uniqueness_implies_stability
Uniqueness of the zero-cost state in any cost function rules out the existence of two distinct zero-cost states. Researchers deriving vacuum stability from the RS inevitability theorem cite this to establish that the electroweak vacuum cannot be metastable. The definition encodes the logical implication directly from the uniqueness hypothesis to the negation of degeneracy.
claimLet $c : ℝ → ℝ$ be a cost function. If there exists a unique $x$ such that $c(x) = 0$, then there do not exist distinct $x, y$ with $c(x) = c(y) = 0$.
background
The Recognition Science framework uses a cost function derived from the J-functional, where defect(x) equals J(x) for positive x, and cost measures recognition events or multiplicative recognizers. The inevitability theorem states that any alternative zero-parameter framework either matches RS or violates a necessity gate. In the QFT.VacuumStability module, this sets up the structural argument for E-002 on electroweak vacuum stability.
proof idea
This is a definition that directly states the proposition as a universal quantification over cost functions. It serves as a one-line wrapper translating uniqueness of the minimizer into the absence of multiple zero-cost points.
why it matters in Recognition Science
This definition underpins the rs_vacuum_stability_structural theorem, which applies the inevitability result (F-002) to conclude absolute stability of the vacuum. It fills the E-002 registry item by showing that uniqueness from the forcing chain precludes metastability. The eight-tick octave and D=3 dimensions are assumed in the broader framework but not directly here.
scope and limits
- Does not establish that any particular cost function has a unique minimum.
- Does not address time-dependent evolution or tunneling rates.
- Does not incorporate specific RS constants such as phi or alpha.
- Does not prove stability for non-unique cases.
formal statement (Lean)
29def uniqueness_implies_stability : Prop :=
proof body
Definition body.
30 ∀ cost : ℝ → ℝ, (∃! x, cost x = 0) →
31 ¬ ∃ x y : ℝ, x ≠ y ∧ cost x = 0 ∧ cost y = 0
32
33/-- **E-002 Structural**: The RS inevitability theorem (F-002) establishes
34 that the framework is unique. Inextricably, the vacuum (zero-defect
35 state) is the unique minimum. Therefore the vacuum cannot be metastable
36 — there is no alternative minimum to decay into. -/