V2.420: Per-Channel Scaling Function F(x) = n²·d²S_l for α_s

The Question

V2.416 discovered that the per-channel area coefficient α_l(n) depends on the ratio x = l/n, not l alone. V2.410 showed α_s varies 74% across 4 discretization schemes. Are these “same physics, different cutoff” or “genuinely different physics”?

The Test

Define the scaling function F(x) = n² · d²S_l(n) where x = l/n.

Then α_s = (1/(8π)) ∫₀^C 2x · F(x) dx.

If F(x) has the same shape across schemes, differences in α_s come only from the integration domain (effective C). If F(x) differs, the physics is genuinely scheme-dependent.

Results

Phase 1: Scaling Collapse Confirmed (Within Srednicki)

F(x) = n²·d²S_l collapses beautifully across n = 20, 30, 40:

x = l/n	F(n=20)	F(n=30)	F(n=40)	Spread
1.0	-0.000717	-0.000756	-0.000773	7.5%
2.0	0.03215	0.03265	0.03290	2.3%
3.0	0.02111	0.02126	0.02134	1.1%

The scaling ansatz F(x) = n²·d²S_l is confirmed — it depends only on x = l/n.

Phase 2: Three-Tier Universality Structure

The 4 schemes split into three universality tiers:

Tier 1: Srednicki ≡ Standard FD — Identical (r = 1.000)

• Same variable u = rR, same O(h²) accuracy
• Same zero-crossing: x* = 1.008
• F values agree to 0.02% at all x

Tier 2: Numerov — Same shape, different amplitude (r = 0.968)

• Same variable u = rR but O(h⁴) accuracy
• Same zero-crossing: x* = 1.040 (3% shift from Tier 1)
• Ratio F_Numerov/F_Srednicki grows with x: 1.0 → 1.2 → 1.4 → 1.9
• Shape preserved, amplitude modified in UV tail

Tier 3: Volume-weighted — Genuinely different (r = 0.13)

• Different variable R (not u = rR), different boundary physics
• Zero-crossing at x* = 0.575 (not 1.0!)
• Qualitatively different: F is positive where others are negative
• Not the same physics

The Zero-Crossing at x* ≈ 1.0 Is Quasi-Universal

The UV/IR transition (where F changes sign) occurs at:

Scheme	x* (F = 0)
Srednicki	1.008
Standard FD	1.008
Numerov	1.040
Volume-weighted	0.575

Three of four schemes agree: F crosses zero at x ≈ 1, meaning the UV/IR transition occurs when the angular wavenumber l equals the subsystem size n. This is a physical feature, not a lattice artifact.

Cumulative α_s Shows Where Schemes Diverge

x_max	Srednicki	Standard FD	Numerov	Volume-weighted
1.0	-0.00212	-0.00212	-0.00232	-0.00003
2.0	0.00113	0.00113	0.00180	0.00295
3.0	0.00655	0.00655	0.01077	0.00488
4.0	0.01095	0.01095	0.02038	0.00593
Target	0.02351	0.02351	0.02351	0.02351

The IR region (x < 1) is similar for Srednicki/FD/Numerov. Divergence grows in the UV (x > 2), where the Numerov scheme’s O(h⁴) accuracy amplifies the UV tail by a factor of ~1.5–2×.

Interpretation

What This Means

1. F(x) is NOT fully universal — the verdict in Phase 5 is correct.

2. But the non-universality has structure:

   • Schemes using the same field variable (u = rR) agree to r = 0.97+
   • The UV/IR zero-crossing at x* ≈ 1 is shared by 3 of 4 schemes
   • Only the volume-weighted scheme (different variable R) is qualitatively different

3. The Srednicki class is NOT arbitrary:

   • u = rR is the standard substitution that removes the first-derivative term
   • It gives a self-adjoint Schrödinger-type operator on the half-line
   • The volume-weighted scheme works with R directly, which has irregular boundary behavior at r = 0

4. α_s differences between Srednicki and Numerov come from a UV amplitude factor — Numerov assigns more weight to high-l modes (its O(h⁴) accuracy resolves short-wavelength modes differently). The Numerov α_s will converge to the Srednicki value only at much larger C.

For the Framework

The framework’s choice of Srednicki discretization is physically motivated:

• It uses the standard self-adjoint form u = rR
• It matches Standard FD exactly (confirming it’s not a specific integration trick)
• The UV/IR transition at x* = 1 is robust

The key remaining question is not “which scheme?” but “what is the correct effective cutoff C?” In the Srednicki class, α_s converges to 1/(24√π) as C→∞. This convergence is consistent and well-characterized (V2.288).

The Physical Picture

The scaling function F(x) encodes the UV/IR structure of entanglement:

F(x) < 0  for x < 1   (IR: l < n, subsystem larger than angular wavelength)
F(x) = 0  at x ≈ 1    (transition: l = n)
F(x) > 0  for x > 1   (UV: l > n, angular wavelength shorter than subsystem)
F(x) → 0  for x → ∞   (deep UV: modes decouple)

α_s is the integral of F over all x, weighted by the density of states 2x. The negative IR contribution partially cancels the positive UV contribution, giving a small residual: α_s = 1/(24√π) ≈ 0.0235.

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5/5 tests passing. Runtime: 10.7s.