IndisputableMonolith.Cosmology.RS_Cosmo_Module_010
Cosmology certificate module that packages a nonnegative domain cost built from the RS J-cost and a strictly positive canonical threshold. It exposes equality and positivity lemmas plus an inhabited certificate record RSCosmo010Cert. Cosmology and forcing-chain consumers cite it when they need a named, checkable cost-threshold pair rather than ad-hoc constants. The module is mostly definitions and short algebraic lemmas over Cost and Constants.
claimDefine a domain cost $C$ on the RS cost structure (nonnegative, with a pointwise equality form) and a canonical threshold $\theta>0$. Package them into an inhabited certificate record asserting $C\ge 0$ and $\theta>0$ for use in RS cosmology arguments.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the T5 step of the unified forcing chain. The Cost import supplies that functional and its elementary inequalities; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the cosmology layer. It does not re-derive $J$; it introduces a domain-level cost functional built from that infrastructure, records that the cost is nonnegative, and fixes a strictly positive canonical threshold used as a comparison scale in later cosmological claims.
Sibling declarations name the cost, its evaluation identity, nonnegativity, the threshold and its positivity, and a certificate bundle RSCosmo010Cert with an inhabited instance so downstream files can assume a concrete witness without reconstructing the data.
proof idea
Definition-heavy module. The cost and threshold are introduced as defs; nonnegativity and positivity are short lemmas discharging against the Cost library and elementary arithmetic. The certificate is a structure packing those facts; inhabitation is a one-line constructor application. No deep tactic scripts or multi-step forcing arguments live here.
why it matters in Recognition Science
Gives cosmology a single named cost-threshold certificate instead of scattered local constants. Downstream cosmology developments that need a nonnegative domain cost and a positive comparison scale can import the inhabited certificate rather than re-proving positivity. In the broader RS picture this is bookkeeping under the J-cost and tick conventions (Cost, Constants), not a new forcing step: it does not claim T5–T8, the eight-tick octave, or $D=3$. With empty used_by in the current graph it is a leaf packaging module ready for later cosmology theorems to cite.
scope and limits
- Does not derive J-uniqueness, phi, eight-tick structure, or D=3.
- Does not state observational cosmology fits or numerical alpha/mass predictions.
- Does not define dynamics, FLRW metrics, or horizon equations.
- Does not claim the threshold equals any particular physical constant beyond positivity.
- Does not discharge sorry-bearing scaffolding elsewhere in the monolith.