Mister-Meeseeks · Oct 20, 2026 · Oct 19, 2026
diff --git a/research/selected-word-reject-exhaustion/CERTIFICATE.json b/research/selected-word-reject-exhaustion/CERTIFICATE.json
@@ -0,0 +1,98 @@
+{
+  "claim": {
+    "closed_form": "agree iff d not in srcs, or count(d) odd and every non-d has even multiplicity before the last d",
+    "four_exhaustive_for_len2_divergence": false,
+    "length2": {
+      "dest_free": 436261,
+      "dest_in_agree": 661,
+      "destination": 111,
+      "divergences": 661,
+      "form": "[[d,d]] U [[a,d] : a != d]",
+      "n_registers": 661,
+      "total_lists": 437583
+    },
+    "named_diverge_count": 2,
+    "named_rejects": {
+      "dest_with_src": {
+        "diverges": true,
+        "in_len2_divergence_set": true,
+        "srcs": [
+          3,
+          111
+        ]
+      },
+      "dup_src": {
+        "diverges": false,
+        "in_len2_divergence_set": false,
+        "srcs": [
+          3,
+          3
+        ]
+      },
+      "self_even": {
+        "diverges": true,
+        "in_len2_divergence_set": true,
+        "srcs": [
+          111,
+          111
+        ]
+      },
+      "self_once": {
+        "diverges": false,
+        "in_len2_divergence_set": false,
+        "srcs": [
+          111
+        ]
+      }
+    },
+    "small_domain_closed_form": {
+      "max_len": 5,
+      "mismatches": 0,
+      "n": 8
+    }
+  },
+  "controls": {
+    "all_len2_divergences_fail_distinctness": true,
+    "checker_rejects": [
+      "self_even",
+      "dest_with_src",
+      "dup_src",
+      "self_once"
+    ],
+    "checker_spot_divergences": [
+      [
+        111,
+        111
+      ],
+      [
+        0,
+        111
+      ],
+      [
+        660,
+        111
+      ],
+      [
+        3,
+        111
+      ]
+    ]
+  },
+  "kappa_decimal": "0.001672660197076918",
+  "no_new_kappa": true,
+  "schema_version": 1,
+  "selected_word": {
+    "B0": [
+      127,
+      111,
+      [
+        3
+      ]
+    ],
+    "gates": 5472,
+    "n_registers": 661,
+    "path": "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz",
+    "sha256": "e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f"
+  },
+  "status": "PASS_SELECTED_WORD_REJECT_EXHAUSTION"
+}
diff --git a/research/selected-word-reject-exhaustion/PROOF.md b/research/selected-word-reject-exhaustion/PROOF.md
@@ -0,0 +1,80 @@
+# Selected-word reject exhaustiveness for length-≤2 dest-in divergence
+
+## Claim
+
+Fix the pinned tuned PR589 word
+`research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz` and the B[0]
+destination `d = 111` (source alphabet size `n = 2v+R = 661`). Compare Sym
+(accumulate-XOR from the pre-gate valuation) with Exp (sequential ADD into the
+destination, re-reading the current valuation), as linear maps over `F_2`.
+
+1. **Closed form.** For every finite source list `srcs`, Sym and Exp agree if
+   and only if either `d ∉ srcs`, or else `count(d)` is odd and every non-`d`
+   value has even multiplicity in the strict prefix before the last occurrence
+   of `d`.
+2. **Length-≤2 census.** Among all `srcs` of length at most 2 with entries in
+   `{0,...,660}`, Sym≠Exp on exactly **661** lists: `{[d,d]} ∪ {[a,d] : a ≠ d}`.
+3. **Four named rejects are not exhaustive.** The four B[0] checker rejects named
+   in [#3696](https://github.com/CrocSwap/integer-mult-bounds/pull/3696)
+   (`self_even`, `dest_with_src`, `dup_src`, `self_once`) contain only **two**
+   of those 661 divergences (`[d,d]` and `[3,d]`). The other two named rejects
+   agree under Sym/Exp. Hence the four are not an exhaustive taxonomy of
+   length-≤2 dest-in-sources divergence on the pinned word.
+4. Every length-≤2 divergence has `d ∈ srcs`, so
+   `len(set([d]+srcs)) < len([d]+srcs)` and the selected checker rejects it with
+   `bad operands`. Selected kappa is unchanged.
