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lemma proved term proof high

sub

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The sub lemma shows that the collection of φ-closed reals is closed under subtraction. Modelers constructing algebraic extensions or verifying field axioms in Recognition Science would reference it when building expressions involving differences of φ-related quantities. The proof reduces immediately to the sub_mem property of the subfield generated by φ.

claimIf $x$ and $y$ lie in the subfield of $ℝ$ generated by $φ$, then $x - y$ lies in the same subfield.

background

In the RecogSpec.Core module, phiSubfield φ is defined as the subfield of the reals generated by adjoining φ, specifically Subfield.closure({φ}). PhiClosed φ x holds precisely when x belongs to this subfield, meaning x can be obtained from φ by field operations. This setup provides the algebraic closure properties needed for the Recognition framework's use of the golden ratio φ as the self-similar fixed point. The lemma relies on the upstream definition of phiSubfield and the fact that subfields are closed under subtraction.

proof idea

The proof is a one-line wrapper applying the sub_mem operation from the Subfield structure of (phiSubfield φ) to the membership hypotheses hx and hy.

why it matters in Recognition Science

This closure under subtraction is a basic field property that underpins many downstream theorems, including the convexity of the J-action on interpolated paths and bounds on ratios in pulsar periods and periodic table constructions. It contributes to the algebraic foundation for the phi-ladder and Recognition Composition Law in the framework. The result ensures consistency in expressions involving differences within the φ-generated subfield.

scope and limits

formal statement (Lean)

  78lemma sub (hx : PhiClosed φ x) (hy : PhiClosed φ y) :
  79    PhiClosed φ (x - y) :=

proof body

Term-mode proof.

  80  (phiSubfield φ).sub_mem hx hy
  81

used by (40)

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