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BURT (Bi-encoder Unified Retrieval Transformer) Formalization
Core retrieval and ranking properties
-/
import Mathlib
noncomputable section
open Real
-- ============================================================
-- BURT State
-- ============================================================
structure BURTState (n d : Nat) where
Q : Matrix (Fin n) (Fin d) Float -- query embeddings
D : Matrix (Fin n) (Fin d) Float -- document embeddings
I : Fin n → Float -- inverted index scores
R : Fin n → Fin n -- ranking permutation
C : Float -- CIFG gate value
E : Float -- energy
-- ============================================================
-- Core Theorems
-- ============================================================
/-- Router entropy is bounded by the entropy ceiling -/
theorem router_entropy_bound {n d : Nat} (s : BURTState n d)
(entropy : (Fin n → Float) → Float)
(router_dist : Fin n → Float)
(h_from_state : ∀ i, router_dist i = s.I i) :
entropy router_dist ≤ 0.20 := sorry
/-- CIFG gate conserves memory during retrieval updates -/
theorem cifg_memory_conservation {n d : Nat}
(s : BURTState n d)
(h_gate : 0 ≤ s.C ∧ s.C ≤ 1)
(mem_prev mem_next : Float)
(h_update : mem_next = s.C * mem_prev + (1 - s.C) * s.E) :
mem_next ≤ max mem_prev s.E := sorry
/-- The ranking function produces a valid permutation -/
theorem ranking_is_permutation {n d : Nat} (s : BURTState n d)
(h_inj : Function.Injective s.R) :
Function.Bijective s.R := sorry
/-- Retrieval has sub-quadratic complexity via inverted index -/
theorem retrieval_complexity {n d : Nat} (s : BURTState n d)
(retrieval_flops : Nat)
(h_indexed : retrieval_flops ≤ n * Nat.log n + d) :
retrieval_flops < n * n := sorry
end
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