spacing_ratio_pos
plain-language theorem explainer
The theorem establishes that the optimal spacing ratio between consecutive learning sessions is strictly positive. Researchers modeling distributed practice from the gap-45 mastery threshold would cite it when confirming that geometric intervals derived from the eight-tick structure exceed zero. The proof is a one-line term reduction that applies the known positivity of phi directly to the definition of optimalSpacingRatio.
Claim. $0 < phi$, where $phi$ is the golden ratio serving as the optimal spacing ratio between consecutive sessions in the gap-45 pedagogy model.
background
The module derives the optimal pedagogy rate by distributing the 45-hour per-rung mastery threshold across an 8-tick octave structure, yielding approximately 5.625 hours per session and a geometric spacing ratio of phi. The definition optimalSpacingRatio : ℝ := phi supplies the ratio, while beats : Nat := Nat.lcm 8 45 gives the minimal joint duration for the 8-beat and 45-fold patterns. Module documentation states: 'Optimal spacing ratio between consecutive sessions = φ.' and notes that distributed practice beats massed practice when spacing follows roughly geometric intervals clustering around phi.
proof idea
The proof is a one-line term wrapper that applies phi_pos to the definition of optimalSpacingRatio.
why it matters
This result supplies the spacing_pos field inside PedagogyOptimalCert, which certifies per-rung hours, session count, per-session length, total equality, and positivity of the spacing ratio. It instantiates the framework's T7 eight-tick octave and T6 phi self-similar fixed point inside the education domain, closing the structural derivation that distributed practice beats massed practice within the gap-45 model.
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