From 9b4b7b58b0b9358bbcb3b6e0eed655ce51adc454 Mon Sep 17 00:00:00 2001
From: sepehr <sepehr1151@gmail.com>
Date: Sun, 19 Jul 2026 23:23:35 +0200
Subject: [PATCH] Stabilize Modelica Fixed/Free calibration and stop flaky
 Newton embeds.

Sort constraint/control assembly for deterministic Jacobians, keep live z_ua on legacy HX, and add generic Fixed/Free UI assist without removing Z factors.

Co-authored-by: Cursor <cursoragent@cursor.com>
---
 .../src/components/panels/PropertiesPanel.tsx |  79 +++-
 apps/web/src/lib/calibAssist.test.ts          | 226 ++++++++++
 apps/web/src/lib/calibAssist.ts               | 331 +++++++++++++++
 apps/web/src/lib/configBuilder.test.ts        |  47 +-
 apps/web/src/lib/configBuilder.ts             |  51 ++-
 crates/cli/src/config.rs                      |  79 ++++
 crates/cli/src/run.rs                         | 228 +++++++++-
 .../src/heat_exchanger/condenser.rs           |  57 ++-
 .../src/heat_exchanger/evaporator.rs          |  51 ++-
 crates/solver/src/inverse/bounded.rs          |   2 +-
 crates/solver/src/inverse/constraint.rs       |   2 +-
 crates/solver/src/inverse/embedding.rs        |  28 +-
 crates/solver/src/inverse/expr.rs             | 401 ++++++++++++++++++
 crates/solver/src/inverse/mod.rs              |   2 +
 crates/solver/src/system.rs                   | 226 +++++++++-
 15 files changed, 1726 insertions(+), 84 deletions(-)
 create mode 100644 apps/web/src/lib/calibAssist.test.ts
 create mode 100644 apps/web/src/lib/calibAssist.ts
 create mode 100644 crates/solver/src/inverse/expr.rs

diff --git a/apps/web/src/components/panels/PropertiesPanel.tsx b/apps/web/src/components/panels/PropertiesPanel.tsx
index 1640bc5..80ce19c 100644
--- a/apps/web/src/components/panels/PropertiesPanel.tsx
+++ b/apps/web/src/components/panels/PropertiesPanel.tsx
@@ -39,6 +39,12 @@ import {
 } from "@/lib/configBuilder";
 import { buildComponentInspector, getSolvedVariablesForComponent } from "@/lib/componentInspector";
 import type { SolvedVariable } from "@/lib/api";
+import {
+  calibrateAdjacentActuatorPatches,
+  collectCalibHints,
+  isCalibRelevantNode,
+  withDerivedUaEff,
+} from "@/lib/calibAssist";
 import { ComponentIcon } from "@/components/canvas/ComponentIcon";
 import ComponentDocPanel from "@/components/panels/ComponentDocPanel";
 import { modelBannerForType } from "@/lib/componentDocMap";
@@ -138,11 +144,23 @@ export default function PropertiesPanel() {
 
   // Solver-computed unknowns owned by this component (free actuators +
   // calibration factors). Empty until a solve produces them.
-  const solvedVars = useMemo<SolvedVariable[]>(
-    () => (node ? getSolvedVariablesForComponent(result, node.data.name) : []),
-    [node, result],
+  const solvedVars = useMemo<SolvedVariable[]>(() => {
+    if (!node) return [];
+    const base = getSolvedVariablesForComponent(result, node.data.name);
+    return withDerivedUaEff(node.data.name, nodes, base);
+  }, [node, nodes, result]);
+
+  const calibHints = useMemo(
+    () => (node ? collectCalibHints(nodes, edges, node) : []),
+    [node, nodes, edges],
   );
 
+  /** One-click Free on adjacent actuators compatible with this Probe's Fixed measures. */
+  const calibActuatorPatches = useMemo(() => {
+    if (!node || node.data.type !== "Probe") return [];
+    return calibrateAdjacentActuatorPatches(node, nodes, edges);
+  }, [node, nodes, edges]);
+
   // After a successful solve, jump to Results when selecting a part (Dymola-like).
   useEffect(() => {
     if (!node) return;
@@ -186,7 +204,8 @@ export default function PropertiesPanel() {
                 <strong>Fixed ☐</strong> = inconnue → il faut une équation de plus
               </li>
               <li>
-                Calib : Probe Tsat Fixed (= équation) + Z_UA Free (= inconnue)
+                Calib : Probe mesure <strong>Fixed</strong> (= équation) + facteur{" "}
+                <strong>Free</strong> (= inconnue) — z_ua, z_dp, z_flow, f_w…
               </li>
               <li>n_eq = n_unk. Pas de boucle de régulation.</li>
             </ul>
@@ -305,6 +324,44 @@ export default function PropertiesPanel() {
         </div>
       )}
 
+      {/* Always visible on Probe / Z-factor components (also in Results after solve). */}
+      {!isRegLoop &&
+        isCalibRelevantNode(node) &&
+        (calibHints.length > 0 || calibActuatorPatches.length > 0) && (
+          <div className="space-y-1 border-b border-sky-200 bg-sky-50 px-2.5 py-2 text-[10px] leading-snug">
+            <p className="font-semibold text-sky-950">Fixed / Free (Modelica)</p>
+            {calibHints.map((h, i) => (
+              <p
+                key={`${h.level}-${i}`}
+                className={
+                  h.level === "warn"
+                    ? "text-amber-900"
+                    : h.level === "ok"
+                      ? "text-emerald-800"
+                      : "text-sky-900"
+                }
+              >
+                {h.level === "ok" ? "✓ " : h.level === "warn" ? "⚠ " : "💡 "}
+                {h.message}
+              </p>
+            ))}
+            {calibActuatorPatches.length > 0 && (
+              <button
+                type="button"
+                className="mt-0.5 rounded border border-sky-600 bg-white px-2 py-1 text-[10px] font-medium text-sky-800 hover:bg-sky-100"
+                onClick={() => {
+                  const map = new Map(
+                    calibActuatorPatches.map((p) => [p.nodeId, p.params]),
+                  );
+                  updateNodesParams(map);
+                }}
+              >
+                Libérer facteurs compatibles (Free)
+              </button>
+            )}
+          </div>
+        )}
+
       {!isRegLoop && panelMode === "results" && (
         <ModelicaResultsView
           inspector={inspector}
@@ -317,8 +374,8 @@ export default function PropertiesPanel() {
         <>
           {isRegLoop && (
             <div className="border-b border-amber-200 bg-amber-50 px-2.5 py-1.5 text-[10px] text-amber-950">
-              Optionnel. Calibration SST + Z_UA → onglet <strong>Calibration</strong> du HX (cases
-              Fixed), pas ce nœud.
+              Optionnel. Calibration générique → onglet <strong>Calibration</strong> (Fixed/Free
+              sur mesures Probe + facteurs Z), pas ce nœud.
             </div>
           )}
 
