X_opt_pos
plain-language theorem explainer
X_opt_pos establishes positivity of the optimal recognition ratio X_opt = φ/π. Inflation modelers in Recognition Science cite it to guarantee that the derived modulation frequency Ω₀ = 2π / ln(1/X_opt) is well-defined and real. The proof is a one-line term application of the division-positivity lemma to the known positivity of φ and π.
Claim. Let $X = φ/π$ be the optimal recognition ratio. Then $0 < X$.
background
The Gravity.Inflation module derives RS inflationary predictions from the cost-functional self-similarity, with α = φ², n_s ≈ 1 - 2/N, r ≈ 12φ²/N², and log-periodic modulation Ω₀ = 2π/ln(π/φ). X_opt is defined as the ratio φ/π at which recognition cost balances geometric constraint. The module documentation states: 'Optimal recognition ratio: X_opt = φ/π'. This positivity result is required before the modulation period Δln(k) = 2π/Ω₀ can be formed.
proof idea
The proof is a direct term-mode application of div_pos to phi_pos and Real.pi_pos. No tactics or intermediate steps appear.
why it matters
The result secures the domain of the log-periodic modulation frequency Ω₀ ≈ 9.47 that appears in the module's core results for the primordial spectrum. It feeds the downstream definitions of power and period used in cosmology calculations. In the framework it rests on the phi fixed-point property (T6) and the eight-tick octave structure. No open questions are closed here.
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