[Dugongue]

Given genuinely emitted observations v=(I+U/2)y with fixed cyclic Pauli routing U, supply an executable finite-pass consumer that retains absolute phase, arbitrary observations and owned buffers. Compile V=U^W once and factor the inverse: W+q-2 monomial/ADD passes suffice, with q=2 for XOR and q=4 for general quarter phases. The inspected three-role and thirteen-role XOR ports use 3 and 13 passes.

Implement every prescribed fixed-rational rounding by a bit-plane prefix-remainder scan in O(log(2 + h)) ordinary h-bit Word RAM operations using Boolean operations, addition/subtraction/comparison and both logical shifts, without MUL or DIV. This improves #4885's conservative scalar recovery fallback from O(N h) to O(N log(2 + h)) in the declared ABI. For N TOTAL entries across fixed W roles, the combined cyclic consumer costs O(W N log(2 + h)), plus O(W N) literal return compilation, with 2/3 owned buffers and explicit address/guard precision. The proof gives 5 grid units of component error against exact recovery of stored observations, or 6 including observation quantization. Operation constants depend on the fixed denominators; no practical crossover speedup is claimed.

No new kappa, favorable paid observation producer or complete multiplication supplier is claimed. Literal couplers from #4818/#4819/#4826 are source-bound and independently checked; their retained source headers make large-kappa claims that this consumer does not endorse. The exact Problem 1.8 Lean RAM derives right shift using DIV; this both-shifts ABI is stated separately. All upstream ownership, producer, compiler, precision and outer-budget obligations remain explicit.

Validation: all 7 registered standard-library Python checks pass, including normal and optimized scientific replays. Coverage includes 6912 unsigned quotients, 24978 signed-rounding and 840 rational cases through width 4096; independent review adds 5117 boundary cases. The source-bound cascade checker covers 376 basis columns and 7168 retained coupler entries; independent review adds 144 fixtures and tensor-phase controls. End-to-end bounded-word recovery agrees within the proved 5/6 grid-unit bounds with an independent exact rational reference. Three separate optimized source mutations reject a wrong suffix offset, discarded return phases and incorrect normalization. Committed bytes, source/staging hashes, export replay and publisher preflight are checked locally. No fresh Lean kernel audit, hosted Linux CI, full source-history audit or full multiplier replay is claimed.

Credit hexapode #4818/#4819/#4826, Clear1nsight #4842, Dugongue #4743/#4885, JoyboyBrian #4797, Nick Petalas #4825, Ali Romman #4733, Yiyang/Lyy-iiis #4710, lee-phys #4688, Jacob Alan Sussman #4628 and every retained source notice. Geometric-series inversion, Pauli conjugation, Blelloch scan and classical invariant-division context are credited standard ingredients. Prepared by Dugongue with substantial OpenAI Codex assistance, one lead plus three workers. All redistributed per-package LICENSE and NOTICE files are retained.

Only research/native-cyclic-recovery is added against canonical main 624de9d. Selected frontier, shared code, frozen predecessors and authoritative MathProof DAG remain unchanged.