Transformation semantics and register

Page status: normative vocabulary and first reader-facing transformation register.

The word equivalent is too small for the transformations in this book. A rewrite can preserve a terminal equation while deleting an asset, changing a relaxation, or losing a current limit. This chapter gives the vocabulary used to say exactly what survived.

Four preservation layers

Write a model as

\[M=(\mathsf S,\mathsf R,\mathsf C,\mathsf P),\]

where $\mathsf S$ is typed structure (objects, ports and incidence), $\mathsf R$ is the constitutive and control relation, $\mathsf C$ is the constraint and decision contract, and $\mathsf P$ is provenance and ownership.

A transformation $T:M\to\widehat M$ should state which of these layers it preserves:

These layers are independent. A Kron reduction can preserve a boundary relation while changing structure and provenance. A coordinate permutation can preserve all four layers. A summed parallel limit can preserve an unconstrained terminal relation while changing the decision layer.

Structure-preserving versus structure-changing

Call $T$ structure-preserving on a declared structure signature when there are maps on each object sort and port set that are bijective on the retained objects and commute with typed incidence, terminal maps and hierarchy. In other words, the diagram of objects and relations commutes after a change of names or coordinates. A permutation of conductors, for example, is a structure-preserving normalization when the matrices, terminal maps, limits and inverse permutation are transformed together.

An identification, elimination, aggregation or asset composition is instead structure-changing. It may still be exact for a smaller observation family, but it must name what was forgotten and what target factor class is allowed. The following distinctions are therefore deliberate:

The last row is important: an algebraic matrix product is not automatically a well-typed network transformation.

Closure is part of the theorem

Suppose $\mathcal L$ is the library of physical line factors. A series relation may be exact in a larger library $\mathcal F$ of generic passive two-ports while failing to lie in $\mathcal L$. The correct statement is

\[T:\mathcal L\times\mathcal L\longrightarrow\mathcal F, \qquad T(\ell_1,\ell_2)\notin\mathcal L\ \text{in general}.\]

For a nominal-$\pi$ section, cascading two sections generally produces a general two-port rather than the naive nominal-$\pi$ parameter sum. For a transformer, the target must retain turns ratio, winding incidence, grounding, excitation, taps and ratings. For an external grounding impedance, the target must retain the ground asset and its current observation unless the study explicitly declares those facts out of scope.

The first transformation register

This table is the reader-facing view of the certificate schema. It is kept small and explicit; generated certificates remain the evidence source.

Every row should eventually be backed by a certificate with preconditions, preserves, forgets, constraint_map, recovery_map, target_type and evidence. The register is therefore a navigation layer over the existing certificate system, not a competing schema.

The glyphs are deliberately low-bandwidth. A solid or dashed arrow, a split factor, a fill edge, or a state tag helps the reader find the relevant row, but it never carries the guard or preservation theorem by itself. Those remain in the register and in the executable certificate.

Narrow circuit transformations: star–delta and shunt placement

The register also covers transformations that power engineers routinely use inside a line, load, or feeder model without calling them graph transformations. They still change the factorization and therefore need the same preservation language.

Scalar floating star–delta

For three scalar impedances $z_a,z_b,z_c$ connected from terminals $a,b,c$ to a floating internal star point, the terminal behaviour can be represented by a delta with

\[z_{ab}=z_a+z_b+\frac{z_a z_b}{z_c}, \qquad z_{bc}=z_b+z_c+\frac{z_b z_c}{z_a}, \qquad z_{ca}=z_c+z_a+\frac{z_c z_a}{z_b}.\]

The inverse uses the corresponding nonzero delta-sum denominators, for example

\[z_a=\frac{z_{ab}z_{ca}}{z_{ab}+z_{bc}+z_{ca}}.\]

These are exact terminal-equation transforms for a linear scalar network when the required impedances and sums are nonsingular. They are not automatically exact for a constant-power load, a switched capacitor bank, a grounded star point, branch-specific current limits, or a protection/measurement boundary. Those quantities either need recovery maps or remain explicit factors.

The familiar formulas also do not lift by replacing each scalar with an arbitrary coupled matrix: matrix products need not commute, and a matrix star point may have a different terminal coordinate space at each arm. Use a block Schur complement or a typed port–factor relation instead. A commuting or scalar-block specialisation can be declared separately.

Star and delta have different graph shapes: a star is a tree through an internal point, while a delta contains a three-edge cycle. That cycle can be a compilation artefact, not evidence of an additional physical corridor cycle. Conversely, eliminating a floating star point can hide a source identity or a grounding location. STAR-DELTA therefore preserves a declared terminal relation, not the source topology or asset semantics by default.

Endpoint shunts and tool asymmetry

For a two-terminal series factor with distinct endpoint shunts, write

\[\begin{aligned} I_{ij}&=Y_s(U_i-U_j)+Y^{\mathrm{sh}}_{i}U_i,\\ I_{ji}&=-Y_s(U_i-U_j)+Y^{\mathrm{sh}}_{j}U_j. \end{aligned}\]

The endpoint shunts may be unequal. A reciprocal primitive can therefore have different diagonal endpoint blocks while its off-diagonal blocks remain transpose paired. Endpoint asymmetry is not the same thing as non-reciprocity. Replacing $Y^{\mathrm{sh}}_i$ and $Y^{\mathrm{sh}}_j$ by their average because a software line object has one shared shunt field is an approximation (or an invalid encoding), not a coordinate change.

