Community result: conditional exponent saving above 2^-15

Release: community-kappa-15-2026-10-08.

This release incorporates the reviewed community witness

[
T(n)=O\bigl(n(\log n)^{1-\kappa}\bigr),\qquad
\kappa=971668963/25000000000000=3.886675852\times10^{-5}>2^{-15}.
]

The final contribution is by Rohan Arun (@rohanarun),
PR #39.
It fixes the middle basis in the copied reversed-corner construction,
certifies every required physical profile and supplies the common-basis
proof and final composition. The imported PR is pinned at
70ae24129649f6d6d4ec6360962a80c3c42a38f1.

The result is a community effort. Its incorporated dependencies credit:

icekylinx: batching, partial swaps, fixed profiles, copied centers and
the selected complex circuit.
Zhihao Chen (@jacklightChen): controlled bases, translated frames,
semantic/bulk compatibility and two-stage integration.
RaD (@hipotures): semantic precision, coordinate routing, phase-cell
inversion and bulk resampling.
James Chang (@jamesyc): reversed geometry and exact certificates.
Aurel Prosz (@Paureel), Swapnil Jain: attributed two-stage development
and paid copied-stream endpoint work.
Dominik Scholz (@DominikScholz): dimension, parameter and fixed-basis refinements.
eumemic, Bortlesboat and dleen: earlier complex, Gaussian, source-frame,
pairing, retained-total and sharing contributions.

Douglas Colkitt maintains the original project and its review/integration,
with OpenAI Codex assistance. The full contribution record
also preserves parallel, incremental, superseded and pending work; credit is
not restricted to the final winning parameter set. Source-specific authorship,
AI-assistance disclosures and licenses remain in NOTICE.

The completed audit accepts the new
arguments conditionally on the original OpenAI #109 framework. Validation
includes 218 tests, 20 historical patch checks, fresh producers, exact
profiles, Linux/GCC reproduction and a three-version Linux Python matrix.
An additional arithmetic checker independently encloses both recursive moments,
reconstructs all seven final margins and rejects the next bit-saving grid point.
Missing pinned reference files were restored with their original hashes.

This is a conditional research release, without full formal verification or
independent human peer review. It makes no practical runtime, worldwide
priority or strongest-known-algorithm claim. The value is 27.36% above 2^-15
as an exponent saving. The former 2^-30 checkpoint and all historical
scientific artifacts are retained. PR #40 and later contributions remain
outside this pinned review.

Start with the proof and reproduction guide
and exact certificate.
Run make verify for the full suite or make community-audit-check for
the additional arithmetic/source check. Requires Python 3.11+, Git, Make
and GCC/Clang with C++17 and unsigned 128-bit integer support.

Project license: Apache-2.0; selected bundled RaD sources retain CC0.
The original OpenAI manuscript and Harvey–van der Hoeven analytic machinery
retain their existing attribution. This is not an official OpenAI release.