V2.718 - Vacuum Energy Non-Gravitation — the Inverted Hierarchy
V2.718: Vacuum Energy Non-Gravitation — the Inverted Hierarchy
The Idea
The cosmological constant problem asks: why is Λ ~ 10^{-122} M_Pl^4? This experiment shows the question is ill-posed. In the framework, Λ is NOT the vacuum energy — it is the ratio of two entanglement entropy coefficients:
R = |δ_total|/(6·α_s·N_eff) = 149√π/384 = 0.6877
This ratio is O(1), has zero free parameters, and is UV-finite. The “hierarchy” of 10^{122} is simply (L_H/l_Pl)^4 — the ratio of the cosmic horizon area to the Planck area. That’s a fact about our universe’s size, not a fine-tuning problem.
Key Results
1. The exact formula decomposes cleanly
| Ingredient | Value | Magnitude | Origin |
|---|---|---|---|
| R = Ω_Λ | 0.6877 | O(1) | Ratio of O(1) QFT numbers |
| δ_total | 149/12 = 12.42 | ||
| α_s | 1/(24√π) = 0.0235 | O(0.02) | Entanglement density (universal) |
| N_eff | 128 | O(100) | SM components + graviton modes |
Every ingredient is O(1)–O(100). There is no hierarchy in the formula itself. The 10^{122} hierarchy is entirely in the conversion to Planck units — specifically in the factor n^4 = (L_H/l_Pl)^4.
2. Sector decomposition of |δ_total| = 149/12
| Sector | |δ| contribution | Fraction | N_eff | |--------|-----------------|----------|-------| | Gauge bosons (12 vectors) | 8.267 | 66.6% | 24 | | Fermions (45 Weyl) | 2.750 | 22.1% | 90 | | Graviton (10 modes) | 1.356 | 10.9% | 10 | | Higgs (4 scalars) | 0.044 | 0.4% | 4 |
Gauge bosons dominate the numerator (66.6%); fermions dominate the denominator (90/128 = 70.3%). This imbalance gives R ≈ 0.69 rather than ~1.
3. Lattice verification: R converges as cutoff → ∞
| C | α_s (lattice) | dev(α) | R = |δ_exact|/(6α·128) | dev(R) | |---|-------------|--------|---------------------|--------| | 2 | 0.01561 | −33.6% | 1.036 | +50.6% | | 3 | 0.01871 | −20.4% | 0.864 | +25.7% | | 4 | 0.02031 | −13.6% | 0.796 | +15.7% | | 5 | 0.02123 | −9.7% | 0.761 | +10.7% | | 6 | 0.02180 | −7.3% | 0.742 | +7.8% | | ∞ (V2.609) | 0.02351 | −0.01% | 0.6877 | +0.01% |
The key insight: δ is exact (it’s a topological invariant from the trace anomaly). Only α_s needs lattice computation, and it converges to 1/(24√π) as C→∞ (confirmed to 0.011% in V2.609). Therefore R converges to 149√π/384 = 0.6877.
4. The “hierarchy” is just universe size
| n = L/l_Pl | ρ_vac ~ α·n² | Λ ~ |δ|·ln(n)/n² | ρ_vac/Λ | |-----------|-------------|----------------|---------| | 10 | 2.35 | 0.286 | 8.2 | | 10³ | 2.4×10⁴ | 8.6×10⁻⁵ | 2.7×10⁸ | | 10⁵ | 2.4×10⁸ | 1.4×10⁻⁸ | 1.6×10¹⁶ | | 10^{30.5} | ~10^{59} | ~10^{-63} | ~10^{122} |
The ratio ρ_vac/Λ grows as n⁴/ln(n). At n ~ 10^{30.5} (the cosmic horizon in Planck units), this gives ~10^{122} — the “cosmological constant problem.” But R = |δ|/(6α·N_eff) = 0.69 at every scale.
5. Why vacuum energy doesn’t gravitate
Traditional assumption: Λ = 8πG·ρ_vac → “what cancels ρ_vac to 122 digits?”
Framework answer: Λ is NOT sourced by ρ_vac. In entanglement entropy S(n) = α·n² + δ·ln(n) + γ:
- The area-law term (α·n²) determines G (Newton’s constant) — UV-divergent, extensive
- The log correction (δ·ln(n)) determines Λ (cosmological constant) — UV-finite, topological
- These are different terms in the entropy. They map to different gravitational constants.
No fine-tuning is needed because Λ was never ρ_vac in the first place.
6. Comparison with other approaches
| Approach | Free params for Ω_Λ | Explains hierarchy? | Falsifiable? |
|---|---|---|---|
| ΛCDM | 1 | No (fits, doesn’t explain) | No |
| SUSY | ≥100 | Partially (still needs tuning) | Partially |
| Anthropic | 0 predictions | ”Explains” everything | No |
| Quintessence | ≥2 | Trades one mystery for another | Partially |
| This framework | 0 | Dissolves the question | Yes |
Honest Assessment
Strengths:
- The formula R = 149√π/384 = 0.6877 has zero free parameters and matches Planck at +0.4σ
- The “hierarchy” is cleanly identified as (L_H/l_Pl)^4 — a unit choice, not physics
- All ingredients are standard QFT quantities (trace anomalies, entanglement density)
- The lattice confirms α_s convergence; δ is exact by the trace anomaly theorem
Weaknesses:
- The claim that “the area-law term maps to G and the log term maps to Λ” is the framework’s central assumption, not a derived result. It’s physically motivated (Clausius relation, QNEC) but not proven from first principles.
- The lattice α_s convergence is slow (~1/C), requiring C > 40 for sub-percent precision. This is a practical limitation, not a fundamental one.
- The argument doesn’t explain WHY ρ_vac doesn’t gravitate — it simply constructs a formula for Λ that doesn’t involve ρ_vac. A complete theory would derive this from a deeper principle.
- The n^4/ln(n) scaling of the hierarchy assumes ρ_vac scales as the area law (n²), which is true for a free field but may receive corrections from interactions (though V2.248 showed these are sub-percent).
What this does NOT claim:
- We do NOT claim to have solved the cosmological constant problem in the traditional sense (explaining why ρ_vac doesn’t gravitate)
- We claim to have DISSOLVED it: the framework produces Λ from a different formula entirely, making the ρ_vac question irrelevant
- Whether this dissolution is satisfying depends on whether the framework’s derivation chain (entanglement → Clausius → Einstein) is accepted
Files
src/vacuum_hierarchy.py: Core computations (lattice entropy, hierarchy analysis, formula decomposition)tests/test_vacuum_hierarchy.py: 7 validation tests (all pass)run_experiment.py: Full analysis with tables and diagramsresults.json: Machine-readable results