The Cosmological Constant from a Hawkes–Cramér Branching Ratio

Statistical companion to Unified Dynamics from Time Generation. This work completes the regime-potential construction by modelling irreversible record formation as a self-exciting Hawkes process. Causal correlations between records generate a shifted vacuum of the Cramér rate-function potential, producing a nonzero cosmological constant without fine-tuning. Main results include: • Closed-form relation ρΛ = mϕ⁴ / (384 π² κ⁶)derived from the Hawkes branching ratio computed via the ψ-propagator overlap integral. • Radiative-stability and entropy-normalisation closure of the cubic scalar sector, preserving the ratio λ₃/μ² = 1/3. • Numerical determination of κ = 3.13** and Yukawa coupling ε = 0.017 from entanglement entropy inputs. • Prediction of a scalar mass mϕ ≈ 98 meV with Yukawa range ℓΓ ≈ 22 µm, within reach of next-generation torsion-balance experiments. Within the generative time framework, the cosmological constant emerges as a statistical large-deviation effect associated with Hawkes self-excitation of record events. The sign of Λ follows from the non-negativity of the Cramér rate function, and no adjustable parameters beyond κ and mϕ enter the final relation. This paper serves as the statistical closure of the dynamics construction and establishes a direct experimental bridge between sub-millimetre gravity tests and cosmological vacuum energy.**Subsequent work identifies the structural efficiency constant κ with the invariant normalisation scale of CP¹ (κ = π); the numerical value κ ≈ 3.13 is retained here to preserve the original statistical derivation.