Self-Consistent Lambda from R(mu) Crossover
Experiment V2.89: Self-Consistent Lambda from R(mu) Crossover
Status: COMPLETE
Summary
The self-consistency ratio R(mu) = |delta(mu)|/(12alpha(mu)) varies with energy scale mu as SM species decouple below their mass thresholds. At high mu (all species active), R_UV = 0.329. At low mu (photon only), R_IR = 1.205. Since R must cross 1 somewhere, this experiment finds the crossover scale mu and asks: does H = mu* give the observed cosmological constant?
Key Result: Crossover Exists but Lambda Prediction Fails
R(mu) crosses 1 at mu* ~ 0.02 eV (near the neutrino mass scale), but the naive identification H = mu* gives Lambda/Lambda_obs ~ 10^63 — approximately 60 orders of magnitude too large.
Phase 1: R(mu) at Key Scales
| Scale | mu (eV) | R(mu) | Active species |
|---|---|---|---|
| Hubble | 1.5e-33 | 1.205 | 1 (photon) |
| Neutrino | 0.05 | 0.630 | 1 (photon + partial nu) |
| Electron | 5.11e5 | 0.336 | 4 |
| QCD | 2.0e8 | 0.276 | 26 |
| Top | 1.73e11 | 0.332 | 55 |
| Planck | 1.22e28 | 0.329 | 61 (all SM) |
R is non-monotonic: it dips to a minimum ~0.276 at the QCD scale (where 8 gluons activate, adding large alpha with relatively small delta), then rises back to ~0.329 at the top/Planck scale.
Phase 2: Crossover Analysis
| Suppression model | log10(mu*) | mu* (eV) | R at crossing |
|---|---|---|---|
| Exponential | -1.71 | 0.0195 | 1.000 |
| Boltzmann | -1.51 | 0.031 | 1.000 |
| Smooth step | -1.84 | 0.014 | 1.000 |
| Step function | -1.30 | 0.050 | 1.000 |
| Lattice | -3.56 | 2.8e-4 | 1.000 |
The crossover is robust: it exists for all suppression models, though the exact scale varies by ~2 decades (dominated by the lattice model outlier). The exponential, Boltzmann, and smooth-step models agree to within 0.5 decades.
Phase 3: Lambda Prediction
For the exponential model crossover at mu* = 0.0195 eV:
| Quantity | Value |
|---|---|
| H = mu*/M_Planck | 1.60e-30 (Planck units) |
| Lambda = 3*H^2 | 7.64e-60 (Planck units) |
| Lambda_obs | 1.1e-122 (Planck units) |
| Lambda/Lambda_obs | 6.95e62 |
| log10(Lambda/Lambda_obs) | 62.8 |
The predicted Lambda overshoots by ~63 orders of magnitude. The crossover scale (~0.02 eV, near neutrino masses) is enormously larger than the actual Hubble parameter (~1.5e-33 eV).
Phase 4: Physical Interpretation
The R(mu) = 1 crossover is physically meaningful: it marks the scale where the entanglement entropy exactly satisfies self-consistency. However, the naive identification mu = H (energy scale = expansion rate) fails catastrophically.
Possible resolutions:
- The relevant mu is not H but some other cosmological scale
- The mapping from mu to Lambda involves running of G or other couplings
- Self-consistency should be imposed at the Planck scale, not at R=1
- The factor-of-3 gap is an artifact of using free-field entropy coefficients
Phase 5: R(mu) Landscape
The R(mu) landscape reveals that the QCD phase transition is the dominant feature: gluon activation drives R down to 0.276, and the slow recovery through heavy quarks and W/Z/Higgs brings it back to 0.329. The neutrino-scale crossing at R=1 is a consequence of neutrino decoupling tipping the balance between delta (dominated by vectors at low mu) and alpha.
Files
| File | Description |
|---|---|
| src/species_database.py | SM species list, masses, alpha/delta per type, suppression functions |
| src/r_mu_crossover.py | R(mu) computation and crossover finder |
| src/lambda_prediction.py | Lambda prediction from crossover scale |
| tests/test_r_mu.py | 12 tests (all pass) |
| run_experiment.py | 5-phase driver |