PedagogyOptimalCert
plain-language theorem explainer
This structure certifies that 45 hours per mastery rung distribute into eight sessions of 5.625 hours each with spacing ratio phi. Education researchers modeling distributed practice would cite it to connect the gap-45 threshold to the eight-tick octave. The definition assembles the equalities and the phi-band inequalities directly from the module's prior definitions.
Claim. A certificate asserting that the per-rung mastery threshold equals 45 hours, the session count per rung equals 8, the duration per session equals $45/8$ hours, the total time satisfies the product equality, the optimal spacing ratio is positive, and satisfies $1 <$ optimal spacing ratio $< 2$.
background
The module deepens the gap-45 mastery threshold by deriving the optimal pedagogy rate. Per-rung hours is defined as 45, the session count as 8, and per-session hours as their quotient in rationals. The optimal spacing ratio is defined as phi, the golden ratio from the self-similar fixed point in the forcing chain. This setting uses the eight-tick octave for session distribution within each rung, with upstream results including the positivity of 4 phi from the EarthBrainResonance module confirming spacing is positive. The local theoretical setting is the E5 deepening of optimal pedagogy rate from mastery threshold, linking to distributed practice literature where geometric spacing clusters around phi.
proof idea
The declaration is a structure definition that bundles the six certification conditions. It directly references the module definitions for per-rung hours, session count, per-session hours, and optimal spacing ratio, along with the positivity theorem for spacing from the upstream EarthBrainResonance result.
why it matters
This certificate feeds the downstream cert definition that instantiates it with reflexivity and the spacing positivity. It fills the gap-45 to optimal rate step in the education module, connecting to the eight-tick octave (T7) and phi uniqueness (T6) in the forcing chain. The module notes a falsifier as any trial showing spacing outside the phi band on the per-session retention curve.
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