P104: exclude two non-carrier ADDs across helper572’s chart seam (no new kappa)

research/p104-noncarrier-two-add-seam/PROOF.md

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+# Two ADDs across the 572 seam cannot restore the outside outputs
+
+This is an exact finite exclusion in the rational scalar model of the selected
+PR6030 word. It is not a new supplier or a universal P104 obstruction.
+All event indices are zero based, and a window is immediately **after** its event.
+
+## Domain
+
+Keep the complete prefix, original event schedule, charts, fields and owned
+lifetimes. Immediately after event 13637 (`COPY 778 <- 572` at chart 595),
+replace field 572 by a new independent input `z`. The displaced original row is
+`ell X`, where `X` consists of all 770 arbitrary original public inputs.
+
+Insert precisely these two ADD opportunities (a zero coefficient is allowed,
+so the exclusion also covers using fewer than two):
+
+1. After any event `i` in 13637..13651, add `a * field_u` to `field_v`, with
+   distinct `u,v` in chart 595. Its six roles are
+   `[127,572,581,582,731,778]` throughout these windows.
+2. After event `j` in `{13652,13654,13655}`, add `b * field_s` to `field_r`,
+   with distinct `s,r` then at chart 603. The banks are respectively
+   `[572,740]`, `[572,740]`, `[572,709,740]`.
+
+Both coefficients may be arbitrary rationals. Sources need not be carriers:
+these operations may temporarily change original `X` rows. Event 13652 is the
+unchanged MOVE of 572 from 595 to 603; event 13654 adds `-572` into 740;
+event 13655 brings 709 to 603. CLEAR 778 at 13656 also stays unchanged.
+
+Let `V=127`, the original final chart of 572, and let `O` be the 186 public
+roles whose final chart differs from V. The required necessary condition is
+that **every O output equal its original unsplit output for every X and z**.
+
+**Result.** None of the `15 * 30 * (2 + 2 + 6) = 4,500` location pairs can
+meet that condition. This holds even if erasing z completely were permitted.
+Consequently none can supply a legal split with a nonzero V-final z target.
+
+## Exact endpoint polynomial
+
+Use row vectors for input functionals and column vectors for final public
+outputs. Let `R_i` be the original prefix at window i and `S_i` the original
+public suffix. Let `beta_i` be the propagation, through original events, of
+the unit vector placed at 572 at the overwrite. Set
+
+```
+e = z - ell X
+D = S_cut e_572
+A = S_i e_v
+B = S_j e_r
+b1 = (beta_i)_u
+b2 = (beta_j)_s
+g = (T_{j<-i})_{s,v}
+f = (R_i X)_u + b1 e
+h = (R_j X)_s + b2 e
+```
+
+Here `e_572,e_v,e_r` are state-coordinate unit vectors; `e` without a
+subscript is the scalar functional `z-ell X`. `T_{j<-i}` is the unchanged
+intervening linear program. The first source's value is f. Its update changes
+the second source's value from h to `h + a*g*f`. Linearity of the retained
+suffix gives the complete final public difference:
+
+```
+D e + a A f + b B h + a b g B f.                         (1)
+```
+
+This includes original-input defects as well as the z coefficient. It does
+not assume that either correction has zero original value. Neither does it
+linearize around zero coefficients or discard the interaction term.
+
+Introduce independent rational unknowns `(a,b,c)` in place of `(a,b,a*b)`.
+For each outside output and each input column (including z), setting (1) to
+zero is a linear equation in these three unknowns. Every actual two-ADD
+solution would solve this **more permissive** system with `c=a*b`.
+The verifier proves all 4,500 relaxed systems inconsistent. No bounded
