VRC Conjecture: Persistent Volumetric State Enables (O(N^3)) Controllability

Conjecture

A Volumetric Recurrent Computing substrate can approach

[ N_{\mathrm{ECD}}=O(N^3) ]

for an (N\times N\times N) lattice only when each volumetric site supports an independently programmable, persistent local state.

The essential material primitive is:

[ \boxed{ \text{persistent} + \text{locally writable} + \text{optically readable} + \text{reconfigurable state} } ]

at every computational site.

Without such local persistence, three intersecting optical control fields generally produce only a low-dimensional instantaneous configuration.

For example,

[ \theta_{ijk}=f(a_i,b_j,c_k) ]

contains (N^3) spatial values but is generated from only (3N) independent controls. Its reachable configuration space therefore has dimension at most approximately

[ O(N), ]

not (O(N^3)).

With line-indexed control fields, the bound may increase to

[ O(N^2), ]

but it still does not yield independent volumetric configuration.

Persistent-site mechanism

Suppose three addressing paths identify a site:

[ X_i\cap Y_j\cap Z_k \longrightarrow v_{ijk}. ]

If their coincidence causes a durable local change,

[ (X_i,Y_j,Z_k) \longrightarrow \theta_{ijk}, ]

and that state remains after the addressing fields are removed, then sites may be programmed sequentially:

[ \Theta

{\theta_{ijk}}_{i,j,k=1}^{N}. ]

The number of addressing channels can remain much smaller than the number of stored states, just as an address bus can select among many memory cells.

Under ideal independent programming,

[ N_{\mathrm{configurable\ sites}}=N^3 ]

and therefore the upper bound becomes

[ \boxed{ N_{\mathrm{ECD}}\leq N^3. } ]

Reaching that upper bound requires that distinct site configurations produce independently distinguishable effects on computation.

Role of CsPbBr(_3)

CsPbBr(_3) nanocrystals embedded in glass are a candidate mechanism for introducing localized, optically active material states into a femtosecond-laser-written volumetric substrate.

Their possible relevance to VRC is not principally that they are quantum emitters. The relevant possibility is that engineered CsPbBr(_3) regions could provide some combination of:

localized optical response;
strong light–matter interaction;
optical nonlinearity;
persistent or metastable state;
optical write and read access;
compatibility with volumetric laser fabrication.

The proposed role is therefore:

[ \text{fs-written paths} \rightarrow \text{addressing and connectivity}, ]

[ \text{CsPbBr}_3\text{ sites} \rightarrow \text{persistent programmable node state}. ]

This would produce a substrate of the form

[ \boxed{ \text{permanent 3-D topology} + \text{persistent programmable volumetric state} + \text{fast circulating optical field}. } ]

At present, CsPbBr(_3) should be treated only as a candidate material. Existing demonstrations of CsPbBr(_3) nanocrystals in glass do not establish independently writable, persistent, high-density (N^3) optical memory nodes.

Effective Controllable Dimension

Raw persistent-site count is not sufficient.

For recurrent behavior

[ B_T(\Theta), ]

define the configuration-response Jacobian

[ J_{\Theta,T}

\frac{\partial B_T}{\partial\Theta}. ]

The Effective Controllable Dimension is the number of singular directions that remain distinguishable above the relevant physical noise floor:

[ \boxed{ N_{\mathrm{ECD}}(T)

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

The strong conjecture is therefore not merely

[ N_{\mathrm{sites}}=O(N^3), ]

but

[ \boxed{ N_{\mathrm{ECD}}(T)=\Omega(N^3) } ]

over a useful range of lattice sizes and recurrence horizons.

Equivalently, controllability efficiency

[ \eta_{\mathrm{ECD}}

\frac{N_{\mathrm{ECD}}}{N^3} ]

must remain bounded away from zero as (N) increases.

Required material properties

A candidate material must demonstrate:

Localization

Writing site (v_{ijk}) must not materially alter nearby sites.
Persistence

The programmed state must survive without continuous control illumination.
Rewritability

Sites must support repeated updates or erasure.
Selectivity

One- and two-path exposure must not cause substantial half-selection.
Readable influence

Different local states must produce distinguishable effects on propagating computational fields.
Low crosstalk

Programming and reading one site must not collapse the independence of neighboring sites.
Adequate retention-to-write ratio

The state must persist much longer than the time required to program it.
Compatible nonlinearity

Local response must support useful state-dependent computation without uncontrolled instability or loss.
Scalable optical access

Addressing, readout, and propagation must remain practical as lattice size increases.

Falsification

The conjecture is weakened if any of the following occurs:

the number of distinguishable configurations scales only as (O(N)) or (O(N^2));
local states cannot be retained after addressing fields are removed;
programming crosstalk grows with lattice size;
singular directions of (J_{\Theta,T}) rapidly collapse below noise;
optical loss prevents deeper sites from influencing output;
independently programmed sites produce strongly redundant computational effects;
(\eta_{\mathrm{ECD}}\rightarrow0) as (N\rightarrow\infty).

It is supported if experiments show that:

[ N_{\mathrm{ECD}} \propto N^3 ]

to useful approximation while maintaining acceptable write energy, retention, loss, noise, and recurrence stability.

Research implication

The primary VRC materials question is not:

Which material has the strongest optical nonlinearity?

It is:

Which material can provide the greatest density of independently programmable, persistent, optically coupled computational state?

The relevant substrate metric is therefore not merely voxel density, but

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

effective controllable dimension per unit volume, together with:

[ \frac{N_{\mathrm{ECD}}}{E_{\mathrm{write}}}, \qquad \frac{N_{\mathrm{ECD}}}{E_{\mathrm{inference}}}, \qquad \frac{N_{\mathrm{ECD}}}{$}, \qquad \text{retention}, \qquad \text{rewrite endurance}. ]

Summary

Three intersecting paths can identify (N^3) locations, but they do not create (N^3) independent computational degrees of freedom by themselves.

The transition from

[ O(N)\ \text{or}\ O(N^2) ]

[ O(N^3) ]

requires local memory:

[ \boxed{ \text{address coincidence} \rightarrow \text{persistent site state} \rightarrow \text{independent computational effect}. } ]

CsPbBr(_3) in glass is one candidate for supplying that local physical state. Whether it can do so with the required persistence, selectivity, rewritability, and scale is an open experimental question.