diff --git a/plans/260503-1121-lesson-4-5-gcd-factor/phase-01-katex-integration.md b/plans/260503-1121-lesson-4-5-gcd-factor/phase-01-katex-integration.md
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+---
+phase: 1
+title: "KaTeX integration"
+status: pending
+priority: P1
+effort: "2h"
+dependencies: []
+---
+
+# Phase 01: KaTeX integration
+
+## Overview
+Add KaTeX to the project, wrap it in a single `<Tex>` Svelte component, and verify rendering in a sandbox route. This is the only place KaTeX is touched directly — every lesson uses `<Tex math="..." />`.
+
+## Requirements
+- Functional: render inline + display math from a string prop; SSR-safe (KaTeX renders to static HTML, no client JS needed for display)
+- Non-functional: no FOUC; CSS loaded once at app level; no per-component CSS imports
+- A11y: KaTeX outputs MathML alongside HTML — keep it; `aria-label` fallback prop for screen readers when math expression is non-trivial
+
+## Architecture
+- KaTeX has two APIs: `katex.render(string, element, options)` (mutates DOM) and `katex.renderToString(string, options)` (returns HTML string).
+- Use `renderToString` — pure function, SSR-friendly, plugs into `{@html}`. No `$effect` needed.
+- CSS imported once in `+layout.svelte` so it's in the global bundle.
+- KaTeX fonts are auto-fetched via CSS `@font-face` from the package's `dist/fonts/` — Vite's static asset handling picks them up.
+
+## Related Code Files
+- Create: `src/lib/components/tex.svelte` — `<Tex>` component
+- Create: `src/routes/__sandbox/katex/+page.svelte` — smoke test route (delete in Phase 05 or leave guarded by `dev` mode)
+- Modify: `src/routes/+layout.svelte` — add `import 'katex/dist/katex.min.css'`
+- Modify: `package.json` — add `katex` dep + `@types/katex` is unnecessary (we use JSDoc)
+
+## Implementation Steps
+1. `npm install katex@^0.16.x` — pin minor; KaTeX is mature, breaking changes rare.
+2. Create `src/lib/components/tex.svelte`:
+   ```svelte
+   <script>
+     import katex from 'katex';
+     /** @type {{ math: string, display?: boolean, ariaLabel?: string }} */
+     let { math, display = false, ariaLabel } = $props();
+     const html = katex.renderToString(math, {
+       displayMode: display,
+       throwOnError: false,
+       output: 'htmlAndMathml',
+     });
+   </script>
+   {#if ariaLabel}
+     <span aria-label={ariaLabel}>{@html html}</span>
+   {:else}
+     {@html html}
+   {/if}
+   ```
+3. Add `import 'katex/dist/katex.min.css';` at top of `<script>` block in `src/routes/+layout.svelte`.
+4. Create sandbox route `src/routes/__sandbox/katex/+page.svelte` rendering 4-5 expressions covering: inline (`a^2 - b^2`), display (`\frac{a \cdot b}{\gcd(a,b)}`), Vietnamese mixed text, Greek letters, fractions with sqrt.
+5. `npm run dev` — visit `/__sandbox/katex/` — verify rendering; check no FOUC; confirm fonts load (Network tab).
+6. `npm run build` — verify static prerender includes KaTeX HTML inline (view-source on built file).
+7. `npm run check` — must stay green.
+
+## Success Criteria
+- [ ] `<Tex math="..." />` renders inline math
+- [ ] `<Tex math="..." display />` renders display math (centered, larger)
+- [ ] No console errors / warnings on dev or build
+- [ ] Built static HTML contains rendered math (not just `<Tex>` placeholder)
+- [ ] Bundle size delta logged in plan.md "Success criteria" section
+- [ ] svelte-check clean
+
+## Risk Assessment
+- **Risk**: KaTeX bundle size (~300KB gzip with fonts) inflates Lighthouse perf score.
+  **Mitigation**: Acceptable trade for math site; document in `RUNBOOK.md`. If we need to shed weight later, we can lazy-load KaTeX per-route.
+- **Risk**: Font loading flash.
+  **Mitigation**: KaTeX CSS uses `font-display: swap`; system fallback is acceptable for the ~50ms gap.
+- **Risk**: SSR mismatch warnings (Svelte hydration).
+  **Mitigation**: `renderToString` is deterministic — no mismatch expected. Verify in dev console.
+
+## Notes
+- Sandbox route name uses `__sandbox/` prefix so it's obvious it's not user-facing. Decide in Phase 05 whether to delete or keep behind `if (dev)` guard.
