66 1. UV value: α_G₂(M_Pl) = |Z₃|/(2π D²_tot) = 1/(24π) (SU(3)₃ MTC,
77 executable in root.py).
88
9- 2. EXACT WZW CANCELLATION down to μ*. One-loop G₂ running with
9+ 2. EXACT WZW CANCELLATION down to μ*. Layer-1 G₂ back-reaction with
1010 C_A = h∨(G₂) = 4 and three Dirac fermions in the 7 (T(7) = 1):
1111 b₀ = (11/3)·4 − (4/3)·3 = 32/3
1212 over the GAUGE lever arm 9π²/2 − 6 (the lepton action with the
2727 is the algebraic coupling of the embedding layer, defined by the
2828 Singh division of the WZW identity, not a running coupling
2929 evaluated at 246.2 GeV.)
30- STATUS: a NON-PERTURBATIVE WZW IDENTITY. The two-loop
30+ STATUS: a NON-PERTURBATIVE WZW IDENTITY. The layer-2
3131 coefficient b₁ = (34/3)C_A² − ((20/3)C_A + 4C₂(7))·T(7)·n_f
3232 = 232/3 > 0 would shift α_s(μ*) up ~14%, consistent with
3333 π/32 being an exact operator identity (Knizhnik-Zamolodchikov
3939 half. The 7 of G₂ is a real representation, consistent with
4040 exactly this non-chiral embedding.
4141
42- 3. THRESHOLD at G₂ → SU(3) (Weinberg 1980 / Hall 1981 matching).
42+ 3. DERIVED THRESHOLD at G₂ → SU(3) (Weinberg 1980 / Hall 1981 matching).
43+ (Standard QFT calls this "threshold correction"; here the matching
44+ scale M_V is derived, not fitted — see below.)
4345 The coset G₂/SU(3) (dim 6) gives six massive vectors in 3 ⊕ 3̄:
4446 1/α_s(μ*) = 1/α_G₂(μ*) − λ₃/(12π),
4547 λ₃ = (C_{G₂} − C_{SU(3)}) − 21·T_V·ln(M_V/μ*),
5860 transmits only scalar-like spectral weight, and no coset
5961 quantum number survives as a conserved charge below matching.
6062 M_V enters ONLY as a matching parameter (exactly like heavy
61- states in GUT threshold corrections ), not as a particle.
63+ states in GUT threshold back-reactions ), not as a particle.
6264
63- 4. SM 2-loop QCD running μ* → M_Z with derived thresholds
65+ 4. SM layer-2 QCD running μ* → M_Z with derived thresholds
6466 → α_s(M_Z) = 0.1184 (+0.33%, +0.43σ of PDG 0.1180(9)).
6567 Without the threshold, π/32 + SM running alone gives 0.1119
6668 (−5.1%): the derived 112 GeV matching is load-bearing.
9193 Used in the lepton vertex factor.
9294
9395The relation between them is NOT a perturbative running approximation;
94- it is a one-loop identity of the (7,26) bridge sector, fixed by three
96+ it is a layer-1 identity of the (7,26) bridge sector, fixed by three
9597independent rational identities in the four integers:
9698
9799 (A) h_bridge = h(n₇,G₂) + h(n₂₆,F₄) = 2/5 + 3/5 = 1 [MARGINAL]
98100 (B) D²_local = 1 [LAGRANGIAN]
99101 (C) c_coset = c(E₈) − c(G₂) − c(F₄) = 0 [TOPOLOGICAL]
100102
101103The worldsheet integral of a marginal (h=1) primary with unit coupling
102- gives the universal h/(2π ) = 1/(2π ); c_coset = 0 forbids higher-loop
103- contributions . The reading is one-loop exact.
104+ gives the universal h/(2pi ) = 1/(2pi ); c_coset = 0 forbids higher-layer
105+ back-reactions . The reading is layer-1 exact.
104106
105107Same bridge, two observables. gravity.py reads the same (7,26) sector
106108as 182 scalar-like heat-kernel channels with non-minimal coupling
107109ξ_bridge = α_G₂·E[v²]·h_bridge = 1/(48π) and derives Newton's constant.
