Foundation Phi Forcing Phi Gt One Point Six One Eight

A tight bound on the golden ratio

The golden ratio is the positive number that satisfies the equation x² = x + 1. Its exact value is (1 + √5)/2, roughly 1.6180339887. It appears throughout classical mathematics, from the regular pentagon to the Fibonacci sequence, where the ratio of successive terms approaches it. The equation itself has two solutions; the other is negative, so the golden ratio is the unique positive one.

The theorem phi_gt_onePointSixOneEight proves that this number is greater than 1.618. The proof works by comparing squares: since (2.236)² = 4.999696 is less than 5, the square root of 5 must be greater than 2.236, and substituting that into (1 + √5)/2 gives a value above 1.618. A companion theorem, phi_lt_onePointSixOneNine, proves it is less than 1.619 by a similar argument with 2.238. Together they bound the golden ratio to three decimal places.

In Recognition Science, the framework models reality as a discrete ledger, a record of events with a forced cost structure. The framework proves that if such a ledger is self-similar, meaning the same cost structure appears at every scale, then the scale ratio must satisfy x² = x + 1. The unique positive solution is the golden ratio, so the framework derives φ as the only possible non-trivial self-similar scale. The bound theorem then confirms that this forced ratio lies in the narrow interval between 1.618 and 1.619.

This result is a piece of pure mathematics, machine-checked in the framework's library of formal theorems. It does not say anything about physical measurements; it establishes a property of a number defined by an equation. The bound is exact and rigorous, but its significance depends on the framework's broader claims about ledgers and self-similarity, which are separate theorems.

/-- φ > 1.618. -/
theorem phi_gt_onePointSixOneEight : φ > (1.618 : ℝ) := by
  simp only [φ]
  have h5 : Real.sqrt 5 > (2.236 : ℝ) := by
    have h : (2.236 : ℝ)^2 < 5 := by norm_num
    rw [← Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2.236)]
    exact Real.sqrt_lt_sqrt (by norm_num) h
  linarith

/-- The golden constraint characterizes φ among positive reals. -/
theorem golden_constraint_unique {r : ℝ} (hr_pos : 0 < r) (hr_eq : satisfies_golden_constraint r) :
    r = φ := by
  -- r² = r + 1 has solutions (1 ± √5)/2
  -- Only (1 + √5)/2 is positive
  simp only [satisfies_golden_constraint] at hr_eq
  have h : r^2 - r - 1 = 0 := by linarith
  -- Use quadratic formula and positivity
  -- The solutions are (1 ± √5)/2, and only (1 + √5)/2 > 0
  have h5 : Real.sqrt 5 > 2 := by
    have h4 : (4 : ℝ) < 5 := by norm_num
    have hsqrt4 : Real.sqrt 4 = 2 := by
      rw [show (4 : ℝ) = 2^2 by norm_num, Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2)]
    calc Real.sqrt 5 > Real.sqrt 4 := Real.sqrt_lt_sqrt (by norm_num) h4
      _ = 2 := hsqrt4
  -- The positive root is (1 + √5)/2
  have hsq5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 5)
  -- Verify φ satisfies the equation
  have hphi_satisfies : φ^2 = φ + 1 := phi_equation
  -- Both r and φ satisfy x² = x + 1, and both are positive
  -- The polynomial x² - x - 1 has exactly two roots
  -- Since r > 0 and φ > 0, and the other root is negative, we have r = φ
  nlinarith [sq_nonneg (r - φ), sq_nonneg (r + φ - 1), phi_pos, hsq5]

/-- **PHI FORCING THEOREM**: In a self-similar discrete ledger, the scale ratio is φ.

If:
1. L is a discrete ledger (from DiscretenessForcing + LedgerForcing)
2. L is self-similar with scale ratio r
3. r satisfies the compositional constraint r² = r + 1

Then: r = φ = (1 + √5)/2 -/
theorem phi_forced (L : DiscreteLedger) (r : ℝ) (hr : is_self_similar L r) : r = φ := by
  rcases hr with ⟨S, rfl⟩
  exact golden_constraint_unique S.ratio_pos (self_similar_forces_golden_constraint S)

What this page does not claim

This page does not claim that any physical object has a golden ratio scale. The bound theorem says nothing about the framework's broader claims regarding the fine-structure constant or particle masses. The proof of the bound does not depend on the framework's cost function; it is a fact about the real numbers.

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/Foundation/PhiForcing.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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