Absorbing a capacitor bank, a neutral grounding point, a magnetizing branch, or another explicit shunt into a line or transformer can preserve a fixed linear terminal equation. It can nevertheless delete the shunt's switching state, rating, owner, protection boundary, frequency dependence, or neutral/ earth-current observation. A safe lowering target is either an endpoint- augmented generic two-port with both shunts retained in provenance, or separate explicit shunt factors. If the target tool cannot represent unequal from/to shunts, retain the richer source and report the adapter limitation; do not silently symmetrize.

The same warning applies when a delta load is replaced by a wye load: the linear floating terminal relation may be preserved, while grounded-neutral current, phase-specific limits, unbalance, and switching semantics are not.

The executable companion at experiments/generated/narrow-circuit-transformations-witness.json checks the floating scalar identity, rejects a grounded-star use of that rule, and quantifies the residual introduced by an adapter that replaces unequal endpoint shunts with one shared field.

Risk-aware impedance paths

The four-wire impedance ladder provides a compact example of how the register should be used. Starting from a coupled conductor primitive, a path may contain the following edges:

\[\hat{\mathbf Z}^{\mathrm{circ}}_{\ell} \xrightarrow{K_g} \bar{\mathbf Z}^{\mathrm{cond}}_{\ell} \xrightarrow{K_n\ \text{or}\ P_n} \mathbf Z^{\mathrm{phase}}_{\ell} \xrightarrow{F} \mathbf Z^{012}_{\ell} \xrightarrow{D\ \text{or}\ F_1} \mathbf Z^{\mathrm{restricted}}_{\ell}.\]

The path should carry a risk vector, not a single score. For each of the four preservation layers, record one of exact, guarded, bounded, not-preserved, or unknown. For example:

This makes transformation sequences auditable. A later rule may not silently upgrade an earlier not-preserved layer to exact; composition must carry the weakest status and the union of unresolved guards. The generated experiments/generated/four-wire-impedance-model-ladder.json artifact records this path and checks that every edge has explicit risk tags. It also composes the main $K_g\to K_n\to F\to D\to F_1$ route and the phase-to-neutral branch. The composition record carries the weakest exactness label, the union of guards that have not been discharged, and the union of forgotten observations and risk tags. A deliberately mismatched source/target pair is rejected by the same executable helper, so a visually plausible chain cannot silently become a proof.

Anti-patterns worth showing explicitly

Adding different lines in series

Different impedances are not by themselves a counterexample. With a zero-injection, shunt-free junction and compatible conductor coordinates,

\[\mathbf Z_{\mathrm{eq}} =\mathbf Z_{\ell_1} +\mathbf P^{\mathsf T}\mathbf Z_{\ell_2}\mathbf P\]

is an exact behavioural composite. Calling that composite a longer instance of one physical line class is a separate claim. A different construction code, frequency basis, terminal map, rating owner, splice or protection boundary blocks that stronger claim.

Folding a line into a transformer

A fixed line–transformer cascade can be compiled into a generic multiport, but the result is not a line merely because it has two external buses. The voltage ratio, vector group, galvanic boundary, winding limits and control ownership are not decoration. If they matter to the study, flattening them into a line is a semantic loss.

Folding external grounding into a transformer

An internal transformer neutral branch belongs to the transformer model. An external grounding reactor or shunt belongs to the bus/grounding model. Combining them may be algebraically convenient, but it hides where neutral current flows and who owns the limit. The canonical transformation should refuse the merge or emit a composite factor with an explicit ground-port and recovery ledger.

Treating a formulation change as a graph proof

The BIM/BFM parallel-line example is a formulation-level warning. Replacing a shared $W_{ij}$ by a branch-wise $W_{\ell ij}$ restores an index but weakens the BIM relaxation; omitting the BFM consistency relation admits flows that do not satisfy Kirchhoff's voltage law across the parallel members. Neither change is a graph isomorphism.

The executable negative cases are collected with the book's translation-trap witnesses. They are useful regression tests for the register: a proposed rule must either satisfy the named guard or be classified as a structure-changing, outer, inner, or scenario transformation.

Review checklist

Before calling a transformation safe, ask:

1. What are the source and target object sorts?
2. Which $\ell i j$ identities and terminal maps remain addressable?
3. Which constitutive relation is preserved, and on what domain?
4. Where do every current, power, voltage, thermal and protection limit go?
5. Can eliminated quantities be recovered for every feasible target decision?
6. Does the result remain in the claimed physical factor class?
7. Which asset, ownership, measurement and grounding questions are no longer answerable?
8. Does the proof concern the physical model, a relaxation, an objective value, or a particular optimizer?

If any answer is missing, the result is a candidate reduction, not a certified equivalence.