IndisputableMonolith.Cosmology.OmegaBaryon4
Cosmology certificate module packaging a non-negative domain cost, a positive canonical threshold, and an inhabited baryon-density certificate tied to the factor four. RS cosmologists cite it when closing the Ω_b ledger against the Cost and Constants layers. Structure is definitional: cost evaluation lemmas, positivity, and a trivial inhabited certificate record.
claimThe module defines a domain cost $C$, proves $C\ge 0$ and an evaluation identity, introduces a canonical threshold $\theta>0$, and supplies an inhabited certificate type asserting the Recognition Science baryon-density relation associated with the factor $4$.
background
Recognition Science cosmology sits on the same Cost layer used for the J-functional $J(x)=(x+x^{-1})/2-1$ and on Constants (RS-native units with time quantum $\tau_0=1$ tick). Domain cost here is the local cost assigned to the baryon sector; the canonical threshold is the positive cutoff against which that cost is compared when forming a certificate.
The module does not re-derive the forcing chain (T0–T8). It assumes the Cost and Constants imports and builds only the bookkeeping objects needed for an $\Omega_b$-style ledger entry labeled by the integer factor four.
Sibling content is elementary: evaluation of the domain cost at a point, non-negativity, positivity of the threshold, and a certificate record with an inhabited instance.
proof idea
Definition-first module. Domain cost and canonical threshold are introduced as defs; non-negativity and positivity are short lemmas over the Cost import. The certificate type is a structure (or Prop bundle) with a single inhabited instance, so downstream code can obtain a cert by cert_inhabited without further proof work. No deep tactic scripts; mostly rfl, positivity, and structure inhabitation.
why it matters in Recognition Science
Fills the cosmology niche for a baryon-density certificate keyed to the factor four, so later RS cosmology developments can quote a named cert rather than inline cost inequalities. Upstream edges are only Constants and Cost; the page lists no downstream consumers yet, so the module is a leaf ledger entry rather than a step inside T0–T8. It does not touch $\phi$-ladder mass formulae, eight-tick structure, or the $\alpha$ band; it only packages cost non-negativity and a threshold for $\Omega_b$-style accounting.
scope and limits
- Does not derive observational $\Omega_b$ or $\Omega_b h^2$ from data.
- Does not prove uniqueness of the factor-four baryon relation.
- Does not connect to T5–T8 forcing (J-uniqueness, $\phi$, eight-tick, $D=3$).
- Does not supply mass-ladder or $\alpha$-band bounds.
- Does not yet feed any recorded downstream theorem.