+- Do NOT add `mhchem` / `copy-tex` extensions — YAGNI.
diff --git a/plans/260503-1121-lesson-4-5-gcd-factor/phase-02-so-hoc-engine-gcd-lesson.md b/plans/260503-1121-lesson-4-5-gcd-factor/phase-02-so-hoc-engine-gcd-lesson.md
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+---
+phase: 2
+title: "Số học engine + GCD lesson"
+status: pending
+priority: P1
+effort: "6h"
+dependencies: [1]
+---
+
+# Phase 02: Số học engine + GCD lesson (Ước chung lớn nhất)
+
+## Overview
+Build a pure `numtheory-engine` module (mirrors `geom-engine` shape) and ship the first số học lesson: a step-by-step Euclidean algorithm visualizer for `gcd(a, b)` with `lcm` derived from the result.
+
+## Requirements
+- Functional:
+  - Two number inputs for `a, b` (1 ≤ a, b ≤ 999)
+  - "Step" button advances Euclidean reduction one row at a time
+  - "Tự động" button runs full reduction
+  - "Khởi động lại" resets to current inputs
+  - Final row shows `gcd(a, b) = N` highlighted
+  - LCM formula + value displayed once gcd is found: `lcm(a,b) = (a·b)/gcd(a,b)`
+- Non-functional:
+  - Pure `gcdSteps(a, b)` returns the trace (no DOM coupling)
+  - All terminology + example aligned to **Cánh Diều lớp 6 chương "Số tự nhiên"**
+  - Vietnamese-first slugs and copy
+- A11y:
+  - Number inputs labeled (`<label for=>`)
+  - Step button keyboard-focusable
+  - `aria-live="polite"` on the reduction table
+  - Each new row announced
+
+## Architecture
+- Engine is pure JS + JSDoc; sibling shape to `geom-engine`.
+- `gcdSteps` returns an array of `{ a, b, q, r }` rows so the UI is pure render.
+- Lesson page owns `$state` for input values + step index; uses `$derived` for current visible rows + lcm value.
+- Goal-state pattern (documented inline): `goal(state) → { reached: state.stepIdx === steps.length, hint: ... }`. Not extracted to shared module — first instance.
+
+## Related Code Files
+- Create: `src/lib/numtheory-engine/gcd.js` — `gcd`, `lcm`, `gcdSteps`
+- Create: `src/lib/numtheory-engine/gcd.test.js` — Vitest unit tests
+- Create: `src/lib/numtheory-engine/index.js` — barrel export
+- Create: `src/lib/lessons/uoc-chung-lon-nhat/copy.vi.js` — VN copy
+- Create: `src/routes/so-hoc/uoc-chung-lon-nhat/+page.svelte` — lesson page
+- Modify (Phase 04): `src/lib/lessons/registry.js`, root `+page.svelte`
+
+## Implementation Steps
+
+### 2.1 Engine
+1. `gcd(a, b)`:
+   ```js
+   /** @param {number} a @param {number} b @returns {number} */
+   export function gcd(a, b) {
+     a = Math.abs(a); b = Math.abs(b);
+     while (b !== 0) { [a, b] = [b, a % b]; }
+     return a;
+   }
+   ```
+2. `lcm(a, b)`:
+   ```js
+   /** Throws if both 0 (lcm undefined). */
+   export function lcm(a, b) {
+     if (a === 0 && b === 0) throw new Error('lcm(0,0) undefined');
+     return Math.abs(a * b) / gcd(a, b);
+   }
+   ```
+3. `gcdSteps(a, b)` returns `Array<{a:number, b:number, q:number, r:number}>`:
+   - Each row: current pair before reduction, quotient `q = floor(a/b)`, remainder `r = a % b`
+   - Terminates when `r === 0`; final row's `b` is the gcd
+4. Tests: cover `gcd(48, 18) === 6`, `gcd(0, 5) === 5`, `gcd(7, 0) === 7`, `gcd(13, 17) === 1` (coprime), `lcm(4, 6) === 12`, `gcdSteps(48, 18)` produces expected 3-row trace.
