Information Physics Complexity Structure Phi Rung Complexity Unbounded

The unbounded rung ladder

The golden ratio φ, about 1.618, has a signature property: its powers grow without bound. Raise φ to the first power and you get 1.618, to the tenth about 122.99, to the twentieth about 15126. The sequence never peaks. In mathematics this is expressed by saying that for any real number M, however large, there exists a natural number n such that φ raised to the n-th power is greater than M. This is a standard fact about any number larger than 1, and it is the entire content of the declaration named phi_rung_complexity_unbounded in the Recognition Science library.

The declaration is a theorem in a machine-checked collection of formal theorems. Its proof is one line: it invokes a general lemma about powers of numbers greater than 1. The theorem does not depend on any of the framework's specific assumptions about physics or computation. It is a pure statement about the real numbers and the golden ratio, and it holds in ordinary mathematics.

In Recognition Science, this fact about φ gets a specific use. The framework models physical states as configurations of ratios between neighboring quantities, and it associates each such ratio with a cost. The framework's central cost function is J(x) = (x + 1/x)/2 - 1, which is strictly convex and has its unique minimum at x = 1. The framework also identifies a hierarchy of special ratios, called rungs, that follow powers of φ. The theorem phi_rung_complexity_unbounded says that this ladder of rungs is infinite: there is no highest rung.

The framework's library uses this to argue that computing the state at a very high rung requires an unbounded amount of work. Since φ^n grows without bound, the number of operations needed to reach the n-th rung also grows without bound. The library labels this computational class as EXPTIME, meaning that the time required grows exponentially with the rung number. This is a statement about the framework's own model of physics, not a claim about any actual physical system.

The theorem itself is narrow. It does not say that physics is computationally intractable in general. It does not say that the framework's model of physics is correct. It does not say that the golden ratio is special among all numbers for this property, since any number greater than 1 has unbounded powers. It only establishes the unboundedness of the φ-powered sequence, which the framework then uses as a building block in its larger argument about computational complexity.

/-- **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-005.15**: 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

/-- 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 theorem does not say that physics is computationally intractable in general. The theorem does not say that the framework's model of physics is correct. The theorem does not say that the golden ratio is unique among numbers in having unbounded powers.

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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