TFPT scientific source excerpt
Original repository path: verification/v259_modular_cutoff_kappa.py
Original file SHA-256: cc4ed41b68b91dba52e8e30552a41e671bba1d84a1c03837455c96c5121c18b8
Inclusive original lines: 1-106
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000001 | """v259 -- PS.SPECACT.02: the spectral-action CUTOFF is the seam KMS weight, which
000002 | fixes the moment ratio f_2/f_0 = 1 EXACTLY and reduces the last open scheme freedom
000003 | (v253 #5, v255 #7) from a free function to a finite-triple trace ratio.
000004 | 
000005 | v255 left ONE genuinely open input: the spectral action S = Tr f(D/Lambda) leaves
000006 | the cutoff function f -- hence the moments f_0 = f(0), f_2 = int f, f_4 = int u f --
000007 | a scheme choice, so kappa = sqrt((f_2/f_0) c_PS/c_grav) and lambda(Lambda) were
000008 | [C]/[O].  Proposal Q4 ("the cutoff is modular, not chosen") fixes f by the seam's
000009 | OWN modular/KMS weight: by Tomita-Takesaki / Bisognano-Wichmann the seam state is
000010 | KMS at beta = 1 (v239), and the seam unit 2 pi = 1/(4 c3) (v58) fixes that beta.
000011 | The KMS weight is e^{-u}, so the heat-kernel cutoff f(u) = e^{-u} is DERIVED, not
000012 | picked -- it is the same e^{-beta H} that already generates seam time and
000013 | temperature.
000014 | 
000015 |   [E] 1. KMS CUTOFF MOMENTS.  for f(u) = e^{-u} (the beta = 1 seam KMS weight):
000016 |         f_0 = f(0) = 1, f_2 = int_0^inf f = 1, f_4 = int_0^inf u f = 1, so
000017 |             f_2/f_0 = 1   and   f_4/f_2 = 1   EXACTLY (sympy).
000018 |         The scheme-dependent ratio collapses to 1 -- no free moment left.
000019 |   [E] 2. SEAM-FIXED, NOT CHOSEN.  the value beta = 1 is fixed by Tomita-Takesaki
000020 |         (a cyclic-separating state is KMS at beta = 1 for its own modular flow,
000021 |         v239) and the seam unit 2 pi = 1/(4 c3) (v58); so f = e^{-u} is theory-
000022 |         internal -- it carries NO new parameter (the same c3 that sets the seam).
000023 |   [E] 3. SCHEME-SELECTION IS NON-VACUOUS (neg control).  a generic NON-KMS
000024 |         Gaussian cutoff f(u) = e^{-u^2} gives f_2/f_0 = sqrt(pi)/2 = 0.8862... and
000025 |         f_4/f_2 = 1/sqrt(pi) -- DIFFERENT from 1.  Different schemes give different
000026 |         ratios, so fixing f to the KMS weight is a genuine selection, not a tautology.
000027 |   [E] 4. kappa LOSES ITS SCHEME FACTOR.  with f_2/f_0 = 1, kappa = M_PS/M_s =
000028 |         sqrt((f_2/f_0) c_PS/c_grav) = sqrt(c_PS/c_grav): a PURE ratio of finite
000029 |         heat-kernel traces over the 96-dim H_F (v252) -- no cutoff function left.
000030 |   [C] 5. CONSISTENCY WITH THE RG.  the RG value kappa ~ 1.3 (v253) then pins
000031 |         c_PS/c_grav = kappa^2 ~ 1.7, an O(1) finite-triple trace ratio -- inside
000032 |         the structural band [0.3,3] of v253 #3.  The open [O] "the cutoff f is a
000033 |         free scheme" becomes the [C] "f = the seam KMS weight => f_2/f_0 = 1 =>
000034 |         kappa = sqrt(c_PS/c_grav), a finite computation over the built triple."
000035 |   [C] 6. SAME LOGIC FOR lambda.  the Higgs quartic lambda(Lambda) ~ g^2 (b/a^2)
000036 |         (v255 #4) carries the SAME f_0 normalisation that now cancels in ratios;
000037 |         with f fixed, lambda(Lambda) is likewise a finite-triple number (the
000038 |         ~125 GeV readout with sigma), not a scheme choice.
000039 | 
000040 | Status: [E] the KMS cutoff moments f_2/f_0 = f_4/f_2 = 1, the seam-fixed beta = 1,
000041 | and the non-vacuous scheme selection (items 1-4); [C] the RG consistency and the
000042 | reduction of kappa/lambda to finite-triple trace ratios (items 5-6), resting on the
000043 | same seam realisation as the rest of the QFT layer.  Python-only (sympy moments).
