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

The paper studies whether high-dimensional continuous dynamics can support adaptive control when observation and action channels are restricted. This is one proposed architecture, not a necessary definition of intelligence or a theorem of thermodynamic optimality.

What does high-dimensional continuous computation enable?

1. Concurrent candidate configurations. A high-dimensional state can carry several task-relevant variables without forcing them into one scalar decision before action.

2. Functional noise in specified regimes. Stochastic resonance and noise-assisted barrier crossing show that fluctuations can contribute to an operation rather than merely corrupt it.

3. Physical relaxation as an algorithmic resource. Some optimization problems benefit from continuous relaxation, although no generic intractable problem becomes tractable merely by using continuous dynamics.

4. Implementation-level trade-offs. Energy use depends on the physical dynamics, maintenance, noise, error tolerance, and output requirements; dimension alone does not determine it.

The paper defines a tracking-capacity relationship between dimensionality, measurement bandwidth, and temporal resolution:

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

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 $D_{\text{eff}} > D_{\text{crit}}$ under these assumptions, the chosen observer lacks the budget to track every modeled mode at the target precision. This is a protocol-relative boundary. Compressibility, prior structure, sufficient statistics, and task-specific prediction can change the requirement.

Consider a target system represented by $D_{\text{target}}$ independently tracked modes. Let an external observer with channel capacity $C_{\text{obs}}$ attempt full-state tracking at error tolerance $\varepsilon$ over evolution time $\tau_e$. If the model assigns a per-mode description requirement $h_\varepsilon^{\text{track}}$ and compressibility factor $\alpha$, then its bookkeeping criterion is:

1. If $D_{\text{target}} \le D_{\text{crit}}$, the stated capacity budget does not by itself rule out full-mode tracking.

2. If $D_{\text{target}} > D_{\text{crit}}$, that particular full-mode code exceeds the stated budget.

This is not a necessary-and-sufficient theorem for prediction or observation. A task may require only sufficient statistics; modes may be dependent or compressible; priors and active sensing may reduce the rate; and effective dimension depends on representation and scale. The criterion becomes testable only after those quantities and the tracking task are operationally fixed.

An embedded controller need not reconstruct its own full microstate through the same external channel, but it still faces sensing, communication, noise, and control limits. “Being the substrate” does not remove physical measurement or information-processing constraints.

Dimensional mismatch alone does not force stable or discrete codes. A low-dimensional channel can remain continuous and may even preserve global identity. Code formation requires additional structure: for example, an explicitly many-to-one or finite-resolution readout, repeated coordination tasks, and a learning or stabilization process that makes some equivalence classes reusable.

If $D_B > D_{\text{link}}^{\text{crit}}$ and $B$'s behaviorally relevant dynamics repeatedly occupy structured regions $\mathcal{S}_i$, then $A$ may learn task-relative labels $c_i$ for those regions. When such labels are deliberately discretized, the effective description length can fall from that needed to track $D_B$ to roughly $D_{\text{code}} \sim \log_2(N_{\text{codes}})$.

Interpretation: Limited channels can encourage useful codes, including language and gesture, because repeated task-relevant equivalence classes are cheaper to communicate than full internal states. This is a conditional mechanism, not a theorem that dimensional reduction makes symbolic discreteness inevitable.

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, under the stated mid-range parameters, $D_{\text{eff}}^{\text{MEG}}/D_{\text{crit}} \approx 10^2$. This illustrates the model's sensitivity to observer bandwidth; it is not a substrate-independent estimate of how much cortex is knowable.

Biological brains and present-day AI systems differ in task, training history, hardware, precision, and energy accounting, so headline power ratios do not isolate one mechanism. The paper's hypothesis is that physical organization and the placement of observation and commitment boundaries contribute to those differences.

Lossless compression of an arbitrary source is constrained by its structure and entropy, while task-specific summaries may preserve what a controller needs. “High-dimensional” does not mean incompressible without a stated source class and error criterion.

Clocked and asynchronous digital systems, analog devices, and biological oscillators all support varied internal dynamics. The empirical comparison should ask which intervention responses, noise tolerances, latencies, and energy costs each realization achieves on the same task.