module module moderate

`IndisputableMonolith.Cosmology.PhaseSaturationVacuum`

Defines the vacuum energy density parameter from phase saturation, tying the measured fine-structure constant to a closed-form Ω_Λ and proving elementary positivity and unit-interval bounds. Cosmologists comparing RS vacuum predictions to ΛCDM cite the bounds package. The argument is definitional plus short real-arithmetic inequalities from the external α anchor.

claimFix the measured fine-structure constant $\alpha>0$ as the sole external input. Define the phase-saturation vacuum density parameter $\Omega_\Lambda$ (in RS-native units) from $\alpha$, and establish $0 < \Omega_\Lambda < 1$ together with the seed comparison $\alpha/\pi < \Omega_\Lambda^{\mathrm{seed}}$ used downstream.

background

Recognition Science quarantines empirical calibration in ExternalAnchors so the cost-first core stays free of CODATA. This cosmology module imports that anchor for one measured quantity: the fine-structure constant $\alpha$, treated as a positive real seed rather than a derived RS constant.

The Cost and Constants layers supply the RS-native units and J-cost infrastructure; DimensionForcing records that spatial $D=3$ is already forced (T8), so the vacuum energy is interpreted in three-space. Phase saturation here means the residual vacuum density left after the recognition ledger closes on the $\phi$-ladder, expressed as a dimensionless density parameter $\Omega_\Lambda$ comparable to the ΛCDM dark-energy fraction.

Sibling lemmas package the elementary analytic facts: positivity of $\alpha$ and of $\alpha/\pi$, $\alpha<1/2$, and the corresponding positivity and upper bounds for $\Omega_\Lambda$.

proof idea

Definition module with a thin inequality layer. $\alpha$ is bound to the external measured anchor; $\Omega_\Lambda$ is introduced by a closed-form definition in terms of that anchor (and the RS unit conventions). Positivity and comparison lemmas are short real-arithmetic tactics: positivity of the seed, division by $\pi$, and the standard $0<x<1$ rearrangements that feed later Friedmann and phase-transition estimates. No deep forcing or RCL algebra appears here.

why it matters in Recognition Science

Feeds IndisputableMonolith.Cosmology.EWPhaseTransition, which builds the electroweak transition temperature and radiation-era Hubble rate on the $\phi$-ladder and needs a controlled vacuum density seed. Downstream status is explicitly MODEL (RS-native-unit scaffold): this module supplies the $\Omega_\Lambda$ bounds that keep that scaffold inside a physically admissible interval.

In the broader framework it sits after T8 ($D=3$) and the external $\alpha$ band (inverse fine structure near $137.03$–$137.04$), converting one measured input into a vacuum fraction usable by cosmology without reopening the forcing chain. It does not derive $\alpha$ from RCL; it only packages the saturation map and its elementary bounds.

scope and limits

Does not derive α from the Recognition Composition Law or J-uniqueness.
Does not prove the observed cosmological constant from first principles alone.
Does not model structure formation, perturbations, or late-time acceleration dynamics.
Does not fix H0, matter densities, or full ΛCDM parameter fits.
Does not discharge the MODEL status of the electroweak phase-transition scaffold.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Cosmology.EWPhaseTransition module_import

depends on (4)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Constants.ExternalAnchors module_import
IndisputableMonolith.Cost module_import
IndisputableMonolith.Foundation.DimensionForcing module_import

declarations in this module (42)

def alpha L52
def Omega_Lambda L56
theorem Omega_Lambda_def L59
lemma alpha_pos_aux L62
lemma alpha_over_pi_pos L67
lemma alpha_lt_half L71
lemma alpha_pos_local L76
theorem alpha_over_pi_lt_seed L79
theorem Omega_Lambda_pos L85
theorem Omega_Lambda_lt_seed L90
theorem Omega_Lambda_lt_one L95
theorem Omega_Lambda_bounds L100
lemma alpha_over_pi_lt_tight L106
theorem Omega_Lambda_gt_05 L115
theorem Omega_Lambda_lt_069 L121
theorem Omega_Lambda_gt_068 L129
theorem Omega_Lambda_band_unconditional L143
def mode_budget L151
def active_modes L155
def passive_modes L159
theorem mode_budget_partition L162
theorem geometric_seed_eq L166
theorem mode_budget_from_D3 L170
theorem active_modes_eq L174
def vertex_ground_states L177
def unexcited_face_modes L180
theorem passive_mode_decomposition L183
theorem vertex_count_from_D3 L188
def Omega_matter L196
theorem omega_closure L199
theorem coincidence_ratio_structural L204
def equation_of_state L217
theorem w_is_minus_one L219
theorem no_dark_energy_evolution L222
def H_CosmicPhaseEquilibrium L242
theorem cosmic_phase_equilibrium_consistent L248
def H_ScaleInvariance L267
theorem scale_invariance_consistent L270
theorem no_vacuum_catastrophe L278
theorem vacuum_energy_is_mode_fraction L285
structure PhaseSaturationVacuumCert L292
theorem phase_saturation_vacuum_cert L302