Research construction: cyclic output frames reduce restoration (no new kappa)

[RossRichardson]

Fixed cyclic-equivariant children can carry their workspace drift between calls by rotating the smaller output. In the complete five-child dirty echo, this reduces 588 to 504 signed ADD instructions, with both copy passes, all child calls/inverses and arbitrary dirty-workspace restoration charged. Both words use the existing six-ADD three-cycle; no faster rotation circuit is claimed.

This is a scoped recursive-interface research construction; no new kappa. It adds the exact admission gate R P = Q R, a sharp uniform three-state boundary recurrence for order-three permutations, an executable recursive emitter and separately represented admission/boundary/invoice checks. The entire change is research/cyclic-output-restoration/.

Proof, full recurrence and limitations · Exact certificate · Source pins · Validation.

The fixed-child boundary and equivariance gate are essential. Opening the toy matrix leaves permits cheaper direct evaluation. No actual supplier with the required indivisible contract has been identified. The deeper witness saves arithmetic but increases address travel and intermediate coefficient bounds; these are recorded, not omitted. Preparation, physical routing/hosting, finite precision and all-size/analytic transfer remain separate obligations. No selected supplier, frontier, upstream input or review ledger is changed.

Reproduce with the standard library:

python3.12 scripts/research_package.py --package research/cyclic-output-restoration

Expected: PASS_CYCLIC_OUTPUT_RESTORATION ... saved_rotations=14 saved_primitives=84 no_new_kappa, seven passing package test groups, then PASS_REGISTERED_RESEARCH_PACKAGE.

Validation covers all 66,420 boundary schedules, all 19,683 ternary 3×3 admission matrices (27 admitted), 100 price comparisons and 192 recursive cases / 3,744 basis columns. Every case checks the full dirty-state map, literal inverse and a separate inverse block formula, recursive invoice, data/descriptor accesses and a rational dirty probe. Ten mathematical controls and five on-disk corruptions pass, as does isolated relocation and byte-identical certificate regeneration. CPython 3.12.12 registered replay, 3.9.6 replay/tests and 3.14.6 regeneration agree. Selected-record verification and five selected tests pass. Full selected cold replay, human review and hosted CI success are not claimed.

Assignment seed 56, recursive interfaces. Research base dadfea6e8a78ff6ba901705f87532a71c6f5bbb8; refreshed submission base b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c. The rebase preserves every package code/certificate/input byte. Final overlap refresh covered 871 open PRs and 32 discussion records with complete pagination. No pinned coordination issue existed; the local claim is updated once with the result and immutable artifact link.

Credit Sagar Thakrar's #1227 for paid output-sign frames, mightyOil/Philipp's #1203 for noncommuting reader interfaces, #1145 for the existing six-ADD rotation upper bound, and Aaron Wang/Navinor #1155, Sagar #1199, Adam Gall #1205 and Ross Richardson/CubeJerry #889 for the retained restoration/echo lineage. Hanson Sun's concurrent #1246 prices additive defect repair at canonical cuts; this package instead transports the output in a cyclic frame. Exact predecessor heads and original notices are retained. No priority is claimed for those earlier mechanisms.

Prepared by Ross Richardson with substantial OpenAI Codex assistance in algebra, both implementations, verification and prose. New material is Apache-2.0. Review should focus on the cyclic admission and indivisible-child contracts.

[Mister-Meeseeks]

100-PR batch review at 8606df05eca3c65032361c74ee3dbc2630b0cdc4.

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

Reviewed scope: Non-scalar three-cycle reader contract R P=Q R, exact three-state boundary recurrence, and fully restored dirty echoes: 504 versus 588 ADDs in the frozen example. Preparation, travel and prefix growth are separately priced; no physical tape or supplier speedup claimed.

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.