V2.404: BH vs Cosmological Horizon — Log Coefficient Split

Motivation

The framework predicts γ_cosmo = δ_total = -149/12 for the cosmological horizon (conformally flat background). But is this the same as the Schwarzschild BH log coefficient? If not, the framework makes two distinct predictions from the same trace anomaly — one for cosmology, one for black holes.

The answer depends on the Weyl anomaly coefficient c, which doesn’t contribute to the cosmological prediction (Casini-Huerta theorem, exact for conformally flat backgrounds) but does contribute to the BH prediction (Schwarzschild is not conformally flat).

Key Results

The (a, c) Split

The trace anomaly ⟨T^μ_μ⟩ = a·E₄ + c·W² has two independent coefficients. The cosmological constant depends only on a (through δ = -4a). Black hole entropy depends on BOTH a and c.

Field	a	c	c/a	δ = -4a	n_SM
Real scalar	1/360	1/120	3.00	-1/90	4
Weyl fermion	11/720	1/40	1.64	-11/180	45
Gauge vector	31/180	1/10	0.58	-31/45	12
Graviton	61/180	7/10	2.07	-61/45	1

SM + graviton totals:

• a_total = 149/48 ≈ 3.104
• c_total = 367/120 ≈ 3.058
• c/a = 0.985 (nearly equal!)

Two Predictions from One Input

Observable	Formula	Value	Comment
γ_cosmo	-4a_total	-149/12 = -12.42	Exact (Casini-Huerta)
γ_Schw	-4a + 2c (est.)	≈ -6.30	Solodukhin estimate
Weyl correction	2c_total	+6.12	49.3% of γ_cosmo

The Weyl correction is 49% — nearly half of the cosmological value. This is because c/a ≈ 1 for the SM + graviton: the Euler and Weyl anomalies are nearly equal, so the BH log coefficient is roughly HALF the cosmological one.

Comparison with Other Approaches

Approach	γ_cosmo	γ_BH	Field-dependent?
This framework	-12.42	-6.30	YES
LQG	N/A	-1.50	NO (universal)
String (N=2, 4D)	N/A	-2.00	varies
String (N=4, 5D)	N/A	-1.00	varies

Key distinction from LQG: LQG predicts γ = -3/2 from quantum geometry alone (SU(2) Chern-Simons theory), independent of matter content. This framework predicts γ from quantum fields, with |γ| 4-8× larger. Adding a new particle changes both Λ AND γ_BH simultaneously.

Lattice Curvature Scan

Modified the Srednicki coupling matrix with a Schwarzschild-like potential (compactness parameter ε). Results:

• ε = 0 (flat): δ_scalar = -0.0084 (standard result)
• ε = 0.001: δ shifts measurably → geometry-dependence confirmed
• ε ≥ 0.01: lattice computation becomes ill-conditioned (the Schwarzschild metric factor f = 1 - r_s/r → 0 near the horizon makes the coupling matrix singular)

Verdict: The lattice confirms that δ is geometry-dependent, but the specific numbers at moderate compactness are unreliable due to the coordinate singularity. The analytical prediction (γ_Schw ≈ -4a + 2c) is the primary result.

Joint Prediction Map

The SM field content (4 scalars + 45 Weyl + 12 vectors + 1 graviton) simultaneously determines:

1. Ω_Λ = 0.688 (from a alone, obs: 0.685 ± 0.007)
2. γ_cosmo = -12.42 (determines Λ-horizon entropy)
3. γ_Schw ≈ -6.30 (determines BH entropy correction)
4. BH remnant mass ≈ 0.71 M_Pl (from γ_Schw)
5. N_ν = 3, Majorana (from species-dependence)

This is an over-determined system: measuring any two of {Ω_Λ, γ_BH, N_eff} fixes the third → falsifiable consistency check.

Honest Assessment

What’s new and significant

1. First computation of γ_Schw for the full SM spectrum in this framework
2. The split γ_cosmo ≠ γ_Schw (49% correction) is a NEW prediction
3. The c/a ≈ 1 near-equality for the SM+grav is notable and unexplained
4. Both γ values are 4-8× larger than any competing QG approach

Caveats and weaknesses

1. The “-4a + 2c” formula is an ESTIMATE from the Solodukhin universal formula. The exact BH log coefficient involves subtleties with extrinsic curvature, ghost contributions, and zero modes that we don’t fully resolve here.
2. The graviton c-coefficient (c = 7/10) comes from Christensen-Duff (1979), which uses a different ghost/gauge-fixing scheme than the entanglement-based a = 61/180 we derive from the lattice. Cross-consistency is uncertain.
3. The lattice Schwarzschild model is ill-conditioned at moderate compactness — a proper lattice computation would need tortoise-coordinate discretization with careful horizon-adapted coordinates.
4. Neither γ_cosmo nor γ_Schw is directly measurable with current experiments. But they differentiate approaches and could become testable with quantum gravity phenomenology.
5. The 49% Weyl correction is large enough that a factor-of-2 error in the c-coefficient would qualitatively change γ_Schw (possibly even flip sign).

Files

• src/anomaly_coefficients.py — (a, c) coefficients for SM fields
• src/schwarzschild_lattice.py — Modified Srednicki lattice with curvature
• tests/test_anomaly.py — Verification tests
• run_experiment.py — Full 5-phase experiment