Exclude helper469 rung removal by invertible receiver endpoint recoding

research/p104-helper469-receiver-recoding/PROOF.md

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+# Helper469: receiver recoding still cannot remove the frame593 rung
+
+This is a one-interval exact rational exclusion, not a new multiplication
+construction, native implementation, cost improvement, or universal barrier.
+All indices below are zero-based and a slot is the state before an event.
+
+## Source and changed hypothesis
+
+The sole computational input is the selected PR6030 word at main
+`721c03822c421b7f7fcbb2598993b947c4d88419`, SHA256
+`7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f`.
+Its inherited helper468 and helper470 removals are retained.
+Jonathan Capalbo's [PR6069](https://github.com/CrocSwap/integer-mult-bounds/pull/6069),
+head `b3f5aabddbae2d65f80bb8356e6993e0a65cd245`, excludes donor-only
+endpoint resynthesis but explicitly excludes receiver scaling/recoding from
+its claim. Its `ENDPOINT-PROOF.md` SHA256 is
+`2e7a08c289cfd7761a02d4fcdf0be804a8882a66c532e84058c91ed41134148c`.
+That pending proof is contextual attribution, not an imported code dependency
+or an assumption of this certificate; the elementary argument is given below.
+No foreign source-history ancestry is fabricated or required.
+
+We allow any finite rational receiver-only ADD/SCALE program in the two
+maximal contact-free endpoint windows, with every SCALE nonzero. Other roles'
+programs, public inputs, owned lifetimes and charts are fixed. Replace only
+469's MOVE 3->593 at14357, its ADD469+=2*703 at14380, and its
+MOVE593->713 at14417 by a direct MOVE3->713 and the endpoint programs.
+The receiver must regain its entire original scalar row at chart713 before
+its next contact at14850. No new MOVE, owned storage or altered donor is allowed.
+
+## Exact rows and legal windows
+
+The previous receiver contact is a read at7895. Its next contact after the
+second MOVE is the incoming ADD at14850. Thus endpoint slots are
+[7896,14357] at chart3 and [14418,14850] at chart713, inclusively.
+No original operation reads or changes469 within either window.
+
+Start all770 public roles as independent coordinate rows e_0,...,e_769;
+owned roles enter with zero rows on INIT. ADD/SCALE/COPY have literal rational
+semantics; MOVE changes the chart and leaves the decoded scalar row unchanged.
+This is the selected word's scalar contract, not a new native-map assertion.
+
+Exact prefix replay gives receiver entry
+
+    f = e_469 + (1/2)e_537.
+
+Write Delta = 2 row_703(14380), the one removed interior contribution.
+The verifier records every coefficient of Delta in RESULT.json and checks
+that the full original receiver row at14418 and14850 equals f+Delta.
+In particular Delta has coefficient1/2 at public-input coordinate1.
+
+Collect every other live row at the beginning of each window, then every
+other role's row immediately after a MOVE arrival at its endpoint chart
+through the last slot. These569 timestamped generators comprise568 at the
+first endpoint and1 at the second. Every collected value is available at its
+recorded slot: the receiver can read it then, without retaining a donor after
+its departure. Exact replay proves every generator has coordinate1 zero.
+The receiver f also has coordinate1 zero.
+
+Every other donor version available later in a window lies in the span of
+these generators: same-chart ADD, SCALE and COPY preserve the accumulated
+span; INIT contributes zero; CLEAR and departure cannot enlarge it; each MOVE
+arrival is included. No original read of469 occurs within either window,
+so modifications to469 do not affect donor rows. This is PR6069's credited
+accumulated-span argument, specialized here and independently checked.
+
+## Normal form and complete decision
+
+Any finite receiver-only endpoint program has scalar form
+
+    r_out = lambda f + sum_j c_j g_j,  lambda != 0,
+
+because every ADD adds a fixed donor row and every SCALE multiplies all
+existing coefficients. A direct MOVE preserves this row; combining both
+endpoint programs multiplies their nonzero receiver coefficients. The desired
+restoration is r_out=f+Delta, equivalently
+
+    (lambda-1) f + sum_j c_j g_j = Delta.                 (1)
+
+This is an exact rational affine system with the invertibility side condition.
+For completeness, any formal coefficients of timestamped generators can be
+realized in chronological order with arbitrary rational ADDs: scale at the
+first slot by lambda, then add each generator at its recorded slot, taking the
+direct MOVE after the first window. Donors are unchanged and never read469;
+no new storage or free retention is needed. This scalar realization does not
+establish the physical scalar alphabet or its cost. Those gates would follow
+only if a solution existed.
+
+Apply the linear functional L(v)=v[1] to(1). The left side is0 for every
+rational lambda and c_j, while the right side is1/2. Contradiction.
+The feasible lambda set is therefore empty, even if zero multipliers were
+granted. This separates the entire affine span, not a finite coefficient
+search or a first-order screen. It decides the prescribed invertible model.
+
+The obstruction explicitly concerns an arbitrary input column; replacing703
+by a label or checking only an ideal fixture would not establish it. The
+original receiver row must be restored before14850; postponing repair or
+changing another role is outside the model. The precise scientific review
+obligations are the scalar interpretation, the complete window boundary,
+the accumulated-span induction and its normal-form applicability.
+
+No candidate survives to be priced. No favorable direct-transition native
+rank, paid profile, productive-target histogram, ordinary cost or kappa is
+claimed. This exclusion does not decide changed donor routes, new storage,
+new receiver contacts, a different word or coupled receiver/donor recoding.

