# DRCT v11 — Gauge, Measurement Ports, and Identifiable Subspaces

## Theorem 1 — Hidden-basis gauge

Consider a two-stage linear mechanism

\[
x \xrightarrow{B} z \xrightarrow{A} y
\]

with external function

\[
y=ABx.
\]

For every invertible hidden-coordinate transform \(S\),

\[
B'=SB,\qquad A'=AS^{-1}
\]

gives

\[
A'B'=AB.
\]

Therefore token/input-to-output behavior can identify at most the product \(AB\),
not the individual hidden operators \(A,B\), without additional structure.

This is a continuous gauge of equivalent internal coordinate systems.

## Theorem 2 — One internal measurement port fixes the gauge when reachable

Suppose experiments can observe

\[
z=Bx
\]

as well as \(y\), and the input basis \(e_1,\ldots,e_n\) can be applied.

Then

\[
B[:,j]=z(e_j),
\]

so \(B\) is recovered directly from \(n\) basis experiments.

Collect

\[
Z=[z(e_1)\cdots z(e_n)],\qquad
Y=[y(e_1)\cdots y(e_n)].
\]

If \(B\) has full row rank, then \(Z\) spans the hidden space and

\[
\boxed{A=YZ^+}.
\]

Thus a single internal measurement port plus spanning experiments makes both
operators individually identifiable in the port's chosen coordinate system.

## Theorem 3 — Reachable-subspace observability

If \(\operatorname{rank}(B)=r<h\), then only the restriction

\[
A|_{\operatorname{im}B}
\]

is identifiable.

For any matrix \(D\) satisfying

\[
DB=0,
\]

the operator

\[
A'=A+D
\]

produces identical outputs for every possible external input:

\[
A'Bx=ABx.
\]

Hence the action of \(A\) on hidden directions never reached by \(B\) is
experimentally meaningless relative to this interface.

This gives DRCT a stronger notion than "recover every stored number":

\[
\boxed{\text{recover every operator value on the experimentally reachable subspace}.}
\]

Values outside the reachable subspace are not determined by the experiment.

## Consequence for transformers

- logits alone leave large internal gauge/equivalence classes;
- component intervention signatures reduce those classes;
- module measurement ports can fix a basis and identify operators;
- if a hidden direction is never activated/reached, its downstream weights
  cannot be inferred from ordinary behavior on that support.

Thus "what does every value do?" must always be qualified by an experimental
support and intervention language.