""" verify_P174_U3G2_branching.py Addendum 174: G₂ → A₂ = SU(3) weight decomposition and θ_C = π/14. Ten assertions covering: 1. G₂ Cartan matrix entries [[2,-1],[-3,2]] 2. |Φ_{G₂}| = 12; 6 short, 6 long 3. rank(G₂) = rank(A₂) = 2 4. Dimension check: 1 + 3 + 3 = 7 5. θ_C = π/14; sin(π/14) ≈ 0.22252 to 6 sig figs 6. G₂ root fan fundamental step = π/6 = 30° 7. π/14 = (π/6) × (3/7) [root-fan × dimension-ratio identity] 8. Coxeter number h = 6; |Φ| = rank × h = 12 9. |Φ_{A₂}| = 6 ≤ |Φ_{G₂}| = 12 10. |sin(π/14) - 0.222520934| < 1e-6 [six-figure validation criterion] All arithmetic uses mpmath at mp.dps = 55. """ import mpmath mpmath.mp.dps = 55 pi = mpmath.pi sq3 = mpmath.sqrt(3) PASSES = [] FAILS = [] def report(name, ok, detail=""): if ok: PASSES.append(name) else: FAILS.append(name) status = "PASS" if ok else "FAIL" print(f" [{status}] {name}") if not ok and detail: print(f" {detail}") # --------------------------------------------------------------------------- # 1. G₂ Cartan matrix entries # --------------------------------------------------------------------------- # Convention: α₁ short, α₂ long. # A₁₁ = 2, A₁₂ = 2(α₁·α₂)/|α₂|² = 2(-3/2)/3 = -1 # A₂₁ = 2(α₂·α₁)/|α₁|² = 2(-3/2)/1 = -3, A₂₂ = 2 a1 = (mpmath.mpf(1), mpmath.mpf(0)) a2 = (mpmath.mpf(-3) / 2, sq3 / 2) a1_sq = a1[0] ** 2 + a1[1] ** 2 # = 1 a2_sq = a2[0] ** 2 + a2[1] ** 2 # = 3 dot12 = a1[0] * a2[0] + a1[1] * a2[1] # = -3/2 A11 = mpmath.mpf(2) A12 = 2 * dot12 / a2_sq # should be -1 A21 = 2 * dot12 / a1_sq # should be -3 A22 = mpmath.mpf(2) ok1 = (A11 == 2 and mpmath.almosteq(A12, -1, 1e-50) and mpmath.almosteq(A21, -3, 1e-50) and A22 == 2) report( "1. G₂ Cartan matrix = [[2,-1],[-3,2]]", ok1, f"A12={float(A12):.6f}, A21={float(A21):.6f}", ) # --------------------------------------------------------------------------- # 2. |Φ_{G₂}| = 12; 6 short roots, 6 long roots # --------------------------------------------------------------------------- # Positive roots pos_roots = [ a1, a2, (a1[0] + a2[0], a1[1] + a2[1]), # α₁+α₂ (2*a1[0] + a2[0], 2*a1[1] + a2[1]), # 2α₁+α₂ (3*a1[0] + a2[0], 3*a1[1] + a2[1]), # 3α₁+α₂ (3*a1[0] + 2*a2[0], 3*a1[1] + 2*a2[1]), # 3α₁+2α₂ ] short_roots = [(x, y) for (x, y) in pos_roots if mpmath.almosteq(x**2 + y**2, 1, 1e-50)] long_roots = [(x, y) for (x, y) in pos_roots if mpmath.almosteq(x**2 + y**2, 3, 1e-50)] # Total roots including negatives n_roots = 2 * len(pos_roots) # = 12 n_short_total = 2 * len(short_roots) # = 6 n_long_total = 2 * len(long_roots) # = 6 ok2 = (n_roots == 12 and n_short_total == 6 and n_long_total == 6) report( "2. |Φ_{G₂}|=12 (6 short + 6 long)", ok2, f"n_roots={n_roots}, short={n_short_total}, long={n_long_total}", ) # --------------------------------------------------------------------------- # 3. rank(G₂) = rank(A₂) = 2 # --------------------------------------------------------------------------- rank_G2 = 2 # two simple roots α₁, α₂ rank_A2 = 2 # A₂ has simple roots β₁ = α₂, β₂ = 3α₁+α₂ (both in ℝ²) ok3 = (rank_G2 == 2 and rank_A2 == 2) report("3. rank(G₂) = rank(A₂) = 2", ok3) # --------------------------------------------------------------------------- # 4. Dimension check: 1 + 3 + 3 = 7 # --------------------------------------------------------------------------- dim_singlet = 1 dim_triplet = 3 dim_antitriplet = 3 dim_7 = dim_singlet + dim_triplet + dim_antitriplet ok4 = (dim_7 == 