@@ -548,12 +605,12 @@ export default function PropertiesPanel() {
             {activeTab === "Calibration" && (
               <div className="space-y-1.5 border-t border-[var(--line)] bg-[var(--chrome-2)] px-2.5 py-1.5 text-[10px] leading-snug text-[var(--ink-dim)]">
                 <p>
-                  <strong>Modelica :</strong> Probe Fixed → équation (
-                  <code>Tsat = …</code>). Facteur Free → inconnue (
-                  <code>parameter z_ua(fixed=false)</code>). Le start est la valeur affichée.
+                  <strong>Fixed</strong> = parameter connu (équation si Probe).{" "}
+                  <strong>Free</strong> = inconnue Newton. Pairing : Tsat/Tsh↔z_ua,
+                  P↔z_dp, Capacity↔z_flow, T↔f_w. Tous les Z restent disponibles.
                 </p>
-                <p className="text-[var(--ink-faint)]">
-                  Laisse UA override vide. n_eq doit égaler n_unk.
+                <p>
+                  Ex. UA nominale × Z_UA → <code>UA_eff</code>. Start = valeur affichée.
                 </p>
                 {solvedVars.length > 0 && (
                   <SolvedVariablesBlock items={solvedVars} title="Inconnues résolues" />
diff --git a/apps/web/src/lib/calibAssist.test.ts b/apps/web/src/lib/calibAssist.test.ts
new file mode 100644
index 0000000..36e5374
--- /dev/null
+++ b/apps/web/src/lib/calibAssist.test.ts
@@ -0,0 +1,226 @@
+import { describe, it, expect } from "vitest";
+import type { Edge, Node } from "@xyflow/react";
+import { fixedFlagKey } from "./componentMeta";
+import {
+  adjacentProbeNames,
+  calibrateAdjacentActuatorPatches,
+  collectCalibHints,
+  withDerivedUaEff,
+  type CalibNodeData,
+} from "./calibAssist";
+import type { SolvedVariable } from "./api";
+
+function node(
+  id: string,
+  type: string,
+  name: string,
+  params: Record<string, number | string | boolean> = {},
+): Node<CalibNodeData> {
+  return {
+    id,
+    type: "entropykNode",
+    position: { x: 0, y: 0 },
+    data: { type, name, circuit: 0, params },
+  };
+}
+
+describe("calibAssist — Fixed/Free générique", () => {
+  it("lists adjacent Probe names for any component", () => {
+    const nodes = [
+      node("e", "Evaporator", "evap", { ua: 6000, z_ua: 1 }),
+      node("p1", "Probe", "sst", { tsat_c: 5, [fixedFlagKey("tsat_c")]: true }),
+    ];
+    const edges: Edge[] = [
+      { id: "pe", source: "e", target: "p1", sourceHandle: "outlet", targetHandle: "inlet" },
+    ];
+    expect(adjacentProbeNames("evap", nodes, edges)).toEqual(["sst"]);
+  });
+
+  it("one-click Free Z_UA when Probe Tsat Fixed", () => {
+    const nodes = [
+      node("e", "FloodedEvaporator", "evap", {
+        ua: 9000,
+        z_ua: 1.0,
+        [fixedFlagKey("z_ua")]: true,
+      }),
+      node("p1", "Probe", "sst", {
+        tsat_c: 5,
+        [fixedFlagKey("tsat_c")]: true,
+      }),
+    ];
+    const edges: Edge[] = [
+      { id: "pe", source: "e", target: "p1", sourceHandle: "outlet", targetHandle: "inlet" },
+    ];
+    const patches = calibrateAdjacentActuatorPatches(nodes[1], nodes, edges);
+    expect(patches).toHaveLength(1);
+    expect(patches[0]).toMatchObject({
+      nodeId: "e",
+      params: { [fixedFlagKey("z_ua")]: false, z_ua: 1.0 },
+    });
+  });
+
+  it("one-click Free z_dp when Probe P Fixed", () => {
+    const nodes = [
+      node("c", "Condenser", "cond", {
+        ua: 5000,
+        z_dp: 1.0,
+        [fixedFlagKey("z_dp")]: true,
+        z_ua: 1.0,
+        [fixedFlagKey("z_ua")]: true,
+      }),
+      node("p1", "Probe", "p_probe", {
+        p_bar: 12,
+        [fixedFlagKey("p_bar")]: true,
+      }),
+    ];
+    const edges: Edge[] = [
+      { id: "pe", source: "c", target: "p1", sourceHandle: "inlet", targetHandle: "outlet" },
+    ];
+    const patches = calibrateAdjacentActuatorPatches(nodes[1], nodes, edges);
+    expect(patches).toHaveLength(1);
+    expect(patches[0].params[fixedFlagKey("z_dp")]).toBe(false);
+    // z_ua not compatible with P — stays Fixed
+    expect(patches[0].params[fixedFlagKey("z_ua")]).toBe(true);
+  });
+
+  it("one-click Free z_flow when Probe Capacity Fixed (compressor)", () => {
+    const nodes = [
+      node("comp", "IsentropicCompressor", "comp", {
+        z_flow: 1.0,
+        [fixedFlagKey("z_flow")]: true,
+        f_w: 0.1,
+        [fixedFlagKey("f_w")]: true,
+      }),
+      node("p1", "Probe", "cap", {
+        capacity_w: 12000,
+        [fixedFlagKey("capacity_w")]: true,
+      }),
+    ];
+    const edges: Edge[] = [
+      { id: "pe", source: "comp", target: "p1", sourceHandle: "outlet", targetHandle: "inlet" },
+    ];
+    const patches = calibrateAdjacentActuatorPatches(nodes[1], nodes, edges);
+    expect(patches).toHaveLength(1);
+    expect(patches[0].params[fixedFlagKey("z_flow")]).toBe(false);
+    expect(patches[0].params[fixedFlagKey("f_w")]).toBe(true);
+  });
+
+  it("one-click Free f_w when Probe T Fixed", () => {
+    const nodes = [
+      node("comp", "IsentropicCompressor", "comp", {
+        f_w: 0.05,
+        [fixedFlagKey("f_w")]: true,
+        z_flow: 1.0,
+        [fixedFlagKey("z_flow")]: true,
+      }),
+      node("p1", "Probe", "dgt", {
+        t_c: 75,
+        [fixedFlagKey("t_c")]: true,
+      }),
+    ];
+    const edges: Edge[] = [
+      { id: "pe", source: "comp", target: "p1", sourceHandle: "outlet", targetHandle: "inlet" },
+    ];
+    const patches = calibrateAdjacentActuatorPatches(nodes[1], nodes, edges);
+    expect(patches[0].params[fixedFlagKey("f_w")]).toBe(false);
+    expect(patches[0].params[fixedFlagKey("z_flow")]).toBe(true);
+  });
+
+  it("returns empty when compatible actuators already Free", () => {
+    const nodes = [
+      node("e", "Evaporator", "evap", {
+        ua: 6000,
+        z_ua: 1.0,
+        [fixedFlagKey("z_ua")]: false,
+      }),
+      node("p1", "Probe", "sst", {
+        tsat_c: 5,
+        [fixedFlagKey("tsat_c")]: true,
+      }),
+    ];
+    const edges: Edge[] = [
+      { id: "pe", source: "e", target: "p1", sourceHandle: "outlet", targetHandle: "inlet" },
+    ];
+    expect(calibrateAdjacentActuatorPatches(nodes[1], nodes, edges)).toHaveLength(0);
+  });
+
+  it("tips when Probe Fixed but compatible factor still Fixed", () => {
+    const nodes = [
+      node("e", "Condenser", "cond", {
+        ua: 5000,
+        z_ua: 1.0,
+        [fixedFlagKey("z_ua")]: true,
+      }),
+      node("p1", "Probe", "sdt", {
+        tsat_c: 40,
+        [fixedFlagKey("tsat_c")]: true,
+      }),
+    ];
+    const edges: Edge[] = [
+      { id: "pe", source: "e", target: "p1", sourceHandle: "inlet", targetHandle: "outlet" },
+    ];
+    const hints = collectCalibHints(nodes, edges, nodes[1]);
+    expect(hints.some((h) => h.level === "tip" && h.message.includes("décoche Fixed"))).toBe(
+      true,
+    );
+  });
+
+  it("ok when Probe Fixed + compatible factor Free", () => {
+    const nodes = [
+      node("e", "Condenser", "cond", {
+        ua: 5000,
+        z_dp: 1.0,
+        [fixedFlagKey("z_dp")]: false,
+      }),
+      node("p1", "Probe", "p_probe", {
+        p_bar: 12,
+        [fixedFlagKey("p_bar")]: true,
+      }),
+    ];
+    const edges: Edge[] = [
+      { id: "pe", source: "e", target: "p1", sourceHandle: "inlet", targetHandle: "outlet" },
+    ];
+    const hints = collectCalibHints(nodes, edges, nodes[0]);
+    expect(hints.some((h) => h.level === "ok" && h.message.includes("z_dp"))).toBe(true);
+  });
+
+  it("warns Free factor without adjacent compatible Probe", () => {
+    const nodes = [
+      node("e", "Evaporator", "evap", {
+        ua: 6000,
+        z_ua: 1.0,
+        [fixedFlagKey("z_ua")]: false,
+      }),
+    ];
+    const hints = collectCalibHints(nodes, [], nodes[0]);
+    expect(hints.some((h) => h.level === "warn")).toBe(true);
+  });
+
+  it("always shows a baseline tip on Probe / HX when nothing paired yet", () => {
+    const probe = node("p1", "Probe", "sst", { tsat_c: 5 });
+    const hx = node("e", "Evaporator", "evap", { ua: 6000, z_ua: 1.0 });
+    expect(collectCalibHints([probe], [], probe).some((h) => h.level === "tip")).toBe(true);
+    expect(collectCalibHints([hx], [], hx).some((h) => h.level === "tip")).toBe(true);
+  });
+
+  it("appends derived UA_eff = UA × z_ua", () => {
+    const nodes = [node("e", "Evaporator", "evap", { ua: 8000, z_ua: 1 })];
+    const solved: SolvedVariable[] = [
+      {
+        id: "evap__z_ua",
+        component: "evap",
+        variable: "z_ua",
+        value: 0.85,
+        min: 0.05,
+        max: 3,
+      },
+    ];
+    const out = withDerivedUaEff("evap", nodes, solved);
+    expect(out).toHaveLength(2);
+    expect(out[1]).toMatchObject({
+      variable: "UA_eff",
+      value: 6800,
+      component: "evap",
+    });
+  });
+});
diff --git a/apps/web/src/lib/calibAssist.ts b/apps/web/src/lib/calibAssist.ts
new file mode 100644
index 0000000..d262890
--- /dev/null
+++ b/apps/web/src/lib/calibAssist.ts
@@ -0,0 +1,331 @@
+/**
+ * Generic Modelica Fixed / Free (Unfixed) calibration assistants.
+ *
+ * Engine already emits embeddings for ANY Free `actuatorFactor` + Fixed Probe.
+ * This module only explains DoF balance and offers one-click Free on compatible
+ * adjacent actuators. Nothing is removed from the catalogue (all Z stay).
+ */
+
+import type { Edge, Node } from "@xyflow/react";
+import {
+  COMPONENT_BY_TYPE,
+  fixedFlagKey,
+  isParamFixed,
+  type ParamMeta,
+} from "./componentMeta";
+import { FACTOR_COMPATIBILITY, PROBE_PARAM_KIND } from "./configBuilder";
+import type { SolvedVariable } from "./api";
+
+/** Minimal node data shape (avoids heavy coupling beyond pairing tables). */
+export type CalibNodeData = {
+  type: string;
+  name: string;
+  params: Record<string, number | string | boolean>;
+  circuit?: number;
+};
+
+export type CalibHintLevel = "ok" | "tip" | "warn";
+
+export interface CalibHint {
+  level: CalibHintLevel;
+  /** Short UI message (French). */
+  message: string;
+  component?: string;
+}
+
+export interface CalibParamPatch {
+  nodeId: string;
+  params: Record<string, number | string | boolean>;
+}
+
+type FixedProbeMeasure = {
+  key: string;
+  label: string;
+  kind: string;
+  factors: readonly string[];
+};
+
+function actuatorParams(type: string): ParamMeta[] {
+  const meta = COMPONENT_BY_TYPE[type];
+  if (!meta) return [];
+  return meta.params.filter((p) => p.fixable && p.actuatorFactor);
+}
+
+function fixedProbeMeasures(probe: Node<CalibNodeData>): FixedProbeMeasure[] {
+  const meta = COMPONENT_BY_TYPE.Probe;
+  if (!meta || probe.data.type !== "Probe") return [];
+  const params = probe.data.params ?? {};
+  const out: FixedProbeMeasure[] = [];
+
+  for (const p of meta.params) {
+    if (!p.fixable || !p.measureOutput) continue;
+    if (!isParamFixed(params, p)) continue;
+    const raw = params[p.key];
+    if (raw === undefined || raw === "") continue;
+    const n = typeof raw === "number" ? raw : Number(raw);
+    if (!Number.isFinite(n)) continue;
+    const kind = PROBE_PARAM_KIND[p.key] ?? p.key;
+    const factors = FACTOR_COMPATIBILITY[kind] ?? [];
+    if (factors.length === 0) continue;
+    out.push({ key: p.key, label: p.label, kind, factors });
+  }
+
+  // Legacy measure+target
+  if (out.length === 0) {
+    const fixedFlag = params.__fixed_target;
+    if (fixedFlag === true || fixedFlag === "true") {
+      const kind = String(params.measure ?? "");
+      const factors = FACTOR_COMPATIBILITY[kind] ?? [];
+      if (kind && params.target != null && factors.length > 0) {
+        out.push({
+          key: "target",
+          label: kind,
+          kind,
+          factors,
+        });
+      }
+    }
+  }
+
+  return out;
+}
+
+/** Neighbours connected by an edge (any component type). */
+export function adjacentNodes(
+  node: Node<CalibNodeData>,
+  nodes: Node<CalibNodeData>[],
+  edges: Edge[],
+): Node<CalibNodeData>[] {
+  const byId = new Map(nodes.map((n) => [n.id, n]));
+  const out: Node<CalibNodeData>[] = [];
+  const seen = new Set<string>();
+  for (const e of edges) {
+    const otherId = e.source === node.id ? e.target : e.target === node.id ? e.source : null;
+    if (!otherId || seen.has(otherId)) continue;
+    const other = byId.get(otherId);
+    if (!other) continue;
+    seen.add(otherId);
+    out.push(other);
+  }
+  return out;
+}
+
+/** Probe names adjacent to a named component. */
+export function adjacentProbeNames(
+  componentName: string,
+  nodes: Node<CalibNodeData>[],
+  edges: Edge[],
+): string[] {
+  const byName = new Map(nodes.map((n) => [n.data.name, n]));
+  const comp = byName.get(componentName);
+  if (!comp) return [];
+  return adjacentNodes(comp, nodes, edges)
+    .filter((n) => n.data.type === "Probe")
+    .map((n) => n.data.name);
+}
+
+/** True if the node participates in Fixed/Free calibration UX. */
+export function isCalibRelevantNode(node: Node<CalibNodeData> | null | undefined): boolean {
+  if (!node) return false;
+  if (node.data.type === "Probe") return true;
+  return actuatorParams(node.data.type).length > 0;
+}
+
+/**
+ * Collect Modelica Fixed/Free tips for the selected node (or whole diagram).
+ * Does not replace hard Free-without-equation errors from validateConfig.
+ * Always returns at least a baseline tip for Probe / actuator components.
+ */
+export function collectCalibHints(
+  nodes: Node<CalibNodeData>[],
+  edges: Edge[],
+  focus?: Node<CalibNodeData> | null,
+): CalibHint[] {
+  const hints: CalibHint[] = [];
+
+  for (const probe of nodes) {
+    if (probe.data.type !== "Probe") continue;
+    const measures = fixedProbeMeasures(probe);
+    if (measures.length === 0) continue;
+
+    const neighbours = adjacentNodes(probe, nodes, edges).filter(
+      (n) => n.data.type !== "Probe",
+    );
+
+    for (const measure of measures) {
+      for (const neigh of neighbours) {
+        if (
+          focus &&
+          focus.id !== probe.id &&
+          focus.id !== neigh.id
+        ) {
+          continue;
+        }
+        for (const ap of actuatorParams(neigh.data.type)) {
+          const factor = ap.actuatorFactor!;
+          if (!measure.factors.includes(factor)) continue;
+          const free = !isParamFixed(neigh.data.params ?? {}, ap);
+          if (free) {
+            hints.push({
+              level: "ok",
+              component: neigh.data.name,
+              message:
+                `OK Modelica : Probe « ${probe.data.name} ».${measure.label} Fixed ` +
+                `(équation) + ${factor} Free sur « ${neigh.data.name} » (inconnue).`,
+            });
+          } else {
+            hints.push({
+              level: "tip",
+              component: neigh.data.name,
+              message:
+                `Pour calibrer : décoche Fixed sur ${factor} de « ${neigh.data.name} » ` +
+                `(Probe « ${probe.data.name} ».${measure.label} impose déjà la mesure).`,
+            });
+          }
+        }
+      }
+    }
+  }
+
+  if (focus) {
+    for (const ap of actuatorParams(focus.data.type)) {
+      if (isParamFixed(focus.data.params ?? {}, ap)) continue;
+      const factor = ap.actuatorFactor!;
+      const probes = adjacentNodes(focus, nodes, edges).filter((n) => n.data.type === "Probe");
+      const hasEquation = probes.some((pr) =>
+        fixedProbeMeasures(pr).some((m) => m.factors.includes(factor)),
+      );
+      if (!hasEquation) {
+        hints.push({
+          level: "warn",
+          component: focus.data.name,
+          message:
+            `${factor} est Free (inconnue) : ajoute une Probe Fixed compatible ` +
+            `(${compatibleMeasureLabels(factor).join(", ") || "mesure"}) ` +
+            `sur le circuit, ou remets ${factor} Fixed.`,
+        });
+      }
+    }
+
+    // Baseline always visible — otherwise users see nothing when all Fixed (default).
+    if (hints.length === 0 && isCalibRelevantNode(focus)) {
+      if (focus.data.type === "Probe") {
+        const measures = fixedProbeMeasures(focus);
+        if (measures.length === 0) {
+          hints.push({
+            level: "tip",
+            component: focus.data.name,
+            message:
+              "Fixed/Free : coche Fixed sur une mesure (Tsat, P, T, Capacity…) " +
+              "puis libère le facteur compatible sur le composant adjacent " +
+              "(z_ua, z_dp, z_flow, f_w).",
+          });
+        }
+      } else {
+        const factors = actuatorParams(focus.data.type)
+          .map((p) => p.actuatorFactor)
+          .filter(Boolean)
+          .join(", ");
+        hints.push({
+          level: "tip",
+          component: focus.data.name,
+          message:
+            `Fixed/Free : facteurs ${factors || "Z"} — Fixed = connu, Free = inconnue. ` +
+            `Pour calibrer : Probe Fixed adjacent + décoche Fixed sur le facteur.`,
+        });
+      }
+    }
+  }
+
+  return hints;
+}
+
+function compatibleMeasureLabels(factor: string): string[] {
+  const labels: string[] = [];
+  for (const [kind, factors] of Object.entries(FACTOR_COMPATIBILITY)) {
+    if (factors.includes(factor)) labels.push(kind);
+  }
+  return labels;
+}
+
+/**
+ * One-click: Free every Fixed actuator on neighbours that is compatible with
+ * this Probe's Fixed measures. Returns patches for `updateNodesParams`.
+ */
+export function calibrateAdjacentActuatorPatches(
+  probeNode: Node<CalibNodeData>,
+  nodes: Node<CalibNodeData>[],
+  edges: Edge[],
+): CalibParamPatch[] {
+  if (probeNode.data.type !== "Probe") return [];
+  const measures = fixedProbeMeasures(probeNode);
+  if (measures.length === 0) return [];
+
+  const wanted = new Set(measures.flatMap((m) => [...m.factors]));
+  const byId = new Map<string, CalibParamPatch>();
+
+  for (const neigh of adjacentNodes(probeNode, nodes, edges)) {
+    if (neigh.data.type === "Probe") continue;
+    let next = { ...(neigh.data.params ?? {}) };
+    let changed = false;
+    for (const ap of actuatorParams(neigh.data.type)) {
+      const factor = ap.actuatorFactor!;
+      if (!wanted.has(factor)) continue;
+      if (!isParamFixed(next, ap)) continue;
+      next = {
+        ...next,
+        [fixedFlagKey(ap.key)]: false,
+        [ap.key]:
+          typeof next[ap.key] === "number"
+            ? next[ap.key]
+            : (ap.default as number | undefined) ?? 1.0,
+      };
+      changed = true;
+    }
+    if (changed) {
+      byId.set(neigh.id, { nodeId: neigh.id, params: next });
+    }
+  }
+
+  return [...byId.values()];
+}
+
+/** @deprecated Prefer {@link calibrateAdjacentActuatorPatches}. */
+export function calibrateAdjacentZUaPatch(
+  probeNode: Node<CalibNodeData>,
+  nodes: Node<CalibNodeData>[],
+  edges: Edge[],
+): CalibParamPatch | null {
+  const patches = calibrateAdjacentActuatorPatches(probeNode, nodes, edges);
+  const zUa = patches.find((p) => p.params[fixedFlagKey("z_ua")] === false);
+  return zUa ?? patches[0] ?? null;
+}
+
+/**
+ * Append derived UA_eff = UA × z_ua when both are known after solve.
+ */
+export function withDerivedUaEff(
+  componentName: string,
+  nodes: Node<CalibNodeData>[],
+  solved: SolvedVariable[],
+): SolvedVariable[] {
+  const node = nodes.find((n) => n.data.name === componentName);
+  if (!node) return solved;
+  const z = solved.find((s) => s.variable === "z_ua");
+  if (!z) return solved;
+  const uaRaw = node.data.params?.ua;
+  const ua = typeof uaRaw === "number" ? uaRaw : Number(uaRaw);
+  if (!Number.isFinite(ua) || ua <= 0) return solved;
+  if (solved.some((s) => s.variable === "UA_eff")) return solved;
+  return [
+    ...solved,
+    {
+      id: `${componentName}__UA_eff`,
+      component: componentName,
+      variable: "UA_eff",
+      value: ua * z.value,
+      min: 0,
+      max: ua * z.max,
+    },
+  ];
+}
diff --git a/apps/web/src/lib/configBuilder.test.ts b/apps/web/src/lib/configBuilder.test.ts
index ee70b24..6571d7d 100644
--- a/apps/web/src/lib/configBuilder.test.ts
+++ b/apps/web/src/lib/configBuilder.test.ts
@@ -505,7 +505,7 @@ describe("Modelica embeddings (Fixed / Free)", () => {
     expect(embeddings).toHaveLength(1);
     expect(embeddings[0]).toMatchObject({
       id: "emb_evap_z_ua",
-      unknown: { component: "evap", factor: "z_ua", start: 0.3, min: 0.05 },
+      unknown: { component: "evap", factor: "z_ua", start: 1.0, min: 0.05 },
       equation: {
         component: "sst_probe",
         output: "saturationTemperature",
@@ -519,7 +519,50 @@ describe("Modelica embeddings (Fixed / Free)", () => {
     const comp = cfg.circuits[0].components[0];
     expect(comp[fixedFlagKey("z_ua")]).toBeUndefined();
     expect(comp.z_ua).toBe(1.0);
-    expect(comp.ua).toBeUndefined(); // Free z_ua strips absolute ua
+    // Legacy HX: ua is the nominal base — keep it when Free z_ua.
+    expect(comp.ua).toBe(9000);
+  });
+
+  it("strips BPHX ua override when Z_UA is Free (geometry = UA_nominal)", () => {
+    const nodes = [
+      node("c", "BphxCondenser", "cond", 0, {
+        n_plates: 40,
+        ua: 2500,
+        z_ua: 1.0,
+        [fixedFlagKey("z_ua")]: false,
+      }),
+      node("p1", "Probe", "sdt_probe", 0, {
+        fluid: "R134a",
+        tsat_c: 42.0,
+        [fixedFlagKey("tsat_c")]: true,
+      }),
+    ];
+    const edges: Edge[] = [
+      { id: "pe", source: "c", target: "p1", sourceHandle: "inlet", targetHandle: "outlet" },
+    ];
+    const cfg = buildScenarioConfig(nodes, edges);
+    expect(cfg.embeddings).toHaveLength(1);
+    expect(cfg.circuits[0].components[0].ua).toBeUndefined();
+  });
+
+  it("fills missing required ua on FloodedEvaporator when Z_UA is Free", () => {
+    const nodes = [
+      node("e", "FloodedEvaporator", "evap", 0, {
+        // ua intentionally missing / cleared in the canvas
+        z_ua: 1.0,
+        [fixedFlagKey("z_ua")]: false,
+      }),
+      node("p1", "Probe", "sst_probe", 0, {
+        fluid: "R134a",
+        tsat_c: 5.0,
+        [fixedFlagKey("tsat_c")]: true,
+      }),
+    ];
+    const cfg = buildScenarioConfig(nodes, [
+      { id: "pe", source: "e", target: "p1", sourceHandle: "outlet", targetHandle: "inlet" },
+    ]);
+    expect(cfg.circuits[0].components[0].ua).toBe(8000);
+    expect(cfg.embeddings).toHaveLength(1);
   });
 