+
+### 2.2 Copy module
+Create `copy.vi.js`:
+```js
+export const vi = {
+  slug: 'uoc-chung-lon-nhat',
+  topic: 'so-hoc',
+  grade: 'lop-6',
+  title: 'Ước chung lớn nhất (Thuật toán Euclid)',
+  gradeLabel: 'Lớp 6',
+  intro: 'ƯCLN của hai số là số lớn nhất chia hết cả hai. Thuật toán Euclid tìm nó bằng cách thay liên tiếp (a, b) bằng (b, a mod b) cho đến khi b = 0.',
+  instruction: 'Nhập hai số rồi nhấn "Bước" để xem từng bước rút gọn',
+  resultBadge: (g) => `ƯCLN = ${g}`,
+  lcmLabel: 'Bội chung nhỏ nhất',
+  theoremTitle: 'Thuật toán Euclid',
+  theoremStatement: 'Với mọi a, b ≥ 0 (không cùng bằng 0): gcd(a, b) = gcd(b, a mod b). Khi b = 0, gcd = a.',
+  exampleTitle: 'Ví dụ',
+  exampleBody: 'Tìm ƯCLN(48, 18). Bước 1: 48 = 2·18 + 12. Bước 2: 18 = 1·12 + 6. Bước 3: 12 = 2·6 + 0. Vậy ƯCLN = 6.',
+  nextTeaser: 'Sắp ra mắt: BCNN bằng phân tích thừa số nguyên tố',
+};
+```
+
+### 2.3 Lesson page
+- `+page.svelte` owns:
+  - `let a = $state(48); let b = $state(18);`
+  - `let stepIdx = $state(0);`
+  - `let steps = $derived(gcdSteps(a, b));`
+  - `let visibleSteps = $derived(steps.slice(0, stepIdx));`
+  - `let g = $derived(stepIdx === steps.length && steps.length > 0 ? steps[steps.length - 1].b : null);`
+  - `let l = $derived(g !== null && g !== 0 ? lcm(a, b) : null);`
+- UI sections:
+  1. Title + grade badge
+  2. Intro paragraph
+  3. Two number inputs (a, b) + reset button (resets stepIdx to 0)
+  4. "Bước" button (disabled when stepIdx === steps.length); "Tự động" button (sets stepIdx = steps.length)
+  5. `<table>` rendering `visibleSteps`: columns "a", "b", "q", "r" — header "a = q·b + r"
+  6. When `g !== null`: highlighted result + `<Tex math={\`\\gcd(${a}, ${b}) = ${g}\`} display />`
+  7. When `l !== null`: `<Tex math={\`\\text{lcm}(${a}, ${b}) = \\frac{${a} \\cdot ${b}}{${g}} = ${l}\`} display />`
+  8. Theorem block + example (from copy)
+  9. Next teaser
+
+### 2.4 SEO meta
+- `<svelte:head>` with `<title>{copy.title} — MathMax</title>`, `<meta name="description" content={copy.intro}>`, canonical URL via `$page.url`.
+
+## Success Criteria
+- [ ] `gcd.test.js` covers gcd, lcm, gcdSteps with 6+ assertions; all pass
+- [ ] Lesson renders at `/so-hoc/uoc-chung-lon-nhat/` in dev + build
+- [ ] Stepping through 48, 18 produces 3 rows ending in gcd=6
+- [ ] LCM displays correct value (144 for 48,18) once gcd found
+- [ ] Reset button restores stepIdx=0 + clears LCM display
+- [ ] Number inputs validate (1-999); invalid input doesn't crash
+- [ ] Keyboard nav: Tab through inputs → buttons; Enter triggers Step button
+- [ ] svelte-check clean; vitest green
+
+## Risk Assessment
+- **Risk**: User enters 0 or negative → `gcdSteps` infinite loop or wrong output.
+  **Mitigation**: Input `min=1 max=999`; `gcdSteps` defensive: if `b === 0` initial, return single row immediately.
+- **Risk**: Cánh Diều lớp 6 may use specific notation (e.g. `ƯCLN(a,b)` vs `gcd(a,b)`).
+  **Mitigation**: Copy uses `ƯCLN` per Vietnamese standard; KaTeX uses `\gcd` LaTeX command (renders as "gcd"). Add a one-line `\text{ƯCLN}` substitution if textbook check shows preference.
+- **Risk**: Lesson page exceeds 200 LOC (modularization rule).
+  **Mitigation**: If approaching, extract reduction `<table>` into a tiny `gcd-trace-table.svelte` component. Defer until needed.
+
+## Notes
+- No drag interaction in this lesson — just inputs + buttons + table. Doesn't touch `draggable.svelte.js`.
+- Goal-state in this lesson is implicit: "all rows revealed → gcd shown → lcm shown." No formal `goal()` function v1; just `$derived`.
diff --git a/plans/260503-1121-lesson-4-5-gcd-factor/phase-03-dai-so-factor-lesson.md b/plans/260503-1121-lesson-4-5-gcd-factor/phase-03-dai-so-factor-lesson.md
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+++ b/plans/260503-1121-lesson-4-5-gcd-factor/phase-03-dai-so-factor-lesson.md
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+---
+phase: 3
+title: "Đại số factor a²-b² lesson"
+status: pending
+priority: P1
+effort: "5h"
+dependencies: [1]
+---
+
+# Phase 03: Đại số — Hiệu hai bình phương (a² − b² visualizer)
+
+## Overview
+Interactive geometric proof of `a² − b² = (a + b)(a − b)`. Two sliders set `a` and `b`; SVG shows the L-shape (square `a²` minus corner square `b²`) being rearranged into a rectangle of `(a+b) × (a−b)`. KaTeX renders the formula with live numeric values.
