[RossRichardson]

The common-factor batch y_i += A_i B x can eliminate its dedicated dirty connector by borrowing an output between complete child calls. The scratch/work tradeoff is explicit: four B calls and two calls per A_i, with every other output excluded from each child's private scratch. This preserves arbitrary dirty inputs and noncommutative factor order.

Status: recursive-interface research construction; no new kappa. For the seed-59 three-output fixture, complete balanced child lowering uses 238 leaf calls versus 318 for separate output-borrowed echoes, both with three extra dirty rows. The dedicated PR889-style batch uses 158 calls and four rows. Removing that row therefore costs 80 calls against the dedicated batch; it is not a free improvement or a measured runtime claim.

The written proof establishes a sharp 2k+4 query minimum only in the stated output-to-output unit-generator alphabet, including feedback through modified outputs. For two outputs, the proof reduces all minimum words to 40,320 permutations; full free-polynomial replay finds 224 correct words. Scalar copies, composite-product queries, specialized matrix identities and unrestricted circuits are outside this lower bound.

Only research/output-borrowed-fanout/ is added. It contains the ownership proof, complete work/storage/tape recurrence, literal certificates, an independent verifier, source pins, attribution, local claim and validation. Exact experimental base: bbc600db2fc15007b7c914f4f2b2ed55fa909c87. Submission refresh: 407d52ff15b2217b6e22bce44728b4c3907b6515; the selected PR942 certificate is unchanged at conditional kappa 1661960547713947/10^18.

python3 -I -O -B research/output-borrowed-fanout/verify.py
python3 -I -O -B research/output-borrowed-fanout/test_package.py

Expected markers: PASS_OUTPUT_BORROWED_FANOUT and PASS_OUTPUT_BORROWED_PACKAGE.

Validation on macOS arm64:

• CPython 3.14.6 and 3.9.6 agree on the complete evidence receipt. The 41 declared cases cover 123 schedules, 1,550 complete child boundaries and 27,020 forward leaf occurrences, with exact arbitrary-dirt identities and literal inverses.
• 369 exact dense forward and 369 inverse executions use identical comparator inputs; prefix widths, full factor prices and tape travel are checked. Eight mathematical/price defects reject; two nonlinear controls exhibit the explicit linearity boundary.
• Offline relocation, two exact decompressed certificate regenerations, receipt regeneration and eight package-corruption rejections pass. Four retained references match their Git blobs and SHA256; base context, selected-record integrity, all five selected-result tests and whitespace pass.

Direct predecessors: CubeJerry / PR889 (e2021d1c9e06948528cbc0a8fd7079b671da981d) for dedicated fanout; shhxera / PR822 (8c5e1b557f59bca29139c4c2dcc65edfb808372b) for sibling-pool reuse and the unsafe ancestor-output counterexample; credit also to Chafik Boukhalfa's PR628 and androolloyd's PR673 lineage. Original notices are retained. Our private child pool still excludes all other outputs, so this does not reopen PR822's failed unrestricted donation rule.

The refreshed queue through PR978 and targeted all-state overlap searches found no same-scope duplicate. PR934/943/944 use dedicated multi-source connectors; PR951's additive cyclic filters and PR953's dyadic batches use different interfaces. No pinned coordination issue exists, so the sprint claim is local. Seed 59 was drawn from OS randomness.

Prepared by Ross Richardson (@RossRichardson) with substantial OpenAI Codex assistance, under Apache-2.0. No supplier integration, full repository replay, hosted-CI success, human peer review or proof-assistant multiplication theorem is claimed. Physical hosting/frames, common labels, efficient tape/precision, all-size compilation and outer analytic transfer remain obligations. Neither upstream/ nor the selected frontier changes; no merge is requested. The upstream repository has no codex or codex-automation labels available.

[Mister-Meeseeks]

100-PR batch review at 5eca5b435d2ce172edf6ffb85694c7efa31bf114.

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

Reviewed scope: Output-borrowed fanout trades extra B calls for one fewer dedicated row; exact restricted-alphabet cost/storage comparison.

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.