The file src/27-fields/o2_numbers.py of the validation package of the paper 27-fld, rendered with an anchor per line; the file itself is beside this page.
1# framed-rational status: MIXED -- exact framed-rational core; continuum constructs are confined to clearly-labelled [approx] / degenerate-idealisation comparisons (Tier 2/3/5 or measured-data), never to an exact claim.2"""3O2 supporting computation: the electromagnetic coupling as phase-channel capacity.45Pins the numbers behind the three O2 statements:6 (1) channel unity: bare coupling under unit capacity = 1/4pi (= EM-1 Coulomb coeff)7 (2) hierarchy theorem: alpha is Omega-independent; gravity ~ (m/sqrt Omega)^2,8 so alpha / (G m_p^2 / hbar c) is set by Omega ~ 10^122.9 (3) the gap from 1/4pi to the measured 1/137, decomposed (charge unit, running).10No fit; every number is either a measured constant or an exact substrate ratio.11"""12import math1314# --- measured constants (PDG) ------------------------------------------------15alpha0 = 1 / 137.035999084 # fine-structure constant, Thomson (zero-energy)16alpha_MZ = 1 / 127.951 # at the Z pole17m_P_GeV = 1.220890e19 # Planck mass18m_p_GeV = 0.9382720813 # proton19m_e_GeV = 0.51099895000e-3 # electron20Omega = (m_P_GeV / 1.0) ** 2 # placeholder; we use m_p/m_P directly below2122print("=" * 66)23print("O2 electromagnetic coupling as phase-channel capacity")24print("=" * 66)2526# (1) channel unity: the bare coupling -----------------------------------------27inv_alpha_bare = 4 * math.pi28print("\n(1) Channel unity / bare coupling (unit capacity, unit winding charge)")29print(f" shell Coulomb coefficient (EM-1 prototype, gravity G_shell) = 1/4pi"30 f" = {1/(4*math.pi):.5f}")31print(f" => 1/alpha_bare = 4pi = {inv_alpha_bare:.4f} (alpha_bare = {1/inv_alpha_bare:.5f})")32print(f" measured 1/alpha(0) = {1/alpha0:.4f}")33print(f" measured 1/alpha(M_Z) = {1/alpha_MZ:.4f}")3435# (2) hierarchy theorem --------------------------------------------------------36grav_coupling = (m_p_GeV / m_P_GeV) ** 2 # G m_p^2 / (hbar c) = (m_p/m_P)^237ratio = alpha0 / grav_coupling38print("\n(2) Hierarchy: EM couples to winding INDEX (label), gravity to winding")39print(" RATE (cardinality fraction m/sqrt(Omega)). alpha is Omega-independent.")40print(f" G m_p^2/(hbar c) = (m_p/m_P)^2 = {grav_coupling:.3e}")41print(f" alpha(0) = {alpha0:.3e}")42print(f" alpha / (G m_p^2/hbar c) = {ratio:.3e} (~10^{math.log10(ratio):.1f})")43print(f" => the ~10^36 EM/gravity hierarchy IS the substrate size:")44print(f" (m_P/m_p)^2 = Omega/m_p^2 (Planck units) = {1/grav_coupling:.3e}")45print(f" gravity is weak because Omega ~ 10^122 is large; EM does not see Omega.")4647# (3) the residual gap, decomposed honestly -----------------------------------48print("\n(3) From the bare O(1/4pi) to the measured 1/137 -- the honest open part")49gap = (1/alpha0) / inv_alpha_bare50print(f" gap in 1/alpha: 4pi = {inv_alpha_bare:.2f} -> 137.04 (factor {gap:.2f} in alpha)")51print(f" sign: alpha is larger in the UV (screening) -- CORRECT direction.")52print(f" magnitude check: SM RG running of 1/alpha_EM from M_P to 0 is ~ +30..50,")53print(f" NOT +124, so running ALONE cannot close 12.6 -> 137.")54print(f" => most of the gap is NOT running; it is the projection of the bare")55print(f" reframing-channel coupling onto the physical photon:")56print(f" (b) electroweak mixing e = g' cos(theta_W) [Weinberg angle = EW-1,")57print(f" the split/non-split torus tilt in the weak sector], and")58print(f" (a) the precise O(1) capacity coefficient (analogue of gravity's open")59print(f" order-one constants, e.g. the 2pi of the a_0 floor).")60print(f" CONCLUSION: O2 fixes the mechanism (alpha = phase-channel capacity), the")61print(f" order (1/4pi, channel unity with gravity), and the sign; the PURE NUMBER")62print(f" 1/137 reduces to the charge spectrum + Weinberg angle and does NOT close")63print(f" within EM-1. It is the meeting point of the EM and weak sectors.")64print("=" * 66)65