# DRCT v13 — Continuous Model Manifold and Invented Experiments

## Model

Let the unknown mechanism parameter be continuous:

\[
\Theta\in\mathbb R^d
\]

with Gaussian posterior

\[
\Theta\mid D\sim\mathcal N(m,\Sigma).
\]

A scalar experiment chooses a continuous unit vector \(a\) and observes

\[
Y=a^\top\Theta+\varepsilon,
\qquad
\varepsilon\sim\mathcal N(0,\sigma^2).
\]

## Theorem 38 — Closed-Form Expected Information Gain

The mutual information between the continuous parameter and the experiment
outcome is

\[
\boxed{
I(\Theta;Y\mid D,a)
=
\frac12\log_2\left(
1+\frac{a^\top\Sigma a}{\sigma^2}
\right).
}
\]

### Proof

The predictive outcome is Gaussian with variance

\[
\operatorname{Var}(Y\mid D,a)=a^\top\Sigma a+\sigma^2.
\]

Conditional on \(\Theta\), only measurement noise remains, with entropy
corresponding to variance \(\sigma^2\). The difference of the two Gaussian
entropies yields the formula.

This is standard linear-Gaussian Bayesian experimental design.

## Theorem 39 — Continuous Experiment Synthesis

Constrain \(\|a\|_2=1\). Since the logarithm is monotone, maximizing EIG is
equivalent to maximizing

\[
a^\top\Sigma a.
\]

By the Rayleigh-Ritz theorem,

\[
\boxed{
a^\star=v_{\max}(\Sigma),
}
\]

a principal eigenvector of the current posterior covariance.

Thus the next experiment direction is **derived continuously** rather than
selected from a pre-enumerated list.

## Posterior update

Let

\[
r=\sigma^2+a^\top\Sigma a.
\]

Then the exact Gaussian posterior update is

\[
m^+
=
m+
\frac{\Sigma a}{r}
(Y-a^\top m),
\]

\[
\boxed{
\Sigma^+
=
\Sigma-
\frac{\Sigma aa^\top\Sigma}{r}.
}
\]

The covariance update does not depend on the realized outcome. Therefore the
planned information gain is known before the experiment.

## Multiple experiment families

If experiment family \(k\) has cost \(c_k\), noise \(\sigma_k\), and access to
subspace covariance \(\Sigma_k\), DRCT chooses

\[
\boxed{
k^\star
=
\arg\max_k
\frac{
\frac12\log_2(1+\lambda_{\max}(\Sigma_k)/\sigma_k^2)
}{
c_k
}.
}
\]

Within that family it chooses the corresponding principal eigenvector.

This is a continuous analogue of the v12 finite hypothesis-class planner.

## Closure at finite resolution

With noisy continuous measurements, exact point identification generally
requires unbounded information. Therefore DRCT must specify a resolution
criterion.

For family \(k\), declare closure at resolution \(\epsilon_k\) when

\[
\boxed{
\lambda_{\max}(\Sigma_k)\le \epsilon_k^2.
}
\]

This is not metaphysical certainty. It is posterior closure relative to a
declared experimental resolution.

## Literature boundary

Bayesian D-optimal design and expected information gain for Gaussian inverse
problems are established mathematics. DRCT does not claim the EIG formula or
Rayleigh-Ritz result as novel.

The DRCT synthesis is the progression:

\[
\text{equivalence class}
\to
\text{posterior manifold}
\to
\text{continuous uncertainty geometry}
\to
\text{automatically synthesized experiment direction}
\to
\text{posterior contraction}
\to
\text{resolution-relative closure}.
\]