# DRCT v10 — Exact Module Tomography and the Observability Ladder

## Result

For the known transformer architecture, an operational interface exposing each
parameterized module's input/output map is sufficient to recover all 1428
scalar parameters.

### Linear Operator Tomography

For an affine map

    f(x)=W x+b,

query

    f(0)=b

and every basis vector e_j. Then

    W[:,j]=f(e_j)-f(0).

Therefore an n-input affine module is exactly identified in n+1 vector queries.

Each recovered W[i,j] has a direct operational meaning:

    W[i,j] = change in output coordinate i caused by a unit change in input
             coordinate j with all other coordinates held at zero.

### LayerNorm Tomography

For

    LN_i(x)=gamma_i z_i(x)+beta_i,

the zero vector has z=0, hence

    beta=LN(0).

Choose any known probe x whose normalized coordinates z_i are nonzero. Then

    gamma_i=(LN_i(x)-beta_i)/z_i.

Thus two vector queries identify all affine LayerNorm parameters when epsilon
and normalization convention are architecture metadata.

### Embeddings

Finite token and positional embeddings are identified directly by enumerating
their discrete indices.

## Observability hierarchy

1. token -> logits:
   identifies only the external function/equivalence class in general.

2. named component ablation -> logits:
   identifies causal component signatures relative to the intervention language.

3. arbitrary module input -> module output:
   identifies affine transformer operators exactly by basis tomography.

The stronger the operational interface, the finer the identifiable equivalence
class.

This is not a claim that ordinary hosted LLM APIs expose module tomography.
They do not. It is a theorem about sufficient experimental access.

## Anti-overclaim

The result does not show that raw weights can be uniquely recovered from token
logits alone. v6 proved the opposite by transformer parameter symmetries.

It shows that once the experimental language contains spanning module probes,
the coordinate symmetry is fixed by that interface and the model becomes
operator-identifiable.