# DRCT v12 — Autonomous Experimental Scientist

Let \(\Theta\) be a finite set of viable mechanisms and let experiment \(e\)
produce outcome \(Y_e\).

## Theorem 35 — Expected Information Gain

\[
H(\Theta\mid D)
-
\mathbb E_{Y_e\mid D,e}H(\Theta\mid D,Y_e,e)
=
I(\Theta;Y_e\mid D,e).
\]

Therefore a myopic information-per-cost experiment is

\[
e^\star=
\arg\max_e
\frac{I(\Theta;Y_e\mid D,e)}{c(e)}.
\]

This is standard Bayesian experimental-design / active-learning mathematics.

For deterministic experiments,

\[
I(\Theta;Y_e\mid D,e)=H(Y_e\mid D,e),
\]

so information gain is exactly the entropy of the partition induced on the
surviving mechanism class.

## Theorem 36 — Experimental-Language Impossibility Certificate

If every allowed experiment has the same outcome for every surviving
hypothesis, then no adaptive sequence of those experiments can distinguish the
survivors. The experimental language must be enlarged.

## Gauge-Fixing Laboratory

Mechanisms are \(\theta=(C,S)\), with 12 external laws \(C\) and
\(S\in GL(2,5)\), where \(|GL(2,5)|=480\).

Define

\[
B=S,\qquad A=CS^{-1},
\]

so

\[
AB=C.
\]

Thus there are \(12\times480=5760\) hidden mechanisms.

External probe:
\[
y=Cx,\qquad c=1.
\]
It contains zero information about \(S\).

Internal-port probe:
\[
z=Sx,\qquad c=4.
\]
It contains zero information about \(C\).

## Theorem 37 — Exact Optimal-Cost Lower Bound for this Laboratory

At least one external experiment is required to distinguish the external law
\(C\), because internal measurements are independent of \(C\).

At least two internal experiments are required to identify the gauge \(S\):
one nonzero vector \(x\) can produce only 24 possible nonzero outputs \(Sx\),
which cannot distinguish 480 possible gauges.

Hence every exact identification policy costs at least

\[
1+4+4=9.
\]

One external probe that separates all 12 candidate \(C\)'s plus two linearly
independent internal probes identifies \(C\) and \(S\) exactly, achieving that
lower bound.

The posterior survivor set is the current experimental equivalence class.
The loop
\[
\text{ambiguity}\to\text{experiment}\to\text{outcome}\to\text{refinement}
\]
is the operational DRCT realization of "find the next missing dimension."