V2.735 - Dual Observable BSM Exclusion Map — Two Predictions, Zero Parameters
V2.735: Dual Observable BSM Exclusion Map — Two Predictions, Zero Parameters
The Idea
The framework makes TWO independent predictions from the SAME trace anomaly coefficients {a_i, c_i}:
- Ω_Λ = 0.6877 — from the Euler channel (a coefficients) on FRW cosmology
- γ_BH = 8.996 — from Euler + Weyl channels (a + c) on Schwarzschild BH
These probe DIFFERENT linear combinations of the anomaly data. Any BSM model shifts BOTH predictions simultaneously, but in a direction determined by its spin content. The 2D constraint (Ω_Λ, γ_BH) is more powerful than either alone.
Why This Matters
No other quantum gravity approach connects dark energy to BH entropy through the same calculable coefficients. LQG predicts γ_BH = -3/2 universally (independent of matter content). This framework predicts γ_BH = 8.996 (matter-dependent). These are structurally incompatible.
Results
The Two Predictions
| Observable | Formula | SM+grav Value | Status |
|---|---|---|---|
| Ω_Λ | |Σ4na_i|/(6α_s N_eff) | 0.6877 | Measured: +0.4σ |
| γ_BH | Σ n_i(a_i + c_i) | 8.996 | To be measured |
| η = γ_BH/Σ(na) | Pure ratio | 2.898 | Convention-independent |
Per-Spin Enhancement Factors
The Weyl-to-Euler ratio differs dramatically by spin:
| Spin | a (Euler) | c (Weyl) | η = (a+c)/a |
|---|---|---|---|
| Scalar | 1/360 | 1/120 | 4.00 |
| Weyl fermion | 11/720 | 1/40 | 2.64 |
| Vector | 31/180 | 1/10 | 1.58 |
| Graviton | 61/180 | 53/15 | 11.43 |
Scalars are enhanced 4× in the BH channel; vectors only 1.6×. This means adding scalars vs vectors moves you in DIFFERENT DIRECTIONS in the 2D plane.
BSM Catalog — Dual Exclusion Map
| Model | R (Ω_Λ) | σ(Ω_Λ) | γ_BH | η | Δγ/γ_SM (%) | Verdict |
|---|---|---|---|---|---|---|
| SM + graviton | 0.6877 | +0.4 | 8.996 | 2.898 | 0.0 | ALLOWED |
| +1 axion | 0.6830 | -0.2 | 9.007 | 2.899 | +0.1 | ALLOWED |
| +1 sterile ν | 0.6805 | -0.6 | 9.036 | 2.897 | +0.4 | ALLOWED |
| +1 complex scalar | 0.6784 | -0.9 | 9.018 | 2.900 | +0.2 | ALLOWED |
| +1 dark photon | 0.7147 | +4.1 | 9.268 | 2.829 | +3.0 | DISFAVORED |
| 4th generation | 0.5983 | -11.8 | 9.600 | 2.880 | +6.7 | EXCLUDED |
| MSSM | 0.4673 | -29.8 | 10.218 | 2.925 | +13.6 | EXCLUDED |
| SU(5) GUT | 0.9311 | +33.8 | 12.329 | 2.377 | +37.1 | EXCLUDED |
Degeneracy Breaking
Among models within 2σ of Ω_Λ(obs), γ_BH still differs:
| Model | R | σ(Ω_Λ) | Δγ/γ_SM (%) |
|---|---|---|---|
| SM + graviton | 0.6877 | +0.4 | 0.0 |
| +1 axion | 0.6830 | -0.2 | +0.1 |
| +1 sterile ν | 0.6805 | -0.6 | +0.4 |
| +1 Dirac fermion | 0.6735 | -1.5 | +0.9 |
| +1 triplet scalar | 0.6738 | -1.5 | +0.4 |
Even models that are degenerate in Ω_Λ can be distinguished by γ_BH at the sub-percent level.
Trajectory Angles in the 2D Plane
Adding particles of different spin moves (R, γ_BH) in different directions:
| Added species | Trajectory angle |
|---|---|
| Scalar | 113° |
| Weyl fermion | 100° |
| Vector | 84° |
These are non-parallel — no BSM model can be hidden by adding offsetting fields of different spin. The 2D map is genuinely two-dimensional.
Measurement Precision Needed
| BSM model | Δγ/γ_SM (%) | Precision for 3σ |
|---|---|---|
| +1 axion | 0.12% | 0.04% |
| +1 sterile ν | 0.45% | 0.15% |
| +1 dark photon | 3.0% | 1.0% |
| 4th generation | 6.7% | 2.2% |
| MSSM | 13.6% | 4.5% |
| SU(5) GUT | 37.1% | 12.4% |
A 1% measurement of γ_BH would test all vector and multi-generation models. A 0.1% measurement would test single-particle additions.
Comparison with Quantum Gravity Approaches
| Approach | Ω_Λ | γ_BH | Connected? |
|---|---|---|---|
| This framework | 0.6877 (calculated) | 8.996 (matter-dep) | YES |
| LQG | Not predicted | -3/2 (universal) | NO |
| String theory | Landscape | Model-dependent | NO |
| Asymptotic Safety | In principle | Unknown | NO |
The framework is the ONLY approach where Ω_Λ and γ_BH are connected through calculable anomaly coefficients. LQG’s universal γ_BH = -3/2 is structurally incompatible with the framework’s matter-dependent prediction.
The Overconstrained System
The key power of the dual prediction:
- 2 independent observables (Ω_Λ, γ_BH)
- From 1 set of coefficients ({a_i, c_i} for each SM field)
- With 0 free parameters
- Probing different combinations (Euler vs Euler+Weyl)
If both agree with SM values → strong evidence for the framework. If either disagrees → framework falsified, and the PATTERN reveals the BSM content.
The enhancement factor η = 2.898 is a pure number:
- Convention-independent (doesn’t depend on α_s or N_eff)
- Sensitive to spin content (η = 4.0 for pure scalar, 1.58 for pure vector)
- A fingerprint that distinguishes this framework from all alternatives
Honest Assessment
Strengths:
- Two independent predictions from one theory, zero parameters
- The 2D constraint is genuinely more powerful than either alone
- Trajectory angles (84°–113°) show the map is two-dimensional
- Structural incompatibility with LQG is a clear distinguishing feature
- η = 2.898 is a calculable, convention-independent pure number
Weaknesses:
- γ_BH is not measurable with current technology
- The graviton’s Weyl anomaly coefficient c = 53/15 is uncertain for physical TT modes (Duff’s value for the full spin-2 field)
- The BH entropy log correction has convention issues across the literature
- The quantitative comparison with LQG’s -3/2 depends on normalization matching
- Among models that pass Ω_Λ, the γ_BH differences are small (0.1–0.9%)
What this establishes: The framework makes an overconstrained prediction: one set of SM anomaly coefficients must give BOTH the correct Ω_Λ AND the correct γ_BH. This is not a coincidence that can be engineered — it follows from the fact that the Euler and Weyl channels are independent linear combinations of {a_i, c_i}. The dual prediction is the framework’s strongest argument against the accusation of numerology.