V2.531 - Dual Observable QG Discriminator — Joint (Ω_Λ, γ_BH) Prediction
V2.531: Dual Observable QG Discriminator — Joint (Ω_Λ, γ_BH) Prediction
Motivation
The entanglement framework uniquely predicts both the cosmological constant Ω_Λ and the black hole entropy log correction γ_BH from the same trace anomaly coefficients {a_i, c_i}. No other quantum gravity approach connects these two observables. This experiment constructs the “entanglement S-T plot” — a 2D zero-parameter test analogous to the Peskin-Takeuchi S-T plot in electroweak physics.
Method
For each field type in the Standard Model, the trace anomaly has two independent coefficients:
- a (Euler density / type-A anomaly): determines the cosmological horizon entropy
- c (Weyl² / type-B anomaly): contributes only where spacetime curvature is non-conformal
The two observables are:
- Ω_Λ = |δ_total| / (6 α_s N_eff) where δ = −4a (de Sitter horizon, conformally flat)
- γ_BH = −4a_total + 2c_total (Schwarzschild horizon, non-zero Weyl curvature)
The Weyl correction (2c) vanishes for de Sitter but not for Schwarzschild — creating a split between the cosmological and BH predictions from the same field content.
Results
Per-field (a, c) decomposition
| Field | a | c | c/a | δ = −4a | γ_BH/field | n_SM |
|---|---|---|---|---|---|---|
| Real scalar | 1/360 | 1/120 | 3.00 | −1/90 | +0.0056 | 4 |
| Weyl fermion | 11/720 | 1/40 | 1.64 | −11/180 | −0.0111 | 45 |
| Gauge vector | 31/180 | 1/10 | 0.58 | −31/45 | −0.4889 | 12 |
| Graviton | 61/180 | 7/10 | 2.07 | −61/45 | +0.0444 | 1 |
SM + graviton predictions (exact fractions)
| Observable | Exact value | Numerical | Status |
|---|---|---|---|
| δ_total (= γ_cosmo) | −149/12 | −12.417 | Exact (trace anomaly) |
| a_total | 149/48 | 3.104 | |
| c_total | 367/120 | 3.058 | |
| c/a | 734/745 | 0.985 | Near-unity! |
| Ω_Λ | 149√π/384 | 0.6877 | +0.42σ from Planck |
| γ_BH (Schwarzschild) | −63/10 | −6.300 | Exact |
| Weyl correction | 367/60 | 6.117 | 49.3% of |
| γ_BH/γ_cosmo | 0.507 | Weyl/Euler ratio |
The key insight: c/a ≈ 1 is a coincidence of the SM
The SM-weighted c/a = 734/745 ≈ 0.985, remarkably close to unity. This means the Weyl and Euler anomalies are nearly equal, causing γ_BH ≈ γ_cosmo/2. This near-equality is not protected by any symmetry — it’s an accident of the SM field content. Adding BSM fields changes it.
Contributions to γ_BH by sector
| Sector | Contribution | Fraction |
|---|---|---|
| Gauge vectors (12) | −5.867 | 93.1% |
| Weyl fermions (45) | −0.500 | 7.9% |
| Scalars (4) | +0.022 | −0.4% |
| Graviton (1) | +0.044 | −0.7% |
| Total | −6.300 |
Gauge vectors dominate γ_BH. This is because vectors have c/a = 0.58 < 1 (Euler dominates Weyl), so their γ_BH ≈ −0.49 per field is large and negative. Scalars and gravitons have c/a > 1, giving small positive contributions.
Comparison with other QG approaches
| Approach | γ_BH | Field-dependent? | |Δ| from framework | |----------|------|-----------------|-------------------| | This framework | −6.30 | YES | — | | LQG (microcanonical) | −1.50 | NO | 4.80 | | LQG (grand canonical) | −0.50 | NO | 5.80 | | Euclidean QG | −4.98 ± 1.0 | YES | 1.32 | | Induced gravity | −3.33 ± 0.5 | YES | 2.97 | | String theory (1/4-BPS) | −4.00 | NO | 2.30 | | Asymptotic safety | −5.0 ± 2.0 | YES | 1.30 |
The framework’s γ_BH = −6.30 is 4.2× larger than LQG’s −1.5. This is a qualitative, not just quantitative, difference.
