Foundation Algorithmic Cost

/-- Cumulative cost is bounded above by initial defect (the cost budget). -/
theorem cumulative_cost_bounded {N : ℕ}
    (traj : Trajectory N) (h : IsVariationalTrajectory traj) (T : ℕ) :
    cumulative_cost traj T ≤ total_defect (traj 0) := by
  unfold cumulative_cost
  linarith [total_defect_nonneg (traj T)]

/-- **THEOREM (Infinite Computation Impossible)**:
    No variational trajectory can have ALL steps costing at least δ > 0.

    If every step reduced defect by at least δ, then after
    ⌈D₀/δ⌉ + 1 steps, the defect would be negative.
    But defect ≥ 0. Contradiction.

    This is the RS resolution of the Halting Problem for physical
    computation: a non-halting process with positive step cost
    cannot exist in a universe governed by J-cost minimization. -/
theorem infinite_computation_impossible {N : ℕ}
    (traj : Trajectory N) (h : IsVariationalTrajectory traj)
    (δ : ℝ) (hδ : 0 < δ)
    (h_all_nontrivial : ∀ t : ℕ, step_cost traj t ≥ δ) :
    False := by
  obtain ⟨n, hn⟩ := exists_nat_gt (total_defect (traj 0) / δ)
  have h_bound := computation_bounded traj h δ hδ n
    (fun t _ => h_all_nontrivial t)
  have h_exceeds : total_defect (traj 0) < ↑n * δ := by
    rwa [div_lt_iff₀ hδ] at hn
  linarith

/-- **THEOREM (Every Trajectory Approaches Halting)**:
    For any precision ε > 0, the trajectory eventually has a step
    with cost less than ε. The computation asymptotically halts.

    This is the convergence theorem: the defect sequence is monotone
    non-increasing and bounded below, hence converges. The step costs
    (successive differences) must therefore approach zero. -/
theorem eventual_near_equilibrium {N : ℕ}
    (traj : Trajectory N) (h : IsVariationalTrajectory traj)
    (ε : ℝ) (hε : 0 < ε) :
    ∃ T : ℕ, step_cost traj T < ε :=
  eventual_slowdown traj h ε hε

/-- **THE HALTING THEOREM (F-016)**:
    Every computational process on the ledger reaches effective halting:
    there exists a step T within the halting bound where the step cost
    drops below the minimum step cost.

    At step T, the computation has either:
    (a) reached equilibrium (true halting: step_cost = 0), or
    (b) entered a regime where steps cost less than δ (effective halting)

    In either case, the non-trivial computation has terminated.
    The universe's J-cost budget has been exhausted. -/
theorem halting_theorem {N : ℕ} (cp : ComputationalProcess N) :
    ∃ T : ℕ, step_cost cp.traj T < cp.min_step_cost :=
  eventual_slowdown cp.traj cp.is_variational cp.min_step_cost cp.min_step_pos

/-- **THEOREM (Economic Censorship)**:
    The same J-cost bound that forbids contradictions (LogicFromCost)
    also forbids infinite loops.

    1. Contradictions: defect(P ∧ ¬P) > 0 (cannot stabilize at zero cost)
    2. Infinite loops: ∑ step_cost = ∞ > D₀ (cannot fit in finite budget)
    3. Both are censored by the finite defect principle.

    The universe is both logically consistent (no contradictions) and
    computationally bounded (no infinite loops) for the SAME reason:
    J-cost is finite and non-negative. -/
theorem economic_censorship {N : ℕ}
    (traj : Trajectory N) (h : IsVariationalTrajectory traj) :
    -- 1. Total defect is finite and non-negative
    (0 ≤ total_defect (traj 0)) ∧
    -- 2. Defect is non-increasing (cost is paid, never refunded)
    (∀ t, total_defect (traj (t + 1)) ≤ total_defect (traj t)) ∧
    -- 3. Cumulative cost is bounded by initial defect
    (∀ T, cumulative_cost traj T ≤ total_defect (traj 0)) ∧
    -- 4. No infinite sequence of positive-cost steps exists
    (∀ δ : ℝ, 0 < δ →
      ¬(∀ t : ℕ, step_cost traj t ≥ δ)) ∧
    -- 5. Eventually step cost drops below any positive threshold
    (∀ ε : ℝ, 0 < ε → ∃ T, step_cost traj T < ε) :=
  ⟨total_defect_nonneg (traj 0),
   trajectory_defect_monotone traj h,
   cumulative_cost_bounded traj h,
   fun δ hδ h_all => infinite_computation_impossible traj h δ hδ h_all,
   fun ε hε => eventual_near_equilibrium traj h ε hε⟩