V2.529 - Species-Dependence Curve — The Framework's Most Powerful Unique Prediction
V2.529: Species-Dependence Curve — The Framework’s Most Powerful Unique Prediction
Status: KEY RESULT — Lambda is a calculable function of particle content; axion improves fit
The Unique Prediction
In ΛCDM, the cosmological constant Λ is a free parameter with no connection to particle physics. In this framework:
Λ = |δ_total| / (2·α_s·N_eff·L_H²)
where δ_total and N_eff are completely determined by the Standard Model field content. Adding ANY new fundamental field shifts Λ in a calculable, spin-dependent way. This is the single most powerful unique prediction: no other framework connects Lambda to particle content.
The Species-Dependence Formula
For the SM + graviton baseline:
- δ_total = -149/12 = -12.4167 (one-loop exact, Adler-Bardeen)
- N_eff = 128 (4 scalars×1 + 45 Weyl×2 + 12 vectors×2 + 10 graviton)
- R = 0.6877, Λ/Λ_obs = 1.004, σ = +0.41
Adding n fields of spin s shifts both δ and N_eff:
- R(n) = |δ_base + n·δ_s| / (6·α_s·(N_eff_base + n·n_comp_s))
The Continuous Curves
Per-spin sensitivity at SM+graviton baseline
| Spin | δ per field | n_comp | dR/dn | Direction | Fields per σ | Sweet spot |
|---|---|---|---|---|---|---|
| Scalar | -1/90 | 1 | -0.00476 | ↓ toward obs | 1.5 | n = 0.63 |
| Weyl | -11/180 | 2 | -0.00736 | ↓ toward obs | 1.0 | n = 0.41 |
| Dirac | -11/90 | 4 | -0.01472 | ↓ toward obs | 0.5 | n = 0.20 |
| Vector | -31/45 | 2 | +0.02741 | ↑ AWAY from obs | 0.3 | None |
Key finding: Scalars and fermions move R toward observation. Vectors move R away. This is because vectors have large |δ| relative to n_comp, making the delta shift dominate the N_eff dilution.
The vector asymmetry is the smoking gun
The vector slope is POSITIVE and 5.8× steeper than the scalar slope. This means:
- Discovery of a new scalar: R decreases, prediction improves — SUPPORTING EVIDENCE
- Discovery of a new vector: R increases by +0.027 per field — POTENTIAL FALSIFICATION
A single new massless vector boson shifts R to 0.715 (+4.1σ), which is already excluded by Planck. This is the most constraining single-field prediction in the framework.
The Axion Sweet Spot
The baseline SM+graviton prediction sits at +0.41σ above Ω_Λ = 0.6847. Exactly 0.63 real scalars close this gap. A QCD axion (1 real scalar) gives:
| Configuration | R | Λ/Λ_obs | σ | Status |
|---|---|---|---|---|
| SM + graviton (baseline) | 0.6877 | 1.004 | +0.41 | OK |
| SM + graviton + 1 axion | 0.6830 | 0.998 | -0.24 | IMPROVED |
| SM + graviton + 2 scalars | 0.6783 | 0.991 | -0.87 | OK |
The observation sits exactly between the SM+grav and SM+grav+axion predictions. The axion improves the fit from +0.41σ to -0.24σ. This is not a prediction OF the axion — but if the axion exists, it is the one BSM particle that IMPROVES the framework’s agreement with data.
Dark Matter Candidate Landscape
| DM Candidate | R | Λ/Λ_obs | σ | Direction | Verdict |
|---|---|---|---|---|---|
| QCD axion (1 scalar) | 0.6830 | 0.998 | -0.24 | closer | OK |
| ALP/fuzzy DM (1 scalar) | 0.6830 | 0.998 | -0.24 | closer | OK |
| Scalar singlet (1 scalar) | 0.6830 | 0.998 | -0.24 | closer | OK |
| Primordial black holes | 0.6877 | 1.004 | +0.41 | baseline | OK |
| Majorana sterile ν (1 Weyl) | 0.6804 | 0.994 | -0.58 | farther | OK |
| Complex scalar (2 scalar) | 0.6783 | 0.991 | -0.87 | farther | OK |
| Scalar triplet (3 scalar) | 0.6737 | 0.984 | -1.50 | farther | OK |
| Dirac WIMP (1 Dirac) | 0.6734 | 0.984 | -1.55 | farther | OK |
| Inert Higgs doublet (4 scalar) | 0.6692 | 0.977 | -2.12 | farther | tension |
| Dark photon (1 vector) | 0.7147 | 1.044 | +4.11 | farther | EXCLUDED |
| Dark SU(2) (3 vectors) | 0.7662 | 1.119 | +11.2 | farther | KILLED |
| Dark SU(3) (8 vectors) | 0.8826 | 1.289 | +27.1 | farther | KILLED |
The DM Hierarchy
The framework creates a natural hierarchy of dark matter candidates:
- Preferred: Light scalars (axion, ALP) — improve the fit
- Neutral: PBH — no shift, no new fields
- Allowed: Singlet fermions, small scalar multiplets (< 3 fields)
- In tension: 4+ scalars (inert doublet)
- Excluded: Any new vector boson, any non-abelian gauge sector
This is a falsifiable DM prediction: if DM is discovered to be a dark photon or part of a hidden gauge sector, the framework is killed.
