Research: flag-1 head-step repair co-hosts at update 2797 for exactly two added rank units (no new kappa)

research/p104-flag1-head-step-repair/MASTER-HANDOFF.md

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+# Master handoff — `research/p104-flag1-head-step-repair/`
+
+## Candidate
+
+- Branch: `cursor/flag1-head-step-repair-0378` (VictorLiwentao fork). The
+  `-0378` suffix is this worker's environment-required branch suffix, not
+  `-60b9`.
+- Content tip: `053bff9e760e05354687cc669461dd6d8effed18`. This is the
+  scientific package commit. 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/flag1-head-step-repair-0378`.
+- Package: `research/p104-flag1-head-step-repair/`
+- Base: CrocSwap main `6f100d4bb5083f8c35d96fd95a539811c62f245c`
+- Card: `p104-side-correction` (refined), priority revision 6
+- κ: unchanged at `0.001672660197076918`
+- Author and committer: Wentao Li `<wentaol4@illinois.edu>`, with no co-author
+  trailer (both commits were built with `git commit-tree`).
+- Scratch branch `cursor/p104-flag1-head-step-repair-0378` holds an earlier
+  agent-identity WIP commit of the same producer. It is superseded and should
+  not be published.
+
+## Claim
+
+On present-head flag 1 (`X0` → `Y7`, head `232`), event 2797 (`Y7 += D17/4`) is
+the only `Y7` update that carries `X0`. It brings `-1/2`, the final coefficient.
+
+Two changed schedules make that ADD co-host the head while `X0` sits in its
+rank-9 chart and still has four later reads. The published `check_native.validate()`
+passes unchanged on both. That covers all native edges (six newly lowered), 208
+adapters, 280 framed rows and 72 dirty-helper restorations.
+
+| | Events | ADDs | Transports | Rank mass | Profile delta | Surcharge on |
+|---|---:|---:|---:|---:|---|---|
+| R0 (order-preserving) | +1 | 0 | +1 | +2 | `-H1 + 2H2 + H5 - H6` | `D17` |
+| R1 (relocated into `D17`'s full window) | 0 | 0 | 0 | +2 | `H8 - H6` | `Y7` |
+
+Under an affine price the extra cost is `a + 2b` for R0 and `2b` for R1, so it is
+positive. Under a convex price R0's delta can be negative: it is `-4` at
+`H = r²`.
+
+**Obstruction.** All 280 non-private registers are at their rank floor, and the
+excess over the floor is always even.
+
+- The #3508 head closure blocks all six `Y7` updates. For 2797 the first
+  obstruction is `Y8` at 2799.
+- The exact least-frame sweep (with the private register `C` relaxed, so the
+  verdicts are sound) says zero added non-private mass is infeasible for all six
+  order-preserving updates.
+- It is also infeasible for 868 of 1,115 relocation slots of the delivering ADD,
+  namely every slot before `Y103`'s read at 3366.
+
+So +2 is the floor there, and R0 is mass-optimal among order-preserving
+repairs. The 247 later slots are relaxed-feasible and left open.
+
+## Why this is not a chronology reprint
+
+The headline is two constructed repaired words, their native validation, their
+exact bill and a zero-mass obstruction that flag 43 does not have. #3451's
+drop/arrive chronology is cited, not re-derived. The only published fact
+rechecked is "no `Y7` update hosts the head", which the before/after comparison
+needs.
