IndisputableMonolith.Cosmology.OmegaBaryon3_FromJCost
Module packaging a J-cost certificate for the baryon density parameter in RS cosmology. It defines a non-negative domain cost, a positive canonical threshold, and an inhabited OmegaBaryon3Cert that ties those quantities together. Cosmologists working the RS forcing chain would cite the certificate when closing the baryon sector from cost geometry rather than from free parameters. The argument is definitional plus elementary positivity lemmas imported from Cost and Constants.
claimThe module introduces a domain cost $C$ built from the RS $J$-cost, proves $C\ge 0$ and evaluates it on the equality locus, fixes a positive canonical threshold $\theta_*>0$, and packages an inhabited certificate $\mathrm{OmegaBaryon3Cert}$ asserting that the baryon density fraction is controlled by that threshold via $J$-cost geometry.
background
Recognition Science forces the cost functional $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) at T5 of the unified forcing chain; the Recognition Composition Law then constrains how costs compose under products and ratios. Cosmology modules import this $J$ together with the RS-native constants ($c=1$, $\hbar=\varphi^{-5}$, etc.) to replace free density parameters by cost thresholds.
This file sits in the Cosmology domain and pulls only Constants and Cost. Sibling definitions name a domain cost (non-negative, with an equality-case evaluation), a canonical positive threshold, and a certificate type OmegaBaryon3Cert together with an inhabited instance. The intended reading is that baryon loading is fixed by where the $J$-cost crosses that threshold on the recognition lattice, not by an external $\Omega_b$ knob.
proof idea
Definition module with light positivity scaffolding. domainCost is introduced from the imported $J$-cost; domainCost_nonneg and domainCost_at_eq are elementary consequences of Cost lemmas. canonicalThreshold is a fixed positive scalar (canonicalThreshold_pos). OmegaBaryon3Cert bundles these into a Prop-level certificate; cert and cert_inhabited supply a concrete witness so downstream cosmology can assume the baryon sector is closed under the $J$-cost threshold without re-deriving non-negativity.
why it matters in Recognition Science
Closes a named gap in the RS cosmology stack: baryon density as a derived $J$-cost threshold rather than a fit parameter. No downstream edges are recorded yet in the mirror graph, so the certificate is presently a leaf intended for later $\Omega_b$ assembly theorems and for comparison against the observed baryon fraction. It sits downstream of T5 $J$-uniqueness and the Cost library, and is consistent with the broader program that dimensionless cosmological ratios should emerge from $\varphi$-ladder and eight-tick structure once the cost geometry is fixed. Until parent theorems consume OmegaBaryon3Cert, the module functions as the formal interface for that claim.
scope and limits
- Does not derive a numerical value for $\Omega_b$ matching observation.
- Does not prove uniqueness of the canonical threshold from first principles.
- Does not connect the certificate to CMB, BBN, or large-scale structure data.
- Does not treat dark-matter or dark-energy density parameters.
- Does not discharge any sorry outside the listed positivity facts.