V2.623 - EW Phase Transition Invariance — Ω_Λ Is the Same at Every Epoch
V2.623: EW Phase Transition Invariance — Ω_Λ Is the Same at Every Epoch
Status: CONFIRMED — δ and N_eff are exactly invariant across all SM phase transitions
The Problem
In standard QFT + gravity, the cosmological constant receives contributions from the vacuum energy of every field: Λ ~ Σ m_i⁴. At each phase transition, vacuum energies shift by ~ T_c⁴, requiring Λ_bare to re-cancel to extraordinary precision:
| Transition | ΔΛ_QFT/Λ_obs | Digits of re-tuning |
|---|---|---|
| Planck scale | ~10¹²³ | 123 |
| GUT scale | ~10¹¹¹ | 111 |
| EW scale (v = 246 GeV) | ~10⁹¹ | 91 |
| Top quark | ~10⁵⁵ | 55 |
| QCD confinement | ~10⁴⁴ | 44 |
This is the hierarchy of fine-tuning problems — not just one 122-digit cancellation, but 11 separate re-tunings at every mass threshold.
The Framework’s Resolution
The framework says Ω_Λ = |δ_total|/(6 α_s N_eff), where:
- δ_total is the Seeley-DeWitt a₂ trace anomaly coefficient
- N_eff counts field components for the area law
Both are UV quantities. They depend on the FIELD CONTENT, not on masses or coupling constants. Phase transitions rearrange IR physics but cannot change UV degrees of freedom.
Result: ΔΛ = 0 at every phase transition. Zero digits of tuning needed.
The Goldstone Absorption Identity
At the EW transition, 3 Goldstone bosons are “eaten” by W±, Z to become their longitudinal modes. The key identity:
δ(massive vector) = δ(massless vector) + δ(scalar)
-7/10 = -31/45 + (-1/90) ✓ (exact)
N_comp(massive V) = N_comp(massless V) + N_comp(scalar)
3 = 2 + 1 ✓ (exact)This means the Goldstone absorption is invisible to both δ and N_eff:
| Phase | Scalars | Massless V | Massive V | Weyl | Grav | δ_total | N_eff |
|---|---|---|---|---|---|---|---|
| Unbroken | 4 | 12 | 0 | 45 | 1 | -149/12 | 128 |
| Broken | 1 | 9 | 3 | 45 | 1 | -149/12 | 128 |
| Confined | 1 | 9 | 3 | 45 | 1 | -149/12 | 128 |
Verified: δ_total = -149/12 and N_eff = 128 in all three phases (exact rational equality).
Cosmic History: R = 0.687749 at Every Epoch
| Epoch | Temperature | g_* (thermal) | Ω_Λ (framework) |
|---|---|---|---|
| Planck era | 10¹⁹ GeV | — | 0.687749 |
| GUT scale | 10¹⁶ GeV | — | 0.687749 |
| EW (above) | 246 GeV | 106.75 | 0.687749 |
| EW (below) | 246 GeV | 86.25 | 0.687749 |
| QCD (above) | 200 MeV | 61.75 | 0.687749 |
| QCD (below) | 200 MeV | 17.25 | 0.687749 |
| BBN | 1 MeV | 10.75 | 0.687749 |
| Recombination | 0.3 eV | 3.36 | 0.687749 |
| Today | 2.7 K | 3.36 | 0.687749 |
g_ drops from 106.75 to 3.36 (32× variation) — Ω_Λ doesn’t budge.*
The contrast is sharp: g_* tracks IR degrees of freedom (which particles are relativistic), while Ω_Λ tracks UV degrees of freedom (trace anomaly). These are completely decoupled.
BSM Phase Transitions: Also Invariant
If a dark sector has its own symmetry breaking:
| Scenario | R (unbroken) | R (broken) | Invariant |
|---|---|---|---|
| Dark U(1) → massive dark photon | 0.705102 | 0.705102 | Yes |
| Dark SU(2) → 3 massive dark W | 0.746371 | 0.746371 | Yes |
The mechanism is universal: Goldstone equivalence preserves both δ and N_eff for ANY gauge symmetry breaking pattern.
Lattice Verification
Massive scalar δ on the Srednicki lattice (N=80, C=2.0):
| Mass | δ_weighted |
|---|---|
| 0.000 | -0.10663 |
| 0.010 | -0.10663 |
| 0.050 | -0.10663 |
| 0.100 | -0.10664 |
CV = 0.0% across masses. Confirms V2.622: δ is mass-independent on the lattice.
What This Means
The 56-Digit Problem Is Dissolved
The EW phase transition shifts vacuum energy by ~(88 GeV)⁴, creating a 10⁹¹ ratio with Λ_obs. In standard physics, this requires Λ_bare to cancel to 91 digits. The framework eliminates this entirely: the entanglement trace anomaly doesn’t see the phase transition, so ΔΛ = 0 exactly.
w = -1 at All Epochs
Since Ω_Λ is the same constant at every temperature, the dark energy equation of state is w = -1 exactly — not approximately, not asymptotically, but identically at every epoch. This is a sharp prediction for DESI.
No Cosmological Moduli Problem
In BSM scenarios with extra phase transitions (e.g., dark sector symmetry breaking), the framework predicts ΔΛ = 0 there too. There is no “cosmological moduli problem” from dark sector dynamics — Λ is immune to all IR rearrangements.
Connection to Other Results
- V2.619: n_grav = 10 gives N_eff = 128 (determines the denominator)
- V2.620: N_gen = 3 selected by R = 0.688 (determines the field content)
- V2.621: Exact formula R(N) = (116+11N)√π/(3(38+30N))
- V2.622: δ mass-independent to CV = 0.72% on lattice
- V2.326: EW transition ΔΛ = 0 (confirmed here with full bookkeeping)
This experiment completes the chain: the framework predicts a SINGLE value of Ω_Λ = 149√π/384 = 0.6877 that is invariant from the Planck epoch to today, consistent with Planck at +0.4σ, requires no fine-tuning at any of the 11 SM mass thresholds, and generalizes to arbitrary BSM symmetry breaking patterns.
Honest Assessment
This Is Not Surprising
The invariance follows from two well-established theorems:
- Adler-Bardeen non-renormalization of the trace anomaly
- Goldstone equivalence theorem
There is no dynamical content here — this is a bookkeeping verification. The REAL content is in the framework itself (that Λ comes from δ rather than from vacuum energy). This experiment verifies that the bookkeeping is self-consistent, which is necessary but not sufficient.
The Deep Question Remains
WHY does the trace anomaly (a UV quantity) determine the cosmological constant (an IR quantity)? This UV-IR connection is the framework’s central mystery. This experiment shows the connection is STABLE, but does not explain it.
Files
src/phase_transition.py: Full analysis (10 modules)tests/test_phase_transition.py: 34 tests, all passingresults.json: Complete numerical results