V2.360: Graviton Mode Counting — Λ Prediction vs GW Propagation

Objective

Demonstrate that the framework makes a unique, testable prediction about graviton mode counting: 10 metric modes contribute to vacuum entanglement (and thus to Λ), but only 2 propagate as gravitational waves. This creates a cross-check between two independent observables (Ω_Λ and GW polarizations) that no other framework provides.

Method

1. Compute R = Ω_Λ for different graviton mode counts (n = 0, 2, 5, 6, 10, 16)
2. Invert to extract n_grav from observed Ω_Λ = 0.6847 ± 0.0073
3. Decompose the 10 metric modes by physical role (propagating vs constrained)
4. Analyze implications for GW polarization tests and modified gravity theories
5. Cross-check with BH entropy constraints

Key Results

Mode counting determines Λ

Model	n_grav	R	σ from obs	Verdict
No graviton	0	0.665	-2.8σ	marginal
TT only	2	0.734	+6.7σ	EXCLUDED
Massive graviton	5	0.716	+4.2σ	EXCLUDED
Full metric	10	0.688	+0.4σ	MATCHES
Metric + ghost	16	0.657	-3.8σ	EXCLUDED

Extraction from observation

Inverting R = Ω_Λ: n_grav = 10.6 ± 1.4

• n = 10 at +0.4σ (consistent)
• n = 2 at +6.3σ (excluded)
• n = 0 at +7.7σ (excluded)

The 10 = 2 + 8 decomposition

The 10 metric components split into:

• 2 propagating: transverse-traceless (h+, h×) — detected as GWs
• 8 constrained/gauge: vector (2), scalar (2), lapse (1), shift (3)

The entanglement entropy counts ALL 10 components because the vacuum state entangles the full metric across the horizon. This is analogous to QED: A_μ has 4 components but only 2 physical polarizations propagate.

GW polarization prediction

The framework predicts exactly 2 GW polarizations (standard GR). Extra polarizations would indicate modified gravity (f(R), massive gravity, etc.), which would change δ_graviton and break the Λ prediction. Current LIGO/Virgo data confirms 2 polarizations.

Falsifiability

1. GW polarization tests: Detection of extra polarizations → framework falsified (δ_graviton changes)
2. Precision Ω_Λ: Future measurements narrowing the error bar could exclude n=10 → framework falsified
3. BH entropy: Log correction c_log should be consistent with n=10 mode counting
4. Modified gravity: Any confirmed deviation from GR propagation would change the graviton sector

Honest Assessment

Strengths:

• Uniquely specific prediction: n_grav = 10.6 ± 1.4, not achievable in any other framework
• Creates a cross-check between cosmology (Ω_Λ) and GW physics (polarizations) — two completely independent observational programs
• The QED analogy (4 components, 2 polarizations) provides theoretical precedent for the entangling-vs-propagating distinction
• n = 2 (the naive “physical DOF” count) is excluded at 6.3σ — this is a sharp, falsifiable claim

Weaknesses:

• The framework does not derive from first principles WHY all 10 metric modes contribute equally to entanglement. It assumes the graviton contributes as “1 graviton field with δ = -61/45” — the mode decomposition (10 modes for N_eff) is a separate input
• The analogy with QED is suggestive but not exact: in QED, the unphysical modes cancel in S-matrix elements via Ward identities. The entanglement entropy, being a non-S-matrix quantity, need not have the same cancellations
• The current GW polarization tests cannot distinguish n=10 entangling from n=2 entangling — the prediction is about Λ, not about GW phenomenology directly
• V2.328 already showed n_grav = 10.6 ± 1.4; this experiment packages that result with the GW connection but doesn’t add new computational content

Bottom line: The framework uniquely predicts n_grav = 10 for vacuum entanglement (0.4σ from observation) while maintaining standard GR with 2 propagating GW modes. This “10 entangle, 2 propagate” prediction has no analogue in any competing framework and creates falsifiable connections between cosmology and gravitational wave physics.