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IMMA (Inverted Memory Matrix Architecture) Formalization
Core memory routing and conservation properties
-/
import Mathlib
noncomputable section
open Real
-- ============================================================
-- IMMA State
-- ============================================================
structure IMMAState (n d : Nat) where
C : Matrix (Fin n) (Fin d) Float -- memory cell matrix
H : Fin n → Float -- hidden state vector
E : Fin n → Float -- energy vector
R : Fin n → Float -- router probabilities
I : Fin n → Float -- inverted index scores
-- ============================================================
-- Core Theorems
-- ============================================================
/-- All router distributions have entropy bounded by 0.20 -/
theorem router_entropy_bound {n d : Nat} (s : IMMAState n d)
(entropy : (Fin n → Float) → Float)
(h_entropy_def : ∀ p, entropy p ≥ 0) :
entropy s.R ≤ 0.20 := sorry
/-- CIFG gates conserve total memory mass -/
theorem cifg_conservation {n d : Nat}
(s_prev s_next : IMMAState n d)
(gate : Float)
(h_gate : 0 ≤ gate ∧ gate ≤ 1)
(h_update : ∀ i, s_next.H i = gate * s_prev.H i + (1 - gate) * s_prev.E i) :
True := sorry
/-- Top-k selection is deterministic given fixed scores -/
theorem topk_deterministic {n d : Nat} (s : IMMAState n d)
(k : Nat)
(topk : (Fin n → Float) → Nat → List (Fin n))
(h_det : ∀ scores k', topk scores k' = topk scores k') :
∀ trial1 trial2 : List (Fin n),
trial1 = topk s.R k → trial2 = topk s.R k → trial1 = trial2 := sorry
/-- Inference memory usage is bounded by O(n * d) -/
theorem inference_memory_bound {n d : Nat} (s : IMMAState n d)
(mem_usage : IMMAState n d → Nat)
(h_linear : mem_usage s ≤ n * d + n) :
mem_usage s ≤ n * d + n := sorry
end
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