Algebra Phi Ring Phi Psi Sum

The golden ratio's companion

The golden ratio φ = (1+√5)/2 ≈ 1.618 is famous for its self-similarity: it is the unique positive number satisfying φ² = φ + 1. Less famous is its algebraic conjugate ψ = (1−√5)/2 ≈ −0.618, the other root of the same quadratic equation. The Recognition Science declaration phi_psi_sum establishes the simple relation between them: φ + ψ = 1. This is a proved theorem in the framework's machine-checked library of formal theorems, derived directly from the definitions of the two numbers.

The relation is not deep numerology; it falls out of the quadratic formula. The two roots of x² − x − 1 = 0 sum to 1 by Vieta's formulas, since the coefficient of x is −1. The same library also proves φ − ψ = √5 and φψ = −1. Together these three identities mean the pair (φ, ψ) behaves like a miniature number system: their sum and product are simple integers, and their difference is the square root of 5. This is why the golden ratio and its conjugate appear together throughout algebraic number theory.

In Recognition Science, the golden ratio is forced by the cost algebra: self-similarity in a discrete ledger, a record of recognition events, requires x² = x + 1, whose unique positive root is φ. The framework models all its native physical constants as algebraic expressions in φ, and the ring ℤ[φ] = {a + bφ : a, b ∈ ℤ} serves as the natural coefficient ring. The conjugate ψ is not a separate physical constant; it is the algebraic shadow of φ under the Galois automorphism that swaps the two roots. The sum identity φ + ψ = 1 is therefore a structural fact about the number system the framework uses, not a new physical prediction.

What the declaration does not claim is just as important. It does not assert that ψ itself is a physical quantity in Recognition Science, nor that the sum φ + ψ = 1 has any direct experimental consequence. The theorem is purely algebraic: it states a relation between two real numbers defined by quadratic equations. The framework's broader claim, that physical constants live in ℤ[φ], is a separate modeling choice, not something this declaration establishes. The sum identity is a lemma in the algebraic toolkit, not a law of physics.

/-- **THEOREM: φ + ψ = 1** (trace). -/
theorem phi_psi_sum : φ + ψ = 1 := by
  unfold φ ψ; ring

/-- **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 conjugate ψ is not a physical quantity in Recognition Science. The sum φ + ψ = 1 has no direct experimental consequence. The theorem does not establish that physical constants live in ℤ[φ]; that is a separate modeling 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.

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