IndisputableMonolith.Cosmology.MatterPert4
Module packaging a four-domain matter-perturbation certificate for RS cosmology: a nonnegative domain cost functional, its evaluation identity, a positive canonical threshold, and an inhabited certificate record. Cosmologists citing RS structure-formation bounds use the cert bundle. Content is definitional plus elementary positivity and equality lemmas over the Cost and Constants imports.
claimThe module defines a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at equality cases, introduces a canonical threshold $\theta_{\mathrm{can}}>0$, and packages these into an inhabited matter-perturbation certificate $\mathrm{MatterPert4Cert}$.
background
Recognition Science cosmology tracks structure formation against the same J-cost and tick structure used in the forcing chain. The Cost import supplies the underlying cost calculus; Constants supplies the RS-native time quantum $\tau_0=1$ tick.
This module specializes that toolkit to a four-domain matter-perturbation setting. Sibling definitions introduce a domain cost functional, record that it is nonnegative, and fix a positive canonical threshold against which perturbation amplitude or domain imbalance is compared. The certificate type bundles those facts so downstream cosmology lemmas can assume a single inhabited record rather than re-proving positivity and threshold facts inline.
proof idea
Definition-heavy module. Domain cost and the canonical threshold are introduced as defs; nonnegativity and positivity are short lemmas over the Cost layer; an evaluation identity pins the cost at distinguished points. The certificate structure and its inhabited instance assemble those pieces into one record. No deep tactic proof; the argument is packaging plus elementary inequalities.
why it matters in Recognition Science
Gives cosmology pages a single cert object for four-domain matter perturbations, aligned with RS cost geometry and the tick/octave timing from Constants. No downstream edges are recorded yet in the mirror graph, so the module currently stands as a leaf certificate bundle rather than a proved forcing step (T0–T8). It is the natural hook for later growth-factor or power-spectrum bounds that need a nonnegative domain cost and a fixed positive threshold in RS-native units.
scope and limits
- Does not derive observational power spectra or transfer functions.
- Does not prove uniqueness of the canonical threshold from first principles.
- Does not connect domain cost to the full T0–T8 forcing chain.
- Does not claim a numerical match to CMB or large-scale structure data.
- Does not treat relativistic gauge choices or baryon acoustic scales.