spacing_below_two
plain-language theorem explainer
The optimal spacing ratio between consecutive pedagogy sessions is shown to lie strictly below 2. Education researchers calibrating distributed-practice schedules within the Recognition Science model would cite this bound to keep geometric intervals from becoming too sparse. The proof is a direct one-line application of the established inequality phi < 2.
Claim. The optimal spacing ratio between consecutive sessions satisfies $phi < 2$.
background
In the module deepening the gap-45 mastery threshold, the optimal spacing ratio is defined as the golden ratio phi. This choice follows from the 8-tick octave structure that distributes 45 hours across eight sessions per rung, with geometric spacing maximizing long-term retention. Upstream lemmas in Constants, PhiForcing, RecognitionEntropy, and Euler each establish phi < 2 by comparing sqrt(5) < 3 or by direct linear arithmetic.
proof idea
The proof is a one-line wrapper that applies the lemma phi_lt_two.
why it matters
The bound feeds directly into the PedagogyOptimalCert definition in the same module, closing the spacing-positivity part of the 8-tick + gap-45 argument. It aligns with the eight-tick octave (T7) in the forcing chain and supports the claim that geometric spacing near phi maximizes mastery probability. The module falsifier remains any large-scale trial showing optimal ratio outside the phi band.
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