Intelligence as High-Dimensional Coherence: The Observable Dimensionality Bound and Computational Tractability

Intelligence arises from high-dimensional continuous dynamics operating in phase space with effective dimensionality $D_{\text{eff}} \sim 10^3$–$10^4$. This is not one implementation among many—it is the thermodynamically favored way to track and control complex environments in real time under bounded measurement capacity.

What does high-dimensional continuous computation enable?

1. Simultaneous exploration of incompatible states. In high-D phase space ($D_{\text{eff}} \gg 1$), orthogonal subspaces allow the system to maintain superpositions of mutually exclusive configurations without forced resolution.

2. Thermal noise as functional dimensionality expansion. Coupling to a thermal bath enables computation through stochastic resonance and noise-assisted barrier crossing.

3. Constraint satisfaction without enumeration. Problems intractable for discrete search become tractable when relaxed to continuous high-D dynamics.

4. Power scaling with behavioral output, not internal complexity. Biological systems dissipate $\sim$20 W regardless of task dimensionality.

We establish a fundamental relationship between dimensionality, measurement capacity, and temporal resolution. There exists a critical dimension:

$D_{\text{crit}} = \frac{C_{\text{obs}} \tau_e}{\alpha h_\varepsilon^{\text{track}}}$

where $C_{\text{obs}}$ is the observation channel capacity (bits/s), $\tau_e$ is the evolution timescale, $h_\varepsilon^{\text{track}}$ is the minimum bits per mode per $\tau_e$ to track geometry, and $\alpha$ captures compressibility.

When $D_{\text{eff}} > D_{\text{crit}}$, the system's constraint geometry reconfigures faster than behavioral measurement can track—it becomes timing-inaccessible. This establishes a physical boundary separating observable computation from timing-inaccessible computation.

Theorem 1 (Measurement-Theoretic Tracking Bound). Consider a target system with effective dimensionality $D_{\text{target}}$. Let an external observer with measurement bandwidth $C_{\text{obs}}$ attempt to predict events with timing precision $\varepsilon$ over coherence time $\tau_e$. Then:

1. If $D_{\text{target}} \le D_{\text{crit}}$: The system is observationally accessible. External measurement can resolve enough dimensions to predict outputs.

2. If $D_{\text{target}} > D_{\text{crit}}$: The system is observationally inaccessible. Insufficient measurement bandwidth to resolve the full manifold of causal influences.

The brain does not face this constraint because it is the high-dimensional substrate. No "measurement" occurs during internal evolution—information is carried in geometric configuration.

Theorem 2 (Code Formation from Dimensional Mismatch). When two high-dimensional systems interact through a low-dimensional communication channel, stable codes must form at the boundary.

If $D_B > D_{\text{link}}^{\text{crit}}$, the full state is observationally inaccessible. However, if $B$'s behaviorally-relevant dynamics are confined to structured regions—recurring subspaces $\mathcal{S}_i$—then $A$ can learn to recognize these regions as discrete codes $c_i$. The effective dimensionality of what must be tracked collapses from $D_B$ to $D_{\text{code}} \sim \log_2(N_{\text{codes}})$.

Implication: This theorem provides a thermodynamic explanation for the emergence of language, gesture, and other symbolic codes in biological systems. When two agents with high internal dimensionality ($D_{\text{brain}} \sim 10^3$–$10^6$) coordinate through low-bandwidth channels (speech $\sim 50$ phonemes/s), stable symbolic codes are thermodynamically necessary.

MEG source reconstruction yields hundreds of cortical parcels ($N \sim 10^2$–$10^3$) coupling across frequency bands ($|B| \sim 3$–$5$). A conservative estimate of effective dimensionality:

$D_{\text{eff}}^{\text{MEG}} \sim \kappa N |B| \approx 0.3 \times 250 \times 4 \approx 300$

For mid-range parameters ($C_{\text{obs}} \sim 10^2$ bits/s, $\tau_e \sim 0.1$ s):

$D_{\text{crit}} = \frac{100 \times 0.1}{1 \times 2} = 5 \text{ modes}$

Therefore: $D_{\text{eff}}^{\text{MEG}}/D_{\text{crit}} \approx 10^2$. Even at MEG-accessible scales, cortex operates $\sim 10^2\times$ above the observability threshold.

Training GPT-scale models ($\sim 10^{11}$ parameters) requires $\sim 10^{24}$ collision events, consuming megawatts. The human brain achieves comparable complexity at $\sim$20 W—six orders of magnitude less. This gap reflects the fundamental thermodynamic cost of dimensional mismatch.

Irreducible complexity is irreducible. Systems with irreducible high-D—ecosystems, vector addition systems with Ackermann-complete reachability, multi-agent dynamics—cannot be faithfully compressed without information loss.

The clocking constraint: Digital systems enforce temporal synchronization via a global clock signal. Every register must settle to a definite state at each clock edge, forcing $D_{\text{eff}}/D_{\text{crit}} \to 1$. Biological systems operate unclocked—oscillations emerge from coupled dynamics without external synchronization, permitting $D_{\text{eff}} \gg D_{\text{crit}}$.