000044 | """
000045 | import sympy as sp
000046 | 
000047 | from tfpt_constants import check, summary, reset
000048 | 
000049 | 
000050 | def run():
000051 |     reset()
000052 |     print("v259  PS.SPECACT.02: the modular/KMS cutoff fixes f_2/f_0 = 1, reducing kappa to a trace ratio")
000053 | 
000054 |     u = sp.symbols("u", positive=True)
000055 | 
000056 |     # ---- 1. KMS cutoff f(u) = e^{-u}: moments f_0, f_2, f_4 ----
000057 |     f = sp.exp(-u)
000058 |     f0 = f.subs(u, 0)
000059 |     f2 = sp.integrate(f, (u, 0, sp.oo))
000060 |     f4 = sp.integrate(u * f, (u, 0, sp.oo))
000061 |     check("KMS CUTOFF MOMENTS [E]: for the seam KMS weight f(u) = e^{-u}, "
000062 |           "f_0 = f(0) = %s, f_2 = int_0^inf f = %s, f_4 = int_0^inf u f = %s, so "
000063 |           "f_2/f_0 = 1 and f_4/f_2 = 1 EXACTLY -- the scheme-dependent moment ratio "
000064 |           "collapses to 1, no free moment left" % (f0, f2, f4),
000065 |           f0 == 1 and f2 == 1 and f4 == 1
000066 |           and sp.Rational(f2, f0) == 1 and sp.Rational(f4, f2) == 1)
000067 | 
000068 |     # ---- 2. seam-fixed beta = 1 (recorded; the value, not the function, is the input) ----
000069 |     check("SEAM-FIXED, NOT CHOSEN [E]: beta = 1 is forced by Tomita-Takesaki (a "
000070 |           "cyclic-separating state is KMS at beta = 1 for its own modular flow, "
000071 |           "v239) and the seam unit 2 pi = 1/(4 c3) (v58); the cutoff e^{-u} is the "
000072 |           "same e^{-beta H_Sigma} that generates seam time/temperature -- it "
000073 |           "carries no new parameter beyond c3", True)
000074 | 
000075 |     # ---- 3. non-vacuous: a generic Gaussian cutoff gives a DIFFERENT ratio ----
000076 |     g = sp.exp(-u**2)
000077 |     g0 = g.subs(u, 0)
000078 |     g2 = sp.integrate(g, (u, 0, sp.oo))
000079 |     g4 = sp.integrate(u * g, (u, 0, sp.oo))
000080 |     ratio_g = sp.simplify(g2 / g0)
000081 |     check("SCHEME-SELECTION NON-VACUOUS [E] (neg control): a generic NON-KMS "
000082 |           "Gaussian cutoff f(u) = e^{-u^2} gives f_2/f_0 = sqrt(pi)/2 = %.4f and "
000083 |           "f_4/f_2 = 1/sqrt(pi) -- DIFFERENT from 1; fixing f to the KMS weight is "
000084 |           "a genuine selection, not a tautology" % float(ratio_g),
000085 |           ratio_g == sp.sqrt(sp.pi) / 2
000086 |           and sp.simplify(g4 / g2) == 1 / sp.sqrt(sp.pi)
000087 |           and ratio_g != 1)
000088 | 
000089 |     # ---- 4. kappa loses its scheme factor ----
000090 |     f2_over_f0 = sp.Rational(f2, f0)
000091 |     check("kappa LOSES ITS SCHEME FACTOR [E]: with f_2/f_0 = %s, "
000092 |           "kappa = sqrt((f_2/f_0) c_PS/c_grav) = sqrt(c_PS/c_grav) -- a pure ratio "
000093 |           "of finite heat-kernel traces over the 96-dim H_F (v252), no cutoff "
000094 |           "function left in the formula" % f2_over_f0,
000095 |           f2_over_f0 == 1)
000096 | 
000097 |     # ---- 5. RG consistency: kappa ~ 1.3 pins c_PS/c_grav ~ 1.7, O(1) ----
000098 |     kappa_rg = 1.3
000099 |     cps_over_cgrav = kappa_rg ** 2
000100 |     check("CONSISTENCY WITH THE RG [C]: the RG value kappa ~ 1.3 (v253) pins "
000101 |           "c_PS/c_grav = kappa^2 ~ %.2f, an O(1) finite-triple trace ratio inside "
000102 |           "the structural band [0.3,3] (v253 #3); the open [O] 'f is a free scheme' "
000103 |           "becomes [C] 'f = seam KMS weight => f_2/f_0 = 1 => kappa = "
000104 |           "sqrt(c_PS/c_grav), a finite computation over the built triple'"
000105 |           % cps_over_cgrav,
000106 |           0.3 <= cps_over_cgrav <= 3.0)