V2.416: α_s from Per-Channel Asymptotics

Goal

Find the asymptotic form of the per-channel area coefficient α_l to analytically derive α_s = 1/(24√π).

Key Discovery: UV/IR Phase Transition in α_l

The per-channel contribution α_l(n) has a sign change at l ≈ n:

n/l	α_l sign	Physical regime
n/l > 1	Negative	n > l: subsystem larger than angular scale (IR)
n/l ≈ 1	Zero crossing	Transition point
n/l < 1	Positive	n < l: subsystem smaller than angular scale (UV)

This was verified directly in Phase 4:

• l=20, n=20 (n/l=1): α_l ≈ 0 (crosses zero)
• l=50, n=50 (n/l=1): α_l ≈ 0 (crosses zero)
• l=100, n=50 (n/l=0.5): α_l > 0 (UV regime)

Structure of the Sum

$\alpha_s = \sum_{l=0}^{\infty} (2l+1) \, \alpha_l(n)$

At fixed n, the sum splits into:

• IR tail (l = 0 to ~n): negative contributions
• UV tail (l = ~n to ∞): positive contributions
• α_s is the residual after near-cancellation

This explains why:

1. The finite-C bias is always negative (cutoff l_max = Cn truncates the positive UV tail)
2. Convergence requires C → ∞ (need all of the UV tail)
3. α_s is 96% UV-dominated (V2.287) — the UV tail dominates the residual

Numerical Results

α_l Is Not a Simple Power Law

The asymptotic form α_l ~ A/l^p gives p ≈ 0.49, far from any clean value. The weighted summand (2l+1)·α_l decays as l^{0.51} — so slowly that the sum may not formally converge at fixed n.

This means α_s cannot be computed by a simple asymptotic expansion of α_l. The per-channel approach hits a fundamental obstacle: α_l is not a function of l alone but of the ratio n/l.

Scaling Function α_l(n) = f(n/l)

Phase 4 reveals that α_l depends on n/l as a scaling variable:

l=5:  α_l(n) = negative for all n (always in IR)
l=20: α_l(n) changes sign at n/l ≈ 1
l=50: α_l(n) changes sign at n/l ≈ 1
l=100: α_l(n) > 0 for n ≤ 55 (still in UV at n/l < 0.55)

The scaling function f(x) = α_l · l^p where x = n/l satisfies:

• f(x) < 0 for x > 1 (IR)
• f(x) > 0 for x < 1 (UV)
• f(x) → 0 as x → 0 (deep UV: modes decouple)

Sum Rule Convergence

At fixed l_max = 250:

n	α_s(l_max=250)	Error vs target
20	0.02163	-8.0%
30	0.01933	-17.8%
40	0.01676	-28.7%
50	0.01414	-39.9%

α_s decreases with n at fixed l_max because more channels enter the IR regime (l < n). The true α_s requires l_max → ∞ with n fixed (or C = l_max/n → ∞).

Implications

Why 1/(24√π) Is Hard to Derive

The per-channel structure reveals why no simple formula captures α_s:

1. α_l is not a function of l alone — it depends on n/l
2. The sum involves a UV/IR cancellation, not a simple tail
3. The asymptotic form of α_l has no clean power-law exponent (p ≈ 0.49)

A closed-form derivation would need the exact scaling function f(n/l) and its integral:

$\alpha_s = \int_0^\infty (2l+1) \, f(n/l) \, dl$

This integral might have a Mellin transform structure, but the scaling function itself is not analytically known.

Connection to V2.311

V2.311 identified a UV/IR phase transition at l* ≈ 0.52·Cn in the QNEC structure. The sign change at l ≈ n found here is consistent: the transition occurs at l/n ≈ 1 (or l* ≈ n), which for the standard cutoff C is at l*/l_max = n/(Cn) = 1/C.

The per-channel α_l and the QNEC mode structure share the same UV/IR boundary. This suggests both phenomena have a common origin in the Williamson normal form of the covariance matrix.

What Would Close the Gap

1. Analytical form of f(n/l): The scaling function near l/n ≈ 1 determines α_s. This might be accessible via WKB analysis of the Srednicki eigenvalue problem at large l.

2. Mellin transform: If f(x) = Σ c_k x^{s_k}, then α_s has a Mellin representation that might simplify to a gamma function expression.

3. Conformal field theory: In the continuum (n → ∞), the entanglement entropy of a sphere in D=4 is constrained by conformal symmetry. The coefficient α might follow from the conformal a-anomaly.

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