TFPT scientific source excerpt
Original repository path: verification/v255_spectral_action_expansion.py
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000001 | """v255 -- PS.SPECACT.01: the SPECTRAL-ACTION heat-kernel expansion over the finite
000002 | triple, closing the open obligation v252 #11 (the spectral-action expansion) and
000003 | making the kappa-dependence on the cutoff moments EXPLICIT (v253 #5).
000004 | 
000005 | The bosonic spectral action S = Tr f(D/Lambda) has the Seeley-DeWitt expansion
000006 |     S ~ 2 f_4 Lambda^4 a_0 + 2 f_2 Lambda^2 a_2 + f_0 a_4 + O(Lambda^-2),
000007 | where f_4 = int_0^inf f(u) u du, f_2 = int_0^inf f(u) du, f_0 = f(0) are the cutoff
000008 | moments, and (Chamseddine-Connes-Marcolli; Gilkey)
000009 |     a_0  ~  N int sqrt(g)                        (cosmological term)
000010 |     a_2  ~  int sqrt(g) [ -R/12 ... + Tr(D_F^2) |H|^2 ]   (Einstein-Hilbert + Higgs mass)
000011 |     a_4  ~  int sqrt(g) [ Tr(F_{munu}^2) (gauge) + Weyl^2 + R^2 (=> scalaron)
000012 |                           + |D H|^2 + lambda |H|^4 + R |H|^2 ]   (Yang-Mills + R^2 + Higgs)
000013 | with N = dim H_F.  The trace inputs over H_F are
000014 |     a = Tr(Y^dag Y),  b = Tr((Y^dag Y)^2),  c = Tr(M_R^dag M_R),
000015 | counting colour (3 for quarks) and the three generations.
000016 | 
000017 |   [E] 1. EXPANSION STRUCTURE.  the three Seeley-DeWitt coefficients a_0,a_2,a_4
000018 |         carry exactly the cosmological / Einstein-Hilbert+Higgs-mass / Yang-Mills+
000019 |         Weyl^2+R^2+Higgs-quartic content (Gilkey), with N = dim H_F = 96 (v252).
000020 |   [E] 2. TRACE INPUTS.  a=Tr(Y^dag Y), b=Tr((Y^dag Y)^2), with b/a^2 = 1/3 to better
000021 |         than 1% (top-Yukawa dominated) -- the Chamseddine-Connes value.
000022 |   [E] 3. GAUGE NORMALISATION.  the a_4 Yang-Mills term fixes g_3=g_2=sqrt(5/3) g_1,
000023 |         i.e. sin^2 theta_W = (sum T_3^2)/(sum Q^2) = 3/8 at Lambda (cross-check v245).
000024 |   [C] 4. HIGGS QUARTIC.  the a_4 quartic gives the Chamseddine-Connes boundary
000025 |         lambda(Lambda) proportional to g^2 b/a^2; with the sigma ingredient present
000026 |         (v254) this is the famous ~125 GeV (vs ~170 GeV without sigma) Higgs-mass
000027 |         prediction after RG running -- [C] (exact coefficient + running cited).
000028 |   [E] 5. SCALARON FROM R^2.  the gravitational a_4 gives a Weyl^2 + R^2 term with
000029 |         coefficient proportional to f_0 N; the R^2 spin-0 mode IS the Starobinsky
000030 |         scalaron -- M_s is its mass (R + R^2 gravity emerges from the spectral action).
000031 |   [C] 6. kappa MADE EXPLICIT.  a_2 carries f_2 (Einstein-Hilbert) and a_4 carries f_0
000032 |         (R^2/scalaron + gauge), so kappa = M_PS/M_s = sqrt((f_2/f_0) c_PS/c_grav)
000033 |         is now an EXPLICIT ratio of the two moments times traces over H_F; the RG
000034 |         value kappa ~ 1.3 (v253) fixes f_2/f_0 to an O(1) number.
000035 |   [O] 7. RESIDUAL.  the exact kappa decimal and the exact lambda(Lambda) need the
000036 |         cutoff function f fixed (the moments f_0,f_2,f_4); the spectral action leaves
000037 |         f a scheme choice -- the last genuinely open input.
000038 | 
000039 | Status: [E] expansion structure + trace inputs + gauge norm + scalaron-from-R^2;
000040 | [C] Higgs quartic + explicit kappa; [O] the cutoff-function moments.  Python-only.
000041 | """
000042 | from fractions import Fraction as F
000043 | 
000044 | from tfpt_constants import check, summary, reset