Research: four grouped cyclic inverse calls with one dirty connector (no new kappa)

[chafreaky]

Replaces PR628/905's quadratic number of first-order dirty inverse calls with four higher-order calls and one arbitrary dirty length-L connector. Data scans use O(rL) scalar arithmetic for r-1 root factors. Source and connector restore bit for bit, and the signed inverse word cancels exactly, including when every incoming lowest working bit is nonzero. No new κ.

The construction splits the roots into two stable polynomials. Each group uses causal integer lifts, a periodic companion-state closure, a homogeneous correction, and reverse source lifts. A four-call dirty commutator combines the groups; both rounded inner calls have separate error charges because their increments may depend on the initial dirt.

The adjugate resolvent bounds the sum of companion-power norms. Together with #862's common-operator complement bound, this gives cubic data precision. Polynomial coefficients, phase windows, initial gauge, output multipliers and rounded binary powering all have explicit error budgets. The dyadic closure uses classical fraction-free integer elimination, with wider intermediates charged.

For prime-power r>=3, target precision p, incoming grid w and magnitude at most 2^m, let A=max(p,w)+2m+48r^3. With exactly supplied rational unit phases of bit size at most B_phi, the neighboring-cycle bill is

soft-O(rL*(A+B_phi) + r^3 log L*(A+log L) + r^4*(A+log L)).

Clean scalar workspace is O(r^3*(A+log L+log r)) bits, in addition to the data arrays. It can exceed a length-L array for short cycles. This trades the prior quadratic data term for a larger scalar preparation term; it is not an unconditional runtime or space improvement. The Python reference is an exact arithmetic verifier, not a measured implementation of the soft-O bound.

Reproduce offline with standard-library Python 3.11+:

python3 -I -O -B research/grouped-dirty-cyclic/verify.py
python3 -I -O -B research/grouped-dirty-cyclic/test_package.py

Validation on the final frozen source:

• macOS arm64 / Python 3.11.15 and Linux aarch64 / Python 3.14.8 produced byte-identical complete mathematical receipts under isolated optimized execution.
• 160 real, 40 weighted complex, 120 mixed-root and 48 original-kernel cases passed. All ten pinned prime-power fixtures through radix 16 meet the original 128-bit target.
• Repeated roots, zero inside roots, empty groups, both directions, L<k, odd lowest working bits, nonzero preparation rounding, exact restoration and inverse cancellation are exercised against separate dense rational oracles.
• 2,313 budget rows, three written all-size polynomial identities, 16 additional mathematical rejection controls and 36 independent fraction-free solver comparisons passed.
• Each platform passed 11 package rejection controls and a full read-only relocated replay in a path with spaces. Linux also ran the supplied package read-only outside a checkout. All 14 retained files match their pinned Git objects; selected-record consistency and staged whitespace passed.

Full receipts, environment records and logs are in research/grouped-dirty-cyclic-validation. They also record the corrected archive sidecars and read-only mutation-fixture issue; both final platform runs were repeated after the harness repair.

Sources are retained byte for byte with SHA256 and Git-object pins: Ali Romman's #621 kernel and uniform inverse proof, Chafik Boukhalfa's #628 lifts and fixtures, #862 complement bound, and #905 arbitrary-dirty cubic baseline. Classical state-space filtering, Mahler/Wiener arguments and Bareiss elimination are credited separately. No priority claim is made for those methods. Related #805 instead supplies determinant-balanced in-place leaves for real coefficients bounded by 1/2, with exact coefficient growth left open; its code is not imported and no savings are combined.

All factors use the same immutable unit-weighted cycle; phase data is distinct from mutable data. This supplies a local classical arithmetic construction. Bit-reversible arithmetic, common-label/Clifford realization, arbitrary-stride tape routing, irrational phase preparation, clean-scalar hosting and the full multiplication reduction remain separate obligations. Selected certificates and upstream/ are unchanged. Hosted CI is configured for complete Python 3.11/3.14 replays; prepublication evidence does not claim hosted success.

Prepared by Chafik Boukhalfa with substantial OpenAI Codex assistance, Apache-2.0. No independent human or delegated-agent review, or proof-assistant verification of the full theorem, is claimed.