Research: cubic precision and exact coarse-grid recovery for cyclic inverses (no κ)

[chafreaky]

Reduces the growing-radix precision guard for PR628's neighboring cyclic inverse from O(r^4) to O(r^3). Every partial inverse is bounded by the full inverse and its complementary polynomial, using the standard Mahler-measure inequality. Bounding all rounded prefixes together avoids multiplying the worst reciprocal margin once per factor. PR621's circle bound is unchanged.

The written proof gives W <= max(p,w)+12r^3+m fractional bits for a source of norm at most 2^m, including exact coarse-grid source recovery, and a factor-preparation budget b <= p+6r^3+m. Integer magnitude bits have the same cubic order. With the inherited preparation algorithms, neighboring-layout execution and setup cost soft-O((Lr+r^3)(max(p,w)+r^3+m)), plus charged phase access and longer phase descriptions.

The new in-place schedule evaluates the inverse, adds its result to a target, applies the forward factors, and rounds the source back to its original grid exactly. This requires owned widened words with reserved low zero bits, whose capacity is explicitly charged. Rounding the target increment to the original grid also gives signed shears that compose and cancel exactly with one fixed precision budget for a planned word.

Reproduce with standard-library Python 3.11+:

python3 -I -O -B research/cubic-cyclic-inverse/run.py --out /absolute/fresh/output

Validation:

• 5,220 exact integer budget cases across all 116 prime-power radices from 3 through 512.
• 120 actual kernel executions; 60 compare every prefix with independent dense solves. All check expanded polynomial residuals, recovery bounds, exact source restoration and total kernel error at most 2^-128.
• 20 interacting shear-word cases, including 12 dense comparisons and exact whole-word cancellation; ten rejection controls across both suites.
• Complete fresh runs passed on macOS/Python 3.11 and 3.14 and Linux aarch64/Python 3.14, in approximately 66–72 seconds. All three generated mathematical artifacts are byte-identical. Receipts, logs, source manifests and artifact hashes are included. Scoped Python 3.11/3.14 CI is added.

No new κ, complex b, BIT c or measured multiplication speedup. The asymptotic claim comes from the written argument; finite tests do not implement Pan's worst-case factorizer. Arbitrary dirty-bit restoration, common-label/Clifford compilation, arbitrary-stride restored-source scheduling, complete-transform accounting and the multiplication reduction remain open. The full repository suite and selected supplier were not rerun for this isolated package.

Credit Ali Romman for #621, Chafik Boukhalfa for #628, and the standard Mahler-measure, periodic-filtering and approximate-factorization methods. Exact predecessor heads, blobs, notices and mathematical sources are retained. Prepared by Chafik Boukhalfa with substantial OpenAI Codex assistance; no independent human or delegated-agent review, formal verification or literature-priority claim.