optimalBlock_in_range
Recognition Science sets the optimal study block to the golden ratio phi hours, and this theorem confirms the strict interval 1 < phi < 2. Educational designers applying the framework to pedagogy would cite the bound to justify sessions near 97 minutes. The term proof reduces the claim by unfolding the definition of optimalBlockHours and invoking the pre-established phi bounds.
claim$1 < phi < 2$, where $phi$ is the golden ratio serving as optimal study block duration in hours.
background
The MasteryDesignFromGap45 module derives pedagogical parameters from the gap-45 threshold of 45 hours per mastery rung. Optimal study block length is defined as phi hours to align with self-similar scaling, yielding approximately 97.6 minutes and matching observed cycles such as Pomodoro. The definition optimalBlockHours := phi supplies the concrete value used throughout the module.
proof idea
The term proof unfolds optimalBlockHours to phi, then constructs the conjunction via the exact constructor applied to one_lt_phi paired with a linarith tactic on phi_lt_onePointSixTwo.
why it matters in Recognition Science
This theorem populates the block_range field of the MasteryDesignCert definition. It realizes the module prediction that optimal block duration equals phi hours, consistent with the phi-ladder and recovery ratio 1/phi. The result closes the interval verification for the educational application of the T6 phi fixed point.
scope and limits
- Does not derive phi from the Recognition Composition Law.
- Does not quantify mastery hours beyond the gap-45 base case.
- Does not model transitions between the five mastery stages.
- Does not supply error bounds for empirical application.
formal statement (Lean)
40theorem optimalBlock_in_range :
41 (1 : ℝ) < optimalBlockHours ∧ optimalBlockHours < 2 := by
proof body
Term-mode proof.
42 unfold optimalBlockHours
43 exact ⟨one_lt_phi, by linarith [phi_lt_onePointSixTwo]⟩
44
45/-- Recovery ratio = 1/φ (inverse of study block). -/