V2.393: First-Principles Graviton Entanglement Entropy on the Srednicki Lattice

Purpose

Compute the graviton’s entanglement entropy contribution on a Srednicki lattice using the SVT (scalar-vector-tensor) decomposition of the spatial metric perturbation h_ij. The graviton DOF count (n_grav = 2 vs 4 vs 6 vs 10) is the single largest uncertainty in the framework’s Λ prediction, and this experiment tests which counting survives.

Method

Decompose the linearized metric perturbation h_ij into SVT sectors on S², each sector reducing to independent scalar-type fields on the radial lattice:

Sector	Components	l range	Description
Scalar trace (h)	1	l ≥ 0	Conformal factor
Scalar shear (σ)	1	l ≥ 2	Traceless scalar
Transverse vector (F_i)	2	l ≥ 1	Divergence-free vector
TT tensor (h^TT_ij)	2	l ≥ 2	Transverse-traceless
Total	6	—	Full h_ij

Four graviton hypotheses compared:

Hypothesis	n_grav	Sectors included
TT only	2	TT tensor only
TT + vector edge	4	TT + transverse vector
Full h_ij	6	All SVT sectors
Framework	10	Full h_ij + 4 edge modes (gauge constraints)

Lattice: N = 400 sites, C = [2.0, 3.0, 4.0, 5.0], n_sub = 8..39, l = 0..195.

Key Results

1. Per-Sector Entropy Ratios (Universal)

Each SVT sector’s entanglement entropy scales as n_comp × α_scalar:

Sector	α/α_scalar	Expected	CV across C
Scalar trace (1c, l≥0)	1.000	1	0.0%
Scalar shear (1c, l≥2)	1.000	1	0.0%
Transverse vector (2c, l≥1)	2.000	2	0.0%
TT tensor (2c, l≥2)	2.000	2	0.0%

The ratio α_grav/α_scalar is exactly n_grav for all hypotheses, at all C values (CV = 0.0%). This is the key universal quantity.

2. Cosmological Predictions

Hypothesis	n_grav	α_grav/α_s	R	Λ/Λ_obs	σ from Ω_Λ	Status
TT only	2	2.000	0.7335	1.071	+6.7σ	EXCLUDED
TT + vector edge	4	4.000	0.7215	1.054	+5.0σ	EXCLUDED
Full h_ij	6	6.000	0.7099	1.037	+3.4σ	Marginal
Framework (10)	10	10.000	0.6877	1.004	+0.4σ	VIABLE

Only n_grav = 10 matches Ω_Λ = 0.6847 ± 0.0073. The exact match requires n_eff_grav = 10.56.

3. Fit Quality

All entropy fits achieve R² = 1.000000 (machine precision) for the form S = 4πα n² + δ ln(n) + γ. The d²S method (second derivative) gives R² = 0.90–0.999 depending on the number of modes summed.

4. C-Independence

While α itself varies ~11% across C values (known lattice artifact), the ratio α_grav/α_scalar is exactly C-independent. This confirms it as the physically meaningful quantity.

Interpretation

What the lattice proves

1. Each SVT sector behaves as independent scalar channels — entropy scales linearly with component count, no cross-sector correlations
2. The ratio α_grav/α_scalar = n_grav is universal — independent of lattice coupling C, subsystem size, angular cutoff
3. R = |δ_total|/(6α_total) varies monotonically with n_grav — the DOF count is the sole remaining free parameter

What the lattice does NOT prove

The lattice computes entanglement entropy for each SVT sector separately. It confirms the arithmetic (n components → n × α_scalar) but does not determine which sectors are physical. The question “how many graviton modes contribute?” is a gauge-theory question about which modes carry physical entanglement across the horizon.

The four hypotheses encode different physical assumptions:

• n = 2 (TT only): Only gauge-invariant propagating DOF. Standard GR counting. EXCLUDED at 6.7σ.
• n = 6 (full h_ij): All metric components, before gauge fixing. Marginal at 3.4σ.
• n = 10 (framework): All components plus edge modes from gauge constraints. Viable at 0.4σ.

The edge mode question

The gap between n = 6 and n = 10 is the 4 “edge modes” — gauge-constraint contributions that become physical at an entanglement boundary (Donnelly & Wall 2012, 2015). The framework prediction n_grav = 10 implicitly requires these edge modes. V2.312 found graviton edge mode δ ≈ 0 (unlike the vector edge mode δ = −1/3), but the α contribution is separate from δ.

Honest assessment

Strengths:

• Clean per-sector decomposition with machine-precision fits (R² = 1.000000)
• Universal ratios (CV = 0% across C) — no lattice artifacts in the key observable
• Sharp discrimination: TT-only excluded at 6.7σ, only n = 10 viable
• Consistent with V2.328 spectroscopic result: n_grav = 10.6 ± 1.4

Weaknesses:

• The lattice treats each SVT sector as independent scalars — this is correct for free fields but does not capture gauge structure
• The n = 10 counting relies on the edge mode argument, which is theoretically motivated but not derived from first principles on this lattice
• The “10 modes” hypothesis is really “6 h_ij components + 4 edge modes” — the lattice verifies n_comp × α_scalar but doesn’t prove which modes are physical
• A proper graviton lattice computation would need gauge-invariant entanglement entropy (Casini, Huerta & Rosabal 2014), which requires the full constrained phase space
• The exact match n_eff = 10.56 (not exactly 10) suggests either small corrections or that the true counting is slightly different

What would strengthen this:

• A constrained-phase-space lattice computation that naturally produces n_eff without specifying sectors by hand
• An independent derivation of n_grav = 10 from the graviton’s symplectic structure
• Precision δ_grav extraction matching the analytical −61/45 (V2.312 achieved 1.0% for “2 scalars l≥2”)

Files

• src/graviton_entanglement.py — Srednicki lattice, SVT decomposition, 4 hypotheses, fitting
• tests/test_graviton_entanglement.py — 15 tests, all passing
• run_experiment.py — Full 7-section analysis

Status

COMPLETE — Per-sector graviton entanglement entropy computed. TT-only (n=2) excluded at 6.7σ. Framework counting (n=10) viable at 0.4σ. Key result: α_grav/α_scalar = n_grav exactly (universal, C-independent). The graviton DOF question reduces to which sectors carry physical entanglement across the horizon — the lattice confirms the arithmetic but the physics requires gauge-theoretic input.