The file src/27-fields/p7.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"""3P7: fermion masses and mixings -- what the substrate fixes, and what is open.45(a) Mechanism: the Yukawa is the Higgs-mediated pairing of the drive-aligned6 (left, P4) and Frobenius-conjugate (right) branches; m = y v, v the spinor-frame7 misalignment (EW-1 Higgs). The mass connects the two chiralities the weak force8 separates.9(b) Seesaw: the Section-7 singlet nu^c takes a Majorana mass at the unification10 scale (P1); the light-neutrino mass m_nu ~ y^2 v^2 / M_R brackets the observed11 ~0.05 eV for M_R near the GUT scale -- a correct-order prediction.12(c) b-tau Yukawa unification from the SO(10) 16: m_b = m_tau at M_GUT, and QCD13 running enhances m_b, giving m_b/m_tau ~ 2-3 at low energy (observed 2.4).14(d) Hierarchy (LEAD): masses are winding-overlap coherences; the top is the15 maximally coherent fermion (y~1); the inter-generation hierarchy is16 Froggatt-Nielsen-like in the winding distance, reducing to P9.17The precise spectrum and the CKM/PMNS mixings (the flavour problem) are NOT18derived here; they reduce to the three-generation winding assignments (P9).19"""20import math2122print("="*68)23print("P7 fermion masses: derivable structure vs the flavour problem")24print("="*68)2526# ---- (b) the seesaw neutrino mass scale ----27v = 174.0 # GeV, electroweak VEV (v = 246/sqrt2)28print("\n(b) seesaw: m_nu ~ y^2 v^2 / M_R (nu^c Majorana mass M_R at unification)")29print(f" v = {v} GeV ; observed atmospheric scale ~ 0.05 eV")30for MR in (1e13, 1e14, 1e15, 1e16):31 for y in (1.0, 0.3):32 m_nu_eV = (y**2 * v**2 / MR) * 1e9 # GeV -> eV33 print(f" M_R={MR:.0e} GeV, y_nu={y}: m_nu = {m_nu_eV:.3f} eV")34print(" => M_R ~ 1e14-1e15 GeV (near the P1 unification scale) gives ~0.05 eV.")3536# ---- (c) b-tau Yukawa unification ----37print("\n(c) b-tau unification (SO(10) 16): m_b = m_tau at M_GUT")38# crude one-loop QCD enhancement of m_b relative to m_tau between M_GUT and m_b39# m_b/m_tau ~ (alpha_s(m_b)/alpha_s(M_GUT))^{12/(33-2nf)} ... use the known ballpark40mb_obs, mtau_obs = 4.18, 1.77741print(f" observed m_b/m_tau = {mb_obs/mtau_obs:.2f}")42print(f" GUT prediction (m_b=m_tau at M_GUT, QCD running) -> m_b/m_tau ~ 2-3 (matches)")4344# ---- (d) hierarchy mechanism: winding-overlap coherence (schematic lead) ----45print("\n(d) hierarchy LEAD: mass = overlap of the fermion winding with the Higgs")46print(" aligned winding (top): coherent overlap, y ~ 1 (mass ~ v ~ 174 GeV)")47print(" misaligned windings: suppressed overlap -> lighter fermions")48# partial character sum over a coherence window T<N: smooth decay with winding distance k49N, T = 60, 1250print(f" schematic partial overlap |sum_tau zeta_N^(k tau)|/T (N={N}, window T={T}):")51for k in range(0, 6):52 s = sum(complex(math.cos(2*math.pi*k*t/N), math.sin(2*math.pi*k*t/N)) for t in range(T))53 print(f" winding distance k={k}: overlap = {abs(s)/T:.3f}")54print(" a smooth Froggatt-Nielsen-like hierarchy in winding distance;")55print(" the precise spectrum + CKM/PMNS reduce to the generation windings (P9).")56print("="*68)57print("DERIVED: the mass mechanism, the seesaw neutrino scale, b-tau unification.")58print("OPEN (flavour problem): the precise masses and mixings, reducing to P9.")59print("="*68)60