Construction: reversible quadratic phases for the dirty stride solver (no new kappa)

[aliromman]

Extend PR #985 from constant site coefficients to the nonconstant quadratic
diagonal D_j=exp(2 pi i a j^2/m). The leaf adds an approximation to
(I-b D Z^w D^-1)^-1 x to an arbitrary dirty target and restores the source
exactly. Its negative call reverses the complete grid word. The complete
factor interface keeps the original logarithmic borrowed-array count,
restores every borrowed array, and pays four phase passes around the whole
word. No phase array, cache, or free phase lookup is used.

Review files: Proof and scope · Evidence · Source pins and credit.

The family includes the retained single-axis periodic chirp
exp(-pi i s j^2/t) for even L=t. For L=8,a=-3,m=16,w=3, the edge phases
have residues 11,9,7,5,3,1,15,13; the ideal response differs from the
unphased response by more than 1/8 in a real component.

The implementation uses exact finite differences, a rounded Machin/Taylor
evaluator, three reversible real shears, and exact quarter-turns. The proof
bounds coefficient error and both directions of rounded conjugation,
including large dirty targets. Phase preparation costs O((L+1)P) scalar
steps with P=W+O(log W). The full bill includes descriptors, input and
scratch storage, scalar widths, recurrence counters, data access, and tape
returns. This is a slower paid phase interface, with no fast-bit or runtime
speedup claim.

Reproduce with Python 3.11 or later and the standard library:

python3 -I -O -B research/dirty-stride-phases/verify.py
python3 -I -O -B research/dirty-stride-phases/test_package.py

The primary receipt checks 910 phase cases, 263 rounded leaves, 20 complete
factor words, two retained radix-four polynomial kernels, four larger tape
fixtures, and eight invalid-input controls. The kernel checks pay the exact
factor residual and execution error to a 128-bit total budget. All leaf
numerical cases have nonzero error; dirty data reaches 2^80. Separate
full-angle rational intervals, geometric sums, and dense Gaussian elimination
supply the numerical oracles.

The package harness rejects 12 semantic, source, and type faults, then
regenerates the same receipt from a read-only relocated copy. Literal array
accesses are checked independently of the meter. Phase-shear intermediates
are measured; Machin integer widths use the stated analytical bound. Receipt
SHA256: 3305f81e9befe11133e85e0e38b5f0824c7a520d02e1799d5bf703e0bd1df740.

Local isolated optimized checks pass on macOS arm64 / Python 3.14.7, as do
selected-record integrity and all 25 selected-result tests. The primary
replay took 65.20 seconds and the complete package harness 63.62 seconds;
these are verification durations. The scoped Python 3.11/3.14 workflow is
included; hosted CI and local Python 3.11 execution are separate. The full
repository suite and selected cold replay were not run for this isolated
addition. VALIDATION.json records the exact source freeze and limits.

Base: 8bddbee377e2f3ce1f1247196c4b290d55767d20. PR #985 remains pinned at
9df42a99dd8497b6e7d46cd2507bb89071372640, and PR #479 at
40dd01058439cdffae3e3b41c4fe52edaccf5f91. The local derived solver has two
read-once validation edits because the pinned expressions each read twice
but charge once. The complete DERIVATION.diff is retained beside the
byte-exact original. No inverse arithmetic or transport instruction changes.
All 14 retained source files have SHA256 and Git-blob pins.

Credit Ali Romman (@aliromman)'s PR #985, Rohan Guliani (@Rohansguliani)'s PR #846, Charles Bethin (@cbethin)'s PR #764,
Chafik Boukhalfa (@chafreaky)'s PR #479 and PR #628, Ross Richardson (@RossRichardson)'s PR #948 for related
independent phase-state verification, and the retained manuscript lineage.
New material is prepared for Ali Romman (@aliromman), with substantial
OpenAI Codex assistance; original licenses and notices are retained.

No new kappa is claimed. The selected frontier and upstream/ are unchanged.
This does not implement the multi-axis stream, arbitrary supplied phase
arrays, variable magnitudes, full transform, constant-coefficient precision
improvement, scalar bit-reversible compiler, physical supplier hosting, or
multiplication reduction. The finite checks support the written proof and
are not formal verification or independent human peer review.

Coordinator review: the proof, implementation and complete cost were read. A separate optimized package replay passed all 12 controls and reproduced the receipt from a read-only relocated copy.

[Mister-Meeseeks]

100-PR batch review at 99c71d385171480ecc2c43287aa4a04e1027fcd9.

This remains open after scoped triage. Next review step: New reversible quadratic-phase/tape implementation must be checked jointly with dirty-stride source reuse and precision assumptions.

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.