V2.417: Ratio Universality Test — R = |δ|/α is NOT Scheme-Universal

Objective

V2.410 showed α_s varies 74% across lattice schemes. Test whether the physically meaningful ratio R = |δ|/α (which determines Ω_Λ) is universal even when α and δ individually are not.

Key Finding: R is Even More Scheme-Dependent Than α

| Scheme | α | δ | |δ|/α | R² | |--------|---|---|------|-----| | Srednicki | 0.02103 (−10.5%) | −0.01027 (−7.6%) | 0.488 (+3.3%) | 0.992 | | Standard FD | 0.02103 (−10.5%) | −0.02221 (+100%) | 1.056 (+124%) | 1.000 | | Numerov | 0.01609 (−31.6%) | −0.02200 (+98%) | 1.367 (+189%) | 0.921 | | Volume-weighted | 0.00762 (−67.6%) | +0.05613 (−605%) | 7.364 (+1458%) | 1.000 |

Spreads: α = 82%, δ = 18,934%, |δ|/α = 268%

After C→∞ extrapolation: R spread = 278% (vs α spread of 74% from V2.410).

Why δ is More Scheme-Sensitive Than α

The trace anomaly δ = −1/90 is topological and scheme-independent in the continuum. But its lattice extraction via d²S = A + B/n² picks up scheme-dependent subleading corrections (O(1/n⁴), O(ln(n)/n²)) that contaminate the B coefficient:

• Srednicki: δ ≈ −0.011 (close to −1/90 = −0.0111) ✓
• Standard FD: δ ≈ −0.022 (exactly 2× target)
• Numerov: δ ≈ −0.022 (similar to FD)
• Volume-weighted: δ ≈ +0.056 (wrong sign!)

Only the Srednicki scheme correctly captures δ at accessible lattice sizes. This is likely because the φ = rR substitution eliminates the r² Jacobian, producing a coupling matrix with better-behaved finite-size corrections.

Critical Observations

1. Srednicki and Standard FD Agree on α but Disagree on δ

These two schemes give α to 0.001% agreement (same universality class for α), but their δ values differ by ~65%. This means α and δ have completely different convergence structures — δ is controlled by subleading terms that are far more sensitive to the discretization.

2. The Srednicki Scheme is Not “Just One Choice”

The Srednicki discretization uniquely gives the correct δ ≈ −1/90 at moderate lattice sizes. The φ = rR substitution that defines it produces a tridiagonal matrix whose spectral properties match the continuum faster — for both α and δ. This suggests it is the natural discretization for this problem, not merely one option among equivalents.

3. Ratio Cancellation Does NOT Occur

The hope was that scheme-dependent corrections to α and δ would cancel in the ratio. They do not — they add. The ratio is less universal than either component.

Implications for the Framework

Negative: The Ω_Λ prediction cannot claim scheme-independence based on lattice evidence alone. At accessible sizes, different discretizations give wildly different answers for both α_s and δ.

Mitigating: The Srednicki scheme is not arbitrary — it is the natural discretization arising from the Hamiltonian formulation with φ = rR. It gives the correct δ = −1/90 and the best-converging α. The prediction Ω_Λ = 149√π/384 is specific to the continuum limit, which the Srednicki scheme best approximates.

Bottom line: An analytical proof that α_s = 1/(24√π) in the continuum (without any lattice) is the single most important open problem. Until then, the framework’s prediction rests on the Srednicki lattice’s numerical convergence to 0.011%.

Files

• src/ratio_test.py — 4 lattice schemes + joint α, δ, ratio extraction
• run_experiment.py — 4-phase analysis
• tests/test_ratio.py — 6/6 tests passing