+
+## Requirements
+- Functional:
+  - Slider for `a` (range 2-10), slider for `b` (range 1-10, capped at `a`)
+  - SVG shows side-by-side: (1) big square with corner cut, (2) rearranged rectangle
+  - Numeric panel: `a² = N`, `b² = N`, `a² - b² = N`, `(a+b)(a-b) = N`
+  - KaTeX displays: `a^2 - b^2 = (a+b)(a-b)` with current numbers substituted
+  - Edge case `b = a` → both sides = 0; show degenerate state without crashing
+- Non-functional:
+  - No new engine module (algebra is `a*a - b*b`; YAGNI)
+  - Aligned to Cánh Diều lớp 7 chương "Hằng đẳng thức đáng nhớ"
+- A11y:
+  - Sliders have `<label>` + `aria-valuetext` (e.g. "a bằng 5")
+  - Numeric panel has `aria-live="polite"`
+  - Reduced motion: no transitions when `prefers-reduced-motion: reduce`
+
+## Architecture
+- Pure render lesson — no draggable, no engine module needed.
+- Math fits inside the lesson page; `a*a - b*b` is too small to extract.
+- SVG viewBox sized to fit `a + b ≤ 20` worst case; use `viewBox` units = math units * 20px.
+- Two SVG groups side-by-side, both inside one `<svg>`. CSS Grid for layout.
+
+## Related Code Files
+- Create: `src/lib/lessons/hieu-hai-binh-phuong/copy.vi.js`
+- Create: `src/routes/dai-so/hieu-hai-binh-phuong/+page.svelte`
+- Read for pattern: `src/routes/hinh-hoc/tam-giac-bang-nhau/+page.svelte` (existing SVG lesson shape)
+- Modify (Phase 04): `src/lib/lessons/registry.js`, root `+page.svelte`
+
+## Implementation Steps
+
+### 3.1 Copy module
+```js
+export const vi = {
+  slug: 'hieu-hai-binh-phuong',
+  topic: 'dai-so',
+  grade: 'lop-7',
+  title: 'Hiệu hai bình phương',
+  gradeLabel: 'Lớp 7',
+  intro: 'Hằng đẳng thức a² − b² = (a + b)(a − b). Kéo thanh trượt để thấy vì sao hình vuông cạnh a, cắt đi góc b, có cùng diện tích với hình chữ nhật (a+b)(a−b).',
+  instruction: 'Kéo hai thanh trượt để chọn a và b (với b ≤ a)',
+  identityTitle: 'Hằng đẳng thức',
+  identityStatement: 'Với mọi số thực a, b: a² − b² = (a + b)(a − b).',
+  exampleTitle: 'Ví dụ',
+  exampleBody: 'Tính 7² − 3². Áp dụng hằng đẳng thức: 7² − 3² = (7+3)(7−3) = 10 · 4 = 40. Cũng có thể tính trực tiếp: 49 − 9 = 40.',
+  numericTitle: 'Giá trị',
+  rearrangeNote: 'Phần hình chữ L (diện tích a² − b²) được cắt và ghép lại thành hình chữ nhật cạnh (a + b) × (a − b).',
+  nextTeaser: 'Sắp ra mắt: (a + b)² và (a − b)²',
+};
+```
+
+### 3.2 Lesson page state
+```js
+let a = $state(5);
+let b = $state(2);
+// keep b ≤ a:
+$effect(() => { if (b > a) b = a; });
+
+let aSq = $derived(a * a);
+let bSq = $derived(b * b);
+let diff = $derived(aSq - bSq);
+let product = $derived((a + b) * (a - b));  // always === diff; shown for pedagogy
+```
+
+### 3.3 SVG layout
+
+Two diagrams in one `<svg>`:
+
+**Left diagram — square minus corner**:
+- Big square: `<rect x=0 y=0 width=a*U height=a*U>` where `U = 20px`
+- Cut corner: `<rect x=(a-b)*U y=0 width=b*U height=b*U fill="white">`
+- Labels: "a" along top + left; "b" on the cut edges
+
+**Right diagram — rearranged rectangle**:
+- Two pieces side-by-side: top piece `(a-b) × (a+b)` rotated/translated to form `(a+b) × (a-b)`
+- v1 simplification: just draw the result rectangle `(a+b) × (a-b)` with the two regions colored differently to show the L-shape parts. Don't animate the rearrangement (WAAPI deferred per port spec).
+- Labels: "a + b" along width; "a − b" along height
+
+Color tokens (extend `tailwind.config.js` if needed):
+- L-shape part 1 (top strip): `pair-1`
+- L-shape part 2 (right strip): `pair-2`
+- Removed corner: white/transparent
+
+### 3.4 KaTeX panel
+```svelte
+<Tex display math={`${a}^2 - ${b}^2 = (${a}+${b})(${a}-${b}) = ${diff}`} />
+```
+Plus a static formula card:
+```svelte
+<Tex display math="a^2 - b^2 = (a+b)(a-b)" />
+```
+
+### 3.5 SEO meta
+- Same pattern as Phase 02: `<svelte:head>` with title/desc/canonical.
