Research: cubic precision for cyclic inverses with arbitrary dirty arrays (no new kappa)

[chafreaky]

PR628's balanced cyclic inverse can use cubic sufficient precision with arbitrary dirty arrays:

max(p, incoming_grid_bits) + O(m + r^3)

The dispatcher, rounded leaves, quadratic leaf count and logarithmic number of borrowed arrays are unchanged. No reserved zero bits or clean length-L array are required. No new κ.

The new free-algebra induction shows that every ideal prefix and inverse prefix is a sum of ordered products with no repeated factor occurrence. Their coefficient row sum is at most 2*3^ceil(log2 n). Applying #862's common-isometry complementary-factor bound to these products, then #877's nonlinear perturbation argument, gives the cubic precision bound. Both source and borrowed arrays still restore exactly; the rounded increment may depend on the initial dirt.

At radix 64, p=128 and initial magnitude at most 2^20, the explicit fractional-bit budget is 1,162,553, versus 73,245,212 in #877. Both schedules have 4,000 leaves and six dirty arrays. These are conservative sufficient bounds, not minimum precision or measured performance. Cubic coefficient preparation uses #862's inherited full-inverse estimate; every leaf pass, inverse call, validation, source access, arithmetic and routing charge remains.

Reproduce with standard-library Python 3.11+:

python3 -I -O -B research/dirty-cubic-precision/verify.py
python3 -I -O -B research/dirty-cubic-precision/test_package.py

Validation on frozen source:

• macOS arm64 Python 3.11.15 and Linux aarch64 Python 3.14.8 produced byte-identical complete mathematical receipts. Linux ran the entire package read-only, outside a repository checkout.
• 5,220 exact integer budgets, checked by a separate recurrence/inequality implementation.
• 142 free noncommuting polynomial traces through 256 factors: 652,654 leaf boundaries, checking prefixes and inverse prefixes. Another 128 words match the frozen dispatcher.
• 18 dense traces, 66 separate Gauss-Jordan inverse checks and 5,104 directly reconstructed intervals.
• 80 actual rounded kernel cases, all with nonzero rounding error and an odd lowest working bit in every initial scalar word, including borrowed storage. They check all 6,032 leaf boundaries, dense exact errors, true-kernel perturbation, source/dirty restoration and inverse cancellation.
• Thirteen mathematical controls, eight package-binding controls per platform, read-only relocation, exact source Git-blob comparisons and unchanged selected-record integrity passed. Full per-platform receipts and logs are included.

Sources are pinned and retained unchanged: #628 0768f97d…, #877 80a60c0d…, and #862 57e06591…, including the original #621 proof/notice lineage. Credit Chafik Boukhalfa, Gareth Davidson (@bitplane), Ali Romman, 0xkilgore and the retained predecessors. #896's concurrent formalization concerns complete nonlinear restoration and ordered products; it explicitly supplies no precision estimate and is not imported here.

All factors must use the same immutable unit-weighted permutation and direction, with phase data distinct from mutable arrays. The conclusion rests on the written universal induction plus inherited analytic, leaf and supplied-preparation hypotheses. It does not supply a bit-reversible compiler, common-label/Clifford word, physical layout, complete transform or multiplication reduction. Selected construction and upstream/ are unchanged. Hosted CI is configured for full Python 3.11/3.14 replays; no hosted success is claimed by the prepublication records.

Prepared by Chafik Boukhalfa with substantial OpenAI Codex assistance, Apache-2.0. Separate arithmetic paths are not independent human review or a formal verification of the full theorem.