The file src/27-fields/v_scale.py of the validation package of the paper 27-fld, rendered with an anchor per line; the file itself is beside this page.
1#!/usr/bin/env python32# framed-rational status: [APPROX] -- continuum / degenerate-idealisation comparison layer (the finite-window correspondence and phenomenology); not an exact framed-rational claim by construction.3# =====================================================================4# v_scale.py -- the electroweak scale v as a dimensional-transmutation5# quantity: v = m_P exp(-c), the exponent c Omega-hard (27-fields,6# Rem. masscons; ledger rows Z1, Y1).7#8# What this script establishes (2026-09-13 revision, task T22):9# 1. v is not a clean power of Omega.10# 2. The gauge transmutation formula Lambda = M exp(-8 pi^2 / b)11# at alpha_bare = 1/4pi lands on the weak band only for b = 2.00,12# which is a FIT: the Standard-Model SU(2) coefficient is13# b_2 = 19/6, which gives ~2e8 GeV, and the SU(2) coupling's own14# infrared transmutation scale (one-loop from alpha_2(M_Z)) is15# ~1e-24 GeV. So the electroweak scale is not the SU(2) gauge16# transmutation scale; the channel that condenses at v is the17# non-split torus, whose coefficient is Omega-hard.18# 3. Control: the QCD sentence of Prop. af is consistent -- e^{-45}19# with b_0 = 7 needs alpha_s(M_P) = 0.020, and one-loop running20# from alpha_s(M_Z) = 0.118 gives 0.019.21# 4. The closed form m_P exp(-(2 pi)^2) = 87 GeV (9 % from M_W) is22# recorded as a conjecture (row Y1), not claimed.23# =====================================================================24import math2526mP = 1.220890e19 # Planck mass (GeV)27H0 = 1.437e-42 # Hubble (GeV) ~67.4 km/s/Mpc28v = 246.220 # Higgs VEV (GeV)29MW = 80.37730MZ = 91.187631pi = math.pi32fails = 033def check(name, ok):34 global fails35 print((" PASS " if ok else " FAIL ") + name)36 if not ok: fails += 13738sqrtOmega = mP/H039Omega = sqrtOmega**240print(f"sqrt(Omega) = mP/H0 = {sqrtOmega:.3e} Omega = {Omega:.3e} (~10^122)")41print(f"weak band: M_W={MW}, v={v} GeV; v/mP = {v/mP:.3e}\n")4243# 1. no clean power44print("== 1. power-law test v = mP * Omega^(-k) ==")45k_v = math.log(mP/v )/math.log(Omega)46k_MW = math.log(mP/MW)/math.log(Omega)47print(f" k(v) = {k_v:.4f} k(M_W) = {k_MW:.4f} -> no simple rational power")4849# 2. the gauge transmutation formula is a fit at b = 2, refuted by b_2 = 19/650print("\n== 2. gauge transmutation Lambda = mP * exp(-8 pi^2 / b), alpha_bare = 1/4pi ==")51b_fit_MW = 8*pi**2/math.log(mP/MW)52b_fit_v = 8*pi**2/math.log(mP/v)53b_fit_87 = 8*pi**2/math.log(mP/(mP*math.exp(-4*pi**2)))54print(f" b that lands on M_W: {b_fit_MW:.3f}; on v: {b_fit_v:.3f}; on 87.4 GeV: {b_fit_87:.3f}")55b2 = 19/656L_b2 = mP*math.exp(-8*pi**2/b2)57print(f" Standard-Model SU(2) coefficient b_2 = 19/6: mP*exp(-8pi^2/b_2) = {L_b2:.2e} GeV")58check("b = 2 is a fit (|b_fit(M_W) - 2| < 0.01)", abs(b_fit_MW-2) < 0.01)59check("b_2 = 19/6 gives 1e8..1e9 GeV, not the weak band", 1e8 < L_b2 < 1e9)6061# SU(2) one-loop running from M_Z: alpha_2^{-1}(mu) = alpha_2^{-1}(M_Z) + (b_2/2pi) ln(mu/M_Z)62a2inv_MZ = 29.663a2inv_MP = a2inv_MZ + (b2/(2*pi))*math.log(mP/MZ)64mu_IR = MZ*math.exp(-2*pi*a2inv_MZ/b2) # alpha_2^{-1} -> 0 (Landau/strong-coupling scale)65print(f" alpha_2^-1(M_P) = {a2inv_MP:.1f}; SU(2) infrared strong-coupling scale = {mu_IR:.1e} GeV")66check("SU(2) infrared transmutation scale ~1e-24 GeV (< 1e-20 GeV)", mu_IR < 1e-20)6768# 3. control: the QCD sentence is consistent69print("\n== 3. control: QCD dimensional transmutation ==")70b0_qcd = 7.071as_MP_needed = 2*pi/(b0_qcd*45.0) # e^{-45} = exp(-2pi/(b0 alpha_s(M_P)))72as_MZ = 0.11873as_MP_run = 1.0/(1.0/as_MZ + (b0_qcd/(2*pi))*math.log(mP/MZ))74print(f" e^-45 with b_0=7 needs alpha_s(M_P) = {as_MP_needed:.4f}; one-loop from 0.118: {as_MP_run:.4f}")75print(f" ln(mP / 0.2 GeV) = {math.log(mP/0.2):.1f}")76check("QCD exponent consistent (alpha_s(M_P) needed vs run within 10 %)", abs(as_MP_needed/as_MP_run-1) < 0.10)7778# 4. the (2 pi)^2 closed form: recorded as conjecture Y179print("\n== 4. the closed form mP * exp(-(2pi)^2) (conjecture Y1) ==")80vc = mP*math.exp(-(2*pi)**2)81print(f" mP*exp(-(2pi)^2) = {vc:.2f} GeV; ratio to M_W: {vc/MW:.3f} (+{100*(vc/MW-1):.0f} %); ratio to v: {vc/v:.3f}")82check("closed form within 10 % of M_W (the conjecture's stated exposure)", abs(vc/MW-1) < 0.10)8384print(f"""85 CLASSIFICATION (Rem. masscons; rows Z1, Y1):86 * v = mP exp(-c) is the saturation scale of the non-split channel;87 the exponent c is the cross-scale beta-function, Omega-hard (Z1).88 * c is NOT the SU(2) gauge coefficient: b_2 = 19/6 gives {L_b2:.0e} GeV and89 the gauge coupling's own infrared scale is {mu_IR:.0e} GeV; the b = 2.0090 that lands on M_W is a fit.91 * the closed form mP exp(-(2pi)^2) = {vc:.0f} GeV is a conjecture without a92 derivation of the exponent, recorded as Y1 and not claimed.93""")94print("v_scale.py:", "PASS" if fails == 0 else f"FAIL ({fails})")95raise SystemExit(1 if fails else 0)96