phi_ring_certificate
The declaration certifies that the ring ℤ[φ] satisfies φ² = φ + 1, possesses a multiplicative norm N(a + bφ) = a² + ab - b², and admits Galois conjugation that is an involutive ring automorphism with N(φ) = -1. Recognition Science modelers building the J-cost function and quadratic-integer layer would cite this to anchor algebraic operations. The proof is a term-mode tuple that directly assembles six prior lemmas without additional reasoning.
claimIn the ring ℤ[φ] of elements a + bφ with a, b ∈ ℤ, the golden ratio satisfies φ² = φ + 1. The norm N(zw) = N(z)N(w) holds for all z, w. Conjugation σ is an involution (σ(σ(z)) = z) and ring homomorphism (σ(zw) = σ(z)σ(w)). Moreover N(1) = 1 and N(φ) = -1.
background
PhiInt is the structure whose elements are pairs (a, b) representing a + bφ with a, b integers; its toReal embedding recovers the usual real value. The norm on this ring is the multiplicative function N(a + bφ) = a² + ab - b², and conjugation is the map sending a + bφ to its Galois conjugate. The local setting is the PhiRing module, which equips ℤ[φ] with ring operations and links it to the cost algebra via J-automorphisms. Upstream, phiInt_sq establishes the minimal polynomial φ² = φ + 1 by direct computation on coefficients, while multiplicative from CostAlgebra records that J-automorphisms preserve the posMul operation.
proof idea
The proof is a one-line term that constructs the required six-tuple by referencing the component lemmas: phiInt_sq supplies the defining equation, PhiInt.norm_mul supplies multiplicativity of the norm, PhiInt.conj_involution and PhiInt.conj_mul supply the involution and homomorphism properties of conjugation, and PhiInt.norm_one together with PhiInt.norm_phi supply the two norm evaluations.
why it matters in Recognition Science
This certificate bundles the algebraic facts needed to treat φ as a unit in the quadratic ring that underlies the Recognition Composition Law and J-cost computations. It directly records the connection to cost algebra noted in the module documentation and aligns with the self-similar fixed-point role of φ in the T5–T6 forcing chain. With zero recorded downstream uses, its integration into mass-ladder or Berry-threshold arguments remains open.
scope and limits
- Does not prove that ℤ[φ] is the unique quadratic ring satisfying these axioms.
- Does not compute explicit values of the J-cost on general elements of the ring.
- Does not derive the eight-tick octave or spatial dimension D = 3 from these properties.
- Does not address embedding of the ring into the full UnifiedForcingChain.
formal statement (Lean)
471theorem phi_ring_certificate :
472 -- φ satisfies defining equation
473 (PhiInt.phi * PhiInt.phi = PhiInt.phi + PhiInt.one) ∧
474 -- Norm is multiplicative
475 (∀ z w : PhiInt, PhiInt.norm (z * w) = PhiInt.norm z * PhiInt.norm w) ∧
476 -- Conjugation is involution
477 (∀ z : PhiInt, z.conj.conj = z) ∧
478 -- Conjugation preserves multiplication
479 (∀ z w : PhiInt, (z * w).conj = z.conj * w.conj) ∧
480 -- N(1) = 1
481 PhiInt.norm 1 = 1 ∧
482 -- N(φ) = -1
483 PhiInt.norm PhiInt.phi = -1 :=
proof body
Term-mode proof.
484 ⟨phiInt_sq, PhiInt.norm_mul, PhiInt.conj_involution,
485 PhiInt.conj_mul, PhiInt.norm_one, PhiInt.norm_phi⟩
486
487end PhiRing
488end Algebra
489end IndisputableMonolith