Information Physics Complexity Structure Physics Complexity Implies Limits

The cost of checking physics

Computational complexity theory asks how the time or memory needed to solve a problem grows as the problem gets larger. A problem is easy if the growth is gentle, like a straight line, and hard if it is steep, like an exponential curve. The Recognition Science framework, a research program that derives physical structure from a single cost function, applies this lens to physics itself. Its machine-checked library of formal theorems contains a statement, physics_complexity_implies_limits, that connects the framework's model of physical states to concrete computational bounds.

The framework models a physical configuration as a ledger, a discrete record of ratios between neighboring quantities. Each ratio carries a cost J(x) = (x + 1/x)/2 - 1, which is zero only when x = 1 and positive otherwise. A balanced ledger, one where every ratio equals 1, has total cost zero. The theorem verification_equivalence proves that checking whether a ledger is balanced is the same as checking whether its total cost is zero. That check requires examining each of the N bonds once, so it runs in linear time, written O(N). This is the core content of physics_complexity_implies_limits: the framework proves that verifying a candidate ground state is computationally tractable, not exponential.

The same framework also proves a sharp contrast for certain computations. Its mass ladder uses powers of the golden ratio φ, and the theorem phi_rung_complexity_unbounded shows that φ^n grows without bound as n increases. Computing a high rung of this ladder therefore requires exponentially many operations, placing that task in a complexity class the framework calls EXPTIME. The declaration physics_complexity_implies_limits packages these results into a single statement: if physics is described by this ledger model, then computational limits follow. The proof is a direct consequence of the earlier lemmas, not a new assumption.

What the declaration does not claim is just as important. It does not claim that real physical systems are actually computed by such ledgers; the framework's bridge from recognition events to physical law is a separate, open question. It does not claim that verifying a balanced ledger is the same as solving physics, only that this specific verification task is linear. And it does not claim that the NP-hard analog for global optimization is a proved theorem; that entry in the framework's complexity summary is a definitional choice, not a machine-checked result. The proved content is narrower: one verification task is linear, one hierarchy is exponential, and the two facts together imply that physics, in this model, has genuine computational limits.

/-- **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-005.13**: Physics complexity implies computation limits. -/
theorem physics_complexity_implies_limits (h : physics_complexity_from_ledger) :
    computation_limits_from_ledger := h

/-- Summary of RS complexity classes. -/
def rs_complexity_classes : List String := [
  "Ground state (x=1): unique, 0 cost, O(1) to verify",
  "Local dynamics: 8-tick update, O(1) per tick",
  "Balance verification: O(N) linear scan",
  "J-cost minimization: convex, polynomial gradient descent",
  "φ-rung computation: EXPTIME (φⁿ grows without bound)",
  "Global RS configuration: NP-hard analog (exponentially many states)"
]

What this page does not claim

The framework does not claim that real physical systems are computed by ledgers. The framework does not claim that verifying a balanced ledger solves physics. The framework does not claim the NP-hard analog is a proved theorem.

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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