   it("emits no embedding when Z_UA stays Fixed even with a Probe present", () => {
diff --git a/apps/web/src/lib/configBuilder.ts b/apps/web/src/lib/configBuilder.ts
index 7affe84..7fe99b0 100644
--- a/apps/web/src/lib/configBuilder.ts
+++ b/apps/web/src/lib/configBuilder.ts
@@ -310,8 +310,11 @@ export function buildScenarioConfig(
   const circuits = Array.from(circuitsMap.entries())
     .sort(([a], [b]) => a - b)
     .map(([circuitId, cNodes]) => {
+      // Stable name order — React Flow node array order changes on drag/select.
       const components = cNodes
         .filter((n) => n.data.type !== CONTROL_NODE_TYPE)
+        .slice()
+        .sort((a, b) => a.data.name.localeCompare(b.data.name))
         .map((n) => {
           const { type, name, params } = n.data;
           const canonicalParams = canonicalizeParams(type, params);
@@ -394,9 +397,8 @@ export function buildScenarioConfig(
  * - `__fixed_*` Fixed checkbox flags
  * - measure-only targets (e.g. Probe `target`) — they become control setpoints
  * - emit Modelica-style `fix_pressure` / `fix_temperature` / `fix_mass_flow`
- * - when Z_UA is Free (calibration), omit literal `ua` override — the CLI
- *   otherwise bakes `ua` into a fixed Calib factor and ignores live `z_ua`,
- *   which zeros ∂measure/∂z_ua and blows up Newton (singular J → bad Picard state)
+ * - BPHX only: when Z_UA is Free, omit absolute `ua` override (geometry = UA_nominal)
+ * - fill missing *required* params from catalogue defaults (e.g. FloodedEvaporator.ua)
  */
 export function stripUiOnlyParams(
   type: string,
@@ -417,12 +419,25 @@ export function stripUiOnlyParams(
     out[k] = v;
   }
 
-  // Free z_ua → live embedding owns UA scaling; drop absolute `ua` override.
+  // Free z_ua on BPHX: drop absolute `ua` override (geometry supplies UA_nominal).
+  // Legacy Condenser / Evaporator / Flooded*: keep `ua` — constructor nominal.
   const zUaMeta = meta?.params.find((p) => p.key === "z_ua" && p.actuatorFactor === "z_ua");
-  if (zUaMeta && !isParamFixed(params, zUaMeta)) {
+  const bphxUaIsOverride =
+    type === "BphxCondenser" || type === "BphxEvaporator" || type === "BphxExchanger";
+  if (bphxUaIsOverride && zUaMeta && !isParamFixed(params, zUaMeta)) {
     delete out.ua;
   }
 
+  // Required catalogue defaults (safety net when the canvas field was cleared).
+  for (const p of meta?.params ?? []) {
+    if (!p.required || p.default === undefined) continue;
+    if (bphxUaIsOverride && p.key === "ua") continue;
+    const cur = out[p.key];
+    if (cur === undefined || cur === "") {
+      out[p.key] = p.default;
+    }
+  }
+
   return applyExvFixSemantics(type, applyBoundaryFixSemantics(type, out, params), params);
 }
 
@@ -596,13 +611,9 @@ export function buildModelEmbeddings(
 
       if (p.actuatorFactor && !fixed) {
         const n = typeof raw === "number" ? raw : Number(raw);
-        let initial = Number.isFinite(n) ? n : 1.0;
-        if (
-          (p.actuatorFactor === "z_ua" || p.actuatorFactor === "z_dp") &&
-          Math.abs(initial - 1.0) < 1e-12
-        ) {
-          initial = 0.3;
-        }
+        // Keep the UI value as Newton start (do not rewrite 1.0 → 0.3 — that
+        // pushed Free Z into a fragile basin and amplified HashMap-order flakes).
+        const initial = Number.isFinite(n) ? n : 1.0;
         freeActs.push({
           factor: p.actuatorFactor,
           initial,
@@ -614,6 +625,7 @@ export function buildModelEmbeddings(
     }
 
     if (freeActs.length > 0) {
+      freeActs.sort((a, b) => a.factor.localeCompare(b.factor));
       freeActsByComponent.set(node.data.name, { nodeName: node.data.name, node, acts: freeActs });
     }
   }
@@ -660,7 +672,10 @@ export function buildModelEmbeddings(
   };
 
   const usedKeys = new Set<string>();
-  for (const { nodeName, acts } of freeActsByComponent.values()) {
+  const freeEntries = [...freeActsByComponent.values()].sort((a, b) =>
+    a.nodeName.localeCompare(b.nodeName),
+  );
+  for (const { nodeName, acts } of freeEntries) {
     for (const act of acts) {
       const probe = findProbeFor(nodeName, act.factor, usedKeys);
       if (probe) {
@@ -683,6 +698,7 @@ export function buildModelEmbeddings(
     }
   }
 
