/-
  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
