working_memory_approx

formal source

 244    capacity = φ³ relative to Level 0 (focal attention, capacity 1). -/
 245noncomputable def working_memory_capacity : ℝ := phi ^ 3
 246
 247theorem working_memory_approx :
 248    4 < working_memory_capacity ∧ working_memory_capacity < 5 := by
 249  unfold working_memory_capacity
 250  constructor
 251  · -- φ³ > 4: use φ > 1.61 and 1.61³ > 4
 252    have hphi : phi > 1.61 := Constants.phi_gt_onePointSixOne
 253    have hphi_pos : (0:ℝ) ≤ 1.61 := by norm_num
 254    nlinarith [sq_nonneg phi, sq_nonneg (phi - 1.61), Constants.phi_pos,
 255               show (1.61:ℝ)^3 > 4 by norm_num]
 256  · -- φ³ < 5: use φ < 1.62
 257    have hphi : phi < 1.62 := Constants.phi_lt_onePointSixTwo
 258    have hphi_pos : (0:ℝ) < phi := Constants.phi_pos
 259    nlinarith [sq_nonneg (1.62 - phi), sq_nonneg phi,
 260               show (1.62:ℝ)^3 < 5 by norm_num]
 261
 262/-! ## Status -/
 263
 264def localCacheStatus : String :=
 265  "═══════════════════════════════════════════════════\n" ++
 266  "    LOCAL CACHE THEOREM — LEAN STATUS\n" ++
 267  "═══════════════════════════════════════════════════\n" ++
 268  "✓ local_cache_benefit: Caching reduces cost (proved)\n" ++
 269  "✓ fibonacci_ratio_forces_golden: Fib + ratio → r²=r+1 (proved)\n" ++
 270  "✓ fibonacci_partition_forces_phi: Fib + ratio → r=φ (proved)\n" ++
 271  "✓ hebb_is_Jcost_gradient: J'(r) = (1-r⁻²)/2 (proved)\n" ++
 272  "✓ hebbian_matches_descent: J' > 0 for r > 1 (proved)\n" ++
 273  "✓ Jcost_min_at_one: J(1) = 0 (proved)\n" ++
 274  "✓ Jcost_pos_of_ne_one: J(r) > 0 for r ≠ 1 (proved)\n" ++
 275  "✓ working_memory_approx: 4 < φ³ < 5 (proved)\n" ++
 276  "✓ Jcost_symmetry_forces_geometric_boundary (proved via Jcost_eq_sq factorization)\n"
 277