Necessary and Sufficient Substrate for Consciousness

In this work, the authors present a mathematical condition on the substrate for consciousness. This has implications for neuroscience and artificial intelligence.

Theorem. A necessary and sufficient minimal substrate for consciousness consists of a firing three-dimensional axon tract with ephaptic coupling, with at least one stage \(S 1\) such that, for \(S=1\), the tract length \(L\) satisfies \(L L_{}\), where \(L_{}\) depends on axonal diameters \(d_{axon}\), extracellular medium resistivity \(_{extracellular}\), ephaptic coupling strength \(g_{ephaptic}\), and conduction velocity \(v\). This ensures both completion of in-phase synchronization and sufficient propagation time for spacetime-mediated informational entropy to contribute to surprise amplification.

Definitions:

1. Surprise. The invariant of consciousness is quantified as Bayesian surprise: \[ {S}(t) = D_{{KL}}(P_{{posterior}}(t) \,\|\, P_{{prior}}(t)), \] or equivalently via Shannon information^†the Shannon entropy or self-information of a random variable can be applied to the inputs to the tract as well as to the outputs to measure the change, encoding the divergence between expected and actual inputs across time.

2. Ephaptic coupling. Non-synaptic, current-mediated interaction among adjacent axons, parameterized by coupling strength \(g_{ephaptic}\), enabling collective phase dynamics and amplification of entropy.

3. Stage number \(S\). Represents sequential or parallel amplification units in the tract. Minimal sufficiency is achieved at \(S=1\) if \(L L_{}\); higher \(S\) can increase amplification but is not required.

4. Tract length \(L_{}\). The minimum length necessary to allow:

   1. Completion of in-phase synchronization of axonal firing, ensuring coherent collective dynamics.

   2. Sufficient temporal window for propagation of spacetime-embedded informational bits contributing to entropy amplification.

5. Spacetime contribution. The causal structure of spacetime supplies additional probabilistic variability (entropy) that integrates with internal neural inputs via ephaptic coupling, increasing the total system surprise: \[ {S}_{total} = {S}_{internal} + (L) {S}_{spacetime}, \] where \((L)\) is a monotonically increasing function of tract length representing integration capacity for spacetime-mediated entropy.

Corollaries and Implications:

1. Minimal sufficiency. The substrate described is sufficient for consciousness because it guarantees nonzero, amplified surprise arising from both internal and spacetime-coupled inputs.

2. Necessary condition. Any system lacking:

   1. Ephaptic coupling,

   2. Sufficient tract length \(L < L_{}\), or

   3. Ability to sustain in-phase synchronization across at least one stage

   cannot achieve the amplification of surprise and therefore cannot manifest consciousness.

3. Exclusion of trivial systems. Simple reflex arcs, feedforward networks, or isolated axonal bundles without sufficient length and coupling fail the criteria, eliminating nonconscious mechanisms.

4. Integration of geometry and biophysics. \(L_{}\) is explicitly dependent on measurable physical parameters: \[ L_{} = f(d_{axon}, _{extracellular}, g_{ephaptic}, v), \] making the theorem empirically testable.

5. Spacetime as entropy source. Longer tracts allow both greater non-spacetime synchronization and additional entropy accumulation from spacetime-mediated causal structure, providing a dual mechanism for surprise amplification.

6. Compression. Consciousness, a high-level cognitive phenomenon, is thereby reduced to a precise, physically and information-theoretically defined mechanism, eliminating the need for extra explanatory entities beyond ephaptically coupled, spacetime-embedded axon dynamics.

Conclusion. This theorem establishes a minimal, mechanistic, and measurable condition for consciousness, grounded in:

• Axonal geometry and biophysics,

• Ephaptic field interactions,

• Temporal dynamics of synchronization, and

• Integration of spacetime-mediated informational entropy.

It provides a concrete, testable, and theoretically compressed framework, analogous in conceptual structure to the reduction of gravity to spacetime curvature.

Acknowledgments

This work was produced with the assistance of language models.