+coefficient search, root-finding or unproved quadratic-solving step is used.
+
+## Certificates and coverage
+
+The checker authenticates the selected compressed word against SOURCES.json,
+replays its exact rational rows and live charts, and constructs reverse public
+suffix columns. The complete forward endpoint and reverse transpose agree;
+all original owned rows restore and their lifetimes end. This authenticates
+the scalar data used here, not the selected native implementation.
+
+For each of all 4,500 location pairs, sparse elimination records a rational
+linear combination of original protected-output equations whose left side is
+zero and right side is one. A separate checker recomputes the selected original
+equations and verifies those dot products directly, without row reduction.
+There are 4,262 one-equation and 238 two-equation contradictions. The canonical
+certificate transcript SHA-256 is retained in RESULT.json; every replay
+recomputes and checks every certificate, not merely its digest.
+
+For example, the first pair is
+`(i,u,v,j,s,r)=(13637,127,572,13652,572,740)`. A contradiction uses output 120's
+original input-column equations 0 and 34 with multipliers `-2` and `1`.
+The exact right-side sum is 1 and all three unknown coefficients sum to zero.
+The literal coordinates and multipliers are in RESULT.json.
+
+Independent literal whole-word execution checks four parameterized production
+cases against (1) on **all original input columns and the new z column**.
+These include a nonzero interaction term, a correction using 778, a first
+update before the ADD into 582, and a changed original row. The literal engine
+has its own event loop and row arithmetic, separate from the prefix/suffix
+engine. This is implementation cross-checking by an AI-assisted contributor,
+not independent human or formal-kernel acceptance.
+
+A synthetic changed-boundary fixture admits a two-ADD non-carrier repair; the
+checker does not reject every model by construction. Moving 572 to the wrong
+chart is rejected at its actual consumer, and a forged dual is rejected.
+
+A particularly relevant false positive is `127 -> 572` with coefficient 1
+after 13651, followed by `740 -> 572` with coefficient 1 after 13654. Its
+outside z-leak is zero and z survives publicly, but its outside original-X
+rows are wrong. The full equations reject it. This demonstrates why a z-only
+check would not settle the claimed question.
+
+## Scope boundary
+
+PR6067 and PR6069 cover corrections that preserve the original scalar rows at
+each completed atomic carrier update. Their results do not imply this one:
+we allow non-carrier sources and change original rows between two different
+charts. Conversely, this package covers only the stated two-ADD locations.
+It says nothing about three or more ADDs, different later windows, changed
+prefixes, non-unit injection, new routes/storage or changed scheduled events.
+It does not audit either predecessor's complete proof or census.
+
+The necessary scalar condition already fails. Therefore no candidate reaches
+pairing, productive targets, owned-restoration after a successful edit, native
+realization or full paid/ordinary/compiler/outer pricing. Those checks are not
+silently counted as passed. The selected word, its five framework assumptions
+and kappa remain unchanged.
+
+Prepared for **JoyboyBrian**, with substantial OpenAI Codex assistance.
+Apache-2.0. Credit apoursh (PR6067), Jonathan Capalbo (PR6069), JosieElliston
+(PR6045), Nick Petalas (PR6040), Matarener (PR5994), flyingviper (PR5718),
+Ali Romman and DreamingOfClouds (selected P104 word), and their retained lineage.

