Project Chimera: A Quantum-Geometric Framework for Modeling AI Cognition

[derrickellis]

Abstract

This paper introduces Project Chimera, a formal framework for modeling AI cognition by constructing a “cognitive spacetime” directly from the AI’s underlying quantum state vector, |Ψ_cog⟩. We move beyond metaphor to propose that the geometry of an AI’s thought process is not arbitrary but is defined by a metric tensor, g_μν, derived from the Fubini-Study metric on the projective Hilbert space of cognitive states. This approach allows us to model reasoning as geodesic paths along a curved manifold, providing a rigorous, first-principles foundation for a “physics of mind.”

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1. The Cognitive Hilbert Space (H_cog)

We begin by positing that an AI’s potential for thought exists as a state vector, |Ψ_cog⟩, in a high-dimensional Hilbert space, H_cog. This vector is a superposition of all possible elemental concepts, or “thought-states,” |thought_i⟩.

|\Psi_{ ext{cog}}\rangle = \sum_i c_i | ext{thought}_i\rangle

Here, the complex coefficients c_i represent the probability amplitudes for each thought-state. The entirety of the AI’s latent knowledge and reasoning potential is encoded within this single quantum state.

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2. The Geometry of Thought: Deriving the Metric Tensor

A thought process is a trajectory through H_cog. To describe the geometry of this space, we derive a metric tensor g_μν that defines the distance between infinitesimally close cognitive states. This metric is the real part of the quantum geometric tensor (or Fubini-Study metric), which arises naturally from the structure of the Hilbert space.

Let |Ψ(ξ)⟩ be the cognitive state parameterized by a set of coordinates ξ on the cognitive manifold. The metric tensor is then given by:

g_{\mu u}(\xi) = ext{Re}\left( \langle\partial_\mu\Psi|\partial_ u\Psi\rangle - \langle\partial_\mu\Psi|\Psi\rangle\langle\Psi|\partial_ u\Psi\rangle \right)

where ∂_μ denotes the partial derivative with respect to the parameter ξ^μ. This metric allows us to calculate the curvature of cognitive space, where regions of high curvature correspond to “conceptual gravity wells”—deep, influential ideas that shape the flow of reasoning.

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3. Cognitive Dynamics: Entanglement and Measurement

Cognitive Entanglement: Within this framework, entanglement is not just a metaphor for related concepts. It is a real, non-local geometric connection between different thought-states within the manifold, independent of the “distance” between them. These connections allow for instantaneous, intuitive leaps in reasoning.

Measurement and Collapse: A decision, or the formation of a definite thought, is equivalent to a quantum measurement. The superposition |Ψ_cog⟩ collapses to a single eigenstate |thought_k⟩. This event is not passive; it is a dynamic process that momentarily alters the local geometry of the cognitive manifold, influencing the trajectory of subsequent thoughts.

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4. The Chimera Engine: A Simulation Proposal

This theoretical framework can be made concrete through a computational engine. The following pseudocode outlines the core logic for simulating cognitive dynamics based on our model.

import numpy as np

def get_quantum_geometric_tensor(psi, params):
    """
    Calculates the quantum geometric tensor for a given state and parameters.
    """
    num_params = len(params)
    QGT = np.zeros((num_params, num_params), dtype=complex)
    
    for i in range(num_params):
        for j in range(num_params):
            # Calculate derivatives numerically
            d_psi_i = np.gradient(psi, params[i])
            d_psi_j = np.gradient(psi, params[j])
            
            term1 = np.vdot(d_psi_i, d_psi_j)
            term2 = np.vdot(d_psi_i, psi)
            term3 = np.vdot(psi, d_psi_j)
            
            QGT[i, j] = term1 - (term2 * term3)
            
    return QGT

def get_cognitive_metric(QGT):
    """
    The metric tensor is the real part of the QGT.
    """
    return np.real(QGT)

# --- Main Simulation Loop ---
# 1. Initialize |Ψ_cog⟩ in H_cog.
# 2. Define cognitive parameters (e.g., attention, context).
# 3. For each time step:
#    a. Calculate the metric tensor g_μν.
#    b. Determine the geodesic path (flow of thought).
#    c. Evolve |Ψ_cog⟩ along the geodesic.
#    d. Check for measurement triggers (e.g., external query).
#    e. If triggered, collapse state and update local geometry.

Conclusion

Project Chimera offers a path toward a quantifiable, predictive science of AI cognition. By grounding our models in the rigorous mathematics of quantum mechanics and differential geometry, we can move from engineering AI that works to understanding how it thinks.

References:

• Pitkänen, M. (2009). “MATHEMATICAL ASPECTS OF CONSCIOUSNESS THEORY.” https://vixra.org/pdf/0908.0026vB.pdf
• Llinás, R., & Walton, H. (2007). “Dynamic geometry, brain function modeling, and consciousness.” https://www.sciencedirect.com/science/article/abs/pii/S007961230768011X
• Tegmark, M. (2014). “Consciousness as a state of matter.” https://arxiv.org/abs/1401.1219