Contact-Hamiltonian Constitutional Gate — Density Matrix Validation

P_M Contact-Preserving vs Standard Liouville Evolution

Mixed-state Bloch sphere evolution under the Contact-Hamiltonian Constitutional Gate.
Validates purity retention, quantum Fisher information advantage, and Bures distance divergence.
Inventor: Joshua Lopez — DCGP.AI LLC — July 4, 2026

⚖ USPTO 19/731,016 — Reduction to Practice

P_M (Contact-Hamiltonian)

Liouville (Standard)

Unitary / Heisenberg Limit

Run simulation to validate

γ₀=0.25 · μ=0.12 · σ=0.08 · ω=5.0 · t∈[0,5]

Simulation Parameters

γ₀ — Dissipation rate

μ — Contact correction

σ_noise — SDE noise

ω — Precession frequency

t_end — Integration horizon

N_steps — Time resolution

SDE Ensemble (50 traj)

Ready

Validation Checks — USPTO 19/731,016

Contact-form fidelity mean

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⬜ PENDING

Target: < 0.002

Bures distance P_M at t=end

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⬜ PENDING

Target: < 0.3

dθ/dt residual mean

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⬜ PENDING

Target: < 0.01

P_M QFI advantage vs Liouville

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⬜ PENDING

Target: ratio > 1.0×

SDE norm P_M vs Liouville

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⬜ PENDING

Target: P_M > Liouville

(A) Bloch Norm |r(t)| — Purity Retention

(B) Polar Angle θ(t) — Stability

(D) Bures Distance to Ideal ρ_ideal

(E) Contact-Form Fidelity ||r(t)| − 1| (log scale)

(F) SDE Ensemble — Final Norm Distribution

Matplotlib Reduction-to-Practice Outputs — USPTO 19/731,016

Python-generated figures from contact_hamiltonian_density_matrix_validation.py · All 5 validation checks PASS · DCGP.AI LLC · Joshua Lopez, Sole Inventor

Fig 1 — Main Comparison

Bloch norm, polar angle, QFI, Bures distance — P_M vs Liouville vs Unitary

Fig 2 — Contact-Form Fidelity Verification

||r_PM| − 1| vs Liouville drift · analytic dθ/dt residual · QFI metric

Fig 3 — Stochastic SDE Ensemble

50-trajectory Euler-Maruyama · P_M norm retention vs Liouville decoherence

Fig 4 — Bloch Sphere Trajectory

3D phase-space · P_M stays on sphere surface · Liouville spirals inward