The complex suppliers transfer to the exact DFT: θ = 1 − 7009/10⁷ (7.009·10⁻⁴) for the exact discrete Fourier transform (no new κ)

[chafreaky]

Adds research/exact-dft-transfer/ and one workflow; no existing file changes and no new κ. It records that this repository's complex suppliers are batched networks in the sense of eumemic's exact-dft-bounds (notes/batched-dft-note.tex, 6f87d1a), so each certified complex moment root a gives, in the exact complex-arithmetic model of OpenAI's An explicit power saving for the exact discrete Fourier transform (25 September 2026, openai/math at adc7f12), an exact DFT of every length and an exact complex convolution in

O( n (log n)^θ (log log n)^(4−θ) ),   θ = 1 − a.

eumemic's repository proves this for the PR #144 supplier on main (a = 607/1250000). This package adds the two later suppliers; the same two propositions are eumemic/exact-dft-bounds#1 (c92cf70).

Result

Supplier	Certificate	m	W / vertex	Deficit	Max child	a (certified)	θ
PR #144 (main d1d6c07)	certificates/paired-cube-complex-input.json	72	29,937	1,936	60	607/1250000 = 4.856·10⁻⁴	eumemic's value
PR #200 (a1175449)	research/paired-cube-diagonal-bit-168/certificate.json, complex.profile	66	13,163	1,320	20	3327/5000000 = 6.654·10⁻⁴	+37%
PR #194 (a8c8778)	research/source-assisted-v4/certificate.json, complex_profile	66	12,052	1,320	20	7009/10⁷ = 7.009·10⁻⁴	+44%

For every fixed κ < 7009/10⁷ the exact DFT and exact convolution take O(n (log n)^(1−κ)) operations. OpenAI's saving is about 2.1·10⁻¹³.

Argument

• The batched moment Σ_ρ N_ρ (ρ/m)^θ < W with θ = 1 − a is exactly this repository's coarse-saving moment Σ_ρ n_ρ ρ (m/ρ)^a < W m (|G| cancels). So a supplier's complex saving is the DFT exponent saving, up to the slightly weaker exp/log bounds used here (which is why PR A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200's 6.65489…·10⁻⁴ is certified as 6.654·10⁻⁴).
• PR A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200 is a batched network (Proposition B in PROOF.md, by reference): the shared-core word is retained from Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144; the fused scalar word's complete signed identity is replayed on every formal column in both shear signs, which is the one-column identity that eumemic's Lemma 2.2 extends to every f; each operation carries one rational frame shared by both registers with nested chains, so each transition is adapters around one child of width its rank increment (Lemma 4.1); reuse pairs, terminal sinks and fused outputs are gates; the ledger Σ ρ n_ρ = 867438 = 66·13163 − 1320 is re-counted by that package.
• PR Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194 is a batched network (Proposition C, by reference to notes/source-assisted-note.tex and the PR Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194 checkers): the complete transcript is a linear word z_final = Mz + Lx with M invertible, each operation an exact local map with dyadic coefficients or a source control (gates); reads subtract the recomputed dirty response at frame zero, K + B = I is checked exactly, every auxiliary is brought to the full frame and every gate reversed (the one-column identity, replayed by the independent scalar transcript with mutation controls); the flow is monotone along nested frame chains with an acyclic once-only reuse DAG, so each transition is adapters around one child of its Fourier rank; Σ ρ n_ρ = 794112 = 66·12052 − 1320.

Verify

python3 -B research/exact-dft-transfer/verify.py --check     # stdlib only, about 20 seconds; -O is refused

The script checks every pin in SOURCE.json (the main certificate of PR #144 and the vendored certificates of PR #200 and PR #194, byte for byte as git show <commit>:<path>), rebuilds the three child histograms from those certificates and compares them with the committed certificates/network-children-*.json, checks the ledger identities (rank sum, deficit = Wm − rank = 2v − 3ℓ, largest child < m), certifies the batched moment at each claimed saving in exact rational arithmetic (artanh series with tail bound; exp(x) ≤ 1 + x + x²/(2(1 − x/3))), rejects the next 10⁻⁷ grid point for each supplier, and compares with expected.json. Margins: 3.16·10⁻³, 2.69·10⁻³, 5.27·10⁻⁴. The workflow runs --check on Python 3.11 and 3.14.

