reciprocalGeneratorCert_holds
plain-language theorem explainer
Packages five proved properties into one certificate: reciprocal inversion ι(x)=x⁻¹ is an involution that generates both J-cost symmetry and the golden-ratio fixed-point equation. Anyone citing the structural unification of cost form and φ-forcing (rather than their mere conjunction) would reference this object. Construction is pure structure assembly: each field is filled by a lemma already proved in the same module.
Claim. There is a certificate asserting that the reciprocal map $\iota(x)=x^{-1}$ on positive reals is an involution; that the recognition cost satisfies $J(\iota(x))=J(x)$; that $\iota(x)=x$ iff $J(x)=0$; that among reals $>1$, the unique fixed point of the shift $g(x)=1+\iota(x)$ is $\varphi$; and that $\varphi$ itself solves $g(\varphi)=\varphi$.
background
The module isolates a single generator beneath two facts that the capstone otherwise only conjoins: the shape of the recognition cost $J$ and the forcing of the golden ratio $\varphi$. That generator is the reciprocal involution $\iota(x)=x^{-1}$ on the positive reals (dual-recognition symmetry exchanging a quantity with its reciprocal).
On the cost side, $J$ is exactly the $\iota$-symmetric cost ($J\circ\iota=J$), and the fixed-point set of $\iota$ coincides with the null set of $J$ (both are ${1}$). On the scale side, $\varphi$ is the unique fixed point $>1$ of the $\iota$-shift $g(x)=1+\iota(x)$. The certificate structure bundles the involution law, both downstream forcings, and the witness that $\varphi$ solves the shift equation.
Upstream lemmas already establish each piece: involutivity of $\iota$; $J(\iota x)=J(x)$; $\iota x=x\leftrightarrow J(x)=0$; uniqueness of the $g$-fixed point above $1$; and $g(\varphi)=\varphi$.
proof idea
One-line structure constructor. Each field of the certificate is discharged by a named lemma from the same module: involutivity by the reciprocal-involution theorem; cost symmetry by the $J\circ\iota=J$ theorem; unit-equals-cost-zero by the fixed-point/null-set coincidence; scale uniqueness by the unique-$g$-fixed-point theorem above $1$; and the $\varphi$-solves witness by the direct fixed-point check for $\varphi$. No new algebra is performed here.
why it matters
This is the structural unification the module advertises: both halves of the cost-and-scale story quantify over the same reciprocal map, so the certificate is a deduction about a shared generator rather than a glued pair of independent facts. Downstream, the public-spine theorem that packages $\varphi$-from-$\iota$ trace closure takes this certificate as its cost_and_scale witness.
In the forcing chain this sits under T5 (J-uniqueness via the symmetric cost form) and T6 ($\varphi$ as the self-similar fixed point). The reciprocal shift $g(x)=1+1/x$ is exactly the classical fixed-point equation for $\varphi$. The certificate does not itself re-prove J-uniqueness or the full T0–T8 chain; it records that one involution generates both the cost symmetry axis and the scale ratio.
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