Algebra Phi Ring Phi Int Sq

The golden ratio ring

The golden ratio φ = (1 + √5)/2 is the positive solution to x² = x + 1, the equation that defines its self-similarity. The set of all numbers of the form a + bφ, where a and b are integers, forms a ring: it is closed under addition, subtraction, and multiplication. This ring, written ℤ[φ], is the ring of integers in the real quadratic field ℚ(√5), and it is a Euclidean domain, meaning it has a division algorithm and unique prime factorization.

The ring's structure is rich. Its units, the elements with multiplicative inverses, are exactly ±φⁿ for integer n, an infinite cyclic group generated by φ. The conjugation map sending φ to ψ = (1 − √5)/2 is the unique nontrivial automorphism, and the norm N(a + bφ) = a² + ab − b² is multiplicative: N(αβ) = N(α)N(β). A classical fact: the Fibonacci recurrence φⁿ⁺² = φⁿ⁺¹ + φⁿ holds for all integer n, connecting the ring directly to the Fibonacci sequence.

In Recognition Science, the framework's cost algebra forces the golden ratio as the unique self-similar scaling (theorem T6). The framework models all its native physical constants as elements of this ring ℤ[φ], so each one is an integer combination of 1 and φ. The declaration phiInt_sq establishes that φ² = φ + 1 within this ring, the algebraic identity that makes the ring work. This means the framework's constants, such as its values for ħ = φ⁻⁵ and G = φ⁵/π, are not arbitrary real numbers but live in this discrete, well-understood algebraic structure.

What phiInt_sq does not claim is that the golden ratio is the only self-similar scaling in all of mathematics. The Cantor set, for instance, is self-similar with a scaling factor of 3, and the Koch curve uses a factor of 4; neither is φ. The theorem applies within the framework's specific model of recognition costs, not to every notion of self-similarity. The declaration also does not claim that the ring ℤ[φ] is the only coefficient ring the framework could use, nor does it derive any specific physical constant's numerical value; it only establishes the algebraic identity φ² = φ + 1 that underpins the ring's structure.

/-- An element of ℤ[φ] is a pair (a, b) representing a + bφ. -/
@[ext]
structure PhiInt where
  /-- The "rational" part -/
  a : ℤ
  /-- The "φ" part -/
  b : ℤ

/-- **THEOREM: φ² = φ + 1** (the defining equation). -/
theorem phi_equation : φ ^ 2 = φ + 1 := by
  unfold φ
  have h5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (5:ℝ) ≥ 0)
  ring_nf
  nlinarith [h5]

/-- An element of ℤ[φ] is a pair (a, b) representing a + bφ. -/
@[ext]
structure PhiInt where
  /-- The "rational" part -/
  a : ℤ
  /-- The "φ" part -/
  b : ℤ

What this page does not claim

The golden ratio is the only self-similar scaling in all of mathematics; the Cantor set and Koch curve use other factors. The declaration derives any specific physical constant's numerical value; it only establishes the algebraic identity φ² = φ + 1. The framework's use of ℤ[φ] as its coefficient ring is the only possible choice.

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/Algebra/PhiRing.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: