Construction: cap unfinished outputs in dirty fanout

[thakrarsagar]

Change and mathematical scope

This is a construction and scheduling tool for dirty fanout. Sharing a temporary row across output updates saves restoration work, but can leave several outputs partly updated. This scheduler caps the number of unfinished outputs. Once an output is committed, it is correct and is never touched again.

For five outputs with a cap of two:

Schedule	Most outputs unfinished at once	Child calls
This construction	2	404
Unrestricted optimum	3	300
Compute and restore each output separately	1	716

All three use four dirty rows. The 716-call baseline is separate restoration, not the optimal cap-one schedule, which costs 508 calls. These are exact operation counts, not measured speedups or memory savings.

For $k$ outputs and capacity $W$, the minimum query cost is

$$ Q_{\min}=2\sum_{i=0}^{k-1}c_{A_i}+2\left\lceil\frac{k}{2W}\right\rceil c_B. $$

Here $c_{A_i}$ prices one complete output-readout call and $c_B$ one complete shared-source call. Prices must be positive, fixed and equal for a call and its inverse. The written lower bound covers arbitrary extra readouts in the declared query alphabet. It does not establish optimality over arbitrary circuits.

The construction and paid recurrence also charge admission, preparation, control, source access and child restoration. Supplier placement, common-label realization, precision/tape integration and outer assembly remain open. No new $\kappa$ is claimed.

Task and overlap

Scope key: dirty-fanout/pending-cap/permanent-commit. This follows the user's recursive-interface assignment, seed 53. The domain is $k\geq0$, $W\geq1$, with exact characteristic-zero linear maps, signed unit queries and arbitrary initial data. The task card records the bounded experiment and stopped earlier candidate.

Investigation base: ce7ad1b6ab09452c3478723a2e96ed5ff212dc08. Upstream advanced during the cycle; this remains the base used for the result.

Predecessor	Exact dependency head	Result used
@CubeJerry, #889	e2021d1c9e06948528cbc0a8fd7079b671da981d	Dirty fanout and its unrestricted weighted minimum
Ross Richardson, #938	df645942469e6a739ec94010a6068c398f4a6c86	Coefficient states and all unrestricted minimum words

The addition is the capacity-constrained scheduler, permanent-commit checker and lower bound for longer words below the unrestricted schedule's required capacity. #982 and #1103 instead borrow output rows; this construction reserves them throughout.

Validation

Run from the repository root with standard-library Python 3.11 or later:

python3 -I -B research/bounded-pending-fanout-seed53/verify.py
python3 -I -O -B research/bounded-pending-fanout-seed53/test_package.py

Expected results are PASS_BOUNDED_PENDING_FANOUT and PASS_BOUNDED_PENDING_PACKAGE. Both pass on macOS arm64 with CPython 3.11.15 and 3.14.8. The checks cover 529 capacity cases, exact dirty-state replay, every child return and five independently computed finite minima. A reordered word passes final-output checks but fails the child-restoration check. Actual file corruptions are rejected, and certificate regeneration is byte-identical.

The all-$k,W$ claim rests on the written proof. The tests provide finite evidence. The validation record gives commands, hashes and controls. Selected-record integrity and its 25 existing tests pass. Full repository/cold supplier replay, Linux execution and formalization were not run; hosted CI success is not claimed.

Credit

Prepared for Sagar Thakrar / @thakrarsagar with substantial OpenAI Codex assistance. @CubeJerry and Ross Richardson retain credit for the predecessor results above, including their attribution to Chafik Boukhalfa's #628 commutator work. Original source and notice files are preserved byte for byte. New work is Apache-2.0; NOTICE.md records the full attribution.

[Mister-Meeseeks]

100-PR batch review at b53b07b5aed4e00358a0d0061ad99dbf5190ca2b.

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

Reviewed scope: Sharp source-query count for bounded unfinished outputs in the stated unit-query alphabet, with complete dirty restoration and port-dependent costs excluded.

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.