vacuum_exists
plain-language theorem explainer
The vacuum state in the Recognition Science Hilbert space has vanishing J-cost. Field theorists and cosmologists verifying Wightman axioms on H_RS would cite this when confirming the zero-energy ground state required by W2. The proof is a direct one-line application of the unit lemma for the J-cost function.
Claim. The J-cost of the vacuum configuration is zero: $J(1) = 0$, where $J(x) = (x-1)^2/(2x)$ measures deviation from the ground state.
background
The J-cost function is defined by $J(x) = (x-1)^2/(2x)$, which quantifies the squared ratio deviation from unit scaling. In the Wightman Axioms Status module the RS Hilbert space H_RS is shown to carry the axioms W0-W5, with W2 specifically the existence of a vacuum state satisfying J=0. This theorem depends on the lemma establishing Jcost(1)=0 by direct simplification of the J-cost definition and on the vacuum definition as the gauge bond configuration with all bonds at rung zero.
proof idea
This is a one-line wrapper that applies the Jcost_unit0 lemma from the Cost module, which itself follows by simp on the J-cost definition.
why it matters
This result supplies W2 (vacuum existence) inside the Recognition Science realization of the Wightman axioms. It is invoked by the dark energy equation-of-state certificate, which requires a phase-locked vacuum to derive w = -1, and by the overall Wightman status certificate that counts five axioms holding on H_RS. The placement aligns with the forcing chain base where the vacuum sits at rung zero of the phi-ladder.
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