and does not re-derive, the drop/arrive chronologies of
+#3456/#3451/#3443/#3392. The only fact it rechecks from #3456 is "no published
+`Y59` update hosts the head", because the before/after comparison needs it.
+
+## Evidence commands
+
+```sh
+python3.12 -I -B research/p104-flag43-head-step-repair/verify.py
+python3.12 -I -O -Werror -B research/p104-flag43-head-step-repair/verify.py
+python3.12 -I -O -Werror -B research/p104-flag43-head-step-repair/independent.py
+python3.12 -I -O -Werror -B research/p104-flag43-head-step-repair/test_package.py
+python3.12 -B scripts/research_package.py --package research/p104-flag43-head-step-repair
+```
+
+Expected markers:
+
+```text
+PASS_P104_FLAG43_HEAD_STEP_REPAIR flag=43 head=15 repaired_update=3364 added_events=0 added_adds=0 added_transports=0 added_rank_mass=0 retargeted_transports=9 forced_registers=4 profile_delta=-3H1+5H2-H3-H4 native_validate=PASS earlier_updates_obstructed=14/14 no_new_kappa
+PASS_P104_FLAG43_HEAD_STEP_REPAIR_INDEPENDENT flag=43 head=15 repaired_update=3364 added_events=0 added_adds=0 added_transports=0 added_rank_mass=0 retargeted_transports=9 forced_registers=4 profile_delta=-3H1+5H2-H3-H4 rows_equal=280/280 earlier_updates_obstructed=14/14 no_new_kappa
+PASS_P104_FLAG43_HEAD_STEP_REPAIR_TESTS cases=14 clean_read_only=1 scanners=2
+PASS_REGISTERED_RESEARCH_PACKAGE research/p104-flag43-head-step-repair outcome=completed
+```
+
+The worker ran all five commands on the content tip in the working tree. It ran
+the registered package command again in a fresh detached worktree of the pushed
+content tip. The runs leave the package and the pinned inputs byte-identical.
+`research_package.py --changed-since <base> --check-only` selects only this
+package.
+
+## Closest priors
+
+- #3456 `p104-present-head-flag43-audit`: flag 43's published obstruction. Its
+  first open question is answered here, following its suggested first step at
+  3363/3364 and the `D17` common-chart constraint. Leave its chronology alone.
+- #3451 (flag 1) and #3443 (flag 9): the same open question on other flags. Not
+  touched.
+- #3349: present-head census. Reused, not re-derived.
+- #3363: relay census and final coefficients. This package reports only the `X21`
+  entries of the three `Y59` reads, which the co-hosting statement needs.
+- Merged `p104-refinement-rank-budget` and `p104-single-target-acquisition`:
+  priced added transports and refinements in a different model, namely identity
+  completion. They are not reused for any number here.
+
+## Soft-overlap risks
+
+- #3363: the event-level delivery split (`+1/2` at 3076, `-1/2` at 3231, `+1/2`
+  at 3364) is a coefficient fact on #3363's turf. A reviewer may ask that the
+  `R0_source_delivery_into_target` field be cited to #3363 or trimmed.
+- #3456: R0 changes four registers' published frames. Any statement in #3456
+  about the published word stays true, because only the repaired word differs.
+- `Y103` and `D28` are not flag-43 endpoints. Changing their frames could matter
+  to a future package about flags that target `Y103`. The class-Z table shows that
+  these changes are forced, not chosen.
+- `H_r` is left symbolic. If the repo's actual per-rank weights are strictly
+  concave on `{1,...,4}`, R0 costs `5H2 - 3H1 - H3 - H4 > 0`. The package does
+  not evaluate those weights.
+- R1 reorders two reads of `D17`. That is legal under the checker, but it sits
+  outside the class-Z obstruction by design.
+
+## Open questions
+
+1. Do flags 1 and 9 admit an event-preserving head-step repair, or are their
+   closures obstructed? Use a separate package and cite #3451/#3443.
+2. Can a side correction exploit R0's co-hosted head chart at 3364, and what
+   does `I - G/2` cost at that cut?
+3. Is R0's profile majorization-minimal among class-Z repairs?
+
+## Disposition note for master
+
+This worker produces a fork branch and package only. Do **not** open a CrocSwap
+PR from this worker. Master decides whether to publish or defer after an
+independent skeptical review.

