# DRCT v14 — Symbolic Sensor Equation Synthesis

## Operational setting

Let primitive transduction channels be

\[
M=(M_1,\ldots,M_p)
\]

and let \(\mathcal G_d\) be the set of physically executable scalar expressions
generated by a fixed operator grammar with complexity at most \(d\).

A synthesized scalar sensor is

\[
Q=g(M),\qquad g\in\mathcal G_d.
\]

The predictive target is \(F\).

## Definition — Grammar Closure Depth

\[
\boxed{
d_\mathcal G
=
\min\left\{
d:\exists g\in\mathcal G_d,\ 
H(F\mid Z,g(M))=0
\right\}.
}
\]

If no such finite \(d\) exists, set \(d_\mathcal G=\infty\).

This is closure relative to an executable expression language, not a statement
that the expression is ontologically unique.

## Theorem 40 — Monotone Grammar Refinement

If

\[
\mathcal G_1\subseteq\mathcal G_2\subseteq\cdots,
\]

then

\[
\boxed{
\min_{g\in\mathcal G_{d+1}}H(F\mid Z,g(M))
\le
\min_{g\in\mathcal G_d}H(F\mid Z,g(M)).
}
\]

Proof: the minimization at depth \(d+1\) is over a superset.

Thus residual predictive entropy can only stay constant or decrease as the
physical expression language becomes richer.

## Theorem 41 — No New Information Beyond Primitive Joint Readout

Because \(Q=g(M)\) is deterministic,

\[
\boxed{
I(F;Q\mid Z)\le I(F;M\mid Z).
}
\]

The synthesized equation does not create information. Its purpose is to find a
compact, executable scalar statistic that preserves the predictively relevant
information already available in the primitive transduction channels.

If

\[
H(F\mid Z,Q)=0,
\]

then \(Q\) is a sufficient compressed readout for the stated prediction target.

## Minimal symbolic sensor

Among all closing expressions, define

\[
\boxed{
g^\star=
\arg\min_{g:H(F\mid Z,g(M))=0}K(g),
}
\]

where \(K(g)\) is expression complexity.

This turns "invent the sensor equation" into a constrained program-synthesis
problem.

## Prospective anti-overfitting requirement

Expression search is performed on a training set. The selected syntax is then
frozen before holdout labels are inspected.

A symbolic sensor earns empirical support only if its predictive closure or
gain survives the untouched holdout.

For a genuinely finite enumerable domain, an additional complete-domain audit
can certify the claimed finite-domain relation after prospective testing.

## Relation to established work

Symbolic regression already searches mathematical expression spaces for
interpretable equations, and modern physics-informed methods explicitly search
operator grammars for scientific descriptors. Active feature acquisition
already values measurements by predictive utility.

DRCT does not claim to invent symbolic regression.

The candidate DRCT synthesis is:

\[
\text{closure defect}
\to
\text{required predictive information}
\to
\text{fixed executable grammar}
\to
\text{minimal symbolic readout}
\to
\text{prospective closure test}
\to
\text{grammar expansion if residual remains}.
\]