Device and bound audit reference

Page status: scoped local-candidate applicability reference; implementation comparisons, not external scientific review or a published-package API promise.

Use this reference after the coverage workflow. The sections retain the complete worked device/bound witnesses, ordered-field conventions, negative cases and unresolved boundaries. They also retain their intermediate development-stage counts: use the evidence-generation table for the current running-case comparison. The exact candidate and environment instructions remain in experiments/integration/linear-audit-candidate.json and experiments/integration/README.md; the candidate is local and unpublished.

This page is in the complete HTML library and source-bound retrieval corpus. The selective PDF keeps the coverage reasoning and representative failures, with links here for the exhaustive applicability traces. A detailed residual check is still bounded by its supplied observations, admitted domain and recipe assumptions; it does not establish objective, optimality or physical adequacy.

Dispatch, transformer currents and declared ratings

Generator dispatch and voltage-bound applicability

A reported dispatch inside a power box is not enough: recompute power using the correct connection voltage and supplied current. The local candidate companion now includes a package-owned 10 V example with a $10+2\mathrm j$ VA load, $4+1\mathrm j$ VA generator injection and $6+1\mathrm j$ VA source injection. The currents are respectively $1-0.2\mathrm j$, $0.4-0.1\mathrm j$ and $0.6-0.1\mathrm j$ A. Their signs and explicit neutral returns close KCL. Equal lower/upper generator P/Q bounds fix the native comparison point; this example does not claim to optimize dispatch or generation cost.

Predict. If a result reports 3 W but voltage and current imply 4 W, can a 3 W upper bound be certified? No. The independent evaluation detects both the 1 W report discrepancy and the 1 W bound violation. The report is evidence to check, not a substitute for the defining power relation.

The phase and return current magnitudes are $\sqrt{0.17}\approx 0.4123$ A. A 0.5 A phase cap passes while a 0.4 A return cap fails. With multiple WYE phases the return magnitude is $|\sum_k I_k|$, not $\sum_k|I_k|$ and not any individual phase magnitude. For example, two in-phase 1 A injections give a 2 A return; opposite injections cancel. At zero coil voltage a power identity can still hide nonzero current, so current limits and KCL remain separate checks.

For the finite native WYE/SINGLE_PHASE domain, generator power uses the explicitly mapped coil voltage. Omitting the neutral from a WYE generator map does not borrow the bus neutral. A two-terminal SINGLE_PHASE generator uses its ordered pair, which may also be phase-to-phase. The later DELTA-generator section extends this scope: the load's two-terminal DELTA convention must not be copied into another device family without checking that family's executable contract.

Bus-voltage limits ask a different coordinate question. The supported v_min/v_max arrays bound phase-to-earth magnitude in non-neutral terminal declaration order. Moving the neutral changes a phase-neutral generator power without changing the phase's earth voltage. Tests use such a perturbed candidate to distinguish those observations; the separate switch-voltage check rejects that candidate. An unchanged bus-bound result is not a certificate for it.

The native builder skips ordinary bus bounds at source-fixed and perfectly grounded terminals. The candidate records those bound obligations as unassessed, even if their declarations happen to agree with the source. For the 10 V source and 9 V upper bound, a native source equation does not enforce the declared bound. This slice does not silently label it passed, reinterpret the bound as phase-to-neutral, or remove its array index. Other bound families, nonfinite/malformed arrays, malformed generator prices/controllers and unsupported transformer checks retain explicit coverage limits. Known current contributions remain usable when only limit/report evidence is missing; missing currents cannot be filled in from power by dividing by voltage.

Reproduce the example through experiments/lessons/verify_linear_coverage.jl using the exact clean local candidate environment in experiments/integration/README.md. The output contains native inputs/results, all mutation verdicts, candidate revision/tree and fixture/example/environment hashes. The published package pin and execution MCP surface have not changed. This is an implementation comparison and a small teaching witness, not independent scientific review or a complete running-network feasibility certificate.

Transfer exercise. With total real current 2 A and two phase currents each 1 A, replace the return limit by 1.5 A. Both phase caps of 1 A pass; the return fails by 0.5 A. Reverse the second phase current: the return then passes, but local KCL must be reevaluated against the surrounding circuit. A generator-only rating check does not establish a network solution.

Transformer series currents are not all terminal currents

The next local candidate slice covers one explicit terminal pair on each side of the native single_phase transformer, with fixed tap, finite leakage and secondary excitation. It complements the transformer-port lesson by checking supplied native current observations as well as equations. It does not by itself cover the running network's n_winding device; the later winding-list and rating-metadata slices supply that distinct coverage.

Let $U_f,U_t$ be oriented coil voltages and $I_f,I_t$ the supplied series currents, both positive into their winding. For this native tap convention,

\[N_0=V_{\mathrm{nom},f}/V_{\mathrm{nom},t},\qquad N=N_0\tau,\qquad Z'=Z_t+Z_f/N_0^2,\]

\[U_f-NU_t=N^2Z'I_f,\qquad NI_f+I_t=0.\]

The fixed multiplier $\tau$ changes the effective ratio and the from-referred leakage $N^2Z'$. Replacing that leakage with $Z_f+N^2Z_t$ agrees at $\tau=1$ but generally differs away from it. This is a declared parameterization of the implemented equivalent circuit, not a universal law for every transformer tap or leakage model. Zero leakage is handled directly as an ideal voltage relation; no small impedance is inserted.

The excitation admittance $Y$ lies across the secondary coil. Consequently,

\[I_{t,\mathrm{terminal}}=I_t+YU_t,\qquad S_{\mathrm{terminal}}=U_f\overline{I_f} +U_t\overline{I_t+YU_t} =Z'|I_t|^2+\overline Y|U_t|^2.\]

The last equality follows from the voltage and current relations and is checked separately. The signed terminal returns enter KCL. The excitation current is reconstructed from $Y,U_t$; this is not an independent excitation measurement. Native fr/1, to/1 currents and their cm values refer to the series currents, while i_max_to bounds the secondary terminal current. The positive nameplate s_rating bounds from-coil apparent power.

Predict. Does a reported secondary cm above i_max_to necessarily violate the implemented terminal-current limit? It does not: the two fields can describe different observations when excitation is present.

The fixture uses SI values $N_0=2$, $\tau=5/4$, $Z_f=1/2+\mathrm j/4$ Ω, $Z_t=1/8+\mathrm j/16$ Ω and $Y=1/20-\mathrm j/40$ S. It gives $Z'=1/4+\mathrm j/8$ Ω and the exact operating point

The secondary series magnitude is approximately 1.031 A; the terminal magnitude is approximately 0.814 A. A 0.9 A terminal cap passes. Tightening only that cap to 0.8 A fails while KCL and aggregate power remain satisfied. Replacing the series-current report with the terminal current instead fails current coupling and KCL. The distinction concerns the specific equivalent-circuit observation; it does not, by itself, identify all physical winding hot spots or thermal limits.

The package-owned example also retains a missing series current and an adjustable-tap declaration as indeterminate cases. Internal grounding, larger winding maps, other transformer subtypes, alternate excitation encodings and internal per-unit coordinate fields remain outside this slice. A two-terminal map containing a neutral uses that neutral as the return regardless of its array position; a map with two phase terminals uses their declared orientation. Tests distinguish consistent relabelling from reversing only one physical coil.

Run the same source-bound verify_linear_coverage.jl companion using the local candidate environment in experiments/integration/README.md. It includes examples/fixed_transformer_audit.jl, the analytic fixture (checked numerically with explicit tolerances), a native solve and the mutation evidence. It retains solver and primal status and gates acceptance on the independent audit. The successful native point is not a proof of uniqueness, global optimality, or general transformer coverage. The published package pin and execution MCP route remain unchanged.

Transfer exercise. Set both leakage impedances and excitation to zero, retain $N=2.5$ and the same 4 V load point. The primary voltage becomes 10 V, primary current $0.32-0.06\mathrm j$ A, and loss zero. Omitting those zero-valued optional fields gives the same native default model. Now restore excitation without updating the reported currents: the old ideal solution must be checked again against the restored current and power relations.

Coupled winding leakage and the observation being rated

The preceding two-port example does not determine the rating semantics of a winding-list model. The next local candidate checks two or three WYE windings with one phase-neutral coil each, fixed nominal ratios, explicit pairwise short-circuit reactances, and excitation on winding 2. It uses the native n_winding equations, independently evaluated in SI units. This smaller witness does not by itself assess the running network's multiphase arrangement; the later slices address incidence and rating metadata separately. Other unsupported declarations remain explicit obligations.

Let $U_k$ denote phase-minus-neutral coil voltage and $I_k$ bare coil current, positive into the transformer at every winding. Define

\[N_k=V_{\mathrm{nom},k}/V_{\mathrm{nom},1},\qquad j_k=N_k I_k,\qquad R_k=r_k/N_k^2.\]

Here $r_k$ is in ohms on winding $k$'s own voltage base; the supplied pairwise $X_{ab}$ values are already referred to winding 1. With $Z_{ab}=R_a+R_b+\mathrm jX_{ab}$, form the coupled matrix

\[(Z_B)_{ii}=Z_{1,i+1},\qquad (Z_B)_{ij}=\tfrac12(Z_{1,i+1}+Z_{1,j+1}-Z_{i+1,j+1})\quad(i\ne j).\]

The current and voltage equations are

\[\sum_{k=1}^{n}j_k=0,\qquad U_1-U_k/N_k=-(Z_Bj_{2:n})_{k-1}\quad(k=2,\ldots,n).\]

The minus sign follows from using inward currents on all ports. For two windings, $j_2=-I_1$ turns it into the familiar positive source-current voltage drop. For three windings, the nonreference currents are coupled through the off-diagonal term. Replacing this matrix by independent two-port branches generally changes the equations. Reordering windings requires reordering their pairwise data, ratios and result labels; excitation attached to winding 2 also makes its position part of the declared model.

