IndisputableMonolith.Cosmology.MatterPert4
Module packaging Recognition Science fourth-order matter-perturbation bookkeeping in cosmology. It defines a domain cost, a canonical positive threshold, and an inhabited certificate MatterPert4Cert bundling nonnegativity and threshold positivity. Cosmologists wiring RS growth or power-spectrum claims would cite the certificate. Content is definitional plus short positivity lemmas from the Cost and Constants imports.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$, the identity for its pointwise evaluation, nonnegativity $C_{\mathrm{dom}} \ge 0$, a canonical threshold $\theta_{\mathrm{can}} > 0$, and a certificate type asserting those facts for fourth-order matter perturbations.
background
Recognition Science fixes cosmology scales from the J-cost $J(x) = (x + x^{-1})/2 - 1$ and the $\varphi$-ladder, with RS-native constants imported from Constants (including the tick $\tau_0$). The Cost library supplies the underlying cost calculus used to score recognition defects.
This Cosmology module restricts that cost to a domain relevant for fourth-order matter perturbations, names a canonical threshold for the onset or bound of growth, and packages both into a certificate structure. Sibling names indicate evaluation identities, nonnegativity, threshold positivity, and an inhabited cert record rather than a deep dynamical derivation.
proof idea
Definition-and-certificate module, not a long forcing argument. domainCost and canonicalThreshold are introduced as defs; domainCost_at_eq is an evaluation identity; domainCost_nonneg and canonicalThreshold_pos are short nonnegativity/positivity lemmas. MatterPert4Cert and cert assemble those facts; cert_inhabited shows the certificate type is inhabited. No multi-step tactic development beyond elementary Cost/Constants facts.
why it matters in Recognition Science
Gives the local certificate interface for RS fourth-order matter-perturbation bounds so downstream cosmology claims can cite cost nonnegativity and a positive canonical threshold in one object. The import graph ties it to Constants and Cost only; the used-by graph is currently empty, so it is a leaf packaging layer rather than a step inside T0–T8. It does not itself force $D=3$, the eight-tick octave, or the $\alpha$ band; it only stages perturbation bookkeeping for later cosmology theorems.
scope and limits
- Does not derive the matter power spectrum or growth factor from the forcing chain.
- Does not claim observational fits to CMB or large-scale structure data.
- Does not prove uniqueness of the canonical threshold.
- Does not connect directly to T5–T8 (J-uniqueness, phi, eight-tick, D=3).
- Does not supply dynamical PDEs for density contrast evolution.