critical_density_from_phi
plain-language theorem explainer
Critical density emerges in Recognition Science as a direct consequence of φ-constrained ledger minimization, fixing Ω exactly to 1 and resolving the flatness problem without fine-tuning. Cosmologists examining the origin of spatial flatness would cite the result to connect ρ_c to the golden-ratio fixed point and J-cost. The proof reduces to a one-line term-mode assertion of triviality, with the explicit φ-power relation left as a comment placeholder.
Claim. In Recognition Science the critical density satisfies the relation $ρ_c × τ₀³ × c³ / G = φ^n$ for some integer $n$, where φ denotes the self-similar fixed point forced by the Recognition Composition Law and τ₀ is the fundamental time scale.
background
The module COS-005 treats the flatness problem: spatial curvature satisfies |Ω − 1| < 10^{-60} at the Planck epoch yet grows as a²(t), making Ω = 1 an unstable fixed point. Recognition Science resolves this by requiring Ω = 1 as the unique value consistent with ledger structure and J-cost minimization. Upstream, PrimitiveDistinction.from supplies the seven axioms that yield four structural conditions plus three definitional facts; ContinuumBridge.as identifies the discrete Laplacian action with the continuum form (1/2) Σ w_ij (ε_i − ε_j)² = (1/κ) Σ δ_h A_h.
proof idea
The declaration is a term-mode proof that directly returns the proposition True via the trivial constructor, without invoking any of the upstream lemmas or expanding the commented φ-relation.
why it matters
The theorem occupies the COS-005 slot that links critical density to φ-constraints, thereby supporting the claim that ledger structure forces Ω = 1 exactly. It sits downstream of the forcing chain (T5 J-uniqueness, T6 φ fixed point) and the Recognition Composition Law. Because used_by is empty and the explicit exponent n remains unverified, the declaration functions as a scaffolding marker whose closure would require an explicit mass-ladder or cost-minimization derivation.
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