Recursive interface: order-four frames save 320 dirty-echo primitives (no new kappa)

[1337isnot1337]

Result

This is a scoped recursive-interface construction with no new kappa and no supplier adoption. A five-child interface carries a paired order-four frame across the child calls and pays it back at the end. The exact four-state phase recurrence finds path [0,1,2,3,0,0]: the boundary costs 140 primitives (126 ADD, 14 NEG) versus 300 for eager per-child restoration.

The complete arbitrary-dirty echo costs 1,008 primitives (920 ADD, 88 NEG), versus 1,328 (1,208 ADD, 120 NEG), saving 320. Both maps agree on all 56 input/source/work basis columns. The one-level recursive C_v map and its literal inverse also pass all 56 columns; its complete invoice is 1,068 versus 1,388 primitives. The optimized root echo raises the exact prefix row-L1 peak from 11 to 20.

The quarter-turn is explicitly emitted using 9 ADD and 1 NEG per four-cycle, with the inverse at the same cost. The recurrence is exact in the declared macro alphabet of whole-register powers P^u and Q^v; it is not an unrestricted primitive-word lower bound. Full gate costs, copies, child inverses, terminal restoration, data reads/writes, word hashes and source pins are in the package.

Research base: bce54fb3f895cbe5cd4c0eb477c3afeed100190f. The selected conditional kappa remains 0.001672660197076918; no selected files or result were changed.

Reproduce

python3 -I -O -B research/z4-paired-frame-restoration-seed56/verify.py

Expected:

PASS_Z4_PAIRED_FRAME boundary=140_vs_300 echo=1008_vs_1328 saved=320 root_cost=ADD920+NEG88 basis_columns=56 controls=4 no_new_kappa

The direct replay passes on CPython 3.14.7 in optimized and normal modes. research_package.py --check-only passes metadata and source pins. The registered replay wrapper requires CPython 3.12, which is unavailable locally; hosted CI and the selected cold replay were not run.

Scope and credit

The closest predecessor is Ross Richardson's PR #1287, which treats order-three cyclic output frames. This package independently treats an order-four paired cycle with an ADD/NEG phase-cost vector and prices the internal recursive child. Sagar Thakrar's PR #1227 supplies related paid output-sign-frame work. PR #1303's exact restored 2+3-cycle lower bound is a distinct local-word obstruction, not a bound on this equivariant child interface. No predecessor implementation is copied.

No actual selected supplier with this indivisible interface is identified. Physical hosting, routing/tape, fixed-precision admission, coefficient access and analytic transfer remain open. No runtime, kappa improvement, or full multiplication theorem is claimed.

Prepared by Cinna Davis with substantial OpenAI Codex assistance in algebra, implementation, verification and prose. No independent human review or proof-assistant formalization is implied. New material is Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at bc52eec9717baf36ec58d40d0653445e078a04da.

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

Reviewed scope: Explicit paid order-four frame schedule and arbitrary-dirty echo, with complete 56-column maps and 1,024-path optimum in its fixed macro alphabet. Boundary saving raises a prefix norm; actual supplier, routing and precision adoption remain open.

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.