diff --git a/components/chain-table.tsx b/components/chain-table.tsx
index c540b08..ec02122 100644
--- a/components/chain-table.tsx
+++ b/components/chain-table.tsx
@@ -21,26 +21,15 @@
  * (vertical column hairlines removed with them, owner ruling: "anything
  * else is better without them").
  *
- * THE CHAIN LINE: every proposition name is a faint-outlined CHIP with
- * the → a clearly separated connector between chips — a lonely chip
- * reads as a standalone proposition (the "(no premises)" annotation is
- * dead in the visible UI; it survives only as the row's aria-label).
- * Icons: ONLY the cluster FolderIcon survives (owner ruling, evening
- * round 3: the chain waypoints icons are not needed — the chips +
- * connectors already read as chains); propositions chip-only. Chevrons
- * rotate with expansion. Statement rows show propositions PLAIN (owner
- * ruling: the derivation is already encoded in the chain line — no
- * conclusion marker down there).
- *
- * THE ALIGNMENT STAIRCASE (owner ruling, round 3): every level's
- * checkbox sits ONE COLUMN under its parent's expander — aligned with
- * the parent's ICON slot: open a folder and the chain checkboxes line
- * up with the FOLDER icon; the statements step one more unit out (the
- * would-be chain-icon column — the chains lost their icons but the
- * pattern continues, owner: "double the indentation"). Geometry: every
- * column is 14px + 4px gap = 18px and each step is TWO columns (36px) —
- * clusters pl-3 (12px), chains pl-12 (48px = the cluster's folder
- * icon), statements pl-21 (84px = where a chain icon WOULD sit).
+ * THE CHAIN LINE: every proposition name is a faint-outlined CHIP
+ * with the LOGIC as the connectives (owner rulings, 2026-09-30):
+ * premises join with the LaTeX LAND (∧) and the conclusion separates
+ * by the LaTeX IMPLICATION (⟹) — `A ∧ B ⟹ C`, never the misleading
+ * `A → B → C`. The glyphs are REAL KaTeX renders (components/
+ * latex-glyph.tsx — the app's math engine, never SVGs), PROMINENT in
+ * the theme foreground (white), not dim gray. A lonely chip (no
+ * premises) carries no glyph — it reads as a standalone proposition;
+ * "(no premises)" survives only as the row's aria-label.
  *
  * SORTING: all three columns sortable (owner ruling — "consistency is
  * key"): Name (A→Z), Props (largest-first), Created (newest-first);
@@ -83,6 +72,7 @@ import type { GraphEdge, GraphNode } from "@/src/core";
 import { TopBar } from "@/components/top-bar";
 import { PracticeDialog } from "@/components/practice-dialog";
 import { DeleteNodeDialog } from "@/components/delete-node-dialog";
+import { LatexGlyph } from "@/components/latex-glyph";
 
 /** The sortable columns (owner ruling: all three — consistency). */
 export type ChainSortCol = "name" | "props" | "created";
@@ -405,20 +395,42 @@ export function ChainTableBody({
 												/>
 												<Expander open={chainOpen} onClick={() => onToggleChain(chain.id)} testid={`chain-toggle-${chain.id}`} label={`Expand chain ${nameOf(chain.conclusion)}`} />
 												{/* The chain line: proposition names as faint-outlined
-												    chips, → the connector BETWEEN them (owner ruling) — a
-												    lonely chip reads as a standalone proposition. */}
+												    chips. THE LOGIC IS THE PAYLOAD (owner rulings,
+												    2026-09-30): premises join with the LaTeX LAND (∧),
+												    the conclusion separates by the LaTeX IMPLICATION
+												    (⟹) — `A ∧ B ⟹ C`, never the misleading `A → B →
+												    C` — the glyphs render through KaTeX (the app's
+												    math engine), PROMINENT in the theme foreground
+												    (white), not the old dim gray. A lonely chip reads
+												    as a standalone proposition (no glyphs at all);
+												    the row's aria-label keeps the plain-text form. */}
