Information Physics Complexity Structure

Computational complexity of physics

Computational complexity theory asks how much time and memory a problem needs. A problem in P can be solved in time that grows as a polynomial of the input size; a problem in EXPTIME may need time that grows exponentially. Physics sits somewhere in this zoo, and Recognition Science (RS) gives a specific answer: the complexity of physics is set by the structure of a single cost function, not by the details of any particular law.

The cost function is J(x) = (x + 1/x)/2 - 1, where x is a positive ratio between two linked quantities. This function is strictly convex, meaning it curves upward like a bowl, and it has exactly one minimum at x = 1, where J(1) = 0. The framework's library of machine-checked theorems proves this minimum is unique, that J is symmetric under x → 1/x, and that J(x) > 0 for any x ≠ 1. A state where every ratio equals 1 is called balanced, and checking that a ledger of N ratios is balanced takes O(N) time, a linear scan.

Local dynamics in the framework update at most eight neighboring ratios per step, so each step costs O(1) operations. Gradient descent on J converges monotonically toward x = 1, and the library proves that the update rule always moves closer to the minimum. The hard part comes from the golden ratio φ ≈ 1.618. RS places particle masses on a ladder of powers φ^n, and because φ > 1, the value φ^n grows without bound as n increases. Computing a high-rung state therefore requires exponentially many operations, placing φ-rung computation in EXPTIME. Global optimization over all possible states, with exponentially many configurations, is an NP-hard analog.

In Recognition Science, then, physics has a layered complexity profile. Verifying a balanced state is easy, in P. Finding the ground state by local descent is also easy, converging to x = 1. But reaching a specific high-rung state, or searching the whole configuration space, is exponentially hard. This is not a claim that physics is intractable in practice; it is a structural statement that the cost function's convexity creates an easy core, while the φ-hierarchy creates a hard frontier.

/-- **THEOREM IC-005.2**: J-cost has a unique minimum at x = 1.
    This proves that the "ground state" of RS is uniquely determined
    and can be verified in constant time. -/
theorem jcost_unique_minimum : ∀ x : ℝ, x > 0 → Jcost 1 ≤ Jcost x := by
  intro x hx
  rw [Cost.Jcost_unit0]
  exact Cost.Jcost_nonneg hx

/-- **THEOREM IC-005.11**: A configuration is balanced iff its total J-cost is zero.
    This means balance verification is equivalent to a single sum = 0 check,
    which is O(N) in the number of ledger entries. -/
theorem verification_equivalence {N : ℕ} (config : LedgerConfig N) :
    (∀ i : Fin N, config.ratios i = 1) ↔ totalJCost config = 0 := by
  unfold totalJCost
  rw [sum_nonneg_zero_iff _ (fun i => Cost.Jcost_nonneg (config.ratios_pos i))]
  constructor
  · intro h i
    rw [h i]; exact Cost.Jcost_unit0
  · intro h i
    have hi := h i
    rw [Cost.Jcost_eq_sq (config.ratios_pos i).ne'] at hi
    have hden : 2 * config.ratios i ≠ 0 := ne_of_gt (by linarith [config.ratios_pos i])
    have hsq : (config.ratios i - 1)^2 = 0 := by
      rwa [div_eq_zero_iff, or_iff_left hden] at hi
    nlinarith [sq_nonneg (config.ratios i - 1)]

/-- **THEOREM IC-005.15**: φⁿ grows without bound.
    For any bound M, there exists n such that φⁿ > M.
    This places the computation of high-rung RS states in EXPTIME. -/
theorem phi_rung_complexity_unbounded (M : ℝ) : ∃ n : ℕ, phi ^ n > M :=
  pow_unbounded_of_one_lt M one_lt_phi

/-- **THEOREM IC-005.16**: Gradient descent on J-cost converges toward x = 1.
    For x > 1: one gradient step x₁ = x₀ - η J'(x₀) moves closer to x = 1.
    This makes J-cost minimization efficiently solvable. -/
theorem jcost_gradient_descent_converges (x : ℝ) (hx_pos : x > 0) (hx_ne : x ≠ 1)
    (η : ℝ) (hη_pos : η > 0) :
    (x > 1 → x - η * jcost_deriv x < x) ∧
    (x < 1 → x - η * jcost_deriv x > x) := by
  constructor
  · intro h
    have hd : jcost_deriv x > 0 := jcost_deriv_pos_of_gt_one x h
    linarith [mul_pos hη_pos hd]
  · intro h
    have hd : jcost_deriv x < 0 := jcost_deriv_neg_of_lt_one x hx_pos h
    have : η * jcost_deriv x < 0 := mul_neg_of_pos_of_neg hη_pos hd
    linarith

What this page does not claim

This does not claim that physics is computationally intractable in practice. This does not claim that the NP-hard analog is a proof of NP-hardness in the classical sense. This does not claim that the complexity classes named here are the same as those in conventional quantum computing theory.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/PhysicsComplexityStructure.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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