feat(backend): extract shared FDI tooth geometry

Voice extraction needs quadrant mapping and adjacency server-side, and
treatment.utils.ts already held a private copy of the tooth set. Lift it into
common/fdi.ts rather than create a second source of truth; treatment.utils now
imports it, behaviour unchanged (existing suites still pass).

toFdi() is the single place the patient-right convention lives: quadrant 1 is
the patient's upper right, so upper+patient_right -> 1x, upper+patient_left ->
2x, lower+patient_left -> 3x, lower+patient_right -> 4x. Getting this backwards
mirrors every quadrant and yields a valid-looking code for the wrong tooth,
which no schema check can catch — so all four quadrants are pinned by tests,
along with out-of-range positions never being clamped and deciduous teeth being
rejected outright (the chart is permanent dentition only).

Adjacency mirrors the frontend's arch-order rule, so the midline pairs 11-21
and 41-31 count as neighbours exactly as the chart treats them.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>

backend/src/common/fdi.ts

@@ -0,0 +1,120 @@
+/**
+ * FDI tooth geometry — permanent dentition only.
+ *
+ * Mirrors `frontend/src/components/treatment/fdiToothMeta.ts` and the adjacency rules in
+ * `toothSelectionGroups.ts`. Adjacency is defined by position in the arch order, so the
+ * midline pairs (11–21, 41–31) are neighbours, exactly as the chart treats them.
+ */
+
+export type Arch = 'upper' | 'lower';
+
+/** Which side of the *patient*, not of the screen. Quadrant 1 is the patient's upper right. */
+export type PatientSide = 'patient_right' | 'patient_left';
+
+/**
+ * Upper arch in chart order: patient's RIGHT (18) → midline → patient's LEFT (28).
+ * That is the drawn left-to-right layout, which is the mirror of the patient's own sides.
+ * Do not read a tooth position off this array by index — use `toFdi()`, which owns the
+ * side convention.
+ */
+export const FDI_UPPER_ARCH_ORDER = [
+  '18',
+  '17',
+  '16',
+  '15',
+  '14',
+  '13',
+  '12',
+  '11',
+  '21',
+  '22',
+  '23',
+  '24',
+  '25',
+  '26',
+  '27',
+  '28',
+] as const;
+
+/** Lower arch, same chart ordering: patient's RIGHT (48) → midline → patient's LEFT (38). */
+export const FDI_LOWER_ARCH_ORDER = [
+  '48',
+  '47',
+  '46',
+  '45',
+  '44',
+  '43',
+  '42',
+  '41',
+  '31',
+  '32',
+  '33',
+  '34',
+  '35',
+  '36',
+  '37',
+  '38',
+] as const;
+
+export const FDI_TOOTH_IDS: ReadonlySet<string> = new Set<string>([
+  ...FDI_UPPER_ARCH_ORDER,
+  ...FDI_LOWER_ARCH_ORDER,
+]);
+
+export function isFdiTooth(value: unknown): value is string {
+  return typeof value === 'string' && FDI_TOOTH_IDS.has(value);
+}
+
+function archOrder(tooth: string): readonly string[] | null {
+  if ((FDI_UPPER_ARCH_ORDER as readonly string[]).includes(tooth))
+    return FDI_UPPER_ARCH_ORDER;
+  if ((FDI_LOWER_ARCH_ORDER as readonly string[]).includes(tooth))
+    return FDI_LOWER_ARCH_ORDER;
+  return null;
+}
+
+export function sameArch(a: string, b: string): boolean {
+  const archA = archOrder(a);
+  const archB = archOrder(b);
+  return Boolean(archA && archB && archA === archB);
+}
+
+export function areArchNeighbors(a: string, b: string): boolean {
+  const arch = archOrder(a);
+  if (!arch || !sameArch(a, b)) return false;
+  return Math.abs(arch.indexOf(a) - arch.indexOf(b)) === 1;
+}
+
+/** Inclusive span between two teeth of the same arch, in arch order. Null if not comparable. */
+export function teethBetweenInclusive(a: string, b: string): string[] | null {
+  const arch = archOrder(a);
+  if (!arch || !sameArch(a, b)) return null;
+  const i = arch.indexOf(a);
+  const j = arch.indexOf(b);
+  if (i < 0 || j < 0) return null;
+  const [from, to] = i <= j ? [i, j] : [j, i];
+  return [...arch.slice(from, to + 1)];
+}
+
+/**
+ * Arch + patient side + position (1 = central incisor … 8 = third molar) → FDI code.
+ *
+ * This function is the single place the patient-right convention lives. Getting it
+ * backwards mirrors every quadrant and produces a valid-looking code for the wrong tooth,
+ * which no schema check can catch — hence the exhaustive test coverage.
+ */
+export function toFdi(
+  arch: Arch,
+  side: PatientSide,
+  position: number,
+): string | null {
+  if (!Number.isInteger(position) || position < 1 || position > 8) return null;
+  let quadrant: number;
+  if (arch === 'upper') {
+    quadrant = side === 'patient_right' ? 1 : 2;
+  } else {
+    quadrant = side === 'patient_left' ? 3 : 4;
+  }
+  const code = `${quadrant}${position}`;
+  return FDI_TOOTH_IDS.has(code) ? code : null;
+}