[aliromman]

This encoded algorithm and caller integration implements the minimum root-prefix search from PR #5688 on the retained physical Count tapes. It supplies all 29 RootPlan fields to the unchanged RootLayer. A complete dirty root with stock six pads four rows to six, returns the exact Walsh endpoint, preserves borrowed data and clears its working state. Complete kappa is unchanged.

Proof, implementation, source pins, validation and limits.

The predecessor proved the numeric optimum but left physical Count integration open. This implementation evaluates the exact predicate floor(log2(F^D(M-q))) <= K*q with serial Count operations, including the valid cases R<S<2R. It does not decode input-dependent Counts into host integers. The search needs at most 14 live owned leases; the inherited root keeps its fixed 256-tape pool. The production parameter law and guard constant are unchanged. Integration uses the existing plan constructor argument and adds no runtime patch or changed source execution body.

python3.12 -B scripts/research_package.py --package research/encoded-minimum-root-prefix

Python 3.12 and the standard library suffice. Expected: PASS_ENCODED_MINIMUM_ROOT_PREFIX_ARITHMETIC, PASS_ENCODED_MINIMUM_ROOT_PREFIX_ROOTS-DIRECT, PASS_ENCODED_MINIMUM_ROOT_PREFIX_ROOTS-TWO, PASS_ENCODED_MINIMUM_ROOT_PREFIX_ROOTS-SIX, then PASS_REGISTERED_RESEARCH_PACKAGE. The checks cover 1,251 minima against direct padded integer rows and the retained reference, including 771 non-power-divisor cases and 316 cases with R<S<2R; six invalid domains; seven complete dirty roots; exact independent Walsh outputs; pool/borrow preservation; repeated use; and nonzero-padding rejection followed by recovery. Three arithmetic cases use the historical 2,642-bit production divisor. Measured search work is 4,273,750 reads, 858,487 writes and 4,087,798 moves, with the measurement boundary stated in the evidence.

Normal Python 3.14 and registered Python 3.12.14 checks pass. The optimized read-only Git-free replay passes with an empty executable path and preserves all 15 exported files. An actual source mutation rejects before execution. The 14-file committed export passes. An initial direct-fixture timeout is recorded; the verifier now observes the plan from the actual call instead of computing it twice. The complete payload checks are retained.

Four direct-root fixtures use the production formulas with a reduced guard. The fast payloads use an explicit reduced parameter law and complete elementary C2 fixture, including three disjoint copies for divisor six. They do not execute the production O73/GL64 payload. The numeric optimum, eventual one-axis theorem and source native/physical contracts remain predecessor results or assumptions. No improved complete QQ16 invoice, production speedup, independent human/formal acceptance or new multiplication theorem is claimed. Hosted CI is pending at publication. Full-history preflight was not run in the shallow checkout; exact source bytes and immediate retained-source ancestry were checked.

Base and refreshed main: df0137efd35e11b4445734bf612bdd4c7ebc330c, priority 7, selected conditional kappa .00310045. Source-retention commit: 244e717163e902558b3e7ce5e767b38fc4413db6; scientific implementation: 1470cc5f2ddf1332701f35d1492c71b5202e17ae; contribution head: 8a3ed4641ed4efd5c0eef70c94dbfc48f95d8cd1. The historical selected record exists only inside the isolated source archive; the live frontier is unchanged.

Credit Ali Romman's PR #5688 for the minimum-prefix theorem, commdata2338's PR #5515 for exact reached depth, Jose Manuel's PR #3385 for the physical controller/Count cohort and complete fixtures, Nick Petalas's PR #4865 for the geometry context, and Sagar Thakrar's address refinement. All retained notices remain. This uses the same assistant family as the minimum-prefix predecessor; no independent human audit is claimed.

Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0. Scoped claim.