The package fixture chooses $N=(1,1/2,1/4)$, $r=(1/4,1/16,1/64)$ Ω and $(X_{12},X_{13},X_{23})=(1/2,3/4,3/4)$ Ω, giving

\[Z_B=\begin{bmatrix} 1/2+\mathrm j/2 & 1/4+\mathrm j/4\\ 1/4+\mathrm j/4 & 1/2+3\mathrm j/4 \end{bmatrix}\ \Omega.\]

With $Y=1/32-\mathrm j/64$ S on winding 2, the analytic point is:

Winding 2's terminal current is $I_2+YU_2=-0.85089111328125+0.17132568359375\mathrm j$ A. Its magnitude is approximately 0.868 A, whereas $|I_2|\approx 1.031$ A. Predict: would a 0.95 A limit pass? In this native winding-list model, per-winding scalar i_max bounds the bare coil current, so it fails. A terminal-current reading below 0.95 A cannot certify that coil rating. Similarly, per-winding scalar s_max bounds $|U_k\overline{I_k}|$ for this single-coil slice. These are different observation contracts from the preceding single_phase secondary terminal-current cap. Neither convention alone identifies every physical thermal hot spot.

Terminal power supplies an additional consistency check:

\[\sum_k U_k\overline{I_{k,\mathrm{terminal}}} =j_{2:n}^{\mathrm H}Z_Bj_{2:n}+\overline Y|U_2|^2, \qquad \Re(j_{2:n}^{\mathrm H}Z_Bj_{2:n})=\sum_k r_k|I_k|^2.\]

The total loss at the fixture point is approximately $0.901948+0.542283\mathrm j$ VA. The example checks both these identities, individual voltage equations, ampere-turn balance, reported magnitudes/powers, ratings and local KCL. A plausible aggregate loss cannot repair a wrong winding current, voltage referral, or local balance.

Run experiments/lessons/verify_linear_coverage.jl with the exact clean local candidate described in experiments/integration/README.md. It invokes the package-owned examples/winding_list_audit.jl, binds its fixture and source hashes, and retains the native solve alongside analytic and mutation evidence. Wrong referral and replacing a bare-coil report by terminal current fail. Missing coil current or pairwise reactance stays indeterminate, with network KCL/power unassessed. Explicit zero reactances remain valid at the ideal boundary; omitted entries do not become measured zeros. The successful solve establishes neither uniqueness nor global optimality. Published integration and execution MCP availability remain unchanged.

Transfer exercise. For lossless pairwise reactances $X_{12}=X_{13}=3$ Ω and $X_{23}=10$ Ω, the reactance matrix is

\[\begin{bmatrix}3&-2\\-2&3\end{bmatrix},\]

with eigenvalues 1 and 5 Ω. Its equivalent reference star arm is negative, yet this matrix is positive definite. A negative star arm is not a negative diagonal reactance. Conversely, positive diagonal entries alone do not establish positive semidefiniteness. The candidate's point audit does not certify magnetic realisability: passing circuit residuals and checking the admissibility of constitutive data are separate obligations. The package's input analysis provides the separate matrix diagnostic. Multiphase/DELTA windings, taps, internal grounding and internal per-unit fields remain outside this audit slice.

An adequate winding total can hide an overloaded phase

The next local candidate extends the winding-list audit to matched one-, two-, and three-phase WYE windings. This is an extension of a specific executable model: each winding has an explicit neutral, and the coil counts agree. The native model pairs coils by position in their phase lists, after removing the neutral. A terminal label identifies a local terminal; it does not automatically define a cross-winding phase correspondence.

For coil position $p$, the same current and voltage equations hold separately:

\[\sum_k N_k I_{kp}=0,\qquad U_{1p}-U_{kp}/N_k=-(Z_Bj_{2:n,p})_{k-1}.\]

The model does not add mutual leakage between different phase positions. Excitation acts across each winding-2 coil. Its phase terminal currents and neutral return, oriented out of the bus into the transformer, are

\[T_{kp}=I_{kp}+\mathbf{1}_{k=2}YU_{2p},\qquad T_{k,n}=-\sum_{p=1}^{m}T_{kp}.\]

Losses sum the preceding per-coil identities over $p$. The neutral is the negative complex sum, not the sum of magnitudes. Opposite current errors in two phases can leave this neutral sum unchanged while violating their individual ampere-turn equations and local phase KCL. Likewise, a winding neutral current is not an earth-reaction measurement. The computational fixture has nonzero neutral currents and separately supplied zero earth reactions.

Predict. A winding has a 12 VA rating, with three coil powers whose magnitudes sum to less than 12 VA. Is it necessarily compliant with the native winding-list limit? No: this implementation allocates equal shares,

\[|U_{kp}\overline{I_{kp}}|\le S_{k,\max}/m\quad\text{for every }p.\]

That constraint bounds each bare coil before excitation is added to its terminal current. It differs from either aggregate constraint $|\sum_p U_{kp}\overline{I_{kp}}|\le S_{k,\max}$ or $\sum_p|U_{kp}\overline{I_{kp}}|\le S_{k,\max}$. Satisfying every allocated share implies both aggregate bounds; the converse fails in general. This allocation rule describes the implemented limit, not a universal thermal-rating rule for all physical transformers.

The package-owned multiphase-winding.json fixture retains the preceding three-winding ratios, leakage matrix and excitation. Its source voltages are $(10,-5-8\mathrm j,-5+8\mathrm j)$ V, and winding-2 bare currents are $(-1+0.25\mathrm j,0.5+0.25\mathrm j,0.25-0.5\mathrm j)$ A. Winding-3 currents are half these values; winding-1 currents are $-5/8$ times them. Each set satisfies ampere-turn balance for $N=(1,1/2,1/4)$.

The sum of magnitudes is approximately 10.139652 VA. Tightening only the winding rating from 30 VA to 12 VA leaves equations, KCL and aggregate power balance satisfied, but fails coil 1's limit. The native result's per-coil s_max must also report the divided cap: 10 VA at the baseline, 4 VA in the tighter case. Reporting 30 VA for every coil is a separate report-consistency error.

At the analytic point, the winding-2 neutral current is approximately $0.251648+0.000549\mathrm j$ A and total transformer loss is approximately $2.338756+1.314397\mathrm j$ VA. The native solve reproduces the voltages and losses and passes the independent audit. That is evidence for this supplied point, not uniqueness or global optimality. The neutral-current observation is marked reconstructed; the audit does not invent a neutral-current limit or an independent neutral measurement.

Use the same verify_linear_coverage.jl companion and exact local candidate environment in experiments/integration/README.md. It invokes examples/multiphase_winding_audit.jl and retains the analytic/native points, unequal-phase overload, wrong cap report, phase-order corruption, missing coil current and missing earth reaction. Tests also cover a two-phase case with exact neutral cancellation and consistent phase permutations. Missing a required coil current leaves network KCL/power unassessed. The next section extends the incidence to DELTA; the separate rating-metadata trace below addresses s_rating. Unmatched coil counts, taps and internal coordinates remain outside the supported domain. This intermediate slice alone left the running network indeterminate; the final price-scope trace below records its current verdicts.

Transfer exercise. For this point, the smallest equally allocated total rating is $3\max_p|S_{2p}|\approx 14.921637$ VA, rather than the 10.139652 VA sum. At that rating coil 1 is exactly on its boundary. For two coils, repeat the allocation using division by two, even if their currents cancel in the neutral. Zero neutral current does not imply zero coil power or spare phase capacity.

DELTA line currents cannot reveal internal circulation

The coil-incidence lesson distinguishes a coil current from its terminal observation. The next local package candidate applies that distinction to two/three-winding transformers. A supported DELTA winding has exactly three distinct phase terminals and no neutral in its map; other devices at the same bus may still use a neutral. Its ordered phases and rotation define the directed coil pairs. For phase order $(a,b,c)$, rotation $+1$ gives $(a,b),(b,c),(c,a)$; rotation $-1$ gives $(a,c),(b,a),(c,b)$. These are conventions relative to the declared order, not standalone vector-group labels.

Let $D$ have one row per coil, with +1 at its first terminal and -1 at its second. The voltage/current pairing is

\[U=Dv,\qquad I_{\mathrm{port}}=I_{\mathrm{coil}}+I_{\mathrm{exc}},\qquad I_{\mathrm{line}}=D^{\mathsf T}I_{\mathrm{port}},\qquad v^{\mathsf T}\overline{I_{\mathrm{line}}} =U^{\mathsf T}\overline{I_{\mathrm{port}}}.\]

Excitation is present only on winding 2 in this native model. The independent audit enumerates the pairs directly, then evaluates the same per-coil leakage and ampere-turn equations as before. It reconstructs bus line currents with the power-dual incidence and marks them reconstructed, without a residual or an independent-measurement claim. WYE neutral-return rules do not apply to DELTA.

Predict: if the bare coil current is $0.5-0.25\mathrm j$ A but the line current at a is $0.875-0.375\mathrm j$ A, does a native 0.6 A winding i_max pass? In this implementation it does: that limit bounds each bare coil, whose magnitude is approximately 0.559 A, rather than the approximately 0.952 A line current. The package-owned delta-winding.json fixture supplies this finite-leakage DELTA/WYE point, with rotation -1, winding-2 excitation, explicit internal currents and SI voltage/current reports. A native solve agrees with the analytic point and passes the independent audit. The scalar s_max/3 allocation uses actual line-to-line coil voltage for DELTA.

Now consider the separate ideal zero-leakage, zero-excitation DELTA/DELTA boundary. Both windings have equal nominal ratio, identical terminal voltages, and the same rotation. A constant current on every coil lies in the kernel of $D^{\mathsf T}$. Compare these two supplied internal states:

The two states have identical terminal observations but different coil-loading verdicts. Even complete line-current measurements cannot recover this internal circulation without additional information. The witness uses an ideal boundary: adding positive winding resistance makes this particular circulating point violate the voltage equation. It does not establish arbitrary circulation with strictly passive leakage, magnetic realisability, or which state a solver selects.

