Tier-4: Lambda Extraction and Regime-Conditioned Dispersion in the Particle Domain

This paper presents the corrected Tier-4 analysis for the particle-domain validation of Unified Recursion Theory (URT). Tier-4 computes the diagnostic proportional parameter lambda for admissible particle decay modes using the canonical URT proportional constraint applied to artifacts produced by earlier validation tiers. This version replaces the initial Tier-4 release following the discovery of an implementation error in the original bandwidth proxy used for lambda extraction. The earlier run incorrectly applied a fixed reference temperature when computing lambda values. The corrected Tier-4 implementation uses the lifetime-derived bandwidth proxy defined for the particle domain: T = ℏ / τ Under this corrected proxy, lambda is computed deterministically from energetic release (DeltaE), entropy change (DeltaH), and particle lifetime (tau). DeltaE and DeltaH are Tier-defined inferred quantities derived from particle decay properties, while the lifetime-derived quantity serves as the declared bandwidth proxy for the particle domain. Re-running Tier-4 with the corrected proxy produces a substantially different structural result. Particle decay modes partition into two completely separated regions of lambda-space corresponding to the persistence regimes established in Tier-3. FAST and SLOW decay modes occupy disjoint regions of log10(lambda) with a gap of approximately 1.3 decades between the regimes. No decay mode lies within this interval. Statistical diagnostics confirm complete regime separation: Kolmogorov–Smirnov statistic: 1.0 p-value: 6.5 × 10⁻⁷⁷ overlap coefficient: 0.0 The FAST and SLOW distributions therefore exhibit complete statistical separation under the corrected proxy. Within-regime dispersion differs as well: FAST decays show broader internal lambda spread while SLOW decays cluster tightly in lambda-space. This corrected Tier-4 result replaces the earlier dispersion-focused interpretation and establishes regime-conditioned structure in lambda distributions under the URT proportional constraint. Lambda remains a diagnostic proportionality coefficient. The observed regime structure is interpreted descriptively and does not assign causal or dynamical status to lambda. The corrected Tier-4 results provide the basis for the subsequent Tier-5 structural evaluation of regime separation and within-regime dispersion characteristics. Update — Tier-4c Boundary Sensitivity Addendum (March 2026) This version includes an additional Tier-4c boundary sensitivity addendum evaluating the robustness of the Tier-3 lifetime threshold used to partition particle decay modes into FAST and SLOW persistence regimes. Tier-4 established that the inferred proportional diagnostic parameter λ\lambdaλ separates particle decay modes into two non-overlapping regions of log-space corresponding to the Tier-3 regime partition. FAST and SLOW modes occupy disjoint ranges of log⁡10(λ)\log_{10}(\lambda)log10(λ) with a separation gap of approximately 1.31 decades, producing maximal statistical distinguishability (KS statistic = 1.0). The Tier-4c addendum evaluates whether this separation depends on the exact placement of the Tier-3 threshold (τ=10−13\tau = 10^{-13}τ=10−13 s). The analysis operates exclusively on locked Tier-4 outputs. Lambda values are not recomputed. Instead, regime labels are reassigned using alternative lifetime thresholds to examine boundary sensitivity. Three thresholds were evaluated: 10−1410^{-14}10−14 s 10−1310^{-13}10−13 s (locked Tier-3 threshold) 10−1210^{-12}10−12 s Results show: The separation is unchanged under a downward perturbation to 10−1410^{-14}10−14 s because no decay modes in the dataset occupy the interval between 10−1410^{-14}10−14 and 10−1310^{-13}10−13 seconds. When the threshold is shifted upward to 10−1210^{-12}10−12 s, 69 decay modes are reassigned from the SLOW regime to FAST. These reassigned modes retain lambda values characteristic of the SLOW distribution, causing complete separation to collapse. The Tier-4c result therefore demonstrates asymmetric boundary sensitivity: the Tier-4 separation is stable under downward perturbation but fails when the boundary is shifted upward sufficiently to reassign structurally SLOW modes. This addendum does not modify any Tier-4 computations or upstream artifacts and serves only as an informational robustness analysis. The Tier-4 result remains unchanged: FAST and SLOW particle decay modes occupy disjoint regions of λ\lambdaλ-space under the locked Tier-3 regime definition. The Tier-4c analysis confirms that the Tier-3 lifetime threshold aligns with a natural partition in the observed particle decay dataset while making no claim that the dataset’s lifetime gap represents a fundamental physical boundary.