VRC Conjecture: Effective Controllable Dimension

Conjecture

The useful computational capacity of a Volumetric Recurrent Computing substrate is determined primarily by the number of independently controllable transformations it can induce, not by its raw number of physical sites.

For

[ S’ = F_\Theta(S), ]

define the configuration-response Jacobian

[ J_\Theta = \frac{\partial F_\Theta(S)} {\partial \Theta}. ]

Its singular spectrum

[ \sigma_1 \ge \sigma_2 \ge \cdots ]

describes how strongly independent changes in physical configuration affect computational behavior.

Because a real analog substrate has noise, drift, fabrication error, and finite measurement resolution, literal matrix rank is not the useful quantity.

Define the Effective Controllable Dimension as

[ \boxed{ N_{\mathrm{ECD}}

#\left{ \sigma_i(J_\Theta)>\sigma_{\mathrm{noise}} \right}. } ]

That is, (N_{\mathrm{ECD}}) counts the number of independently controllable computational directions whose effects remain distinguishable above the relevant physical noise floor.

Capacity conjecture

For VRC substrates of comparable architecture,

[ \boxed{ \text{useful learned capacity} \sim N_{\mathrm{ECD}} } ]

should correlate more strongly with computational capability than

[ N_{\mathrm{physical\ sites}}. ]

A substrate containing (10^{11}) programmable voxels is therefore not meaningfully a (10^{11})-parameter machine unless changes to those voxels produce approximately (10^{11}) independently distinguishable effects on computation.

Controllability efficiency

Define

[ \boxed{ \eta_{\mathrm{ECD}}

\frac{N_{\mathrm{ECD}}} {N_{\mathrm{physical\ sites}}}. } ]

This measures how efficiently physical complexity becomes independently usable computational complexity.

For example,

[ N_{\mathrm{sites}}=10^{11}, \qquad \eta_{\mathrm{ECD}}=10^{-4} ]

implies only

[ N_{\mathrm{ECD}}\sim10^7. ]

Conversely,

[ N_{\mathrm{sites}}=10^9, \qquad \eta_{\mathrm{ECD}}=0.3 ]

implies

[ N_{\mathrm{ECD}}\sim3\times10^8, ]

making the physically smaller substrate potentially much more computationally useful.

Substrate metrics

Candidate VRC materials and architectures should therefore be compared using quantities such as

[ \boxed{ \frac{N_{\mathrm{ECD}}}{V} } ]

effective controllable dimension per unit volume,

[ \boxed{ \frac{N_{\mathrm{ECD}}}{E} } ]

effective controllable dimension per unit operating energy, and

[ \boxed{ \frac{N_{\mathrm{ECD}}}{$} } ]

effective controllable dimension per unit manufacturing cost.

Raw voxel density is insufficient.

Experimental estimation

For a programmable substrate:

  1. choose an operating state ((S,\Theta));
  2. perturb configurable physical parameters;
  3. measure the resulting changes in output or recurrent dynamics;
  4. estimate the action of

[ J_\Theta= \frac{\partial F_\Theta}{\partial\Theta}; ]

  1. estimate its singular spectrum using matrix-free methods;
  2. determine which singular directions remain distinguishable above the measured physical noise floor.

For large systems, explicitly constructing (J_\Theta) should not be required. Use:

  • Jacobian-vector products;
  • randomized SVD;
  • power iteration;
  • experimentally sampled perturbations;
  • low-rank spectral estimation.

The same methodology should apply to simulated and physical VRC substrates.

Falsification

The conjecture is weakened if computational capability continues scaling strongly with raw physical site count after

[ N_{\mathrm{ECD}} ]

has saturated.

Conversely, if substrates with comparable (N_{\mathrm{ECD}}) exhibit comparable learned capacity despite very different raw site counts, that would support ECD as a fundamental VRC capacity measure.

Relation to recurrence

VRC has two independent scaling variables:

[ \boxed{ N_{\mathrm{ECD}} } \qquad\text{and}\qquad \boxed{ T } ]

where

  • (N_{\mathrm{ECD}}) measures the effective size of the configurable machine;
  • (T) measures how long its transient state is allowed to evolve through that machine.

This suggests the VRC analogue of conventional parameter count × inference compute:

[ \boxed{ \text{effective controllable dimension} \times \text{recurrence time}. } ]

The two central VRC questions therefore become:

How many independently useful computational directions does the physical substrate provide?

and

How much additional computation can the same substrate obtain by evolving for longer?