+
+## Success Criteria
+- [ ] Lesson renders at `/dai-so/hieu-hai-binh-phuong/`
+- [ ] Sliders update both diagrams + numeric panel + KaTeX in real time
+- [ ] `b > a` is auto-corrected via `$effect` (not visible to user as glitch)
+- [ ] `a = b` degenerate case renders without overlap or NaN
+- [ ] Reduced-motion respected (no transitions if user prefers it)
+- [ ] svelte-check clean
+
+## Risk Assessment
+- **Risk**: SVG rearrangement might be confusing without animation.
+  **Mitigation**: Static side-by-side with text annotation `rearrangeNote`. Animation in v2.
+- **Risk**: Sliders on touch devices may be too small.
+  **Mitigation**: Native `<input type="range">` is fine on mobile; add `accent-color` Tailwind class for theming.
+- **Risk**: Slider $effect to clamp `b ≤ a` may cause feedback loop.
+  **Mitigation**: `$effect` only writes when `b > a` (guard); Svelte 5 batches; tested manually.
+- **Risk**: Page exceeds 200 LOC.
+  **Mitigation**: Likely tight; if it grows, extract `<square-cut-svg>` and `<rectangle-svg>` as small components. Defer until needed.
+
+## Notes
+- Goal-state for this lesson: implicit (sandbox). User experiments freely. No `goal()` function needed.
+- KaTeX `<Tex>` is the only place we render math text. Don't manually compose math with HTML `<sup>`.
+- Slider primitive stays raw `<input type="range">` — extracting `<Slider>` is YAGNI at first use.
diff --git a/plans/260503-1121-lesson-4-5-gcd-factor/phase-04-registry-landing-glue.md b/plans/260503-1121-lesson-4-5-gcd-factor/phase-04-registry-landing-glue.md
new file mode 100644
index 0000000..306736a
--- /dev/null
+++ b/plans/260503-1121-lesson-4-5-gcd-factor/phase-04-registry-landing-glue.md
@@ -0,0 +1,107 @@
+---
+phase: 4
+title: "Registry + landing glue"
+status: pending
+priority: P2
+effort: "2h"
+dependencies: [2, 3]
+---
+
+# Phase 04: Registry + landing glue
+
+## Overview
+Wire the two new lessons into the lesson registry, enable the Số học and Đại số tiles on the root landing, and create topic landing pages at `/so-hoc/` and `/dai-so/`.
+
+## Requirements
+- Functional:
+  - `lessons` registry includes both new lessons (slug, topic, grade, copy)
+  - Root `/` page: Số học + Đại số tiles linked + clickable (no longer "Sắp ra mắt")
+  - `/so-hoc/` lists Ước chung lớn nhất; placeholder text for upcoming lessons
+  - `/dai-so/` lists Hiệu hai bình phương; placeholder text for upcoming lessons
+  - Hình học landing already exists; check it still works
+- Non-functional:
+  - Topic landing pages reuse the same component structure as `/hinh-hoc/`
+  - SEO meta on every new page
+
+## Architecture
+- Registry stays a single `lessons` array with helper `lessonsByTopic`. No restructuring.
+- Topic landing pages are nearly identical except for the topic filter; copy lives in `src/lib/i18n/site.vi.js` (extend if needed).
+- Root tile state derived from `lessonsByTopic('so-hoc').length > 0` etc. — auto-enables when first lesson per topic ships.
+
+## Related Code Files
+- Modify: `src/lib/lessons/registry.js` — import + push two new copies
+- Modify: `src/routes/+page.svelte` — root landing tile state
+- Modify: `src/lib/i18n/site.vi.js` — topic page chrome strings (if not already present)
+- Create: `src/routes/so-hoc/+page.svelte` — số học topic landing
+- Create: `src/routes/dai-so/+page.svelte` — đại số topic landing
+- Read for pattern: `src/routes/hinh-hoc/+page.svelte`
+
+## Implementation Steps
+
+### 4.1 Registry update
+```js
+// src/lib/lessons/registry.js
+import { vi as sssCopy } from './tam-giac-bang-nhau/copy.vi.js';
+import { vi as similarityCopy } from './tam-giac-dong-dang/copy.vi.js';
+import { vi as inscribedCopy } from './goc-noi-tiep/copy.vi.js';
+import { vi as gcdCopy } from './uoc-chung-lon-nhat/copy.vi.js';
+import { vi as diffSquaresCopy } from './hieu-hai-binh-phuong/copy.vi.js';
+
+export const lessons = [
+  gcdCopy,           // số học
+  diffSquaresCopy,   // đại số
+  sssCopy,           // hình học
+  similarityCopy,
+  inscribedCopy,
+];
+```
+Order: by topic → grade ascending. Nav order will follow the array.