 												<span className="flex min-w-0 flex-1 flex-wrap items-center gap-x-0.5 gap-y-1 text-[13px] leading-tight">
-													{memberIds.map((id, i) => (
-														<span key={id} className="flex items-center">
-															{i > 0 && <span data-testid={`chain-arrow-${chain.id}-${i}`} className="px-1.5 text-muted-foreground/50">→</span>}
-															<span
-																data-testid={`prop-chip-${id}`}
-																className="rounded border border-border/70 bg-muted/30 px-1.5 py-0.5"
-															>
-																{nameOf(id)}
+													{memberIds.map((id, i) => {
+														const isConclusion = i === memberIds.length - 1;
+														return (
+															<span key={id} className="flex items-center">
+																{i > 0 && (isConclusion ? (
+																	<LatexGlyph
+																		cmd="\implies"
+																		testid={`chain-implies-${chain.id}`}
+																		className="px-1.5 text-foreground"
+																	/>
+																) : (
+																	<LatexGlyph
+																		cmd="\land"
+																		testid={`chain-and-${chain.id}-${i}`}
+																		className="px-1.5 text-foreground"
+																	/>
+																))}
+																<span
+																	data-testid={`prop-chip-${id}`}
+																	className="rounded border border-border/70 bg-muted/30 px-1.5 py-0.5"
+																>
+																	{nameOf(id)}
+																</span>
 															</span>
-														</span>
-													))}
+														);
+													})}
 												</span>
 												<span className="w-14 shrink-0 text-right text-xs tabular-nums text-muted-foreground">
 													{memberIds.length}
diff --git a/components/latex-glyph.tsx b/components/latex-glyph.tsx
new file mode 100644
index 0000000..d00cb90
--- /dev/null
+++ b/components/latex-glyph.tsx
@@ -0,0 +1,53 @@
+import katex from "katex";
+
+/**
+ * THE LaTeX glyph seam (owner ruling, 2026-09-30): UI connectors that
+ * want math symbols render the REAL LaTeX glyphs through KaTeX — the
+ * app's math engine, already shipped (app/layout.tsx carries the
+ * katex stylesheet) — never hand-rolled SVGs or font-dependent
+ * unicode. Born for the chain table's logical connectives: premises
+ * join with \land (∧), the conclusion separates with \implies (⟹) —
+ * `A ∧ B ⟹ C`, not the misleading `A → B → C`.
+ *
+ * Prominent by the same ruling: the connectives render in the
+ * theme's FOREGROUND (white in the dark theme) — the chain line's
+ * logic is the payload, not decoration. Callers pass the class.
+ */
+
+/** Cached katex HTML per command — the glyph set is tiny and
+ *  renderToString is pure; one render per command per process. */
+const cache = new Map<string, string>();
+
+/** The raw katex HTML for a LaTeX command (pure, sync — SSR-safe). */
+export function latexGlyphHtml(cmd: string): string {
+	let html = cache.get(cmd);
+	if (html === undefined) {
+		html = katex.renderToString(cmd, { throwOnError: false, output: "html" });
+		cache.set(cmd, html);
+	}
+	return html;
+}
+
+/** One rendered LaTeX glyph. `cmd` is the LaTeX command ("\land",
+ *  "\implies"); the output is aria-hidden — callers carry the textual
+ *  aria-label (formatChainLine keeps the plain-text form). */
+export function LatexGlyph({
+	cmd,
+	testid,
+	className,
+}: {
+	cmd: string;
+	testid?: string;
+	className?: string;
+}) {
+	return (
+		<span
+			data-testid={testid}
+			aria-hidden="true"
+			className={className}
+			// katex.renderToString is the library's own trusted output
+			// (the same thing rehype-katex injects for markdown math).
+			dangerouslySetInnerHTML={{ __html: latexGlyphHtml(cmd) }}
+		/>
+	);
+}
diff --git a/docs/architecture.md b/docs/architecture.md
index e609484..acf6567 100644
--- a/docs/architecture.md
+++ b/docs/architecture.md
@@ -382,7 +382,7 @@ Practice turns the graph into problem sets the learner reasons through out loud.
