V2.74: Area-Law Coefficient Convergence to C -> infinity

Goal

Extrapolate the area-law coefficient alpha to the scheme-independent C -> infinity limit for scalar, Dirac, and vector species, removing the ~18% cutoff-convention ambiguity identified in V2.73.

Method

For each species, compute alpha at multiple global cutoff values C = l_max/n_max. Fit the convergence model alpha(C) = alpha_inf + A/C^p to extrapolate.

• Scalar: N=400, C = [5, 8, 10, 13, 15, 20, 30, 50]
• Dirac: N=200, C = [5, 10, 15, 20, 30]
• Vector: N=400, C = [5, 10, 15, 20, 30]
• n_values: range(15, 61, 3) -> 16 points per fit
• 5-param fit: S = alpha4pin^2 + betan + delta*ln(n) + gamma + eps/n

Results

Scalar alpha(C) convergence

C	alpha	R^2
5	0.01160	0.99999976
8	0.01747	0.99999993
10	0.01926	0.99999996
13	0.02075	0.99999998
15	0.02134	0.99999999
20	0.02218	1.00000000
30	0.02285	1.00000000
50	0.02324	1.00000000

Extrapolation: alpha_scalar_inf = 0.02376 +/- 0.00008, p = 1.45 +/- 0.03

Vector alpha(C) convergence

C	alpha	R^2
5	0.02320	0.99999976
10	0.03853	0.99999996
15	0.04267	0.99999999
20	0.04436	1.00000000
30	0.04570	1.00000000

Extrapolation: alpha_vector_inf = 0.04764 +/- 0.00018, p = 1.44 +/- 0.03

Vector/scalar ratio

alpha_vector_inf / alpha_scalar_inf = 2.005

This matches the heat kernel prediction of exactly 2.0 to three significant figures.

Dirac alpha(C) — DOES NOT CONVERGE

C	alpha	R^2
5	-0.01367	0.99999996
10	0.08885	0.99999995
15	0.17088	0.99999996
20	0.23793	0.99999996
30	0.34432	0.99999996

The Dirac alpha grows nearly linearly with C, showing no sign of convergence. The nonlinear fit hits the lower bound (p=0.1) with huge uncertainty (alpha_inf = 2.1 +/- 8.0). This is almost certainly due to fermion doubling on the naive lattice (wilson_r=0): the doublers contribute additional area-law entropy that grows with the cutoff.

Note: At C=10, alpha_Dirac/alpha_scalar = 0.089/0.019 = 4.7, consistent with V2.72’s finding of 4.6. But this ratio grows with C (15.0 at C=30), invalidating the Dirac extrapolation.

Corrected SM Prediction

Since the Dirac alpha doesn’t converge on our lattice, we use the heat kernel prediction for the Dirac/scalar ratio (4.0) combined with the well-determined scalar extrapolation:

• alpha_scalar_inf = 0.02376 (measured)
• alpha_Dirac_inf = 4 x 0.02376 = 0.09504 (heat kernel)
• alpha_Weyl = alpha_Dirac/2 = 0.04752
• alpha_vector_inf = 0.04764 (measured)

alpha_SM = 4(0.02376) + 45(0.04752) + 12(0.04764) = 0.0950 + 2.138 + 0.572 = 2.806

With delta_SM = -11.06 (corrected from code bug that used -9.69):

R_SM = |delta_SM| / (12 x alpha_SM) = 11.06 / 33.67 = 0.329

This is a factor of 3.0 from unity (vs. 2.7 with C=10 values).

Lambda_SM / Lambda_obs = |delta_SM| / (4 x alpha_SM x L_H^2 x Lambda_obs) = 1.15

Bug Found

The original code used delta_Weyl = -11/360 instead of -11/180. This is a factor-of-2 error: the correct value is delta_Weyl = -4a_Weyl = -4 x 11/720 = -11/180. The code has been corrected. The analytical delta values used in the SM sum are:

• delta_scalar = -1/90 = -0.01111
• delta_Weyl = -11/180 = -0.06111
• delta_vector = -31/45 = -0.68889

Key Findings

1. Scalar and vector alpha converge cleanly to C -> infinity with power law p ~ 1.45
2. Vector/scalar ratio = 2.005 — matches heat kernel exactly
3. Dirac alpha does NOT converge due to fermion doubling (wilson_r=0)
4. Using heat kernel for Dirac and extrapolated scalar/vector: R_SM = 0.33, Lambda/Lambda_obs = 1.15
5. The C -> infinity extrapolation changes alpha_scalar from 0.019 (C=10) to 0.024, a 25% increase that slightly worsens R_SM from 0.36 to 0.33

Runtime

153 seconds total.