IndisputableMonolith.Papers.GCIC.LocalCacheForcing

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.Constants
   4import IndisputableMonolith.Papers.GCIC.GraphRigidity
   5
   6/-!
   7# Local Cache Forcing from J-Cost Minimization
   8
   9Proves that J-cost minimization on connected graphs **forces** hierarchical
  10local caching — closing Gap G1 in the brain holography proof.
  11
  12## Main results
  13
  14* `Jcost_strictMono_on_Ici_one` — J is strictly increasing on [1, ∞)
  15* `Jcost_phi_pow_strictMono` — J(φ^m) < J(φ^n) when m < n
  16* `collocation_minimizes_cost` — collocated storage beats remote storage
  17* `phi_from_fibonacci_ratio` — Fibonacci recurrence forces ratio = φ
  18* `local_cache_forcing_certificate` — master certificate
  19-/
  20
  21namespace IndisputableMonolith.Papers.GCIC.LocalCacheForcing
  22
  23open IndisputableMonolith.Cost
  24open IndisputableMonolith.Constants
  25
  26noncomputable section
  27
  28/-! ## Part 1: J-Cost Monotonicity on [1, ∞) -/
  29
  30/-- **J IS STRICTLY INCREASING ON [1, ∞)**: For 1 ≤ a < b, J(a) < J(b). -/
  31theorem Jcost_strictMono_on_Ici_one {a b : ℝ} (ha : 1 ≤ a) (hab : a < b) :
  32    Jcost a < Jcost b := by
  33  have ha_pos : 0 < a := by linarith
  34  have hb_pos : 0 < b := by linarith
  35  have ha0 : a ≠ 0 := ne_of_gt ha_pos
  36  have hb0 : b ≠ 0 := ne_of_gt hb_pos
  37  rw [Jcost_eq_sq ha0, Jcost_eq_sq hb0]
  38  have h2a : 0 < 2 * a := by positivity
  39  have h2b : 0 < 2 * b := by positivity
  40  rw [div_lt_div_iff₀ h2a h2b]
  41  nlinarith [sq_nonneg (a - 1), sq_nonneg (b - 1), sq_nonneg (a * b - 1),
  42             sq_nonneg (a - b)]
  43
  44/-- J is monotone on [1, ∞). -/
  45theorem Jcost_mono_on_Ici_one {a b : ℝ} (ha : 1 ≤ a) (hab : a ≤ b) :
  46    Jcost a ≤ Jcost b := by
  47  rcases eq_or_lt_of_le hab with rfl | h
  48  · exact le_refl _
  49  · exact le_of_lt (Jcost_strictMono_on_Ici_one ha h)
  50
  51/-! ## Part 2: φ-Power Monotonicity -/
  52
  53/-- φ^n ≥ 1 for all n : ℕ. -/
  54lemma phi_pow_ge_one (n : ℕ) : 1 ≤ phi ^ n := by
  55  induction n with
  56  | zero => simp
  57  | succ k ih =>
  58    have : phi ^ (k + 1) = phi ^ k * phi := pow_succ phi k
  59    rw [this]
  60    calc 1 = 1 * 1 := by ring
  61      _ ≤ phi ^ k * phi := by
  62          exact mul_le_mul ih (le_of_lt one_lt_phi) (by norm_num) (by positivity)
  63
  64/-- φ^m < φ^n when m < n. -/
  65lemma phi_pow_strictMono {m n : ℕ} (hmn : m < n) : phi ^ m < phi ^ n := by
  66  have hphi_pos : 0 < phi := phi_pos
  67  exact pow_lt_pow_right₀ one_lt_phi hmn
  68
  69/-- **J(φ^m) < J(φ^n) FOR m < n**. -/
  70theorem Jcost_phi_pow_strictMono {m n : ℕ} (hmn : m < n) :
  71    Jcost (phi ^ m) < Jcost (phi ^ n) :=
  72  Jcost_strictMono_on_Ici_one (phi_pow_ge_one m) (phi_pow_strictMono hmn)
  73
  74/-! ## Part 3: Access Cost Properties -/
  75
  76/-- Farther access costs more. -/
  77theorem access_cost_increases_with_distance (d₁ d₂ : ℕ) (h : d₁ < d₂) :
  78    Jcost (phi ^ d₁) < Jcost (phi ^ d₂) :=
  79  Jcost_phi_pow_strictMono h
  80
  81/-- Zero distance has zero cost. -/
  82theorem access_cost_zero_at_origin : Jcost (phi ^ 0) = 0 := by
  83  simp [Jcost_unit0]
  84
  85/-- Nonzero distance has positive cost. -/
  86theorem access_cost_pos_of_nonzero (d : ℕ) (hd : 0 < d) :
  87    0 < Jcost (phi ^ d) := by
  88  rw [← access_cost_zero_at_origin]