 
 **The jobs pattern — registry, not JobRecords.** Practice runs are NOT Workbench jobs: no `JobStore` records, no steer messages, nothing ever wakes the teacher — a practice run isn't chat-scoped (a set outlives its birth session and has no lesson to re-enter; its result is itself, persisted in its own store and read by its own page). What it shares with the Workbench is the observability surface: a `globalThis` run registry (the module-instance idiom — `__lhPracticeRuns`) with PER-PROBLEM ROWS (owner ruling, 2026-09-29 — matching the extraction per-chunk pattern): `setId:partition` while the partitioner runs ("Partitioning — title"), then one `setId:gen:problemId` per problem ("queued"→"generating", label "Problem i/N — title"), and one `setId:grade:problemId` per graded answer ("queued"→"grading", "Grading i/N — title") — each row its OWN abort controller and its OWN usage accumulation; the row key IS the feed id (the dock dup-key law) and the abort target. ABORT ROUTES BY ROW SHAPE: a gen-row abort kills THAT problem only (siblings continue; the set still settles `ready` with one errored problem), a partition-row abort ends the whole run, a grade-row abort drops that grade (answer kept — one attempt, the problem stays done); set-level abort (delete) composes from row aborts plus the synchronous persisted marks. Model calls — partitioner included — run under the SHARED background-LLM semaphore (`bg-llm-semaphore.ts`): one process-wide pool of 3 permits for ALL background LLM jobs, extraction chunks included (total background LLM concurrency bounded at 3, whoever calls). A boot sweep marks orphaned `partitioning`/`generating` sets as `error "Interrupted (server restarted)"` — once per process, and only when the set's `updatedAt` heartbeat is >120s stale (a live run's 30s FILE watchdog keeps the heartbeat fresh even across dev's per-worker processes — the registry is not shared between them; the guard trio closed a live-stamping bug from the first live test) — the job runner's crash-honesty at set grain. Store writes serialize behind a per-set mutex; the pipeline is the sole writer of `data/practice`.
 
-**API + UI.** Creation is GRAPH-DOMAIN, not chat-domain (owner ruling 2026-09-29: "problem sets always take in nodes — graph objects"): `POST /api/practice/sets` takes `{nodeIds, params, title?, note?}` ONLY — no chatId, no clusterId; the UI resolves a picked cluster to its member node ids and passes the cluster's label as the title. The `note` is the learner's optional free-form creation input (owner, same evening: more learner control, no other purpose): trimmed, capped at 2000 characters, persisted on the set, and carried to the partitioner and every generator as `THE LEARNER'S NOTE` — an additional input the agents WEIGH but are never obliged to follow (the doctrine pins the stance: it never moves a problem off its rung or out of its band, and the partition laws always win). `GET /api/graph/clusters` serves the picker: the birth-session partition itself (`birthSessionPartition`, same ordering — size desc, ties by id) — EVERY group regardless of size (owner ruling, same evening: MIN_CLUSTER_SIZE is a hull-drawing rule; practice availability must not inherit a visual threshold), each as `{key, kind: "chat"|"notes", label, nodeIds, nodeCount}`; a cluster whose naming record is gone keeps `label: null` — honest, never guessed. 202 `{id}` with generation fire-and-forget; 400 empty selection (empty intersection with live nodes — the vanished-nodes edge) / bad params; 503 model/key refusal. `GET /api/practice/sets` · `GET/DELETE /api/practice/sets/[id]` (delete aborts live runs first) · `POST /api/practice/sets/[id]/answers` (202; 409 when not ready; 400 re-answering a submitted problem — one attempt). Surfaces: `/practice` (the list — THE shared **TopBar** (`components/top-bar.tsx`, standardization ruling 2026-09-29: one element, every surface — h-15 matching the sidebar's wordmark band, `lh-drag` where NATIVE interactive children carve out AUTOMATICALLY (the globals.css auto carve-out: inside any drag band, `button/input/textarea/select/a/summary/[contenteditable]` are no-drag islands — owner standardization ruling 2026-09-29 night; manual `lh-no-drag` is only for clickable non-natives), `select-none` so drags never highlight, the SidebarToggle reopen with its Electron traffic-light gutter, `px-3` + one border line; chat's rename mode passes `drag={false}` — the one escape hatch). The overlay twin of the mask law rides with it: app-region subtraction resolves by DOM order, not z-index, so any surface containing a TopBar that COVERS other content (a full-bleed overlay) must be the LAST child of its container — born from the chains-overlay bug, where the graph chrome's lh-no-drag column (later in DOM, geometrically under the overlay) punched dead zones through the bar's drag band over the counts/search/close area with the New action at the right edge, status/counts rows, delete confirm, the new-set dialog with the named-level controls (a 5-detent ladder slider, a 3-detent cue slider, a Theory|Application toggle — each showing the level's name + definition), the cluster picker, and the optional free-form note textarea (typing-only). The SECOND creation entry (owner redesign, same evening): the graph view's **chain table** — a `Chains` button in the graph chrome (empty