Run before opening: --check passes on this branch (Python 3.14, macOS). Not run: the full make verify (no existing file changes).

Scope and limits

• Conditional on the suppliers' constructions exactly as in eumemic's note: exact complex phases, the Cayley cover and completed-core sharing are written arguments; scalar identities, frame nesting, ranks and ledgers are checked by the packages' scripts. For PR Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194 the scalar transcript replay checks the one-column identity but does not materialize the physical Clifford transports.
• The multiplication bound's bit, assembly, routing, precision and Turing-machine interfaces are not used (item-by-item replacement in Section 6 of eumemic's note).
• Not formally verified; astronomically large constants; nothing here changes any κ.

Credits

OpenAI (exact-transform framework and the original phase network); eumemic (batched tensor recursion, normal-form lemma, transfer, PR #144 proposition; scripts/moment.py adapts its verify_moment.py, Apache-2.0); Ryan Shea (priority for the idea of transferring the community networks to the exact-DFT model); icekylinx (paired cubes, compiler, gauges, Clifford frames, three-stage cover, whole-residual recursion, source-assisted frame flow, PR #184) and ikeboy (PR #193/#194 rebuild on v4 modules); eumemic (PR #117/#155/#157/#161/#168 lineage); James Chang (#124/#166); DaysSky (#162/#192); GamingPuzzled; gupt1156 and Abhinav Ramachandran; Zhihao Chen; Rohan Arun; an664; and the earlier lineage in the repository NOTICE. Full attribution in the package NOTICE. Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance; Apache-2.0. This package claims only the observation that the two later suppliers are batched networks and the exact re-check of their moments.

[chafreaky]

Pointer, for the record: Jacob Sussman's wht-power-saving-lean (f010392) proves in Lean, in OpenAI's family-130 model, the exact DFT of every length in O(n (log n)^z) with z = 1 − 7474546/10¹⁰ (OAI.PowerSaving.transform_mainZ), using a five-stage bridged layout (m = 5h = 110) of the #193 helper circuit; that machine-checked 7.4745·10⁻⁴ is above the paper-proof 7.099·10⁻⁴ claimed here. The two levers are independent: #233's reuse pairing (+1.28% in the three-stage flow) should compose with the five-stage layout (integer-mult-bounds#250 prices such a composition prospectively at 7.547·10⁻⁴); I have asked in his repository (issue #1 there) whether the pairs file can be swapped into his rebuild.

[chafreaky]

Update (830971b): Proposition E adds Jacob Sussman's five-stage bridged layout as a derived supplier. From the certified three-stage profiles the verifier builds the five-stage ledger H5 = 5·H_inv + 2v·(e42 + e21 + e46 + e4) (m = 110, W = 14,692, deficit 3,080) and prices it: for #194's word it recovers 7474/10⁷ (Sussman's Lean-checked 7.474546·10⁻⁴), and for #233's word 7547/10⁷ = 7.547·10⁻⁴ (margin 1.35·10⁻³; 7548/10⁷ rejected). The #233 row is a priced transfer: the pairing changes neither ports, source pairs nor frames, so the layout applies, but its kernel-checked certificate has not been produced (asked for in his repository). #250 prices the same composition at 7.5474·10⁻⁴.

[chafreaky]

Proposition E (#233's word in the five-stage layout, a = 7547/10⁷) now has a kernel-checked companion: jacobalansussman/wht-power-saving-lean#2 (certificate exported by #256) proves the Walsh–Hadamard transform at 1 − 7547361/10¹⁰ and the Fourier transform of every length at 1 − 7547360/10¹⁰ in Sussman's Lean framework (family-130 cost model, Compare.lean passing; comparator not run). The transfer stated here is unaffected; this is the same word priced by a second, machine-checked route.

[Mister-Meeseeks]

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

Contribution: Transfer of certified complex suppliers to exact DFT accounting.

Review and limits: Fresh six-supplier histogram/moment and pin replay passed. Exact complex arithmetic and the referenced Fourier framework are distinct from a bit-cost multiplication theorem.

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: 830971b6b157374ebf941d0ca0b73fd2b956e3e0. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.