V2.403: Phase-Transition Invariance — Lambda Unchanged Through EW/QCD Transitions

The Problem

The cosmological constant problem has two parts:

1. Why is Λ so small? (our framework: R = |δ|/(6αN_eff) = 0.6877, matching Ω_Λ)
2. Why doesn’t Λ change during phase transitions? (the fine-tuning problem)

Part 2 is the harder challenge. The EW phase transition shifts the vacuum energy by ~(123 GeV)⁴ = 2.3 × 10⁸ GeV⁴. In the traditional picture, this requires cancellation against Λ_bare to 55 decimal places. Including zero-point energies, the total cancellation is 123 digits.

The Key Insight

In our framework, Λ is determined by:

R = |δ_total|/(6 α_s N_eff)

All three ingredients are UV-topological quantities, invariant under IR phase transitions:

Quantity	Value	Why invariant
δ_total = -149/12	Trace anomaly	One-loop exact (non-renormalization theorem). Mass-independent.
α_s = 1/(24√π)	Area coefficient	UV entanglement property. Physical masses satisfy m/M_Pl < 10⁻¹⁶.
N_eff = 128	DOF count	UV field content. Same fields above and below transitions.

Phase transitions change the vacuum energy ⟨ρ_vac⟩, but our Λ doesn’t depend on ⟨ρ_vac⟩. It depends on entanglement structure, which is UV and topological.

Results

R is exactly invariant through all phase transitions

Phase	Gauge Group	δ_total	N_eff	R	σ from Ω_Λ
Above EW	SU(3)×SU(2)×U(1)	-12.417	128	0.6877	+0.42
Below EW	SU(3)×U(1)_em	-12.417	128	0.6877	+0.42
Below QCD	SU(3)_confined×U(1)_em	-12.417	128	0.6877	+0.42

R spread = 0. Exactly zero, not approximately.

Why the field content is unchanged

• EW transition: The Higgs mechanism gives masses to W±, Z, and fermions. But it doesn’t remove any fields. The 3 Goldstone bosons become longitudinal W/Z modes — they’re still there, just in a different gauge. UV field content: 4 scalars + 45 Weyl fermions + 12 vectors = unchanged.

• QCD transition: Quarks confine into hadrons below Λ_QCD ≈ 300 MeV. But for the UV trace anomaly (computed in dim-reg at the UV scale), the relevant degrees of freedom are quarks and gluons, not hadrons. UV field content: unchanged.

Why physical masses don’t affect α_s

Physical SM masses in Planck units:

• m_W/M_Pl = 6.6 × 10⁻¹⁸
• m_top/M_Pl = 1.4 × 10⁻¹⁷
• m_Higgs/M_Pl = 1.0 × 10⁻¹⁷
• Λ_QCD/M_Pl = 2.5 × 10⁻²⁰

Mass corrections to α_s are O(m²/M_Pl²) ~ 10⁻³². The entanglement cutoff (Planck scale) is 17 orders of magnitude above any SM mass. Phase transitions are invisible to the UV entanglement structure.

Non-renormalization of δ

The trace anomaly coefficient is protected by:

• One-loop exactness: No higher-loop corrections (Duff 1994)
• Mass independence: δ is the same for m=0 and m≠0 (computed in dim-reg)
• Topological nature: Counts degrees of freedom via the a-coefficient
• Wess-Zumino consistency: Algebraic constraint prevents renormalization

References:

• Duff (1994) — Twenty years of the Weyl anomaly
• Deser & Schwimmer (1993) — Geometric classification of conformal anomalies
• Komargodski & Schwimmer (2011) — On RG flows in 4D (a-theorem proof)

Fine-tuning scorecard

	Traditional	Our Framework
Zero-point energy	1.8 × 10⁷⁶ GeV⁴	Not an input
EW transition	2.3 × 10⁸ GeV⁴	Doesn’t affect R
QCD transition	1.6 × 10⁻⁴ GeV⁴	Doesn’t affect R
Observed Λ	2.8 × 10⁻⁴⁷ GeV⁴	R = 0.6877
Cancellation required	123 digits	0 digits

Why This Matters

This experiment addresses the #1 objection to any CC solution: “What about phase transitions?”

The traditional CC problem assumes Λ = Λ_bare + ⟨ρ_vac⟩, so when ⟨ρ_vac⟩ changes by (100 GeV)⁴ during the EW transition, Λ_bare must be fine-tuned to compensate.

Our framework dissolves this problem:

• Λ is NOT determined by ⟨ρ_vac⟩
• Λ IS determined by entanglement structure: R = |δ|/(6αN_eff)
• δ, α, N_eff are all UV quantities, invariant under IR phase transitions
• Therefore no fine-tuning is needed — zero digits of cancellation

The resolution is conceptual: gravity couples to entanglement entropy, not energy density. The vacuum energy ⟨ρ_vac⟩ is real, but it doesn’t gravitate in the way the CC problem assumes. What gravitates is the entanglement structure encoded in {δ, α, N_eff}.

Lattice Verification

On the Srednicki lattice, mass deformation confirms the UV nature of the entanglement coefficients:

• For m² ≲ 0.001 (in lattice units), both α and δ are unchanged
• For m² ≳ 1 (lattice-scale masses), entropy decreases toward zero
• Physical SM masses correspond to m²_lattice ~ 10⁻³⁴ — utterly negligible

Connection to Other Experiments

• V2.250: QNEC completeness argument for Λ_bare = 0
• V2.256: Bisognano-Wichmann excludes Λ_bare slot in modular Hamiltonian
• V2.267: RG monotonicity — SM mass corrections to R are 3.3 × 10⁻³³
• V2.326: EW phase transition ΔΛ = 0 exactly (brief note, now fully demonstrated)
• V2.400: Interaction corrections close the 0.4σ gap (c₁ = O(1))

Honest Caveats

1. The lattice shows α and δ DO vary with mass at finite lattice size when m ~ O(1) in lattice units. This is expected: large mass freezes out the field. The physical point is that SM masses are m/M_Pl ~ 10⁻¹⁷, which is 10⁻³⁴ in lattice units — effectively zero.

2. “Gravity couples to entanglement, not energy” is a strong claim. It requires justification beyond this experiment. The QNEC derivation (V2.250) and modular Hamiltonian analysis (V2.256) provide this, but a complete proof would need a UV-complete theory of quantum gravity.

3. The non-renormalization theorem for δ is for the conformal anomaly in curved spacetime. Connecting this to the entanglement entropy log coefficient requires the Solodukhin–Fursaev correspondence, which is established for free fields but not rigorously proven for interacting theories.

4. We have not addressed dynamical mechanisms for why Λ_bare = 0 from first principles. V2.250 shows it’s QNEC-required, but the physical mechanism (if any) remains open.

Files

• src/phase_transition.py — SM field content through phases, vacuum energy shifts, fine-tuning
• tests/test_phase_transition.py — 31 tests, all passing
• run_experiment.py — 7-phase analysis
• results/summary.json — Numerical results

Status

COMPLETE — R = |δ|/(6αN_eff) = 0.6877 is exactly invariant through all SM phase transitions (EW, QCD). The traditional CC problem requires 123-digit fine-tuning; our framework requires 0. The resolution: Λ is determined by UV entanglement structure (δ, α, N_eff), not by vacuum energy ⟨ρ_vac⟩. Phase transitions change ⟨ρ_vac⟩ but cannot change the topological/UV quantities that determine Λ.