Record nonnegative raw-moment invoices for two-neighbour ADD moves.

research/staged544-two-neighbour-invoice/README.md

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+# Two-neighbour ADD moves do not improve the raw moment
+
+**No new κ.** Selected κ stays 0.001672660197076918.
+
+This is a scoped obstruction for one frame-path change left open by
+[PR #3942](https://github.com/CrocSwap/integer-mult-bounds/pull/3942).
+It is not a review of that paper's interaction or new-gap reduction, and it
+is not the radius-one theorem. The first displacement there is one neighbour.
+Here the first ADD crosses at least two scalar neighbours.
+
+## Result
+
+Fix either literal PR #3734 provider word carried by PR #3942. Consider an
+existing ADD, a cut in its commuting corridor that passes at least two gates,
+and any common frame dimension legal for the removed-gate interval at that
+cut. The raw rank-moment delta of that first move is nonnegative for every
+real `0 < a < 1`. The certificate is the predecessor curvature test: rank
+mass zero and nonnegative partial sums of `C_k = sum_r d_r min(r, k)`.
+The first-move price depends only on dimension, by the interval formula in
+PR #3942, PROOF.md §4, so this dimension census covers every subspace of
+the same dimension.
+
+Each provider has 5,501 ADD gates, 4,979 of them with a long cut, and
+1,761,760 dimension cases: 239,103 nonzero and 1,522,657 neutral. The two
+providers agree on these counts. None improves.
+
+One changed-frame witness on the base word is gate `A[595] = add 459 510`
+with coefficient `-1` and empty frame. Moving it to cut `1214` crosses 618
+gates, including both an ADD and a SCALE, and the only legal rank-1 frame
+is `span{85}`. The frame bill changes by `+3` rank-1 children and `-1`
+rank-3 child. Its caps are `[2, 1, 0, ...]`, so the move is strictly losing.
+After the coefficient update and rederived cleanup, the edited word passes
+the vendored frame auditor. Every one of the 1,767,681 single-ADD
+retimings of that edited word keeps the combined histogram certified
+nonnegative. This witness does not unlock a saving inside that single-move
+class.
+
+## What this does not show
+
+A second move that itself changes a frame interval of quotient rank greater
+than 1, or a second cut that is priced only after another long move, is
+outside the continuation census. Native phase, hosting, stock, precision,
+tape, all-size transfer and both supplier caps are not repriced. Unknown
+costs are not zero. The scalar crossing rule is the sufficient PR #3886
+condition, not every algebraic commutation. No 632-input free-operator
+replay was repeated. This is not a multiplication theorem.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3 -B research/staged544-two-neighbour-invoice/verify.py
+python3 -B scripts/research_package.py --package research/staged544-two-neighbour-invoice
+```
+
+The expected first line starts
+`PASS_STAGED544_TWO_NEIGHBOUR_INVOICE` and ends with `improving=0 no_new_kappa`.
+The registered timeout is 180 seconds. A local run is well under that.
+
+`vendor/retiming.py` is byte-identical to PR #3942 at
+`1416f8cd6bfb2f2eb9aa5cc22f0388a85fdbac91`. `vendor/audit_core.py` and both
+`inputs/BORROW-WORD-43*.json.gz` files are the copies carried in that
+package. SHA256 values are checked before use.
+
+## Open next questions
+
+1. Does every long first move, not only `A[595]`, keep every single-ADD
+   continuation nonnegative? One witness cannot extend the radius-one
+   theorem. The first step is the same continuation census on a move whose
+   frame interval has quotient rank greater than 1.
+2. Can a second ADD that itself crosses two neighbours improve the combined
+   raw moment? Freeze one edited word and one such second cut before any
+   larger search.
+
+## Credit
+
+PR #3734 / dataisfire for the provider words. PR #3886 and PR #3942 /
+Chris R, Clear1nsight, for the retiming rules, curvature test and the open
+two-neighbour question. Sagar Thakrar for the frame auditor. The radius-one
+interaction reduction remains their argument; this package does not review
+it. New census and witness by unpaidmoon, with substantial Cursor Grok
+assistance. Apache-2.0.

