module module high

`IndisputableMonolith.Education.MasteryThresholdFromGap45`

This module sets the per-rung baseline mastery cost at 45 hours, identified with consciousnessGap 3. Education researchers in the Recognition Science framework cite it to anchor learning thresholds on the phi-ladder. The module supplies a short chain of definitions and elementary monotonicity lemmas built directly on the imported Cost and Constants primitives.

claimDefine per-rung baseline mastery cost as $45$ hours with consciousnessGap $3$, where the fundamental time quantum satisfies $τ_0 = 1$ tick. Let masteryCost$(r)$ be the cumulative cost to rung $r$ on the phi-ladder, with perRungCost the constant increment satisfying masteryCost$(r+1) = $ masteryCost$(r) + 45$.

background

Recognition Science measures learning progress via the phi-ladder whose rungs are spaced by the self-similar fixed point phi. The Cost module supplies the defectDist and J-cost primitives that convert rung gaps into time costs; Constants fixes the RS-native time unit $τ_0 = 1$ tick. This module calibrates the baseline increment to 45 hours, corresponding to consciousnessGap 3, and records the resulting ordering and positivity properties for the masteryCost function.

proof idea

This is a definition module, no proofs. It introduces perRungCost as a numeric constant, defines masteryCost recursively from that constant, and states the immediate consequences (positivity, strict monotonicity, successor rule) as one-line wrappers over the imported Cost lemmas.

why it matters in Recognition Science

The module supplies the concrete numerical anchor required by any downstream education theorem that invokes mastery thresholds. It directly implements the gap-45 calibration referenced in the Recognition Science chain (T5–T7) and feeds the rung-ordering lemmas used by higher-level statements on expert and world-class rungs.

scope and limits

Does not derive the numerical value 45 from the J-cost equation.
Does not treat variable consciousness gaps other than the fixed baseline 3.
Does not include empirical data or external calibration checks.

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Cost module_import

declarations in this module (15)

def perRungCost L45
theorem perRungCost_eq L47
def masteryCost L52
theorem masteryCost_pos L55
theorem masteryCost_strict_mono L60
theorem masteryCost_succ L68
def subMasteryRung L77
def expertRung L80
def masterRung L83
def worldClassRung L87
theorem rung_ordering L91
theorem masteryCost_rung_ordering L101
structure MasteryThresholdCert L110
def masteryThresholdCert L119
theorem mastery_one_statement L128