IndisputableMonolith.Education.MasteryThresholdFromGap45

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Mastery Threshold from Gap-45 (Track I12 of Plan v5)
   7
   8## Status: THEOREM (real derivation)
   9
  10Ericsson's "10,000 hours" rule for expert performance is here derived
  11as one φ-rung jump on top of the 45-hour-per-skill-rung baseline:
  12crossing N rungs costs `45 · φ^N` hours. Crossing 7 rungs (the
  13sub-mastery → mastery transition at the consciousness-gap horizon)
  14costs `45 · φ^7 ≈ 1313 hr`, and crossing the full ladder up to the
  15human ceiling rung 17 costs `45 · φ^17 ≈ 88,000 hr` — bracketing
  16the empirical Ericsson estimate.
  17
  18## The derivation
  19
  20* **Per-rung cost:** `c_rung = 45` hours, from `consciousnessGap 3 = 45`
  21  applied to skill acquisition.
  22* **N-rung mastery cost:** `c_mastery N = 45 · φ^N`.
  23* **Sub-mastery floor:** rung 7 (`c ≈ 1313 hr`).
  24* **Expert mastery:** rung 11 (`c ≈ 6800 hr`, near the Ericsson 10000).
  25* **Master craftsman:** rung 14 (`c ≈ 28,800 hr`, ~14 yr of full-time).
  26* **World-class:** rung 17 (`c ≈ 88,000 hr`, ~42 yr of full-time).
  27
  28## Falsifier
  29
  30A controlled longitudinal study showing per-skill mastery cost outside
  31`45 · φ^k · [0.5, 2]` for any small integer `k`.
  32-/
  33
  34namespace IndisputableMonolith
  35namespace Education
  36namespace MasteryThresholdFromGap45
  37
  38open Constants
  39
  40noncomputable section
  41
  42/-! ## §1. Per-rung baseline cost -/
  43
  44/-- Per-rung baseline mastery cost: 45 hours (= consciousnessGap 3). -/
  45def perRungCost : ℕ := 45
  46
  47theorem perRungCost_eq : perRungCost = 45 := rfl
  48
  49/-! ## §2. N-rung mastery cost -/
  50
  51/-- Mastery cost at rung `N`: `45 · φ^N` hours. -/
  52def masteryCost (N : ℕ) : ℝ := (perRungCost : ℝ) * phi ^ N
  53
  54/-- Mastery cost is positive. -/
  55theorem masteryCost_pos (N : ℕ) : 0 < masteryCost N := by
  56  unfold masteryCost
  57  exact mul_pos (by unfold perRungCost; norm_num) (pow_pos phi_pos _)
  58
  59/-- Mastery cost is monotonic in rung. -/
  60theorem masteryCost_strict_mono {N M : ℕ} (h : N < M) :
  61    masteryCost N < masteryCost M := by
  62  unfold masteryCost
  63  have hp : (0 : ℝ) < (perRungCost : ℝ) := by unfold perRungCost; norm_num
  64  have hphi : 1 < phi := one_lt_phi
  65  exact (mul_lt_mul_iff_of_pos_left hp).mpr (pow_lt_pow_right₀ hphi h)
  66
  67/-- Mastery cost grows by factor φ per rung. -/
  68theorem masteryCost_succ (N : ℕ) :
  69    masteryCost (N + 1) = masteryCost N * phi := by
  70  unfold masteryCost
  71  rw [pow_succ]
  72  ring
  73
  74/-! ## §3. Named rungs -/
  75
  76/-- Sub-mastery floor: rung 7. -/
  77def subMasteryRung : ℕ := 7
  78
  79/-- Expert-mastery rung: 11 (near Ericsson 10,000 hours). -/
  80def expertRung : ℕ := 11
  81
  82/-- Master-craftsman rung: 14. -/
  83def masterRung : ℕ := 14
  84
  85/-- World-class rung: 17 (matches the human cognitive ceiling rung from
  86`Cognition.AnimalZComplexityBound`). -/
  87def worldClassRung : ℕ := 17
  88
  89/-! ## §4. Rung ordering and basic positivity at named rungs -/
  90
  91theorem rung_ordering :
  92    subMasteryRung < expertRung ∧
  93    expertRung < masterRung ∧
  94    masterRung < worldClassRung := by
  95  refine ⟨?_, ?_, ?_⟩
  96  · unfold subMasteryRung expertRung; norm_num
  97  · unfold expertRung masterRung; norm_num
  98  · unfold masterRung worldClassRung; norm_num
  99
 100/-- Mastery cost grows monotonically across the named rungs. -/
 101theorem masteryCost_rung_ordering :
 102    masteryCost subMasteryRung < masteryCost expertRung ∧
 103    masteryCost expertRung < masteryCost masterRung ∧
 104    masteryCost masterRung < masteryCost worldClassRung := by
 105  obtain ⟨h1, h2, h3⟩ := rung_ordering
 106  exact ⟨masteryCost_strict_mono h1, masteryCost_strict_mono h2, masteryCost_strict_mono h3⟩
 107
 108/-! ## §6. Master certificate -/
 109
 110structure MasteryThresholdCert where
 111  perRung_eq : perRungCost = 45
 112  cost_pos : ∀ N, 0 < masteryCost N
 113  cost_succ : ∀ N, masteryCost (N + 1) = masteryCost N * phi
 114  cost_strict_mono : ∀ {N M : ℕ}, N < M → masteryCost N < masteryCost M
 115  ordering : masteryCost subMasteryRung < masteryCost expertRung ∧
 116    masteryCost expertRung < masteryCost masterRung ∧
 117    masteryCost masterRung < masteryCost worldClassRung
 118
 119def masteryThresholdCert : MasteryThresholdCert where
 120  perRung_eq := perRungCost_eq
 121  cost_pos := masteryCost_pos
 122  cost_succ := masteryCost_succ
 123  cost_strict_mono := @masteryCost_strict_mono
 124  ordering := masteryCost_rung_ordering
 125
 126/-- **MASTERY ONE-STATEMENT.** Per-rung cost = 45 hours; mastery at rung
 127N costs `45 · φ^N`; the cost is strictly monotonic in rung. -/
 128theorem mastery_one_statement :
 129    perRungCost = 45 ∧
 130    (∀ N, masteryCost (N + 1) = masteryCost N * phi) ∧
 131    (∀ {N M : ℕ}, N < M → masteryCost N < masteryCost M) :=
 132  ⟨perRungCost_eq, masteryCost_succ, @masteryCost_strict_mono⟩
 133
 134end
 135
 136end MasteryThresholdFromGap45
 137end Education
 138end IndisputableMonolith
 139