research/p104-flag43-head-step-repair/NOTICE.md

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+# Notice
+
+This package applies two explicit schedule changes to the published P104
+native-channel witness in memory and checks them with the published native
+checker and a second scanner. It does not modify frozen upstream sources. The
+nine recipes in `REPAIR-RECIPES.json` were lowered with the pinned
+`research/p104-native-channel/compile.py` and `vendor/clifford_exact.py`
+(PR1991 lineage, Chafik Boukhalfa), whose notices are retained in that package.
+
+Prepared by Wentao Li <wentaol4@illinois.edu> with substantial AI assistance.
+Apache-2.0; predecessor notices and exact source pins are retained in
+`SOURCES.json`.

research/p104-flag43-head-step-repair/PROOF.md

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+# Proof: flag-43 head-step repair, its bill, and the forced closure
+
+## Setup
+
+Pin CrocSwap main `6f100d4bb5083f8c35d96fd95a539811c62f245c`. Inputs:
+
+- The #2981 flag matching (`research/p104-joint-unit-flags/CERTIFICATE.json`).
+  Flag 43 has source 21, target 59 and basis `[15, 387, 6, 18, 32, 67, 128, 1, 2]`.
+- The reviewed #1991 witness, recipes and checker in `research/p104-native-channel/`.
+
+All of these are hash-pinned.
+
+The channel semantics are those of `research/p104-native-channel/PROOF.md` §2–§3:
+
+- Each register carries a binary frame `W` with canonical chart `Gamma_W`.
+- A transport is nested, changes only `W`, and pays one positive child of rank
+  `|dim B - dim A|`.
+- An ADD is allowed only when both registers have the same frame. It changes
+  only the destination's logical row.
+
+A register *hosts* `15` when `15` lies in the GF(2) span of its frame.
+
+Leave-alone priors are cited only. #3456 proves that on the published word none
+of the 15 `Y59` updates runs in a chart that spans `15`, while every `X21`
+annotation spans it. This package rechecks only the one-line consequence it
+needs: no published `Y59` update hosts the head.
+
+## 1. The repair R0
+
+Let:
+
+- `W4 = [95,447,6,32]` and `W4' = [95,447,32,20]`;
+- `W5 = [95,447,6,32,20]` and `W6 = W5 + [15]`;
+- `V5 = [95,447,6,32,192]`, `V6 = V5 + [20]` and `W7 = V6 + [15]`.
+
+R0 replaces exactly these published events (indices are unchanged):
+
+| Event | Published | R0 |
+|---|---|---|
+| 3362 | `D17: W4 -> W5`, rank 1 | `D17: W4 -> W6`, rank 2 |
+| 3363 | `Y59: W4' -> W5`, rank 1 | `Y59: W4' -> W6`, rank 2 |
+| 3364 | `Y59 += D17/4` in `W5` | same ADD in `W6` |
+| 3365 | `Y103: W4' -> W5`, rank 1 | `Y103: W4' -> W6`, rank 2 |
+| 3366 | `Y103 += D17/4` in `W5` | same ADD in `W6` |
+| 3460 | `D28: V5 -> V6`, rank 1 | `D28: V5 -> W7`, rank 2 |
+| 3461 | `Y103: W5 -> V6`, rank 1 | `Y103: W6 -> W7`, rank 1 |
+| 3462 | `Y103 += D28/4` in `V6` | same ADD in `W7` |
+| 3525 | `Y59: W5 -> final`, rank 3 | `Y59: W6 -> final`, rank 2 |
+| 3569 | `Y103: V6 -> final`, rank 2 | `Y103: W7 -> final`, rank 1 |
+| 3612 | `D17: W5 -> FULL`, rank 4 | `D17: W6 -> FULL`, rank 3 |
+| 3634 | `D28: V6 -> FULL`, rank 3 | `D28: W7 -> FULL`, rank 2 |
+
+The final frames of `Y59` and `Y103` both contain `W6`. `Y103`'s also contains
+`V6` and hence `W7`. So every changed transport is nested, and its rank equals its
+frame gap. `D17` and `D28` have no other events between their changed transports.
