vacuum_mode
plain-language theorem explainer
The vacuum mode definition constructs the phase-locked state at ratio one. Derivations of the dark energy equation of state cite it to anchor the constant energy density with zero recognition cost. The construction is a direct structure instantiation that sets the ratio field to one, applies reflexivity to the vacuum condition, and invokes the unit cost lemma.
Claim. Define the vacuum mode as the phase-locked structure satisfying ratio $=1$, with the vacuum condition holding by reflexivity and J-cost equal to zero.
background
Phase-locked modes are structures with a committed ledger entry at ratio one whose J-cost is zero and independent of the tick counter. The J-cost function is J(x) = (x-1)^2/(2x), so it vanishes at x=1. This module derives w = -1 from such modes, replacing a prior definitional assignment with a theorem, as stated in Dark_Energy_Mode_Counting.tex §7.1. Upstream, the Jcost unit zero lemma establishes Jcost 1 = 0 by direct simplification.
proof idea
This is a one-line structure instantiation. It sets the ratio field to 1, uses reflexivity to discharge the at-vacuum condition, and applies the Jcost unit zero lemma to establish the cost zero property.
why it matters
This definition supplies the vacuum mode instance required by the dark energy equation of state certificate. It fills the phase-locked mode existence step in the derivation of w = -1 from zero J-cost in phase-locked recognition modes, per paper theorem 7.1. It connects to the Recognition Science framework where phase-locked modes produce tick-independent energy density satisfying p = -ρ.
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