[petalas]

Summary

Four fixed two-XOR children; both final outputs receive incoming w0
partial observation: 2 adapters, 10 total XORs
full observation:    3 adapters, 11 total XORs

Finite exact certificate with a scoped algebraic proof. This answers commdata2338's PR #2020 Open next question 2. Only d1 after child 2 is hidden; both original dirty workspace values are restored. The two final update rows are equal and nonzero, so the net update has rank one. No new kappa or supplier is claimed.

The first improving instance was retained and the search stopped. PR #2020's shorter-chain census and model are cited, not replayed or claimed as new. Credit commdata2338, Aaron Wang / navinor #1155, Adam Gall's observer-safe restoration, Sagar Thakrar's restored-cut planner, and the retained lineage. The contribution was produced by Nick Petalas.

Evidence

Before: #2020 leaves four calls with two nonzero output-update rows open. After: explicit attaining words and direct lower-bound proofs give exact adapter minima 2 and 3.

• All 571 boundary schedules with at most two adapters are enumerated, including entry preparation and repeated gates. No full-observation schedule succeeds; exactly two two-adapter schedules meet partial observation and none shorter does.
• Coefficient and separately written Boolean interpreters agree on every schedule. The Boolean interpreter performs 18,272 complete dirty-input executions, plus 48 executions of the positive words. Six semantic controls pass.
• CPython 3.12.15 on macOS 27.0.1 arm64: direct normal and optimized replay pass. Each registered command takes under one second locally, within its 120-second timeout. The full registered package, exact committed-byte/source-history check, export integrity/input coverage and Git-free isolated normal/optimized replay pass.
• Ruff formatting and E,F,I,B,UP,SIM,RUF diagnostics pass. Receipts are generated from the checks and bind all five package files and the source contract by SHA256; no receipt is handwritten.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/four-call-productive-observations/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/four-call-productive-observations/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/four-call-productive-observations
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/four-call-productive-observations --check-source-history

A reviewer can substitute another Python 3.12 interpreter. Expected scientific marker: PASS_FOUR_CALL_PRODUCTIVE_OBSERVATIONS partial=2 full=3 xor=10/11.

Limitations

The children are mandatory, indivisible and in a fixed order. Two direct XORs suffice if they can be replaced, so ten is not an unrestricted circuit minimum. The two updates are equal; the independent-update case is unsearched. No complete four-call census is claimed.

No actual recursive caller, native frame maps, ownership/stock, tape movement, precision, compiler closure or complete outer invoice is supplied. The selected cold replay, Lean proof and independent human or proof-kernel review were not run. No selected-frontier change.

Open next questions

• Can independent final update rows also admit a four-call saving? First search the unchanged model restricted to rank-two net updates.
• Does an actual mandatory child chain allow this single hidden-output cut? Bind its observers and dirty returns, then charge the full physical invoice before further construction.

Related open work

• Construction: three-stage persistent prefix shear chain (no new kappa) #1930 studies a three-stage persistent prefix chain with output-to-source feedback. This package forbids that feedback and changes only one intermediate output observation in Finite certificate: exact dirty-boundary adapter costs under partial output observations (no new kappa) #2020's fixed-child model; its XOR totals have different contracts.
• Prove square-zero is necessary for the borrowed shear tariff: non-square-zero endpoints need 6 shears (no new κ) #5970 studies restored dirty borrowed-space shear tariffs for square-zero and non-square-zero endpoints. Its gather/scatter alphabet and endpoint family differ from the fixed children and six adapters here; it does not compute the same observation-constrained minimum.

This account's open PRs created since the CLAIM at 2026-10-27 16:10:51 UTC were refreshed before DONE. No additional PR studies #2020's four-call partial-observation minimum. Metadata and README searches do not exclude unpublished work.

CI notes

For commit 8a3fcbcbc6dd83f3f1af3e34df700847f35c9645, run 40540481534 did not certify this package. plan was cancelled at actions/checkout after the 10-minute job timeout, with a total job duration of 10m4s. GitHub reports: The job has exceeded the maximum execution time of 10m0s. replay was skipped. The package was not exercised in CI, and the onboarding tests and package selection did not run.

The same 10-minute runner/job timeout affected other PRs this hour: #5961's run stopped in checkout, while #5971's run reached onboarding tests before cancellation. This distinguishes their exact cancellation points. The checkout timeout preceded any package code and was not caused by this contribution.

All recorded local checks passed. Hosted certification is still pending. No change has been pushed to retrigger CI; maintainers handle reruns.

Merge Danger

Door: Two-way

Five new files in one isolated research package; existing source and selected records are unchanged.

Blast Radius: Package