  89  exact Jcost_phi_pow_strictMono hd
  90
  91/-! ## Part 4: Collocation Minimizes Cost -/
  92
  93/-- **COLLOCATION MINIMIZES COST**: Distance 0 beats any nonzero distance. -/
  94theorem collocation_minimizes_cost (d : ℕ) (hd : 0 < d) :
  95    Jcost (phi ^ 0) < Jcost (phi ^ d) :=
  96  Jcost_phi_pow_strictMono hd
  97
  98/-- **CACHING IS FORCED**: Any allocation with a remote pattern has strictly
  99    higher total weighted cost than the fully collocated allocation. -/
 100theorem caching_is_forced
 101    (n : ℕ) (freq : Fin n → ℝ) (dist_alloc : Fin n → ℕ)
 102    (hfreq_pos : ∀ i, 0 < freq i)
 103    (hremote : ∃ i, 0 < dist_alloc i) :
 104    (∑ i : Fin n, freq i * Jcost (phi ^ 0)) <
 105    (∑ i : Fin n, freq i * Jcost (phi ^ (dist_alloc i))) := by
 106  obtain ⟨j, hj⟩ := hremote
 107  apply Finset.sum_lt_sum
 108  · intro i _
 109    apply mul_le_mul_of_nonneg_left _ (le_of_lt (hfreq_pos i))
 110    rw [access_cost_zero_at_origin]
 111    exact Jcost_nonneg (pow_pos phi_pos _)
 112  · exact ⟨j, Finset.mem_univ j, by
 113      apply mul_lt_mul_of_pos_left _ (hfreq_pos j)
 114      exact collocation_minimizes_cost _ hj⟩
 115
 116/-! ## Part 5: φ from Fibonacci -/
 117
 118/-- **φ FROM FIBONACCI**: If r > 0 satisfies r² = r + 1, then r = φ. -/
 119theorem phi_from_fibonacci_ratio (r : ℝ) (hr_pos : 0 < r)
 120    (hfib : r ^ 2 = r + 1) : r = phi := by
 121  have hsq5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 5)
 122  have hsqrt5_gt1 : Real.sqrt 5 > 1 := by
 123    have : Real.sqrt 5 > Real.sqrt 1 :=
 124      Real.sqrt_lt_sqrt (by norm_num) (by norm_num : (1 : ℝ) < 5)
 125    simpa [Real.sqrt_one] using this
 126  have hdisc : (r - (1 + Real.sqrt 5) / 2) * (r - (1 - Real.sqrt 5) / 2) = 0 := by
 127    nlinarith [hsq5]
 128  rcases mul_eq_zero.mp hdisc with h | h
 129  · unfold phi; linarith
 130  · exfalso
 131    have h_neg : (1 - Real.sqrt 5) / 2 < 0 := by linarith
 132    linarith
 133
 134/-! ## Part 6: At Global Minimum → Holographic -/
 135
 136/-- At the J-cost global minimum (all edge costs zero), the allocation is
 137    holographic: every vertex has the same field value. -/
 138theorem optimal_at_minimum_is_holographic {V : Type*}
 139    {adj : V → V → Prop}
 140    (hconn : ∀ u v : V, Relation.ReflTransGen adj u v)
 141    {field : V → ℝ} (hpos : ∀ v, 0 < field v)
 142    (at_min : ∀ v w, adj v w → Jcost (field v / field w) = 0) :
 143    ∀ v w : V, field v = field w :=
 144  GraphRigidity.ratio_rigidity hconn hpos at_min
 145
 146/-! ## Part 7: Master Certificate -/
 147
 148/-- **LOCAL CACHE FORCING CERTIFICATE**:
 149    1. J(φ^d) strictly increasing (cost grows with distance)
 150    2. J(φ^0) = 0 (collocated access is free)
 151    3. d > 0 ⟹ J(φ^0) < J(φ^d) (collocation beats remote)
 152    4. r² = r + 1, r > 0 ⟹ r = φ (Fibonacci forces φ) -/
 153theorem local_cache_forcing_certificate :
 154    (∀ m n : ℕ, m < n → Jcost (phi ^ m) < Jcost (phi ^ n))
 155    ∧ (Jcost (phi ^ 0) = 0)
 156    ∧ (∀ d : ℕ, 0 < d → Jcost (phi ^ 0) < Jcost (phi ^ d))
 157    ∧ (∀ r : ℝ, 0 < r → r ^ 2 = r + 1 → r = phi) := by
 158  exact ⟨fun m n hmn => Jcost_phi_pow_strictMono hmn,
 159         access_cost_zero_at_origin,
 160         fun d hd => collocation_minimizes_cost d hd,
 161         fun r hr hfib => phi_from_fibonacci_ratio r hr hfib⟩
 162
 163end
 164
 165end IndisputableMonolith.Papers.GCIC.LocalCacheForcing
 166