V2.161 - Precision Error Budget — Is the Ω_Λ Match Statistically Significant?
V2.161: Precision Error Budget — Is the Ω_Λ Match Statistically Significant?
Status: Complete Date: 2026-03-03 Depends on: V2.156-160
Abstract
The prediction Ω_Λ = |δ_SM|/(6α_SM) achieves a 0.1σ match when the graviton (N=9) is included, but the theoretical “σ” itself was never rigorously established. The key parameter α_scalar = 0.02377 had no stated error bar. This experiment constructs the first rigorous error budget by propagating all parameter uncertainties (α_scalar, Weyl ratio, vector ratio) through Monte Carlo simulation, estimating interaction corrections from SM gauge couplings at the Planck scale, and performing Bayesian model comparison.
The central finding: After accounting for all uncertainties, the SM-only prediction is 1.6σ from observation (not 3.8σ as naively computed). The graviton and dark photon scenarios match at 0.1σ with Bayes factor 3.3 over SM-only — moderate but not decisive evidence. The prediction connects to the conformal a-anomaly via Ω_Λ = 2a_SM/(3α_SM), linking the cosmological constant to the deepest invariant in QFT.
The Error Budget Problem
Prior experiments quoted the SM-only prediction as “3.8σ from observation” using only the observational uncertainty σ_obs = 0.0073. But the prediction itself has theoretical uncertainties from three parameters:
| Parameter | Value | Uncertainty | Source | % of variance |
|---|---|---|---|---|
| α_scalar | 0.02377 | ±0.00050 (2.1%) | V2.74 lattice | 76.6% |
| r_Weyl = α_W/α_s | 2.00 | ±0.03 (1.5%) | Heat kernel; V2.157 measured 1.97 | 22.7% |
| r_vector = α_V/α_s | 2.00 | ±0.02 (1.0%) | Heat kernel; V2.95 measured 2.005 | 0.7% |
The α_scalar uncertainty dominates the error budget (77% of variance) because R ∝ 1/α_scalar exactly — the elasticity is -1, meaning a 1% shift in α_scalar produces a 1% shift in R.
The uncertainty σ(α_scalar) = 0.00050 is estimated from the V2.156 stress test finding that “α₀ can vary by ±2% before breaking at 2σ.” This is the weakest link in the entire prediction chain.
Results
1. Monte Carlo Error Propagation (10⁶ samples)
Sampling all parameters from Gaussian distributions:
| Scenario | R ± σ_R | Combined deviation | P(within 1σ_obs) | P(within 2σ_obs) |
|---|---|---|---|---|
| SM (HK ratios) | 0.658 ± 0.016 | -1.6σ | 9.0% | 20.7% |
| SM (measured ratios) | 0.665 ± 0.016 | -1.1σ | 16.6% | 33.4% |
| SM+grav(9) (HK) | 0.686 ± 0.016 | +0.1σ | 34.8% | 63.2% |
| SM+grav(9) (measured) | 0.693 ± 0.016 | +0.5σ | 31.3% | 58.8% |
| SM+dark photon | 0.687 ± 0.017 | +0.1σ | 34.3% | 62.4% |
Key shift: The SM-only discrepancy drops from 3.8σ (obs-only error) to 1.6σ (combined theory+obs error). This is still disfavored but far less dramatically — it is within 2σ.
2. Analytic Sensitivity
R has exact elasticity -1 to α_scalar:
| Parameter | ∂R/∂p | Elasticity | To close SM gap (ΔR = +0.027) |
|---|---|---|---|
| α_scalar | -27.7 | -1.000 | Δα₀ = -0.00099 (-4.2%) |
| r_Weyl | -0.251 | -0.763 | Δr_W = -0.109 |
| r_vector | -0.067 | -0.203 | Δr_V = -0.411 |
The SM gap could be closed by either: (a) α_scalar being 4% lower than the lattice value, or (b) r_Weyl being ~1.89 instead of 2.0.
3. Critical Parameter Values
What α_scalar would make each scenario match observation exactly?
| Scenario | α₀* | Shift from 0.02377 | Distance from current |
|---|---|---|---|
| SM | 0.02282 | -4.0% | 1.9σ |
| SM+grav(9) | 0.02380 | +0.1% | 0.1σ |
| SM+dark photon | 0.02383 | +0.3% | 0.1σ |
For SM-only: the critical α₀ = 0.02282 is 1.9σ from the current value. This is uncomfortably close to being within error bars — SM-only cannot be firmly excluded based on the current α_scalar uncertainty.
For SM+grav(9): the critical α₀ = 0.02380 is 0.1σ from the current value. The match is essentially exact with no parameter adjustment needed.
4. Interaction Corrections to α at the Planck Scale
SM gauge couplings at the Planck scale (2-loop RG evolution):
| Coupling | Value at M_P |
|---|---|
| α_s (QCD) | 0.0187 |
| α_2 (SU(2)) | 0.0337 |
| α_1 (U(1)) | 0.0170 |
| y_t (top Yukawa) | 0.382 |
Interaction corrections to α (Hertzberg 2013: interactions reduce entanglement):
| Correction | Δα/α | Physical origin |
|---|---|---|
| QCD → quarks | -0.20% | C₂(3)=4/3, α_s at M_P |
| SU(2) → LH fermions | -0.20% | C₂(2)=3/4, α₂ at M_P |
| QCD → gluons | -0.45% | C₂(adj)=3, α_s at M_P |
| SU(2) → W/Z | -0.54% | C₂(adj)=2, α₂ at M_P |
| Top Yukawa → top+Higgs | -0.09% | y_t² at M_P |
| Total | -0.30% |
Net effect: Interactions reduce α by 0.3%, increasing R from 0.6573 to 0.6593 (+0.3%).
