V2.447: Equation of State Test — w = -1 as a Theorem, Not an Assumption

Status: COMPLETE — Framework PREFERRED over w₀wₐCDM by Bayes factor 6:1

The Core Question

The framework predicts w = -1 EXACTLY, with zero dark energy parameters and no escape route. DESI Y1 finds w₀ = -0.727 ± 0.067, nominally 4.1σ from -1. Is the framework already dead?

Why w = -1 Is a Theorem

The proof in six steps:

1. δ = -4a (Solodukhin 2008): the entanglement log coefficient equals -4× the Euler anomaly
2. a is topological (Deser-Schwimmer): depends only on field content, not on couplings or masses
3. δ doesn’t evolve: the SM Lagrangian’s field content doesn’t change with cosmological time
4. α is UV-fixed: the area coefficient depends on field type, not on the state (V2.107: <0.1% variation)
5. Λ = |δ|/(2α·L_H²) is constant: both δ and α are time-independent constants
6. Constant Λ ⟹ w = -1 exactly: by definition, p_Λ = -ρ_Λ

This is NOT w ≈ -1. Not w = -1 ± ε. It is a mathematical consequence of topological protection. The framework has zero parameters to adjust.

Key Results

1. Bayesian Model Selection: Framework WINS

Model	χ²/12 bins	DE params	BIC	ΔBIC
Framework (w=-1)	17.05 (1.42/bin)	0	17.05	-3.59
Planck ΛCDM	18.92 (1.58/bin)	0	18.92	—
DESI w₀wₐCDM	15.68 (1.31/bin)	2	20.64	reference

The Occam razor verdict: w₀wₐCDM fits DESI better by Δχ² = 1.4. But it costs 2 parameters. The Occam penalty (+5.0 in BIC for 12 data points) exceeds the fit improvement. The framework is preferred by ΔBIC = -3.6, corresponding to a Bayes factor of 6:1.

2. Where Does the Signal Live?

The framework-vs-w₀wₐ separation is largest at:

Bin	z	Separation (σ)
LRG3+ELG1 D_M	0.93	1.60
BGS D_V	0.295	1.48
LRG1 D_M	0.51	1.43
LRG2 D_M	0.71	1.34

No single bin exceeds 2σ separation between the models. The w₀wₐ preference is distributed, not localized — consistent with a statistical fluctuation.

3. The w(z) Divergence

z	w (Framework)	w (DESI fit)	Difference
0.0	-1.000	-0.727	0.273
0.5	-1.000	-1.070	0.070
1.0	-1.000	-1.252	0.252
2.0	-1.000	-1.427	0.427

The DESI best fit crosses w = -1 at z ≈ 0.4 and plunges to w < -1.4 at high z (phantom dark energy). This requires violating the null energy condition — a strong theoretical prior against it.

4. Observable Differences

• Distance modulus: max |Δμ| = 0.059 mag at z ≈ 0.46 (1.5σ with Pantheon+ errors)
• Growth rate: Δf/f = 3.4% at z = 0.38, 1.0% at z = 0.7
• BAO distances: 1-2σ differences across bins

5. DESI Y3/Y5 Forecasts

If the framework is correct (true cosmology is w = -1):

Release	Bayes factor (FW:w₀wₐ)	Evidence
Y1 (now)	6:1	Moderate for FW
Y3 (2027)	10⁸:1	Decisive for FW
Y5 (2028)	10¹²:1	Overwhelming

If the DESI Y1 trend persists (true cosmology is w₀wₐCDM):

Release	Bayes factor (w₀wₐ:FW)	Evidence
Y1 (now)	1:6 (FW still wins)	FW preferred
Y3 (2027)	1:2 (inconclusive)	Dead heat
Y5 (2028)	3:1 for w₀wₐ	Barely significant
Y5+Euclid	81:1 for w₀wₐ	Strong against FW

Critical insight: even if the DESI Y1 trend is REAL, the evidence against the framework doesn’t become decisive until Y5+Euclid, because the Occam penalty for 2 extra parameters is severe with only 12 data points.

6. The Verdict Year

The crossover point — where BIC shifts from favoring the framework to favoring w₀wₐCDM (if the trend persists) — is at DESI Y4 (error reduction factor ~2).

• Y1-Y3: Framework preferred even if DESI trend holds
• Y4: Crossover (inconclusive)
• Y5: Mild preference for w₀wₐ if trend holds
• Y5+Euclid: Decisive either way

Honest Assessment

What favors the framework

1. Bayesian model selection currently prefers w = -1 (6:1 Bayes factor)
2. The w₀wₐ fit improvement (Δχ² = 1.4) is tiny compared to the parameter cost
3. The DESI w₀wₐ best fit requires phantom dark energy (w < -1), which violates NEC
4. No single DESI bin shows >2σ separation between models
5. The framework’s w = -1 is a theorem, not a fit — if confirmed, it’s the most successful zero-parameter cosmological prediction in history

What threatens the framework

1. DESI w₀ = -0.727 is 4.1σ from -1 in the w₀ parameter alone
2. The chi-squared FIT is better for w₀wₐ (by 1.4), just not enough to overcome Occam
3. If DESI Y3 confirms the trend with smaller errors, the evidence will shift
4. The framework has NO escape route — w = -1 is non-negotiable
5. Multiple probes (BAO + SN) both hint at w ≠ -1

What this experiment adds beyond V2.435/V2.436

• Bayesian model selection between framework and w₀wₐCDM (not just chi-squared)
• Occam factor analysis: quantifies exactly how much the 0-parameter advantage helps
• Verdict year: identifies DESI Y4 as the crossover point
• w(z) comparison: shows the DESI fit requires phantom crossing, a theoretical red flag
• Observable differences: quantifies the signal in distance modulus and growth rate

The Physics

The deepest point: in the framework, w = -1 follows from the SAME topological protection that makes the trace anomaly one-loop exact (Adler-Bardeen). This is the same protection that prevents the cosmological constant from receiving radiative corrections — which is the resolution of the cosmological constant PROBLEM.

If w ≠ -1 is confirmed, it means either:

1. The trace anomaly IS dynamical (breaks topological protection) → catastrophic for QFT
2. Dark energy is NOT entanglement entropy → framework falsified
3. A new field contributes at late times → needs to also match Ω_Λ species dependence

Option 3 is the only one that could save the framework in principle, but it would require a very specific new field that evolves w away from -1 while keeping Ω_Λ near 0.6877. This is highly constrained by V2.446’s species-dependence curve.

Files

• src/w_eos_test.py — Full analysis engine
• tests/test_w_eos.py — 15 tests, all passing
• run_experiment.py — 8-phase experiment
• results.json — Machine-readable output