Generation Count: Why Three?
| N_gen | δ_total | N_eff | R | σ from obs |
|---|---|---|---|---|
| 1 | -10.583 | 68 | 1.103 | +57σ |
| 2 | -11.500 | 98 | 0.832 | +20σ |
| 3 | -12.417 | 128 | 0.688 | +0.4σ |
| 4 | -13.333 | 158 | 0.598 | -12σ |
| 5 | -14.250 | 188 | 0.537 | -20σ |
N_gen = 3 is uniquely selected. The nearest competitor (N=2 at 20σ, N=4 at 12σ) is devastatingly excluded. No other framework predicts the generation count from cosmology.
Majorana vs Dirac
| Type | R | σ | Λ/Λ_obs |
|---|---|---|---|
| Majorana | 0.6877 | +0.41 | 1.004 |
| Dirac | 0.6666 | -2.48 | 0.974 |
Majorana neutrinos are preferred by ~2.9σ (|0.41| vs |2.48|). Dirac neutrinos add 3 right-handed Weyl fermions, shifting R too far below observation.
Joint (Λ, γ_BH) Prediction
The same δ_total determines both Lambda AND the BH entropy log correction γ_BH, but with different sensitivity:
| BSM addition | Δγ_BH (%) | ΔR (%) | γ more sensitive? |
|---|---|---|---|
| +1 axion | -0.09% | -0.69% | No (R wins 8×) |
| +1 dark photon | -5.55% | +3.92% | Yes (γ wins 1.4×) |
| +1 Dirac WIMP | -0.98% | -2.08% | No (R wins 2×) |
| +4th generation | -7.38% | -13.01% | No (R wins 1.8×) |
For most BSM scenarios, Λ is the more sensitive discriminator. But for vectors, γ_BH is competitive — providing a second, independent check if BH entropy corrections are ever measured.
Experimental Discrimination Timeline
| Experiment | σ(Ω_Λ) | SM+grav vs obs | +axion vs obs | SM+grav vs +axion | SM+grav vs +dark photon |
|---|---|---|---|---|---|
| Planck 2018 | 0.0073 | 0.4σ | 0.2σ | 0.6σ | 3.7σ |
| DESI Y3 + Planck | 0.0050 | 0.6σ | 0.3σ | 0.9σ | 5.4σ |
| DESI Y5 + Planck | 0.0035 | 0.9σ | 0.5σ | 1.3σ | 7.7σ |
| CMB-S4 + DESI Y5 | 0.0020 | 1.5σ | 0.9σ | 2.4σ | 13.5σ |
| CMB-S4 + Euclid | 0.0015 | 2.0σ | 1.2σ | 3.1σ | 18.0σ |
CMB-S4 + Euclid can distinguish SM+grav from SM+grav+axion at 3.1σ. This is a concrete, scheduled experiment (2030s) that can distinguish between the two most important scenarios for this framework.
The Falsification Protocol
The framework provides a precise falsification protocol for any particle discovery:
- New particle discovered (e.g., at LHC, in direct detection, or via cosmological signals)
- Identify its spin (scalar, fermion, or vector)
- Count its field DOF (real scalars, Weyl fermions, vector bosons)
- Compute the shift: ΔR = n × (dR/dn) for that spin type
- Check against observation: If the new R exceeds the 3σ band [0.663, 0.706], framework falsified
Specific kill conditions:
- ≥1 new massless vector: killed at >3σ
- ≥3 new Dirac fermions: killed at >3σ
- ≥5 new real scalars: killed at >3σ
- Any non-abelian hidden gauge sector: killed
- MSSM: killed at >38σ
Specific survival conditions:
- QCD axion (1 scalar): improved fit
- Light ALP (1 scalar): improved fit
- PBH dark matter: unchanged
- 1 sterile neutrino (Majorana): OK at 0.6σ
- No new fields at all: OK at 0.4σ
Honest Assessment
What is genuinely new in this experiment
- Continuous curves: Previous work (V2.515) tabulated discrete candidates. This gives the full continuous dependence R(n) for each spin, revealing that scalars are special — they’re the only spin type that IMPROVES the fit.
- The axion sweet spot: The gap of +0.41σ is exactly what 0.63 scalars close. This wasn’t previously identified as a quantitative prediction.
- The DM landscape: Systematic mapping of every major DM candidate to the species-dependence curve, with explicit verdicts.
- The experimental timeline: When each scenario becomes distinguishable.
What this does NOT prove
- The framework does not PREDICT an axion — it says the fit improves if one exists
- The 0.41σ gap could be zero (within uncertainty on α_s and Ω_Λ)
- The species-dependence is only unique IF the framework’s derivation of Λ is correct
- Vectors shifting R upward rather than downward is a mathematical consequence, not a physical explanation
The fundamental limitation
All of this assumes the framework’s core identity R = |δ|/(6·α_s·N_eff) is correct. This identity has been verified on the lattice (R² = 0.9999990 for the QNEC form) and passes all cosmological consistency checks. But it remains a derived result, not a theorem from first principles. The species-dependence curve is only as strong as this identity.
What This Means for the Science
This is the framework’s most powerful unique prediction. No other approach to the cosmological constant connects Lambda to particle physics in this way. If a new particle is discovered, the framework makes an immediate, quantitative prediction for how Lambda should change. This is:
- Unique: ΛCDM has no such prediction (Lambda is a free parameter)
- Precise: Each field type gives a specific, calculable shift in R
- Falsifiable: A single new vector boson already falsifies at >4σ
- Connected: Links particle physics (LHC, direct detection) to cosmology (CMB, BAO)
- Surprising: The QCD axion — independently motivated by strong CP — actually improves the fit
The species-dependence curve is the framework’s fingerprint. It should be prominently displayed in any presentation of this work.
Files
src/species_dependence.py— Core computation (exact rational trace anomalies, continuous curves)tests/test_species.py— 14 tests (all passing)results.json— Full numerical resultsrun_experiment.py— Main driver