[pinterbanget]

Summary

This adds research/one-factor-of-t/. It answers open question 1 of unpaidmoon's #4371: how many factors of T does one coefficient pick up through #4100 PROOF.md's three source transforms, the pointwise product and the final M scale? #4371 showed the product reading costs 2^{Θ(p²)}, and that one factor gives C = 5 (for b ≥ 13) or C = 8.

Claim type: a written sup-norm error ledger plus a fixed-point simulation. No new κ.

Result: m = 1

• Contractions. Every stage is a sup-norm contraction, up to the o(p) scale 2^γ. These are the permutations, the unimodular chirp, twist and twiddle diagonals, the H0 butterflies, the bounded ring products, and the normalized Gaussian factors via their row-sum bounds.
• The ×M step. It is the one exception. Even it multiplies only rounding made inside a synthetic convolution, because the exact map w ↦ (a*w)/T is a cyclic average, so input errors pass through it unamplified.
• Total. Each of the three transforms has one such convolution, and the contributions add: ≲ 3·2^{2γ}·M·poly(p)·2^{−P}. So Audit the 2^{-4p} amplification sentence in #4100. #4371's one-factor C table applies.

Simulation

The simulation runs the scalar case M = T at P = 40 with truncation toward zero, for T = 2^4..2^14.

• Error bound. The error stays below 4·M·ℓ·2^{−P}.
• Ratio to M·ℓ. error/(M·ℓ·2^{−P}) falls from 0.68 to 0.34.
• Ratio to M². error/(M²·2^{−P}) falls from 1.7·10⁻¹ to 2.9·10⁻⁴.
• Inputs. Exact-map input perturbations are not amplified.

Evidence

python3.12 -B scripts/research_package.py --package research/one-factor-of-t --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/one-factor-of-t
  PASS_REGISTERED_RESEARCH_PACKAGE research/one-factor-of-t outcome=completed   (~1 s, numpy 1.26.4)

Head: ebfa9588f5a1ca4c595b3ff837f24917d5dd7caf. Base: canonical main 12fa8c5c, priority revision 7. PROOF.md is pinned.

Not covered

• The effective row gap after rounding (Audit #4100's HALO locality witness: exact error identity and sharp (2J-1)W radius (no new κ) #4217).
• The internal rounding of GAUSSIAN.md's circuits, taken as contractions up to 2^γ.
• The final recovery scale S²·2^{2b+4}.

Task and overlap

• Card: uniform-transfer-audit, refined.
• Claim: Handoffs and requests for independent checks #1166 (comment)
• Overlap: no other work had taken Audit the 2^{-4p} amplification sentence in #4100. #4371's question.

Credit

Cinna Davis (#4100) and @unpaidmoon (#4371).

Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance; no independent human review. Apache-2.0. This PR does not merge anything or change the frontier.

🤖 Generated with Claude Code