108- The conformal weight h_bridge = 1 sets BOTH the QED one-loop fraction
110+ The conformal weight h_bridge = 1 sets BOTH the QED layer-1 fraction
1091111/(2π) and the gravitational coupling 1/(48π). EM renormalisation and
110112induced gravity are two readings through one bridge.
111113
@@ -135,32 +137,31 @@ def _rk4_run(beta_func, a0, t0, t1, n_steps=10000):
135137 return _shared_rk4 (beta_func , a0 , t0 , t1 , n_steps )
136138
137139
138- def _run_SM_2loop (a0 , mu0 , mu1 , nf ):
139- """Two-loop SM QCD running in d/d(ln mu) convention.
140+ def _b0 (nf ):
141+ """Layer-1 QCD beta coefficient, derived from SU(d₁₁) Casimirs."""
142+ C_A = d11
143+ return (11 * C_A - 2 * nf ) / (6 * math .pi )
140144
141- All coefficients derived from SU(d₁₁) gauge theory with nf Dirac fermions:
142- C_A = d₁₁ = 3, C_F = (d₁₁²−1)/(2d₁₁) = 4/3, T_F = 1/2
143145
144- One-loop :
145- b₀ = (11·C_A − 2nf)/(6π) = (11d₁₁ − 2nf)/(6π)
146+ def _b1 ( nf ) :
147+ """Layer-2 QCD beta coefficient, derived from SU(d₁₁) Casimirs.
146148
147- Two-loop:
148- b₁ = (34·C_A² − 10·C_A·nf − 6·C_F·nf) / (24π²)
149- = (34d₁₁² − 10d₁₁·nf − 6·(d₁₁²−1)/(2d₁₁)·nf) / (24π²)
150- = (34·9 − (10·3 + 3·4/3)·nf) / (24π²)
151- = (306 − (30+4)nf) / (24π²)
152- but standard MS-bar two-loop for SU(N) gives:
153- b₁ = (34C_A² − (10C_A + 6C_F)nf) / (24π²) = (306 − 38nf)/(24π²)
154- = (153 − 19nf) / (12π²)
149+ b₁ = (34·C_A² − (10·C_A + 6·C_F)·nf) / (24π²)
150+ = (153 − 19·nf) / (12π²) for SU(3)
155151
156- nf = d₁₀ × d₁₁ = 6 active flavours (2 types × 3 gen).
157152 No external numerical input: all numbers trace to d₁₀, d₁₁.
158153 """
159- # C_A = d₁₁, C_F = (d₁₁²−1)/(2d₁₁) = 4/3 for SU(d₁₁)
160154 C_A = d11
161155 C_F = (d11 ** 2 - 1 ) / (2.0 * d11 )
162- b0 = (11 * C_A - 2 * nf ) / (6 * math .pi )
163- b1 = (34 * C_A ** 2 - (10 * C_A + 6 * C_F )* nf ) / (24 * math .pi ** 2 )
156+ return (34 * C_A ** 2 - (10 * C_A + 6 * C_F )* nf ) / (24 * math .pi ** 2 )
157+
158+
159+ def _run_SM_2loop (a0 , mu0 , mu1 , nf ):
160+ """Layer-2 SM QCD running in d/d(ln mu) convention.
161+
162+ nf = d₁₀ × d₁₁ = 6 active flavours (2 types × 3 gen).
163+ """
164+ b0 , b1 = _b0 (nf ), _b1 (nf )
164165 return _rk4_run (lambda a : - b0 * a ** 2 - b1 * a ** 3 ,
165166 a0 , math .log (mu0 ), math .log (mu1 ))
166167
@@ -188,12 +189,12 @@ def _derive_bridge():
188189 3. Coset central charge:
189190 c_coset = c(E₈) − c(G₂) − c(F₄) = 8 − 14/5 − 26/5 = 0
190191 Vanishing c_coset makes the bridge sector TOPOLOGICAL:
191- no propagating degrees of freedom → one-loop exact (no higher
192- loop corrections to the self-interference integral).