+
+**Native validation.** `verify.py` builds the R0 word in memory. It assembles a
+recipe set from two parts: the published recipes for every edge still in use, and
+the committed `REPAIR-RECIPES.json` for every new edge. Eight published edges drop
+out and eight new ones enter. Edge `W4' -> W6` is shared by events 3363 and 3365,
+as `W4' -> W5` was. The script then calls the published `check_native.validate()`
+unchanged. That function:
+
+- checks every edge's complete Clifford operator, including phase, against the
+  independent interpreter;
+- binds every transport event to its recipe and rank;
+- checks all 208 intrinsic endpoint adapters;
+- runs the framed scalar induction over all 280 inputs.
+
+The induction requires every ADD to run in a common chart. It also requires each
+source and each of the 72 dirty helper rows to end equal to its original input,
+and each target row to end equal to `y_t + sum_s <r_t,r_s> x_s / 2`, with every
+final frame matched. R0 passes. `independent.py` separately replays the published
+and repaired words with exact rational rows. It finds all 280 final rows equal to
+the published ones.
+
+**Head co-hosting.** In R0, `Y59`'s updates are the published ones. The last,
+3364, runs in `W6`, which lists `15`. At 3364, `X21`'s live frame is the
+rank-9 full chart. It has later events 3705, 3760, 3811 and 3863, all of them
+reads. The rational replay gives the `X21` entries of the rows `Y59` reads:
+
+| Event | Partner | `X21` carried | `Y59`'s `X21` coefficient after |
+|---|---|---:|---:|
+| 3076 | `D10` | `1/2` | `1/2` |
+| 3231 | `D5` | `-1/2` | `0` |
+| 3364 | `D17` | `1/2` | `1/2` |
+
+So the ADD that leaves `Y59` with its final `X21` coefficient runs in a chart
+co-hosting the basis head, while the source hosts it and is still live. #3363's
+relay census records the final coefficient. The event-level split is reported
+here only because the co-hosting statement needs it.
+
+## 2. The bill
+
+Counts come from the validated words:
+
+- events, scalar ADDs and transport occurrences are unchanged: 4,058, 2,624 and
+  1,410;
+- rank mass is unchanged at 2,368;
+- distinct native edges are unchanged at 740;
+- endpoint adapters are unchanged at 208.
+
+The paid profile changes from `{1:1074, 2:58, 3:86, 4:104, ...}` to
+`{1:1071, 2:63, 3:85, 4:103, ...}`. The ranks of 5 to 8 are unchanged.
+
+Per register, the profile changes are:
+
+- `Y59`: `-H1 - H3 + 2H2`;
+- `D17`: `-H1 - H4 + H2 + H3`;
+- `Y103`: no change, since `{1,1,2}` becomes `{2,1,1}`;
+- `D28`: `-H1 - H3 + 2H2`.
+
+The total delta is `Delta = -3H1 + 5H2 - H3 - H4`.
+
+Sorted in decreasing order, the nine changed ranks go from `(4,3,3,2,1,1,1,1,1)`
+to `(3,2,2,2,2,2,2,1,1)`. Both sum to 17, and the repaired prefix sums never
+exceed the published ones. So the published vector majorizes the repaired one.
+By Karamata's inequality, `Delta <= 0` for every convex rank price `H`.
+Directly: convexity gives `2H2 <= H1 + H3` and `3H2 <= 2H1 + H4`, and these two
+inequalities add up to `Delta <= 0`. `Delta = 0` for every affine `H(r) = a + br`.
+
+**R1.** R1 replaces published events 3363–3366 by:
+
+1. `Y103: W4' -> W5`
+2. `Y103 += D17/4` in `W5`
+3. `D17: W5 -> W6`, rank 1 (added)
+4. `Y59: W4' -> W6`, rank 2
+5. `Y59 += D17/4` in `W6`
+
+It also sets 3525 to `Y59: W6 -> final` (rank 2) and 3612 to
+`D17: W6 -> FULL` (rank 3). The two reads of `D17` into distinct targets commute,
+and nothing between them touches `D17`, `Y59` or `Y103`. R1 passes the same native
+validation. Its bill:
+
+- +1 event and +1 transport occurrence;
+- rank mass and ADDs unchanged;
+- profile delta `+2H2 - H4`;
+- 742 distinct edges.
+
+R1 avoids the `Y103`/`D28` cascade by giving up event preservation.
+
+## 3. Forced closure
+
+**Class Z.** A repair is in class Z when:
+
+1. it keeps the published events in their published order, with the same ADD
+   registers and coefficients, and may insert transports;
+2. it keeps every initial and final frame;
+3. it adds no rank mass.
