I. THE SINGULARITY HYPOTHESIS
The fundamental failure of standard quantitative models lies in the assumption of Brownian Ergodicity.
The Ma-Chao Equation re-characterizes the order book as a Non-Hermitian Information Manifold.
Liquidity is treated as a fluid density governed by a complex Hamiltonian.
A "crash" is not a statistical outlier — it is a Topological Symmetry Breaking event at an Exceptional Point (EP),
where the spectral discriminant Δ → 0 and resolvent diverges as χ ∝ Δ⁻¹/².
Eq. (9.3): Two-mode reduced operator L₁₂. EP criterion: Δ = 0 (Theorem 9.5).
PHYSICAL INVARIANTS DETECTED:
- • Atomic Integrity AI_log: Manifold structural density = log₁₀(ΣVol / (|ΔP|+ε))
- • Pressure Gradient ∇AI: Velocity of spectral gap closure (∂AI/∂t)
- • Imbalance Curvature Γ: ∂²A/∂x² — 3-point central difference operator
- • Spectral Discriminant Δ_eff: EP early-warning metric (Theorem 9.9)
- • Topological Winding Ŵ_N: Discrete Hilbert-transform winding count
- • Q_topo = δ̂_min√n_min: Bootstrap-calibrated confidence metric
II. THREE-VALUED TOPOLOGICAL AUDIT PROTOCOL (Theorem 9.12 / Appendix O)
EP-CONFIRMED
Ŵ_N = 1 ∧ Q_topo ≥ Q* ∧ |ρ̂_A − ρ̂_B| ≤ 0.05 → Full reactor glow. FPR ≤ 10⁻⁴ (bootstrap Q*).
EXCLUDED
Ŵ_N = 0 ∧ Q_topo ≥ Q* → Standard tracking. No EP crossing detected.
UNDETERMINED
Q_topo < Q* OR Aliasing Veto triggered. SNR insufficient to resolve topological invariant.
Dual pipeline: T_A (Hawkes pre-whitening, parametric) ∩ T_B (CPSD coherence, non-parametric) = ∅.
Pipeline divergence |ρ̂_A − ρ̂_B| > 0.05 triggers [PIPELINE ALIASING] warning.