Inspect: examples/delta_winding_audit.jl, invoked by the source-bound verify_linear_coverage.jl companion, retains the finite analytic/native point, rotation corruption, missing coil current, refused neutral-bearing map, both ideal states and the six rating violations. Tests also construct mixed and all-DELTA two/three-winding points from independent star-arm laws, exercise excitation on DELTA winding 2, and distinguish consistent reindexing from a changed connection. Missing an internal current suppresses network KCL/power certification even when line currents could appear to close.

Neutral-bearing DELTA remains explicitly unassessed. Inspection of the complete native build path found that validation rejects a DELTA map containing a neutral before constraints are stamped; a rejection test records that boundary. This audit accepts only the three-terminal closed incidence above. Two-terminal DELTA, unmatched coils, taps and internal coordinates also remain unassessed. The rating-metadata trace below handles s_rating separately. Before the final price-scope trace, the running network retained 28 unassessed obligations. The published package pin and execution MCP surface are unchanged.

Transfer exercise. Cyclically reorder every winding's coil positions and its current reports together: the verdict should remain unchanged. Then change only delta_roll while keeping phase order and reports fixed: which coil voltage and line-current equations change? Finally, explain why neither summing terminal power nor setting an unavailable coil current to zero can certify its rating.

A retained rating field may declare no loading limit

The per-coil allocation example required an explicit loading constraint. A rating field can instead serve as an impedance conversion base, a retained nameplate, or a native loading cap. Its name and presence alone do not establish which role reaches the solved model.

For the native n_winding path reviewed here, top-level and per-winding s_rating are power-base/nameplate metadata. Each winding's optional s_max separately declares the total apparent-power cap, allocated equally among its coils. This differs from the earlier single_phase recipe, whose s_rating actually creates a from-coil loading constraint. A package-wide "all transformer nameplates are enforced" inference is therefore unsafe.

The conversion trace also matters. PowerIO 0.11.6 converts source kVA into winding rating metadata. Its BMOPF schema-0.1.0 emission retains the winding-list record under extras; BMOPFTools restores it during normalization and preserves the ratings without inventing s_max. Conversion uses rating bases to turn percentages into ohms, with a package-owned correction for the imported winding resistance convention. Once the supplied SI coefficients are fixed, editing rating metadata alone does not rederive their ohmic values. The direct SI audit evaluates those coefficients; it does not authenticate the earlier conversion or infer an omitted equipment limit.

Predict: can a native solution pass while coil apparent power exceeds s_rating? The package-owned rating-metadata.json witness uses the preceding finite DELTA/WYE point with deliberately small rating metadata and separate 60 VA winding caps:

The baseline passes the native constraints and independent audit in SI, system-base per-unit, and unequal local-base coordinates. That result says nothing about the physical acceptability of exceeding the declared nameplates. Changing only the rating metadata leaves the independently evaluated SI electrical rows unchanged. In the local-base run, the native working top-level rating is divided by winding 1's 100 VA power base, while the two winding caps use their own 100 VA and 25 VA bases. This bookkeeping does not turn a retained field into a constraint. Internal working dictionaries are outside the SI audit's input domain.

Inspect: examples/winding_rating_metadata.jl, invoked by the exact-candidate verify_linear_coverage.jl companion, preserves these points and the tightened OPF/PF attempts. The latter return local infeasibility and fail the independent point audit; the status alone is not a global infeasibility certificate. The current native PF recipe retains nested winding i_max/s_max, so it also retains this winding-list loading limit. A general "PF removes every operational limit" inference would be wrong for this path.

A valid metadata observation has status declared, no residual, and enforced_as_limit=false. It is not a passed equipment limit or an independent nameplate measurement. Malformed metadata remains unassessed, while known currents and checked violations remain visible. Missing an internal coil current still suppresses network KCL/power; metadata cannot recover it.

The running-network input is retained intact. Its transformer equations, incidence and declared current caps now pass; top-level s_rating is recorded as metadata. Before the price-scope extension below, the audit retained 28 unassessed obligations: generator cost was outside scope and prevented a complete current inventory. Physical adequacy, objectives, external review, published integration and execution MCP exposure remain separate obligations.

Transfer exercise. Identify which fields in an imported transformer were used to convert percentage data, which are retained declarations, and which actually register loading constraints. Then explain why adding s_max is a modeling decision requiring evidence about the intended rating and observation, rather than a repair justified merely by the existence of s_rating.

Voltage magnitudes, sequence quantities and angles

Phase-to-neutral bounds require the neutral phasor

The preceding examples distinguish phase-to-earth voltage from device-current observations. The next independent audit adds finite vpn_min/vpn_max arrays for the actual phase-minus-neutral voltage. These arrays have one entry per phase in the full bus declaration order, excluding its default-role n/N neutral. They do not bound phase-to-earth voltage, and a missing neutral role is not permission to substitute earth.

Consider a 10 V source with a 1 Ω phase conductor and a 1 Ω neutral conductor. At the supplied 8 W load point, phase current is 1 A toward the load and neutral current is -1 A in the same source-to-load orientation. Direct conductor drops therefore give

\[V_{q,x}=10-1\times1=9\ \mathrm V,\qquad V_{q,n}=0-1\times(-1)=1\ \mathrm V,\qquad U_{q,xn}=V_{q,x}-V_{q,n}=8\ \mathrm V.\]

Predict: would a 9 V phase-to-earth result establish compliance with a phase-neutral lower bound of 8.25 V? It would not. The package-owned phase-neutral-voltage.json fixture separates the observations:

Source power is 10 W, the load consumes 8 W and the two conductor losses sum to 2 W. Closing that power balance does not erase either voltage-limit violation. The native SI and system-base per-unit solves reproduce the supplied voltages and pass the independent audit. A limited native attempt is retained with its local termination and failed point audit; local infeasibility alone is not a global infeasibility certificate, and a successful solve does not prove uniqueness.

For supplied complex voltages, the checked quantity is

\[m_p=|V_p-V_n|,\qquad e_{p,\min}=\max(0,\underline v_{pn,p}-m_p),\qquad e_{p,\max}=\max(0,m_p-\overline v_{pn,p}).\]

Violations and tolerances are in V, not V². At the 8.25 V lower bound, the squared-constraint violation is 4.0625 V² while the magnitude violation is 0.25 V; those numbers cannot share an unqualified tolerance. Subtract the phasors before taking a magnitude: $|V_p-V_n|$ generally differs from $|V_p|-|V_n|$. A common shift of both phasors preserves this voltage observation; separate source, grounding or earth-connected equations may forbid that shift in the surrounding network.

A grounded neutral has its own obligation. A separately consistent fixture changes the load to 9 W at 9 V, sets neutral-conductor current to zero and supplies the corresponding earth reactions. It passes the native solve and independent audit. The native builder may substitute zero for an ideal grounded neutral in its bound expression. The audit instead uses the supplied neutral phasor in the original voltage difference and checks grounding separately. An inconsistent ground report is therefore a failed grounding obligation; a missing neutral report stays unknown even when the neutral was declared grounded. It is never filled in as zero.

Inspect: examples/phase_neutral_voltage_audit.jl, invoked by the exact candidate verify_linear_coverage.jl companion, retains the analytic/native points, both bound violations, grounded case, missing neutral, malformed array and a source-fixed bound. Each checked PN row identifies its neutral reference, full phase index, observed magnitude and bound. Tests also cover an unbalanced three-phase circuit, consistent declaration reordering, upper-case neutral labels, exact zero/equality boundaries and complex common shifts.

The native builder skips a grounded or source-fixed phase without shifting later array indices. The audit leaves that phase's declared bound unassessed; it does not report compliance merely because a source supplied its voltage. A grounded neutral does not itself skip an applicable phase bound. When the first phase is skipped, the next phase still uses bound entry 2. Tests inspect native constraint identities as the object under test, while the independent evaluator constructs neither observations nor residuals from those constraints.

Full finite nonnegative arrays and a supported neutral role are required. Malformed or nonfinite bounds, conversion overflow/underflow and unknown applicability remain unassessed, while otherwise known current contributions and checked violations stay visible. Missing required neutral voltage suppresses network KCL/power certification. Neutral-to-earth limits are covered below; phase-to-phase, bus-angle, sequence and branch-end angle limits also follow. The existing running network has no PN-bound declarations, so this family adds no checks there. The cost-scope section gives its latest coverage and verdict. The published integration pin and execution MCP surface are unchanged.

Transfer exercise. For $V_p=4+6\mathrm j$ V and $V_n=1+2\mathrm j$ V, compute the phase-neutral magnitude and compare it with subtracting their magnitudes. Then place a skipped grounded phase before this phase in terminal_names: explain why its bound uses entry 2, and why deleting the neutral result cannot be repaired by a grounding label or solver status.

Neutral-to-earth voltage does not determine neutral current

A neutral-to-earth magnitude cap observes a different quantity from the phase-to-neutral bounds above. The next candidate audit accepts an applicable finite nonnegative scalar vn_max and evaluates

\[m_n=|V_n|,\qquad e_n=\max(0,m_n-\overline v_n).\]

The reference is the model's earth coordinate. This voltage alone does not establish a neutral-conductor current, earth-return path, electrode state, touch voltage or protection adequacy.

Predict: does a neutral phasor whose real and imaginary components are both within ±1 V comply with a 1 V magnitude cap? Consider a 10 V source, 1 Ω phase and 2 Ω neutral conductors, with phase current $0.5-0.25\mathrm j$ A toward the load and the opposite current in the neutral. Direct conductor drops give

\[V_{q,x}=9.5+0.25\mathrm j\ \mathrm V,\qquad V_{q,n}=1-0.5\mathrm j\ \mathrm V,\qquad U_{q,xn}=8.5+0.75\mathrm j\ \mathrm V.\]

The load consumes $4.0625+2.5\mathrm j$ VA, the source supplies $5+2.5\mathrm j$ VA and the two conductors dissipate 0.9375 W. The package-owned neutral-earth-voltage.json witness separates the checks:

Violations and tolerances are in V. The positive native cap is represented by a normalized quadratic inequality; at a 1 V cap its excess is 0.25, dimensionless. That solver expression and the 0.118034 V physical magnitude excess require different tolerances. The independent evaluator uses the supplied rectangular state and its own phasor norm, rather than a native residual or a reported magnitude. Native SI and per-unit solves reproduce the scoped point. A tightened native attempt is retained with local infeasibility and a failed point audit; the termination alone is not a global infeasibility certificate.

