Research: per-helper-sector census of the literal K_equals_D defect on retained R439 helper-star words (no new kappa)

research/r439-star-kdefect-sectors/README.md

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+# R439 star K-defect sectors: a complete per-helper census of the literal `K_equals_D` failure
+
+**Finite certificate on three named words; no new κ.** The selected result, the selected
+bit supplier and `upstream/` are unchanged; the bit side does not bind.
+
+The retained R439 helper-star words all carry the two false syntactic flags
+`H2_D_frame0_targets_from_helpers` and `K_equals_D` (see PR967 lineage
+`research/r439-helper-stars-seed19` and PR1062 `research/r439-bsection-stars-seed01`).
+PR1194 retains a semantic replacement that proves `N = K` globally for a virtual spliced D.
+This package quantifies the exact size and shape of the syntactic failure those words carry,
+under the pinned owner checker's own definitions
+(`research/e8-bit-p111-tuned/code/xcheck_p.py`, sections quoted below), with an
+independent GF(2) replay that cross-agrees with the checker on every derived boolean.
+
+| word (v = 111, R = 439, 661 columns) | D gates | defect bits (K ⊕ N) | defective helper sectors | KS(h) pass | single-gate repairs | single-bit repair |
+|---|---:|---:|---:|---:|---:|---:|
+| `helper_star` = seed19 candidate (PR967 lineage) | 1,165 | **17,636** | 439 / 439 | 0 / 439 | 0 | no |
+| `composition` (PR1062, +spliced phase-B blocks) | 1,165 | **17,636** | 439 / 439 | 0 / 439 | 0 | no |
+| `candidate` (PR1062, star rerun) | 1,195 | **17,653** | 439 / 439 | 0 / 439 | 0 | no |
+
+`K` = helper-column part of each target's raw-word (A, scatter, B) output under symbolic
+all-column replay; `N` = helper columns named by D's target writes; defect = `K ⊕ N`;
+`KS(h)` = sector-local condition "no target carries a defect in helper column h".
+
+## What this establishes (finite, scoped to the three words)
+
+1. **The failure is global, not local.** All 439 helper sectors are defective; the
+   per-operation condition `KS(h)` passes for **zero** star helpers on every word
+   (sector sizes range from 4 to 68 defective targets, 17,636–17,653 bits total). A
+   per-helper
+   sector-local replacement of H2/K cannot lift to the global `K_equals_D`: any such
+   replacement must account for cross-sector terms — consistent with the retained
+   PR1194 route, which proves `N = K` globally rather than per star.
+2. **No single-operation edit repairs it.** Exhaustive enumeration over all 1,165/1,195
+   single-D-gate removals, and over all single-bit toggles (possible only at defect size
+   one), yields zero repairs on every word.
+3. **Localization of the failure source.** The raw helper part `K` is bit-identical across
+   all three words (0 differing bits), even though the phase-B splice rewrites B
+   (712 → 578 gates) and cleanup C (2,574 → 2,440 gates), and identical D (byte-equal N)
+   between `helper_star` and `composition` leaves the defect profile exactly equal at
+   17,636 bits. The star rerun (+30 D gates, N differing in 173 bits) shifts the defect
+   by 17 bits (17,636 → 17,653). The literal-K failure lives entirely in D versus the
+   fixed raw helper component; phase-B replacement neither creates nor removes it.
+4. **H2 clause decomposition.** On all three words the frame clauses are clean (0 D gates
+   at nonzero frame, 0 target/helper start-frame violations) and every uncanceled N entry
+   is helper-column-only; H2 fails purely through destination/read clauses: 442 D gates
+   write non-target registers (472 on the rerun candidate) and 612 D gates read non-helper
+   registers.
+
+## Relationship to adjacent open work
+
+PR #3533 gives the abstract three-gate restored-edge mechanism (the literal mask names
+`h0` while `y` receives `h0 ⊕ h1`); this package counts the consequence across a full
+retained word. PR #3904 pins one per-target example (target 112, helper 260) measured on
+a **D-inclusive** replay against the literal mask. Under the pinned checker's official
+definition — raw `W0 = (A, scatter, B)` with D excluded, used here and cross-agreeing
+with the checker flag — that particular bit pair is **not** in the defect set (sector of
+column 260 has 57 defective targets, not including 112). The two statements use different
+raw-word definitions; neither contradicts the other, and the census here is on the
+official W0 definition.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3 -I -B research/r439-star-kdefect-sectors/verify.py
+python3 -I -O -B research/r439-star-kdefect-sectors/verify.py
+```
+
+Expected (~2 s each):
+
+```text
+PASS_R439_STAR_KDEFECT_SECTORS words=3 defect_bits=17636/17636/17653 defective_sectors=439/439/439 KS_pass=0/439/0/439/0/439 single_op_repairs=0 controls=5 no_new_kappa
+```
+
+`verify.py --json OUT` additionally writes the full sector/target table, per-word flag
+maps and cross-word bit counts. Five controls run: SHA-256 source bindings; the
+checker-agreement and retained-flag-pattern control on all three words; the multi-defect
+no-single-repair control; a one-bit-flip localization sanity control; and a substitution
+control showing
+`N := K` satisfies the flag trivially (the honest claim is only that no N derived from D
+does). Inputs are bound by SHA-256 in `SOURCES.json`.
+
+## Open next questions
+
+1. **Which weakened K contract admits these words?** Per sector, test whether the defect
+   equals the sum of restored-edge cross terms introduced inside D (the PR3533 mechanism
+   counted on the real word). First step: compare the JSON sector table against a D-internal
+   propagation tally; blocker: a maintained policy decision on admissible weakened K
+   contracts; no paid-profile gain is implied by either answer.
+2. **Does the census pattern persist on other helper-mixing lineages?** The claim here is
+   scoped to three named words; extension requires rerunning `verify.py`'s census function
+   against new pinned words, one lineage per bounded experiment.
+
+## Scope — what this is not
+
+A finite certificate on three named words at the stated base. Not a proof that no
+per-operation replacement can ever satisfy a weakened interface (a changed contract with
+named cross-sector hypotheses escapes this census); not a supplier admission; no κ,
+profile or selected-frontier change; no native, hosted or all-size replay. Full exact
+legality replay (`full()`) runs as a positive control, so the words are confirmed legal
+while carrying the quantified syntactic failure.
+
+## Credit and disclosure
+
+Credit: Nick Petalas. Produced by Nick Petalas with AI assistance.
+Predecessor results retained and unchanged: PR967 (helper-star word), PR1062
+(composition line), the retained PR1194 semantic route, and the pinned e8-bit-p111-tuned
+owner checker, each with their own attribution. Apache-2.0.