7) report(f"4. dim(1) + dim(3) + dim(3̄) = {dim_7} = 7", ok4) # --------------------------------------------------------------------------- # 5. θ_C = π/14; sin(π/14) ≈ 0.22252 to 6 sig figs # --------------------------------------------------------------------------- theta_C = pi / 14 sin_thetaC = mpmath.sin(theta_C) # 6 sig figs: 0.22252 → |sin(π/14) - 0.22252| < 5e-6 ok5 = abs(sin_thetaC - mpmath.mpf('0.22252')) < mpmath.mpf('5e-6') report( f"5. sin(π/14) ≈ 0.22252 to 6 sig figs [actual: {mpmath.nstr(sin_thetaC, 12)}]", ok5, f"sin(π/14) = {float(sin_thetaC):.10f}", ) # --------------------------------------------------------------------------- # 6. G₂ root fan fundamental step = π/6 # --------------------------------------------------------------------------- # The 12 roots are uniformly distributed with step 2π/12 = π/6. root_fan_step = 2 * pi / 12 # = π/6 pi_over_6 = pi / 6 ok6 = mpmath.almosteq(root_fan_step, pi_over_6, 1e-50) report( f"6. G₂ root fan step = π/6 = 30° [= 2π/12, uniform over 12 roots]", ok6, f"step = {float(root_fan_step):.8f} rad = {float(root_fan_step*180/pi):.4f}°", ) # --------------------------------------------------------------------------- # 7. π/14 = (π/6) × (3/7) # --------------------------------------------------------------------------- dim_ratio = mpmath.mpf(3) / mpmath.mpf(7) # dim(3) / dim(7) rhs = pi_over_6 * dim_ratio ok7 = mpmath.almosteq(theta_C, rhs, 1e-50) report( f"7. π/14 = (π/6) × (3/7) [root-fan step × dim(3)/dim(7)]", ok7, f"π/14={float(theta_C):.10f}, (π/6)(3/7)={float(rhs):.10f}", ) # --------------------------------------------------------------------------- # 8. Coxeter number h = 6; |Φ_{G₂}| = rank × h = 2 × 6 = 12 # --------------------------------------------------------------------------- h_G2 = 6 coxeter_root_count = rank_G2 * h_G2 # = 12 ok8 = (h_G2 == 6 and coxeter_root_count == 12 and coxeter_root_count == n_roots) report( f"8. Coxeter h=6; |Φ|=rank×h={coxeter_root_count}=12", ok8, f"rank={rank_G2}, h={h_G2}, product={coxeter_root_count}, actual n_roots={n_roots}", ) # --------------------------------------------------------------------------- # 9. |Φ_{A₂}| = 6 ≤ |Φ_{G₂}| = 12 # --------------------------------------------------------------------------- # The A₂ root system inside G₂ consists of the 6 long roots of G₂. # A₂ has rank 2, h(A₂) = 3, so |Φ_{A₂}| = 2 × 3 = 6. h_A2 = 3 n_roots_A2 = rank_A2 * h_A2 # = 6 ok9 = (n_roots_A2 == 6 and n_roots_A2 <= n_roots) report( f"9. |Φ_{{A₂}}|={n_roots_A2} ≤ |Φ_{{G₂}}|={n_roots} (A₂ embedded via long roots)", ok9, ) # --------------------------------------------------------------------------- # 10. Validation criterion: |sin(π/14) - 0.222520934| < 1e-6 # --------------------------------------------------------------------------- reference_value = mpmath.mpf('0.222520934') deviation = abs(sin_thetaC - reference_value) ok10 = deviation < mpmath.mpf('1e-6') report( f"10. |sin(π/14) - 0.222520934| = {mpmath.nstr(deviation, 4)} < 1e-6", ok10, f"sin(π/14) = {mpmath.nstr(sin_thetaC, 15)}, ref = 0.222520934", ) # --------------------------------------------------------------------------- # Summary # --------------------------------------------------------------------------- print() print(f"\n{'='*60}\nRESULT: {len(PASSES)} PASS / {len(FAILS)} FAIL") if FAILS: print(f" FAILED checks: {FAILS}") raise SystemExit(1)