IndisputableMonolith.Cosmology.OmegaLambdaDerivation
Derives the dark-energy density fraction Ω_Λ from eight-tick DFT mode counting plus a small positive electromagnetic correction. Cosmologists and RS auditors comparing the prediction to Planck cite the resulting strict band 0.683 < Ω_Λ < 0.686. The module fixes total and saturated phase-mode counts on the 8-tick cycle, forms their ratio, then shifts by a bounded EM term.
claimFrom the eight-tick DFT structure one fixes total and saturated phase-mode counts $N_{\mathrm{tot}}$ and $N_{\mathrm{sat}}$, sets $\Omega_{\mathrm{raw}} = N_{\mathrm{sat}}/N_{\mathrm{tot}}$, and applies a small positive electromagnetic correction $\varepsilon_{\mathrm{EM}}$ to obtain $\Omega_\Lambda = \Omega_{\mathrm{raw}} + \varepsilon_{\mathrm{EM}}$ with $0.683 < \Omega_\Lambda < 0.686$.
background
Recognition Science forces an eight-tick octave (forcing-chain T7): discrete time is organized in periods of length $2^3 = 8$, and phase modes of the associated DFT live on that cycle. This cosmology module imports only the RS constants layer and the quarantined external-anchor module used for empirical comparison; the cost-first core stays free of fitted cosmological parameters.
The raw dark-energy fraction is the ratio of saturated modes to total modes under tick addressing. A separate electromagnetic correction, proved positive and small, shifts that raw ratio into the narrow observational window. Downstream certificates then attach the band to the cosmological-constant problem and to Planck likelihood scoring.
proof idea
Definition-first module with supporting numeric inequalities. Total and saturated mode counts are fixed from the 8-tick DFT / tick-addressing structure; the raw ratio is defined from them, with an equality lemma and an evaluated numeric value. The EM correction is introduced as a positive quantity with an upper-bound lemma. Ω_Λ is the corrected sum; two comparison lemmas establish the strict bounds 0.683 < Ω_Λ < 0.686 by arithmetic on those definitions. No deep tactic development beyond the mode counts and the correction bounds.
why it matters in Recognition Science
Supplies the Ω_Λ band consumed by four parents: CosmologicalConstantDerivation (registry C-010: what fixes Λ, against the 10^120 QFT vacuum mismatch), Track4ACert (Track 4.A master certificate bundling the Ω_Λ band with η_B and Planck consistency), VacuumFluctuationStructural (Track 4.B structural address of the vacuum-fluctuation discrepancy), and OmegaLambdaPlanckLikelihood (dataset-specific likelihood certificate for the §7 Ω_Λ falsifier row). The mode count is the direct cosmology-side use of the eight-tick octave (T7). The narrow band is the RS-native target against which Planck data are scored.
scope and limits
- Does not derive absolute Λ in SI units; only the density fraction Ω_Λ.
- Does not claim a full QFT vacuum-energy cancellation beyond structural mode counting.
- Does not fit free cosmological parameters; the band is fixed by modes plus EM correction.
- Does not itself attach Planck likelihoods; that lives downstream.
- Does not treat matter or radiation fractions Ω_m, Ω_r.
used by (4)
depends on (2)
declarations in this module (24)
-
def
N_modes_total -
def
N_modes_saturated -
def
tick_addressing -
def
omega_raw -
theorem
omega_raw_eq -
theorem
omega_raw_val -
def
em_correction -
theorem
em_correction_small -
theorem
em_correction_pos2 -
def
omega_lambda -
theorem
omega_lambda_lt_686 -
theorem
omega_lambda_gt_683 -
theorem
omega_lambda_interval -
def
omega_lambda_planck2018 -
def
omega_lambda_planck_err -
theorem
rs_consistent_with_planck -
def
gray_code_flip_counts -
theorem
gray_flip_total -
theorem
q3_mode_count -
theorem
tick_addressing_is_power2 -
theorem
omega_lambda_one_measured_input -
theorem
omega_lambda_canonical_form -
structure
OmegaLambdaCert -
theorem
omegaLambdaCert