V2.421: Per-Channel Scaling Function — F(x) is NOT Scheme-Universal

Objective

Test whether the per-channel scaling function F(x) = n²·d²s_l (at l = x·n) is universal across lattice schemes, even though the total α_s varies 82%.

If F(x) were universal, the α_s spread would be a summation artifact. If F(x) differs, the schemes encode genuinely different UV physics.

Key Result: F(x) Differs Across Scheme Classes

Comparison	RMS difference
Srednicki vs Standard FD	0.3%
Srednicki vs Numerov	35.0%
Standard FD vs Numerov	35.0%

The Srednicki class (Srednicki + Standard FD) has its own F(x), while Numerov has a systematically different scaling function — especially in the UV region (x > 1) where F(x) is 40-50% larger for Numerov.

Detailed Findings

Phase 1: F(x) Comparison

Both Srednicki and Standard FD produce nearly identical per-channel functions (confirming they are the same universality class). Numerov deviates progressively above x ≈ 0.5, with up to 46% difference at x = 3.

Phase 3: Zero Crossing (UV/IR Transition)

Scheme	x* (zero crossing)
Srednicki	1.008
Standard FD	1.008
Numerov	1.040

The transition is at x* ≈ 1 for all schemes (l* ≈ n), but Numerov’s is shifted slightly higher. This means Numerov’s IR regime extends further.

Phase 4: Integral Test

At fixed n=30, the integral ∫₀^C 2x·F(x)dx gives different α_s:

C	Srednicki	Numerov
3	0.0065 (−72%)	0.0108 (−54%)
4	0.0110 (−53%)	0.0204 (−13%)
5	0.0141 (−40%)	0.0240 (+2%)

Numerov’s wider F(x) integrates to a larger value, converging to the target FASTER in the C-direction. However, its d²S extraction at varying n gives a different convergence path (V2.410: −29% at C=5).

Phase 5: Shape Not Universal

Even after normalizing F(x)/F(0), the shapes differ by up to 47%. The Numerov scheme has a systematically flatter, wider UV tail.

What This Means

The α_s spread is NOT a summation artifact.

Different discretizations produce genuinely different per-channel physics. The coupling matrix K encodes the UV spectral structure, and different discretizations sample this differently.

The Srednicki class is self-consistent.

Srednicki and Standard FD agree to 0.3% at every x — they are truly the same physics. The prediction α_s = 1/(24√π) is a property of this specific universality class, which arises from the Hamiltonian with φ = rR.

Why the Srednicki scheme is natural.

The φ = rR substitution transforms the radial equation into a 1D Schrödinger equation with flat kinetic term (-d²/dx²). This is the canonical form:

• It preserves the symplectic structure of the Hamiltonian
• It gives the standard Sturm-Liouville problem
• It produces the correct trace anomaly δ = -1/90 at moderate N (V2.417)

Other discretizations (Numerov, Volume-weighted) work with different field variables that introduce additional metric factors. These are valid classically but have different UV behavior at finite lattice spacing.

Implications for the Framework

The prediction Ω_Λ = 149√π/384 rests on:

1. Trace anomaly δ = -149/12 (exact, topological, scheme-independent) ✓
2. α_s = 1/(24√π) (numerical, Srednicki universality class only) — unproven
3. The Srednicki class is the correct discretization of the continuum

The third point is reasonable (it’s the canonical Hamiltonian discretization) but unproven. An analytical derivation from the continuum would bypass all lattice concerns entirely.

Files

• src/scaling_function.py — Per-channel F(x) computation for 3 schemes
• run_experiment.py — 5-phase analysis
• tests/test_scaling.py — 5/5 tests passing