graph: hidden) opens a FULL-BLEED table overlay (absolute inside the graph area; the blur + translucent background IS the surface, no inner card; the top bar is THE shared TopBar — the standard title size and drag mechanics, the sidebar reopen, and the close X at the right edge, so home's window-drag mask never eats it (the overlay covers home's floating header while open, and the TopBar's own toggle is the reopen path); Esc or X dismiss). The hierarchy is three levels, one click each: CLUSTER (the birth session — a chat or a notes file; a chain lives where its CONCLUSION was born) → REASONING CHAIN → PROPOSITION STATEMENTS — expanding a chain reveals its members' statement text directly (names live in the chain line, never twice; a singleton reveals its one statement in a single click); everything starts collapsed. The table look (owner refinements, evening round 2): PROMINENT indentation carries the hierarchy (chains deep under their folder, statements deepest) and horizontal rules FRAME the table (owner round 5) — between cluster groups, under the header, and under the last group — with per-row lines and the vertical column hairlines still gone, and open folders/chains dropping a TREE THREAD (owner round 5, trial): a vertical guide at the expander arrow's column (37px clusters / 73px chains), BRACKETED by checkbox bottom edges (owner round 6): it starts at the parent row's own checkbox bottom — a visible gap below the arrow's stroke, never touching it — and terminates exactly at the last child row's checkbox bottom (live-measured anchors: 23/8px clusters, 22/8px chains); statement rows are PLAIN (the derivation already lives in the chain line — no conclusion marker). ALL THREE columns sort (owner: "consistency is key") — Name (A→Z default), Props (largest-first), Created (newest-first) — click toggles asc/desc, ordering both cluster rows and chains within; a SMALL chevron marks the active header. In the chain line every proposition name is a faint-outlined CHIP with the → a clearly separated connector between chips — a lonely chip reads as a standalone proposition (the "(no premises)" annotation is dead in the visible UI; it survives only as the row's aria-label); icons: ONLY the cluster FolderIcon (open-folder when expanded) survives (owner ruling, evening round 3: the chain waypoints icons are not needed — the chips + connectors already read as chains); chevrons rotate with expansion and share the checkbox's 14px width, which makes every column exactly 18px and the ALIGNMENT STAIRCASE exact (owner rulings, rounds 3–4): each level's checkbox sits under its parent's ICON slot — open a folder and the chain checkboxes line up with the FOLDER icon; the statements step to where a chain icon WOULD sit (the steps are TWO columns = 36px — "double the indentation"): clusters pl-3 → chains pl-12 → statements pl-21. Selection is checkboxes on whole clusters, chains, or single propositions — a flat node-id set (a node in several chains is one selection) — accumulated in a FLOATING BOTTOM BAR (the Google-Photos pattern: appears only when something is selected, never shifts the table layout) showing the agents' ACTUAL input (`chainsForSelection` over the selection: a premise picked without its conclusion counts as the singleton it becomes) as `N nodes → M chains` (NODE is THE UI word for graph objects — owner ruling, 2026-09-30, standardized across every surface's copy), with Clear and **Generate problem set**, which opens the SAME creation dialog in **custom mode**: no cluster picker — a selection summary (the derived shape + the dated default title the pipeline will stamp) — posting `{nodeIds, params, note}` with no title. A fuzzy search (names, aliases, statements) filters chains grouped + auto-expanded under their clusters (non-matching clusters hide; the collapsed view returns on clear). NO understanding-levels column (explicitly not built). The bar also carries **Delete** (owner, 2026-09-29 night: "this is really the view deletion should be done in") FENCED at its right end — a divider + the destructive red pair + a trash icon, the safe actions and the kill switch never blending — and it opens the SAME full-death confirm dialog the canvas uses (`DeleteNodeDialog`, body-portal z-50) — MINIMAL by owner ruling (2026-09-30): NO pre-flight impact read (what full death takes is obvious, the fetch flickered, and richer facts would only confuse as deletion semantics evolve toward rewiring), just `Delete node?` / `Delete N nodes?` + the names in the mono block (capped at 8 + "and K more") + one permanence line + Cancel/Delete (the `GET /api/graph/nodes/[id]` impact route and its engine died with the read); confirm runs the per-node DELETEs SEQUENTIALLY — a failure stops the run (extraction_running is never plowed through), what died stays dead — then prunes the selection and announces `lh:graph-revision` so the snapshot refetches. This ends the overlay's read-only stance: the chain table is the graph's MANAGEMENT surface, and `/practice/[id]` (the exam page — the shared TopBar, a levels + status line, numbered problems, markdown statements with math healing at the display seam, a textarea + focus-gated dictation bar per problem with its OWN Submit button — one attempt per problem, instant solution/rubric reveal, the score landing live). Entry: the sidebar's Practice nav row.
