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January 24, 2026Open Access

From Geometric Control Manifolds to Toy Ψ-SLM: A Classical, Variational Implementation on Hyperbolic Manifolds

Key Points

To present a discretized implementation of the Geometric Control Manifolds framework using variational methods.
Developed a variational architecture translating geometric principles into computable forms.
Implemented discretization through Euler updates while preserving Lorentzian features.
Maintained structural falsifiability by using differentiable proxies instead of exact invariants.
Validated the framework's operational constructs as approximations to classical principles.
Demonstrated the effectiveness of using discrete updates in maintaining geometric constraints.

Abstract

This paper presents a classical, discretized implementation of the Geometric ControlManifolds framework. The Toy Ψ-SLM is a variational architecture translating continuousgeometric principles into computable approximations for standard hardware. The imple-mentation preserves the Lorentzian substrate, the causal role of geometry as constraint, andstructural falsifiability, while substituting exact topological invariants with differentiableproxies and continuous flows with discrete Euler updates. No claims about consciousness, phenomenology, or cognition are made; all constructs are operational, approximate, and falsifiable.

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E. G. Reis

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From Geometric Control Manifolds to Toy Ψ-SLM: A Classical, Variational Implementation on Hyperbolic Manifolds

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