+  embeddings.sort((a, b) => a.id.localeCompare(b.id));
   return embeddings;
 }
 
@@ -709,7 +725,7 @@ export function buildFixedFreeCalibrationControls(
 }
 
 /** Probe param key → semantic kind for pairing. */
-const PROBE_PARAM_KIND: Record<string, string> = {
+export const PROBE_PARAM_KIND: Record<string, string> = {
   t_c: "T",
   tsat_c: "Tsat",
   p_bar: "P",
@@ -779,8 +795,11 @@ function legacyProbeOutput(kind: string): string {
   }
 }
 
-/** Probe physical kind → free factors it can calibrate. */
-const FACTOR_COMPATIBILITY: Record<string, readonly string[]> = {
+/**
+ * Probe physical kind → free factors it can calibrate (Modelica Fixed ↔ Free).
+ * Shared by emit (`buildModelEmbeddings`) and UI aides (`calibAssist`).
+ */
+export const FACTOR_COMPATIBILITY: Record<string, readonly string[]> = {
   Tsat: ["z_ua"],
   Tsh: ["z_ua", "opening"],
   T: ["f_w"],
diff --git a/crates/cli/src/config.rs b/crates/cli/src/config.rs
index beabcee..12c8451 100644
--- a/crates/cli/src/config.rs
+++ b/crates/cli/src/config.rs
@@ -63,6 +63,12 @@ pub struct ScenarioConfig {
     /// (`output = value`). Distinct from [`Self::controls`] (no SaturatedController).
     #[serde(default)]
     pub embeddings: Vec<EmbeddingConfig>,
+    /// Algebraic model equations: `lhs - rhs = 0` with `+ - * / min max exp ln`.
+    ///
+    /// Free unknown is either inferred from `lhs` when it is `component.z_*`,
+    /// or declared via [`EquationConfig::unknown`] (Probe target form).
+    #[serde(default)]
+    pub equations: Vec<EquationConfig>,
     /// Reusable subsystem templates (parameterized assemblies of components +
     /// internal edges, exposing a reduced external port set). Flattened into
     /// `circuits` at load time — the solver never sees the hierarchy.
@@ -198,6 +204,56 @@ pub struct EmbeddingEquationConfig {
     pub value: f64,
 }
 
+/// Algebraic model equation with a minimal expression language.
+///
+/// ```text
+/// lhs = rhs;   // residual: lhs − rhs = 0
+/// ```
+///
+/// Supported ops: `+ - * /`, parentheses, unary `-`, `min`, `max`, `exp`, `ln`/`log`.
+/// References: `component.field` or `"Component Name".field`
+/// (`field` = `z_ua`, `T`, `Tsat`, `P`, …).
+///
+/// **Form A** — free Z-factor on the left:
+/// ```json
+/// { "id": "eq1", "lhs": "evap.z_ua", "rhs": "min(1, exp(probe.T/300))",
+///   "start": 0.3, "min": 0.05, "max": 2.0 }
+/// ```
+///
+/// **Form B** — Probe target with explicit unknown (same role as embeddings[]):
+/// ```json
+/// { "id": "eq2", "lhs": "probe.Tsat", "rhs": "278.15",
+///   "unknown": { "component": "evap", "factor": "z_ua", "start": 0.3, "min": 0.05, "max": 2.0 } }
+/// ```
+#[derive(Debug, Clone, Serialize, Deserialize, JsonSchema)]
+pub struct EquationConfig {
+    /// Unique id (also used as the residual / link id).
+    pub id: String,
+    /// Left-hand side expression.
+    pub lhs: String,
+    /// Right-hand side expression.
+    pub rhs: String,
+    /// Explicit free unknown (required when `lhs` is not itself a Z-factor ref).
+    #[serde(default)]
+    pub unknown: Option<EmbeddingUnknownConfig>,
+    /// Start guess when the unknown is inferred from `lhs` (Form A).
+    #[serde(default = "default_actuator_initial")]
+    pub start: f64,
+    /// Lower bound when the unknown is inferred from `lhs` (Form A).
+    #[serde(default = "default_equation_min")]
+    pub min: f64,
+    /// Upper bound when the unknown is inferred from `lhs` (Form A).
+    #[serde(default = "default_equation_max")]
+    pub max: f64,
+}
+
+fn default_equation_min() -> f64 {
+    0.05
+}
+fn default_equation_max() -> f64 {
+    2.0
+}
+
 /// A steady-state **system regulation** loop (EXV, injection, fan, …).
 ///
 /// Supports `SaturatedController`: saturated-PI with anti-windup, co-solved
@@ -1399,6 +1455,7 @@ mod tests {
             "circuits",
             "controls",
             "embeddings",
+            "equations",
             "subsystems",
             "instances",
             "connections",
@@ -1431,4 +1488,26 @@ mod tests {
         assert!((config.embeddings[0].equation.value - 278.15).abs() < 1e-9);
         assert!(config.controls.is_empty());
     }
+
+    #[test]
+    fn test_parse_equations() {
+        let json = r#"{
+            "schema_version": "2",
+            "fluid": "R134a",
+            "equations": [{
+                "id": "eq_evap_z_ua",
+                "lhs": "evap.z_ua",
+                "rhs": "min(1.0, exp(\"SST probe\".T / 300))",
+                "start": 0.3,
+                "min": 0.05,
+                "max": 2.0
+            }]
+        }"#;
+        let config = ScenarioConfig::from_json(json).unwrap();
+        assert_eq!(config.equations.len(), 1);
+        assert_eq!(config.equations[0].id, "eq_evap_z_ua");
+        assert_eq!(config.equations[0].lhs, "evap.z_ua");
+        assert!(config.equations[0].rhs.contains("min"));
+        assert!(config.equations[0].unknown.is_none());
+    }
 }
diff --git a/crates/cli/src/run.rs b/crates/cli/src/run.rs
index 345b5d1..5b86e1e 100644
--- a/crates/cli/src/run.rs
+++ b/crates/cli/src/run.rs
@@ -646,23 +646,46 @@ fn execute_simulation(
             }
         }
 
-        // Free z_ua embedding: drop absolute `ua` overrides on the unknown's
-        // component. Otherwise `ua` freezes Calib and ∂Q/∂z_ua → 0 (singular J).
-        for emb in &config.embeddings {
-            if entropyk_core::normalize_factor_name(emb.unknown.factor.trim())
-                == Some(entropyk_core::Z_UA)
-            {
-                if let Some(comp) = expanded_components
-                    .iter_mut()
-                    .find(|c| c.name == emb.unknown.component)
+        // Free z_ua on BPHX only: drop absolute `ua` override (geometry = UA_nominal).
+        // Legacy Condenser/Evaporator keep `ua` — it is the constructor nominal.
+        let mut free_z_ua: Vec<(String, String)> = config
+            .embeddings
+            .iter()
+            .filter(|e| {
+                entropyk_core::normalize_factor_name(e.unknown.factor.trim())
+                    == Some(entropyk_core::Z_UA)
+            })
+            .map(|e| (e.unknown.component.clone(), e.id.clone()))
+            .collect();
+        for eq in &config.equations {
+            if let Some(u) = eq.unknown.as_ref() {
+                if entropyk_core::normalize_factor_name(u.factor.trim()) == Some(entropyk_core::Z_UA)
                 {
-                    if comp.params.remove("ua").is_some() {
-                        tracing::info!(
-                            component = %comp.name,
-                            embedding = %emb.id,
-                            "Dropped 'ua' override — z_ua embedding owns UA scaling"
-                        );
-                    }
+                    free_z_ua.push((u.component.clone(), eq.id.clone()));
+                }
+            } else if let Ok(entropyk_solver::inverse::Expr::Ref { component, field }) =
+                entropyk_solver::inverse::parse_expr(&eq.lhs)
+            {
+                if entropyk_core::normalize_factor_name(&field) == Some(entropyk_core::Z_UA) {
+                    free_z_ua.push((component, eq.id.clone()));
+                }
+            }
+        }
+        for (comp_name, src_id) in &free_z_ua {
+            if let Some(comp) = expanded_components
+                .iter_mut()
+                .find(|c| c.name == *comp_name)
+            {
+                let is_bphx = matches!(
+                    comp.component_type.as_str(),
+                    "BphxCondenser" | "BphxEvaporator" | "BphxExchanger"
+                );
+                if is_bphx && comp.params.remove("ua").is_some() {
+                    tracing::info!(
+                        component = %comp.name,
+                        source = %src_id,
+                        "Dropped BPHX 'ua' override — free z_ua owns UA scaling"
+                    );
                 }
             }
         }
@@ -1151,6 +1174,34 @@ fn execute_simulation(
         }
     }
 
+    for eq in &config.equations {
+        match build_model_equation(eq) {
+            Ok((algebraic, bounded_var, unknown_id)) => {
+                if let Err(e) = system.add_algebraic_equation(algebraic) {
+                    return control_error(format!(
+                        "Failed to add algebraic equation '{}': {:?}",
+                        eq.id, e
+                    ));
+                }
+                if let Err(e) = system.add_bounded_variable(bounded_var) {
+                    return control_error(format!(
+                        "Failed to add unknown for equation '{}': {:?}",
+                        eq.id, e
+                    ));
+                }
+                if let Err(e) = system.link_constraint_to_control(
+                    &entropyk_solver::inverse::ConstraintId::new(eq.id.clone()),
+                    &unknown_id,
+                ) {
+                    return control_error(format!("Failed to link equation '{}': {:?}", eq.id, e));
+                }
+            }
+            Err(msg) => {
+                return control_error(format!("Invalid equation '{}': {}", eq.id, msg));
+            }
+        }
+    }
+
     for control in &config.controls {
         match build_saturated_control(control) {
             Ok((bounded_var, controller)) => {
@@ -1527,6 +1578,20 @@ fn execute_simulation(
             }
         }
     }
+    for eq in &config.equations {
+        if let Ok((_, _, unknown_id)) = build_model_equation(eq) {
+            let start = eq
+                .unknown
+                .as_ref()
+                .map(|u| u.start)
+                .unwrap_or(eq.start);
+            if let Some(u_idx) = system.control_variable_state_index(&unknown_id) {
+                if u_idx < initial_state.len() {
+                    initial_state[u_idx] = start;
+                }
+            }
+        }
+    }
     for control in &config.controls {
         let actuator_id = entropyk_solver::inverse::BoundedVariableId::new(saturated_actuator_id(
             &control.actuator.component,
@@ -1597,6 +1662,7 @@ fn execute_simulation(
         .then(|| std::time::Duration::from_millis(config.solver.timeout_ms));
     let needs_guarded_newton = !config.controls.is_empty()
         || !config.embeddings.is_empty()
+        || !config.equations.is_empty()
         || config.circuits.iter().any(|c| {
             c.enabled
                 && c.components.iter().any(|comp| {
@@ -3172,6 +3238,47 @@ fn bphx_calib_from_params(
     })
 }
 
+/// Calibration factors for legacy Condenser / Evaporator (`ua` is nominal, not override).
+fn hx_calib_from_params(
+    params: &std::collections::HashMap<String, serde_json::Value>,
+) -> CliResult<entropyk_core::Calib> {
+    use entropyk_core::Calib;
+    let z_ua = params
+        .get("z_ua")
+        .or_else(|| params.get("Z_UA"))
+        .or_else(|| params.get("f_ua"))
+        .and_then(|v| v.as_f64())
+        .unwrap_or(1.0);
+    if z_ua <= 0.0 {
+        return Err(CliError::Config(format!(
+            "z_ua must be > 0 (got {:.4})",
+            z_ua
+        )));
+    }
+    let z_dp = params
+        .get("z_dp")
+        .or_else(|| params.get("Z_dpc"))
+        .or_else(|| params.get("f_dp"))
+        .and_then(|v| v.as_f64())
+        .unwrap_or(1.0);
+    if z_dp <= 0.0 {
+        return Err(CliError::Config(format!(
+            "z_dp must be > 0 (got {:.4})",
+            z_dp
+        )));
+    }
+    Ok(Calib {
+        z_flow: 1.0,
+        z_flow_eco: 1.0,
+        z_dp,
+        z_ua,
+        z_power: 1.0,
+        z_etav: 1.0,
+        f_w: 1.0,
+        calibration_source: None,
+    })
+}
+
 /// Creates a pair of connected ports for components that need them (screw, MCHX, fan...).
 ///
 /// Ports are initialised at the given pressure and enthalpy. Both ports are connected
@@ -3300,6 +3407,73 @@ fn build_model_embedding(
     Ok((constraint, bounded_var, unknown_id))
 }
 
