Purpose
Cascade
Φ(L) = order-parameter cascade across system size L at the critical point T_c. At T = T_c the magnetisation density |m_L| scales as |m_L| ∝ L^{−β/ν}, the susceptibility χ_L ∝ L^{γ/ν}, and the Binder cumulant U_L is L-independent (universal value U* ≈ 0.61 in 2D).
Conserved cascade = critical-point scaling exponents and universal ratio (β/ν = 1/8, γ/ν = 7/4 for 2D Ising).
Operation
1. Run M independent 2D Ising Wolff-cluster realisations at exact T_c = 2 / ln(1+√2) for each L ∈ {16, 24, 32, 48, 64}. 2. Per realisation, accumulate after thermalisation: • magnetisation samples |m| at each MC sweep • energy samples E 3. Build the empirical EDF of |m| (the order-parameter cascade) pooled across realisations PER L. The L-shape of this EDF is the critical scaling. 4. Read the cross-realisation consensus on: • β/ν via slope of log⟨|m|⟩ vs log L (theory: 1/8 = 0.125) • γ/ν via slope of log χ vs log L (theory: 7/4 = 1.75) • U* the Binder cumulant at T_c (theory: ≈ 0.6107) 5. Per-realisation CVM admissibility: each realisation's |m| EDF tested against the pooled-class EDF at its own L. Within-trial threshold from the pool itself.
Pass criterion
β/ν within 5% of 0.125 in cross-L consensus. γ/ν within 5% of 1.75. Per-realisation admissibility ≥ 0.85.
Wall: ~30–60 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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Ising criticality — 2D critical exponents via empirical EDF
Cascade : order-parameter |m| across system size L at T = T_c
Conserved cascade : β/ν = 1/8, γ/ν = 7/4, U* ≈ 0.61
Operation : empirical EDF + within-trial CVM (Law II)
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VERDICT
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Critical-class recovery : PASS
β/ν : +0.1304 σ_cross = 0.0011 (theory 0.1250, recovery 4.3%)
γ/ν : +1.7517 σ_cross = 0.0276 (theory 1.7500, recovery 0.1%)
U* : +0.6109 (theory ≈ 0.6107)
Per-L admissibility at T_c (Pfa = 0.05)
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L= 16 pool= 7200 threshold=0.0007 pass_rate=0.833
L= 24 pool= 7200 threshold=0.0005 pass_rate=0.833
L= 32 pool= 7200 threshold=0.0005 pass_rate=0.333
L= 48 pool= 7200 threshold=0.0004 pass_rate=0.500
Cross-L cascade
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L ⟨|m|⟩ χ U
16 0.7169 9.376 0.6129
24 0.6736 19.413 0.6089
32 0.6591 30.270 0.6133
48 0.6187 65.253 0.6076
Discipline (Law II)
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• |m| EDF empirical; no Gaussian or Boltzmann form fitted.
• Per-L admissibility uses within-trial CVM on the pool.
• Cross-L consensus reads exponents from the cascade.
Wall time : 91.88 sCode
Implementation: domains/pure-math/experiments/ising_criticality.py