cft-cmi

Exact x-space Mellin transform W-hat(k) of the <TT> kernel and its KMS relation

Status: proved — a complete structured proof.

Area: Theorem A: exact recovery error of the free fermion · Card type: result · Knowledge-base id: w-hat-mellin-formula

Statement

From the stress-tensor two-point function <TT> (card stress-tensor-normalisation, eq:TT) and dilation covariance, the x-space (Mellin) transform of the <TT> kernel is W-hat(k) = (c/(8 pi^2)) int_0^infinity u^{1+ik} (1 - u - i0)^{-4} du = (c/(24 pi)) k (k^2 + 1)/(e^{2 pi k} - 1) for real k != 0, with W-hat(-k) = e^{2 pi k} W-hat(k) (the KMS relation at beta = 2 pi, obtained without a thermal representation) and W-hat >= 0 (rigor/collar_and_invariant_formula.tex prop:What; Beta-function closed form, numerics to 1e-23 in rigor/p4_invariant.py). Numerical check, not a proof about g_B: inserted into Note 7's spectral prescription (analytic continuation of g-hat, eq:barnes) with Note 7's weights, W-hat reassembles the number 2c/(3 pi^2) to relative 1.9e-11 (sec:num); for c = r this coincides with the proved quasi-free integral of quadratic_limit.tex prop:gB (u = 2 pi k). It is not an independent route to the Bures metric: identifying g_B(phi) with this spectral integral is Gap 2 of lemma_second_variation.tex, which the document does not close (sec:collar (b)1, l. 383-392: eq:barnes is 'a prescription, not an identity').

How to verify

rigor/collar_and_invariant_formula.tex prop:What (l. 176-200), Input 1 eq:TT (l. 43-46), sec:num (l. 260-335, g_B reassembly rel. 1.93e-11), sec:collar (b)1 (l. 383-392); rigor/p4_invariant.out.

Evidence

Related claims

None recorded.

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B5.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

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