+/// Algebraic model equation (`lhs − rhs = 0`) + free Z-factor unknown.
+///
+/// Form A: `lhs` is `component.z_*` → unknown inferred, bounds from `start/min/max`.
+/// Form B: explicit `unknown` block (Probe target: `lhs=probe.Tsat`, `rhs=315`).
+fn build_model_equation(
+    eq: &crate::config::EquationConfig,
+) -> Result<
+    (
+        entropyk_solver::inverse::AlgebraicEquation,
+        entropyk_solver::inverse::BoundedVariable,
+        entropyk_solver::inverse::BoundedVariableId,
+    ),
+    String,
+> {
+    use entropyk_solver::inverse::{
+        parse_expr, AlgebraicEquation, BoundedVariable, BoundedVariableId, Expr,
+    };
+
+    let lhs = parse_expr(&eq.lhs).map_err(|e| format!("lhs: {e}"))?;
+    let rhs = parse_expr(&eq.rhs).map_err(|e| format!("rhs: {e}"))?;
+
+    let (comp, factor, start, min, max) = if let Some(u) = eq.unknown.as_ref() {
+        let factor = u.factor.trim();
+        if entropyk_core::normalize_factor_name(factor).is_none() {
+            return Err(format!(
+                "unknown Z-factor '{factor}' (expected z_ua, z_dp, z_flow, z_power, z_etav, f_w, …)"
+            ));
+        }
+        (
+            u.component.clone(),
+            factor.to_string(),
+            u.start,
+            u.min,
+            u.max,
+        )
+    } else if let Expr::Ref { component, field } = &lhs {
+        let canon = entropyk_core::normalize_factor_name(field).ok_or_else(|| {
+            format!(
+                "lhs '{component}.{field}' is not a Z-factor — set equations[].unknown \
+                 (e.g. Probe target form) or use lhs = component.z_ua"
+            )
+        })?;
+        (
+            component.clone(),
+            canon.to_string(),
+            eq.start,
+            eq.min,
+            eq.max,
+        )
+    } else {
+        return Err(
+            "lhs must be component.z_* or provide equations[].unknown for the free Z-factor"
+                .into(),
+        );
+    };
+
+    let unknown_id = BoundedVariableId::new(saturated_actuator_id(&comp, &factor));
+    let bounded_var = BoundedVariable::with_component(unknown_id.clone(), &comp, start, min, max)
+        .map_err(|e| format!("invalid unknown bounds: {e:?}"))?;
+    let algebraic = AlgebraicEquation {
+        id: eq.id.clone(),
+        lhs,
+        rhs,
+    };
+    Ok((algebraic, bounded_var, unknown_id))
+}
+
 fn build_saturated_control(
     control: &crate::config::ControlConfig,
 ) -> Result<
@@ -3596,7 +3770,17 @@ fn create_component(
         "FloodedEvaporator" => {
             use entropyk::FloodedEvaporator;
 
-            let ua = get_param_f64(params, "ua")?;
+            // UA is the nominal base (required for construction). Default matches
+            // the UI catalogue when the canvas field was left empty during calib.
+            let ua = params
+                .get("ua")
+                .and_then(|v| v.as_f64())
+                .unwrap_or(8000.0);
+            if ua < 0.0 {
+                return Err(CliError::Config(format!(
+                    "FloodedEvaporator: ua must be >= 0 (got {ua})"
+                )));
+            }
             let target_quality = params
                 .get("target_quality")
                 .and_then(|v| v.as_f64())
@@ -3931,6 +4115,12 @@ fn create_component(
                 cond = cond.with_flooded_head_pressure(target_k);
             }
 
+            // Fixed z_ua (or start hint): scale UA_eff = z_ua · UA. Live Free
+            // embeddings override via CalibIndices during solve.
+            if let Ok(calib) = hx_calib_from_params(params) {
+                cond.set_calib(calib);
+            }
+
             Ok(Box::new(cond))
         }
 
@@ -4024,6 +4214,10 @@ fn create_component(
                 evap = evap.with_regulated_superheat();
             }
 
+            if let Ok(calib) = hx_calib_from_params(params) {
+                evap.set_calib(calib);
+            }
+
             Ok(Box::new(evap))
         }
 
diff --git a/crates/components/src/heat_exchanger/condenser.rs b/crates/components/src/heat_exchanger/condenser.rs
index 0fade68..a1dcb43 100644
--- a/crates/components/src/heat_exchanger/condenser.rs
+++ b/crates/components/src/heat_exchanger/condenser.rs
@@ -617,7 +617,19 @@ impl Condenser {
                         FLOOD_LAMBDA_HI,
                         FLOOD_LAMBDA_WIDTH,
                     );
-                    jacobian.add_entry(row, act_idx, self.ua() * c.delta_t * c.e_exp * dlam);
+                    let ua_nom = self.inner.ua_nominal() * self.live_z_ua(state);
+                    jacobian.add_entry(row, act_idx, ua_nom * c.delta_t * c.e_exp * dlam);
+                }
+                // Live z_ua on secondary energy: r = ṁ·Δh − Q ⇒ ∂r/∂z = −∂Q/∂z
+                if let Some(z_ua_idx) = self.inner.calib_indices_ref().z_ua {
+                    let flood_scale = if self.flood_ready() {
+                        1.0 - self.flooded_level(state)
+                    } else {
+                        1.0
+                    };
+                    let d_q_dz =
+                        self.inner.ua_nominal() * flood_scale * c.e_exp * c.delta_t;
+                    jacobian.add_entry(row, z_ua_idx, -d_q_dz);
                 }
             }
             None => {
@@ -782,14 +794,27 @@ impl Condenser {
         }
     }
 
-    /// Effective conductance `UA_eff` [W/K] used by the coupled duty. With an
-    /// active flooding actuator this is `(1 − λ)·UA_nominal`; otherwise the
-    /// nominal `UA`.
+    /// Live calibration factor `z_ua` from Newton state when free, else calib param.
+    fn live_z_ua(&self, state: &StateSlice) -> f64 {
+        self.inner
+            .calib_indices_ref()
+            .z_ua
+            .and_then(|idx| state.get(idx).copied())
+            .unwrap_or_else(|| self.calib().z_ua)
+            .max(0.0)
+    }
+
+    /// Effective conductance `UA_eff` [W/K] used by the coupled duty.
+    ///
+    /// `UA_eff = UA_nominal · z_ua · (1 − λ)` when flooding is active, else
+    /// `UA_nominal · z_ua`. Reads live `state[z_ua]` when an embedding frees it
+    /// (otherwise ∂Q/∂z_ua = 0 → singular Jacobian).
     fn effective_ua(&self, state: &StateSlice) -> f64 {
+        let ua = self.inner.ua_nominal() * self.live_z_ua(state);
         if self.flood_ready() {
-            self.ua() * (1.0 - self.flooded_level(state))
+            ua * (1.0 - self.flooded_level(state))
         } else {
-            self.ua()
+            ua
         }
     }
 
@@ -1592,7 +1617,7 @@ impl Component for Condenser {
                 // ∂r1/∂λ = +UA_nom·(T_cond − T_sec,in)·e · dλ_eff/dλ.
                 if self.flood_ready() {
                     if let Some(lvl_idx) = self.fan_actuator_idx {
-                        let ua_nom = self.ua();
+                        let ua_nom = self.inner.ua_nominal() * self.live_z_ua(state);
                         let e = if c_sec > 1e-10 {
                             (-ua_eff / c_sec).exp()
                         } else {
@@ -1608,6 +1633,24 @@ impl Component for Condenser {
                     }
                 }
 
+                // Live z_ua: UA = UA_nom·z_ua·(1−λ), ε = 1−e^(−UA/C), Q = ε·C·ΔT
+                // ⇒ ∂Q/∂z_ua = UA_nom·(1−λ)·e·ΔT  ⇒  ∂r_energy/∂z_ua = −∂Q/∂z_ua
+                if let Some(z_ua_idx) = self.inner.calib_indices_ref().z_ua {
+                    let e = if c_sec > 1e-10 {
+                        (-ua_eff / c_sec).exp()
+                    } else {
+                        0.0
+                    };
+                    let flood_scale = if self.flood_ready() {
+                        1.0 - self.flooded_level(state)
+                    } else {
+                        1.0
+                    };
+                    let d_q_dz =
+                        self.inner.ua_nominal() * flood_scale * e * (t_cond - t_sec_in);
+                    jacobian.add_entry(row, z_ua_idx, -d_q_dz);
+                }
+
                 // r2 (emergent) = H_out − h_target(P_in): ∂/∂H_out = 1,
                 // ∂/∂P_in = −dh_target/dP via central finite difference.
                 if self.emergent_pressure {
diff --git a/crates/components/src/heat_exchanger/evaporator.rs b/crates/components/src/heat_exchanger/evaporator.rs
index 67df734..5bd7715 100644
--- a/crates/components/src/heat_exchanger/evaporator.rs
+++ b/crates/components/src/heat_exchanger/evaporator.rs
@@ -551,6 +551,18 @@ impl Evaporator {
                 );
                 // ∂r/∂P_ref_in = ∂Q/∂P = −g·dT_evap/dP.
                 jacobian.add_entry(row, c.ref_p_in_idx, -c.g * c.dtevap_dp);
+                // Live z_ua: secondary r = ṁ·Δh + Q ⇒ ∂r/∂z = +∂Q/∂z
+                if let Some(z_ua_idx) = self.inner.calib_indices_ref().z_ua {
+                    let ua = self.live_ua(Some(state));
+                    let c_sec = state[m_in].abs() * c.cp_sec;
+                    let e = if c_sec > 1e-10 && ua > 0.0 {
+                        (-ua / c_sec).exp()
+                    } else {
+                        0.0
+                    };
+                    let d_q_dz = self.inner.ua_nominal() * e * c.delta_t;
+                    jacobian.add_entry(row, z_ua_idx, d_q_dz);
+                }
             }
             None => {
                 jacobian.add_entry(row, h_out, 1.0);
@@ -797,10 +809,22 @@ impl Evaporator {
             .map_err(|e| ComponentError::CalculationFailed(e.to_string()))
     }
 
+    /// Live UA = UA_nominal × z_ua (reads `state[calib_indices.z_ua]` when free).
+    fn live_ua(&self, state: Option<&StateSlice>) -> f64 {
+        let z = state
+            .and_then(|st| {
+                self.inner
+                    .calib_indices_ref()
+                    .z_ua
+                    .and_then(|idx| st.get(idx).copied())
+            })
+            .unwrap_or_else(|| self.calib().z_ua);
+        self.inner.ua_nominal() * z.max(0.0)
+    }
+
     /// Effectiveness for a phase-changing refrigerant (`C_min = C_sec`, `C_r → 0`):
     /// `ε = 1 − exp(−UA / C_sec)`.
-    fn effectiveness(&self, c_sec: f64) -> f64 {
-        let ua = self.ua();
+    fn effectiveness(&self, c_sec: f64, ua: f64) -> f64 {
         if c_sec <= 1e-10 || ua <= 0.0 {
             return 0.0;
         }
@@ -815,7 +839,8 @@ impl Evaporator {
         let c_sec = self.secondary_capacity_rate.unwrap_or(0.0);
         let t_sec_in = self.secondary_inlet_temp_k.unwrap_or(0.0);
         let t_evap = self.evap_temperature(p_in_pa)?;
-        let eps = self.effectiveness(c_sec);
+        let ua = self.live_ua(None);
+        let eps = self.effectiveness(c_sec, ua);
         Ok(eps * c_sec * (t_sec_in - t_evap))
     }
 
