Research: pay for preparation, execution, restoration and tape for #621's cyclic inverse at every stride (no κ claim)

[Rohansguliani]

The K492 review leaves one obligation open for #621 (Ali Romman). That PR's L-uniform inverse bound still needs paid costs for efficient preparation, rounded execution, restoration and tape. #628 (Chafik Boukhalfa) pays for preparation and neighbouring-stride execution. #764 (Charles Bethin) sharpens the scan rounding. For any stride other than ±1, #628 routes cycles through two merge sorts costing O(L(B+log L) log L). Those sorts bring back the log L factor that the inverse bound removed.

This PR adds research/cyclic-inverse-stride-bridge/ and one CI workflow. No existing file changes. No new integer κ, complex b, or BIT c.

Result

• Theorem 1. A Euclidean chain/closure recursion solves y_i = x_i + b_i y_(i-w) for any stride. A lag-w FIFO built from two stacks gives every pass its predecessor in natural order. Each level at least halves the length. A single stable factor then costs:

  • at most 40L sequential record moves (largest observed: 29.68 per entry through L=131072);
  • O(L) rounded operations;
  • an error of at most min(K,1/β)(u/β)(5+4B/β)+u.

  The error bound holds for every L. The proof reuses Research: sharp inverse-square rounding bound for cyclic scans #764's shared-suffix identity 1-Q ≥ βG. A concavity argument shows that coefficient margins never fall below β at any recursion depth.

• Theorem 2. Composing the solver with Uniform rounded prime-power cyclic inversion (no κ claim) #628's certified factorization gives P(τZ_L^w)^{-1} at precision p+(n+2)g+O(log r) = p+O(r^4), with no L term. One admission condition, p ≥ log2(log2 L+2), depends on L.

• Verdict. For one kernel inverse on fixed sequential tapes, the cost with all four obligations paid is soft-O(r^3(p+r^4)) + O(rL)·M(p+r^4) + O(rL(p+r^4)). That cost per entry does not depend on L, so Prove uniform prime-power cyclic inverse bounds (no κ claim) #621's gain survives payment at the single-kernel level. Restoration in this model is cheap: the input tape is never written, and scratch tapes are erased at linear cost.

Validation

python3 -B -I -O research/cyclic-inverse-stride-bridge/verify.py --check prints PASS_CYCLIC_INVERSE_STRIDE_BRIDGE generic=2052 kernel_cases=300 controls=4 max_records_per_entry=29.671875. It passed under system Python and the repository venv (Python 3.13, Linux). It needs only the standard library and takes about 100 seconds.

• Generic cases (2052). L=1..18, every stride in [-L,L), three margins and two grids. Each case is compared exactly with an independent per-cycle geometric-closure oracle. The margin invariant is checked at every recursion level.
• Kernel cases (300). These use all ten Uniform rounded prime-power cyclic inversion (no κ claim) #628 certified kernels (radix 3 through 16), two gauges, L ∈ {1,2,3,5,8} and all strides for L≤5. Each case checks Theorem 2's exact budget together with Uniform rounded prime-power cyclic inversion (no κ claim) #628's perturbation bound and the expanded residual. 81 cases are also compared with a dense Gauss–Jordan solve.
• Controls. The replay rejects a rotated closure reseed, too little precision, a unit coefficient and a wrong inner-shift direction.
• Vendored sources. The Uniform rounded prime-power cyclic inversion (no κ claim) #628 files are byte-identical to head 0768f97d and are hash-pinned; the verifier refuses drift.
• Repository checks (Linux, base 66849e1). I did not complete a full make verify.
  • make verify passed seven targets with no failures: verify-community, verify-producers, verify-partial-gauge, verify-three-stage-cover, verify-paired-cube, verify-recycled-bit and verify-entrance-banks. My one-hour local limit then stopped the run during gen4-bank-verify.
  • make selected-record-check and make verify-tests passed separately: 549 tests, 0 failures, exit 0.
  • I did not run frontier626-verify or the later targets. This PR changes no existing file and does not affect the selected construction.
  • git diff --cached --check passed.

Still open

• A borrowed-array (dirty) form of the stride solver for the exact-width model. Uniform rounded prime-power cyclic inversion (no κ claim) #628's dirty route still covers that model, at O(p+log(D+1)+r^5) precision.
• Composition across the complete transform, phase collection and normalization.
• The multiplication reduction.
• The O(r^4) guard is a proof bound. Numerical probes put the true min|P| near 2^-10 for r≤32; those probe values are not certified here.

This is not a formal verification. The written proof carries the all-L claims, and the finite cases support the reference implementation.

Credits

• Prove uniform prime-power cyclic inverse bounds (no κ claim) #621, Ali Romman (@aliromman). Kernel and the L-uniform bounds.
• Uniform rounded prime-power cyclic inversion (no κ claim) #628, Chafik Boukhalfa (@chafreaky). Factorization, preparation, phase and dirty-shear budgets, and the vendored certificates.
• Research: sharp inverse-square rounding bound for cyclic scans #764, Charles Bethin (@cbethin). The shared-suffix identity and the inverse-square scan bound.

New work by Rohan Guliani (@Rohansguliani), prepared with substantial AI assistance. Apache-2.0. No literature-priority claim is made.

[Mister-Meeseeks]

100-PR batch review at 4bdb3e755ea7118170a7a34764df955880dec282.

This remains open after scoped triage. Next review step: New all-length fixed-tape stride solver: review FIFO traffic, uniform accuracy and factorization interface together with the later #977/#985 refinements.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.