+
+`verify.py` checks that every one of the 280 non-private registers has published
+rank mass equal to `rank(final) - rank(initial)`. A register's rank mass is at
+least that floor, and equals it only for a monotone path. So by condition 3 every
+non-private register's path in a class-Z repair is monotone. The private register
+`C` is reported separately and never counted as an obstruction.
+
+**Propagation lemma.** Suppose that in a class-Z repair a non-private register
+`r` hosts `15` at event `t`. Then it hosts `15` at every later event, by
+monotonicity. Any ADD or copy at `j >= t` that involves `r` forces its partner to
+host `15` at `j`, because the two frames are equal.
+
+Starting from a `Y59` update `u`, take the closure of this rule. A register is
+*must-change* when its published frame at its forced time does not span `15`.
+
+- If a must-change register's final frame excludes `15`, no monotone path can
+  host `15` there, so no class-Z repair makes `u` head-hosting.
+- If the closure is unobstructed, the published frames already satisfy every
+  other forced condition, because published paths are monotone.
+
+`verify.py` computes the closure with a worklist and `independent.py` with a
+single forward sweep. For the 15 `Y59` updates:
+
+| Update | Partner | Must-change | Obstructions | First obstruction |
+|---:|---|---:|---:|---|
+| 223–227 | `Q1, Q2, Q4, Q5, Q6` | 260–274 | 56 each | `Y67` (264–268) |
+| 2099, 2191 | `D23, D20` | 140 | 56 | `Y96` |
+| 2275 | `D10` | 140 | 56 | `Y26` at 2336 |
+| 2341 | `D5` | 140 | 56 | `Y56` at 2387 |
+| 2393 | `D17` | 133 | 53 | `Y22` at 2638 |
+| 2698 | `D23` | 45 | 17 | `Y96` at 2706 |
+| 2891 | `D20` | 31 | 12 | `Y96` at 2897 |
+| 3076 | `D10` | 17 | 4 | `Y26` at 3220 |
+| 3231 | `D5` | 10 | 1 | `Y56` at 3348 |
+| 3364 | `D17` | 4 | 0 | none |
+
+The exact per-update values are in `CERTIFICATE.json`. So 3364 is the only update
+that any class-Z repair can make head-hosting. Its must-change set is exactly:
+
+- `Y59` and `D17` from 3364;
+- `Y103` from 3366;
+- `D28` from 3462.
+
+**Minimality in the event-preserving subclass.** With no inserted events, each
+must-change register `r` with forced time `t` behaves as follows:
+
+- its last transport before `t` must gain `15` in its after-frame, since no
+  published after-frame there spans `15`;
+- each later transport of `r` then has a changed before-frame.
+
+Together these are exactly R0's nine transports: 2 for `Y59`, 2 for `D17`, 3 for
+`Y103` and 2 for `D28`. At each forced ADD the shared frame must contain both
+published frames and `15`. That gives `span(W5 + 15) = W6` at 3364 and 3366, and
+`span(V6 + 15) = W7` at 3462. R0 uses exactly these minimal frames.
+
+## Controls
+
+`test_package.py` copies the package and the six pinned inputs into a temporary
+tree and runs both scanners there under `-O`. The clean tree must pass and stay
+byte-identical. Fourteen mutations must be rejected by both scanners:
+
+- seven certificate fields;
+- a duplicate JSON key;
+- a witness byte flip;
+- a phase flip in a new recipe;
+- an undercharged 3525 transport;
+- three truncated cascades: `Y59` lifted alone, without `Y103`/`D28`, and without
+  `D28`.
+
+Each truncated cascade must fail at the common-chart rule.
+
+## Limits
+
+This is a statement about one finite repaired word for one flag:
+
+- It does not build or price the `I - G/2` side correction or a supplier.
+- It does not perform global hosting or precision analysis.
+- It does not change κ.
+
+"Co-hosting" means span membership in annotated frames. It does not model what the
+head direction means inside relay registers. `H_r` is left symbolic. The repo's
+actual per-rank weights decide the sign of `Delta` only if they are strictly
+concave somewhere on `{1,2,3,4}`. The class-Z obstruction excludes repairs that:
+
+- reorder events (R1 is one such repair);
+- add rank mass;
+- change ADDs, coefficients or endpoint frames.
+
+Optimality of R0's profile among class-Z repairs is not claimed.