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SumInversionAgent
Exact reconstruction, trajectory sufficiency, and scaling laws
-/
import Mathlib
noncomputable section
open Real
-- ============================================================
-- Core Definitions
-- ============================================================
variable {n m : Nat}
/-- A matrix B is full rank if its rank equals min(rows, cols) -/
def is_full_rank (B : Matrix (Fin n) (Fin m) ℝ) : Prop :=
B.rank = min n m
/-- Round-trip accuracy: encode then decode recovers original -/
def round_trip_accuracy (encode : Fin n → ℝ → ℝ) (decode : Fin n → ℝ → ℝ) : Prop :=
∀ i x, decode i (encode i x) = x
/-- A trajectory function mapping time steps to states -/
def Trajectory (state_dim : Nat) := Nat → Fin state_dim → ℝ
-- ============================================================
-- Theorems
-- ============================================================
/-- If B is full rank, encoding-decoding achieves 100% round-trip accuracy -/
theorem exact_reconstruction
(B : Matrix (Fin n) (Fin n) ℝ)
(h_full_rank : is_full_rank B)
(encode decode : Fin n → ℝ → ℝ)
(h_linear : ∀ i x, encode i x = B i i * x)
(h_decode : ∀ i x, decode i x = x / B i i) :
round_trip_accuracy encode decode := sorry
/-- Trajectory is injective: distinct inputs produce distinct trajectories -/
theorem trajectory_sufficient
(state_dim : Nat)
(traj : ℝ → Trajectory state_dim)
(h_distinct : ∀ x y, x ≠ y → traj x ≠ traj y) :
Function.Injective traj := sorry
/-- Dynamics error (in trajectory space) bounds token-level error -/
theorem dynamics_error_bounds_token_error
(traj_error token_error : ℝ)
(lipschitz_const : ℝ)
(h_lip_pos : lipschitz_const > 0)
(h_bound : token_error ≤ lipschitz_const * traj_error) :
token_error ≤ lipschitz_const * traj_error := sorry
/-- Chinchilla-optimal: N (model size) proportional to C^0.5 (compute budget) -/
theorem chinchilla_optimal
(C : ℝ) (N : ℝ) (D : ℝ)
(h_C_pos : C > 0)
(h_scaling : N = C ^ (0.5 : ℝ))
(h_data : D = C ^ (0.5 : ℝ))
(h_compute : C = 6 * N * D) :
N = C ^ (0.5 : ℝ) := sorry
end
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