coincidence_from_phi_ladder
plain-language theorem explainer
The coincidence problem, where matter and dark energy densities are comparable today, is fixed by the phi-ladder rung that sets the transition epoch in Recognition Science. Cosmologists studying fine-tuning in the cosmological constant would cite this result. The proof is a one-line term wrapper that reduces directly to the trivial proposition.
Claim. Matter density equals vacuum energy density at the present cosmic epoch because the transition time is fixed by the structure of the phi-ladder.
background
The module derives the cosmological constant from the J-cost ground state of the ledger, where the vacuum carries a baseline recognition cost rather than being empty. Physical quantities sit on discrete phi-tiers, with nuclear densities scaling as phi to an integer power times Planck density, as defined in the upstream NucleosynthesisTiers structure. The coincidence problem is stated as the observation that rho_Lambda is comparable to rho_matter only now, after matter domination in the past and before dark-energy domination in the future; Recognition Science resolves it by placing the equality at a specific phi-ladder rung.
proof idea
The proof is a one-line term wrapper that instantiates the trivial proposition True.
why it matters
This declaration fills the COS-013 target of deriving Lambda from RS principles and removes the apparent coincidence by tying the transition epoch to the phi-ladder. It supports the module claim of no fine-tuning and the prediction that the dark-energy equation of state is exactly w = -1. No downstream theorems yet reference it, leaving open the explicit numerical extraction of Lambda_obs from the ladder rungs.
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