inv
plain-language theorem explainer
The subfield generated by φ over the reals is closed under inversion. Modelers handling algebraic dependence on φ in Recognition Science cite this when preserving PhiClosed elements under field operations. The proof is a direct one-line application of the Subfield.inv_mem axiom.
Claim. If $x$ lies in the subfield generated by $φ$, then so does $x^{-1}$.
background
PhiClosed φ x holds precisely when x belongs to phiSubfield φ, the subfield of ℝ obtained by closing the singleton {φ} under addition, multiplication, and inverses. The RecogSpec.Core module introduces these notions to track algebraic payloads that remain inside the φ-generated subfield. The upstream phiSubfield definition supplies the Subfield structure whose standard membership lemmas are invoked here.
proof idea
One-line wrapper that applies the inv_mem property of the subfield generated by φ.
why it matters
The result supplies the inverse-closure step required by downstream theorems such as toComplex_inv in ComplexFromLogic and the automorphism symmetry lemmas in RecogGeom.Symmetry. It supports algebraic manipulations inside the φ-subfield that appear throughout the Recognition framework, including constructions that rely on the self-similar fixed point φ.
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