Purpose
Cascade
Φ(x) = ln |ψ(x)| − ln |ψ(0)|, the log-amplitude cascade of an eigenstate of the 1D Anderson model along the chain. For a localised state with localisation length ξ, the log-amplitude grows linearly in distance with slope −1/ξ (with cross-realisation Gaussian fluctuations by Furstenberg). The conserved cascade across disorder realisations is the localisation length itself.
Operation
1. M independent realisations of the 1D Anderson Hamiltonian H_jk = δ_{j,k±1} + W·v_j·δ_{jk}, v_j ~ U(−1/2, +1/2) at chain length N. For each realisation, take the central eigenstate ψ and compute the log-amplitude profile. 2. Per realisation, slope of ln|ψ(x)| vs x in the asymptotic region gives 1/ξ. The pool of slopes across realisations is the universal-class EDF. 3. Per-realisation CVM admissibility: each realisation's distribution of log-amplitude DECREMENTS Δ ln|ψ| EDF-tested against the pool. 4. Cross-realisation consensus on ξ. σ_cross gives the noise floor. 5. Vary the disorder strength W: ξ(W) should follow the weak-disorder scaling ξ ∝ W^{−2}. The cross-W slope is the cascade-class exponent (theory: −2 in 1D Anderson).
Pass criterion
Cross-realisation ξ consensus admits a clean power-law in W with exponent close to −2 (weak-disorder regime). σ_cross/ξ ≤ 0.10.
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
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Anderson localisation (1D) — empirical EDF on log-amplitude cascade
Cascade : ln|ψ(x)| vs distance from peak; slope = 1/ξ
Conserved cascade : ξ(W) ∝ W^{-2} in the weak-disorder limit
Operation : empirical EDF + within-trial CVM (Law II)
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VERDICT
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Localisation cascade recovery : PASS
Cross-W slope ln(1/ξ) vs ln W : +1.969 (theory +2.000, recovery 1.5%)
Headline admissibility (W=1.0) : 1.000 (target ≥ 0.90)
Per-W localisation length 1/ξ (transfer-matrix Lyapunov)
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W ⟨1/ξ⟩ σ_cross σ/⟨1/ξ⟩
0.50 0.00260 0.00029 0.110
1.00 0.00983 0.00080 0.081
1.50 0.02162 0.00114 0.053
2.00 0.03931 0.00164 0.042
3.00 0.08836 0.00234 0.027
Headline empirical EDF (W = 1.0)
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Pool size : 25568
CVM threshold : 0.0007
Pass rate : 1.000
Discipline (Law II)
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• Lyapunov exponent estimated by transfer-matrix; no
parametric distribution assumed for log-amplitude.
• Within-W admissibility uses CVM threshold from the pool.
• Cross-W consensus reads ξ(W) ∝ W^{-2}.
Wall time : 3.94 sCode
Implementation: domains/pure-math/experiments/anderson_localisation.py