sub
plain-language theorem explainer
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.
Claim. If $x$ 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
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.
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