Research: exact fixed-option census for PR589 A/B cross-pairs (no kappa claim)

research/bit589-fixed-option-cross-pairs-seed16/README.md

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+# PR589 fixed-option cross-section A/B transport-pair census
+
+## Question and scope
+
+PR #1345's [transport atlas](https://github.com/CrocSwap/integer-mult-bounds/pull/1345), at head `b54a856aa79857d91cd5d65f8f7144e26c8c084f`, fixes the PR589 word and checks single transports and every within-section shared-register pair. It explicitly leaves out a simultaneous A transport and B transport that share a register. This package checks the subset formed from PR #1345’s independently generated fixed-boundary option lists on the fixed PR589 word. It does not claim to close the entire cross-section class, because a pair could make additional frame choices feasible through the changed boundary.
+
+The domain contains one movable A operation and one movable B operation per assignment, with at least one register shared by the two operations. The slot/frame option lists are generated against the predecessor atlas’s fixed counterpart-stage boundary; this census does not expand a move’s frame options when the paired move changes that boundary. It freezes the gate multiset and operands, scatter, start/final frames, D section, rebuilt literal-reverse C section, and retained-output COPY frames. Each move uses the exact commutation window and the legal catalogue/closure frame options from the predecessor transport atlas. The combined A→B register chains are then rechecked, including the boundary climb, before scoring.
+
+The precomputed domain is 336 shared registers, 6,151 cross-section operation pairs, and 88,181 non-home frame/slot assignments. An assignment touches at most four register chains. This price makes a full exact census feasible. It does not cover three or more simultaneous transports, disjoint cross-section pairs, other gate orders, or structural rewrites.
+
+## Exact objective
+
+The score is the full per-register paid frame-chain moment at the exact PR589 bit saving
+
+`c = 1733427039589844234749 / 10^24`.
+
+The pinned engine returns outward rational intervals; no floating-point comparison decides a result. An assignment is labeled improving or worse only when the entire interval has the corresponding strict sign. Identical paid profiles are reported as ties; interval overlaps for different profiles are reported as unresolved. Illegal A→B nesting and retained COPY-frame changes are counted separately.
+
+The selected assembly at this base is complex-limited (`a_c = 0.0016754627 < a_b = 0.0017334270`), so this bit-side audit cannot itself establish a larger selected κ. It closes only the first-order cross-section pair search within the predecessor’s fixed-boundary option sets. Whether a paired boundary change enables additional frame options remains open.
+
+## Result
+
+The exhaustive fixed-option census found 76,840 assignments with the baseline paid profile and 11,341 strictly worse assignments; it found zero improving, illegal, or unresolved assignments. The exact counts and representative cases are in [`result.json`](result.json). Two combined assignments were applied to the literal word, C was rebuilt, and the paid profiles matched the frame-chain predictions: one profile tie and one strict loss. The scope claim is not a claim about boundary-adaptive frame options, larger interaction sets, or a global minimum.
+
+## Reproduce
+
+From the repository root, run:
+
+```sh
+bash research/bit589-fixed-option-cross-pairs-seed16/verify.sh
+```
+
+Expected final line:
+
+```text
+PASS_BIT589_FIXED_OPTION_CROSS_PAIRS seed=16 pairs=6151 assignments=88181
+```
+
+The verifier uses the standard library and the exact source pins in [`SOURCES.json`](SOURCES.json). The interval engine must match its recorded SHA-256 before the program runs.
+
+## Attribution
+
+This extension adapts the exact move generation and outward-rational pricing method from PR #1345, prepared by Qwen3.8-Flash. It credits the selected bit supplier from PR #589 and its predecessors PR #555 and PR #589's cited construction ancestry. The cross-section combined-chain scorer, census, package, and verification were prepared with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Open next questions
+
+- Recompute the legal A and B frame options jointly when a paired move changes the last-A/first-B boundary on a shared register. This is the immediate unclosed case and could add assignments absent from the fixed option lists used here.
+- Extend the joint option model to multiple simultaneous moves only after pricing the boundary-dependent state space.
+- Independently review whether the full paid moment, including unchanged scatter tops, remains the right objective for any proposed extension; the current census binds the exact PR589 saving and fixed profile model.