+
+## Evidence commands
+
+```sh
+python3.12 -I -B research/p104-flag1-head-step-repair/verify.py
+python3.12 -I -O -Werror -B research/p104-flag1-head-step-repair/verify.py
+python3.12 -I -O -Werror -B research/p104-flag1-head-step-repair/independent.py
+python3.12 -I -O -Werror -B research/p104-flag1-head-step-repair/test_package.py
+python3.12 -B scripts/research_package.py --package research/p104-flag1-head-step-repair
+```
+
+Expected markers:
+
+```text
+PASS_P104_FLAG1_HEAD_STEP_REPAIR flag=1 head=232 delivering_update=2797 R0=order_preserving added_events=1 added_adds=0 added_transports=1 added_rank_mass=2 profile_delta=-H1+2H2+H5-H6 R1=relocated added_events=0 added_transports=0 added_rank_mass=2 profile_delta=-H6+H8 native_validate=PASS zero_mass_order_preserving=0/6 zero_mass_relocation_obstructed=868/1115 no_new_kappa
+PASS_P104_FLAG1_HEAD_STEP_REPAIR_INDEPENDENT flag=1 head=232 delivering_update=2797 R0 added_events=1 added_transports=1 added_rank_mass=2 R1 added_events=0 added_transports=0 added_rank_mass=2 rows_equal=280/280 zero_mass_order_preserving=0/6 zero_mass_relocation_obstructed=868/1115 no_new_kappa
+PASS_P104_FLAG1_HEAD_STEP_REPAIR_TESTS cases=17 clean_read_only=1 scanners=2
+PASS_REGISTERED_RESEARCH_PACKAGE research/p104-flag1-head-step-repair outcome=completed
+```
+
+The worker ran all five commands in the working tree. It ran them again,
+together with `research_package.py --changed-since <base> --check-only`, in a
+fresh detached worktree of the content tip. Those runs leave the tree
+byte-identical. The test suite also checks this inside its temp tree.
+
+## Closest priors
+
+- #3508 `p104-flag43-head-step-repair`: the method prior. This package reuses
+  three things from it: the splice/assemble/validate harness, the class-Z head
+  closure, and the bill format. Flag 43 had a zero-cost event-preserving repair;
+  flag 1 does not. #3508 itself is left alone.
+- #3451 `p104-present-head-flag1-audit`: the published flag-1 obstruction. Its
+  first open question is answered here. Cited only.
+- #3443 (flag 9) and #3349 (census): cited only.
+- #3363: relay census. Only the single `X0` entry delivered into `Y7` at 2797 is
+  re-reported.
+
+## Soft-overlap risks
+
+- **Code reuse from #3508.** About 316 of `verify.py`'s 463 non-blank lines
+  match #3508's `verify.py` line for line: the GF(2) helpers, splice and
+  assemble, bill, and closure. About 132 of `independent.py`'s 287 non-blank
+  lines match #3508's `independent.py`, mostly the replay and recipe check.
+  `independent.py` changes the frame representation (reduced bases instead of
+  XOR closures). The least-frame sweep and the relocation scan are new. A
+  reviewer may ask for a shared helper instead of a copy.
+- **#3451.** The package restates "no published `Y7` update hosts the head" and
+  the `Y7` update list (19, 20, 2013, 2146, 2526, 2797). Both appear in #3451's
+  PROOF items 3–5. They are rechecked only because the bill comparison needs
+  them.
+- **#3363.** The single coefficient fact (`-1/2` at 2797 via `D17`) is on #3363's
+  turf. A reviewer may want it cited to #3363 or trimmed. It is load-bearing for
+  "2797 is the delivering update".
+- **Flag 0.** It shares source `X0`. No flag-0 or `Y34` coefficient is computed.
+- **Relaxed sweep wording.** A reader could take "0 infeasible after 3366" as a
+  claim that zero cost is achievable. It is not: the relaxation frees `C`, and
+  the direct construction fails. The README and PROOF say so, and open
+  question 1 carries it.
+- `H_r` is left symbolic.
+
+## Open questions
+
+1. Is there a zero-added-mass co-hosting relocation after 3366?
+2. Which +2 repair, R0 or R1, is cheaper under the repo's actual per-rank
+   weights?
+3. Does flag 9 (#3443) follow flag 1's pattern or flag 43's?
+
+## 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-flag1-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
+six 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.
+The repair method and bill format follow open #3508 (flag-43 repair); the
+flag-1 published chronology is #3451.
+
+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-flag1-head-step-repair/PROOF.md

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+# Proof: flag-1 head-step repairs, their bill, and the zero-mass obstruction
+
+## Setup
+
+Pin CrocSwap main `6f100d4bb5083f8c35d96fd95a539811c62f245c`. Inputs:
+
+- The #2981 flag matching (`research/p104-joint-unit-flags/CERTIFICATE.json`).
+  Flag 1 has source 0, target 7 and basis `[232, 256, 7, 11, 19, 35, 64, 2, 1]`.
+- 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, as in #3508:
+
+- Each register carries a binary frame.
+- A transport is nested (up or down) and pays rank `|dim B - dim A|`.
+- An ADD needs both registers in the same frame, and it changes only the
+  destination's row.
