vacuum_stability_implies_schema
The theorem asserts that the uniqueness-implies-stability property directly supplies the structural no-decay marker for the vacuum. Researchers examining electroweak vacuum metastability under Recognition Science would cite it when connecting framework uniqueness to absolute stability. The proof is a one-line term that returns the input hypothesis without further reduction.
claimIf a cost function $C:ℝ→ℝ$ admits a unique state $x$ with $C(x)=0$, then no distinct states $x,y$ exist with $C(x)=C(y)=0$.
background
The module develops the RS structural case for electroweak vacuum stability under registry item E-002. The central definition states that a unique global cost minimum at zero precludes any pair of distinct zero-cost states, because metastability would require at least two consistent minima. Upstream results supply the cost as the J-cost of a recognition event and the vacuum as the gauge-bond configuration with every bond at rung zero.
proof idea
The proof is a one-line term that returns the supplied hypothesis directly.
why it matters in Recognition Science
This declaration supplies the structural half of E-002 by showing that uniqueness of the cost minimum (inherited from the inevitability theorem) immediately forbids decay channels. It therefore supports the claim that the RS framework produces an absolutely stable vacuum. The result links to the Yang-Mills vacuum definition but currently has no downstream applications.
scope and limits
- Does not derive uniqueness of the minimum from the forcing chain.
- Does not compute tunneling rates or lifetimes.
- Does not specify the explicit form of the cost function.
- Does not incorporate finite-temperature or external-field corrections.
formal statement (Lean)
55theorem vacuum_stability_implies_schema (h : uniqueness_implies_stability) :
56 uniqueness_implies_stability :=
proof body
Term-mode proof.
57 h
58
59/-- Unique vacuum immediately excludes two distinct zero-cost vacua. -/