equation_of_state
plain-language theorem explainer
This definition computes the equation of state parameter for any constant energy contribution as the negative ratio of its energy density to itself. Cosmologists working in the Recognition Science framework cite it when replacing an ad-hoc assignment of w with an explicit computation from phase-locked modes. The body is a direct algebraic expression that evaluates to negative one.
Claim. For a constant energy contribution with positive energy density $ρ$, the equation of state parameter is $w = -ρ/ρ$.
background
The module derives the dark energy equation of state from phase-locked recognition modes. ConstantEnergyContribution is a structure holding an energy density that stays fixed across ticks and satisfies the thermodynamic relation dE = -p dV. When energy is constant this forces ρ + p = 0, hence w = p/ρ = -1 for a Lorentz-invariant vacuum. Phase-locked modes have J(1) = 0 at every tick, so their energy density is spatially uniform and temporally constant. The local setting replaces a prior direct assignment w := -1 with this explicit definition, as stated in the module documentation: 'Derives w = -1 from the physics of phase-locked recognition modes, replacing the previous definitional w := -1 with a theorem.' Upstream results supply the fundamental tick as the RS time quantum equal to 1 and the 8-tick phases kπ/4.
proof idea
The definition body is the direct expression -energy_density / energy_density. This one-line algebraic form is unfolded in the downstream theorem that establishes equality to -1, using neg_div and div_self on the positive energy density.
why it matters
This definition supports the certificate structure and the theorem asserting that the equation of state parameter equals -1 for every constant energy contribution. It implements the derivation in Dark_Energy_Mode_Counting.tex §7.1, Theorem 7.1, that links phase-locked modes with zero J-cost at the fixed point to the vacuum condition ρ + p = 0. The result aligns with the eight-tick octave and phase locking in the foundation.
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