+
+### 4.2 Root landing
+- Read current `src/routes/+page.svelte` first (likely uses hard-coded "Sắp ra mắt" labels)
+- Replace per-topic disabled state with derived check:
+  ```js
+  let hasSoHoc = $derived(lessonsByTopic('so-hoc').length > 0);
+  let hasDaiSo = $derived(lessonsByTopic('dai-so').length > 0);
+  let hasHinhHoc = $derived(lessonsByTopic('hinh-hoc').length > 0);
+  ```
+- Tile component:
+  - Enabled → `<a href="/so-hoc/">…</a>` with lesson count badge
+  - Disabled → `<div>` with "Sắp ra mắt" overlay (kept for future-empty topics, none currently)
+
+### 4.3 Topic landing pages
+Pattern for both `/so-hoc/+page.svelte` and `/dai-so/+page.svelte`:
+- Import `lessonsByTopic` from registry
+- Render a list of lesson cards (title, grade badge, intro snippet, link)
+- Order: by grade ascending, then by registry position
+- One placeholder card "Sắp ra mắt" for future lessons (optional; or omit)
+- `<svelte:head>` with topic-level SEO
+
+### 4.4 i18n site chrome
+- Check `src/lib/i18n/site.vi.js` for existing topic strings
+- Add if missing:
+  ```js
+  topics: {
+    'so-hoc': { title: 'Số học', tagline: '...' },
+    'dai-so': { title: 'Đại số', tagline: '...' },
+    'hinh-hoc': { title: 'Hình học', tagline: '...' },
+  }
+  ```
+
+## Success Criteria
+- [ ] Root page tiles for Số học + Đại số are clickable (linked to topic pages)
+- [ ] `/so-hoc/` shows Ước chung lớn nhất card → links to lesson
+- [ ] `/dai-so/` shows Hiệu hai bình phương card → links to lesson
+- [ ] `/hinh-hoc/` still shows all 3 existing lessons (regression check)
+- [ ] All routes prerender at build (no SSR/CSR mismatch)
+- [ ] svelte-check clean
+
+## Risk Assessment
+- **Risk**: Existing hinh-hoc landing pattern may not directly generalize.
+  **Mitigation**: Read it first; refactor to `lessonsByTopic(topic)` pattern if it isn't already.
+- **Risk**: Lesson order in registry affects nav consistency.
+  **Mitigation**: Document ordering rule in registry.js comment.
+
+## Notes
+- next/prev lesson navigation footer is **deferred** (port spec §13 Q4) — revisit at lesson 6+.
+- Grade-filter UI on topic pages also deferred (Sub-project §3.4).
diff --git a/plans/260503-1121-lesson-4-5-gcd-factor/phase-05-test-a11y-verify.md b/plans/260503-1121-lesson-4-5-gcd-factor/phase-05-test-a11y-verify.md
new file mode 100644
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--- /dev/null
+++ b/plans/260503-1121-lesson-4-5-gcd-factor/phase-05-test-a11y-verify.md
@@ -0,0 +1,104 @@
+---
+phase: 5
+title: "Test + a11y verify"
+status: pending
+priority: P2
+effort: "3h"
+dependencies: [4]
+---
+
+# Phase 05: Test + a11y verify
+
+## Overview
+Run full quality gate before declaring lessons 4 + 5 shippable: Vitest, svelte-check, build, and manual keyboard/screen-reader pass on both new lessons.
+
+## Requirements
+- Functional: build passes; all tests green; routes prerender to static HTML
+- Non-functional: bundle size delta logged; KaTeX impact documented
+- A11y: keyboard nav verified; ARIA-live announcements work; reduced-motion respected
+
+## Architecture
+N/A — verification phase.
+
+## Related Code Files
+- Modify (if issues found): any lesson files, draggable action, Tex component
+- Modify: `README.md` — add new lessons to status section
+- Possibly modify: `RUNBOOK.md` — note KaTeX bundle impact
+
+## Implementation Steps
+
+### 5.1 Automated checks
+1. `npm run check` — must be 0 errors, 0 warnings
+2. `npm run test` — must be all green; verify new `gcd.test.js` runs
+3. `npm run build` — must succeed; check `build/` for:
+   - `/so-hoc/uoc-chung-lon-nhat/index.html` exists, contains rendered KaTeX HTML
+   - `/dai-so/hieu-hai-binh-phuong/index.html` exists, contains rendered KaTeX HTML
+   - `/so-hoc/index.html`, `/dai-so/index.html`, root `index.html` present
+4. `npm run preview` — smoke-test all routes manually
+
+### 5.2 Bundle size delta
+- Note `build/_app/` total size before this plan vs after
+- Specifically: KaTeX JS chunk + CSS + fonts
+- Expected: ~300KB gzipped delta
+- Document in `plan.md` Success Criteria
+
+### 5.3 Manual a11y pass — GCD lesson
+1. Tab through page: number a → number b → reset → bước → tự động
+2. Enter on bước advances one row; aria-live announces new row
+3. Screen reader (use VoiceOver or NVDA) announces final gcd correctly
+4. Increase browser font size 200% — layout doesn't break
+
+### 5.4 Manual a11y pass — Diff-of-squares lesson
+1. Tab to slider a → arrow keys change value → aria-valuetext announces "a bằng N"
+2. Tab to slider b → same
+3. Numeric panel updates announced (aria-live)
+4. Enable `prefers-reduced-motion: reduce` (DevTools rendering tab) → no transitions visible
+5. SVG has `aria-label="Sơ đồ hiệu hai bình phương"` (or similar)
+
+### 5.5 Mobile / touch check
+1. Open dev URL on phone or DevTools mobile emulation (iPhone 12)
+2. GCD lesson: tap number inputs, native keypad shows; buttons big enough to tap
+3. Factor lesson: drag sliders with finger; numbers update smoothly
+4. No horizontal scroll on either page
+
+### 5.6 Sandbox cleanup
+- Decide: delete `/__sandbox/katex/` route OR guard with `import { dev } from '$app/environment'; if (!dev) error(404)`
+- Recommend: delete; it served Phase 01 purpose
+
+### 5.7 README update
+Update `README.md` Status section:
+```md
+## Status
+
+3 bài hình học + 2 bài đầu tiên cho số học và đại số đã ra mắt.
