Information Computation Limits Structure No Exact Phi Computation

The exact-value barrier

The golden ratio, usually written φ, is the number that solves r² = r + 1, so it is about 1.618. A classical fact about it is that it is irrational: it cannot be written as a fraction of two whole numbers. The standard proof goes through its defining equation, since the only rational candidates for a root of x² - x - 1 are 1 and -1, and neither works. This means any decimal expansion of φ is an approximation, never the number itself.

In Recognition Science, this classical fact becomes a statement about what a computer can do. The framework models reality as a ledger, a discrete record of events, and its fundamental time unit is the tick, the smallest possible step. The framework's library of machine-checked theorems contains the declaration no_exact_phi_computation. It states that for every rational number q, q is not equal to φ. In plain terms, no calculation that uses only fractions and whole numbers can ever output the exact value of the golden ratio.

The declaration is a direct consequence of φ being irrational, which the library proves separately. It does not depend on the size of the numbers involved or the speed of the machine. The claim is absolute: every rational number misses φ by some nonzero amount. This is not a practical limitation that better hardware could overcome; it is a structural one, baked into the definition of rational numbers.

What the declaration does not claim is just as important. It does not say that φ cannot be approximated. Algorithms can produce rational numbers arbitrarily close to φ, and the library itself proves that powers of φ grow without bound. It also does not say that a physical computer could not, in principle, represent φ through some other means, such as a geometric construction or a symbolic expression. The theorem is specifically about rational arithmetic, the kind of calculation that uses only fractions.

In the framework's account, this irrationality is one of three sources of computational limits, alongside the discrete tick and thermodynamic floors like the Landauer cost of erasing a bit. The consequence is that any simulation of the framework's own dynamics, which involve φ, cannot be exact on a machine that works with rational numbers. The limit is not a wall against progress; it is a precise boundary on what a certain kind of computation can achieve.

/-- **THEOREM IC-002.7**: There is no finite-precision algorithm that exactly computes
    φ in the sense that any rational number differs from φ. -/
theorem no_exact_phi_computation (q : ℚ) : (q : ℝ) ≠ phi := by
  intro heq
  apply phi_irrational
  exact Set.mem_range.mpr ⟨q, heq⟩

theorem computation_limits_structure : computation_limits_from_ledger := phi_irrational

/-- **THEOREM IC-002.15**: φ-based costs grow without bound as exponents increase.
    This means RS dynamics at high rung numbers require exponentially growing resources. -/
theorem phi_powers_unbounded (M : ℝ) : ∃ n : ℕ, phi ^ n > M := by
  obtain ⟨n, hn⟩ := pow_unbounded_of_one_lt M one_lt_phi
  exact ⟨n, hn⟩

/-- **THEOREM IC-002.4**: φ is irrational.
    This is the core structural constraint on RS computation:
    exact representation of RS constants requires transcendental arithmetic. -/
def computation_limits_from_ledger : Prop := Irrational phi

What this page does not claim

This does not claim that φ cannot be approximated arbitrarily well by rational numbers. This does not claim that a physical computer cannot represent φ through non-rational means. This does not claim that the theorem applies to all forms of computation, only to rational arithmetic.

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/ComputationLimitsStructure.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.

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