diff --git a/.gitignore b/.gitignore
index 9f7550b..14a1b3f 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,2 +1,3 @@
 __pycache__
 .venv
+*.pdf
diff --git a/index.html b/index.html
index 9fa757a..a0c8f80 100644
--- a/index.html
+++ b/index.html
@@ -162,10 +162,6 @@
             display: flex;
         }
 
-        .card.log-on .linear-only {
-            opacity: .4;
-        }
-
         /* Output tables: a CSS grid kept inside a scroll container so it never
      stretches the page wider than it should. */
         .gtable-wrap {
@@ -406,13 +402,20 @@
         }
 
         // --- cities ---
+        // Pick a name not already used by an existing city: the lowest non-negative
+        // integer (as a string) that's free, matching the seeded "0", "1", … names.
+        function uniqueCityName() {
+            const taken = new Set(
+                [...document.getElementById("cities").children].map(r => r._get().name));
+            for (let n = 0; ; n++) if (!taken.has(String(n))) return String(n);
+        }
         function addCity(c = {}) {
             const card = el("div", {class: "card"});
-            const name = el("input", {value: c.name || ""});
+            const name = el("input", {value: c.name || uniqueCityName()});
             const type = selectEl(CITY_TYPES, c.type || "hub");
             const renown = num(c.renown ?? "", {min: 1, placeholder: "auto"});
-            const vs = num(c.vat_steel || 0, {min: 0}), vb = num(c.vat_brass || 0, {min: 0}),
-                ve = num(c.vat_electrum || 0, {min: 0});
+            const vs = num(c.vat_steel || 1, {min: 0}), vb = num(c.vat_brass || 1, {min: 0}),
+                ve = num(c.vat_electrum || 1, {min: 0});
             const reno = el("input", {type: "checkbox"}); reno.checked = c.can_renovate !== false;
             const adjacent = el("input", {
                 value: (c.adjacent || []).join(", "),
@@ -515,7 +518,6 @@
             };
             card.append(
                 field("Agent", type),
-                field("Name", name),
                 field("Effect", desc),
                 field("Bastions (Baron only)", bastions),
                 field("Forced city (turn:city, csv)", forced),
@@ -535,6 +537,7 @@
             const toggle = () => card.classList.toggle("log-on", isLog.checked);
             isLog.onchange = toggle;
             const expr = el("textarea", {rows: 1, placeholder: "(x) => Math.log2(x + 1)"});
+            expr.innerText = "(x) => Math.log2(x + 1)";
             if (t._expr) expr.value = t._expr;
             card._get = () => {
                 const o = {resource: res.value, scalar: +scalar.value};
@@ -555,6 +558,10 @@
             document.getElementById("terms").append(card);
             toggle();
         }
+        function addTerms(terms = {}) {
+            Object.entries(terms).forEach(([resource, scalar]) =>
+                addTerm({resource, scalar}));
+        }
 
         // Eval the expression ONCE into a function, then call it over the amounts
         // the lookup table needs (0..max_resource).
@@ -666,7 +673,7 @@
             out.append(el("h2", {}, "Solution"));
             out.append(el("p", {
                 html:
-                    `<b>Status:</b> ${s.status} &nbsp; <b>Objective:</b> ${s.objective_value} &nbsp; ` +
+                    `<b>Status:</b> ${s.status} &nbsp; <b>Objective:</b> ${s.objective_value ?? "—"} &nbsp; ` +
                     `<b>Final renown total:</b> ${s.final_renown_total}`
             }));
 
@@ -711,19 +718,19 @@
         }
 
         // --- seed with the example problem ---
-        addCity({name: "Aridias", type: "hub", renown: 2, adjacent: ["Bearhearth"]});
-        addCity({name: "Bearhearth", type: "foundry", renown: 2, vat_steel: 3, vat_brass: 2, vat_electrum: 1, adjacent: ["Kingsland"]});
-        addCity({name: "Kingsland", type: "metropolis", renown: 4, can_renovate: false, adjacent: ["Roseward"]});
-        addCity({name: "Roseward", type: "monument", renown: 2});
+        addCity({name: "0", type: "hub", renown: 2});
+        addCity({name: "1", type: "foundry", renown: 2, vat_steel: 1, vat_brass: 1, vat_electrum: 1});
+        addCity({name: "2", type: "hub", renown: 2});
+        addCity({name: "3", type: "foundry", renown: 2});
+        addCity({name: "4", type: "monument", renown: 2});
         addAgent({kind: "Planner"});
-        Object.entries({
+        addTerms({
             "renown": 0,
             "luxuries": 1,
             "steel": 2,
             "brass": 1,
             "electrum": 2
-        }).forEach(([resource, scalar]) =>
-            addTerm({resource, scalar}));
+        })
     </script>
 </body>
 
diff --git a/solve.py b/solve.py
index 011754e..5eda9d9 100644
--- a/solve.py
+++ b/solve.py
@@ -394,7 +394,7 @@ class CityTurnPlan:
 @dataclass
 class Solution:
     status: str
-    objective_value: float
+    objective_value: float | None
     final_resources: dict[str, float]
     final_renown_total: int
     plan: list[CityTurnPlan] = field(default_factory=list)
@@ -1277,7 +1277,7 @@ def solve(problem: Problem, max_time_seconds: float = 30.0,
 
     if status not in (cp_model.OPTIMAL, cp_model.FEASIBLE):
         return Solution(
-            status=status_name, objective_value=float("nan"),
+            status=status_name, objective_value=None,
             final_resources={}, final_renown_total=0, plan=[],
         )