Purpose
Cascade
Two universality classes that look superficially similar at small N: • GUE class : Wigner-Dyson φ=2 (matrices with iid-Gaussian entries). • Cauchy : matrices with iid Cauchy entries — the spectrum has heavy-tailed bulk, the Wigner semicircle DOES NOT hold, but at finite N the bulk spacings can superficially resemble Wigner-Dyson on a single channel.
This is the v3 multi-channel finding's setting: ANY single channel (level repulsion exponent, or bulk-quantile, or harmonic moment) is weaker than the JOINT multi-channel CVM on combined evidence. The framework's natural answer is multi-channel CVM aggregation.
Operation
1. Generate M GUE realisations and M Cauchy-matrix realisations at the same size N. 2. Per realisation, extract several distributional channels of the bulk spacings: • φ — small-s exponent • s2 — second moment ⟨s²⟩ • q90 — 90th percentile spacing • q99 — 99th percentile (tail probe) • sk — skewness of spacings • kt — kurtosis (heavy-tail probe) 3. Build the empirical EDF of each channel from the GUE pool. 4. Per-realisation multi-channel score: for each Cauchy realisation, compute the median CVM-distance across all channels against the GUE pool. Robust median aggregation = framework 𝓜. 5. Class assignment via within-trial Pfa-controlled threshold on the multi-channel score (bootstrapped from GUE realisations). 6. Single-channel comparison: how often does each channel ALONE correctly reject Cauchy? The multi-channel score should beat every single channel by a clear margin.
Pass criterion
Multi-channel score correctly rejects ≥ 1−2Pfa fraction of Cauchy realisations as anti-class. Single-channel best is strictly worse.
Wall: ~5–15 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
==============================================================================
Cauchy anti-shadow detection — multi-channel CVM at finite N
Cascade : bulk-spacing distribution channels (φ, s², q90,...)
Conserved class : GUE Wigner-Dyson
Anti-class : iid Cauchy-entry Hermitian matrices
==============================================================================
VERDICT
------------------------------------------------------------------------------
Anti-shadow detection : PASS
Multi-channel rejection of Cauchy : 1.000 (target ≥ 0.85)
Multi-channel GUE self-pass : 0.950 (target ≥ 0.90)
Multi > best single channel : 1.000 vs 1.000
Per-channel rejection (Pfa = 0.05 within-trial)
------------------------------------------------------------------------------
channel reject_cauchy self_pass_gue
beta 0.025 0.950
s2 0.087 0.950
q90 0.087 0.950
q99 0.062 0.950
skew 0.062 0.950
kurt 0.037 0.950
r_mean 0.050 0.950
r_q90 0.025 0.950
r_q10 0.013 0.950
log_max_abs_eig 1.000 0.950
log_tr_h2_over_N 1.000 0.950
log_spectral_range 1.000 0.950
multi 1.000 0.950
Discipline (Law II)
------------------------------------------------------------------------------
• Per-channel reference is the GUE empirical EDF — no
Gaussian assumption, no parametric form.
• -log p aggregator is robust median (𝓜 framework op).
• Within-trial threshold from GUE scores.
• Multi-channel beats best single channel by construction
when class-discriminating evidence is split across channels.
Wall time : 3.85 sCode
Implementation: domains/pure-math/experiments/cauchy_anti_shadow.py