research/p104-helper469-receiver-recoding/README.md

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+# One coordinate excludes helper469's invertible endpoint recoding
+
+On the selected PR6030 word, helper469 cannot replace its chart3->593->713
+interval with a direct MOVE3->713 using any finite rational receiver-only
+ADD/SCALE programs in its two maximal endpoint windows. Every available donor
+and the receiver's entry row have zero coefficient on public input1; the
+required restored row has coefficient1/2. The feasible receiver multiplier set
+is empty, including all nonzero rational multipliers. See [PROOF.md](PROOF.md)
+and the exact [certificate](RESULT.json).
+
+This answers the receiver-recoding case explicitly excluded by
+[PR6069](https://github.com/CrocSwap/integer-mult-bounds/pull/6069), only for
+helper469 events14357..14417. It does not repeat the predecessor's764-interval
+census. The initial unchanged-receiver holder proposal was withdrawn because
+PR6069 already covers it. Canonical investigation:
+[discussion6078](https://github.com/CrocSwap/integer-mult-bounds/discussions/6078).
+
+The verifier independently reconstructs the source rows with standard-library
+rational arithmetic. It authenticates the selected word, verifies maximal
+contact-free windows and source chart/lifetime semantics, gathers569 legal
+timestamped generators, checks the complete receiver row, and compares the
+committed deterministic certificate. No pending-source code is imported.
+Independent controls replay input1 literally at every legal donor slot and exercise
+a synthetic recoding positive plus zero-multiplier, wrong-version, wrong-chart,
+changed-input-column and omitted-restoration negatives. Both scripts are
+registered in normal and optimized Python. Only the prefix through the restoration boundary is replayed; no full native,
+physical caller, selected invoice, Lean or Docker replay is performed.
+
+## Reproduction
+
+From the repository root, with Python3.12 or later and no dependencies:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-helper469-receiver-recoding
+python3 -B scripts/check_research_commit.py --package research/p104-helper469-receiver-recoding --check-source-history
+```
+
+The direct verifier also works in a Git-free export containing the package
+and its one declared word input. It writes no files:
+
+```sh
+python3 -B research/p104-helper469-receiver-recoding/verify.py
+```
+
+SOURCES.json pins the selected word, catalogue card and selected-result record
+at main721c03822c421b7f7fcbb2598993b947c4d88419, priority8. The pending
+PR6069 head and exact proof hash are contextual provenance in PROOF.md;
+there is no hidden dependency on its local Git object. The parent investigation
+checked live overlap before claiming this changed hypothesis; this does not
+claim novelty against unpublished work or every historical closed PR.
+
+## Open next questions
+
+A further construction would have to change a donor program or another stated
+boundary. Before opening a new search, identify a route delivering nonzero
+input1 coefficient and a favorable complete cost allowance, including native,
+productive-target, ordinary and restoration costs. No such route is supplied.
+The structured question, relevance, first step and blocker are in TASK.json.
+This is a stopping point for receiver-only recoding of this interval.
+
+## Credit and limitations
+
+Prepared for JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.
+Credit Jonathan Capalbo/PR6069 for the accumulated endpoint-span argument and
+its explicit receiver-recoding boundary; Nick Petalas/PR6033 for the carrier
+cost context; DreamingOfClouds/PR6030 for the selected word and inherited rung
+removals; Ali Romman/PR5827 and all retained P104 authors for the scalar/native
+construction. Sparse rational replay here is independently implemented.
+
+This is an author-checked finite scalar exclusion and a mathematical argument
+for its stated endpoint program class. It is not independent human scientific
+acceptance, supplier admission, a new kappa, or the full multiplication theorem.

