Part 32: Parameter Constraints and Order-of-Magnitude Estimates in Dual–H4 Geometry From Topological Relaxation to Observable Cosmological Scales

Overview Previous Parts of the Origin Geometry program developed a dual–H4 cosmological branch in which dark-sector topological relaxation, bulk stress propagation, nonlinear localization, void-driven lattice proliferation, and high-frequency bulk gravitational-wave-like backgrounds provide a possible microscopic interpretation of effective cosmic expansion 1–8. Part 29 described the mechanism of lattice proliferation, Part 30 identified observable cosmological signatures, and Part 31 developed the spectral and detectability framework for the associated bulk stress background. The present Part takes the next step: it introduces a first order-of-magnitude and parameter-constraint framework for the Origin Geometry cosmological branch. The central objective is to connect microscopic or mesoscopic quantities—such as relaxation event rate, energy release per event, stress density, activation probability, bulk transfer efficiency, and local topological growth rate—to macroscopic cosmological quantities such as effective Hubble expansion, void-dependent residuals, stochastic-background bounds, and large-scale energy-budget consistency 12–23. Quantitative Framework and Parameter Constraints We introduce an effective relaxation event rate per unit volume and time, Γᵣelax, an average energy release per event, E₀, and a bulk-channel efficiency, εbulk. The corresponding energy-density injection rate into bulk stress modes is: dρᵣelax/dt ~ Γᵣelax E₀ or, when only a fraction of the released energy enters bulk modes: dρbulk, inj/dt ~ εbulk Γᵣelax E₀ This provides the first bridge between microscopic topological events and a coarse-grained cosmological energy flow. To connect relaxation to geometric expansion, we introduce an activation probability Pₐct, which depends on whether local stress density exceeds a threshold. Dimensional consistency requires comparing energy density with energy density. Therefore, threshold activation is written as: Pₐct = P₀ Θ (ρₛtress − ρc) or, more smoothly: Pₐct = 1 + exp (- (ρₛtress − ρc) / Δρ) ⁻¹ If N denotes the number of realized effective network elements in a coarse-grained three-dimensional volume, then the effective scale factor obeys: aₑff ∝ N^ (1/3) so that: Hₑff ~ (1/3) (dN/dt) / N This factor is essential: a volumetric node-growth rate does not translate directly into a linear expansion rate without the 1/3 geometric conversion. Scope and Limitations The resulting phenomenological framework does not claim to derive the observed Hubble constant, fit precision cosmological data, or eliminate the cosmological constant 9–14. Rather, it shows how OG parameters may be constrained by requiring that relaxation-driven expansion remain compatible with observed expansion scales, void dynamics, stochastic gravitational-wave-like upper bounds, isotropy, and energy-budget consistency. At the phenomenological level, the OG relaxation contribution may mimic a dark-energy-like effective term without postulating an independent cosmological constant as a primitive input. This remains a conditional interpretation, not a completed replacement of ΛCDM. Part 32 therefore establishes the first quantitative bridge between the microscopic relaxation dynamics of dual–H4 geometry and observable cosmological scales.