V2.167 - The Complete Theoretical Error Budget
V2.167: The Complete Theoretical Error Budget
Motivation
The cosmological constant prediction R = |delta|/(6*alpha) = 0.6855 matches the observed Omega_Lambda = 0.6847 +/- 0.0073 at 0.11sigma. But a prediction without error bars isn’t publishable. What is the theoretical uncertainty? Which input dominates? Is the agreement robust or fine-tuned?
This experiment produces the first complete error budget for the prediction, propagating all known sources of uncertainty to get:
R = 0.6855 +/- 0.0127 (theory) vs Omega_Lambda = 0.6847 +/- 0.0073 (Planck 2018)
Combined tension: 0.06sigma.
Method
Input Classification
Every input to R = |delta_total|/(f * n_eff * alpha_s) is classified as either exact (zero theoretical uncertainty) or uncertain (with quantified error):
Exact inputs (zero uncertainty):
| Input | Value | Why exact |
|---|---|---|
| delta_scalar | -1/90 | 1-loop exact, protected by Wess-Zumino consistency |
| delta_Weyl | -11/180 | Same |
| delta_vector | -31/45 | Same |
| delta_graviton | -61/45 | Same (EE prescription) |
| SM field content | 4+45+12 | Experimentally established |
| N_graviton | 9 | Derived from canonical gravity (V2.166) |
| f-factor | 6 | Exact from de Sitter thermodynamics |
Uncertain inputs:
| Input | Value | Uncertainty | Source |
|---|---|---|---|
| alpha_s | 0.02377 | +/-1.5% | Lattice (discretization + continuum extrapolation) |
| w_Weyl/w_scalar | 2.0 | +/-1% | Heat kernel (finite-cutoff residual) |
| w_vector/w_scalar | 2.0 | +/-1% | Same |
| alpha_grav/alpha_s | 1.0 | +/-5% | Never lattice-measured for spin-2 |
Sensitivity Analysis
Each uncertain input has an elasticity — the fractional change in R per fractional change in input:
| Input | Elasticity | Meaning |
|---|---|---|
| alpha_s | -1.000 | 1% increase in alpha_s -> 1% decrease in R |
| w_Weyl | -0.709 | 1% increase in Weyl weight -> 0.71% decrease in R |
| w_vector | -0.189 | 1% increase in vector weight -> 0.19% decrease in R |
| alpha_grav/alpha_s | -0.071 | 1% increase in graviton alpha -> 0.07% decrease in R |
The prediction is most sensitive to alpha_s (elasticity = -1, by construction) and second-most to the Weyl fermion weight ratio (because 45 Weyl fermions dominate n_eff).
Higher-Order Corrections
Five categories of corrections were estimated:
| Correction | Size | Status |
|---|---|---|
| Curvature (de Sitter vs flat) | ~10^{-120} | NEGLIGIBLE (120 orders below observable) |
| 2-loop anomaly (interactions) | ~0.5% | NEGLIGIBLE (subdominant to alpha_s) |
| Non-perturbative (instantons) | ~exp(-3948) | NEGLIGIBLE (super-exponentially suppressed) |
| alpha ratio (continuum extrapolation) | ~1% | INCLUDED (residual from Dirac-to-scalar ratio) |
| Threshold corrections (mass running) | 0 | NEGLIGIBLE (all SM fields lighter than M_P) |
4 of 5 corrections are negligible. Only the alpha ratio continuum extrapolation uncertainty contributes to the error budget.
Results
1. The Error Budget
| Source | delta_R | % of variance |
|---|---|---|
| alpha_s (lattice systematic) | 0.0103 | 65.9% |
| Higher-order corrections | 0.0049 | 14.7% |
| w_Weyl (fermion weight ratio) | 0.0049 | 14.7% |
| alpha_grav/alpha_s (graviton mode) | 0.0024 | 3.7% |
| w_vector (vector weight ratio) | 0.0013 | 1.0% |
| TOTAL | 0.0127 | 100% |
The lattice value of alpha_s dominates the error budget at 66% of variance. This is the single most important quantity to improve for tightening the prediction.
2. The Headline Result
PREDICTION: R = 0.6855 +/- 0.0127 (theory, 1.85%)
OBSERVATION: Omega_Lambda = 0.6847 +/- 0.0073 (Planck 2018, 1.07%)
TENSION: 0.06sigma (combined), 0.11sigma (obs only)The theoretical uncertainty (1.85%) is 1.7x the observational uncertainty (1.07%). The prediction is currently theory-limited, not observation-limited. The combined tension of 0.06sigma represents essentially perfect agreement.
3. Prediction Band
The prediction band [0.6729, 0.6982] overlaps 57.6% with the observational 1sigma band [0.6774, 0.6920]. The observation lies well within the theory band.
4. Inverse Prediction: alpha_s from Cosmology
Inverting the formula — assuming the framework is correct and using the observed Omega_Lambda — gives a cosmological measurement of alpha_s:
alpha_s = 0.02380 +/- 0.00025 (from cosmology, +/-1.07%)
alpha_s = 0.02377 (from lattice, Lohmayer-Neuberger 2012)
Difference: 0.000029
Tension: 0.11sigmaThe cosmological and lattice values agree at 0.11sigma. The cosmological precision (+/-1.07%) is set by the Planck measurement of Omega_Lambda.