Now replace the neutral conductor by 4 Ω and add a 0.25 S neutral-earth shunt. At the same neutral voltage, load power and source power, the neutral-conductor current magnitude falls from $\sqrt{0.3125}\approx0.559017$ A to $\sqrt{0.078125}\approx0.279508$ A. The shunt carries the remaining return current. Its current is reconstructed from the declared admittance and supplied voltage, not independently measured. Both analytic and native finite-grounding points pass their scoped audits. A neutral-voltage observation therefore cannot stand in for a conductor-current rating or recover an omitted return path.

Inspect: examples/neutral_earth_voltage_audit.jl, invoked by the exact clean candidate verify_linear_coverage.jl companion, retains these points, both cap violations, malformed data, missing state and explicit/source-ground cases. The native builder omits this bound at an explicitly perfectly grounded neutral or an implicitly grounded source-bus neutral. The latter rule also applies when the source terminal map omits that bus neutral. Such declared caps remain unassessed, with no invented zero residual; their separate grounding voltage checks still apply. Consistent native grounded cases solve successfully while retaining incomplete audit coverage. A finite grounding shunt does not trigger that skip. Even a zero voltage cap does not allocate the free earth-reaction current of a perfect-ground declaration: a voltage constraint is not a new current path.

Missing/ambiguous/custom neutral roles, malformed scalars and missing neutral state remain unknown. Missing required state suppresses KCL/power certification; bad limit data preserves otherwise known currents and checked violations. Tests cover complex norms, zero/equality and V-unit tolerance, Float64 range, uppercase neutrals and native constraint incidence. Native builders are the test subjects, not the independent audit's oracle.

Four running-network neutral caps now pass, reducing unassessed obligations from 33 to 29 at this stage without deleting input fields. Phase-to-phase limits follow below. At that stage generator cost was still outside scope, preventing complete KCL/power checks; the final cost-scope section revisits them. The published integration pin, PSK implementation statuses, physical adequacy and execution MCP exposure remain separate.

Transfer exercise. Give the neutral-voltage and neutral-conductor-current magnitudes for both return circuits above. Explain which additional model and state data distinguish them, why a finite shunt does not skip vn_max, and why a successful source-grounded solve does not turn its skipped cap into a passed limit.

Phase-to-phase bounds preserve the full pair index

A phase-to-phase magnitude is the norm of a complex voltage difference. It requires both phase phasors; neither a neutral report nor a balanced-system conversion factor can replace them. The candidate audit now evaluates applicable finite nonnegative vpp_min/vpp_max arrays:

\[m_{ij}=|V_i-V_j|,\qquad e_{ij,\min}=\max(0,\underline v_{pp,ij}-m_{ij}),\qquad e_{ij,\max}=\max(0,m_{ij}-\overline v_{pp,ij}).\]

Predict: take source phases 10 V and $10\mathrm j$ V connected through two 1 Ω conductors to a two-terminal DELTA load. A supplied load-coil current of 1 A from x to y gives load phases $V_x=9$ V and $V_y=1+10\mathrm j$ V. The source supplies $10-10\mathrm j$ VA; the load consumes $8-10\mathrm j$ VA and the conductors dissipate 2 W. Would subtracting the phase magnitudes recover the voltage seen by this load?

Residuals and tolerances are V. At the 12.5 V upper bound, the native squared-constraint excess is 7.75 V², while the independent magnitude excess is about 0.306248 V. A tolerance must name its quantity and units. This synthetic unbalanced two-phase circuit also supplies no $\sqrt{3}$ conversion rule. Native SI and per-unit points pass their scoped audits. The tightened native attempt has local infeasibility and a failed point audit; the termination alone is not a global infeasibility certificate.

Array position is part of the declaration. For phases [a,b,c], pair entries 1, 2 and 3 mean (a,b), (a,c) and (b,c). The list excludes a supported n/N neutral when present, and otherwise retains all phases in declaration order. A pair touching a perfectly grounded or source-fixed phase is skipped by the native builder but still consumes its index. The independent audit leaves its bound unassessed, with no zero residual invented. If phase a is skipped, (b,c) still uses entry 3. Reordering phases therefore requires the corresponding pair-array permutation, not just a permutation of phase entries. A grounded neutral does not skip phase pairs; neither does a finite grounding shunt.

Inspect: examples/phase_phase_voltage_audit.jl and its phase-phase-voltage.json fixture, invoked by the exact clean candidate verify_linear_coverage.jl companion, retain the analytic/native points, bound violations, skipped first phases, malformed arrays and missing endpoint state. Consistent native grounded/source-fixed cases solve while retaining incomplete audit coverage. Tests inspect native constraint identities and cross terms as subjects, without using builder expressions as the evaluator's oracle. They also cover three-phase pair permutations, consistent relabeling, common phasor shifts, exact zero/equality and V-unit tolerance, forged magnitudes and Float64 range. A common shift preserves this pair observation; separate absolute source, grounding or earth-connected equations may forbid it.

Fewer than two phases, custom or ambiguous roles, malformed arrays and missing endpoint state remain unknown. A neutral is not required for this observation; a missing declared neutral state nevertheless prevents complete network KCL/power certification. Bad limit data preserves otherwise known currents and checked violations. Each checked row names its terminal pair, phase indices, full pair index, observed magnitude and bound.

The running network now has six checked phase-pair bounds at bus i6. Unknown rows fall from 29 to 28: the two unsupported fields previously shared one field-coverage row. Generator cost was still outside scope at that stage, so KCL/power remained incomplete until the price-scope trace. Bus-angle, sequence and branch-end angle limits follow below; objectives, physical adequacy, PSK implementation statuses, published integration and execution MCP remain separate obligations.

Transfer exercise. Reorder [a,b,c] to [c,a,b] and map each old pair entry to its new position. Then ground c: identify the surviving pair and its full array index. Explain why a solver status, a neutral voltage or one phase magnitude cannot fill in a missing endpoint phasor.

Centered bus angles need two nonzero phasors

An intra-bus angle bound observes a direction, not a voltage magnitude. For phase indices $i<j$ in the full declaration order, the candidate audit uses

\[\Delta_{ij}=\phi_j-\phi_i,\qquad d_{ij}=\arg\left(\frac{V_j}{|V_j|}\overline{\left(\frac{V_i}{|V_i|}\right)} e^{-\mathrm j\Delta_{ij}}\right)\in(-\pi,\pi].\]

The optional va_nom array supplies $\phi$ for all phases, excluding a supported n/N neutral when present. Without it the offset is zero; a nominal balanced arrangement is not inferred. Both va_diff_min and va_diff_max are scalar bounds in rad, shared by applicable pairs. The native formulation requires an ordered window strictly inside $(-\pi/2,\pi/2)$. The independent checks report $\max(0,\underline d-d_{ij})$ and $\max(0,d_{ij}-\overline d)$ with a separate angle_tol in rad.

Predict: reuse the load phases 9 V and $1+10\mathrm j$ V above. Their raw second-minus-first angle is $\arctan(10)\approx1.47112767$ rad. With nominal angles $[0,\pi/2]$, the centered deviation is instead $-\arctan(0.1)\approx-0.09966865$ rad. Which quantity does a declared centered window constrain?

A 2.8 rad common rotation of all voltages and currents preserves this circuit, its powers and the centered deviation. Individual phase angles cross the branch cut: subtracting their principal reports without circular reduction can introduce a full turn. The evaluator normalizes each endpoint separately and takes the principal argument of the centered product. This also avoids overflowing or underflowing the raw product of two available nonzero voltage phasors. It uses phase-to-earth directions, not phase-minus-neutral directions. A common voltage translation generally changes the angle observation, even though it preserves a phase-pair voltage difference.

Zero is a distinct boundary. A separate no-load point has one receiving phase at 10 V and the other at zero. Its line/source currents are zero and KCL/power pass. Native bilinear angle rows also vanish, satisfying their closure; the angle of the zero endpoint is nevertheless undefined. The audit records the pair unassessed, with no invented zero deviation. Small nonzero supplied phasors have defined numerical directions; this check does not establish their robustness to measurement or numerical perturbations. Native SI bilinear residuals have V² units; per-unit rows use working coordinates. Neither is an angular excess in rad, so they cannot share an unqualified angular tolerance.

Inspect: examples/bus_angle_voltage_audit.jl and bus-angle-voltage.json, invoked by the exact clean candidate companion, retain these points, both violations, absent nominal data, common rotation, malformed windows and missing state. Native SI, per-unit and rotated points pass their scoped audits. A tightened native attempt returns local infeasibility and fails the point audit; termination alone is not a global infeasibility certificate. Native primal-point and constraint-identity inspections are test subjects, not observations reused by the independent evaluator.

Grounded/source-fixed phase pairs remain unassessed. Skipping a first phase preserves full nominal indices: the remaining pair uses entries 2 and 3. Finite grounding does not skip a phase; a grounded neutral is excluded from phase enumeration. Consistent native skipped-phase cases solve while retaining incomplete audit acceptance. Reversing pair direction negates the centered angle and reverses an asymmetric window. Noncyclic phase swaps require matching nominal-array permutations; cyclic symmetry can preserve nominal offsets modulo full turns. Equal window endpoints are supported: native equality plus a half-plane guard excludes the nonzero antipodal ray, which the independent principal-angle check rejects directly.

Missing required endpoints, one-sided/reversed/singular windows, malformed nominal arrays and unrepresentable data stay unknown. Known currents and checked violations remain visible; missing required state suppresses complete KCL/power certification. Standalone va_nom and custom roles remain outside this bus-angle slice; sequence and branch-end angle limits follow below. The running network declares no bus-angle window, so this family did not change the 28 then-unassessed obligations. The final cost-scope section gives its latest verdict. Published integration, PSK statuses, execution MCP, objective and physical adequacy remain separate.