research/r439-star-kdefect-sectors/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "bb4bccbf42844d2801487450727c2f63e10d1d0f",
+  "files": [
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "research/e8-bit-p111-tuned/code/xcheck_p.py",
+      "usage": "input",
+      "git_blob": "184f5dcf94a486cc7db91ca4db5865916bf26009",
+      "sha256": "af3decc877e27c00001cf56f50c6c09566e8e101297ab34c7860aed96a06d80e"
+    },
+    {
+      "path": "research/r439-bsection-stars-seed01/candidate.json.gz",
+      "usage": "input",
+      "git_blob": "9a17b4920334878d6c949191f6931bd97126fe3c",
+      "sha256": "b452d2c2b0b2a546d1105b1da9769aa78e5c54d6f00e97b30c77e3f54ce445ee"
+    },
+    {
+      "path": "research/r439-bsection-stars-seed01/composition.json.gz",
+      "usage": "input",
+      "git_blob": "d95fc2bf795fd9ef8d8bae9d70693c5e64948cec",
+      "sha256": "0203fea9fa442ca206fe683aa5b6860adac680d63b9fb8be6bd42b39aa7b5cce"
+    },
+    {
+      "path": "research/r439-helper-stars-seed19/candidate.json.gz",
+      "usage": "input",
+      "git_blob": "8b0210d8b8e9524182e5f54f9bf93c28f4719735",
+      "sha256": "d8b08885e8ce9868200b2645189d15ed18a27fc2742e54990b3257a0ca4df6b8"
+    }
+  ],
+  "notes": "Word files and the pinned owner checker are read unmodified from canonical main. Predecessor attribution lives in the cited packages."
+}