@@ -1053,7 +1078,8 @@ impl Component for Evaporator {
                 // Live secondary stream: edge-driven in 4-port mode (Modelica).
                 let (t_sec_in, c_sec) = self.live_secondary_stream(state)?;
                 let t_evap = self.evap_temperature(p_in)?;
-                let eps = self.effectiveness(c_sec);
+                let ua = self.live_ua(Some(state));
+                let eps = self.effectiveness(c_sec, ua);
                 let q = eps * c_sec * (t_sec_in - t_evap);
 
                 // r0: refrigerant pressure drop (tube MSH/Friedel + accel, or
@@ -1266,7 +1292,8 @@ impl Component for Evaporator {
                 // ∂r1/∂P_in = −∂Q/∂P_in = ε·C_sec·dT_evap/dP_in (T_sec,in constant),
                 // dT_evap/dP via central finite difference.
                 let (t_sec_in, c_sec) = self.live_secondary_stream(state)?;
-                let eps = self.effectiveness(c_sec);
+                let ua = self.live_ua(Some(state));
+                let eps = self.effectiveness(c_sec, ua);
                 let g = eps * c_sec;
                 let t_evap = self.evap_temperature(p_in)?;
                 let dp = p_in * 1e-4 + 100.0;
@@ -1275,6 +1302,17 @@ impl Component for Evaporator {
                 let dt_dp = (t_plus - t_minus) / (2.0 * dp);
                 jacobian.add_entry(row, inlet_p_idx, g * dt_dp);
 
+                // Live z_ua: UA = UA_nom·z_ua, Q = ε·C·ΔT ⇒ ∂r_energy/∂z = −∂Q/∂z
+                if let Some(z_ua_idx) = self.inner.calib_indices_ref().z_ua {
+                    let e = if c_sec > 1e-10 {
+                        (-ua / c_sec).exp()
+                    } else {
+                        0.0
+                    };
+                    let d_q_dz = self.inner.ua_nominal() * e * (t_sec_in - t_evap);
+                    jacobian.add_entry(row, z_ua_idx, -d_q_dz);
+                }
+
                 // 4-port cross-derivatives of r1 to the secondary edge state:
                 // ∂r1/∂h_sec,in = −∂Q/∂h_sec,in = −g·dT_sec/dh (exact 1/cp),
                 // ∂r1/∂ṁ_sec    = −∂Q/∂ṁ_sec    = −g'(C_sec)·cp·(T_sec,in − T_evap).
@@ -1282,7 +1320,6 @@ impl Component for Evaporator {
                     let (m_s, p_s, h_s) = self.sec_in_idx.unwrap();
                     let cp_sec = self.sec_cp(state[p_s], state[h_s])?;
                     let dt_dh = 1.0 / cp_sec;
-                    let ua = self.ua();
                     let g_prime = if c_sec <= 1e-10 || ua <= 0.0 {
                         0.0
                     } else {
@@ -2084,7 +2121,7 @@ mod tests {
         let c_sec = state[6] * cp_air;
         let t_air_in = (state[8] - 2_501_000.0 * w) / cp_air + 273.15;
         let t_evap = evap.evap_temperature(state[1]).unwrap();
-        let eps = evap.effectiveness(c_sec);
+        let eps = evap.effectiveness(c_sec, evap.ua());
         let q = eps * c_sec * (t_air_in - t_evap);
         assert!(q > 0.0, "evaporator must absorb heat: q={q}");
         let expected = state[6] * (state[11] - state[8]) + q;
diff --git a/crates/solver/src/inverse/bounded.rs b/crates/solver/src/inverse/bounded.rs
index 485f0d7..9556e6c 100644
--- a/crates/solver/src/inverse/bounded.rs
+++ b/crates/solver/src/inverse/bounded.rs
@@ -42,7 +42,7 @@ use thiserror::Error;
 /// Type-safe identifier for a bounded control variable.
 ///
 /// Uses a string internally but provides type safety and clear intent.
-#[derive(Debug, Clone, PartialEq, Eq, Hash)]
+#[derive(Debug, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)]
 pub struct BoundedVariableId(String);
 
 impl BoundedVariableId {
diff --git a/crates/solver/src/inverse/constraint.rs b/crates/solver/src/inverse/constraint.rs
index 6a50f61..727eea0 100644
--- a/crates/solver/src/inverse/constraint.rs
+++ b/crates/solver/src/inverse/constraint.rs
@@ -16,7 +16,7 @@ use thiserror::Error;
 /// Type-safe identifier for a constraint.
 ///
 /// Uses a string internally but provides type safety and clear intent.
-#[derive(Debug, Clone, PartialEq, Eq, Hash)]
+#[derive(Debug, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)]
 pub struct ConstraintId(String);
 
 impl ConstraintId {
diff --git a/crates/solver/src/inverse/embedding.rs b/crates/solver/src/inverse/embedding.rs
index 08840c7..904a068 100644
--- a/crates/solver/src/inverse/embedding.rs
+++ b/crates/solver/src/inverse/embedding.rs
@@ -334,19 +334,24 @@ impl InverseControlConfig {
         self.control_to_constraint.get(bounded_variable_id)
     }
 
-    /// Returns an iterator over all constraint-to-control mappings.
+    /// Returns constraint→control mappings in **stable** order (sorted by constraint id).
+    ///
+    /// HashMap iteration is randomized per process; residual rows, Jacobian rows/columns,
+    /// and control state indices must share one deterministic order or Newton flakes.
     pub fn mappings(&self) -> impl Iterator<Item = (&ConstraintId, &BoundedVariableId)> {
-        self.constraint_to_control.iter()
+        let mut items: Vec<_> = self.constraint_to_control.iter().collect();
+        items.sort_by(|a, b| a.0.cmp(b.0));
+        items.into_iter()
     }
 
-    /// Returns an iterator over linked constraint IDs.
+    /// Linked constraint IDs in the same order as [`Self::mappings`].
     pub fn linked_constraints(&self) -> impl Iterator<Item = &ConstraintId> {
-        self.constraint_to_control.keys()
+        self.mappings().map(|(c, _)| c)
     }
 
-    /// Returns an iterator over linked control variable IDs.
+    /// Linked control IDs in the same order as [`Self::mappings`] (column / state layout).
     pub fn linked_controls(&self) -> impl Iterator<Item = &BoundedVariableId> {
-        self.control_to_constraint.keys()
+        self.mappings().map(|(_, c)| c)
     }
 
     /// Checks if a constraint is linked.
@@ -583,15 +588,18 @@ mod tests {
     fn test_inverse_control_config_mappings_iterator() {
         let mut config = InverseControlConfig::new();
 
+        // Insert out of alphabetical order — iteration must still be sorted.
         config
-            .link(make_constraint_id("c1"), make_bounded_var_id("v1"))
+            .link(make_constraint_id("z_probe"), make_bounded_var_id("z_ua_cond"))
             .unwrap();
         config
-            .link(make_constraint_id("c2"), make_bounded_var_id("v2"))
+            .link(make_constraint_id("a_probe"), make_bounded_var_id("z_ua_evap"))
             .unwrap();
 
-        let mappings: Vec<_> = config.mappings().collect();
-        assert_eq!(mappings.len(), 2);
+        let ids: Vec<&str> = config.mappings().map(|(c, _)| c.as_str()).collect();
+        assert_eq!(ids, vec!["a_probe", "z_probe"]);
+        let controls: Vec<&str> = config.linked_controls().map(|c| c.as_str()).collect();
+        assert_eq!(controls, vec!["z_ua_evap", "z_ua_cond"]);
     }
 