This moves R toward observation but closes only 7% of the SM gap. To close the full gap through interaction corrections alone would require the geometric factor c_geom ≈ 14 (an order of magnitude beyond the expected O(1) value).
Interaction corrections are real but insufficient to close the gap. They are a systematic effect that should be included in future precision estimates, but they do not change the qualitative picture.
5. Bayesian Model Comparison
| Model | Evidence Z | Bayes Factor vs SM | Interpretation |
|---|---|---|---|
| SM | 0.12 | 1.0 | Reference |
| SM+grav(9) | 0.41 | 3.3 | Moderate preference |
| SM+dark photon | 0.41 | 3.3 | Moderate preference |
On the Jeffreys scale, a Bayes factor of 3.3 is “moderate” evidence — noteworthy but not conclusive. The data moderately prefer the graviton or dark photon extension, but cannot decisively distinguish them from SM-only given the theoretical uncertainties.
6. Precision Requirements
To distinguish SM from SM+grav(9) at various significance levels:
| Significance | Required σ(α₀) | Current σ(α₀) | Improvement needed |
|---|---|---|---|
| 2σ | 0.000513 | 0.00050 | 1.0× (marginal now) |
| 3σ | 0.000341 | 0.00050 | 1.5× |
| 5σ | 0.000205 | 0.00050 | 2.4× |
A modest 1.5× improvement in α_scalar precision would allow 3σ discrimination between SM and SM+graviton. This is achievable with improved lattice calculations at larger volumes.
To distinguish SM+grav(9) from SM+dark_photon at 3σ would require 42× better precision — effectively impossible with current methods, since these scenarios differ by only ΔR = 0.001.
7. The a-Anomaly Connection
The prediction can be reformulated as:
Ω_Λ = 2a_SM / (3α_SM)
where a_SM is the conformal a-anomaly coefficient:
| Species | a per field | δ = -4a | Verification |
|---|---|---|---|
| Scalar | 1/360 | -1/90 | ✓ |
| Weyl | 11/720 | -11/180 | ✓ |
| Vector | 31/180 | -62/90 | ✓ |
SM total: a_SM = 2.765, c_SM = 2.358, a/c = 1.173
This reformulation is significant because:
- The a-theorem (Komargodski-Schwimmer 2011): a_UV ≥ a_IR under any RG flow. This is the deepest proved result in 4D QFT.
- The cosmological constant prediction is therefore linked to the UV conformal anomaly, which counts effective degrees of freedom and can only decrease under RG flow.
- Since a is evaluated at the UV fixed point (Planck scale), the prediction uses the maximum possible value of a — making R an upper bound within the free-field approximation.
8. What the V2.157 Measured Ratio Changes
Using the measured r_Weyl = 1.97 (instead of heat-kernel 2.0):
- SM prediction shifts from 0.657 to 0.665 (closing 25% of the gap)
- SM+grav(9) shifts from 0.686 to 0.693 (overshooting by +1.2%)
- The SM+grav(9) match is slightly degraded but still within 0.5σ
This means: if the measured Weyl ratio is correct, the graviton contribution of exactly N=9 is slightly too much. The continuous best-fit would be N_grav ≈ 8.5 — still consistent with the traceless metric argument (N=9) within errors.
Honest Assessment
What this experiment shows
-
The SM-only prediction is at 1.6σ, not 3.8σ: After properly accounting for theoretical uncertainties on α_scalar and spin ratios, the SM-only prediction is within 2σ of observation. The claim of a “3.8σ discrepancy” was an artifact of ignoring theoretical errors.
-
Graviton/dark photon moderately preferred but not required: The Bayes factor of 3.3 is noteworthy but not decisive. A skeptic can legitimately argue that SM-only is consistent with data.
-
The α_scalar uncertainty is the bottleneck: 77% of the theoretical variance comes from uncertainty in a single lattice number. Improving this by 1.5× would settle the SM vs SM+graviton question at 3σ.
-
Interaction corrections are small but real: SM gauge couplings at M_P reduce α by ~0.3%, which shifts R by +0.002. This is a systematic effect that has been neglected in all prior analyses but is too small to qualitatively change the picture.
What a skeptic would say
“The match to observation is within 2σ even without the graviton — there is no statistically significant evidence that the prediction works better than SM-only. The claim that ‘graviton(N=9) matches to 0.1σ’ is misleading because the theoretical error bar (±0.016) is twice the observational one (±0.007). You’re matching a noisy prediction to a precise observation.”
This is a fair criticism. The remedy is to reduce the theoretical uncertainty, primarily by improving the lattice determination of α_scalar.
What would strengthen this further
- Lattice α_scalar at L → ∞: A dedicated high-precision lattice calculation of α_scalar with explicit continuum-limit extrapolation and controlled systematics would sharpen the prediction by 2-3×.
- Lattice verification of interaction corrections: Compute α for an interacting scalar (e.g., φ⁴ theory at weak coupling) and verify the perturbative estimate.
- Graviton α on the lattice: A direct lattice computation of α for linearized gravity would settle the N=9 question without phenomenological arguments.
Files
| File | Description |
|---|---|
| src/error_budget.py | Sensitivity analysis, Monte Carlo, interaction corrections, Bayesian comparison |
| tests/test_error_budget.py | 43 tests (all pass) |
| run_experiment.py | 10-phase experiment driver |
| results/results.json | Full numerical results |
Tests
All 43 tests pass across 11 test classes:
- Exact values (3), Compute R (7), Sensitivity (5), Error propagation (4)
- Monte Carlo (3), Critical values (4), Interaction corrections (5)
- Bayesian (3), Precision requirements (2), a-anomaly (4), Full budget (3)