192+ no propagating degrees of freedom → layer-1 exact (no higher
193+ back-reaction layers in the self-interference integral).
193194
194195 4. Self-interference integral:
195196 A marginal primary (h=1) with unit coupling (g²=1) on a genus-0
196- worldsheet gives the universal correction :
197+ worldsheet gives the universal back-reaction :
197198 δ(1/α) = g²·h/(2π) = 1/(2π)
198199 This is a standard 2D CFT result for the integrated two-point
199200 function of a dimension-1 primary on the sphere.
@@ -211,7 +212,7 @@ def _derive_bridge():
211212
212213 g2 = D2_local # bridge coupling² = 1
213214 h = float (h_bridge ) # = 1 (marginal)
214- delta_inv = g2 * h / (2.0 * math .pi ) # = 1/(2π) (one-loop exact, c_coset=0)
215+ delta_inv = g2 * h / (2.0 * math .pi ) # = 1/(2π) (layer-1 exact, c_coset=0)
215216 inv_alpha_alg = 2 ** 9 / math .pi # = 512/π (emergence value)
216217
217218 # Depth-3 echo: the mutual electron↔quark loop through the EM channel.
@@ -313,7 +314,7 @@ def derive(R, masses):
313314 alpha_MZ_no_th = _run_SM_2loop (a_no_th_mt , m_t_GeV , M_Z_PDG , n_f - 1 )
314315 err_no_th = 100 * (alpha_MZ_no_th - alpha_s_PDG ) / alpha_s_PDG
315316
316- print (f"\n SM 2-loop running (no threshold):" )
317+ print (f"\n SM layer-2 running (no threshold):" )
317318 print (f" alpha_s(M_Z) = { alpha_MZ_no_th :.4f} { err_no_th :+.1f} )
318319
319320 # ── G₂ -> SU(3) threshold (matched at μ*) ──
@@ -350,13 +351,22 @@ def derive(R, masses):
350351 print (f" Delta = { abs (alpha_MZ_thresh - alpha_s_PDG ):.4f}
351352 f", { abs (alpha_MZ_thresh - alpha_s_PDG )/ 0.0009 :.1f} )
352353
354+ # ── RK4 convergence self-certification ──
355+ # Re-run the full chain at 2× step count; assert agreement.
356+ a_mt_2x = _rk4_run (lambda a : - _b0 (n_f )* a ** 2 - _b1 (n_f )* a ** 3 ,
357+ a_thresh , math .log (mu_star ), math .log (m_t_GeV ), 20000 )
358+ a_MZ_2x = _rk4_run (lambda a : - _b0 (n_f - 1 )* a ** 2 - _b1 (n_f - 1 )* a ** 3 ,
359+ a_mt_2x , math .log (m_t_GeV ), math .log (M_Z_PDG ), 20000 )
360+ assert abs (alpha_MZ_thresh - a_MZ_2x ) < 1e-6 , \
361+ f"RK4 not converged: { alpha_MZ_thresh } { a_MZ_2x }
362+
353363 # ── Bridge self-interference: alpha(0) ──
354364 bridge = _derive_bridge ()
355365
356366 print (f"\n Bridge self-interference -> alpha(0):" )
357367 print (f" h_bridge = h({ n7 } { n26 } { float (h_7 )} { float (h_26 )} { float (h_bridge )} )
358368 print (f" D²_local = { bridge ['D2_local' ]:.1f} )
359- print (f" c_coset = { bridge ['c_coset' ]:.0f} one-loop exact)" )
369+ print (f" c_coset = { bridge ['c_coset' ]:.0f} layer-1 exact)" )
360370 print (f" depth 1: 1/alpha = (2⁹/pi)(1 - 1/(2pi)) = 137.036439" )
361371 print (f" depth 3: e↔q loop, 2 orientations (S,S† rule), self-consistent:" )
362372 print (f" 1/alpha(0) = (2⁹/pi)(1 - 1/(2pi) - 2(alpha/2pi)²)" )
0 commit comments