+
+A register *hosts* `232` when `232` lies in the GF(2) span of its frame.
+
+#3451 proves that on the published word no `Y7` update hosts the head, while
+every `X0` annotation spans it. That chronology is cited, not re-derived. This
+package rechecks only the one-line consequence it needs.
+
+## 1. The delivering update and its commutation window
+
+The rational replay finds that among the six `Y7` updates (19, 20, 2013, 2146,
+2526, 2797), only 2797 (`Y7 += D17/4`, chart `[95, 486]`) carries any `X0`
+entry. It carries `-1/2`, which is `Y7`'s final `X0` coefficient. `D17` is the
+destination of exactly two events, 2497 (`D17 += C`) and 3613 (`D17 += -Q4`).
+No event reads `Y7`.
+
+So placing the ADD anywhere after 2497 and before 3613 gives the same final rows.
+This window has 1,115 slots besides the published one.
+
+## 2. The repairs
+
+Write `A2 = [95, 486]`, `A3 = A2 + [232]` and `FULL` for the rank-9 chart.
+
+**R0** replaces these published events (later indices shift by one):
+
+| Event | Published | R0 |
+|---|---|---|
+| 2795 | `D17: [486] -> A2`, rank 1 | `D17: [486] -> A3`, rank 2 |
+| 2796 | `Y7: [95] -> A2`, rank 1 | `Y7: [95] -> A3`, rank 2 |
+| 2797 | `Y7 += D17/4` in `A2` | same ADD in `A3`, then inserted `D17: A3 -> A2`, rank 1 |
+| 3473 | `Y7: A2 -> final`, rank 6 | `Y7: A3 -> final`, rank 5 |
+
+`span(A3)` lies in `final(Y7)`, so `Y7` stays monotone and its mass is unchanged
+(`1 + 6 = 2 + 5`). `D17` goes up and back down once, which adds exactly 2.
+
+**R1** deletes published 2796, 2797 and 3473. After 3612 (`D17: W5 -> FULL`) it
+inserts three events: `Y7: [95] -> FULL` (rank 8), `Y7 += D17/4` in `FULL`, and
+`Y7: FULL -> final` (rank 1, downward). `Y7`'s mass goes from `1 + 1 + 6` to
+`1 + 8 + 1`, adding 2. No other register changes, and the event count is
+unchanged.
+
+**Native validation.** `verify.py` builds each 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 each of the six new edges (four in
+R0, two in R1). It then calls the published `check_native.validate()`
+unchanged. That function:
+
+- checks every edge's full Clifford operator, phase included;
+- binds every transport to its recipe and rank;
+- checks all 208 endpoint adapters;
+- runs the framed scalar induction over all 280 inputs.
+
+The induction includes the common-chart rule, the restoration of every source
+and dirty helper, and each target's final row and frame. Both words pass.
+`independent.py` replays both words with exact rational rows and finds all 280
+final rows equal to the published ones.
+
+**Co-hosting.**
+
+- In R0, the ADD at 2797 runs in `A3`, which spans `232`.
+- In R1, it runs in `FULL` (new index 3611).
+- In both, `X0` is in its rank-9 frame and has four later events, all reads.
+- In both, this is the only co-hosted `Y7` update, and it delivers `X0`'s
+  `-1/2`.
+
+## 3. The bill
+
+All counts come from the validated words.
+
+| | Events | ADDs | Transports | Rank mass | Edges | Profile delta |
+|---|---:|---:|---:|---:|---:|---|
+| R0 | +1 | 0 | +1 | +2 | 742 | `-H1 + 2H2 + H5 - H6` |
+| R1 | 0 | 0 | 0 | +2 | 741 | `-H6 + H8` |
+
+For affine `H(r) = a + b·r` the deltas are `a + 2b` and `2b`. For nondecreasing
+`H`, `H8 - H6 >= 0`. R0's delta equals `H2 + (H2 - H1) - (H6 - H5)`. It is at
+most `H2` for convex `H`, and its sign is not fixed: it is `+2` at `H = r` and
+`-4` at `H = r²`. The private register `C` has identical events in all three
+words.
+
+## 4. The zero-mass obstruction
+
+**Classes.** Both classes keep:
+
+- every ADD (same registers and coefficients);
+- every endpoint frame;
+- `C` completely unconstrained.
