economic_inevitability
plain-language theorem explainer
The theorem collects four statements establishing that the defect function is non-negative with a unique minimum at one, that costs remain finite, and that the golden ratio satisfies its characteristic equation. Researchers deriving physics from a unique cost-minimization principle would cite it when closing the selection and self-similarity gates. The proof proceeds by direct term construction from four upstream lemmas on defect properties and the phi equation.
Claim. Let $d$ be the defect function. Then $d(x)geq 0$ for every $x>0$, with equality if and only if $x=1$. For any real number $C$ there exists $varepsilon>0$ such that $d(x)>C$ whenever $0<x<varepsilon$. The golden ratio $phi$ satisfies $phi^2=phi+1$.
background
The module formalizes the inevitability structure under a cost-as-foundation approach. Here the defect function plays the role of the J-cost, defined via the unique function satisfying analyticity, symmetry, convexity and normalization, which yields $J(x)=frac12(x+x^{-1})-1$. Selection occurs by driving this defect to zero under coercive projection. From the module documentation: Selection happens by minimizing a unique cost. Any alternative theory must violate one of the CPM/cost necessities including cost uniqueness (T5), selection rule, discreteness, ledger structure, self-similarity to phi, and dimension. This theorem packages the cost non-negativity, uniqueness of minimum, finiteness, and the phi equation from the PhiRing module, where phi_equation states that $phi^2=phi+1$.
proof idea
The proof is a term-mode construction of the conjunction by supplying four explicit terms. It applies the non-negativity lemma for the defect function, the lemma characterizing when the defect vanishes, the lemma asserting that nothing cannot exist, and the theorem establishing the phi equation.
why it matters
This declaration assembles the core necessities for the cost-based foundation, directly supporting the module's claim that alternatives must break one of the listed gates. It cites the inevitability theorem by packaging cost uniqueness from T5 and the forcing of phi as the self-similar fixed point. The result closes the economic inevitability choke point, leaving open the closure of dimension forcing and framework exclusivity as noted in the module summary.
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