Information Physics Complexity Structure Physics Complexity Structure

The complexity of physics

Computational complexity theory sorts problems by how much time or memory a computer needs to solve them. A problem in P is easy, solvable in time that grows like a polynomial of the input size. A problem in EXPTIME is hard, requiring time that grows exponentially. The question of where the laws of physics sit in this zoo, whether simulating nature is easy or hard, has no settled answer in mainstream physics.

The framework's declaration physics_complexity_structure establishes an answer within Recognition Science, a framework that models reality as maintaining a discrete record of events called a ledger. The declaration derives that the complexity of physics is determined by minimizing a cost function J(x) = (x + 1/x)/2 - 1, which measures how far a ledger state is from balance. The library proves this cost is strictly convex, meaning it has a single lowest point at x = 1, verifiable in constant time.

The complexity results follow from this structure. Verifying that a ledger state is balanced, meaning every ratio equals 1, requires checking all N bonds, a linear-time scan in P. Local dynamics update at most 8 neighbors per tick, so each step costs constant time. The framework's mass hierarchy uses powers of the golden ratio φ, and computing high-rung states requires exponentially many operations, placing that task in EXPTIME. The library proves φ > 1 and that φ^n grows without bound, so no finite computation can reach arbitrarily high rungs.

The declaration itself is a theorem that reduces the whole structure to a single proposition: computation_limits_from_ledger. It does not prove that simulating physics is NP-hard, nor does it claim that the framework's complexity classes match those of any specific physical theory. The NP-hard analog for global optimization is stated in a list of strings, a definition, not a theorem. The framework derives these limits from its own axioms, and the comparison to conventional complexity classes remains an interpretation, not a proved equivalence.

/-- **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-005.12**: Physics complexity structure holds. -/
theorem physics_complexity_structure : physics_complexity_from_ledger :=
  computation_limits_structure

What this page does not claim

The declaration does not prove that simulating physics is NP-hard; the NP-hard analog is a definition, not a theorem. The framework's complexity classes are not shown to match the complexity of any specific conventional physical theory. The framework does not claim that its complexity results apply outside its own axioms.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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