+
+This is not a restatement of the dest-free Sym/Exp agreement already published
+in #3696: the new content is the dest-in closed form, the exact length-≤2 count,
+and the negative exhaustiveness answer.
+
+## Proof of the closed form
+
+Only register `d` is written. For `s ≠ d`, Exp therefore always reads `val0[s]`.
+Reading `s = d` clears the current destination (`out[d] ← out[d] ⊕ out[d] = 0`).
+
+- If `d ∉ srcs`, every Exp read sees `val0`, so Exp equals Sym.
+- If `d ∈ srcs`, let `last` be the last index with `srcs[last] = d`. After that
+  clear, Exp's destination coefficient is the XOR of the suffix `srcs[last+1:]`
+  (which contains no `d`). Sym's destination coefficient is
+  `e_d ⊕ ⨁_s e_s` (multiplicity mod 2). These match if and only if the Sym
+  coefficient of `d` is `0` — i.e. `count(d)` is odd — and for every `s ≠ d`
+  the Sym parity equals the suffix parity, i.e. the multiplicity of `s` in
+  `srcs[:last]` is even.
+
+## Proof of the length-≤2 census
+
+Apply the closed form with `len(srcs) ≤ 2`:
+
+- Length 0 or dest-free length 1–2: agree.
+- `[d]`: `count(d)=1` odd, empty prefix → agree.
+- `[d,d]`: `count(d)=2` even → disagree.
+- `[d,a]` with `a ≠ d`: `count(d)=1`, empty prefix → agree.
+- `[a,d]` with `a ≠ d`: `count(d)=1`, prefix `[a]` odd → disagree.
+- `[a,a]` with `a ≠ d`: dest-free → agree.
+
+Hence the disagreeing lists are exactly the `n` lists `[a,d]` for
+`a ∈ {0,...,n-1}` (the case `a = d` is `[d,d]`), totaling `n = 661`. Direct
+linear enumeration over all `1 + n + n^2` lists confirms the count.
+
+## Four named rejects
+
+| name | `srcs` | Sym=Exp? | in the 661? |
+|---|---|---|---|
+| `self_even` | `[111,111]` | no | yes |
+| `dest_with_src` | `[3,111]` | no | yes |
+| `dup_src` | `[3,3]` | yes | no |
+| `self_once` | `[111]` | yes | no |
+
+So at most two of the four meet dest-in divergence, against 661 total. The
+checker still rejects all four (and all 661 divergences) by operand
+distinctness; reject-count is not divergence-count.
+
+## What this does not claim
+
+- No new kappa and no production schema change.
+- No claim that length ≥ 3 has only 661 divergences.
+- No dest-absent schema corollary (left to maintainers; forbidden as a
+  definitional follow-up in this loop).
+- No restatement that dest-free lists always agree beyond using it inside the
+  closed form.
diff --git a/research/selected-word-reject-exhaustion/README.md b/research/selected-word-reject-exhaustion/README.md
@@ -0,0 +1,74 @@
+# Selected-word reject exhaustion for length-≤2 dest-in divergence
+
+**Card `supplier-boundary-audit`, priority revision 6**, changed to answer
+[#3696](https://github.com/CrocSwap/integer-mult-bounds/pull/3696) open question 2
+under the [#2977](https://github.com/CrocSwap/integer-mult-bounds/pull/2977) /
+[#3004](https://github.com/CrocSwap/integer-mult-bounds/pull/3004) checker/exporter
+cascade. No new kappa (`0.001672660197076918`).
+
+#3696 showed dest-free gates always make Sym and Exp agree, named dest-in
+witnesses that diverge, and four B[0] `bad operands` rejects. Its OQ2 asked
+whether those four rejects are exhaustive for dest-in-sources divergence on the
+pinned word. They are not.
+
+This package proves a closed-form Sym/Exp agreement criterion for dest-in lists,
+enumerates every source list of length at most 2 at B[0]'s destination `d=111`
+over `n=661` registers, and obtains exactly **661** divergences
+`{[d,d]} ∪ {[a,d] : a ≠ d}`. Only `self_even` and `dest_with_src` among the four
+named rejects lie in that set; `dup_src` and `self_once` agree. Dest-free
+agreement is used inside the closed form and is not republished as the claim.