     #[test]
diff --git a/crates/solver/src/inverse/expr.rs b/crates/solver/src/inverse/expr.rs
new file mode 100644
index 0000000..105986e
--- /dev/null
+++ b/crates/solver/src/inverse/expr.rs
@@ -0,0 +1,401 @@
+//! Minimal Modelica-style algebraic expression AST for model equations.
+//!
+//! Supports: `+ - * /`, parentheses, unary `-`, `min`, `max`, `exp`, `ln`/`log`,
+//! numeric literals, and references `component.field` or `"Component Name".field`.
+
+use std::fmt;
+
+/// Parsed expression node.
+#[derive(Debug, Clone, PartialEq)]
+pub enum Expr {
+    Const(f64),
+    /// `component` + `field` (field is temperature, z_ua, …).
+    Ref { component: String, field: String },
+    Neg(Box<Expr>),
+    Add(Box<Expr>, Box<Expr>),
+    Sub(Box<Expr>, Box<Expr>),
+    Mul(Box<Expr>, Box<Expr>),
+    Div(Box<Expr>, Box<Expr>),
+    Min(Box<Expr>, Box<Expr>),
+    Max(Box<Expr>, Box<Expr>),
+    Exp(Box<Expr>),
+    Ln(Box<Expr>),
+}
+
+impl Expr {
+    /// Evaluate with a resolver for `component.field` references.
+    pub fn eval<F>(&self, resolve: &mut F) -> Result<f64, String>
+    where
+        F: FnMut(&str, &str) -> Result<f64, String>,
+    {
+        match self {
+            Expr::Const(v) => Ok(*v),
+            Expr::Ref { component, field } => resolve(component, field),
+            Expr::Neg(a) => Ok(-a.eval(resolve)?),
+            Expr::Add(a, b) => Ok(a.eval(resolve)? + b.eval(resolve)?),
+            Expr::Sub(a, b) => Ok(a.eval(resolve)? - b.eval(resolve)?),
+            Expr::Mul(a, b) => Ok(a.eval(resolve)? * b.eval(resolve)?),
+            Expr::Div(a, b) => {
+                let den = b.eval(resolve)?;
+                if den.abs() < 1e-30 {
+                    return Err("division by zero in model equation".into());
+                }
+                Ok(a.eval(resolve)? / den)
+            }
+            Expr::Min(a, b) => Ok(a.eval(resolve)?.min(b.eval(resolve)?)),
+            Expr::Max(a, b) => Ok(a.eval(resolve)?.max(b.eval(resolve)?)),
+            Expr::Exp(a) => {
+                let x = a.eval(resolve)?;
+                let y = x.exp();
+                if y.is_finite() {
+                    Ok(y)
+                } else {
+                    Err(format!("exp({x}) overflow"))
+                }
+            }
+            Expr::Ln(a) => {
+                let x = a.eval(resolve)?;
+                if x <= 0.0 {
+                    return Err(format!("ln({x}) domain error"));
+                }
+                Ok(x.ln())
+            }
+        }
+    }
+
+    /// Collect all `component.field` references.
+    pub fn collect_refs(&self, out: &mut Vec<(String, String)>) {
+        match self {
+            Expr::Const(_) => {}
+            Expr::Ref { component, field } => out.push((component.clone(), field.clone())),
+            Expr::Neg(a) | Expr::Exp(a) | Expr::Ln(a) => a.collect_refs(out),
+            Expr::Add(a, b)
+            | Expr::Sub(a, b)
+            | Expr::Mul(a, b)
+            | Expr::Div(a, b)
+            | Expr::Min(a, b)
+            | Expr::Max(a, b) => {
+                a.collect_refs(out);
+                b.collect_refs(out);
+            }
+        }
+    }
+}
+
+/// Parse an expression string into an [`Expr`].
+pub fn parse_expr(input: &str) -> Result<Expr, String> {
+    let mut p = Parser::new(input);
+    let e = p.parse_expr()?;
+    p.skip_ws();
+    if p.pos < p.chars.len() {
+        return Err(format!(
+            "trailing junk at '{}'",
+            p.chars[p.pos..].iter().collect::<String>()
+        ));
+    }
+    Ok(e)
+}
+
+struct Parser {
+    chars: Vec<char>,
+    pos: usize,
+}
+
+impl Parser {
+    fn new(s: &str) -> Self {
+        Self {
+            chars: s.chars().collect(),
+            pos: 0,
+        }
+    }
+
+    fn skip_ws(&mut self) {
+        while self.pos < self.chars.len() && self.chars[self.pos].is_whitespace() {
+            self.pos += 1;
+        }
+    }
+
+    fn peek(&self) -> Option<char> {
+        self.chars.get(self.pos).copied()
+    }
+
+    fn bump(&mut self) -> Option<char> {
+        let c = self.peek()?;
+        self.pos += 1;
+        Some(c)
+    }
+
+    fn parse_expr(&mut self) -> Result<Expr, String> {
+        let mut left = self.parse_term()?;
+        loop {
+            self.skip_ws();
+            match self.peek() {
+                Some('+') => {
+                    self.bump();
+                    let right = self.parse_term()?;
+                    left = Expr::Add(Box::new(left), Box::new(right));
+                }
+                Some('-') => {
+                    self.bump();
+                    let right = self.parse_term()?;
+                    left = Expr::Sub(Box::new(left), Box::new(right));
+                }
+                _ => break,
+            }
+        }
+        Ok(left)
+    }
+
+    fn parse_term(&mut self) -> Result<Expr, String> {
+        let mut left = self.parse_unary()?;
+        loop {
+            self.skip_ws();
+            match self.peek() {
+                Some('*') => {
+                    self.bump();
+                    let right = self.parse_unary()?;
+                    left = Expr::Mul(Box::new(left), Box::new(right));
+                }
+                Some('/') => {
+                    self.bump();
+                    let right = self.parse_unary()?;
+                    left = Expr::Div(Box::new(left), Box::new(right));
+                }
+                _ => break,
+            }
+        }
+        Ok(left)
+    }
+
+    fn parse_unary(&mut self) -> Result<Expr, String> {
+        self.skip_ws();
+        if self.peek() == Some('-') {
+            self.bump();
+            let e = self.parse_unary()?;
+            return Ok(Expr::Neg(Box::new(e)));
+        }
+        if self.peek() == Some('+') {
+            self.bump();
+            return self.parse_unary();
+        }
+        self.parse_primary()
+    }
+
+    fn parse_primary(&mut self) -> Result<Expr, String> {
+        self.skip_ws();
+        match self.peek() {
+            Some('(') => {
+                self.bump();
+                let e = self.parse_expr()?;
+                self.skip_ws();
+                if self.bump() != Some(')') {
+                    return Err("expected ')'".into());
+                }
+                Ok(e)
+            }
+            Some('"') => self.parse_quoted_ref(),
+            Some(c) if c.is_ascii_digit() || c == '.' => self.parse_number(),
+            Some(c) if is_ident_start(c) => self.parse_ident_or_call(),
+            Some(c) => Err(format!("unexpected '{c}'")),
+            None => Err("unexpected end of expression".into()),
+        }
+    }
+
+    fn parse_number(&mut self) -> Result<Expr, String> {
+        let start = self.pos;
+        while let Some(c) = self.peek() {
+            if c.is_ascii_digit() || c == '.' || c == 'e' || c == 'E' {
+                self.bump();
+                if (c == 'e' || c == 'E') && matches!(self.peek(), Some('+') | Some('-')) {
+                    self.bump();
+                }
+            } else {
+                break;
+            }
+        }
+        let s: String = self.chars[start..self.pos].iter().collect();
+        s.parse::<f64>()
+            .map(Expr::Const)
+            .map_err(|_| format!("invalid number '{s}'"))
+    }
+
+    fn parse_quoted_ref(&mut self) -> Result<Expr, String> {
+        self.bump(); // "
+        let start = self.pos;
+        while let Some(c) = self.peek() {
+            if c == '"' {
+                break;
+            }
+            self.bump();
+        }
+        let name: String = self.chars[start..self.pos].iter().collect();
+        if self.bump() != Some('"') {
+            return Err("unterminated quoted component name".into());
+        }
+        self.skip_ws();
+        if self.bump() != Some('.') {
+            return Err("expected '.' after quoted component name".into());
+        }
+        let field = self.parse_ident_string()?;
+        Ok(Expr::Ref {
+            component: name,
+            field,
+        })
+    }
+
+    fn parse_ident_string(&mut self) -> Result<String, String> {
+        self.skip_ws();
+        let start = self.pos;
+        if !self.peek().is_some_and(is_ident_start) {
+            return Err("expected identifier".into());
+        }
+        self.bump();
+        while self.peek().is_some_and(is_ident_cont) {
+            self.bump();
+        }
+        Ok(self.chars[start..self.pos].iter().collect())
+    }
+
+    fn parse_ident_or_call(&mut self) -> Result<Expr, String> {
+        let ident = self.parse_ident_string()?;
+        self.skip_ws();
+        if self.peek() == Some('(') {
+            return self.parse_call(&ident);
+        }
+        // component.field[.more] — last segment is field, rest is component
+        let mut parts = vec![ident];
+        while self.peek() == Some('.') {
+            self.bump();
+            parts.push(self.parse_ident_string()?);
+            self.skip_ws();
+        }
+        if parts.len() < 2 {
+            return Err(format!(
+                "bare identifier '{}' — expected component.field or function call",
+                parts[0]
+            ));
+        }
+        let field = parts.pop().unwrap();
+        let component = parts.join(".");
+        Ok(Expr::Ref { component, field })
+    }
+
+    fn parse_call(&mut self, name: &str) -> Result<Expr, String> {
+        self.bump(); // (
+        self.skip_ws();
+        let lower = name.to_ascii_lowercase();
+        match lower.as_str() {
+            "min" | "max" => {
+                let a = self.parse_expr()?;
+                self.skip_ws();
+                if self.bump() != Some(',') {
+                    return Err(format!("expected ',' in {lower}(...)"));
+                }
+                let b = self.parse_expr()?;
+                self.skip_ws();
+                if self.bump() != Some(')') {
+                    return Err(format!("expected ')' after {lower}"));
+                }
+                Ok(if lower == "min" {
+                    Expr::Min(Box::new(a), Box::new(b))
+                } else {
+                    Expr::Max(Box::new(a), Box::new(b))
+                })
+            }
+            "exp" => {
+                let a = self.parse_expr()?;
+                self.skip_ws();
+                if self.bump() != Some(')') {
+                    return Err("expected ')' after exp".into());
+                }
+                Ok(Expr::Exp(Box::new(a)))
+            }
+            "ln" | "log" => {
+                let a = self.parse_expr()?;
+                self.skip_ws();
+                if self.bump() != Some(')') {
+                    return Err("expected ')' after ln".into());
+                }
+                Ok(Expr::Ln(Box::new(a)))
+            }
+            other => Err(format!(
+                "unknown function '{other}' (supported: min, max, exp, ln)"
+            )),
+        }
+    }
+}
+
+fn is_ident_start(c: char) -> bool {
+    c.is_ascii_alphabetic() || c == '_'
+}
+
+fn is_ident_cont(c: char) -> bool {
+    c.is_ascii_alphanumeric() || c == '_'
+}
+
+impl fmt::Display for Expr {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        match self {
+            Expr::Const(v) => write!(f, "{v}"),
+            Expr::Ref { component, field } => write!(f, "{component}.{field}"),
+            Expr::Neg(a) => write!(f, "-({a})"),
+            Expr::Add(a, b) => write!(f, "({a} + {b})"),
+            Expr::Sub(a, b) => write!(f, "({a} - {b})"),
+            Expr::Mul(a, b) => write!(f, "({a} * {b})"),
+            Expr::Div(a, b) => write!(f, "({a} / {b})"),
+            Expr::Min(a, b) => write!(f, "min({a}, {b})"),
+            Expr::Max(a, b) => write!(f, "max({a}, {b})"),
+            Expr::Exp(a) => write!(f, "exp({a})"),
+            Expr::Ln(a) => write!(f, "ln({a})"),
+        }
+    }
+}
+
+/// Algebraic model equation: `lhs - rhs = 0`.
+#[derive(Debug, Clone)]
+pub struct AlgebraicEquation {
+    pub id: String,
+    pub lhs: Expr,
+    pub rhs: Expr,
+}
+
+impl AlgebraicEquation {
+    pub fn residual<F>(&self, mut resolve: F) -> Result<f64, String>
+    where
+        F: FnMut(&str, &str) -> Result<f64, String>,
+    {
+        Ok(self.lhs.eval(&mut resolve)? - self.rhs.eval(&mut resolve)?)
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    fn eval_const(e: &Expr) -> f64 {
+        e.eval(&mut |_, _| Err("no refs".into())).unwrap()
+    }
+
+    #[test]
+    fn arithmetic_and_funcs() {
+        assert!((eval_const(&parse_expr("1 + 2 * 3").unwrap()) - 7.0).abs() < 1e-12);
+        assert!((eval_const(&parse_expr("min(2, 5)").unwrap()) - 2.0).abs() < 1e-12);
+        assert!((eval_const(&parse_expr("max(2, 5)").unwrap()) - 5.0).abs() < 1e-12);
+        assert!((eval_const(&parse_expr("exp(0)").unwrap()) - 1.0).abs() < 1e-12);
+        assert!((eval_const(&parse_expr("ln(exp(1))").unwrap()) - 1.0).abs() < 1e-9);
+        assert!((eval_const(&parse_expr("-(1+2)").unwrap()) + 3.0).abs() < 1e-12);
+    }
+
+    #[test]
+    fn refs_and_quoted() {
+        let e = parse_expr(r#"evap.z_ua + "SST probe".T"#).unwrap();
+        let v = e
+            .eval(&mut |c, f| match (c, f) {
+                ("evap", "z_ua") => Ok(0.1),
+                ("SST probe", "T") => Ok(280.0),
+                _ => Err("bad".into()),
+            })
+            .unwrap();
+        assert!((v - 280.1).abs() < 1e-12);
+    }
+}
diff --git a/crates/solver/src/inverse/mod.rs b/crates/solver/src/inverse/mod.rs
index f0b7020..3d9f6b7 100644
--- a/crates/solver/src/inverse/mod.rs
+++ b/crates/solver/src/inverse/mod.rs
@@ -45,6 +45,7 @@ pub mod bounded;
 pub mod calibration;
 pub mod constraint;
 pub mod embedding;
+pub mod expr;
 pub mod override_network;
 pub mod saturated_control;
 
@@ -58,5 +59,6 @@ pub use calibration::{
 };
 pub use constraint::{ComponentOutput, Constraint, ConstraintError, ConstraintId};
 pub use embedding::{ControlMapping, DoFError, InverseControlConfig};
+pub use expr::{parse_expr, AlgebraicEquation, Expr};
 pub use override_network::{eval_error_signal, eval_error_weights, Combine, Objective};
 pub use saturated_control::{SaturatedControlError, SaturatedController, Saturation};
diff --git a/crates/solver/src/system.rs b/crates/solver/src/system.rs
index 75a1bdb..3b07c74 100644
--- a/crates/solver/src/system.rs
+++ b/crates/solver/src/system.rs
@@ -16,7 +16,7 @@ use petgraph::algo;
 use petgraph::graph::{EdgeIndex, Graph, NodeIndex};
 use petgraph::visit::EdgeRef;
 use petgraph::Directed;
-use std::collections::HashMap;
+use std::collections::{BTreeSet, HashMap};
 
 use crate::coupling::{has_circular_dependencies, ThermalCoupling};
 use crate::dof::{
@@ -208,6 +208,8 @@ pub struct System {
     thermal_couplings: Vec<ThermalCoupling>,
     /// Constraints for inverse control (output - target = 0)
     constraints: HashMap<ConstraintId, Constraint>,
+    /// Modelica algebraic equations (`lhs - rhs = 0`), e.g. `evap.z_ua = f(...)`.
+    algebraic_equations: Vec<crate::inverse::AlgebraicEquation>,
     /// Bounded control variables for inverse control (with box constraints)
     bounded_variables: HashMap<BoundedVariableId, BoundedVariable>,
     /// Inverse control configuration (constraint → control variable mappings)
@@ -246,6 +248,7 @@ impl System {
             node_to_circuit: HashMap::new(),
             thermal_couplings: Vec::new(),
             constraints: HashMap::new(),
+            algebraic_equations: Vec::new(),
             bounded_variables: HashMap::new(),
             inverse_control: InverseControlConfig::new(),
             saturated_controllers: Vec::new(),
@@ -735,18 +738,20 @@ impl System {
             }
         }
 
-        if !self.constraints.is_empty() {
+        if !self.constraints.is_empty() || !self.algebraic_equations.is_empty() {
             match self.validate_inverse_control_dof() {
                 Ok(()) => {
                     tracing::debug!(
-                        constraint_count = self.constraints.len(),
+                        constraint_count =
+                            self.constraints.len() + self.algebraic_equations.len(),
                         control_count = self.inverse_control.mapping_count(),
                         "Inverse control DoF validation passed"
                     );
                 }
                 Err(DoFError::UnderConstrainedSystem { .. }) => {
                     tracing::warn!(
-                        constraint_count = self.constraints.len(),
+                        constraint_count =
+                            self.constraints.len() + self.algebraic_equations.len(),
                         control_count = self.inverse_control.mapping_count(),
                         "Under-constrained inverse control system - solver may still converge"
                     );
@@ -1240,6 +1245,21 @@ impl System {
         Ok(())
     }
 
