Information Computation Limits Structure Phi Minimal Polynomial No Rational Root

An irrational root

The golden ratio, usually written φ, is the positive number that satisfies the equation x² = x + 1. Solving it gives φ = (1 + √5)/2 ≈ 1.618. A classical fact about this number is that it is irrational, meaning it cannot be written as a fraction p/q of two whole numbers. One way to prove that is to note that if a rational number solved x² - x - 1 = 0, then by the rational root theorem it would have to be either 1 or -1, and neither works. The equation has no rational roots at all.

This fact is what the declaration phi_minimal_polynomial_no_rational_roots establishes in the machine-checked library of formal theorems. Its statement is precise: for every rational number q, the value q² - q - 1 is not zero. In plainer terms, no fraction, no matter how complicated, plugs into that polynomial to give zero. The declaration is a theorem, proved in the library's formal system, and it rests on the same reasoning just described: check the two candidates the rational root theorem allows, and both fail.

What follows from this theorem is a limit on exact computation. Because φ is irrational, no finite sequence of rational arithmetic operations can represent it exactly. The library draws this conclusion in a separate theorem, no_exact_phi_computation, which states that for every rational q, q is not equal to φ. The practical meaning is that any computer working with rational numbers, which is what ordinary floating-point arithmetic approximates, cannot hold the exact value of φ. Exact simulation of any process that depends on φ would require infinite precision.

In Recognition Science, this observation connects to the framework's account of computation limits. The framework models time as a discrete sequence of ticks, and it derives φ as the unique self-similar scaling of its recognition ledger, a discrete record of events. Because that scaling is irrational, the framework argues, exact simulation of its dynamics is impossible with finite rational arithmetic. The theorem about rational roots is the algebraic engine of that argument: it certifies that no rational shortcut exists.

The declaration does not claim that no algorithm can approximate φ. Approximating it to any desired accuracy is easy and classical; the claim is only about exact representation. It also does not claim that φ is transcendental. φ is algebraic, since it solves a polynomial with integer coefficients, but it is not rational. The distinction matters: a transcendental number like π is not the root of any polynomial with rational coefficients, while φ is the root of this one, just not a rational root.

/-- **THEOREM IC-002.6**: The golden ratio satisfies an irreducible quadratic.
    φ is a root of x² - x - 1 = 0, which has no rational roots (by rational root theorem,
    any rational root would be ±1, but 1² - 1 - 1 = -1 ≠ 0 and (-1)² - (-1) - 1 = 1 ≠ 0). -/
theorem phi_minimal_polynomial : phi ^ 2 - phi - 1 = 0 := by
  have := phi_sq_eq
  linarith

/-- The quadratic x² - x - 1 has no rational root (roots are φ and 1-φ). -/
theorem phi_minimal_polynomial_no_rational_roots :
    ∀ q : ℚ, (q : ℝ)^2 - (q : ℝ) - 1 ≠ 0 := by
  intro q hq
  have hφprod : phi * (1 - phi) = -1 := by
    have hsq := phi_sq_eq
    linarith
  have hfactor :
      ((q : ℝ) - phi) * ((q : ℝ) - (1 - phi)) =
        (q : ℝ)^2 - (q : ℝ) - 1 := by
    calc ((q : ℝ) - phi) * ((q : ℝ) - (1 - phi))
        = (q : ℝ)^2 - (q : ℝ) * (phi + (1 - phi)) + phi * (1 - phi) := by ring
      _ = (q : ℝ)^2 - (q : ℝ) * 1 + phi * (1 - phi) := by ring
      _ = (q : ℝ)^2 - (q : ℝ) + (-1) := by rw [hφprod]; ring
      _ = (q : ℝ)^2 - (q : ℝ) - 1 := by ring
  have hzero : ((q : ℝ) - phi) * ((q : ℝ) - (1 - phi)) = 0 := by
    rw [hfactor, hq]
  rcases mul_eq_zero.mp hzero with h | h
  · exact phi_not_rational q (sub_eq_zero.mp h)
  · have hφ : (phi : ℝ) = 1 - (q : ℝ) := by
      have := sub_eq_zero.mp h
      linarith
    exact phi_not_rational (1 - q) (by simp [hφ])

/-- **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⟩

/-- The fundamental tick: minimum time quantum in RS. -/
def fundamental_tick : ℝ := τ₀

What this page does not claim

No claim that φ is transcendental; it is algebraic, solving a polynomial with integer coefficients. No claim that approximation of φ is impossible; only exact rational representation is ruled out. No claim that this theorem alone establishes the framework's full account of computation limits.

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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