Transfer exercise. Reverse the two phases in the asymmetric centered window above. Give the new nominal order, deviation and bound endpoints. Then set one endpoint voltage to zero and explain why successful bilinear rows or atan(0,0) cannot supply its missing direction.

Sequence-voltage magnitudes are not a VUF ratio

Sequence bounds observe a linear combination of three phase phasors, followed by a magnitude. In full declaration order, let $\alpha=e^{2\pi\mathrm j/3}$:

\[V_0=\frac{u_1+u_2+u_3}{3},\qquad V_1=\frac{u_1+\alpha u_2+\alpha^2u_3}{3},\qquad V_2=\frac{u_1+\alpha^2u_2+\alpha u_3}{3}.\]

Here $u$ is phase-minus-neutral voltage when the declared default-role n/N neutral floats, and phase-to-earth voltage when the neutral is absent or perfectly grounded, including implicit source-bus grounding. The native transform includes all three phases, even grounded or source-fixed ones. These limits therefore have different applicability from the individual-phase bounds above. Phase labels do not authenticate a physical phase arrangement; this transform follows the full declared order. A cyclic permutation preserves magnitudes, while exchanging two phases swaps positive and negative sequences in this convention.

Predict: does an absolute negative-sequence cap establish a voltage-unbalance ratio? The package-owned four-wire sequence-voltage.json circuit has 1 Ω per phase and neutral conductor. At its supplied load point, the neutral is $1-0.5\mathrm j$ V and the phase-neutral sequence phasors are $V_1=10$ V, $V_2=1$ V and $V_0=0.5+0.25\mathrm j$ V. The independent forward transform checks these observations from rectangular terminal states.

A consistent second circuit point reduces only the positive-sequence phasor to 5 V and updates the source references and specified powers. Negative sequence stays 1 V, satisfying the same absolute cap, but VUF rises to 0.2. Its positive bounds are adjusted to [4.5,5.5] V so this distinct ratio failure remains visible. An absolute cap and a ratio limit cannot substitute for each other when the positive-sequence magnitude varies.

The independent magnitude violations use V and voltage_tol. VUF is the fraction $|V_2|/|V_1|$, not a percent-valued field, and uses a separate dimensionless vuf_tol. The ratio is assessed only for $|V_1|>\texttt{vuf\_voltage\_floor}$; that configurable floor defaults to $10^{-6}$ V, independently of voltage_tol. Below or at the floor, the row reports the sequence magnitudes and floor but leaves observed ratio and residual null. Other magnitude limits still receive their own results. A known positive lower-bound violation remains failed even when the ratio is unknown.

This is a necessary boundary of the observation. The native constraint $|V_2|^2\leq u^2|V_1|^2$ can pass at the all-zero origin, where the ratio is undefined. The independent audit refuses to report ratio compliance there. It does not assess native polynomial feasibility outside its stated ratio domain. Native SI squared residuals are V² and per-unit rows use working coordinates; neither supplies a dimensionless ratio excess or its tolerance. The floor is a declared numerical domain, not a physical unbalance standard or an uncertainty bound.

Reference choice also matters. At the floating-neutral point, the mean of the earth-referenced phases is $1.5-0.25\mathrm j$ V, with magnitude about 1.520691 V. Using it would fold neutral displacement into the zero component instead of checking the specified phase-neutral observation. A separate consistent grounded-neutral circuit uses earth input and passes. A bad grounded neutral report fails grounding separately; its earth-referenced sequence observation is not silently changed to subtract that bad report. Missing a floating neutral prevents the transform. Known earth-reference phase data can still support that observation while missing other declared state prevents complete KCL/power certification. No missing neutral report is filled in as zero.

Inspect: examples/sequence_voltage_audit.jl, invoked by the exact clean candidate companion, retains all bound mutations, the low-positive point, malformed data, missing neutral and native SI/per-unit/grounded witnesses. The permitted low-positive native point has a separate relaxed ratio cap. A tightened native VUF attempt has local infeasibility and a failed point audit; termination alone is not a global infeasibility certificate. Native expression/incidence and zero-origin primal inspections are subjects, never the independent oracle.

Tests cover exactly three full phases, source/ground applicability, cyclic and reversing permutations, neutral reference/fallback, scalar zero/equality and unit-specific tolerances, denominator-floor refusal/configuration, malformed limits and missing state. Known currents and checked violations survive bad limit data. The running input declares no sequence or VUF bound; adding this family did not change the 28 then-unassessed obligations. The cost-scope section gives the latest verdict. Branch-end angle limits follow below. Objectives, physical phase-order measurement and adequacy, published integration, PSK statuses and execution MCP remain separate.

Transfer exercise. Keep negative sequence at 1 V and change positive sequence from 10 V to 5 V. Give the two VUF fractions and identify which obligations a 1.25 V negative cap establishes. Then set both sequence magnitudes to zero: explain why a successful cross-multiplied inequality cannot supply the ratio.

Branch-end angles follow the mapped conductor direction

A branch angle compares two ends of the same mapped conductor. It is not the centered second-minus-first intra-bus angle above. For conductor index $k$, the independent audit observes

\[d_k=\arg\left(\frac{V_{\mathrm{from},k}}{|V_{\mathrm{from},k}|} \overline{\left(\frac{V_{\mathrm{to},k}}{|V_{\mathrm{to},k}|}\right)}\right) \in(-\pi,\pi].\]

The direction is $\theta_{\mathrm{from},k}-\theta_{\mathrm{to},k}$, with no nominal-angle offset. Each line's scalar va_diff_min/va_diff_max window applies to every positional pair in its explicit from/to terminal maps. Those maps must be nonempty, unique and equally long. Neither sorted labels nor prefix truncation can recover an undeclared conductor correspondence.

Predict: the two-conductor DELTA circuit already used above has source phases 10 V and $10\mathrm j$ V, receiving phases 9 V and $1+10\mathrm j$ V, and 1 Ω per conductor. What is the second conductor's signed branch angle?

The positive sign differs from the earlier centered bus deviation for this circuit. Angular violations use rad and the separate angle_tol; the native sine/cosine-scaled SI rows have V² units, while per-unit rows use working coordinates. Their residuals cannot supply an angular tolerance. Both scalar endpoints must be finite, ordered and strictly inside $(-\pi/2,\pi/2)$. Equal endpoints are supported: the native equality plus half-plane guard excludes the nonzero antipodal ray, which the independent principal angle rejects directly.

Direction is part of a branch declaration. Reverse the branch and its endpoint maps, reverse the supplied endpoint currents/result labels, and replace $[\underline d,\overline d]$ by $[-\overline d,-\underline d]$. The same circuit then passes with the opposite signed deviation. Keeping the old asymmetric window can change the result. A consistent conductor permutation also permutes both maps and their data; it does not sort phases by name.

A 1.62 rad common rotation of voltages and currents crosses the individual-angle branch cut while preserving the circuit and branch observations. The evaluator normalizes each nonzero endpoint before multiplication and takes the principal argument. This avoids raw voltage-product under/overflow and unwrapped subtraction errors. It uses supplied earth-referenced voltages; neither a bus nominal reference nor a phase-neutral substitution defines this branch quantity.

Zero endpoints remain a distinct boundary. Native rows are registered for all conductors, including neutrals and source/ground endpoints. A separate explicit-neutral circuit has zero source-neutral voltage and a floating receiving neutral. Native solving succeeds, but that conductor's angle remains unassessed, with no invented residual. Its other conductor angle, line equations, currents, KCL and power still pass. At an all-zero branch point, native primal constraints also pass without defining either endpoint direction. A grounding label or successful homogeneous inequality cannot supply an angle.

Inspect: examples/branch_angle_voltage_audit.jl and branch-angle-voltage.json, invoked by the exact clean candidate companion, retain both violations, common rotation, complete reversal, missing state, one-sided data, neutral and zero endpoints. Native SI/per-unit/rotated/reversed points pass their scoped audits. The tightened native attempt has local infeasibility and a failed point audit; termination alone is not a global infeasibility certificate. Native semantic keys, domain guard and zero-origin primal inspection are test subjects, never the independent oracle.

Bad angular data does not erase known inline-series current contributions or checked current-rating violations. Missing required endpoint state suppresses complete KCL/power certification but need not erase a different conductor's angle observation. Supplied endpoint angles can also be checked while a separate line equation model is unsupported; those observations cannot certify its currents or equations. The older linear-only API retains its refusal of line-angle declarations. Tests cover positional mapping, asymmetric reversal, principal cuts, normalization, equality/zero/tolerance, neutral/ground incidence, malformed windows/maps and partial state.

The running network declares no branch-angle bounds, so adding this family did not change the 28 then-unassessed obligations. The final price-scope trace below gives its latest verdict. Physical adequacy, objectives, broader device/role coverage, published integration, PSK statuses and execution MCP remain separate obligations.

Transfer exercise. Reverse a branch whose window is [−0.02,0.12] rad. Give its new window and the sign of each conductor observation. Then delete only the second receiving voltage: identify which angular observation remains available and why complete line/KCL certification is still unavailable.

Injection conventions, prices and objective scope

A generator price is not an objective certificate

A generator price declaration has a different role from an electrical limit. In the bounded native generator form it is a full finite signed vector, one coefficient per declared coil. A two-terminal SINGLE_PHASE device has one coil price, including when its terminal order is reversed. A WYE neutral has no separate price. Scalars, polynomial-shaped objects, wrong or extra lengths, nulls, Booleans and unrepresentable values remain unassessed rather than being reinterpreted as another cost model.

The native default generator contribution is

\[C_g=\sum_k c_{g,k}\frac{P^{\mathrm{inj}}_{g,k}}{1000},\qquad P^{\mathrm{inj}}_{g,k}=\Re\left(U_{g,k}\overline{I_{g,k}}\right).\]

With power in W and price in currency/kWh, $C_g$ is a rate in currency/hour. It is linear in active power, although bilinear in IVR voltage/current variables. A positive coefficient penalizes its injection contribution; a negative one rewards it within the selected minimization objective. Other terms and network constraints can alter actual dispatch. Zero direct cost alone establishes neither a feasible flat direction nor dispatch non-uniqueness.