+**API + UI.** Creation is GRAPH-DOMAIN, not chat-domain (owner ruling 2026-09-29: "problem sets always take in nodes — graph objects"): `POST /api/practice/sets` takes `{nodeIds, params, title?, note?}` ONLY — no chatId, no clusterId; the UI resolves a picked cluster to its member node ids and passes the cluster's label as the title. The `note` is the learner's optional free-form creation input (owner, same evening: more learner control, no other purpose): trimmed, capped at 2000 characters, persisted on the set, and carried to the partitioner and every generator as `THE LEARNER'S NOTE` — an additional input the agents WEIGH but are never obliged to follow (the doctrine pins the stance: it never moves a problem off its rung or out of its band, and the partition laws always win). `GET /api/graph/clusters` serves the picker: the birth-session partition itself (`birthSessionPartition`, same ordering — size desc, ties by id) — EVERY group regardless of size (owner ruling, same evening: MIN_CLUSTER_SIZE is a hull-drawing rule; practice availability must not inherit a visual threshold), each as `{key, kind: "chat"|"notes", label, nodeIds, nodeCount}`; a cluster whose naming record is gone keeps `label: null` — honest, never guessed. 202 `{id}` with generation fire-and-forget; 400 empty selection (empty intersection with live nodes — the vanished-nodes edge) / bad params; 503 model/key refusal. `GET /api/practice/sets` · `GET/DELETE /api/practice/sets/[id]` (delete aborts live runs first) · `POST /api/practice/sets/[id]/answers` (202; 409 when not ready; 400 re-answering a submitted problem — one attempt). Surfaces: `/practice` (the list — THE shared **TopBar** (`components/top-bar.tsx`, standardization ruling 2026-09-29: one element, every surface — h-15 matching the sidebar's wordmark band, `lh-drag` where NATIVE interactive children carve out AUTOMATICALLY (the globals.css auto carve-out: inside any drag band, `button/input/textarea/select/a/summary/[contenteditable]` are no-drag islands — owner standardization ruling 2026-09-29 night; manual `lh-no-drag` is only for clickable non-natives), `select-none` so drags never highlight, the SidebarToggle reopen with its Electron traffic-light gutter, `px-3` + one border line; chat's rename mode passes `drag={false}` — the one escape hatch). The overlay twin of the mask law rides with it: app-region subtraction resolves by DOM order, not z-index, so any surface containing a TopBar that COVERS other content (a full-bleed overlay) must be the LAST child of its container — born from the chains-overlay bug, where the graph chrome's lh-no-drag column (later in DOM, geometrically under the overlay) punched dead zones through the bar's drag band over the counts/search/close area with the New action at the right edge, status/counts rows, delete confirm, the new-set dialog with the named-level controls (a 5-detent ladder slider, a 3-detent cue slider, a Theory|Application toggle — each showing the level's name + definition), the cluster picker, and the optional free-form note textarea (typing-only). The SECOND creation entry (owner redesign, same evening): the graph view's **chain table** — a `Chains` button in the graph chrome (empty graph: hidden) opens a FULL-BLEED table overlay (absolute inside the graph area; the blur + translucent background IS the surface, no inner card; the top bar is THE shared TopBar — the standard title size and drag mechanics, the sidebar reopen, and the close X at the right edge, so home's window-drag mask never eats it (the overlay covers home's floating header while open, and the TopBar's own toggle is the reopen path); Esc or X dismiss). The hierarchy is three levels, one click each: CLUSTER (the birth session — a chat or a notes file; a chain lives where its CONCLUSION was born) → REASONING CHAIN → PROPOSITION STATEMENTS — expanding a chain reveals its members' statement text directly (names live in the chain line, never twice; a singleton reveals its one statement in a single click); everything starts collapsed. The table look (owner refinements, evening round 2): PROMINENT indentation carries the hierarchy (chains deep under their folder, statements deepest) and horizontal rules FRAME the table (owner round 5) — between cluster groups, under the header, and under the last group — with per-row lines and the vertical column hairlines still gone, and open folders/chains dropping a TREE THREAD (owner round 5, trial): a vertical guide at the expander arrow's column (37px clusters / 73px