Tier-5: Structural Characterization of λ Regimes in the Particle Domain

This update replaces the original Tier-5 release with a corrected and expanded structural analysis of the particle-domain λ field. The revision does not change the Tier-4 λ extraction, operator definitions, or proxy construction. Instead, it clarifies the statistical interpretation and expands the diagnostic tests used to evaluate regime structure. The updated Tier-5 includes three substantive changes relative to the original release: 1. Corrected boundary interpretation.The original draft treated near-threshold SLOW modes as potentially indistinguishable from deeper SLOW modes. The revised analysis shows the opposite: near-threshold modes form a statistically distinct lower-λ sub-band within the SLOW regime. This distinction is supported by Cliff’s δ ≈ −0.99 and KS ≈ 0.96 (p ≈ 2.6×10⁻⁵¹). 2. Expanded perturbation stability testing.The updated version introduces a full perturbation-stability analysis including: bootstrap resampling (5,000 replicates), and subset-removal sensitivity tests. All separation metrics remain positive in 100% of bootstrap replicates, demonstrating that the FAST/SLOW regime structure is not driven by outliers or dataset composition. 3. Explicit separation geometry characterization.The revised paper reports the complete geometry of the λ regime structure, including support intervals, interquartile ranges, centroid distance, and bandwidth ratios. The FAST and SLOW regimes remain strongly separated, with: a 1.31-decade support gap, a 7.53-decade IQR gap, and an 8.96-decade centroid distance in log₁₀(λ) space. Together, these revisions strengthen the conclusion that the FAST/SLOW partition identified in Tier-4 corresponds to a stable structural organization of the λ field rather than a boundary artifact. The updated Tier-5 therefore establishes two structural results: The FAST/SLOW λ regimes are geometrically separated and perturbation-stable. The SLOW regime contains internal boundary-adjacent structure in the form of a lower-λ sub-band. These findings motivate the next stage of analysis (Tier-6), which will test whether the observed λ sub-band structure correlates with particle properties such as interaction class, particle composition, or mass scale.