V2.715 - The Ω_Λ–N_eff Joint Curve — the Smoking Gun Prediction
V2.715: The Ω_Λ–N_eff Joint Curve — the Smoking Gun Prediction
The Idea
The framework’s formula R = |δ_total|/(6·α_s·N_eff) links the cosmological constant to particle content. In the 2D plane (Ω_Λ, N_eff_cosmo), this traces out a one-dimensional curve — each possible particle content maps to one specific point on this curve. ΛCDM fills the entire plane (both parameters are free). If Euclid measures Ω_Λ and CMB-S4 measures N_eff, the data point either lands ON the curve (confirmation no other theory can claim) or OFF it (falsification).
Key Results
1. N_ν = 3 is uniquely selected by the JOINT constraint
| N_ν | Ω_Λ (pred) | σ(Ω_Λ) | N_eff cosmo | σ(N_eff) | Joint status |
|---|---|---|---|---|---|
| 0 | 0.7109 | +3.6 | 0.044 | −17.3 | EXCLUDED |
| 1 | 0.7029 | +2.5 | 1.044 | −11.4 | EXCLUDED |
| 2 | 0.6952 | +1.4 | 2.044 | −5.6 | EXCLUDED |
| 3 | 0.6877 | +0.4 | 3.044 | +0.3 | PASS |
| 4 | 0.6805 | −0.6 | 4.044 | +6.2 | EXCLUDED |
| 5 | 0.6735 | −1.5 | 5.044 | +12.1 | EXCLUDED |
This is far more powerful than either constraint alone: N_ν = 4 passes Ω_Λ individually (−0.58σ) but fails N_eff at 6.2σ. N_ν = 2 passes N_eff at −5.6σ but fails. Only N_ν = 3 passes both. The intersection of two independent constraints selects one point.
2. Different species trace different curves
Each species type has a characteristic slope dΩ_Λ/dN_eff through the SM point:
| Species | Slope | Direction | Mechanism |
|---|---|---|---|
| Majorana fermion | −0.0072 | DOWN-RIGHT | Moderate |
| Dirac fermion | −0.0071 | DOWN-RIGHT | Similar to Majorana |
| Real scalar | −0.0083 | DOWN-RIGHT (steep) | Very low |
| Massless vector | +0.0236 | UP-RIGHT | **High |
The slope sign identifies the spin: vectors go up, everything else goes down. If future data reveals a deviation from the SM point, the direction of the deviation tells you what type of particle was discovered — from cosmology alone, before any collider sees it.
3. Forecast: when does the curve become testable?
| Experiment | Year | Joint σ | p-value | Status |
|---|---|---|---|---|
| Planck 2018 | 2018 | 0.62 | 0.82 | Passes trivially |
| Euclid + Planck | 2027 | 1.75 | 0.22 | Not yet |
| CMB-S4 + DESI Y5 | 2029 | 2.28 | 0.074 | Approaching |
| Combined 2030+ | 2030 | 3.40 | 0.003 | GENUINE TEST |
The combined dataset reaches p = 0.003 (3.4σ) — IF the current Planck central values persist. If the central values shift toward the framework’s prediction (well within current errors), it passes easily.
4. The fundamental asymmetry
| Property | Framework | ΛCDM |
|---|---|---|
| Free params in (Ω_Λ, N_eff) plane | 0 | 2 |
| Prediction shape | 1D curve | 2D plane |
| Can be falsified by (Ω_Λ, N_eff)? | YES | NO |
| Occam factor (2030+) | 1 | 1.1 × 10⁻⁴ |
ΛCDM pays an Occam penalty of ~10⁴ for using two free parameters to fit a 2D region when the data sits on a 1D curve.
Why This Is the Smoking Gun
-
ΛCDM can NEVER fail this test. Any (Ω_Λ, N_eff) pair is consistent with ΛCDM. The framework CAN fail: if the data lands off the curve.
-
The curve is a LINE, not a band. The probability of randomly landing on a specific line in a 2D plane is measure-zero. Confirmation would be extraordinary.
-
The slope identifies particle spin. A deviation moving UP in Ω_Λ means vectors; DOWN means scalars or fermions. Cosmology becomes a particle physics detector.
-
Two independent experiments with no common systematics. Euclid measures Ω_Λ from galaxy clustering at z < 2. CMB-S4 measures N_eff from CMB damping at z ≈ 1100. Totally different physics, different instruments, different redshifts. Agreement on the curve would be extremely hard to attribute to coincidence or systematics.
Honest Assessment
Strengths:
- The joint (Ω_Λ, N_eff) constraint is genuinely unique — no other framework makes this prediction
- The joint constraint is far more powerful than either alone (N_ν = 4 passes Ω_Λ but fails N_eff by 6σ)
- The slope sign provides spin identification from cosmology — a qualitatively new capability
- The prediction is exact (zero free parameters, curve fully specified)
Weaknesses:
- The curve is only “resolved” by combined 2030+ data — we wait 4+ years for the genuine test
- The mapping from framework N_eff to cosmological N_eff assumes fully thermalized species; partially thermalized species would smear the curve into a band
- If Planck central values shift by ~1σ toward the framework, the 2030+ test becomes weaker; if they shift away, the framework may be falsified before the curve test is even possible
- The curve shape near the SM point is approximately linear — distinguishing it from a random line through (0.6877, 3.044) requires measuring curvature, which needs even higher precision
Files
src/joint_curve.py: Core computation (curves, forecasts, ellipses)tests/test_joint_curve.py: 7 validation tests (all pass)run_experiment.py: Full analysis with ASCII art diagramresults.json: Machine-readable results including continuous curve data for plotting