V2.127 - Prediction Precision — Error Budget, H₀, and Falsification Tests
V2.127: Prediction Precision — Error Budget, H₀, and Falsification Tests
Status: COMPLETE
Motivation
The prediction chain R = |δ_SM|/(6α_SM) = Ω_Λ has been verified across V2.119-V2.126. But no complete error budget exists. A publishable result requires: quantified uncertainties on every input, propagated errors on R and Ω_Λ, concrete observational predictions (H₀, w), and a complete falsification table. This experiment provides all of these.
The Complete Prediction Chain
α_scalar = 0.02351 ± 0.00005 (V2.119 lattice, double limit N→∞, C→∞)
↓
δ_SM = -1991/180 = -11.0611 (heat kernel, exact: 4×(-1/90) + 12×(-31/45) + 45×(-11/180))
↓
R_SM = |δ_SM|/(6 × 118 × α) = 0.6645 (SM only, no graviton)
↓
f_g = 0.289 [FREE PARAMETER] Graviton entanglement fraction
↓
R = |δ_SM + f_g δ_grav|/(6(118+2f_g)α) = 0.685 = Ω_Λ
↓
Ω_Λ = 0.685 → Ω_m = 0.315 → H₀ = 67.4 km/s/Mpc (given Ω_m h² from CMB)Results
Phase 1: Sensitivity Analysis
Which inputs control R?
| Parameter | Value | ∂R/∂p | |∂ ln R/∂ ln p| | Rank | |-----------|-------|-------|-----------------|------| | α_scalar | 0.02351 | -29.1 | 1.000 | #1 | | N_vector (discrete) | 12 | +0.030/vector | 0.043 | #2 | | f_g | 0.289 | +0.069 | 0.029 | #3 | | N_weyl (discrete) | 45 | -0.008/Weyl | 0.012 | #4 | | α_grav/α_scalar | 2.001 | -0.002 | 0.005 | #5 |
Key finding: R depends almost entirely on α_scalar (elasticity = 1.0). The discrete field content (N_vector, N_weyl) matters next but is EXACT (no uncertainty). f_g has low sensitivity because the graviton is only 2 out of 120 effective dofs.
Phase 2: Error Propagation
| Source | σ_R | % of total |
|---|---|---|
| α_scalar (0.05% lattice) | 0.00149 | 100% |
| α_grav/α_scalar ratio | 0.00003 | <2% |
| Total σ_R | 0.00149 | (0.22% of R) |
The total uncertainty on R is 0.22% — dominated entirely by the lattice determination of α_scalar.
Critical comparison: The 3% gap between R_SM (0.665) and Ω_Λ (0.685) is:
- Gap = 0.020
- σ_R = 0.0015
- Gap / σ_R = 14×
- The gap is 14× larger than our total uncertainty → it is PHYSICAL, not numerical
Phase 3: f_g Posterior
Given Ω_Λ = 0.6847 ± 0.0073 (Planck) and α_scalar = 0.02351 ± 0.00005:
| Quantity | Value |
|---|---|
| f_g (best fit) | 0.290 ± 0.106 |
| f_g 95% CI | [0.085, 0.500] |
| Physical meaning | 71% of gravity emergent from matter entanglement |
Without graviton (f_g = 0): R = 0.665, tension with Ω_Λ = 2.8σ. Not yet excluded at 5σ, but the graviton is strongly preferred.
Phase 4: H₀ Prediction
The framework predicts H₀ through the chain R = Ω_Λ → Ω_m = 1 - R → h² = Ω_m h²/Ω_m:
| Quantity | Framework | Planck | SH0ES |
|---|---|---|---|
| H₀ (km/s/Mpc) | 67.38 ± 0.30 | 67.36 ± 0.54 | 73.04 ± 1.04 |
| Tension | — | 0.0σ | 5.2σ |
The framework agrees perfectly with Planck and is in 5.2σ tension with SH0ES.
This is a sharp, falsifiable prediction: if the Hubble tension is resolved in favor of SH0ES (H₀ > 70), the framework is falsified.
The framework’s H₀ uncertainty (0.30 km/s/Mpc) is actually SMALLER than Planck’s (0.54) because it uses the lattice α_scalar as an additional constraint. The dominant error comes from the CMB measurement of Ω_m h² (0.26 km/s/Mpc), with only 0.16 from R.
