This memorandum serves as a predictive roadmap for the application of the universal bivariate scaling law, V_eff(μ, σ) = μ^γ * F(σ / μ^κ), across divergent topological routes to chaos. By isolating the universal scaling architecture from class-specific geometric constants, we identify the solved eigenvalues for established universality classes and explicitly define the unmapped stochastic eigenvalues required to resolve higher-order and non-unimodal critical transitions. A validation taxonomy is provided to delineate confirmed empirical baselines from theoretical predictions.

This document records a mathematical scaling relationship describing how stochastic perturbations interact with proximity to critical transitions in nonlinear dynamical systems. An empirically observed bivariate scaling form is stated, along with a derived perturbation threshold relationship and a hub-dependent geometric correction factor. A validation taxonomy is provided to delineate confirmed empirical baselines from theoretical predictions. This memorandum places the mathematical relationships into the public scientific record to establish definitive prior art and ensure open accessibility.

This memorandum documents the computational extraction of the stochastic scaling exponent governing the spectral gap of a nonlinear dynamical system transitioning to criticality via the quasiperiodic route. Utilizing a discretized transfer operator methodology on the critical sine circle map, we observe a highly stable, parameter-independent power-law scaling in the subdominant eigenvalue under varying stochastic perturbation amplitudes. This document places the empirically derived scaling exponent into the public scientific record to establish a timestamped baseline for quasiperiodic stochastic vulnerability.

This memorandum places into the public scientific record an empirical validation of the bivariate scaling law Vₑₐₒ(μ, σ) = μ^γ • F(σ/μ^κ) in human cardiac dynamics, using open-access electrocardiographic data from the PhysioNet repository. Using a proprietary extraction mechanism calibrated for biological oscillators, the scaling relationship was applied to ten subjects drawn from two clinically distinct cohorts: five healthy individuals and five individuals diagnosed with congestive heart failure (CHF). Across 444,118 healthy beats and approximately 537,000 CHF beats, the bivariate scaling law was found to hold in both populations, with healthy subjects exhibiting a mean goodness-of-fit of R² = 0.9279 ± 0.017 and CHF subjects exhibiting R² = 0.9004 ± 0.013. A statistically significant reduction in geometric integrity (R²; p = 0.031) was observed in the diseased cohort, alongside a trend toward exponent migration in the scaling exponent p (p = 0.066). These results are consistent with the theoretical prediction that diseased biological oscillators undergo a measurable shift in their stochastic scaling structure as system integrity degrades. This document establishes a timestamped empirical baseline for the cardiac application of the RGL framework and connects these findings to the Biological Feigenbaum Spectrum (BFS) established in prior work (Romberger, 2026).

This memorandum places into the public scientific record a compact, independently testable empirical finding: a hypothesis-testing cohort of biological period-doubling systems exhibits systematic, mechanism-dependent deviations from the classical Feigenbaum constant (δ ≈ 4.669), while physical and mathematical control systems remain tightly concentrated near δ ≈ 4.669. The dataset consists of 16 systems (10 biological + 6 controls) with canonical δ values extracted from the literature or digitizable bifurcation evidence and recorded under fixed selection rules. Two non-parametric tests show complete separation (i) between biological deviations and control deviations (Mann-Whitney U = 60/60, exact p ≈ 1.25 × 10⁻⁴) and (ii) between two biological mechanism-defined tiers (Tier 1 vs Tier 2: U = 0/21, exact p = 0.00833), with a 2.479 δ-unit tier gap and zero overlap. This document is not a final manuscript. The cohort and statistics may change as additional sources are incorporated and the extraction ledger is finalized. A full protocol-forward manuscript including the complete dataset, extraction images, and full reference ledger will follow.