+
+- Lớp 6 — Ước chung lớn nhất (Euclid): `/so-hoc/uoc-chung-lon-nhat/`
+- Lớp 7 — Hiệu hai bình phương: `/dai-so/hieu-hai-binh-phuong/`
+- Lớp 7 — Tam giác bằng nhau (SSS): `/hinh-hoc/tam-giac-bang-nhau/`
+- Lớp 8 — Tam giác đồng dạng: `/hinh-hoc/tam-giac-dong-dang/`
+- Lớp 9 — Góc nội tiếp: `/hinh-hoc/goc-noi-tiep/`
+```
+
+### 5.8 RUNBOOK note
+Add KaTeX bundle impact note to `RUNBOOK.md` under a "Bundle" or "Performance" section (create if absent).
+
+## Success Criteria
+- [ ] All automated checks green (check, test, build, preview)
+- [ ] Bundle delta logged with actual KB number
+- [ ] Manual a11y pass complete on both new lessons (no findings, or findings filed in next plan)
+- [ ] Mobile/touch smoke test passes
+- [ ] Sandbox route removed or dev-guarded
+- [ ] README + RUNBOOK updated
+- [ ] No regressions in 3 existing geometry lessons (open each, verify drag still works)
+
+## Risk Assessment
+- **Risk**: KaTeX SSR may produce hydration warnings.
+  **Mitigation**: Captured in Phase 01; if it surfaces here, escalate.
+- **Risk**: Bundle delta exceeds 400KB gzipped (KaTeX is heavier than estimated).
+  **Mitigation**: Acceptable for math site; note in RUNBOOK; consider per-route lazy load in a future plan if Lighthouse perf drops below 90.
+- **Risk**: Manual a11y findings = blockers.
+  **Mitigation**: If found, fix in this phase; don't punt to next plan.
+
+## Notes
+- This phase is pure verify — no new features. Stay disciplined.
+- Don't add tests "for completeness" beyond gcd-engine. Lesson page integration tests are deferred (port spec §5: `@testing-library/svelte` is dropped).
diff --git a/plans/260503-1121-lesson-4-5-gcd-factor/plan.md b/plans/260503-1121-lesson-4-5-gcd-factor/plan.md
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+---
+title: "Lesson 4 + 5: GCD Euclidean game + factor a²-b² visualizer"
+status: in-progress
+created: 2026-05-03
+priority: P2
+---
+
+# Lesson 4 + 5 — GCD Euclidean + Factor a²-b²
+
+## Context
+- Brainstorm survey: `plans/reports/brainstorm-260503-1121-curriculum-logic-survey.md` §3.1
+- Port spec (conventions): `plans/reports/brainstorm-260430-2207-improved-port-spec.md`
+- Strategy: C (lesson-driven, infra emerges)
+- Textbook alignment: **Cánh Diều** (lớp 6 + lớp 7)
+
+## Lessons
+1. **Ước chung lớn nhất** (số học, lớp 6) — Euclidean algorithm visualizer at `/so-hoc/uoc-chung-lon-nhat/`
+2. **Hiệu hai bình phương** (đại số, lớp 7) — `a²-b² = (a+b)(a-b)` interactive at `/dai-so/hieu-hai-binh-phuong/`
+
+## New infra (only what these lessons force)
+- **KaTeX** — Tex.svelte component (algebra needs `\frac{a \cdot b}{\gcd(a,b)}` and `a^2-b^2`)
+- **`numtheory-engine/`** — pure module: `gcd`, `lcm`, `gcdSteps` (mirrors geom-engine shape)
+- **Goal-state pattern** — per-lesson `goal(state) → { reached, hint? }` (documented, not abstracted)
+
+## Explicit non-goals (YAGNI gate)
+- ❌ algebra-engine module (`a*a - b*b` is too small; defer until 2nd algebra lesson)
+- ❌ shared NumberInput / Slider components (use raw inputs)
+- ❌ shared GoalState type or quiz primitives (Sub-project §3.2 territory)
+- ❌ scoring, persistence, hint-cost, undo/redo (Sub-projects §3.2/§3.3)
+- ❌ EN locale (Sub-project §3.5)
+
+## Phases
+
+| # | Phase | Status | Owns |
+|---|-------|--------|------|
+| 01 | KaTeX integration | completed | `tex.svelte`, deps, smoke test (sandbox removed) |
+| 02 | Số học engine + GCD lesson | completed | `numtheory-engine/` (12 tests), `/so-hoc/uoc-chung-lon-nhat/` |
+| 03 | Đại số factor a²-b² lesson | completed | `/dai-so/hieu-hai-binh-phuong/` |
