rs_vacuum_stability_structural
plain-language theorem explainer
The structural theorem shows that uniqueness of the zero-cost vacuum state precludes any pair of distinct degenerate minima, closing off metastability in the Recognition Science ledger. Physicists examining electroweak vacuum stability would cite it to rule out decay channels once the inevitability theorem is granted. The proof is a direct contradiction that extracts the unique minimizer and forces the supposed degenerate pair to coincide.
Claim. Let $cost : ℝ → ℝ$ be the cost function induced by a multiplicative recognizer. If there exists a unique $x_0$ such that $cost(x_0) = 0$, then there do not exist distinct $x, y$ with $cost(x) = 0$ and $cost(y) = 0$.
background
In Recognition Science the defect functional equals the J-cost on positive reals and vanishes at unity. The cost function is the derived cost of a comparator acting on positive ratios. The module formalizes E-002, the structural half of the electroweak vacuum stability question, by showing that the RS inevitability theorem forces the zero-defect ledger state to be the unique minimum.
proof idea
The term proof assumes a cost function together with a uniqueness hypothesis asserting a single $x_0$ at zero cost and a degeneracy hypothesis asserting two distinct points at zero cost. It extracts $x_0$ from uniqueness, equates both points to $x_0$, and obtains an immediate contradiction with the distinctness assumption.
why it matters
The result supplies the core implication used by unique_vacuum_forbids_degenerate_minima, which directly excludes two distinct zero-cost vacua. It completes the structural component of E-002 and ties to the inevitability theorem F-002. Within the framework it reinforces that J-uniqueness and the phi-ladder admit only a single minimum, consistent with the absence of alternative ledger states.
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