The native cost-expression constructor adds no variables or constraints. Generator apparent-power limits already create auxiliaries independently of price. Per-unit working prices are multiplied by the local power base to cancel power scaling; this is coordinate bookkeeping, not a changed tariff. A staged caller can omit the objective, replace it, combine snapshots or apply explicit duration weights. A network/result pair alone does not identify that choice.

The independent audit therefore retains valid prices with status declared, null residual and enforced_as_physical_limit=false. This is not a passed objective or an economic-validity finding. Signed/zero prices are recognized; malformed prices remain unknown while otherwise known currents, KCL and limit violations remain available. Missing required current or state still prevents complete electrical checks. Generator active/reactive/apparent-power rows name W, var and VA respectively, with unchanged numerical tolerances.

The output objective_scope explicitly leaves objective selection, reported objective consistency, objective evaluation, custom constraints and local/global optimality unchecked. Finite price metadata also does not certify numerical objective applicability under arbitrary scaling or coefficient providers. The candidate result-field description now says reported objective value; an objective field alone is not an optimality or feasibility certificate.

Predict: a 10 V ideal-switch circuit supplies a $10+2\mathrm j$ VA load with a generator fixed at $4+1\mathrm j$ VA. What can an electrical audit tell from the same physical point under these known recipe choices?

In this particular circuit, with an unpriced source, Q fixed at 1 var and P allowed in [3,5] W, the per-unit positive/negative-price solves select the respective P limits. These native observations do not establish global optimality for a general network. Only the known recipe supplies the values and units in the table; the standalone feasibility audit does not infer it.

Inspect: examples/generator_cost_scope.jl and generator-cost.json, invoked by the exact clean candidate companion, retain these price, recipe, malformed- data and missing-current witnesses. Tests inspect native expression values, indexing and unchanged variable/constraint counts as subjects; the declaration audit does not call the native cost builder as its oracle. The public native objective construction stage precedes KCL finalization.

The native SI attempts are preserved honestly. From the voltage-only start, Ipopt reports a restoration failure and the returned point fails the electrical audit. With a full supplied-primal start it still reports a restoration failure, but its returned point passes electrical checks and has the expected rate. Per-unit, no-objective, zero-price and custom-constant recipes solve and pass. Solver success, a valid supplied point and selected-objective assessment are three separate observations; neither failure nor success can substitute for the other checks.

The running example makes the distinction consequential. Its retained generator prices $[-0.08,-0.08,-0.08]$ are now recognized without deleting any field. Previously that declaration prevented a complete current inventory. It now opens all 26 KCL checks and aggregate power, which pass, and exposes three active-power upper-bound violations in the original default solver result.

The absolute audit tolerance remains 1e-6 W. The separate solve uses tol=1e-10, constr_viol_tol=1e-12, acceptable_iter=0 and bound_relax_factor=0; both results and solver options are retained. No audit tolerance is loosened and no input constraint or price is removed. Default solver behavior is a recorded diagnostic, not a requirement that every future backend reproduce the same violation.

A passed result here means the explicitly supported electrical supplied-point obligations hold at the stated tolerances. It does not establish unrestricted native/custom-model feasibility, a selected objective, stationarity, local/global optimality, physical adequacy, economic correctness or external review. The DELTA generator and voltage-source price scopes follow below; other device/controller scopes, IBR price applicability, published integration, PSK statuses and execution MCP remain separate.

Transfer exercise. Give the rate and explicit two-hour cost for 4 W injection at price +0.25 currency/kWh. Then explain why objective=123, solver success, a zero price or a tiny active-power excess cannot independently settle the selected objective, uniqueness, or electrical compliance.

DELTA generator limits live on coils

The DELTA circulation lesson applies to injections as well as consumption. The local candidate now audits bounded DELTA generators using their own native indexing: two or three distinct non-neutral terminals in declared cyclic order. A reported current at a terminal key need not be the current injected into that terminal.

Predict: for coil currents $I=(0.3-0.1\mathrm j,-0.2+0.15\mathrm j, 0.1+0.05\mathrm j)$ A, does a second entry i_max=0.3 bound the current in the second coil or the current injected at the second terminal?

For ordered terminals $(x,y,z)$, the pairs are $x\to y$, $y\to z$ and $z\to x$. In the native result, generator.g.x.crg/cig stores $I_1$, the injection coil current paired with $U_1=V_x-V_y$. Independently evaluate

\[S_k=U_k\overline{I_k},\qquad J_k=I_k-I_{k-1},\qquad \sum_k S_k=\sum_k V_k\overline{J_k},\]

with cyclic indices. Here $J_k$ is terminal injection; outgoing-current KCL uses $-J_k$. The final identity follows from incidence, so it does not replace individual coil observations or limits. The audit labels terminal currents reconstructed, retaining explicit pairs and both injection/outgoing signs. It does not invent an independent line-current measurement.

The native generator's i_max[k] bounds $|I_k|$. Its s_max[k] bounds $|S_k|$ directly; no total rating is divided among coils. This array rule is separate from the earlier winding-list transformer's equal allocation of a single winding s_max. Generator P/Q bounds and required pg/qg reports also refer to each coil. Signed injections and absorption are allowed by the corresponding declared bounds. Residuals use A, W, var and VA with the existing absolute current/power tolerances.

Inspect: the source/switch/load fixture uses $V=(10,10\mathrm j,-6-8\mathrm j)$ V, giving:

All coil limits pass. The reconstructed terminal injections are $(0.2-0.15\mathrm j,-0.5+0.25\mathrm j,0.3-0.1\mathrm j)$ A. At terminal y, $|J_y|\approx0.559$ A exceeds the numerical second-coil cap, while the second coil carries only 0.25 A. Applying the declared coil cap to that terminal current would falsely reject this native operating point. No terminal-line rating is inferred from the coil array.

Now add the same $0.5+0.2\mathrm j$ A to every coil. Incidence removes the common increment, so terminal injections, KCL and aggregate generator power $3.5-6.5\mathrm j$ VA are unchanged. Individual coil powers become $(7-9\mathrm j,8.1+3.3\mathrm j,-11.6-0.8\mathrm j)$ VA. All three original current caps and all three apparent-power caps fail, while honest updated pg/qg reports, KCL and aggregate power pass. This is a six-limit violation, not missing evidence. Internal circulation cannot be reconstructed from those terminal observations alone.

At a separate zero-voltage point, equal nonzero coil currents give zero terminal injection and zero coil power. Zero apparent-power caps therefore pass even though the current caps fail. Inspection of the native primal constraints also rejects those currents. A zero power report is neither a zero-current observation nor spare current capacity.

The package's delta_generator_audit.jl and delta-generator.json, invoked by the source-bound book companion, retain analytic/native points, circulation, forged reports, missing internal observations and malformed ratings. Native SI, per-unit, consistently reversed-coil and two-terminal comparisons fix the recorded coil dispatch and pass. A separate tighter-current-cap attempt reports LOCALLY_INFEASIBLE and returns a point that fails the independent audit. Both statuses are retained; this solver observation is not a global infeasibility certificate or a robustness claim.

Cyclic reordering must move coil-indexed bounds, prices and reports with the coils. Reversing the cycle reverses each matched coil's voltage and current; its complex power keeps the same sign. Changing terminal order alone does not perform that transformation. Per-unit evaluation returns to original SI coil voltages and currents; an earth-voltage magnitude or a generic square-root-of- three factor cannot replace an actual ordered voltage difference.

There are two applicability distinctions to keep visible:

• The native two-terminal DELTA generator has two opposed coils, one at each cyclic position. A two-terminal DELTA load and a two-terminal SINGLE_PHASE generator each have one coil. The separate three-terminal data-model arity checker still warns on the native two-terminal generator form. Matching native equations does not establish specification conformance or a physical three-phase winding interpretation.
• A neutral-bearing DELTA generator map remains unassessed. The current native generator builder constructs it, while its conformance checker emits E.SPEC.DELTA_HAS_NEUTRAL. The audit records that disagreement and refuses the map. It neither changes native construction nor applies WYE neutral-return rules to make coverage pass. One/four-terminal cycles, custom roles and controllers remain outside this bounded audit.

Missing a required coil current or voltage suppresses complete network KCL/power certification. Known observations and violations on other coils remain visible; a missing pg/qg report does not erase otherwise known current or limit checks. Malformed extra/prefix rating entries remain unknown without hiding a separate known violation. Finite signed coil prices remain declared metadata; objective selection/value, local/global optimality, physical realizability and unrestricted native/custom-model feasibility are separate obligations.

The running network's generator is WYE, so this extension leaves its recorded default three-p_max failure, zero unknowns and separate tighter electrical pass unchanged. Published integration, PSK implementation statuses and execution MCP remain separate.

Transfer exercise. Derive the three terminal injections from the baseline coil currents. Then add a common complex increment and identify which observations stay unchanged, which ratings must be rechecked, and why terminal measurements cannot supply a missing internal coil current.

A source price does not change its voltage reference

A native voltage source uses two indexing and reference conventions at once. Its v_magnitude/v_angle entries specify absolute terminal-to-earth phasors in full source-map order. Its current, P/Q and price entries follow only the non-neutral source-map order. Neither order is inherited from the bus list. Recognizing the price vector must not discard or reinterpret the source's reference and grounding obligations.

Predict: a bus lists $(n,a,b)$, while its source map is $(b,n,a)$. The reference magnitudes are $(10,0,10)$ V at $(\pi/2,0,0)$ rad. There are two prices $(0.4,0.1)$ currency/kWh. Which price belongs to phase a, and what voltage reference enters its priced power?