chains), BRACKETED by checkbox bottom edges (owner round 6): it starts at the parent row's own checkbox bottom — a visible gap below the arrow's stroke, never touching it — and terminates exactly at the last child row's checkbox bottom (live-measured anchors: 23/8px clusters, 22/8px chains); statement rows are PLAIN (the derivation already lives in the chain line — no conclusion marker). ALL THREE columns sort (owner: "consistency is key") — Name (A→Z default), Props (largest-first), Created (newest-first) — click toggles asc/desc, ordering both cluster rows and chains within; a SMALL chevron marks the active header. In the chain line every proposition name is a faint-outlined CHIP and THE LOGIC IS THE CONNECTIVE (owner ruling, 2026-09-30): premises join with the LaTeX LAND (∧) and the conclusion separates by the LaTeX IMPLICATION (⟹) — `A ∧ B ⟹ C`, never the misleading `A → B → C` — the glyphs are REAL KaTeX renders (`components/latex-glyph.tsx`, `katex.renderToString` — the app's math engine, never SVGs or font-dependent unicode; the katex stylesheet already rides app-wide), PROMINENT in the theme foreground (white in the dark theme — the logic is the payload, not decoration); a lonely chip reads as a standalone proposition with no glyph (the "(no premises)" annotation is dead in the visible UI; it and the plain-text `→` survive only as the row's aria-label); icons: ONLY the cluster FolderIcon (open-folder when expanded) survives (owner ruling, evening round 3: the chain waypoints icons are not needed — the chips + connectors already read as chains); chevrons rotate with expansion and share the checkbox's 14px width, which makes every column exactly 18px and the ALIGNMENT STAIRCASE exact (owner rulings, rounds 3–4): each level's checkbox sits under its parent's ICON slot — open a folder and the chain checkboxes line up with the FOLDER icon; the statements step to where a chain icon WOULD sit (the steps are TWO columns = 36px — "double the indentation"): clusters pl-3 → chains pl-12 → statements pl-21. Selection is checkboxes on whole clusters, chains, or single propositions — a flat node-id set (a node in several chains is one selection) — accumulated in a FLOATING BOTTOM BAR (the Google-Photos pattern: appears only when something is selected, never shifts the table layout) showing the agents' ACTUAL input (`chainsForSelection` over the selection: a premise picked without its conclusion counts as the singleton it becomes) as `N nodes → M chains` (NODE is THE UI word for graph objects — owner ruling, 2026-09-30, standardized across every surface's copy), with Clear and **Generate problem set**, which opens the SAME creation dialog in **custom mode**: no cluster picker — a selection summary (the derived shape + the dated default title the pipeline will stamp) — posting `{nodeIds, params, note}` with no title. A fuzzy search (names, aliases, statements) filters chains grouped + auto-expanded under their clusters (non-matching clusters hide; the collapsed view returns on clear). NO understanding-levels column (explicitly not built). The bar also carries **Delete** (owner, 2026-09-29 night: "this is really the view deletion should be done in") FENCED at its right end — a divider + the destructive red pair + a trash icon, the safe actions and the kill switch never blending — and it opens the SAME full-death confirm dialog the canvas uses (`DeleteNodeDialog`, body-portal z-50) — MINIMAL by owner ruling (2026-09-30): NO pre-flight impact read (what full death takes is obvious, the fetch flickered, and richer facts would only confuse as deletion semantics evolve toward rewiring), just `Delete node?` / `Delete N nodes?` + the names in the mono block (capped at 8 + "and K more") + one permanence line + Cancel/Delete (the `GET /api/graph/nodes/[id]` impact route and its engine died with the read); confirm runs the per-node DELETEs SEQUENTIALLY — a failure stops the run (extraction_running is never plowed through), what died stays dead — then prunes the selection and announces `lh:graph-revision` so the snapshot refetches. This ends the overlay's read-only stance: the chain table is the graph's MANAGEMENT surface, and `/practice/[id]` (the exam page — the shared TopBar, a levels + status line, numbered problems, markdown statements with math healing at the display seam, a textarea + focus-gated dictation bar per problem with its OWN Submit button — one attempt per problem, instant solution/rubric reveal, the score landing live). Entry: the sidebar's Practice nav row.
