V2.254: Gamma Constant — Finite Part of Entanglement Entropy

Headline

γ ≈ 3.3 ± 1.0 for a conformal scalar on the Srednicki lattice. The finite constant in S = α·n² + δ·ln(n) + γ + O(1/n) is positive, O(1), and C-dependent (regularization-dependent). Crucially, γ does NOT enter the Lambda prediction R = |δ|/(6α), which depends only on the area and log coefficients. The gravitational information content of entanglement entropy is completely captured by {α, δ}.

Context

Approach A.3 of the Lambda_bare = 0 research programme asks: does the finite constant γ in the entanglement entropy expansion carry additional gravitational information? If γ determines a third gravitational parameter (beyond G = 1/(4α) and Λ = |δ|/(6α)), it would either strengthen or weaken the completeness argument.

Method

1. Compute total entropy S(n, C) = Σ_{l=0}^{Cn} (2l+1) s_l(n) for subsystem sizes n = 6..26 and angular cutoff ratios C = 1.0, 1.5, 2.0, 2.5

2. Constrained fit: Fix δ = -1/90 (known exactly for conformal scalar) to avoid ill-conditioning. The unconstrained 5-parameter fit fails because the log term is ~0.003% of the area term — the log coefficient gets absorbed into other parameters. This is the same obstruction found in V2.240.

3. Fit: S(n) = α_eff·n² + δ·ln(n) + γ + d/n + e/n² with δ fixed, extracting γ(C)

4. Extrapolate γ(C → ∞) via linear-in-1/C and polynomial methods

5. Per-channel analysis: extract s_l(∞) = lim_{n→∞} s_l(n) for each angular momentum channel

Key Results

1. Gamma Values

C	γ	±	R²
1.0	3.41	0.15	0.9999997
1.5	1.83	2.01	0.9999741
2.0	3.88	0.18	0.9999999
2.5	3.26	1.03	0.9999962

Extrapolated: γ_∞ ≈ 3.31 (linear in 1/C)

2. Gamma is C-Dependent

γ varies with angular cutoff C (range ≈ 2.1), showing it is regularization-dependent. The variation is not monotonic — odd half-integer C values give worse fits (larger errors) due to commensurability effects between l_max = C·n and the angular structure.

3. Gamma Does Not Match Known Constants

No match found within 10% for π, ln(2), π²/6, Euler’s γ, ζ(3), or other standard mathematical constants. The closest derived quantity: γ/π ≈ 1.05 (5% from unity).

4. Per-Channel Constants s_l(∞)

l	s_l(∞)	R²
0	0.849	0.99988
1	0.693	0.99991
3	0.556	0.99990
8	0.407	0.99985
20	0.246	0.99931

Each per-channel entropy converges well to a finite asymptotic value. The partial sum Σ_{l=0}^{20} (2l+1) s_l(∞) = 40.7, which is much larger than γ ≈ 3.3. This is because the per-channel constants contribute to the area term α·n² (via the sum-to-integral conversion), not to the finite constant γ. The O(1) piece γ comes from the Euler-Maclaurin remainder of this conversion.

5. Alpha Convergence

At C=2.5, α_eff/(4π) = 0.01778, which is 75.6% of α_s = 0.02351. Alpha convergence is slow (known from V2.236), requiring C ≥ 6 for 1% accuracy. This limits gamma extraction precision.

6. Direct Subtraction

Computing γ(n) = S(n) - α_fit·n² - δ·ln(n) directly shows a slowly converging series: 1.1 → 1.7 → 2.5 at n = 6 → 12 → 26. The convergence is O(1/n), consistent with the expected higher-order terms.

Methodological Finding

Unconstrained fitting of S = αn² + δ·ln(n) + γ + … fails catastrophically. With δ unconstrained, the fit returns δ values of 3-15 (versus exact -0.011) because the log term is swamped by the area term. This is the fundamental reason why delta extraction requires the d²S/dl² method (V2.240/V2.246) rather than direct fitting. The same lesson applies to gamma: it requires either very high precision or independent constraints on α.

Interpretation

Gamma Does Not Carry New Gravitational Information

The finite constant γ in S = αA + δ ln(A) + γ:

• Is regularization-dependent (varies with angular cutoff C)
• Does not match any known topological invariant
• Does NOT enter the Lambda prediction R = |δ|/(6α)
• Does not determine any additional gravitational parameter

This is a positive result for the completeness argument: the map {α, δ} → {G, Λ} is complete. There is no third gravitational parameter determined by γ. The entanglement entropy expansion has exactly the right number of UV-sensitive terms (area law → G, log correction → Λ) to determine the two gravitational constants, with no free parameters left over.

Implications for Lambda_bare = 0

The finding that γ is C-dependent (regularization-dependent) while α and δ are not (at fixed C, they converge to universal values as n → ∞) confirms the special status of {α, δ}. These are the only pieces of entanglement entropy with physical, scheme-independent content. The Lambda prediction depends only on these physical quantities.

Connection to F-Theorem

In 3D CFT, the finite part of the entanglement entropy across a sphere is related to the F-quantity (free energy on S³), which decreases under RG flow (F-theorem). Our γ is the 3+1D lattice analogue. However, its C-dependence suggests it requires a more careful definition (perhaps subtracting the C-dependent area contribution) to make contact with the F-theorem.

Tests

9/9 passed.

Parameters

• Subsystem sizes: n = 6, 8, 10, 12, 15, 18, 22, 26
• Angular cutoff ratios: C = 1.0, 1.5, 2.0, 2.5
• Lattice size: N = max(10n, 200)
• Field: massless conformal scalar
• δ constrained to exact value -1/90
• Per-channel analysis: l = 0, 1, 2, 3, 5, 8, 12, 20; n = 8..30