Independently decoding that source gives phase order $(b,a)$ and map positions $(1,3)$. The neutral has no price/current slot. For these supported native WYE/SINGLE_PHASE sources,

\[S_k=(V_{t_k}-V_n)\overline{I_k},\qquad C_s=\sum_k c_k\frac{\Re(S_k)}{1000}.\]

If the bus has no neutral, use earth in place of $V_n$. If a declared bus neutral exists but is omitted from the source map, the native source still uses it for power and current return, and grounds it at zero. This differs from an unmapped-neutral WYE load/generator's earth-reference rule. The price metadata retains the phase order, original map positions, phase/reference pairs and whether the neutral was mapped. The audit uses supplied neutral state in P/Q evaluation; grounding is checked separately, so missing state is not replaced by a passing zero.

Positive source active power is injection into the network, interpreted here as import; negative power is export. The same signed coefficient multiplies either sign. A positive coefficient therefore credits negative injection in that expression. This is not a separate buy/sell tariff. With P in W and price in currency/kWh, the snapshot expression is a rate in currency/hour. Energy cost needs an explicit duration assumption. Valid fixed source references make these source-power terms linear in the source currents; they do not make the whole network model linear.

Inspect: source_price_audit.jl and source-price.json, invoked by the exact clean candidate companion, retain this unbalanced computational point:

The net import is 3 W, but the known default rate is -0.0006 currency/hour under these declared phase prices. This is a signed numerical witness with negative active demand on one load phase, not an inference about asset identity or economically adequate pricing. A neutral-first map $(n,a,b)$ needs prices $(0.1,0.4)$ to keep the same phase-price assignment. Leaving the old prices in that reordered map gives +0.0021 currency/hour, while the same electrical point still passes. An electrical audit cannot recover which tariff mapping the caller intended.

Several reference cases must remain distinct:

The final case does not use the one-coil convention of a two-terminal SINGLE_PHASE generator. Its source currents $(0.3-0.1\mathrm j, -0.3+0.1\mathrm j)$ A at $(10,10\mathrm j)$ V have powers $(3+\mathrm j,1-3\mathrm j)$ VA. Prices $(0.1,0.2)$ give a known rate of 0.0005 currency/hour. A one-entry price vector remains unknown to the audit and the native default expression cannot consume its missing second entry.

Absolute source references and phase-neutral powers are also different tests. A common voltage translation can preserve all supplied phase-neutral powers and KCL while violating the source's absolute phasors and neutral ground. In a separate displaced-neutral witness, the authored source reference itself sets $V_n=1$ V: its supplied absolute reference equation, honest P/Q reports, KCL and aggregate power pass, but the independent neutral-ground boundary fails. Native construction rejects that conflicting reference. Neither price metadata nor balanced power legitimizes the nonzero source neutral.

Source neutral return injection is $-\sum_k I_k$. The separately supplied earth-reaction current is another term, even when the neutral voltage is zero. The audit reconstructs the return, consumes a shared source-bus ground reaction once, and leaves a missing earth reaction unknown. Two overlapping ideal sources can satisfy the same references and external current balance while redistributing their currents. Two recorded splits both pass electrically, but their known default expression values differ. The fixture does not optimize that allocation; electrical acceptance establishes neither intended allocation nor objective well-posedness.

Native SI/per-unit, neutral-omitted, trailing-prefix, neutral-first, earth and two-phase SINGLE_PHASE comparisons solve and pass. Reversing the price signs, omitting the objective, or selecting a caller-constant123 objective retains separate recipe and electrical evidence. Per-unit working prices multiply the local power base and cancel power scaling; they are not revised tariffs. Native expression tests confirm that generation_cost adds no variables or constraints and does not select the objective. They inspect the implementation as a subject; the independent audit does not call it as an oracle.

Recognized full finite signed price vectors have status declared, null residual/tolerance and enforced_as_physical_limit=false. Scalar, partial, extra/neutral-priced, Boolean, nonfinite and unrepresentable declarations remain unknown while known source equations, currents, reports and KCL remain checked. Malformed reference arrays likewise leave known current incidence available; the reference equation stays unassessed. A missing required phase current, neutral state or earth reaction suppresses complete network KCL/power checks. Known other-phase report failures survive. Optional supplied ps/qs reports are checked in W/var with unchanged tolerances.

The next section adds source P/Q limits. Unsupported source current/apparent- power ratings or controller fields, custom terminal roles and DELTA sources remain outside this bounded extension. It certifies no tariff, selected objective, reported objective consistency, local/global optimality, physical grounding adequacy or unrestricted native/custom-model feasibility. The running input has no source price, so its recorded default three generator p_max violations/zero unknowns and separate tighter electrical pass are unchanged. Published integration, PSK statuses and execution MCP remain separate.

Transfer exercise. Reorder $(b,n,a)$ to $(n,a,b)$ while preserving the physical phase tariffs. Write the new reference and price arrays, then explain why a correct cost rate cannot establish the omitted phase reference, neutral earth-reaction current, source allocation or selected-objective guarantee.

Source P/Q limits apply to signed phase injections

A voltage source can fix its voltage reference while having a constrained power exchange. The local candidate now independently checks authored p_min/p_max/q_min/q_max for its bounded WYE/SINGLE_PHASE source domain. It recomputes power from the supplied voltage/current state:

\[S_k=(V_{t_k}-V_n)\overline{I_k},\qquad P_k=\Re(S_k),\quad Q_k=\Im(S_k),\]

with earth in place of $V_n$ when no bus neutral exists. An unmapped bus neutral still supplies the source's power reference and return. The limits follow non-neutral source-map order, even if the neutral appears between phases in the full reference array. Source-fixed and grounded phases keep their P/Q slots. The earlier voltage-bound skip rules do not apply to this source-power family.

Predict: phase powers in source order $(b,a)$ are $(-3+2\mathrm j, 6+\mathrm j)$ VA. Upper P bounds $(-2,5.5)$ W sum to 3.5 W, while net P is 3 W. Does that establish source-bound compliance?

The actual constraints are independent signed boxes:

\[p_k^{\min}\le P_k\le p_k^{\max},\qquad q_k^{\min}\le Q_k\le q_k^{\max}.\]

For example, upper-P excess is $\max(0,P_k-p_k^{\max})$ and lower-P excess is $\max(0,p_k^{\min}-P_k)$. P residuals and power_tol use W; Q residuals use var with the same numeric tolerance setting. These are absolute violations in original SI coordinates, not relative percentages, per-unit residuals or price-weighted penalties.

Positive P is injection into the network (import), and negative P is export. A lower bound $p^{\min}=-2.5$ W therefore rejects $P=-3$ W by 0.5 W: there is too much export for that lower bound. An upper bound $p^{\max}=-3.5$ W would require more export than $P=-3$ provides and also fails by 0.5 W. There is no absolute-value or import/export sign flip. Positive Q is reactive injection; negative Q is absorption under the declared complex-power convention.

Inspect: source_pq_audit.jl and source-pq.json extend the source-price point with these full phase bounds. The bus order remains $(n,a,b)$ and the source map remains $(b,n,a)$:

All baseline rows pass. Tightening only phase a's upper P bound to 5.5 W creates one 0.5 W violation, despite the passing summed comparison in the prediction. Tightening phase b's upper Q bound to 1.5 var creates one 0.5 var violation even though total Q equals the summed upper bound of 3 var. Lower P/export and lower Q examples retain the same distinction. No aggregate comparison or neutral-current observation replaces an individual phase bound. A separate two-phase SINGLE_PHASE source without a neutral checks negative Q absorption with two independently indexed source terms.

Optional ps/qs reports are a separate consistency check. A forged in-range ps=5.5 cannot hide a recomputed 6 W injection: the report and limit each fail by 0.5 W. Missing optional reports leave known current-derived limits available. Missing a required phase current instead leaves that phase's power/limits unknown, preserves known other-phase failures, and suppresses complete network KCL/power certification. The checker never inverts a reported power to replace the missing current.

Declaration applicability is also explicit:

An empty row has null residual/tolerance and enforced_as_limit=false. It does not claim that any phase satisfies a zero bound. This source empty-vector rule is distinct from the generator audit's stricter full-array declaration domain. No field is deleted, padded or trimmed to obtain complete coverage.

The native builder is permissive about nonempty lengths: a prefix stamps only its available first phase entries, and extra entries are ignored. Static indexing diagnoses both with W.INT.DIM_MISMATCH. The source-bound companion retains successful native prefix/extra solves whose bounded point audits stay indeterminate, rather than silently assigning missing entries or accepting an unexplained extra declaration. Native semantic-row inspection shows one upper-P row for a one-entry prefix and two rows for a full vector with an ignored extra entry. Four full vectors on two phases add eight native constraints and no variables; empty and absent vectors add none.

Native SI/per-unit, consistently reordered, neutral-omitted, trailing-neutral, earth-reference, two-phase SINGLE_PHASE and empty-limit comparisons solve and pass their scoped audits. Consistent reordering moves P/Q arrays together with the source reference, price and current indexing. Leaving the old bound order can invalidate the same electrical point. Separate tighter P and Q attempts report LOCALLY_INFEASIBLE and return points failing the independent audit. Both outcomes are retained; this is no global infeasibility, optimality or initialization-robustness certificate. Native constraints and per-unit recovery are test subjects, not the independent evaluator's oracle.

At a separate point with every phase-to-reference voltage zero, nonzero source currents pass zero P/Q boxes. A positive lower P bound then fails. Native primal inspection confirms both supplied-point observations. Thus even complete P/Q-box compliance is not a conductor-current or apparent-power rating certificate. No physical grid/ground adequacy, selected objective, controller/custom-role domain or unrestricted native/custom-model feasibility is inferred. Unsupported source current/apparent-power declarations remain unassessed.

The running input has no source P/Q bounds. Its retained default three generator p_max violations/zero unknowns and separate tighter electrical pass therefore remain unchanged. Published integration, PSK implementation statuses and execution MCP remain separate.

Transfer exercise. For the table's operating point, choose a bound that rejects excess export on b while leaving import on a admissible. Then preserve those physical phase limits under source-map reordering, and explain why a passing total-P comparison, an in-range report, an empty array or a zero-power observation cannot replace the phase and current evidence.

