free-recall explain tool: the quizzing-mechanic switch (owner 2026-09-27)

Sycophancy mitigation as a structural commitment: the model writes its
full correctAnswer (plus explanation, in that order) BEFORE the learner
sees the question; the commitment rides the persisted call args, is
revealed on submit, and grades his free text in the model's next reply —
the client never grades free recall.

- tools.explain setting (segmented 'Quizzing — Multiple choice | Free
  recall' in Agent settings' Tools group): free arms explain and drops
  quiz for BOTH agents (prober variant carries no nodeId, same pattern
  as the probe quiz); mechanics keys never pass filterEnabledTools —
  quiz stays unwritable (the 2026-09-19 wedge ruling stands)
- doctrine lives pi-natively in the tool descriptions (owner ruling —
  no conditional system-prompt chunks): explain carries the commitment
  rule; the MC option-building craft moved out of the prompt into the
  quiz description; shared/20-question-tools.md slims to the
  quiz-agnostic core
- display seams: parseExplainInput (LaTeX-debris restoration included),
  dock text-mode with an 'I don't know' row (kbd row 0), AnsweredCard
  reveal (The answer / Why, no verdict chips), [Explain answer] format
- phases/plan seams recognize both quizzing tools' nodeId tags; explain
  answers deliberately feed no ledger evidence (no client grading to
  key on — a future ruling can parse the model's grade)

1049/1049 tests, typecheck/lint/build green.

components/settings/agent-section.tsx

@@ -66,6 +66,7 @@ import {
 } from "@/src/core";
 import type { ThinkingLevel } from "@earendil-works/pi-agent-core";
 import { prettyModel } from "@/components/usage-format";
+import { cn } from "@/lib/utils";
 import { SettingsGroup, SettingsRow, Toggle } from "./controls.tsx";
 import { ModelMenuList, ThinkingMenuList } from "./model-menu";
 
@@ -110,6 +111,15 @@ export function composeTool(agent: AgentSettings | undefined, id: string, on: bo
 	return { ...agent, tools: { ...agent?.tools, [id]: on } };
 }
 
+/** The quizzing mechanics (owner 2026-09-27): a segmented control, not a
+ *  toggle — exactly one mechanic is ever active. `explain: true` arms
+ *  free-recall (explain in, quiz out); anything else is multiple choice.
+ *  `quiz` itself is never written (mechanics-managed — see quizMechanics
+ *  in agent/tools.ts). Pure — exported for tests. */
+export function composeQuizMechanics(agent: AgentSettings | undefined, mechanics: "choice" | "free"): AgentSettings {
+	return { ...agent, tools: { ...agent?.tools, explain: mechanics === "free" } };
+}
+
 /** Compose the prober graph-tools toggle (wave 7): its own key in the
  *  agent section — never in the tool-toggles map (owner ruling). Absent =
  *  ON (the toggle discipline); the write is explicit either way, so an
@@ -725,6 +735,46 @@ export function AgentSection({
 						</SettingsRow>
 					);
 				})()}
+				{/* The quizzing mechanic (owner 2026-09-27): segmented — exactly one
+				    of Multiple choice / Free recall is active; the choice swaps which
+				    quizzing tool BOTH agents carry (prober checks + teach checks).
+				    Persisted as agent.tools.explain; absent = Multiple choice. */}
+				{(() => {
+					const free = agent?.tools?.explain === true;
+					return (
+						<SettingsRow
+							label="Quizzing"
+							hint="How both tutors check your understanding"
+							testid="quiz-mechanics-row"
+						>
+							<div
+								role="radiogroup"
+								aria-label="Quizzing mechanics"
+								className="flex rounded-lg border border-border/60 bg-muted/30 p-0.5"
+								>
+								{(["choice", "free"] as const).map((m) => {
+									const active = m === "free" ? free : !free;
+									return (
+										<button
+											key={m}
+											type="button"
+											role="radio"
+											aria-checked={active}
+											data-testid={`quiz-mechanics-${m}`}
+											onClick={() => onSectionChange(composeQuizMechanics(agent, m))}
+											className={cn(
+												"h-8 rounded-md px-3 text-sm font-medium transition-colors",
+												active ? "bg-background text-foreground shadow-sm" : "text-muted-foreground hover:text-foreground",
+											)}
+										>
+											{m === "choice" ? "Multiple choice" : "Free recall"}
+										</button>
+									);
+								})}
+							</div>
+						</SettingsRow>
+					);
+				})()}
 				{TOGGLE_TOOLS.map((id) => {
 					// Research needs a live search engine; the row + toggle gray
 					// on the same signal the registry drops the tool on (see