A registered result in its source context

GRAV.ENTROPIC.SCALEFLOW.01

Original ledger classification: Formal/Identity [E] (scale-flow + moment dictionary + closure identity, exact) + [C] (measure is TFPT input) + [O] (R^2 gate unchanged)

Original claim text

THE SCALE-FLOW REPRESENTATION OF THE ENTROPIC ACTION (2026-07-13; the follow-up to GRAV.ENTROPIC.SCALARON.01/v475) -- the surviving interpretation 3 (KMS-spectral renormalisation) typed PRECISELY: the (72/17)(8 pi)^9 kill-test gap collapses to ONE moment condition on the scale weight, and that condition is satisfied by the SAME KMS moment TFPT already fixes (v36's f0); zero new dials; closes NOTHING. [E] SCALE-FLOW IDENTITY: -ln a = int_0^inf dt/t (e^{-ta} - e^{-t}) (symbolic exact) = 2 int_0^inf dchi/chi (e^{-a/chi^2} - e^{-1/chi^2}) (t = 1/chi^2; verified to 1e-20) -- Bianconi's log action IS the FLAT scale-integral of relative heat-kernel actions, while TFPT's S_rel,chi evaluates ONE KMS scale with a weight f: the bridge between the two action principles is a choice of SCALE MEASURE. [E] MOMENT DICTIONARY: with a general weight w(t) dt/t the maximally-symmetric expansion (v475 eigenvalues {1; 1/4 x4; 1/6 x6}) gives S_w = 3 beta R mu1 + (17/24) beta^2 R^2 mu2 + O(R^3), mu1 = int w e^{-t} dt, mu2 = int w t e^{-t} dt; the flat Frullani weight has mu1 = mu2 = 1 and reproduces the v475 raw coefficients identically -- the raw action is the unit-moment point of a moment family. [E] THE ONE MOMENT CONDITION: the weight-dressed EH pin beta' mu1 = c3/6 gives m^2 = (4608 pi^2/17)(mu1^2/mu2) Mbar^2; demanding m^2 = c3^7 Mbar^2 forces EXACTLY mu2/mu1^2 = (72/17)(8 pi)^9 -- the v475 kill-test number IS a moment ratio -- and the closure identity (4608 pi^2/17)/((72/17)(8 pi)^9) = c3^7 holds IDENTICALLY: the '13 orders of magnitude' are not a mismatch of the bridge but the statement that the R^2 coefficient is a SCALE-MEASURE datum which the flat measure fixes wrongly and the TFPT KMS measure fixes correctly. [E] v36 CONSISTENCY: the same physics as f0 = 6 (4 pi)^2 / c3^7 <=> M^2/Mbar^2 = c3^7 -- the moment-condition route and the spectral-action route land on the SAME scalaron mass: one KMS moment, two equivalent parametrisations, zero new dials. [C] HONEST TYPING: a consistency statement, NOT a derivation -- the KMS weight (hence mu2/mu1^2) is TFPT input; the entropic action supplies the Einstein + Lambda level and the scale-flow FORM, TFPT supplies the measure; interpretation 3 stays [C]. NEGATIVE SIDE-RECORD: a first attempt to tie the AP2 state-side reading (v476) to an EXACT lattice Bisognano-Wichmann commutator (Peschel-Eisler simple tridiagonal forms) did NOT hold on the half-filled open chain -- kept out of the suite, logged in the research notes. [O] UNCHANGED: the R^2 gate and v359's equation-of-state typing; no status marker moves anywhere

Original classification detail

Scale-flow representation [E]/[C]/[O]: the entropic action = flat scale-integral of relative heat-kernel actions; moment dictionary (flat = unit moments = the raw v475 coefficients); the kill-test gap IS the moment ratio mu2/mu1^2 = (72/17)(8 pi)^9 with the exact closure (4608 pi^2/17)/((72/17)(8 pi)^9) = c3^7 => m^2 = c3^7 Mbar^2 identically, consistent with v36's f0 -- one KMS moment, zero new dials; consistency not derivation; R^2 gate stays [O]

Dependencies as recorded

GRAV.ENTROPIC.ACTION.01; GRAV.ENTROPIC.SCALARON.01; PS.SPECACT.02; v36; v259; v359; v473; v475