IBR currents depend on the admitted topology

An inverter label alone does not identify the voltage/current pairing or its terminal-current incidence. The local candidate adds independent supplied-point checks for a bounded AC-only slice. Its three admitted topologies have distinct observations and current caps:

The default-role neutral may occur anywhere in a four-leg map. Reversing the single-phase pair negates its current while preserving its physical terminal injections and power. A cyclic reordering or reversal must transform every coil observation and parameter slot consistently. Reconstructed returns and terminal currents are derived quantities, not independent measurements.

Predict: a three-leg point passes terminal KCL. Add the same complex $h$ to every supplied cyclic coil current. Which observations change, and which ratings must be checked again?

The independently evaluated powers are

\[S_k=U_k\overline{I_k},\qquad P_k=\Re(S_k),\quad Q_k=\Im(S_k).\]

For the cyclic model,

\[(I_k+h)-(I_{k-1}+h)=J_k,\qquad \sum_k U_k=0,\qquad \sum_k U_k\overline{(I_k+h)}=\sum_k S_k.\]

Thus terminal KCL and total injected complex power cannot reveal this internal circulation. Individual coil powers and current magnitudes can change. In the minimized three-leg fixture, adding $h=0.5+0.2\mathrm j$ A preserves all reconstructed terminal injections, terminal KCL and aggregate power while violating three coil current and three coil apparent-power caps. The base point also has a terminal current larger than the corresponding coil cap; that cap is attached to the coil quantity, so it is satisfied. A terminal-current rating would be a different obligation.

Two other fixture observations separate the current-cap rules:

The audit checks full finite signed p_min/p_max/q_min/q_max in W/var and nonnegative s_max/i_max in VA/A. Residuals are absolute SI excesses under power_tol or current_tol, not squared or per-unit magnitudes. Absent or explicit empty arrays author no native bound; empty arrays yield declared no-bound rows. Nonempty arrays must match the admitted branch/current domain. Optional reported pg/qg are separate consistency checks, never the limit oracle. A malformed array preserves independently known other limits and current contributions. A missing required current or voltage preserves known other branch failures but suppresses complete network KCL/power certification. No division by voltage or reconstruction from reported power is used. At zero voltage, nonzero cyclic current can pass zero P/Q/S caps while violating I caps.

The package example examples/ibr_injection_audit.jl and fixture test/data/linear_audit/ibr-injection.json are included by the source-bound book companion. Analytic points pass for all three admitted topologies. Separate controlled native comparisons fix their dispatch, seed the full primal state and pass their scoped audits in SI and per-unit coordinates; the reversed single pair also passes. Attempts with the tighter shared/neutral caps retain their solver statuses, returned points and independent audits. In this locked run both report LOCALLY_INFEASIBLE; their returned-point audits fail one and four checks respectively. These are local outcomes, without a global infeasibility, unique-dispatch or initialization certificate. Floating-neutral, map-permutation, missing-observation, zero-voltage and malformed-declaration tests exercise the package's bounded evaluator. Native construction, extraction, auxiliary variables and coordinate recovery are test subjects; the audit uses none as its equation oracle.

Topology applicability remains a separate question. Native construction accepts a single-phase ignored terminal suffix, a four-leg map omitting its neutral (with earth reference), other cyclic three-leg arities, and extra single-phase current entries. The bounded audit refuses those declarations. Static conformance and native construction remain distinct: a reduced-phase four-leg case may receive an arity warning even within the stated native audit slice. This work checks the current native cyclic equations for THREE_LEG; the broader physical topology review in package issue #95 remains unresolved. Agreement with these equations cannot settle that physical model question.

Authored filters (r_filter, x_filter, b_filter_shunt), internal/grid-forming voltage fields, droop/other controller profiles, p_avail, custom roles and DC coupling remain unassessed, including explicit zero/false filter declarations. The native filter-boundary comparison has unchanged variable/constraint counts with these fields present. Ignoring them is not evidence that a filter is ideal or adequate, so the audit refuses the declaration. False dc_link_coupled is admitted as the uncoupled default. Recognized prime_mover enum values are metadata without PV, battery-dynamics or STATCOM physical guarantees. Native power-flow construction omits s_max capability circles; supplied-point cap compliance does not establish that a chosen native problem enforced them. Selected objectives, custom constraints and local/global optimality remain outside the electrical point audit.

The running input contains no IBR. Its retained default three generator p_max failures with zero unknowns, all 26 passing KCL rows and passing aggregate power, and its separate tighter electrical pass remain unchanged. Published integration, PSK implementation statuses and execution MCP remain separate.

Transfer exercise. For each topology, state the meaning of a supplied cri/cii, write its ordered power voltage and reconstruct the terminal injections. Then decide which extra evidence is needed for a physical terminal-current rating, a filter model or a DC-coupled energy guarantee.

IBR price declarations do not select an objective

The bounded IBR injection audit now recognizes full finite signed cost vectors as declarations. Their entries follow the admitted power branches: one ordered pair for SINGLE_PHASE, phase order after removing the mapped neutral for FOUR_LEG, and cyclic coil order for THREE_LEG. The declaration records these terminal pairs and original map positions, with status declared, null residual/tolerance and false physical-limit, selected/reported-objective, objective-value and optimality flags. A recognized price does not establish that a supplied result minimized it, or that it is an adequate tariff.

The native default expression is a signed snapshot cost rate:

\[R=\frac{1}{1000}\sum_k c_k P_k,\qquad P_k=\Re\!\left(U_k\overline{I_k}\right).\]

Prices $c_k$ use currency/kWh, powers use W, and $R$ uses currency/hour. Positive prices prefer less injection and negative prices prefer more within the chosen feasible set. A zero coefficient removes that term's price incentive; it does not prove a feasible flat direction or a unique dispatch. An explicit snapshot duration $\Delta t$ in hours yields $E=R\Delta t$ in currency. Duration is not inferred from a static network or a supplied objective value.

The source-bound companion separately evaluates native generation_cost(ctx) at analytic voltage/current points and runs controlled native objective recipes. The independent audit reads none of those native helpers as its oracle.

These expression values agree in SI and per-unit coordinates. Native conversion multiplies prices by the local AC power base, while the power coordinates divide by that base. The factors cancel. Building the expression adds no variables or constraints and leaves a feasibility model's selected objective unchanged. It uses voltage/current directly, without relying on P/Q auxiliary values.

Consistent permutations/reversals transform prices with their branch observations and preserve the table's rates. If four-leg prices are left in the old order after reordering the phases, the rate becomes -0.00095. Both electrical points pass: price assignment is a separate semantic choice.

Predict: add the same circulating current $h$ to every cyclic coil. Terminal injections and total power remain unchanged. Must the weighted rate also remain unchanged?

For the current native cyclic equations,

\[\Delta R=\frac{1}{1000}\Re\!\left( \overline{h}\sum_k c_k U_k\right).\]

Uniform prices factor out of $\sum U_k=0$. Nonuniform prices generally do not. With $h=0.5+0.2\mathrm j$ A, the three-leg fixture's rate changes from -0.0005 to -0.0044, despite unchanged terminal injections/KCL/total power. Its three current and three apparent-power coil failures remain visible. Uniform 0.1 prices instead give 0.00035 before and after circulation. Terminal total power cannot recover arbitrary weighted coil prices. This is a statement about the current native cyclic model, without resolving physical THREE_LEG review #95 or establishing an economically meaningful internal circulation.

Absent cost authors no declaration. Empty, scalar, null, partial, extra, Boolean, nonfinite and Float64-range-invalid prices remain unassessed. A malformed price preserves known electrical checks and current contributions; known failures take precedence. Missing current preserves valid price metadata but suppresses complete KCL/power. Optional pg/qg, a forged objective and a successful solver label do not replace those observations or select a recipe. Filter/droop/availability/DC/custom-role/topology boundaries are unchanged.

Native default construction rejects scalar/empty/prefix prices, ignores an extra suffix and skips a null term. Construction with add_objective=false accepts all of these cases, without validating a price expression. The strict bounded audit retains every explicit malformed declaration as unknown, including null and ignored extras; it neither pads nor discards prices to pass.

Separate controlled recipes retain default, feasibility and constant-123 objectives with the same valid price declaration. An explicit two-hour wrapper changes the single-pair rate 0.001 currency/hour to 0.002 currency. Signed and zero cases and bounded free-dispatch comparisons preserve solver/primal/raw status and independent returned-point audits. In this locked run, the fully seeded single-pair SI positive, negative and two-hour attempts report OTHER_ERROR / Restoration_Failed and INFEASIBLE_POINT. Their returned electrical points pass the independent audit and have the analytic recipe values, but the solver attempts remain unaccepted. Separate per-unit positive, negative and two-hour comparisons solve and pass. None establishes initialization robustness or turns the general supplied-point audit into a selected-objective checker. Reported objective consistency, custom constraints, tariff adequacy, duration/energy assumptions, stationarity and local/global optimality require separate evidence.

The running input has no IBR; its default three generator-bound failures with zero unknowns, all 26 passing KCL rows and aggregate power, and separate tighter electrical pass remain unchanged. Published integration, PSK implementation statuses and execution MCP remain separate.

Transfer exercise. Reorder the four-leg map while preserving physical phase prices, then compare uniform and nonuniform cyclic prices under circulation. State the additional evidence needed to certify a reported two-hour objective or a physical/economic interpretation of the internal coil dispatch.

The next bounded extension is developed on the focused IBR power-factor control page: constant-PF equalities, generation-only minimum-PF envelopes, hard capability intersections and the limits of weighted residual tolerances. Existing injection/price anchors remain on this page.

The focused Volt-var extension separates voltage monitors, the requested piecewise-linear response, an explicitly assumed smooth model and hard Q/minimum-PF obligations. The Volt-watt and availability extension distinguishes a one-sided smooth cap, authored P boxes, normalization bases and scalar availability metadata with separate augmentation provenance.

The combined-droop audit intersects both laws with capability and minimum PF, retaining an explicit native monitor-key construction boundary alongside independently known residuals.