OBSERVABILITY INDEX · Ω(N) = 1 − H_irr / H_path
0.00
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Target: Ω > 0.60 · Reducible regime · Ω → 1 fully observable
0.00
H_path · Cumulative (nats)
0.00
H_irr · Irreducible (nats)
0.00
H_red · Recoverable (nats)
0.00
Δ(L) · Entropic Leakage
📊 Layer-wise Path Entropy · H_path(l)
🏗️ Architecture Comparison · Estimated Ω
MLP (2L)20.310.91
MLP (6L)60.680.74
MLP (12L)121.140.61
CNN (8L)80.870.68
Transformer (12L)121.420.52
Transformer (24L)242.050.39
🎯 Ω Threshold Reference · Reducibility Regimes
Ω > 0.60🟡 REDUCIBLE · Path entropy recoverable from external measurements
0.40 – 0.60🟠 BORDERLINE · Partial recoverability
Ω < 0.40🔴 IRREDUCIBLE · Path entropy cannot be recovered
📐 Core Equations
Ω(N)
Observability Index
= 1 − H_irr / H_path ∈ [0,1]
H_path^(L)
Cumulative Path Entropy
= Σ H(P_l)
Δ(L)
Entropic Leakage
= H_path^(L) − I(x; h_L)
δ*
Reducibility Threshold
= ε · max H(P_l), ε ∈ (0,1)