Cosmological Constant Density Omega_Lambda = 11/16

1. Setting

The cosmological constant problem is usually a mismatch between vacuum energy estimates and observation. RS replaces that framing with a finite recognition-band count. The key number is 11/16: eleven passive Q3 channels out of sixteen recognition slots. The observational comparison is not exact Planck LCDM fitting, but the fact that the observed dark-energy fraction is near 0.685 while 11/16 = 0.6875.

2. Equations

(E1)

$$ \Omega_\Lambda^{\mathrm{RS}}=\frac{11}{16}=0.6875 $$

RS dark-energy density fraction.

(E2)

$$ \Delta\Omega_\Lambda = \Omega_\Lambda^{\mathrm{obs}}-\frac{11}{16} $$

Direct observational residual.

3. Prediction or structural target

• Omega_Lambda: predicted 11/16 = 0.6875 (dimensionless); empirical 0.6847 +/- 0.0073. Source: Planck 2018 TT,TE,EE+lowE+lensing+BAO

Current Planck-era values are consistent with 11/16 within the observational error bar.

4. Formal anchor

The primary anchor is Cosmology.CosmologicalConstant..

5. What is inside the Lean module

Key theorems:

• cosmological_constant_problem
• jcost_cancellation
• dark_energy_w
• coincidence_from_phi_ladder

Key definitions:

• lambda_observed
• rho_lambda_observed
• dark_energy_scale_eV
• hypothesis1
• t_universe
• hypothesis2
• lambda_exponent
• hypothesis3

6. Derivation chain

• Cosmology.CosmologicalConstant - Cosmological constant module
• Cosmology.CosmologicalConstantDerivation - Cosmological constant derivation
• Cosmology.OmegaLambdaBITKernelBand - Omega_Lambda BIT band
• Cosmology.VacuumUniformity - Vacuum uniformity

7. Falsifier

A future CMB+BAO+supernova consensus placing Omega_Lambda outside the RS band around 11/16, after accounting for curvature and Hubble-tension systematics, refutes the dark-energy count.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

9. Reading note

The minimal way to audit this page is to open the first Lean anchor and then walk the supporting declarations listed above. If the primary theorem is a module-level anchor, the key theorems section names the internal declarations that carry the mathematical load. This keeps the public derivation readable without severing it from the proof object.

10. Audit path

To audit omega-lambda-eleven-sixteenths, start with the primary Lean anchor Cosmology.CosmologicalConstant. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.