feat(tabs): polish new-chat button — stuck-only square edge, New Chat labelHEADmain

- isStuckToEnd (pure): square the sticky '+' right edge only while it floats
  over scrolled tabs; rounded at rest. Edge-measured in TabBar via a disposed
  scroll + ResizeObserver effect (RO guarded for non-browser envs).
- Show a temporary 'New Chat' title when the draft is selected, with the '+'
  moved to the trailing close-button slot for consistency with real tabs.

4 files changed, 123 insertions, 3 deletions

diff --git a/src/features/tabs/tabs.test.ts b/src/features/tabs/tabs.test.ts
index 3034e76..5effa9b 100644
--- a/src/features/tabs/tabs.test.ts
+++ b/src/features/tabs/tabs.test.ts
@@ -6,6 +6,7 @@ import {
 	createTab,
 	deriveTitle,
 	initialState,
+	isStuckToEnd,
 	newDraft,
 	selectTab,
 	setModel,
@@ -189,3 +190,26 @@ describe("deriveTitle", () => {
 		expect(result).toBe(`${"a".repeat(40)}\u2026`);
 	});
 });
+
+describe("isStuckToEnd", () => {
+	it("is false when the strip does not overflow", () => {
+		expect(isStuckToEnd({ scrollLeft: 0, clientWidth: 500, scrollWidth: 500 })).toBe(false);
+		expect(isStuckToEnd({ scrollLeft: 0, clientWidth: 500, scrollWidth: 400 })).toBe(false);
+	});
+
+	it("is true when overflowing and scrolled to the left", () => {
+		expect(isStuckToEnd({ scrollLeft: 0, clientWidth: 500, scrollWidth: 1000 })).toBe(true);
+	});
+
+	it("is true when overflowing and scrolled to the middle", () => {
+		expect(isStuckToEnd({ scrollLeft: 250, clientWidth: 500, scrollWidth: 1000 })).toBe(true);
+	});
+
+	it("is false when overflowing but scrolled fully to the right", () => {
+		expect(isStuckToEnd({ scrollLeft: 500, clientWidth: 500, scrollWidth: 1000 })).toBe(false);
+	});
+
+	it("treats a 1px subpixel gap at the end as at-rest (epsilon)", () => {
+		expect(isStuckToEnd({ scrollLeft: 499, clientWidth: 500, scrollWidth: 1000 })).toBe(false);
+	});
+});
diff --git a/src/features/tabs/tabs.ts b/src/features/tabs/tabs.ts
index 9af522f..a4db6f3 100644
--- a/src/features/tabs/tabs.ts
+++ b/src/features/tabs/tabs.ts
@@ -66,6 +66,26 @@ export function activeTab(state: TabsState): Tab | null {
 	return state.tabs.find((t) => t.conversationId === state.activeConversationId) ?? null;
 }
 
