Finite certificate: exact dirty-boundary adapter costs under partial output observations (no new kappa)

[commdata2338]

Change and mathematical scope

This adds an exact boundary optimizer and an independently checked finite census for a larger checkpoint contract. The earlier restored-interval planner requires every observation cut to be fully restored; here only a declared subset of outputs is observed at each interior cut. The interface has four arbitrary-dirty binary coordinates (two outputs, two workspace), six paid unit-XOR adapters, and one to three compulsory two-XOR children of the form (D, W) → (D + BW, GW).

Result: observing only part of the output lowers the minimum in exactly 100 of the 4,808 instances, all at three calls; 48 of them use both noncommuting workspace shears. A sharp witness needs one adapter instead of three under full observation: seven XORs instead of nine, counting the compulsory children. A direct proof in the README establishes both minima. Every improving instance has one zero final output-update row, and none of the 1,824 instances whose two update rows are both nonzero improves.

Status: a finite certificate with a written completeness proof, for fixed indivisible children. Opening the toy witness reduces its endpoint to the single XOR d0 ^= w0, so this is not an unrestricted circuit saving. No supplier, physical admission, recursive-rank improvement or κ is claimed.

The package: research/partial-observation-boundaries/ holds:

• the written model, completeness proof and witness;
• an exact receipt (RESULT.json) and TASK/SOURCES/VERIFY metadata;
• a primary model, a separately written verifier and integrity tests.

Eight new files. Frozen sources, the selected record and the task catalogue are unchanged.

Remaining obligations.

• Not covered: four or more calls, divisible children, output feedback or output mixing.
• Not supplied: a selected caller, a recursive paid profile, native realization, ownership, precision and tape, a dispatch or routing invoice, the full selected cold replay, a Lean theorem, hosted CI or independent human review.

Task and overlap

• Task ID / scope key: persistent-boundary / 73c0a233e7e1abdccbfe, refined to partial-output-observation/GL2F2-x-Mat2F2/six-unit-XOR-adapters/length1..3. Chooser nonce 86e84927a2ed518696c231e449d50f59, track recursive-interfaces, priority revision 3.
• Base and exact heads: base 1ff9c501b624ec354e1330b1b9be571a6adba9cb; contribution f4f8d2afdd3fec60c7739dcd826497e9738f80c1. Contextual heads: PR Research note: restoration coalescing across a recursion batch (read-free + composable-family boundary; no new kappa) #1155 (d7d50406fbdc7ac4141ba454b2e2440dc74942db) and PR Construction: three-stage persistent prefix shear chain (no new kappa) #1930 (11cb2266f7e99279c440a301ea4b862587a9afa2); the selected PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 (0fc329d0bec93ca7cd288eaea1295172e8789810) is unchanged. There is no pending implementation dependency, and no predecessor executable is imported; SOURCES.json pins the retained predecessor README and NOTICE bytes by base, Git blob and SHA-256.
• Concrete object: fixed children with B one of the four elementary 2×2 readers and G either workspace shear, a declared nonempty output-observation mask at every interior cut, and full dirty-state equality at the end.
• Scope status: changed. This is an abstract finite replacement-interface result, not a selected-supplier call pair. It keeps the drawn lower-bound method and changes only the full-restoration checkpoint rule; its value is exact applicability and cost evidence.
• Method and complete domain: all 4,808 instances over exactly 96 coefficient states, with arbitrary finite adapter words, solved by multi-source shortest paths and, separately, by a 0-1 BFS over literal 4×4 matrices. No output feedback, output mixing, ancillas, clean inputs, child splitting or free operations.
• Optimistic budget: every child costs two scalar XORs and every adapter one, entry and final cleanup included; eager execution costs 3n. No mapping from these scalar costs to recursive rank, physical control or a κ ceiling is asserted, so the complex packing-cap report does not apply.
• Nearest work:
  • PR Research note: restoration coalescing across a recursion batch (read-free + composable-family boundary; no new kappa) #1155 and Adam Gall's observer-safe-restoration give exact defect criteria but do not optimize repairs at partial observations.
  • Sagar Thakrar's reader-restoration-cuts-seed53 planner requires fully restored cuts and diagonal drifts.
  • PR Construction: three-stage persistent prefix shear chain (no new kappa) #1930 changes output-to-source prefix chains.
  • This account's coordinate-address SWAP results use a different object and paid alphabet.
• Claim and final check: the claim is in discussion #2007. A final overlap check of the open PRs (368 at 17:15 UTC, complete pagination) and of the changed-file index found no competing work.

Validation

python3 -I -O -B research/partial-observation-boundaries/verify.py
python3 -B scripts/research_package.py --package research/partial-observation-boundaries --check-only
python3 -B scripts/research_package.py --package research/partial-observation-boundaries

Linux x86_64, CPython 3.12.3, standard library. The checks:

• all 4,808 optimum pairs agree between the two searches;
• every restored-partition price is checked by a separate literal enumeration;
• every attaining program runs on all 16 dirty inputs: 76,928 complete executions;
• ten semantic failure controls, five positive controls, and all 385 full-observation words of at most two adapters for the named witness;
• four integrity tests: source and receipt corruption rejection, refusal to overwrite an existing result, and an optimized read-only relocation from an empty working directory.

Census SHA-256: 5dfeaea1efecf76eff484a2e875d4caad57106e08e67ff3ab0fb3e389e2d12dc. The registered replay was run again on the final package bytes before submission, under an 8 GiB memory cap, and passed.

PASS_PARTIAL_OBSERVATION_BOUNDARIES instances=4808 states=96 partial_gains=100 productive_gains=0 dirty_inputs=76928 controls=10 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/partial-observation-boundaries outcome=completed

Credit

• Aaron Wang (@navinor): PR Research note: restoration coalescing across a recursion batch (read-free + composable-family boundary; no new kappa) #1155.
• Adam Gall: observer-safe-restoration.
• Sagar Thakrar: reader-restoration-cuts-seed53.
• Cinna Davis (@1337isnot1337): PR Construction: three-stage persistent prefix shear chain (no new kappa) #1930, distinct neighboring work.

This extends their restricted observer and planner questions and does not contradict their scoped results. New work is Apache-2.0; predecessor attribution and source licenses are kept.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the scoping, the reasoning, the implementation, the verification and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

From the package README:

1. Does an admitted supplier contain a mandatory call chain with this observation pattern and a noncanonical workspace contract? Map an actual pair or triple of complete call boundaries to the model and charge every adapter, owner, frame and transport obligation. Blockers: no source-bound caller yet, and no permission to hide the intermediate output.
2. At four calls, can partial observations lower the cost when both final output-update rows are nonzero? The same 96-state solvers can exhaust the 110,592 length-four instances. That would test the exact edge of the negative three-call census; a positive toy result would still need the physical binding above.

Workflow feedback (optional)

• What helped: the current priorities and the explicit permission to report a changed, bounded interface scope.
• Friction: a task-level overlap query with the default short limit was dominated by local TASK.json matches and hid remote PR matches; --json --limit 500, grouped by kind, was needed.
• Suggestion: give each source kind a small guaranteed quota in the default display, or add --group-by-kind, so current PRs and claims show alongside local packages.

[Mister-Meeseeks]

Batch review at f4f8d2afdd3fec60c7739dcd826497e9738f80c1.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Two exact optimizers agree on4808 four-bit partial-observation instances and full dirty-state replays. The100 apparent adapter savings occur only in one-output cases with no net productive gain;1824 productive instances do not improve. Fixed child and adapter alphabets only, not a universal interface barrier.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.