Overview This test contains the locked Tier-3 validation of the Unified Recursion Theory (URT) proportional constraint applied to particle decay data from the Particle Data Group (PDG). Tier-3 evaluates whether admissible solutions under the proportional constraint form statistically significant connected structure in log-efficiency space. The proportional relation tested is: ΔE = λ k_B T ΔH Tier-2 previously established interval admissibility across 310 exclusive decay modes within a declared efficiency envelope: λ ∈ [2.757, 4.763×10^17] Tier-3 tests a stricter condition: not merely satisfiability, but whether admissible solutions produce non-random connected topology in Ω = log10(λ). Dataset 310 exclusive particle decay modes (PDG) Energy release ΔE (GeV) Informational proxy ΔH (nats) Lifetime τ (s) Regime partition (defined a priori by lifetime threshold 10^-13 s): FAST: τ ≤ 10^-13 s (N=82) SLOW: τ > 10^-13 s (N=228) ALL: N=310 Eligibility filters applied prior to topology extraction: ΔE > 0 ΔH > 0 Finite values only Method Summary Tier-3 constructs a log-spaced grid over the declared efficiency envelope and evaluates admissibility under the fixed temperature interval: T ∈ [10^-20, 10^2] GeV At each gridpoint, the fraction of decay modes satisfying the declared temperature bounds is computed. Gridpoints with admissible fraction ≥ 0.95 are retained. Connected components are defined as contiguous admitted regions in log-efficiency space. Primary structure metric: maximum contiguous component width (in decades). Two validation layers are applied: Bootstrap stability (N=200 resamples; deterministic seed) Permutation null testing (N=500 realizations) Null types: Shuffle ΔH across modes Shuffle ΔE across modes Separation metric: Z = (W_obs − μ_null) / σ_null Locked robustness threshold: Z ≥ 3.0 Results SLOW regime (τ > 10^-13 s) Component width: 6.781 decades Bootstrap survival: 1.0 Median IoU: 0.997 Null separation (ΔH shuffle): Z = 5.153 Null separation (ΔE shuffle): Z = 4.895 Status: Pass (null-separated topology) FAST regime (τ ≤ 10^-13 s) Component width: 5.056 decades Null separation: Z < 0 under both null types Status: Null-consistent (no certified topology) ALL dataset (N=310) Component width: 6.091 decades Null separation: Z = 2.11 (ΔH), Z = 1.94 (ΔE) Status: Below robustness threshold Topology exceeding null expectation appears exclusively in the SLOW regime. FAST functions as an internal control demonstrating that the pipeline does not trivially manufacture structure across partitions. Interpretation Tier-3 establishes regime-selective constraint geometry under the URT proportional mapping. The result is geometric and statistical. No causal mechanism is asserted and no modification of established particle physics is claimed. Under the declared constraint and fixed temperature envelope, slow particle decays occupy a statistically non-random connected admissible manifold in log-efficiency space, while fast decays do not. Falsification Criteria Tier-3 fails if: Bootstrap stability gates are not met, or Null separation Z < 3.0 in all tested regimes and null types. FAST null-consistency provides an internal negative control. Contents of This Deposit Tier-3 manuscript (publication draft) Tier-3R robustness extension outputs Locked manifest files with SHA256 hashes Configuration files (deterministic, pre-registered) All thresholds, partitions, and gates were declared prior to execution and were not tuned post-analysis. Subsequent tiers evaluate selector conditioning (Tier-4) and cross-domain scaling / structural characterization (Tier-5).