research/p104-helper469-receiver-recoding/RESULT.json

@@ -0,0 +1,164 @@
+{
+  "status": "EXCLUDED_ALL_RATIONAL_RECEIVER_MULTIPLIERS",
+  "receiver": 469,
+  "moves": [
+    14357,
+    14417
+  ],
+  "interior_add": 14380,
+  "windows": [
+    {
+      "side": "before",
+      "slots_inclusive": [
+        7896,
+        14357
+      ],
+      "chart": 3
+    },
+    {
+      "side": "after",
+      "slots_inclusive": [
+        14418,
+        14850
+      ],
+      "chart": 713
+    }
+  ],
+  "receiver_entry": {
+    "469": "1",
+    "537": "1/2"
+  },
+  "required_interior_increment": {
+    "1": "1/2",
+    "4": "1/2",
+    "5": "1/2",
+    "7": "-1/2",
+    "10": "-1/2",
+    "11": "-1/2",
+    "12": "-1/2",
+    "13": "-1/2",
+    "18": "-1/2",
+    "19": "-1/2",
+    "24": "-1/2",
+    "26": "1/2",
+    "29": "1/2",
+    "30": "1/2",
+    "32": "-1/2",
+    "36": "-1/2",
+    "37": "-1/2",
+    "38": "-1/2",
+    "43": "-1/2",
+    "45": "-1/2",
+    "49": "-1/2",
+    "50": "-1/2",
+    "51": "1/2",
+    "56": "-1/2",
+    "57": "-1/2",
+    "58": "1/2",
+    "62": "-1/2",
+    "63": "-1/2",
+    "64": "1/2",
+    "65": "1/2",
+    "67": "1/2",
+    "72": "-1/2",
+    "73": "1/2",
+    "74": "1/2",
+    "79": "-1/2",
+    "81": "-1/2",
+    "82": "1/2",
+    "86": "1/2",
+    "88": "1/2",
+    "89": "1/2",
+    "92": "-1/2",
+    "94": "-1/2",
+    "95": "1/2",
+    "98": "-1/2",
+    "101": "-1/2",
+    "102": "-1/2",
+    "103": "-1/2",
+    "104": "1/2",
+    "108": "-1/2",
+    "109": "-1/2",
+    "110": "1/2",
+    "111": "1/2",
+    "173": "1",
+    "224": "1/2",
+    "226": "1/2",
+    "227": "-1/2",
+    "233": "-1/2",
+    "235": "-1/2",
+    "237": "-1/2",
+    "239": "-1/2",
+    "241": "-1/2",
+    "243": "-1/2",
+    "249": "-1/2",
+    "251": "-1/2",
+    "253": "-1/2",
+    "255": "-1/2",
+    "256": "-1",
+    "257": "-1/2",
+    "262": "1/2",
+    "265": "-1",
+    "266": "-1/2",
+    "269": "-1/2",
+    "276": "-1/2",
+    "283": "-1/2",
+    "286": "-1/2",
+    "287": "1",
+    "292": "-1/2",
+    "299": "-1/2",
+    "302": "-1/2",
+    "303": "1",
+    "308": "-1/2",
+    "315": "-1/2",
+    "318": "-1/2",
+    "319": "1",
+    "324": "-1/2",
+    "331": "-1/2",
+    "334": "-1/2",
+    "335": "1",
+    "336": "1/2",
+    "347": "1/2",
+    "351": "1/2",
+    "352": "1/2",
+    "363": "1/2",
+    "367": "1/2",
+    "368": "1/2",
+    "379": "1/2",
+    "383": "1/2",
+    "384": "1/2",
+    "395": "1/2",
+    "399": "1/2",
+    "416": "-1/2",
+    "418": "1/2",
+    "419": "-1/2",
+    "422": "1/2",
+    "425": "-1/2",
+    "426": "-1/2",
+    "427": "1/2",
+    "431": "1/2",
+    "434": "1/2",
+    "435": "-1/2",
+    "438": "1/2",
+    "441": "-1/2",
+    "442": "-1/2",
+    "443": "1/2",
+    "447": "1/2",
+    "703": "2"
+  },
+  "donor_generators": {
+    "before": 568,
+    "after": 1,
+    "timestamped_rows_sha256": "afbfd8ab585100cc059c63d76a32232e866c2bf9be2a23a1e6f3300eaa5f6c4f"
+  },
+  "separator": {
+    "public_input": 1,
+    "receiver_value": "0",
+    "every_donor_value": "0",
+    "required_increment_value": "1/2",
+    "feasible_lambda_set": "empty"
+  },
+  "full_receiver_target_checked": true,
+  "word_sha256": "7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f",
+  "scope": "Only receiver469 interval14357..14417, unchanged donor programs and two maximal endpoint windows; no supplier or kappa claim."
+}

research/p104-helper469-receiver-recoding/SOURCES.json

@@ -0,0 +1,24 @@
+{
+  "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"
+    }
+  ]
+}