optimal_at_minimum_is_holographic

formal source

 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