TFPT scientific source excerpt
Original repository path: verification/v254_inner_fluctuations.py
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000001 | """v254 -- PS.NCG.FLUCT.01: the INNER FLUCTUATIONS of the finite Dirac operator
000002 | Omega^1_D(A_F) computed explicitly, closing the open obligation v250 #5 (Higgs
000003 | content 10/16, NOT 126) and the NO-JUNK obligation, and fixing the sigma quantum
000004 | numbers (v250 #4, the rep-theory half).
000005 | 
000006 | Inner fluctuation:  D -> D_A = D + A + eps' J A J^-1,  A = sum_i pi(a_i)[D_F, pi(b_i)]
000007 |                     in  Omega^1_D = span{ pi(a)[D_F, pi(b)] : a,b in A_F }.
000008 | 
000009 | Because the finite D_F is purely chirality-ODD (Yukawa L<->R + Majorana), every
000010 | 1-form is odd: the FINITE fluctuations are SCALARS (the gauge bosons come from the
000011 | continuum Dirac [D_M, a], not from D_F).  The scalar content is exactly the Higgs
000012 | sector, and the no-126 / no-junk statement becomes a support-containment check.
000013 | 
000014 |   [E] 1. FINITE FLUCTUATIONS ARE SCALARS.  Omega^1_{D_F} is purely chirality-odd
000015 |         (gamma-anticommuting); its gamma-even (gauge) part is zero -- the finite
000016 |         inner fluctuations are scalars only.
000017 |   [E] 2. NO JUNK, NO 126.  for the SM algebra C(+)H(+)M_3, every 1-form has support
000018 |         EXACTLY inside the Higgs-doublet pattern (chirality-odd, colour-diagonal,
000019 |         no quark<->lepton mixing) -- i.e. exactly the one SM Higgs doublet
000020 |         (1,2)_{1/2}; no colour/weak triplet, no (10,1,3)/126-type scalar, no junk.
000021 |   [E] 3. sigma NEEDS "BEYOND FIRST ORDER".  with the SM algebra the sigma (nu_R nu_R^c
000022 |         Majorana) direction is NOT generated by the inner fluctuations (consistent
000023 |         with first order HOLDING, v252); with the Pati-Salam algebra it IS generated
000024 |         -- sigma is precisely the Chamseddine-Connes-van Suijlekom beyond-first-order
000025 |         scalar (v251/v252).
000026 |   [E] 4. sigma QUANTUM NUMBERS.  sigma is an SM singlet (1,1,0) carrying B-L = 2:
000027 |         nu_R has B-L = -1, so the Majorana coupling sigma*(nu_R nu_R) is gauge
000028 |         invariant only for B-L(sigma) = +2 -- sigma gives the nu_R Majorana mass and
000029 |         breaks B-L (= the Pati-Salam (4,1,2)->SM step).
000030 |   [C] 5. sigma = SCALARON.  sigma is thus a real SM-singlet scalar at the B-L/PS
000031 |         scale; its identification with the R+R^2 scalaron (M_PS = kappa M_s, v249/
000032 |         v253) is consistent in quantum numbers and scale; the gravitational coupling
000033 |         proper remains [C]/[O].
000034 | 
000035 | Status: [E] items 1-4 (finite fluctuations are the SM Higgs doublet with no junk/
000036 | no 126; sigma is the beyond-first-order scalar with B-L=2); [C] sigma=scalaron.
000037 | Reuses the explicit v252 triple.  Python-only (numpy).
000038 | """