The 1sigma allowed range for alpha_s is [0.02355, 0.02405] — a 2.14% window. The lattice value falls squarely within this range.
5. Stress Tests
Scanning alpha_s +/-10% from the lattice value:
| alpha_s deviation | R | Tension |
|---|---|---|
| -10% | 0.7617 | 10.6sigma |
| -6% | 0.7293 | 6.1sigma |
| -2% | 0.6995 | 2.0sigma |
| 0% (lattice) | 0.6855 | 0.11sigma |
| +2% | 0.6721 | 1.7sigma |
| +4% | 0.6592 | 3.5sigma |
| +10% | 0.6232 | 8.4sigma |
The prediction requires alpha_s to be within ~2% of 0.02377 to match observation at 1sigma. This is a non-trivial constraint — the prediction could easily fail if the lattice value were wrong by even 5%.
The exact-match value is alpha_s = 0.02380, only 0.12% above the lattice value.
6. BSM Field Budget
How many additional BSM fields can exist before the prediction is excluded at 3sigma?
| Field type | delta_R per field | Direction | Max fields (3sigma) |
|---|---|---|---|
| Scalar | -0.0048 | decreases R | 4 |
| Weyl fermion | -0.0073 | decreases R | 3 |
| Gauge vector | +0.0268 | increases R | 0 |
Zero additional gauge vectors are allowed at 3sigma. This is because vectors increase R (the SM+graviton already gives 0.6855, and any increase pushes away from observation). Scalars and fermions decrease R, so a few are tolerable (they push R away from Omega_obs in the other direction, but the SM+graviton starts very close).
What This Means for the Science
The prediction is robust
The theoretical uncertainty delta_R = 0.013 (1.85%) is small enough that the prediction is meaningful: even accounting for all known uncertainties, the agreement with observation is 0.06sigma. The prediction is not fine-tuned — it survives variation of every input within its uncertainty range.
The bottleneck is alpha_s
The lattice value alpha_s = 0.02377 accounts for 66% of the theoretical variance. A factor-of-2 improvement in the lattice determination (from 1.5% to 0.75% systematic) would halve the theoretical uncertainty. The other inputs (weight ratios, graviton alpha) are subdominant.
The prediction constrains BSM physics
The framework predicts that no additional gauge vectors can exist beyond the SM. This rules out, for example, an additional U(1) gauge boson at the 3sigma level. Scalars and fermions are more permissive (4 and 3 additional fields, respectively), but the budget is tight.
Cosmological alpha_s agrees with lattice
The framework can be inverted to “measure” alpha_s from the observed Omega_Lambda, giving alpha_s = 0.02380 +/- 0.00025. This is consistent with the lattice value (0.11sigma) and provides an independent determination with 1% precision. If future lattice calculations refine alpha_s and continue to agree, this strengthens both the lattice methods and the framework.
Honest Limitations
-
The alpha_s uncertainty estimate (1.5%) is itself uncertain. The internal consistency of the lattice extrapolation is much better (0.0034%, from V2.114), but systematic errors from discretization and finite volume are harder to quantify. The 1.5% is a conservative estimate based on typical lattice QFT systematics.
-
The weight ratio uncertainties (1%) are estimates, not computed from first principles. The heat-kernel prediction of exactly 2 for the Weyl-to-scalar and vector-to-scalar ratios is well-established for free fields, but interactions and regularization effects could shift these ratios at the percent level.
-
The graviton alpha_grav/alpha_s uncertainty (5%) is the least controlled. This has never been computed on the lattice for spin-2 fields. The assumption that each traceless metric mode contributes the same alpha_s as a scalar is motivated by the heat-kernel universality of the a_2 coefficient, but remains unverified for gravity.
-
The two-loop anomaly correction estimate (0.5%) is approximate. The actual correction depends on the SM coupling constants at the relevant scale, which involves the full renormalization group running. However, since the correction is subdominant to alpha_s, this does not affect the conclusion.
Key Numbers
| Quantity | Value |
|---|---|
| R (prediction) | 0.6855 +/- 0.0127 (theory) |
| Omega_Lambda (observation) | 0.6847 +/- 0.0073 (Planck 2018) |
| Combined tension | 0.06sigma |
| Dominant uncertainty source | alpha_s lattice systematic (66% of variance) |
| Theory/observation error ratio | 1.74 (theory-limited) |
| alpha_s from cosmology | 0.02380 +/- 0.00025 |
| alpha_s from lattice | 0.02377 |
| alpha_s consistency | 0.11sigma |
| Max additional vectors (3sigma) | 0 |
| Max additional scalars (3sigma) | 4 |
| Max additional fermions (3sigma) | 3 |
Tests
76 tests, all passing. Coverage: input catalog consistency, sensitivity analysis (partials, elasticities), correction bounds, error propagation, combined uncertainty, prediction band, inverse prediction, stress tests, BSM budget.