A Unified Framework for Witness Cosmology: Information, Cosmology, and the Cosmological Constant (Version 2)

We introduce the witness number density n_w as a physical quantity and propose the first unified equation linking the dark energy density to the thermodynamic cost of information preservation: ρ_Λ = λ(a) n_w (k_B T ln 2). This equation is dimensionally consistent, it constrains the present-day witness density to n_w^0 ≈ 2.3 × 10^13 m^-3 from observations, and it factorizes the cosmological constant problem into a thermal factor T_0/T_Pl ~ 10^-32 and a density factor N_w/N_max ~ 10^-30, whose product matches the observed value to within 20%. The framework rests on a foundational axiom: the universe is a witnessed contract with sixteen structural elements organized into five functional groups. From this axiom, we deduce: (i) the unified equation above; (ii) a beta function for the information–geometry coupling, β(λ) = 4λ, yielding λ(a) ∝ a^4; and (iii) a complete mapping of the sixteen structural elements onto physical counterparts. This Version 2 extends Version 1 with three new hypotheses: (A) The Consistency Bit: the logical content preserved by the universe is a single bit representing its self-consistency, with entropy log 2. (B) The Payment Mechanism: the running coupling λ(a) ∝ a^4 is the physical realization of a payment mechanism for information preservation. (C) The Information Hierarchy: information flows through three levels — logical, physical, and dynamical. The framework generates 12 testable predictions, including a Z_4 symmetry in w(z), a flat entanglement spectrum at large angular scales, and a dark matter particle mass in the range m ∈ [0.05, 0.5] meV. We discuss the predictions, limitations, and open questions. Main contributions:(1) Introduction of the witness number density n_w as a physical quantity.(2) The unified equation ρ_Λ = λ(a) n_w (k_B T ln 2).(3) Derivation of the beta function β(λ) = 4λ.(4) Exact factorization of the cosmological constant problem.(5) Complete mapping of the sixteen structural elements.(6) The consistency bit hypothesis.(7) The payment mechanism hypothesis.(8) The information hierarchy hypothesis.