IndisputableMonolith.Cosmology.OmegaMatter3_FromJCost
Cosmology module that packages the RS derivation of the matter density parameter from the J-cost functional. It defines a domain cost, a canonical positive threshold, and an inhabited certificate OmegaMatter3Cert asserting the forced value. Cosmologists working the RS forcing chain would cite the certificate when closing the Ω_m claim. The module is definition-and-certificate scaffolding over Constants and Cost; no deep tactic proof lives here.
claimDefine a nonnegative domain cost from the RS $J$-cost, a positive canonical threshold, and a certificate asserting that the cosmological matter density parameter $\Omega_m$ is forced to the RS value associated with three spatial dimensions (the $\Omega_m\sim 3$ claim) by that cost structure.
background
Recognition Science derives continuum physics from the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), fixed by the Recognition Composition Law and the T5 uniqueness step of the forcing chain. Cosmological density parameters are not free fits: they are read off as dimensionless ratios of recognition costs on the causal domain fixed by the eight-tick octave and $D=3$ (T7–T8).
This module sits in the Cosmology domain and imports only Constants (RS-native units, including the tick $\tau_0$) and Cost (the $J$-cost API). Sibling definitions introduce a domain cost functional, its evaluation identity and nonnegativity, a canonical positive threshold, and a certificate type OmegaMatter3Cert with an inhabited witness. The local claim is that the matter fraction is the value selected by comparing that domain cost to the threshold, not an independent FLRW parameter.
proof idea
Definition-and-certificate module, not a long tactic development. It introduces domainCost and records domainCost_at_eq plus domainCost_nonneg so the cost is a well-typed nonnegative real built from $J$. It defines canonicalThreshold with a positivity lemma. The payload is the structure OmegaMatter3Cert together with a concrete cert and cert_inhabited, packaging the equality (or bound) that identifies $\Omega_m$ with the RS three-dimensional cost ratio. Downstream users discharge cosmology goals by inhabiting or projecting that certificate rather than re-deriving the cost algebra.
why it matters in Recognition Science
In RS, $\Omega_m$ is supposed to be forced once $J$, $\varphi$, the eight-tick period, and $D=3$ are fixed; this module is the Cosmology-side packaging of that forcing for the matter sector (the name flags the dimension-three link). It does not itself appear in the supplied used-by list, so it is a leaf certificate module meant to be imported by higher cosmology assembly theorems (density closure, FLRW parameter tables, or observational band checks). It connects the Cost layer to a named cosmological observable and gives a single inhabitance point for the $\Omega_m$ claim instead of scattering cost inequalities through later files. Landmarks touched: T5 $J$-uniqueness, T8 $D=3$, and the general RS pattern that dimensionless cosmology is cost geometry rather than fitted $\Lambda$CDM knobs.
scope and limits
- Does not derive dark-energy density, curvature, or $H_0$; only the matter-sector certificate.
- Does not prove observational error bars or match to Planck posteriors.
- Does not re-prove $J$-uniqueness or $D=3$; those are imported upstream.
- Does not supply dynamics of structure formation or Boltzmann hierarchy.
- Does not claim a numeric Lean float for $\Omega_m$ beyond the certificate's stated relation.