The Reactive Universe: Differential Geometry and the Birth of Consciousness

Mathematical Foundations of Reactive Substrate Theory (RST)

Donald Hoffman’s Conscious Agent Theory argues that consciousness is the fundamental building block of reality. Reactive Substrate Theory (RST) takes a different approach: it treats reactivity as the true primitive, with consciousness emerging only when a reactive system becomes self-referential and internally modelled. Below is a more technical, equation-level sketch of how RST formalizes this idea.

1. The Reactive Substrate as a Continuous Manifold

RST begins by modelling reality as a smooth manifold 𝓜 equipped with a state field: 

Φ : 𝓜 × ℝ → ℝⁿ

This field assigns each point in the substrate a dynamic internal state. A structure tensor or metric g governs how reactivity propagates across the manifold.

2. Core Evolution Equation

The substrate evolves according to a general reactive equation: 

∂Φ/∂t = D Δg Φ + F(Φ) + S(x,t)

where D is a diffusion tensor, Δg is the Laplace–Beltrami operator, F(Φ) encodes nonlinear feedback, and S represents perturbations. This is not a metaphor: in RST, this is the underlying physics.

3. Objects as Attractors

Stable “things” in the world correspond to attractors of the substrate dynamics. An attractor 𝓐 satisfies: 

Φ(x,t₀) ∈ 𝓐 ⇒ Φ(x,t) ∈ 𝓐  for all t ≥ t₀

Objects are not fundamental particles but self-maintaining patterns in the reactive field.

4. Organisms as High-Order Reactive Subsystems

An organism is a bounded region Ω ⊂ 𝓜 with specialized internal dynamics: 

∂ΦΩ/∂t = DΩ ΔgΩ ΦΩ + FΩ(ΦΩ) + B(...)

The boundary term B couples the organism to the surrounding substrate, enabling perception and action.

5. Internal Models and Self-Reference

Consciousness requires internal modelling. RST defines internal and external encodings: 

Ψext(t) = Eext(Φ𝓜 \ Ω(t))
Ψint(t) = Eint(ΦΩ(t))

These evolve according to: 

dΨ/dt = G(Ψ)

When this system develops a stable, self-referential attractor, RST identifies this as a conscious state.

6. Perception as Structure-Preserving Mapping

Unlike Hoffman’s “interface” metaphor, RST treats perception as a structure-preserving compression: 

Π : Φoutside → Ψext

Similar substrate states map to similar internal states, preserving relational structure even if compressed.

7. Space-Time as Emergent Geometry

RST derives effective geometry from reactive propagation costs: 

deff(x,y) = infγ ∫ L(Φ(γ(s)), γ'(s)) ds

Space‑time is not fundamental; it is an emergent geometry. .

Conclusion

RST provides a mathematically grounded alternative to idealist models like Hoffman’s. Instead of consciousness creating reality, RST shows how consciousness emerges from the deeper, continuous dynamics of a reactive substrate. This framework unifies objects, organisms, perception, and even space-time under a single dynamical principle.

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