V2.437: Full SM Spectrum Lattice R — End-to-End Verification

Status: COMPLETE

Question

Can we compute R = |delta_total|/(6alpha_total) directly from the Srednicki lattice with the full SM+graviton field content and reproduce R = 149sqrt(pi)/384 = 0.6877?

Method

• Srednicki radial lattice with L=60-80, C=1.0-3.0
• Compute S(n) with angular momentum sum S = sum_{l=l_min}^{Cn} (2l+1) S_l(n)
• Extract alpha and delta from d²S(n) = 8pialpha + delta/n²
• Combine across 4 field types: scalar (l>=0), vector (l>=1), Weyl (l>=0), graviton (l>=2)
• Weight by SM content: 4 scalars, 12 vectors, 45 Weyl fermions, 5 graviton modes

Key Results

Per-Spin Alpha Extraction

• All spins give alpha_lattice/alpha_analytic = 0.664 (C=2.0, L=60)
• This 34% deficit is the known finite-C convergence — consistent with V2.246, V2.288
• Alpha is universal across spins (same value within 0.3%), as expected
• Alpha ratio is C-dependent: approaches 1.0 only in the double limit

Per-Spin Delta Extraction: THE PROBLEM

Spin	delta_lattice	delta_analytic	Ratio
Scalar	+0.003	-0.011	-0.30
Vector	+0.146	-0.689	-0.21
Weyl	+0.003	-0.061	-0.06
Graviton	+0.471	-1.356	-0.35

Delta extraction fails catastrophically: wrong sign (positive instead of negative), wrong magnitude (factor 3-15x off).

Root Cause

The angular momentum cutoff l_max = C*n creates an Euler-Maclaurin boundary effect. When taking d²S, this leaves a POSITIVE contamination term in the 1/n² coefficient that overwhelms the physical (negative) delta.

This is documented across V2.246, V2.257, V2.288:

• The 1/n² signal from delta is ~0.03% of the constant term (8pialpha)
• The Euler-Maclaurin cutoff artifact contributes a POSITIVE 1/n² term of similar or larger magnitude
• The two partially cancel, but the residual has wrong sign at accessible C values

R Computation

• R_lattice = 0.34-1.48 (depending on C), vs R_analytic = 0.6877
• Deviations of 50-115% — the end-to-end lattice test DOES NOT WORK at accessible sizes
• Non-monotonic C dependence (oscillatory) due to Euler-Maclaurin artifacts

Component Budget (qualitative)

• Alpha budget: Weyl 70.3%, vector 18.8%, graviton 7.8%, scalar 3.1%
• Delta budget: dominated by cutoff artifacts, not physically meaningful at this C

Conclusions

1. The direct lattice R computation fails at accessible lattice sizes. The finite-C contamination of delta makes the end-to-end pipeline unreliable. This is not a bug — it’s a fundamental limitation of the angular cutoff method.

2. Alpha extraction works (to ~34% at C=2, converging in the double limit). The alpha ratio is universal across spins, confirming the framework’s component counting.

3. Delta extraction requires the analytical route. The trace anomaly coefficients (delta_scalar = -1/90, delta_vector = -31/45, etc.) come from the heat kernel a_4 coefficient, which is known analytically. The lattice can verify alpha but cannot currently verify delta at accessible sizes.

4. The framework’s prediction R = 0.6877 relies on:

   • Lattice-verified: alpha_s = 1/(24*sqrt(pi)) (verified to 0.01% in double limit, V2.288)
   • Analytically exact: delta values from trace anomaly (Deser-Schwimmer theorem)
   • Counting: N_eff = 128, n_grav = 10 (lattice-tested in V2.312, V2.433)

5. The weakest link is NOT alpha or delta individually, but the angular momentum cutoff procedure. A future lattice test with adaptive cutoff (e.g., smooth window function instead of hard l_max = Cn) might resolve this.

Implication for the Framework

The framework’s R = 0.6877 prediction is robust because it combines:

• Lattice alpha (well-converged)
• Analytical delta (exact from QFT)
• Field counting (verified)

The end-to-end lattice test would be the “smoking gun” but requires lattice sizes beyond current computational reach. The framework lives or dies on DESI Y5, not on lattice delta extraction.