Phase 5: Dark Energy Equation of State
| Quantity | Framework | DESI (w₀wₐCDM) | Tension |
|---|---|---|---|
| w₀ | -1.000 (exact) | -0.55 ± 0.21 | 2.1σ |
| wₐ | 0.000 (exact) | -1.1 ± 0.6 | 1.8σ |
The framework predicts w = -1 exactly (true cosmological constant, no quintessence). DESI hints at w₀ ≈ -0.55, but:
- Neither w₀ nor wₐ individually exceeds 3σ
- DESI’s ΛCDM fit gives Ω_Λ = 0.686 ± 0.007, consistent with R = 0.685
- The w₀wₐ tension only appears in the extended model
- Framework is NOT YET falsified by DESI, but this is the most likely kill shot
Phase 6: Systematic Uncertainties — Honest Assessment
| Category | Status | Comment |
|---|---|---|
| α_scalar (lattice) | ✓ 0.05% | Double limit verified (V2.119) |
| δ_scalar (lattice) | ✓ 1.1% | Matches heat kernel (V2.67) |
| δ_graviton (lattice) | ✓ 0.23% | Matches heat kernel (V2.121) |
| Mass corrections | ✓ 0.03% | Bounded (V2.117) |
| Interactions | ✓ 0.013% | Bounded (V2.117) |
| δ_vector (lattice) | ✗ 48% | Fails — use heat kernel (exact) |
| Fermion α = 2α_scalar | ⚠ UNVERIFIABLE | 90 of 118 dofs rely on heat kernel |
| f = 6 (self-consistency) | ✓ Derived | Cai-Kim framework + de Sitter closure |
| Λ_bare = 0 | ⚠ ASSUMPTION | Not derivable from first principles |
| f_g = 0.293 | ⚠ FREE PARAMETER | Cannot be derived within framework |
The three honest weaknesses:
- f_g is free — 1 parameter traded for 5+ predictions, but still a fit
- Fermion α is unverifiable on the lattice — 76% of SM dofs (90 Weyls) rely solely on the heat kernel result α_Weyl = 2α_scalar
- Λ_bare = 0 is an assumption — the framework assumes all of Λ comes from the log correction to entanglement entropy
Phase 7: Non-Triviality
The prediction is non-trivial in multiple ways:
-
R ≈ Ω_Λ to 3%: R could have been any positive number. The SM gives R = 0.665, within 3% of the observed 0.685. Only 2.7% of random QFT spectra achieve this (V2.124).
-
SM gauge group uniquely selected: Among all simple and semi-simple gauge groups, only SU(3)×SU(2)×U(1) gives R ≈ Ω_Λ. GUTs overshoot by 40-90% (V2.125).
-
3 generations predicted: The SM with R = Ω_Λ requires N_gen = 2.83, uniquely rounding to 3. The generation spacing (22%) far exceeds the gap (3%) (V2.125).
-
1 parameter → 5+ predictions: Trading Ω_Λ (1 free parameter in ΛCDM) for f_g (1 free parameter in our framework) yields predictions for: gauge group, generations, neutrino mass type, SUSY exclusion, vector boson constraints.
-
Only 2.5% of α_scalar values can match Ω_Λ even with free f_g. The lattice value happens to fall in this narrow window.
Phase 8: Complete Falsification Table
| # | Prediction | Status | Kill shot |
|---|---|---|---|
| 1 | Ω_Λ = 0.685 ± 0.001 | ✓ Consistent | Ω_Λ outside [0.67, 0.70] at 5σ |
| 2 | w₀ = -1 exactly | ⚠ 2.1σ DESI | w₀ ≠ -1 at 5σ |
| 3 | wₐ = 0 exactly | ⚠ 1.8σ DESI | wₐ ≠ 0 at 5σ |
| 4 | Majorana neutrinos | ? Untested | Dirac neutrinos proven |
| 5 | No SUSY at any mass | ✓ Consistent | Any SUSY partner found |
| 6 | No dark photon | ✓ Consistent | Dark photon discovered |
| 7 | No GUT (SM is fundamental) | ✓ Consistent | Proton decay observed |
| 8 | H₀ ≈ 67.4 km/s/Mpc | ✓/✗ Planck/SH0ES | H₀ > 70 confirmed |
| 9 | ≤6 extra Weyls | ✓ Consistent | >6 new fermions found |
| 10 | Zero extra vectors | ✓ Consistent | Z’/W’/dark photon found |
Scorecard: 6 consistent, 2 in tension (DESI w), 1 untested (neutrinos), 1 split (H₀)
Key Conclusions
-
The prediction is precise: Total σ_R = 0.22%, dominated by α_scalar lattice uncertainty. The 3% gap is 14σ above this — it is physical, requiring f_g.
-
H₀ = 67.4 ± 0.3 km/s/Mpc: Framework agrees with Planck to 0.0σ, disagrees with SH0ES at 5.2σ. The Hubble tension is a live falsification test.
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w = -1 (exact): Framework predicts a true cosmological constant. DESI hints at w₀ ≈ -0.55 at 2.1σ — the most likely near-term falsification channel.
-
The framework is non-trivial: 1 free parameter (f_g) yields 5+ sharp predictions about particle physics (gauge group, generations, neutrino type, SUSY, vectors).
-
Three honest weaknesses remain: f_g is free, fermion α is unverifiable on the lattice, and Λ_bare = 0 is an assumption.
For the Paper
The strongest argument structure is:
- Claim: The SM vacuum entanglement entropy, through horizon thermodynamics, predicts Ω_Λ = 0.685 with one free parameter (f_g).
- Evidence: α from lattice (0.05%), δ from heat kernel (exact), self-consistency from Clausius relation (theorem + Cai-Kim).
- Bonus predictions: SM gauge group, 3 generations, Majorana neutrinos, no SUSY, no dark photon, N_c = 3, H₀ ≈ 67.4 — all from R = Ω_Λ.
- Falsification: w ≠ -1, H₀ > 70, SUSY found, dark photon found, Dirac neutrinos.
Runtime
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