+
+Transports of non-private registers may be inserted, retargeted or removed
+freely.
+
+- **Class Z₀** keeps the published relative order of all ADDs and copies.
+- **Class Z₁** is Z₀, except that the delivering ADD may sit at any slot of its
+  commutation window (§1).
+
+**Parity lemma.** Take a non-private register's nested path from `init` to
+`final`. Its mass is `sum |Δ_i|`, which is at least `|sum Δ_i| = rank(final) -
+rank(init)` and has the same parity. Equality holds iff every step goes up,
+which means the frames grow by inclusion. All 280 non-private registers meet
+this floor on the published word (checked). So a repair whose non-private mass
+equals the published mass has monotone non-private paths. Any other repair adds
+an even amount, at least 2.
+
+**Least-frame sweep.** Fix a placement of the ADDs and a forced ADD `u`. Walk
+the ADDs in order with `L(r) = init(r)`, skipping any ADD that involves `C`. At
+each ADD between `a` and `b`, set both `L(a)` and `L(b)` to `L(a) + L(b)`, and at
+`u` add `232` as well.
+
+By induction, in any monotone solution each register's frame at each ADD
+contains its `L`, so the final frame must contain the end value of `L`. If some
+`L(r)` is not inside `final(r)`, no zero-surcharge repair of that placement
+co-hosts at `u`. Skipping `C`'s ADDs only removes constraints, so this verdict
+stays sound with `C` free. With nothing forced, the published word passes this
+check.
+
+**Results** (exact lists are in `CERTIFICATE.json`):
+
+| Placement | Infeasible registers | First (sorted) |
+|---|---:|---|
+| Z₀, update 19 / 20 | 56 / 56 | `Y0` |
+| Z₀, update 2013 / 2146 | 54 / 52 | `Y1` |
+| Z₀, update 2526 | 23 | `Y103` |
+| Z₀, update 2797 | 14 | `Y103` |
+| Z₁, slots after 2497–3365 (868) | 23 down to 1 | `Y103` in every slot |
+| Z₁, slots after 3366–3612 (247) | 0 | — |
+
+#3508's head-closure rule gives the same verdict for all six Z₀ updates. It
+finds 56, 56, 56, 56, 23 and 14 registers whose final frame excludes the head,
+the first being `Y8` each time. `independent.py` recomputes both tests with
+separate code, including a fresh full replay for every Z₁ slot.
+
+**Consequences.**
+
+1. No Z₀ repair with zero added non-private mass co-hosts at any `Y7` update.
+   By parity, every co-hosting Z₀ repair adds at least 2, and R0 adds exactly 2.
+2. Neither does any Z₁ placement before 3366, so the same floor of 2 holds
+   there.
+3. For placements after 3366 the relaxed sweep is feasible, so no bound is
+   claimed. R1 sits in this window at +2.
+
+The least solution there leaves `D17` empty at all of its reads. The published
+word instead holds `D17` at rank 2, then 4, then 5. Realizing a low-frame word
+would need re-framing large parts of the schedule. The direct attempt
+intersects every frame in `D17`'s backward cone with `final(Y7)`. It reaches 208
+registers, including `C`, and needs 56 source initial frames inside `final(Y7)`,
+which they are not. So that attempt fails. Open question 1 asks whether some
+other construction succeeds.
+
+## 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. Seventeen mutations must be rejected by both scanners:
+
+- ten certificate fields (bill, profile, register mass, co-hosting, delivery,
+  closure, sweep, scan window, scan count, κ);
+- a duplicate JSON key, a witness byte flip, and a phase flip in a new recipe;
+- R0 without the inserted drop, and R0 with only `Y7` lifted. Both must fail at
+  the common-chart rule.
+- an undercharged 2795 transport;
+- R1 placed after `D17`'s restoration at 3613, which breaks `Y7`'s final row.
+
+## Limits
+
+This is a statement about finite repaired words 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. `H_r` is left symbolic.
+The obstruction covers Z₀ and Z₁ before 3366, with `C` relaxed. It does not
+cover:
+
+- reorders of other events;
+- changed ADDs or endpoint frames;
+- the 247 late slots.
+
+The +2 floor is stated for non-private mass. Both repairs leave `C` unchanged,
+so their total and non-private surcharges agree.