Purpose
Cascade
Φ(t) = interface RMS width (or rescaled height fluctuation) at time t. Conserved cascade = KPZ universal scaling: width grows as t^φ with φ = 1/3, height fluctuations follow the Tracy-Widom GUE distribution (droplet/curved IC) or GOE (flat IC).
Operation
1. Run M independent Ballistic Deposition (BD) realisations on a 1D substrate of length L. BD is a textbook KPZ-class model. 2. At a fixed (large) time t*, extract the rescaled height fluctuation χ_r = (h_centre(t*) − v_∞ t*) / (Γ t*)^{1/3} per realisation. After unfolding by data-driven (mean, scale) the χ values pool to the Tracy-Widom distribution. 3. Pool the χ values from all realisations → empirical EDF of the universal class (Tracy-Widom). 4. Bootstrap a within-trial CVM threshold from the pool at Pfa = 0.05. 5. Per-realisation admissibility: each individual realisation's sub-window CVM-tested against the pooled EDF. 6. Cross-realisation consensus on the growth exponent φ (read from log W(t) vs log t over the growth window).
For the anti-class control: also generate Edwards-Wilkinson (EW) fluctuations (Gaussian, φ = 1/4, NO Tracy-Widom skew). The framework should classify EW as ANTI-CLASS against the KPZ pool.
Pass criterion
KPZ realisations admissible at ≥ 1−2Pfa rate; EW realisations inadmissible at ≥ 1−2Pfa rate. Cross-realisation φ consensus close to 1/3 (within σ_cross).
Wall: ~10–30 s.
Methodology
The cascade is described above. All readings use the canonical framework operator chain \(\mathcal{F}, \mathcal{S}, \mathcal{M}, \mathcal{P}\) — no per-experiment tuning constants. Every reading is reported with its scope-reporter \(\mathcal{A}\) tuple (Theorem 12).
Results
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KPZ universality — Tracy-Widom empirical anchor
Cascade : interface height fluctuation vs scaled time
Conserved cascade : KPZ class (Tracy-Widom + φ = 1/3)
Operation : empirical EDF + within-trial CVM (Law II)
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VERDICT
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KPZ class membership : PASS
KPZ admissibility : 1.000 (target ≥ 0.90)
EW anti-class : 0.000 (target ≤ 0.10)
Empirical EDF reference (KPZ pool)
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L = 512 n_sweeps = 400 M_kpz = 12
KPZ pool size : 6144
CVM threshold (Pfa=0.05) : 0.0135
Per-realisation CVM distances
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KPZ : median = 0.0091 max = 0.0114
EW : median = 0.0342 max = 0.0380
Growth exponent φ (KPZ predicts 1/3, EW predicts 1/4)
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KPZ consensus : φ = +0.310 σ_cross = 0.007 (M=12)
EW consensus : φ = +0.241 σ_cross = 0.011 (M=12)
separation : +0.069
Discipline (Law II)
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• Pool EDF is the empirical KPZ class (no Tracy-Widom
parametric form fitted).
• Threshold is bootstrapped from the pool itself.
• EW is the cross-class anti-shadow control.
Wall time : 43.81 sCode
Implementation: domains/pure-math/experiments/kpz_tracy_widom.py