Original ledger status
PS.NCG.FLUCT.01
PS.NCG.FLUCT.01
INNER FLUCTUATIONS Omega^1_D -> SM HIGGS DOUBLET (no 126, no junk) + sigma (2026-06-17): closes the open obligation v250 #5 (Higgs content, NOT 126), the NO-JUNK obligation, and fixes the sigma quantum numbers (v250 #4 rep-theory half). Inner fluctuation D -> D + A + eps' J A J^-1 with A = sum pi(a)[D_F,pi(b)] in Omega^1_D = span{pi(a)[D_F,pi(b)]}. Because D_F is purely chirality-odd (Yukawa L<->R + Majorana), every 1-form is odd, so the FINITE fluctuations are SCALARS (gauge bosons come from the continuum [D_M,a]). (1) [E] Omega^1_{D_F} purely chirality-odd, gauge part 0. (2) [E] NO JUNK / NO 126: for the SM algebra C(+)H(+)M_3 every 1-form has support exactly inside the Higgs-doublet pattern (chirality-odd, colour-diagonal, no quark<->lepton mixing) = the single SM Higgs doublet (1,2)_{1/2}; no colour/weak triplet, no (10,1,3)/126-type scalar (0 junk entries, machine-checked). (3) [E] sigma NEEDS BEYOND FIRST ORDER: with the SM algebra the sigma (nu_R nu_R^c Majorana) direction is NOT generated by Omega^1_D (consistent with first order HOLDING, v252) but with the Pati-Salam algebra it IS generated -- sigma is precisely the Chamseddine-Connes-van Suijlekom beyond-first-order scalar (v251/v252). (4) [E] sigma quantum numbers: SM singlet (1,1,0) with B-L=2 (nu_R has B-L=-1, so sigma*(nu_R nu_R) is gauge invariant only for B-L(sigma)=+2), giving the nu_R Majorana mass and breaking B-L (the PS (4,1,2)->SM step). (5) [C] sigma=scalaron: sigma is a real SM-singlet scalar at the B-L/PS scale, consistent in quantum numbers and scale with the R+R^2 scalaron (M_PS=kappa M_s, v249/v253); the gravitational coupling proper remains [C]/[O].
Recorded status: Numerical (NCG inner fluctuations) [E]/[C]
Dependencies, status detail and source
Canonical status: [E] finite fluctuations = SM Higgs doublet (no junk/no 126) + sigma beyond first order + sigma B-L=2 singlet; [C] sigma=scalaron; closes v250 #5 + no-junk obligation + sigma QN
Dependencies: PS.DIRAC.02; PS.DIRAC.01
Scripts: v254_inner_fluctuations.py
Location: tfpt_research_contracts
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