Empirical Evidence for a Universal Threshold Structure Across Eight Physical and Ecological Domains — Multi-Domain Validation of the Δ-Gate Formalism

We present the first cross-disciplinary empirical test of a single mathematical structure — a sigmoid function on a logarithmic scale, P(R=1|X) = σ(w · log10(X/Xcr) − τ) — across eight independent systems spanning 21 orders of magnitude in energy and, for the first time, an ecological domain. Systems tested: (I) Nuclear gamma decay (31 transitions, ENSDF), (II) Ferromagnetic phase transition (Fe, Tc = 1043 K), (III) 2D Ising model (exact Onsager solution), (IV) Na-cluster quantum interference (170 kDa, Pedalino et al., Nature 2026), (V) C70 thermal decoherence (Hackermüller et al., Nature 2004), (VI) XLPE dielectric breakdown (80 specimens, Chauvet & Laurent 1993), (VII) Laser-induced air breakdown (1064 nm, Gaabour et al. 2012), (VIII) Phosphorus-chlorophyll response in an AQUATOX lake model (365 daily time steps, EPA). Key results: ΔAIC uniformly favors the sigmoid-on-log form over domain-specific alternatives in all eight systems (ΔAIC = +0.047 to +166.3). The fitted sharpness parameter w spans 1.15–43.4 and scales monotonically with transition cooperativity: w = 1–3 (single-particle, gamma decay), w = 8–9 (phase accumulation, Na-cluster and ecological), w = 15–17 (thermal/3D, C70 and Fe), w = 27 (critical 2D, Ising), w = 37–43 (avalanche, XLPE and air breakdown). Temperature-dependent gate modulation in the ecological system demonstrates that external variables deform the gate continuously without breaking its structure — the first empirical demonstration of a gate-modulation mechanism within the Δ-Gate framework. Five falsification controls confirm discriminating power: (1) C70 collisional decoherence — no threshold, correctly rejected; (2) Photoelectric effect — exact theory preferred, correctly identified; (3) XLPE Type III specimens — distributed defects, correctly excluded; (4) Supercritical fluid crossover — multi-dimensional threshold, 1D Gate correctly fails (R² < 0); (5) Air breakdown near-tie (ΔAIC = +0.047) — model does not falsely claim superiority. These results constitute the first empirical validation of Axiom A4 of the Δ-Gate formalism. All data and code available under CC BY 4.0. Part of the Δ-Gate Paper Series (DOI: 10.5281/zenodo.19103765).