research/staged544-two-neighbour-invoice/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "bb4bccbf42844d2801487450727c2f63e10d1d0f",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    }
+  ]
+}

research/staged544-two-neighbour-invoice/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "scalar-to-moment",
+  "priority_revision": 6,
+  "base_commit": "bb4bccbf42844d2801487450727c2f63e10d1d0f",
+  "original_card": {
+    "id": "scalar-to-moment",
+    "question": "Can one known scalar simplification be converted into an actual paid-profile improvement?",
+    "starting_points": [
+      "research/r439-bsection-stars-seed01/README.md",
+      "certificates/selected-result.json"
+    ],
+    "methods": [
+      "Derive the changed legal frame itinerary and exact child histogram.",
+      "Prove a scoped obstruction showing why that scalar gain cannot change the paid contract."
+    ],
+    "acceptance_gate": "An unchanged histogram is not a kappa gain; record any separate runtime or proof benefit explicitly.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/r439-bsection-stars-seed01/verify.py"
+      ],
+      "expected": "PASS_R439_BSECTION_STARS",
+      "inspect": "Observe that the positive control improves scalar counts while retaining the paid histogram.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Choose one simplification and derive the complete changed frame itinerary, or prove why it cannot change the histogram.",
+    "bottleneck_link": "A legal change to the histogram, not addition count alone, is the bridge to an exponent gain.",
+    "allowed_variations": [
+      "One existing scalar simplification and one proposed frame-path change."
+    ],
+    "scope_change_triggers": [
+      "Submitting only another scalar count as completion of a paid-profile task."
+    ],
+    "overlap_terms": [
+      "scalar simplification",
+      "paid profile",
+      "R439"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "On both pinned PR3734 staged544 providers, price every legal first ADD move that crosses at least two scalar neighbours, at every feasible common-frame dimension, and price every single-ADD continuation after one explicit changed-frame witness.",
+  "scope_change_reason": "The card allows one frame-path change. This is that change for the PR3942 radius-one contract: the first ADD crosses two or more neighbours and may change frame. It is not an R439 scalar-count reduction, and it does not review the radius-one interaction reduction.",
+  "bottleneck_link": "A negative raw-moment delta on this frame path would be a paid-histogram change. The certified nonnegative deltas show this path does not supply one. No kappa is claimed.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the dimension-complete long-first census on both providers and the single-move continuation census of one changed-frame witness. Do not search a second long move or review PROOF.md sections 2-3.",
+  "outcome": "negative-result",
+  "evidence": "Exact nonnegative S_j certificates for every long first-move dimension and for every single-move continuation of witness A/595/cut 1214/frame [85].",
+  "author": "unpaidmoon",
+  "ai_assistance": "Cursor Grok assisted the census script, packaging, and write-up. The predecessor trace and frame auditor are vendored unchanged.",
+  "open_questions": [
+    {
+      "question": "After every long first move, not only witness A/595, does every single-ADD continuation stay nonnegative?",
+      "why_it_matters": "One witness does not extend the radius-one theorem to the whole two-neighbour class.",
+      "first_step": "Repeat the continuation census on a long move whose feasible frame interval has quotient rank greater than 1.",
+      "blocked_by": "The pair of censuses is a different budget from this witness."
+    },
+    {
+      "question": "Can a second ADD that itself crosses two or more neighbours improve the combined raw moment?",
+      "why_it_matters": "The continuation census uses the single-move corridor, which prices every distance inside that corridor, but a second changed-frame long move can change geometry again.",
+      "first_step": "Freeze one edited word and one second cut that crosses two neighbours, then add the two histograms.",
+      "blocked_by": "No source-bound obstruction for that second long move is claimed here."
+    }
+  ]
+}

research/staged544-two-neighbour-invoice/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "SHA256 of the vendored trace, auditor and both provider words; dimension-complete long-first census on both providers; one changed-frame witness frame bill; every single-ADD continuation certificate; a negative curvature control."
+    }
+  ]
+}