+export interface ScrollMetrics {
+	readonly scrollLeft: number;
+	readonly clientWidth: number;
+	readonly scrollWidth: number;
+}
+
+const STUCK_EPSILON = 1;
+
+/**
+ * True when a right-pinned sticky element is floating over scrolled content — the
+ * strip overflows horizontally AND is not scrolled fully to the right. When it is
+ * at rest (no overflow, or scrolled to the end so it sits at its natural position)
+ * this returns false. Pure: layout measurements in, boolean out.
+ */
+export function isStuckToEnd(m: ScrollMetrics): boolean {
+	const overflows = m.scrollWidth > m.clientWidth + STUCK_EPSILON;
+	const notAtEnd = m.scrollLeft + m.clientWidth < m.scrollWidth - STUCK_EPSILON;
+	return overflows && notAtEnd;
+}
+
 export function deriveTitle(message: string, max: number = DEFAULT_MAX_TITLE_LENGTH): string {
 	const trimmed = message.trim().replace(/\s+/g, " ");
 	if (trimmed.length === 0) return DEFAULT_TITLE;
diff --git a/src/features/tabs/ui.test.ts b/src/features/tabs/ui.test.ts
index 53df0be..1ae18c8 100644
--- a/src/features/tabs/ui.test.ts
+++ b/src/features/tabs/ui.test.ts
@@ -145,4 +145,34 @@ describe("TabBar", () => {
 		const newChat = screen.getByRole("button", { name: "New chat" });
 		expect(newChat).toHaveClass("sticky");
 	});
+
+	it("shows visible 'New Chat' text when activeConversationId is null", () => {
+		render(TabBar, {
+			props: {
+				tabs: sampleTabs,
+				activeConversationId: null,
+				onSelect: vi.fn(),
+				onClose: vi.fn(),
+				onNewDraft: vi.fn(),
+			},
+		});
+
+		const newChat = screen.getByRole("button", { name: "New chat" });
+		expect(newChat).toHaveTextContent("New Chat");
+	});
+
+	it("does not show 'New Chat' text when a real tab is active", () => {
+		render(TabBar, {
+			props: {
+				tabs: sampleTabs,
+				activeConversationId: "c1",
+				onSelect: vi.fn(),
+				onClose: vi.fn(),
+				onNewDraft: vi.fn(),
+			},
+		});
+
+		const newChat = screen.getByRole("button", { name: "New chat" });
+		expect(newChat).not.toHaveTextContent("New Chat");
+	});
 });
diff --git a/src/features/tabs/ui/TabBar.svelte b/src/features/tabs/ui/TabBar.svelte
index 76fab05..eb5b5e5 100644
--- a/src/features/tabs/ui/TabBar.svelte
+++ b/src/features/tabs/ui/TabBar.svelte
@@ -1,5 +1,6 @@
 <script lang="ts">
 	import type { Tab } from "../tabs";
+	import { isStuckToEnd } from "../tabs";
 
 	let {
 		tabs,
@@ -14,9 +15,47 @@
 		onClose: (conversationId: string) => void;
 		onNewDraft: () => void;
 	} = $props();
+
+	// The new-chat button is `position: sticky; right: 0`. It floats over the tabs
+	// only while the strip overflows and isn't scrolled fully right; we square its
+	// right edge only in that "stuck" state. Pure decision (`isStuckToEnd`) +
+	// DOM-measurement at the edge here.
+	let scrollEl = $state<HTMLDivElement>();
+	let stuck = $state(false);
+
+	function recompute(): void {
+		const el = scrollEl;
+		if (el === undefined) {
+			stuck = false;
+			return;
+		}
+		stuck = isStuckToEnd({
+			scrollLeft: el.scrollLeft,
+			clientWidth: el.clientWidth,
+			scrollWidth: el.scrollWidth,
+		});
+	}
+
+	$effect(() => {
+		const el = scrollEl;
+		if (el === undefined) return;
+		// Re-evaluate when the tab set changes (overflow may appear/disappear).
+		void tabs;
+		recompute();
+
+		el.addEventListener("scroll", recompute, { passive: true });
+		const ro =
+			typeof ResizeObserver !== "undefined" ? new ResizeObserver(recompute) : undefined;
+		ro?.observe(el);
+
+		return () => {
+			el.removeEventListener("scroll", recompute);
+			ro?.disconnect();
+		};
+	});
 </script>
 
-<div class="overflow-x-auto border-b border-base-300">
+<div bind:this={scrollEl} class="overflow-x-auto border-b border-base-300">
 	<div class="tabs tabs-border min-w-max">
 		{#each tabs as tab (tab.conversationId)}
 			<div
@@ -43,12 +82,19 @@
 			</div>
 		{/each}
 		<button
-			class="tab sticky right-0 z-10 bg-base-200 shadow-[-2px_0_4px_-1px_rgba(0,0,0,0.2)]"
+			class="tab sticky right-0 z-10 bg-base-200 shadow-[-2px_0_4px_-1px_rgba(0,0,0,0.2)] {stuck
+				? '!rounded-se-none !rounded-ee-none'
+				: ''}"
 			class:tab-active={activeConversationId === null}
 			aria-label="New chat"
 			onclick={() => onNewDraft()}
 		>
-			+
+			{#if activeConversationId === null}
+				<span class="max-w-[120px] truncate">New Chat</span>
+				<span class="btn btn-ghost btn-xs ml-1" aria-hidden="true">+</span>
+			{:else}
+				+
+			{/if}
 		</button>
 	</div>
 </div>