#!/usr/bin/env python3 """ verify_P037.py -- Addendum 37: gauge group derivation. This verifier checks the group/dimension arithmetic in 37_Addendum_GaugeGroup.tex and audits which identifications are proved versus conjectural. Many standard dimensions and the Weinberg-angle arithmetic pass. The flagged issues are mostly proof-status problems, plus two concrete group facts: Aut(C) as a real algebra is Z2 rather than U(1), and Aut(H) is SO(3) (equivalently SU(2)/Z2), not SU(2) itself. """ from __future__ import annotations import math import sys from pathlib import Path sys.path.insert(0, str(Path(__file__).resolve().parent)) class Verifier: """Output shim: identical tolerance semantics to verify_common.Verifier, modern [PASS]/[FAIL] check-line output format.""" def __init__(self, name): self.PASS = 0 self.FAIL = 0 self.n = 0 print(name) def _mark(self, ok, desc): self.n += 1 if ok: self.PASS += 1 else: self.FAIL += 1 print(f" [{'PASS' if ok else 'FAIL'}] {self.n:>2}. {desc}") def check(self, label, computed, claimed, *, rel=1e-3, abs_tol=None, detail=""): if abs_tol is not None: ok = abs(computed - claimed) <= abs_tol err_detail = f"abs err={abs(computed - claimed):.6g}, tol={abs_tol:.6g}" else: if claimed == 0: ok = abs(computed) <= (rel or 1e-12) err_detail = f"abs value={abs(computed):.6g}, tol={rel:.6g}" else: err = (computed - claimed) / abs(claimed) ok = abs(err) <= (rel or 0) err_detail = f"rel err={100 * err:+.6g}%, tol={100 * (rel or 0):.6g}%" self._mark(ok, label + (f" -- {detail}" if detail else "")) print(f" computed: {computed} claimed: {claimed} ({err_detail})") return ok def record(self, label, ok, computed="", claimed="", detail=""): ok = bool(ok) self._mark(ok, label + (f" -- {detail}" if detail else "")) if computed != "" or claimed != "": print(f" computed: {computed} claimed: {claimed}") return ok def summary(self): print(f"\n{'='*60}\nRESULT: {self.PASS} PASS / {self.FAIL} FAIL") return 0 if self.FAIL == 0 else 1 v = Verifier("P037 -- Gauge Group Derivation") ROOT = Path(__file__).resolve().parents[1] TEX = (ROOT / "37_Addendum_GaugeGroup.tex").read_text() PI = math.pi MU0 = 4 * PI**3 + PI**2 + PI v.check("mu0 from rho integral", MU0, 137.036, rel=3e-6) v.check("Hurwitz real dimensions sum endpoint O", 8, 8, rel=1e-12) v.check("bulk coefficient 16=2dim(O)", 16, 2 * 8, rel=1e-12) v.check("dim G2", 14, 14, rel=1e-12) v.check("rank G2", 2, 2, rel=1e-12) v.check("dim SU(3)", 8, 8, rel=1e-12) v.check("rank SU(3)", 2, 2, rel=1e-12) v.check("coset dimension G2/SU3", 14 - 8, 6, rel=1e-12) v.check("dim SU(2)", 3, 3, rel=1e-12) v.check("dim U(1)", 1, 1, rel=1e-12) v.record( "Aut(C) is U(1) as an R-algebra automorphism group", False, computed="Aut_R(C) has two elements: identity and complex conjugation", claimed="Aut(C) superset/equals U(1) in the gauge chain table", detail="Expected group-fact fail: U(1) is the unit group / Hopf structure group, not the algebra automorphism group.", ) v.record( "Aut(H) is SU(2) exactly", False, computed="unit quaternions SU(2) act by conjugation with kernel {+/-1}, giving Aut(H)=SO(3)=SU(2)/Z2", claimed="Aut(H)=SU(2)", detail="Expected group-fact fail.", ) sin2_tree = 1.0 / (1.0 + 3.0) sin2_full = 1.0 / (1.0 + 3.0 + 8.0) sin2_sector = PI / (PI + PI**2) v.check("tree Weinberg angle from dim count", sin2_tree, 0.25, rel=1e-12) v.check("full U1+SU2+SU3 dimension count", sin2_full, 1.0 / 12.0, rel=1e-12) v.check("sector energy ratio Ee/Emu", PI / PI**2, 1.0 / PI, rel=1e-12) v.check("sector energy ratio Ee/Ebulk", PI / (4.0 * PI**3), 1.0 / (4.0 * PI**2), rel=1e-12) v.check("Weinberg sector-energy conjecture value", sin2_sector, 0.241, rel=2e-3) v.check("J3(O) dimension", 3 + 3 * 8, 27, rel=1e-12) v.check("J3(O) diagonal generation slots", 3, 3, rel=1e-12) v.check("J3(O) off-diagonal octonion components", 3 * 8, 24, rel=1e-12) gamma_shift = (9.0 / 16.0) / (PI**2 - PI) muon_total = 7.0e-4 + 4.3e-3 + gamma_shift lambda_mu = PI**2 / (PI + PI**2) v.check("muon gamma T second-order term", gamma_shift, 0.0836, rel=5e-4) v.check("muon perturbative total", muon_total, 0.089, rel=5e-3) v.check("muon fixed-point lambda", lambda_mu, 0.759, rel=6e-4) v.record( "SU(2)_L rather than SU(2)_R follows from S3 orientation alone", False, computed="orientation distinguishes left/right conventions, but physical weak chirality is an additional identification", claimed="TOE boundary carries SU(2)_L and not SU(2)_R", detail="Expected proof-status fail.", ) v.record( "Hopf U(1) is proved to be hypercharge rather than a generic fiber phase", False, computed="Hopf bundle proves a U(1) structure group; identifying it with hypercharge uses the Paper 20 dictionary", claimed="U(1)_Y from Hopf fiber is proven", detail="Expected physical-identification fail.", ) v.record( "Hurwitz theorem forbids any additional gauge factor", False, computed="Hurwitz forbids further normed division algebras, not arbitrary additional gauge sectors unless the division-algebra ansatz is assumed complete", claimed="no fourth gauge group/layer exists", detail="Expected proof-status fail.", ) v.record( "physical color assignment is marked conjectural", "Conjecture" in TEX and "SU(3)_c" in TEX and "Needs quark sector" in TEX, computed="color assignment status table marks it as conjecture", claimed="not fully proved", ) v.record( "off-diagonal 8 real components are SU(3) color generators", False, computed="under SU(3), O decomposes as 1+1+3+3bar; quark colors live in 3/3bar, while 8 generators are the adjoint Lie algebra", claimed="8-dimensional x_ij is natural carrier of 8 color generators", detail="Expected representation-theory fail.", ) v.record( "complete J3(O) Standard Model decomposition is proved here", "Conjecture" not in TEX or "Needs rep theory" not in TEX, computed="summary marks full J3(O) SM spectrum as conjecture needing representation theory", claimed="J3(O) as master algebra proved", detail="Expected status fail.", ) sys.exit(v.summary())