research/r439-star-kdefect-sectors/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "bit-helper-contract",
+  "priority_revision": 6,
+  "base_commit": "bb4bccbf42844d2801487450727c2f63e10d1d0f",
+  "original_card": {
+    "id": "bit-helper-contract",
+    "question": "Can one precisely specified helper-mixing word satisfy a replacement for the inherited H2/K interface?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/r439-bsection-stars-seed01/README.md",
+      "research/r439-copy-path-sharing-barrier/README.md"
+    ],
+    "methods": [
+      "Derive a replacement interface lemma and price its implementation.",
+      "Seek an exact dirty-input or chronological-cut counterexample to the proposed replacement."
+    ],
+    "acceptance_gate": "A full boundary contract and paid profile; unchanged scalar counts alone are insufficient. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/r439-bsection-stars-seed01/verify.py"
+      ],
+      "expected": "PASS_R439_BSECTION_STARS",
+      "inspect": "Compare the two false H2/K flags with the unchanged paid profile in its README.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Write one replacement H2/K implication for a single helper-star operation, then try to falsify it on dirty inputs.",
+    "bottleneck_link": "A legitimate replacement could admit a cheaper bit circuit; scalar savings alone do not alter the selected moment.",
+    "allowed_variations": [
+      "Choose one helper-star or quartet instance; narrow to one boundary implication."
+    ],
+    "scope_change_triggers": [
+      "Changing only source count or scalar addition count without an interface argument."
+    ],
+    "overlap_terms": [
+      "H2_D_frame0_targets_from_helpers",
+      "K_equals_D",
+      "helper-star"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Quantitative finite census of the syntactic K_equals_D failure on the three retained R439 helper-star words (PR967 lineage seed19 candidate; PR1062 composition and candidate): per-helper-sector defect decomposition, exhaustive single-operation repair enumeration, H2 clause decomposition, and cross-word localization; the tested per-operation condition is KS(h) = 'sector h alone satisfies K = N'.",
+  "scope_change_reason": "The card's single replacement implication is settled here as a quantified necessity statement: zero of 439 sector-local KS(h) hold and no single-operation edit repairs the flag, so any per-operation replacement must add cross-sector hypotheses. A full boundary contract and paid profile stay open; nothing is claimed as a supplier.",
+  "bottleneck_link": "Bit-side interface diagnosis; no scalar count or paid-profile change. A future replacement interface lemma for this lineage must absorb all 439 sectors at once (as the retained PR1194 global route does), not per star.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the three-word census, controls and cross-word localization; do not extend to other lineages or attempt supplier admission.",
+  "outcome": "completed",
+  "evidence": "verify.py PASS line plus JSON sector/target tables; independent GF(2) replay agrees with the pinned owner checker on K_equals_D, J_identity and U_block_form for all three words; four controls; SHA-256 input bindings in SOURCES.json.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Does a weakened K contract with named cross-sector hypotheses (delivering K up to restored-edge cross terms) admit these words without the global PR1194 route?",
+      "why_it_matters": "The census shows no sector-local or single-operation route exists on these words; the minimal honest weakening is exactly characterized by the 17,636-bit defect table.",
+      "first_step": "Enumerate, per sector, whether the defect equals the sum of restored-edge cross terms of D (the PR3533 mechanism counted on the real word).",
+      "blocked_by": "A maintained policy decision on admissible weakened K contracts; no paid-profile gain is implied."
+    }
+  ],
+  "scope_key": "R439/helper-star/K-defect-sectors/seed19+1062-composition+candidate",
+  "predecessors": [
+    {
+      "pr": 967,
+      "package": "research/r439-helper-stars-seed19"
+    },
+    {
+      "pr": 1062,
+      "package": "research/r439-bsection-stars-seed01"
+    },
+    {
+      "pr": 1194,
+      "package": "research/r439-semantic-d-contract-seed10",
+      "role": "retained global semantic route, cited only"
+    },
+    {
+      "checker": "research/e8-bit-p111-tuned/code/xcheck_p.py"
+    }
+  ],
+  "related_open_prs": [
+    {
+      "pr": 3533,
+      "relation": "abstract three-gate restored-edge K-mask mechanism; this package counts the consequence across full retained words"
+    },
+    {
+      "pr": 3904,
+      "relation": "pins one D-inclusive per-target example (target 112, helper 260); under the checker's official W0 definition used here that bit pair is not in the defect set - different raw-word definitions, no contradiction"
+    }
+  ],
+  "discussion": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19046223",
+  "validation": {
+    "verifier": "PASS_R439_STAR_KDEFECT_SECTORS words=3 defect_bits=17636/17636/17653 defective_sectors=439/439/439 KS_pass=0/439/0/439/0/439 single_op_repairs=0 controls=5 no_new_kappa",
+    "environment": "CPython 3.12.15; standard library",
+    "runtime_seconds": 2
+  },
+  "not_checked": [
+    "Native 5/9/11-stage execution",
+    "Hosted emission/audit and integer invoice",
+    "Selected full cold replay",
+    "All-size/compiler/physical/precision/tape/recovery/analytic obligations",
+    "Supplier admission or any new exponent"
+  ]
+}

research/r439-star-kdefect-sectors/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/r439-helper-stars-seed19/candidate.json.gz",
+    "research/r439-bsection-stars-seed01/composition.json.gz",
+    "research/r439-bsection-stars-seed01/candidate.json.gz",
+    "research/e8-bit-p111-tuned/code/xcheck_p.py"
+  ],
+  "commands": [
+    {
+      "argv": ["{python}", "-I", "-B", "{package}/verify.py"],
+      "timeout_seconds": 120,
+      "checks": "Per-helper-sector census of the literal K_equals_D defect on three retained R439 helper-star words; independent GF(2) replay cross-agreement with the pinned owner checker; exhaustive single-operation repair enumeration; H2 clause decomposition; cross-word localization; five controls including SHA-256 source bindings. No native or hosted replay."
+    }
+  ]
+}