 
 **Model wiring.** Three roles — `partitioner`, `problemGenerator`, `grader` — all pinned to the **strong** slot (quality-critical, one settings edit from cheap) and in `BACKGROUND_MODEL_ROLES` (per-role overrides). The stage agents are single-submit loops on the shared `runSubmitAgent` seam (`submit-agent.ts`, extracted from extraction for exactly this): a `validate` hook turns a bad submit into a tool-input error the model retries against, within `maxTurns 4`; exhaustion fails the stage honestly (the set says so, the chains re-run cleanly on a new set).
 
diff --git a/tests/render-chain-table.test.tsx b/tests/render-chain-table.test.tsx
index 740a967..e93e64e 100644
--- a/tests/render-chain-table.test.tsx
+++ b/tests/render-chain-table.test.tsx
@@ -5,7 +5,9 @@
  * surface), cluster → chain → statement hierarchy (a chain lives in its
  * CONCLUSION's birth cluster; expanding a chain reveals statements
  * directly — names live in the line, never twice), proposition CHIPS +
- * → connectors (a lonely chip = a standalone proposition; the "(no
+ * ∧/⟹ LaTeX connectives (premises join by land, the conclusion
+ * separates by implication — REAL KaTeX renders, prominent white; a
+ * lonely chip = a standalone proposition; the "(no
  * premises)" annotation is dead), rotating chevrons, folder/waypoints
  * icons, a floating bottom selection bar that never shifts layout, and
  * a fuzzy search that auto-expands matching clusters.
@@ -164,14 +166,22 @@ describe("ChainTableBody — THE shared TopBar + the lines law (owner, round 2)"
 });
 
 describe("ChainTableBody — chain rows (chips + connectors)", () => {
-	it("names render as outlined chips with → connectors BETWEEN them", () => {
+	it("names render as outlined chips; premises join by ∧, the conclusion by ⟹ — REAL KaTeX glyphs in the foreground", () => {
 		const html = render({ expandedClusters: new Set(["chat1"]) });
 		assert.ok(html.includes('data-testid="chain-row-c1"'));
 		assert.ok(html.includes('data-testid="prop-chip-a"'), "premise chip");
 		assert.ok(html.includes('data-testid="prop-chip-c1"'), "conclusion chip");
 		assert.ok(html.includes("Premise one"), "chip text");
-		assert.ok(html.includes('data-testid="chain-arrow-c1-1"'), "the connector");
-		assert.ok(html.includes("→"), "the arrow glyph");
+		// the connectives (owner ruling, 2026-09-30): premises ∧, conclusion
+		// ⟹ — KaTeX renders (class katex present), PROMINENT foreground
+		assert.ok(html.includes('data-testid="chain-and-c1-1"'), "the land between premises");
+		assert.ok(html.includes('data-testid="chain-implies-c1"'), "the implication before the conclusion");
+		const and = html.match(/data-testid="chain-and-c1-1"[^>]*>/)?.[0] ?? "";
+		assert.match(and, /text-foreground/, "prominent — the theme's white, not dim gray");
+		assert.ok(html.includes("katex"), "a REAL KaTeX render, not a unicode arrow");
+		// the old arrow is dead as a VISIBLE connector — it survives only in
+		// the row's aria-label (formatChainLine, the plain-text form)
+		assert.doesNotMatch(html, />\s*→\s*</, "no textual arrow connector remains");
 		// owner ruling (round 3): the chain waypoints icons are GONE — the
 		// chips + connectors already read as chains; only folders keep icons
 		assert.ok(!html.includes("lucide-waypoints"), "no chain icon");