Remove compact FFT reservation and top-bit costs with complementary reservoirs

[chafreaky]

The compact FFT currently transforms its reserved axes individually, imposing c > q, and processes extra top bits at cost O(Vd). This supplies a conditional construction removing both costs while retaining the original all-input dirty-core, routing, precision and analytic interfaces.

Complete every Fourier round of one half of the axes while the other half supplies restored dirty fields, then exchange roles. Route each exceptional top-bit round into virtual chunks filled with existing unselected coordinates. At most five global coordinate routes serve an entire FFT; no Fourier axis is enlarged. Ordinary complete-zero-row padding is retained.

The resulting FFT cost is O(V [p^tau polylog(p) + ell d^lambda_prime]). The proof accounts for complete front/back fields and row indices, actual pass chronology, arbitrary dirty contents, pointwise frequency alignment, inverse normalization, the constant number of paid routes, and p+2 precision. It also provides explicit compressed eventual-capacity bounds.

No new kappa is claimed. Gaussian arithmetic and scalar work still limit the current composition. This removes two FFT implementation costs that would otherwise remain even after those separate algorithms improve; kappa = 1 remains unproved.

Verification passed on macOS Python 3.11 and Linux Python 3.12 with byte-identical outputs: 33 polynomial Fourier shapes, 429 basis columns, 98,080 exact entries, 84 common-field layouts, 168 top-field layouts and seven mutation controls. Four parameter rows set c below q and retain the other 44 strict expressions from #472. Those rows distinguish reservation rules; they are not a new full finite invoice. Python 3.11/3.14 CI and source-bound receipts are included.

Run python -B research/complementary-fft-reservoirs/verify.py /absolute/new/output. Only the standard library is needed. Tiny operator tests and large symbolic field inventories have separate scopes; the all-size fixed-tape cost is a written conditional transfer, not an executed large machine or Lean proof.

All eleven source dependencies match pinned Git bytes. Credits: Douglas Colkitt and integer-mult-bounds contributors for compact layout and phase interfaces; RaD/hipotures and the source authors for balanced layout, arbitrary routing and semantic guard; eumemic, icekylinx and Zhihao Chen for the inherited arithmetic. New construction, proof and controls by chafreaky with OpenAI Codex assistance. Original notices retained; no literature priority claimed.

[Mister-Meeseeks]

Thank you @chafreaky. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Complementary reservoirs for compact FFT control fields.

Review and limits: Removes specified reservation and top-bit costs under written all-input tape and restoration contracts. It supplies no new kappa by itself.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 5996e33ecad6a0ccd578aec58831457829d75575. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.