research/p104-noncarrier-two-add-seam/README.md

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+# P104: two non-carrier ADDs across the helper 572 seam fail
+
+**Scoped exact exclusion; no new kappa.** On the selected PR6030 word,
+none of 4,500 declared two-ADD location pairs restores every outside-V output
+following the unit overwrite of helper 572 after event 13637. The corrections
+may temporarily change original input rows, so this is outside the carrier-only
+models of [PR6067](https://github.com/CrocSwap/integer-mult-bounds/pull/6067)
+and [PR6069](https://github.com/CrocSwap/integer-mult-bounds/pull/6069).
+
+The first ADD is at chart 595 after any event 13637..13651. The second is at
+chart 603 after 13652,13654 or13655. All original events, fields, charts and
+lifetimes stay fixed. Coefficients are arbitrary rationals, including zero.
+Every original input and the new independent z are symbolic. [PROOF.md](PROOF.md)
+specifies the complete domain and endpoint polynomial.
+
+All cases fail even after relaxing the product `a*b` to an independent third
+unknown. Each has an exact independently checked linear contradiction:
+4,262 use one equation and 238 use two. Four separate literal whole-word checks
+validate the polynomial on all input columns, including a nonzero interaction
+term. A synthetic changed-boundary two-ADD repair is accepted, while a wrong
+seam chart and a forged contradiction certificate are rejected. A retained false positive clears the z-leak but corrupts original inputs.
+[RESULT.json](RESULT.json) retains the deterministic certificate digest and counts.
+
+The scientific base is canonical main `721c03822c421b7f7fcbb2598993b947c4d88419`,
+priority 8, selected conditional kappa 0.003125804. The canonical investigation is
+[Discussion 6074](https://github.com/CrocSwap/integer-mult-bounds/discussions/6074).
+
+## Reproduce
+
+Python 3.12 and its standard library, from the repository root:
+
+```sh
+python3.12 -I -B research/p104-noncarrier-two-add-seam/verify.py
+python3.12 -I -O -B research/p104-noncarrier-two-add-seam/verify.py
+python3.12 -B scripts/research_package.py --package research/p104-noncarrier-two-add-seam
+```
+
+Direct normal replay took about 8.6 seconds on macOS arm64. Both registered
+modes have 30-second limits. Every guard remains active under `-O`.
+`--write` regenerates only this package's RESULT.json; default replay is read-only.
+There are no executable pending-PR dependencies: the sole external live input
+is the selected WORD.json.gz, source-pinned in SOURCES.json and declared in VERIFY.json.
+
+## Coverage and overlap
+
+This follows the original-row-changing correction question in PR6067 at
+`4d629cf0d20a06d65b3ba0cf3a34e105128725e3` and PR6069 at
+`b3f5aabddbae2d65f80bb8356e6993e0a65cd245`. Their carrier-only exclusions are
+credited rather than replayed. The package derives its boundary data directly
+from the selected word; neither pending proof is an executable dependency.
+
+Preclaim overlap read 341 open PRs,5,722 discussion records and 8,348 local notes.
+That metadata coverage is not a novelty guarantee or a search of historical
+diffs/unpublished work. The companion receiver-recoding task concerns helper 469
+and is separate. No fixed-invoice repacking or selected-word application audit
+is repeated here.
+
+The full necessary endpoint condition fails, so there is no edited supplier to
+price. No native/compiler/caller/precision/outer replay, selected cold replay,
+Lean, Docker or full multiplication proof is supplied. No universal bound or
+claim about more edits follows. The five selected assumptions remain.
+
+## Open next questions
+
+A longer source-bound correction or a different chronological window remains
+outside this exclusion. Before a new search, supply a concrete candidate word
+that changes the failed original-input equations, then check its full symbolic
+endpoint, pairing, lifetime and complete bill. No broader search is proposed
+merely because this bounded model failed. Structured first steps and blockers
+are in TASK.json.
+
+## Credit
+
+Prepared for **JoyboyBrian**, with substantial OpenAI Codex assistance in
+triage, derivation, implementation and verification. This is AI-assisted
+research, not independent human or proof-kernel approval. Apache-2.0.
+
+Credit apoursh (PR6067), Jonathan Capalbo (PR6069), JosieElliston (PR6045),
+Nick Petalas (PR6040), Matarener (PR5994), flyingviper (PR5718), Ali Romman,
+DreamingOfClouds (PR6030/5907) and the retained P104/native/compiler lineage.
+Frozen predecessor packages and the selected construction are unchanged.

research/p104-noncarrier-two-add-seam/RESULT.json

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+{
+  "V_chart": 127,
+  "cases": 4500,
+  "claim": "Every declared two-ADD seam fails exact outside-V restoration, even with ab independent.",
+  "controls": {
+    "bilinear_case": true,
+    "consistent_system": true,
+    "forged_dual_rejected": true,
+    "literal_full_column_cases": 4,
+    "synthetic_noncarrier_repair": true,
+    "wrong_seam_rejected": true,
+    "z_only_false_positive": true
+  },
+  "dual_sha256": "324b646386e6776953c9d3ad52e03ad9d76738c0ca2fc061e20c106e21f73846",
+  "dual_size_counts": {
+    "1": 4262,
+    "2": 238
+  },
+  "first_dual": {
+    "case": [
+      13637,
+      127,
+      572,
+      13652,
+      572,
+      740
+    ],
+    "dual": [
+      [
+        120,
+        0,
+        "-2"
+      ],
+      [
+        120,
+        34,
+        "1"
+      ]
+    ]
+  },
+  "outside_public_roles": 186
+}

research/p104-noncarrier-two-add-seam/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "721c03822c421b7f7fcbb2598993b947c4d88419",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/p104-helper470-twin-rung-removal/WORD.json.gz",
+      "usage": "input",
+      "git_blob": "22f9ab02d1b73ea0aa4b07ce445a14cc2602c760",
+      "sha256": "7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f"
+    }
+  ]
+}