The file src/27-fields/p8.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: EXACT -- integer / modular-F_p / cyclotomic arithmetic; no float in any asserted check (float, if present, only formats an exact rational for display).2"""3P8: simple unification -- is the rank-5 frame gauged as a simple group?45Architectural argument (lead): matter is the single spinor of the single rank-56internal frame (Section 7). The spinor = exterior algebra Lambda^* C^5 = the7SO(10) chiral spinor 16, IRREDUCIBLE under SO(10) (simple) but REDUCIBLE under the8product SU(3)xSU(2)xU(1) into 1 + 5bar + 10. Matter being ONE irreducible spinor9forces the simple group; the product is the broken-phase remnant. SO(10) (not just10SU(5)) is favoured: the nu^c singlet lives in the SO(10) 16 but not the SU(5)115bar+10, and Section 7 derived nu^c. Predictions: proton decay suppressed if the12unification (X,Y) scale is the substrate/Planck scale; the P1 near-miss is a13prediction of completing structure.14"""15import math1617print("="*68)18print("P8 is unification forced? the spinorial argument and its predictions")19print("="*68)2021# ---- (1) irreducibility: one spinor -> simple group ----22print("\n(1) matter = one spinor of one frame -> simple group")23print(" SO(10) chiral spinor 16: IRREDUCIBLE (one matter multiplet)")24print(" under product SU(3)xSU(2)xU(1): 16 = 1 + 5bar + 10 (REDUCIBLE)")25print(f" dims: 1 + 5 + 10 = {1+5+10} = 16 (the generation pieces)")26print(" => 'matter is a spinor' (one irreducible 16) forces the SIMPLE group;")27print(" the product is the subgroup the drive+Higgs breaking selects.")2829# ---- (2) SO(10) vs SU(5): the nu^c ----30print("\n(2) SO(10) vs SU(5): the right-handed neutrino")31print(" SU(5): 5bar + 10 = 15 fermions (no nu^c)")32print(" SO(10): 16 = 15 + nu^c (the singlet)")33print(" Section 7 derived nu^c (the Lambda^0 singlet) => SO(10) favoured,")34print(" the spinor group of the rank-10 (=2x5) orthogonal frame.")3536# ---- (3) proton lifetime vs the unification scale ----37print("\n(3) proton decay: tau_p ~ M_X^4 / (alpha_GUT^2 m_p^5)")38m_p = 0.938 # GeV39alpha_GUT = 1/40.040GeVinv_to_yr = 6.582e-25 / 3.156e7 # hbar/GeV in seconds, to years41for label, M_X in [("standard GUT", 1e16), ("substrate ~10^18", 1e18),42 ("Planck/substrate", 1.22e19)]:43 tau_GeVinv = M_X**4/(alpha_GUT**2 * m_p**5)44 tau_yr = tau_GeVinv*GeVinv_to_yr45 print(f" M_X = {M_X:.1e} GeV ({label:18s}): tau_p ~ {tau_yr:.1e} yr")46print(" Super-K bound ~ 1e34 yr. Unification at the substrate/Planck scale")47print(" gives tau_p ~ 1e44-1e47 yr: proton effectively stable, safely above bounds.")4849# ---- (4) the near-miss (from P1) as a prediction ----50print("\n(4) the P1 coupling near-miss (~13%) is then a PREDICTION:")51print(" exact unification (forced by one frame) requires completing structure")52print(" between M_Z and the unification scale -- new physics or substrate-scale")53print(" threshold corrections; falsifiable.")54print("="*68)55print("LEAD: unification (SO(10)) forced by the spinorial architecture; product is")56print("the broken remnant; nu^c favours SO(10); proton stable if M_X~Planck.")57print("OPEN: a rigorous a-priori forcing, the precise scale, the near-miss resolution.")58print("="*68)59