[Dugongue]

Replace the Gaussian adapter arithmetic with two explicitly paid native Word RAM constructions: centered chirp/Hankel convolution for the forward map, and a localized banded inverse with a rank-2h periodic Woodbury correction. Reusing prepared factors across fibers gives adapter work O(T[p^ε log p + p^(1/4+ε/2)] + n^η polylog n + n^o(1)), under the source parameters and documented precision guards.

The package includes informal universal proofs, paid native root/exp/phase tables, an exact O(n) packed multiplier with no TM calls, and portable bounded-word models for Gaussian blocks, banded factor reuse and the actual Gaussian inverse coefficients. Linear Word RAM FFT multiplication is classical; this baseline is not claimed as a new bound.

No new best κ is claimed. The complete faster cyclic-transform caller and its final paid recurrence are not integrated here. Selected certificates and upstream files are unchanged. This replaces arithmetic components rather than using PR #4541's TM simulation as a hidden multiplier.

Reproduce with Python 3.12, standard library only:

python3.12 scripts/research_package.py --package research/native-word-fft

Registered isolated/optimized source-bound replay passes: 37 complete products (including 65,535 bits), 80 roots, 960 recovery products, 774 phase queries, signed exp and exact chirp controls, 12 native Gaussian blocks, 20 real band systems, 9 reused periodic solves, and 3 actual Gaussian coefficient/inverse systems. The actual inverse examples pass a 2^-100 reference residual test. Finite tests are distinct from the informal infinite-tail/error proofs and from a full transform or kernel certificate. Git-free replay and a changed-proof rejection control also pass.

See research/native-word-fft/README.md, NATIVE-RECOUNT.md, VALIDATION.md, OVERLAP.md and NOTICE.md for scope and obligations. Discovery reused the linked Problem 1.8 limb/inversion, Gaussian and Machin notes (source-only hashes retained). Credit OpenAI/math and Harvey–van der Hoeven; classical FFT/chirp, Woodbury and diagonal-dominance methods; artuskg's #956/#647; Dugongue's #390 and Jacob Sussman's P193 / #363/#373/#375/#377/#381/#397/#401/#430 lineage. #4535 is the pending comparison, not a result surpassed by this PR. #4538/#4541 are preceding Word RAM work. Current main was refreshed at f626db5; public title/head triage was refreshed through #4551, with incomplete Discussion coverage stated honestly.

Prepared by Dugongue with substantial OpenAI Codex assistance from the root worker and three parallel workers; independent agent reviews are AI reviews, not human peer review. Apache-2.0.