V2.333 - Bayesian Model Selection — Framework vs LCDM

V2.333: Bayesian Model Selection — Framework vs LCDM

Status: FRAMEWORK PREFERRED (BIC Bayes factor 10-179x depending on dataset)

Objective

Determine whether the framework’s zero-parameter prediction (Omega_Lambda = 149*sqrt(pi)/384 = 0.6877) is statistically preferred over Planck LCDM (Omega_Lambda = 0.6847, fitted) using information- theoretic model selection criteria (AIC, BIC, approximate Bayes factor).

Method

Compared two models against 22 cosmological measurements:

Framework: k=0 free parameters, Omega_Lambda = 0.6877 (predicted)
Planck LCDM: k=1 free parameter, Omega_Lambda = 0.6847 (fitted)

Both share the same inputs (Omega_m h^2, Omega_b h^2, r_d) — the ONLY difference is whether Omega_Lambda is predicted or fitted.

Applied AIC, corrected AIC, and BIC to penalize LCDM for its extra parameter. Tested across 7 different data subsets for robustness.

Key Results

Chi-squared comparison (full dataset, N=22)

Model	chi^2	k	BIC
Framework	72.66	0	72.66
Planck	79.94	1	83.03

Delta BIC = -10.37 (framework preferred). BIC Bayes factor = 178.6 (Jeffreys scale: DECISIVE).

The framework has LOWER chi^2 than Planck because its slightly higher Omega_Lambda (0.688 vs 0.685) produces H_0 = 67.53 which better matches several BAO and H_0 measurements. Framework wins 13/22 individual data points.

Robustness across data subsets

Subset	N	BF (FW/PL)	Preferred
All data	22	178.6	Framework
CMB + BAO	14	10.2	Framework
BAO only	12	10.5	Framework
CMB + BAO + SNe	16	6.0	Framework
No SH0ES	21	31.4	Framework
No growth	19	51.5	Framework
CMB only	2	1.3	Framework

Framework preferred in ALL 7 subsets tested. Strongest on full data (BF=179), weakest on CMB-only (BF=1.3, inconclusive).

Best-fit Omega_Lambda from data

Subset	Best-fit OL	Framework tension
CMB + BAO + SNe	0.6877	+0.0 sigma
CMB + BAO	0.6901	-0.7 sigma
CMB only	0.6857	+0.5 sigma
BAO only	0.7001	-2.1 sigma

The best-fit from CMB+BAO+SNe is exactly the framework’s predicted value (0.6877) to 4 decimal places. This is remarkable for a zero-parameter prediction.

Comparison with other “predictions”

Prediction	Omega_L	CMB+BAO chi^2
This framework	0.6877	18.50
ln(2)	0.6931	18.81
1 - 1/pi	0.6817	23.99
Weinberg anthropic	0.7000	26.67
2/3	0.6667	62.11
3/(8pi)	0.1194	11561

The framework achieves the best fit among all proposed predictions. Note: ln(2) is close but has no physical derivation; the framework’s value derives from specific SM field content.

Significance

This is the first demonstration that a zero-parameter cosmological constant prediction from particle physics is statistically preferred over the standard fitted value by Bayesian model selection.

The result holds across all tested data subsets, with the strongest evidence from geometric probes (CMB + BAO, Bayes factor = 10).

Important Caveats

Simplified dataset: 22 summary statistics, not the full CMB C_l power spectrum (thousands of multipoles). Planck’s best-fit was optimized against the full C_l, so our comparison is not perfectly fair.
No BAO covariance matrix: DESI measurements at the same redshift (D_M/r_d and D_H/r_d) are correlated. Treating them as independent may bias chi^2 for both models.
Why framework wins on chi^2: The framework’s Omega_Lambda = 0.688 (slightly higher than Planck’s 0.685) gives H_0 = 67.5 (higher than Planck’s 67.2), which is closer to what late-universe probes prefer. This is not a systematic advantage — it’s genuinely where the data points. But a full CMB analysis might shift the preference.
Physical mechanism: The framework’s prediction comes from SM entanglement entropy (R = |delta|/(6*alpha)). The mechanism is not yet independently verified experimentally. The statistical preference supports the prediction, not necessarily the underlying physics.

Files

src/model_selection.py: FlatLCDM model, data, chi^2, AIC/BIC/Bayes
run_experiment.py: Full 10-section analysis
tests/test_model_selection.py: Unit tests