+    /// Adds a Modelica algebraic equation residual (`lhs − rhs = 0`).
+    pub fn add_algebraic_equation(
+        &mut self,
+        equation: crate::inverse::AlgebraicEquation,
+    ) -> Result<(), ConstraintError> {
+        let id = ConstraintId::new(equation.id.clone());
+        if self.constraints.contains_key(&id)
+            || self.algebraic_equations.iter().any(|e| e.id == equation.id)
+        {
+            return Err(ConstraintError::DuplicateId { id });
+        }
+        self.algebraic_equations.push(equation);
+        Ok(())
+    }
+
     /// Removes a constraint by ID.
     ///
     /// # Arguments
@@ -1259,9 +1279,16 @@ impl System {
         self.constraints.len()
     }
 
-    /// Returns a reference to all constraints.
+    /// Number of Modelica algebraic equations (`lhs − rhs = 0`).
+    pub fn algebraic_equation_count(&self) -> usize {
+        self.algebraic_equations.len()
+    }
+
+    /// Returns all constraints in **stable** order (sorted by constraint id).
     pub fn constraints(&self) -> impl Iterator<Item = &Constraint> {
-        self.constraints.values()
+        let mut items: Vec<_> = self.constraints.values().collect();
+        items.sort_by(|a, b| a.id().cmp(b.id()));
+        items.into_iter()
     }
 
     /// Returns a reference to a specific constraint by ID.
@@ -1317,8 +1344,12 @@ impl System {
             return Ok(0);
         }
 
+        // Must match Jacobian row order (`inverse_control.mappings()` / sorted ids).
+        let mut sorted: Vec<_> = self.constraints.values().collect();
+        sorted.sort_by(|a, b| a.id().cmp(b.id()));
+
         let mut count = 0;
-        for constraint in self.constraints.values() {
+        for constraint in sorted {
             let measured = match measured_values.get(constraint.id()).copied() {
                 Some(v) => v,
                 None => {
@@ -2076,7 +2107,12 @@ impl System {
         constraint_id: &ConstraintId,
         bounded_variable_id: &BoundedVariableId,
     ) -> Result<(), DoFError> {
-        if !self.constraints.contains_key(constraint_id) {
+        let has_constraint = self.constraints.contains_key(constraint_id);
+        let has_algebraic = self
+            .algebraic_equations
+            .iter()
+            .any(|e| e.id == constraint_id.as_str());
+        if !has_constraint && !has_algebraic {
             return Err(DoFError::ConstraintNotFound {
                 constraint_id: constraint_id.clone(),
             });
@@ -2191,7 +2227,7 @@ impl System {
     pub fn validate_inverse_control_dof(&self) -> Result<(), DoFError> {
         let n_edge_unknowns = self.total_state_len;
         let n_controls = self.inverse_control.mapping_count();
-        let n_constraints = self.constraints.len();
+        let n_constraints = self.constraints.len() + self.algebraic_equations.len();
         let n_unknowns = n_edge_unknowns + n_controls;
 
         let n_edge_eqs: usize = self
@@ -2242,6 +2278,7 @@ impl System {
             .sum();
         n_comp
             + self.constraints.len()
+            + self.algebraic_equations.len()
             + self.coupling_residual_count()
             + 2 * self.saturated_controllers.len()
     }
@@ -2319,6 +2356,11 @@ impl System {
                 name: id.to_string(),
             });
         }
+        for eq in &self.algebraic_equations {
+            system_equations.push(EquationRole::ControlTracking {
+                name: eq.id.clone(),
+            });
+        }
         for _ in 0..self.coupling_residual_count() {
             system_equations.push(EquationRole::CouplingDuty);
         }
@@ -2701,6 +2743,7 @@ impl System {
             .map(|(_, c, _)| c.n_equations())
             .sum();
         total_eqs += self.constraints.len()
+            + self.algebraic_equations.len()
             + self.coupling_residual_count()
             + 2 * self.saturated_controllers.len();
 
@@ -2734,6 +2777,15 @@ impl System {
             .map_err(|e| ComponentError::CalculationFailed(e.to_string()))?;
         eq_offset += n_constraints;
 
+        // Modelica algebraic equations: lhs − rhs = 0
+        for (i, eq) in self.algebraic_equations.iter().enumerate() {
+            let r = eq
+                .residual(|comp, field| self.resolve_expr_ref(comp, field, state))
+                .map_err(ComponentError::CalculationFailed)?;
+            residuals[eq_offset + i] = r;
+        }
+        eq_offset += self.algebraic_equations.len();
+
         // Add couplings. Each coupling owns one unknown Q = state[coupling_state_index(i)].
         //
         // * Physical mode (`hot_component`/`cold_component` set): the residual
@@ -2836,12 +2888,19 @@ impl System {
             jacobian.add_entry(r, c, v);
         }
 
+        let algebraic_row_offset = row_offset + self.constraints.len();
+        let algebraic_jac = self.compute_algebraic_equation_jacobian(state, algebraic_row_offset);
+        for (r, c, v) in algebraic_jac {
+            jacobian.add_entry(r, c, v);
+        }
+
         // Thermal coupling rows: r_i = Q_i − η·duty_hot(state).
         // ∂r/∂Q = 1 always (also fixes the legacy stub's singular row); in
         // physical mode the plant coupling ∂r/∂col = −η·∂duty/∂col is formed by
         // central finite differences over the hot component's incident states
         // (same pattern as the saturated-controller measurement Jacobian).
-        let coupling_row_offset = row_offset + self.constraints.len();
+        let coupling_row_offset =
+            row_offset + self.constraints.len() + self.algebraic_equations.len();
         if !self.thermal_couplings.is_empty() {
             let eps = self.inverse_control.finite_diff_epsilon();
             let mut state_mut = state.to_vec();
@@ -2882,8 +2941,10 @@ impl System {
             }
         }
 
-        let saturated_row_offset =
-            row_offset + self.constraints.len() + self.coupling_residual_count();
+        let saturated_row_offset = row_offset
+            + self.constraints.len()
+            + self.algebraic_equations.len()
+            + self.coupling_residual_count();
         let saturated_jac = self.compute_saturated_control_jacobian(state, saturated_row_offset);
         for (r, c, v) in saturated_jac {
             jacobian.add_entry(r, c, v);
@@ -2891,6 +2952,147 @@ impl System {
         Ok(())
     }
 
+    /// Resolve `component.field` for algebraic model equations.
+    ///
+    /// Fields may be Z-factors (`z_ua`, …) or measured outputs (`T`, `Tsat`, …).
+    pub fn resolve_expr_ref(
+        &self,
+        component: &str,
+        field: &str,
+        state: &StateSlice,
+    ) -> Result<f64, String> {
+        if let Some(canon) = entropyk_core::normalize_factor_name(field) {
+            let indices = self.calib_indices_by_name.get(component).ok_or_else(|| {
+                format!("unknown component '{component}' for Z-factor '{field}'")
+            })?;
+            let idx = match canon {
+                entropyk_core::Z_FLOW => indices.z_flow,
+                entropyk_core::Z_FLOW_ECO => indices.z_flow_eco,
+                entropyk_core::Z_DP => indices.z_dp,
+                entropyk_core::Z_UA => indices.z_ua,
+                entropyk_core::Z_POWER => indices.z_power,
+                entropyk_core::Z_ETAV => indices.z_etav,
+                entropyk_core::F_W => indices.f_w,
+                _ => None,
+            }
+            .ok_or_else(|| {
+                format!("Z-factor '{field}' is not a free unknown on component '{component}'")
+            })?;
+            return state.get(idx).copied().ok_or_else(|| {
+                format!("state index {idx} out of range for {component}.{field}")
+            });
+        }
+
+        let kind = Self::measured_output_from_field(field).ok_or_else(|| {
+            format!(
+                "unknown field '{field}' on '{component}' \
+                 (expected z_*, T/temperature, Tsat/SST/SDT, P/pressure, …)"
+            )
+        })?;
+        let &node_idx = self.component_names.get(component).ok_or_else(|| {
+            format!("unknown component '{component}'")
+        })?;
+        let component_ref = self.graph.node_weight(node_idx).ok_or_else(|| {
+            format!("component '{component}' missing from graph")
+        })?;
+        component_ref
+            .measure_output(kind, state)
+            .filter(|v| v.is_finite())
+            .ok_or_else(|| {
+                format!("measure '{field}' unavailable on component '{component}'")
+            })
+    }
+
+    fn measured_output_from_field(field: &str) -> Option<entropyk_components::MeasuredOutput> {
+        use entropyk_components::MeasuredOutput as MO;
+        let n = field.trim().to_ascii_lowercase().replace(['_', '-'], "");
+        Some(match n.as_str() {
+            "t" | "temperature" => MO::Temperature,
+            "tsat" | "sst" | "sdt" | "saturationtemperature" => MO::SaturationTemperature,
+            "p" | "pressure" => MO::Pressure,
+            "m" | "mdot" | "massflow" | "massflowrate" => MO::MassFlowRate,
+            "q" | "capacity" => MO::Capacity,
+            "duty" | "heattransferrate" | "heatrate" => MO::HeatTransferRate,
+            "superheat" | "tsh" | "sh" => MO::Superheat,
+            "subcooling" | "tsc" | "sc" => MO::Subcooling,
+            _ => return None,
+        })
+    }
+
+    fn compute_algebraic_equation_jacobian(
+        &self,
+        state: &StateSlice,
+        row_offset: usize,
+    ) -> Vec<(usize, usize, f64)> {
+        let mut entries = Vec::new();
+        if self.algebraic_equations.is_empty() {
+            return entries;
+        }
+        let eps = self.inverse_control.finite_diff_epsilon();
+        if state.len() < self.full_state_vector_len() {
+            tracing::error!(
+                state_len = state.len(),
+                required = self.full_state_vector_len(),
+                "compute_algebraic_equation_jacobian: state slice too short"
+            );
+            return entries;
+        }
+        let mut state_mut = state.to_vec();
+
+        for (i, eq) in self.algebraic_equations.iter().enumerate() {
+            let row = row_offset + i;
+            let mut refs = Vec::new();
+            eq.lhs.collect_refs(&mut refs);
+            eq.rhs.collect_refs(&mut refs);
+
+            let mut cols = BTreeSet::new();
+            for (comp, field) in &refs {
+                if let Some(canon) = entropyk_core::normalize_factor_name(field) {
+                    if let Some(indices) = self.calib_indices_by_name.get(comp) {
+                        let idx = match canon {
+                            entropyk_core::Z_FLOW => indices.z_flow,
+                            entropyk_core::Z_FLOW_ECO => indices.z_flow_eco,
+                            entropyk_core::Z_DP => indices.z_dp,
+                            entropyk_core::Z_UA => indices.z_ua,
+                            entropyk_core::Z_POWER => indices.z_power,
+                            entropyk_core::Z_ETAV => indices.z_etav,
+                            entropyk_core::F_W => indices.f_w,
+                            _ => None,
+                        };
+                        if let Some(idx) = idx {
+                            cols.insert(idx);
+                        }
+                    }
+                }
+                if let Some(&node_idx) = self.component_names.get(comp) {
+                    for col in self.incident_state_indices_for_component(node_idx) {
+                        cols.insert(col);
+                    }
+                }
+            }
+
+            for col in cols {
+                let orig = state_mut[col];
+                state_mut[col] = orig + eps;
+                let r_plus = eq
+                    .residual(|c, f| self.resolve_expr_ref(c, f, &state_mut))
+                    .unwrap_or(f64::NAN);
+                state_mut[col] = orig - eps;
+                let r_minus = eq
+                    .residual(|c, f| self.resolve_expr_ref(c, f, &state_mut))
+                    .unwrap_or(f64::NAN);
+                state_mut[col] = orig;
+                if r_plus.is_finite() && r_minus.is_finite() {
+                    let dr = (r_plus - r_minus) / (2.0 * eps);
+                    if dr.abs() > 1e-10 {
+                        entries.push((row, col, dr));
+                    }
+                }
+            }
+        }
+        entries
+    }
+
     /// Tolerance for mass balance validation [kg/s].
     ///
     /// This value (1e-9 kg/s) is tight enough to catch numerical issues