research/bit589-fixed-option-cross-pairs-seed16/SELECTION.md

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+# Seed 16 task selection
+
+- OS-random assignment seed: `16` (0–59).
+- Track: bit supplier (seed range 00–19).
+- Exact base: `bce54fb3f895cbe5cd4c0eb477c3afeed100190f`.
+- Selection rule: list three narrow candidates in order, choose zero-based `seed mod 3`; `16 mod 3 = 1` selects candidate 2.
+
+## Three candidates
+
+1. **Boundary-adaptive cross-section options.** For one A and one B transport, allow the paired move to enlarge the other's frame set through the changed A→B boundary. This is a focused follow-up to PR #1345's stated boundary coupling, but requires deriving joint option feasibility.
+2. **Fixed-option cross-section A/B pairs.** For every pair of independently legal A/B transports sharing a register, enumerate every pair of exact slot/frame choices and score the combined register chains. This is the selected candidate.
+3. **Within-section shared-register triples.** Extend PR #1345's pair census to triples on a single section. Its review notes a 1,924-operation A component and several large B components; a full product is much larger than the priced cross-section pair class.
+
+The seed selected candidate 2. Candidate 1 remains an explicit boundary-adaptive limitation of this result; candidate 3 is a different interaction order.
+
+## Overlap and predecessor checks
+
+- The current selected certificate at this base remains the conditional complex-limited frontier; this task does not modify it.
+- PR #1345 was open at head `b54a856aa79857d91cd5d65f8f7144e26c8c084f`. Its exact atlas covers singles and within-section shared-register pairs and expressly says cross-section A/B transports are not enumerated. This work reuses its pinned word, moment engine and option generation, and extends the score to combined A→B chains.
+- The fresh overlap scan at 2026-10-13 23:51 UTC covered 852 open PRs and 100 discussions. The narrow fixed-option query returned no possible matches; its report warns that overlap results are leads, not a novelty certificate. A prior broader query matched PR #1345, the direct predecessor described above.
+- PR #1345 was rechecked at the same head immediately before submission. The latest coordination comments also showed a different-track candidate explicitly deferred because seed 16 had claimed the shared-register subset. No competing claim covers this result. Seed-16 claim: [discussion comment](https://github.com/CrocSwap/integer-mult-bounds/discussions/1163#discussioncomment-18911411).
+- PR #1344's frame-509 matrix/splice replay and PR #1351's physical-family realization concern different objects. PR #1127 searches 3→2 z-program substitutions, a separate structural move class.
+
+## Declared limit
+
+This contribution exhausts the product of the predecessor's per-section option lists with the other stage's boundary held at its base value during option generation. It rescored each resulting combined assignment against the full A→B chain, so any invalid resulting boundary chain would be rejected. It does not generate new slot/frame choices that become feasible only because the paired move changes that external boundary. It also does not cover triples, disjoint cross-section pairs, other gate orders or structural changes.

research/bit589-fixed-option-cross-pairs-seed16/SOURCES.json

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+{
+  "base_commit": "bce54fb3f895cbe5cd4c0eb477c3afeed100190f",
+  "track": "bit supplier",
+  "seed": 16,
+  "predecessors": {
+    "PR589": "a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46",
+    "PR1345": "b54a856aa79857d91cd5d65f8f7144e26c8c084f"
+  },
+  "files": {
+    "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz": "e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f",
+    "research/e8-bit-p111-tuned/tune/wlib.py": "a37fdf3cdf5f62945c2a90193401c0e9edc9dd3cda8d8cb36a3b9c329ebe4dbc",
+    "research/complex-k492-owner-repair/vendor/multicold/engines/base_two_moment.py": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098"
+  },
+  "scope": "One movable A transport and one movable B transport sharing at least one register; all other PR589 data fixed. No new kappa claim."
+}