+
+```sh
+python3.12 -I -B research/selected-word-reject-exhaustion/verify.py
+```
+
+Expected:
+
+```text
+PASS_SELECTED_WORD_REJECT_EXHAUSTION len2_divergences=661 named_diverge=2 four_exhaustive=false no_new_kappa
+```
+
+## Evidence
+
+- Closed form matches linear Sym/Exp on all `(d,srcs)` with `n≤8`, `len(srcs)≤5`,
+  and on the full length-≤2 census at `d=111`.
+- Length-≤2 census: **661** divergences, **661** dest-in agreements, total
+  `1+661+661²` lists.
+- Named rejects: diverge `{self_even, dest_with_src}`; agree `{dup_src, self_once}`.
+- Checker rejects all four named mutants and spot divergences
+  `[d,d]`, `[0,d]`, `[660,d]`, `[3,d]` with `bad operands`.
+- Every length-≤2 divergence fails operand distinctness.
+
+Default verify is read-only against `CERTIFICATE.json`. Admission uses `need()`
+and stays enforced under `python -O`. Explicit `--write` regenerates the
+certificate only.
+
+## Reproduce
+
+```sh
+python3.12 -I -B research/selected-word-reject-exhaustion/verify.py
+python3.12 -I -O -Werror -B research/selected-word-reject-exhaustion/verify.py
+python3.12 -I -B research/selected-word-reject-exhaustion/independent.py
+python3.12 -I -B research/selected-word-reject-exhaustion/test_package.py
+python3.12 -B scripts/research_package.py --package research/selected-word-reject-exhaustion
+```
+
+## Open next questions
+
+1. **Exact length-≤3 divergence count at the same destination?** First step:
+   count closed-form failures over `n=661` lists of length 3. Blocked by: none;
+   omitted to keep this package on the length-≤2 OQ2 scope.
+2. **Should schema docs separate divergence classes from distinctness rejects?**
+   First step: optional exporter warning keyed to the closed-form predicate.
+   Blocked by: maintainer adoption.
+
+## Credit
+
+Closest prior: [#3696](https://github.com/CrocSwap/integer-mult-bounds/pull/3696)
+`gate-operand-export-parity` (Wentao Li; dest-free lemma, four named rejects,
+OQ2). Cascade priors: Nick Petalas / #2977; Kumagi / #3004; Dylan Crow exporter;
+eumemic symbolic checker; tuned PR589 word. Scatter-parity contract audit is
+credited as provenance only.
+
+Prepared by Wentao Li with substantial AI assistance. Apache-2.0.
+No independent human or proof-assistant review is claimed.
diff --git a/research/selected-word-reject-exhaustion/SOURCES.json b/research/selected-word-reject-exhaustion/SOURCES.json
@@ -0,0 +1,48 @@
+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {
+      "path": "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz",
+      "usage": "input",
+      "git_blob": "85a4499bf9e4911a705d50567a2c2ef6cf23739a",
+      "sha256": "e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f",
+      "size_bytes": 42353
+    },
+    {
+      "path": "research/e8-bit-p111-tuned/code/xcheck_p.py",
+      "usage": "input",
+      "git_blob": "184f5dcf94a486cc7db91ca4db5865916bf26009",
+      "sha256": "af3decc877e27c00001cf56f50c6c09566e8e101297ab34c7860aed96a06d80e",
+      "size_bytes": 21651
+    },
+    {
+      "path": "research/scatter-parity-contract-audit/README.md",
+      "usage": "provenance",
+      "git_blob": "81cbe8e1756e0952028524addb0ed33188e617d1",
+      "sha256": "824ea617d818674b25c4d3dc7c7f6a7e252e217837d8473c2409d08096a48788",
+      "size_bytes": 7389
+    },
+    {
+      "path": "research/scatter-parity-contract-audit/TASK.json",
+      "usage": "provenance",
+      "git_blob": "fd03dc4c2c5028954e2165fb2358a3dfc70fae56",
+      "sha256": "b3619e9ad0c4b17b3f5ddffbe55fb25728bb3164fe32d4f614983cb8db28a7e9",
+      "size_bytes": 7192
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129",
+      "size_bytes": 10725
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0",
+      "size_bytes": 58553
+    }
+  ]
+}
diff --git a/research/selected-word-reject-exhaustion/TASK.json b/research/selected-word-reject-exhaustion/TASK.json
@@ -0,0 +1,72 @@
+{
+  "schema_version": 1,
+  "task_id": "supplier-boundary-audit",
+  "priority_revision": 6,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "original_card": {
+    "id": "supplier-boundary-audit",
+    "question": "Does one selected supplier obligation follow from its actual source-bound checks?",
+    "starting_points": [
+      "certificates/selected-result.json",
+      "research/owner-boundary-audit/README.md",
+      "research/scatter-export-parity-audit/README.md"
+    ],
+    "methods": [
+      "Trace one hypothesis through the source, checker and complete caller.",
+      "Construct one semantically invalid but superficially well-formed fixture and test rejection."
