IndisputableMonolith.Cosmology.RS_COS_Structural_004
Structural certificate module for Recognition Science cosmology item 004. It packages a domain cost functional built from the RS J-cost, proves that cost is nonnegative, and introduces a strictly positive canonical threshold against which the cost is compared. Cosmology auditors cite it when wiring structural positivity into larger RS-COS certificate stacks. The module is mostly definitional with short algebraic nonnegativity and positivity lemmas.
claimDefine a domain cost $C$ on the RS cost geometry (via the $J$-cost), prove $C \ge 0$, and fix a canonical threshold $\theta > 0$. Package these into an inhabited structural certificate for RS-COS item 004.
background
Recognition Science derives physics from a single cost functional $J$ forced by the Recognition Composition Law, with $J(x) = (x + x^{-1})/2 - 1$. The Cost import supplies that $J$-geometry; Constants supplies the RS-native tick $\tau_0 = 1$.
In the cosmology layer, structural certificates record elementary positivity and threshold facts that later dynamical or observational claims may assume. This module introduces a domain cost (a real-valued functional on the relevant domain, evaluated pointwise via the cost geometry) and a canonical threshold scalar intended as a comparison level for that cost.
Sibling lemmas record equality of the domain cost at a point with its defining expression, nonnegativity of the cost, and strict positivity of the threshold. The certificate bundle simply assembles those facts into a single inhabited record.
proof idea
Definition-first module. The domain cost is introduced as a thin wrapper around the imported $J$-cost geometry; an equality lemma unfolds the definition at a point. Nonnegativity follows from the standard nonnegativity of $J$ (or an immediate consequence thereof). The canonical threshold is a positive constant definition; its positivity is a one-line arithmetic fact. The certificate type is a structure packing these properties, with an explicit inhabitant constructed from the lemmas above.
why it matters in Recognition Science
RS cosmology certificates are meant to sit under later claims about horizons, expansion, or recognition thresholds in the forced $D=3$, eight-tick setting. This module supplies the elementary cost-and-threshold positivity block labeled structural 004. No downstream consumers are wired in the current graph (used_by is empty), so it presently stands as a leaf certificate ready for import by higher RS-COS stacks. It does not itself touch the forcing chain T0–T8, the mass ladder, or the fine-structure band; it only freezes a nonnegative cost and a positive threshold in the cosmology namespace.
scope and limits
- Does not derive dynamical cosmological equations or Friedmann-type evolution.
- Does not identify the domain cost with observed dark energy, curvature, or entropy densities.
- Does not prove uniqueness of the canonical threshold among possible positive scales.
- Does not connect to the mass ladder, alpha band, or forcing steps T5–T8.
- Does not supply observational bounds or numerical cosmology fits.