Conductor-coordinate normalization
Page status: guarded exact coordinate transformation with an executable certificate; independent mathematical review remains open.
Conductor order is representation, not physics. It is nevertheless part of the meaning of every vector, matrix, and componentwise constraint. A rewrite that changes an order without changing all dependent objects is therefore a model error, not a harmless formatting operation.
Normalization in this chapter is an invertible rewrite into a declared conductor order. It is not per-unit conversion, schema normalization, or ML feature scaling. Those operations require their own maps, units, inverses where applicable, and preservation claims.
Rule
Consider an element $\ell i j$ with source conductor order $\gamma_\ell=(n,a)$ and a requested order $\widehat\gamma_\ell=(a,n)$. Let $\mathbf P_\ell$ be the unique permutation for which
\[\widehat{\mathbf x}_{\ell i j} =\mathbf P_\ell\mathbf x_{\ell i j}.\]
Intrinsic element data retain only the element index. In particular, the coordinate-normalized impedance is
\[\widehat{\mathbf Z}_\ell =\mathbf P_\ell\mathbf Z_\ell\mathbf P_\ell^{\mathsf T},\]
and a vector of componentwise limits becomes
\[\widehat{\mathbf I}^{\max}_\ell =\mathbf P_\ell\mathbf I^{\max}_\ell.\]
The terminal map at $j$ is reordered by preserving each original from–to conductor pairing. Thus a coordinate position can move, but conductor identity cannot silently change.
Exactness and inverse
A permutation matrix satisfies $\mathbf P_\ell^{-1}=\mathbf P_\ell^{\mathsf T}$. Hence every normalized state and every transformed intrinsic matrix has a unique recovery:
\[\mathbf x_{\ell i j} =\mathbf P_\ell^{\mathsf T}\widehat{\mathbf x}_{\ell i j}, \qquad \mathbf Z_\ell =\mathbf P_\ell^{\mathsf T}\widehat{\mathbf Z}_\ell\mathbf P_\ell.\]
The rule is therefore an exact normalization (TR-COORD-001). It forgets no declared source semantics. This statement depends on permuting every indexed quantity consistently; permuting only $\mathbf Z_\ell$ or only a terminal list is not the same rule.
Guards and rejection
The executable rule requires:
- unique conductor labels in both orders;
- equal source and requested conductor sets;
- equal arity; and
- preservation of the original paired terminal map.
A missing, duplicated, or new conductor label produces a structured rejection. The implementation does not guess whether, for example, g and n are interchangeable.
Executable example
For the order reversal used by the running series example,
\[\mathbf P_\ell= \begin{bmatrix}0&1\\1&0\end{bmatrix}.\]
The rule moves limits $(80,110)$ A in order $(n,a)$ to $(110,80)$ A in order $(a,n)$ and permutes both axes of $\mathbf Z_\ell$. Run:
julia experiments/test/coordinate_normalization.jl
julia --project=experiments experiments/run_coordinate_series_composition.jlThe machine-readable result is experiments/generated/coordinate-normalization-certificate.json. The next transformer chapter applies the shared coordinate action to a winding's full terminal-to-coil relation. The series chapter then composes line-coordinate normalization with degree-two elimination.