particleHorizonRungCount
plain-language theorem explainer
The ΛCDM particle horizon sits 294 rungs above the substrate on the φ-ladder. That integer fixes the vacuum-energy suppression factor φ^(-588) relative to the Planck density. Anyone comparing particle, Hubble, and de Sitter horizon predictions in the vacuum-horizon forcing argument cites this constant. It is a bare integer definition, not a derived equality.
Claim. The particle-horizon rung count is the integer $s = 294$. With vacuum density scaling $\rho_\Lambda = \rho_{\mathrm{Pl}}\,\varphi^{-2s}$, this choice yields the exponent $-588$, i.e. $\rho_\Lambda/\rho_{\mathrm{Pl}} = \varphi^{-588}$.
background
In this module the vacuum energy is written $\rho_\Lambda = \rho_{\mathrm{Pl}}\cdot\varphi^{-2s}$, where $s$ is the rung count on the $\varphi$-ladder from the substrate scale out to a chosen cosmological horizon. Three standard horizons give three integers: particle horizon $s=294$, Hubble radius $s=289$, de Sitter event horizon $s=290$. Only $s=294$ lands near the observed density ratio.
The selection rule is the causal-accumulation principle. The recognition ledger $\mathcal{L}(i,j)$ is defined only for substrate cells that have already exchanged a comparison; two cells compare iff a signal has traveled between them since the initial condition. The vacuum ledger cost is therefore the ground-state total over the maximal causally connected region. By definition that region is bounded by the particle horizon (maximum comoving distance from which a signal has reached the observer since the Big Bang).
The Hubble radius is only the instantaneous $v=c$ surface and therefore undercounts past contacts still present in the ledger. The de Sitter event horizon requires future knowledge and is likewise excluded. The present definition simply records the particle-horizon integer used throughout the forcing chain.
proof idea
One-line constant definition: the integer $294$ is assigned directly. No lemmas, tactics, or algebraic reduction are involved. Downstream equalities such as $-2\cdot 294=-588$ and $2\cdot(294-289)=10$ are proved by unfolding this definition and applying ring.
why it matters
This constant is the numerical anchor of the vacuum-horizon forcing package. It is read by vacuumExponent_particleHorizon (exponent $-588$), by hubble_vs_particle_rung_gap (the $\varphi^{10}$ factor that accounts for the $\sim 12.7\times$ Hubble discrepancy), by the certificate record vacuumHorizonForcingCert, and by the one-statement theorem that packages selection plus numerics: particle horizon selected, rung count $294$, exponent $-588$.
Within Recognition Science the $\varphi$-ladder and the eight-tick / $D=3$ forcing chain already fix the discrete rung structure; cosmology then asks which horizon supplies the outer rung. Causal accumulation answers "particle horizon," and this definition supplies the matching integer so that $\rho_\Lambda/\rho_{\mathrm{Pl}}=\varphi^{-588}$ can be compared with observation. It does not itself close an open gap in the forcing chain; it is the shared numeric input those later statements need.
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