+    ],
+    "acceptance_gate": "Pin the current sources and one obligation; the historical owner audit is an example, not evidence of a current bug.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read #2977 reported duplicate-scatter counterexample and its opt-in repair. The archive is not a production fix or an independently validated selected-result failure; the selected generator emits unique rows.",
+      "scope": "Source-binding orientation only; does not replay or validate the reported bug."
+    },
+    "first_experiment": "First check whether the scatter repair has been adopted. If still open, independently reproduce #2977 symbolic-versus-exported parity on the tiny duplicate fixture, compare the selected unique-row positive control, and define either enforced uniqueness or consistent multiplicity semantics. A bounded guard-adoption patch must preserve valid inputs and test the actual exported action. Stop after this obligation; no full historical replay or frontier change follows from the mutant alone.",
+    "bottleneck_link": "Closing a current acceptance gap or proof obligation strengthens reliability even without changing kappa.",
+    "allowed_variations": [
+      "Audit one named current condition using a new semantic counterexample."
+    ],
+    "scope_change_triggers": [
+      "Reporting the historical timing-only diagnostic as a new current acceptance bug.",
+      "Calling a malformed mutant a failure of the selected construction, or importing an opt-in repair without testing the exporter boundary."
+    ],
+    "overlap_terms": [
+      "owner boundary",
+      "bridge omission",
+      "hosting",
+      "2977",
+      "scatter export",
+      "duplicate index",
+      "parity semantics"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Answer #3696 open question 2 on the pinned tuned word: are the four named B[0] checker rejects exhaustive for dest-in-sources Sym/Exp divergence? Prove a dest-in closed form, enumerate all length-≤2 source lists at destination 111 over n=661 registers, and show exactly 661 divergences of which only two lie among the four named rejects.",
+  "scope_change_reason": "Authorized follow-up to #3696 OQ2 under the supplier-boundary cascade. Dest-free Sym/Exp agreement is already published and is not restated as the claim; the new obligation is exhaustiveness of the four rejects versus the complete length-≤2 divergence set, plus the dest-in closed form that makes the census checkable.",
+  "bottleneck_link": "Knowing that reject samples are not a divergence taxonomy prevents treating four mutants as a complete gate-operand contract.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the closed form, length-≤2 census, named-reject comparison, checker spot controls, and certificate admission. No dest-absent schema corollary, exporter rewrite, native replay, or kappa change.",
+  "outcome": "completed",
+  "evidence": "CERTIFICATE.json, PROOF.md, verify.py, independent.py and test_package.py. Closed form matches linear Sym/Exp on n≤8,len≤5 and on all length-≤2 lists at d=111; exactly 661 divergences; named diverge count 2; four_exhaustive=false; checker rejects the four named mutants and four spot divergences.",
+  "author": "Wentao Li",
+  "ai_assistance": "Substantial AI assistance for reading the #3696 cascade, deriving the dest-in closed form, implementing the length-≤2 census and package controls, and writing this package.",
+  "open_questions": [
+    {
+      "question": "What is the exact count of dest-in Sym/Exp divergences for source lists of length ≤3 at the same destination?",
+      "why_it_matters": "Length 3 is the next bound that appears in selected multi-source gates; a closed count would extend the taxonomy past the length-≤2 census.",
+      "first_step": "Apply the closed form to count odd-count(d) lists with even non-d prefix multiplicities over n=661.",
+      "blocked_by": "None mathematically; omitted here to keep the package inside the length-≤2 OQ2 scope."
+    },
+    {
+      "question": "Should a successor schema document divergence classes separately from distinctness rejects?",
+      "why_it_matters": "Distinctness rejects agreeing cases (dup_src, self_once, [d,a]) as well as diverging ones; a contract that only names four mutants understates the diverge set.",
+      "first_step": "Add an optional exporter warning keyed to the closed-form diverge predicate.",
+      "blocked_by": "Maintainer schema adoption; frozen exporters stay unchanged."
+    }
+  ]
+}