E-LAB-07 · EntropyLab · April 2026

ENTRO-QUANTUM

Entropic Collapse in Probabilistic State Spaces of Artificial Intelligence.
From classical deterministic entropy to quantum-inspired probabilistic mechanics.

Ψ-W(s,t) · Entropic Wavefunction · Quantum Probability Amplitude

0.07

⟨Ψ⟩ (expectation value)

QUANTUM · Wavefunction evolving

GitHub Repository DOI: 10.5281/zenodo.19478805

01 · Core Innovation

Entropic Wavefunction · Uncertainty · Entanglement

ENTRO-QUANTUM extends classical entropy control to quantum-inspired probabilistic mechanics. The system's stability is represented as a wavefunction — a complex-valued probability amplitude over possible futures — that collapses to a classical scalar only at inference time.

Entropic Wavefunction

Ψ-W(s,t) : S × R⁺ → ℂ

P(s,t) = |Ψ-W(s,t)|² · Born rule · ⟨Ψ⟩(t) = ∫ s·|Ψ|² ds

Entropic Schrödinger Equation

i·ħ_E · ∂Ψ/∂t = H_E · Ψ

H_E = T_E + V_AEW + V_ext · T_E = -(ħ_E²/2m_E)·∂²/∂s²

Entropic Uncertainty Principle

ΔΨ · ΔM ≥ κ_E / 2

κ_E = ħ_E / m_E · Fundamental monitoring bound

Informational Entanglement

E_ij = I(S_i;S_j) / min[H(S_i), H(S_j)]

Non-local correlations · Simultaneous multi-node collapse

Quantum Jump Operator

Ψ-W(s, t+dt) = Q_k·Ψ-W(s,t) / ||Q_k·Ψ-W(s,t)||

H_eff = H_E - (iħ/2)·Σ Q_k†Q_k · Discontinuous collapse

Silent Observer Protocol

α* = 1/(2 - P_collapse)

G = α²·|⟨Ψ|Π_s|Ψ⟩|² · D = α·(1-α)·Var[S]

02 · Validation Results

Quantum Tests · 55/55 Passed

Test Module Tests Passed Equations

test_wavefunction.py 10 10 Eq 3.1-3.7

test_uncertainty.py 14 14 Eq 3.8-3.11, 4.1-4.3

test_entanglement.py 12 12 Eq 3.12-3.16

test_quantum_jump.py 7 7 Eq 3.17-3.19

test_silent_observer.py 12 12 Section 4

TOTAL 55 55 (100%) 19+ Equations

Entropic Wavefunction Ψ-W(s,t)

⟨Ψ⟩ → 0.339

BORN RULE

P(s,t) = |Ψ|²

EXPECTATION

∫ s·|Ψ|² ds

NORMALIZATION

∫ |Ψ|² ds = 1

COLLAPSE PROB

P(s ≥ s_critical)

Entropic Uncertainty Principle

ΔΨ·ΔM ≥ κ_E/2

ħ_E (Planck)

0.1

m_E (Inertia)

1.0

κ_E = ħ_E/m_E

0.1

BOUND

0.05

Silent Observer Protocol

α* = 1/(2 - P_collapse)

AMBIENT

α = 0.8

ADVISORY

α = 0.3

SILENT WATCH

α = α*

COLLAPSE

α → 0

# pip install entro-quantum
from entro_quantum import EntropicWavefunction, EntropicHamiltonian

psi = EntropicWavefunction(resolution=100)
psi.gaussian_initial(mean=0.339, variance=0.05)

H = EntropicHamiltonian(hbar_E=0.1, m_E=1.0, target=0.339)

# Evolve wavefunction
real, imag = psi.get_amplitude()
h_real, h_imag = H.apply(real, imag, psi.s_grid, psi.ds)

# → Output
⟨Ψ⟩ = 0.339 · Variance = 0.05 · P_collapse = 0.01

"True intelligence is not the elimination of uncertainty — it is the art of acting optimally within it.
ENTRO-QUANTUM gives intelligent systems the mathematical framework to embrace uncertainty as a resource."

— Samir Baladi · ENTRO-QUANTUM · April 2026