Correlation slopes dγ_BH/dΩ_Λ (the unique signature)
As BSM fields are added, both Ω_Λ and γ_BH shift. The slope of the resulting curve in (Ω_Λ, γ_BH) space differs dramatically per spin:
| Field type | dΩ_Λ/dn | dγ_BH/dn | dγ/dΩ | Direction |
|---|---|---|---|---|
| Scalar | −0.00476 | +0.00556 | −1.2 | Toward obs (good) |
| Weyl fermion | −0.00736 | −0.01111 | +1.5 | Toward obs (good) |
| Gauge vector | +0.02741 | −0.48889 | −17.8 | Away from obs (kills) |
| Dirac fermion | −0.01472 | −0.02222 | +1.5 | Toward obs (good) |
Vectors create a dramatic bend in the (Ω_Λ, γ_BH) plane: they push Ω_Λ upward (away from observation) while making γ_BH much more negative. A single dark photon shifts Ω_Λ by +4.1σ and γ_BH by −0.49 — an unmistakable signature.
The Entanglement S-T Plot
See figures/st_plot.png. The framework’s prediction is a curve parameterized
by field content. Other QG approaches appear as horizontal lines (γ_BH independent
of Λ). The intersection of the framework curve with the observed Ω_Λ band uniquely
determines γ_BH = −6.30.
This is the framework’s most powerful discriminating feature: it correlates two observables that all other approaches treat as independent.
BSM scenarios
| Scenario | Ω_Λ | σ | γ_BH | Δγ_BH |
|---|---|---|---|---|
| SM only (no graviton) | 0.6646 | −2.76 | −6.344 | −0.044 |
| SM + graviton | 0.6877 | +0.42 | −6.300 | 0.000 |
| SM + grav + 1 axion | 0.6830 | −0.23 | −6.294 | +0.006 |
| SM + grav + sterile ν | 0.6805 | −0.58 | −6.311 | −0.011 |
| SM + grav + dark photon | 0.7147 | +4.11 | −6.789 | −0.489 |
| SM + grav + 4th gen | 0.5983 | −11.84 | −6.467 | −0.167 |
| MSSM + graviton | 0.4728 | −29.02 | −6.206 | +0.094 |
| Dirac neutrinos | 0.6667 | −2.47 | −6.333 | −0.033 |
Honest assessment
Strengths
- Unique correlation: No other framework connects Ω_Λ to γ_BH. This is a genuine, falsifiable, zero-parameter prediction.
- Exact fractions: γ_BH = −63/10 is an exact rational number from SM field content.
- Per-spin diagnostic: The slope dγ/dΩ differs by 15× between vectors and scalars, creating an unmistakable spin-dependent signature.
- LQG clearly distinguished: Factor 4.2× difference, with completely different physical origin (field-dependent vs universal).
Weaknesses
- γ_BH is not directly measurable with current technology. The log correction to BH entropy is a sub-leading quantum effect (~ln A vs A/4).
- Schwarzschild prediction uses Solodukhin’s formula (γ = −4a + 2c), which assumes the standard heat kernel on the Euclidean BH background. The framework’s lattice computation has only been done for de Sitter (spherical) geometry.
- The c coefficients (Weyl anomaly) have not been independently verified by the framework’s lattice methods — they are taken from standard QFT literature.
- Analog BH tests could in principle measure γ for a single scalar (γ = +0.006), but the predicted value is extremely small.
What this means
The (Ω_Λ, γ_BH) correlation is the framework’s cleanest unique prediction that differentiates it from all other QG approaches right now, even without new observations. It should be prominently featured in any paper claiming the framework makes distinguishing predictions. The visual (Figure 1) — curves vs horizontal lines — is the single most compelling argument for the framework’s uniqueness.
The main caveat is measurability: γ_BH requires quantum gravity phenomenology that is beyond current experimental reach. But the theoretical discriminating power is immediate — if LQG (or strings, or asymptotic safety) computes γ_BH = −1.5 (or −4), they cannot simultaneously explain Ω_Λ = 0.685 within this formalism. The two observables are locked together.