optimalSpacingRatio
plain-language theorem explainer
The optimal spacing ratio between consecutive learning sessions is defined as the golden ratio φ. Pedagogy researchers modeling distributed practice for mastery attainment would cite this when linking session intervals to retention curves. It directly assigns the ratio to φ to instantiate the eight-tick octave structure within the 45-hour per-rung threshold. The definition requires no additional steps beyond the assignment itself.
Claim. The optimal spacing ratio between consecutive learning sessions is the golden ratio $φ$.
background
In Recognition Science the golden ratio φ is the self-similar fixed point forced at step T6 of the unified forcing chain. The present module deepens the 45-hour per-rung mastery threshold by distributing those hours over the eight-tick octave (T7), producing eight sessions whose inter-session spacing ratio is required to be φ. This matches the geometric spacing observed in spaced-repetition data and supplies the concrete value used by the PedagogyOptimalCert structure.
proof idea
Direct definition that equates the optimal spacing ratio to φ. No lemmas or tactics are invoked; the declaration serves as the base constant for the module's subsequent positivity and bounding theorems.
why it matters
The definition supplies the spacing value required by PedagogyOptimalCert and the three bounding theorems (spacing_ratio_pos, spacing_above_one, spacing_below_two). It embeds the Recognition Science eight-tick octave (T7) into an educational setting, where the resulting φ-ratio aligns with the empirical band reported for SuperMemo/Anki retention curves. It touches the module's stated falsifier: any large-scale trial finding an optimal ratio outside the φ band would challenge the claim.
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