+| 04 | Registry + landing glue | completed | `registry.js`, `site.vi.js`, topic landings, root tiles |
+| 05 | Test + a11y verify | partial | check ✓, test 46/46 ✓, build ✓ — **manual a11y/mobile pending user verify** |
+
+## Outcomes
+- 5 routes added: `/so-hoc/`, `/so-hoc/uoc-chung-lon-nhat/`, `/dai-so/`, `/dai-so/hieu-hai-binh-phuong/` (and reuses `/hinh-hoc/`)
+- 12 new unit tests (numtheory-engine), all green; total 46/46 passing
+- KaTeX integrated SSR-style via `<Tex>` component; ~280KB CSS+fonts bundle delta
+- All three topic tiles on landing now live
+- README + RUNBOOK updated
+
+## Dependencies
+- Phase 01 blocks Phase 03 (algebra lesson needs `<Tex>`)
+- Phase 01 blocks Phase 02 (LCM formula uses `<Tex>`)
+- Phase 04 blocks Phase 05 (need glue done before final verify)
+- Phases 02 + 03 can run in parallel after 01
+
+## Success criteria
+- Both new lessons live at correct URLs, rendered server-side via SvelteKit prerender
+- KaTeX renders inline + display math with no FOUC
+- `npm run check` clean (svelte-check + JSDoc strict)
+- `npm run test` green (existing tests + new numtheory-engine tests)
+- Keyboard nav + ARIA work for all new interactive elements
+- Bundle size delta documented (KaTeX is ~300KB gzipped; expected)
+
+## Open questions (resolve during implementation)
+- Cánh Diều lớp 6 GCD example numbers — use canonical (48, 18) → 6 default?
+- Đại số lesson: slider range for `a` and `b` — recommend `1 ≤ b ≤ a ≤ 10` to keep visual clean
+- KaTeX font subset — load full or VN-friendly subset?
diff --git a/plans/reports/brainstorm-260503-1121-curriculum-logic-survey.md b/plans/reports/brainstorm-260503-1121-curriculum-logic-survey.md
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+++ b/plans/reports/brainstorm-260503-1121-curriculum-logic-survey.md
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+# Brainstorm — Curriculum Logic Survey
+
+- Date: 2026-05-03
+- Mode: brutal-honesty advisory. No code.
+- Scope: all logics applicable across MathMax (lớp 6-9, số học + đại số + hình học × 4 dimensions).
+- Predecessor: `plans/reports/brainstorm-260430-2207-improved-port-spec.md` (port spec, already shipped).
+- Conventions: tag = `[grade] [now|next|later|YAGNI]`. `now` = something a current lesson needs. `next` = next 1-2 lessons likely need. `later` = real but not soon. `YAGNI` = stop, don't build.
+
+## TL;DR
+
+1. Pick **Strategy C** (lesson-driven, infra emerges). Stop catalog-shopping; ship lessons.
+2. Next concrete sub-project: **one số học lesson + one đại số lesson** (you pick which). Brainstorm each separately.
+3. Don't build quiz/scoring/persistence/i18n-en until 2+ lessons demand them. Hard rule.
+4. Three things you almost-certainly need before lesson 4: KaTeX (algebra needs `\frac`/`\sqrt`), an "expression input" interaction primitive, and goal-state detection (sandbox-only stops working when you have homework-shaped lessons).
+5. Three things to **never** build: spaced-repetition, accounts/auth, custom analytics.
+
+---
+
+## 1 · Catalog
+
+### 1.1 Math content logic (pure modules; sibling of `geom-engine/`)
+
+**Số học (lớp 6-7)**
+- `[6][next]` divisibility tests (2/3/5/9/10/11) — pure predicates
+- `[6][next]` GCD/LCM via Euclidean — pure
+- `[6][next]` prime check + sieve up to N — pure
+- `[6][next]` prime factorization — pure
+- `[6][later]` modular arithmetic (mod, congruence) — pure
+- `[6][next]` fraction normalize / simplify / common-denominator — pure
+- `[6][next]` mixed ↔ improper fraction — pure
+- `[6][later]` decimal ↔ fraction (terminating + repeating) — pure
+- `[6][later]` percent / ratio / proportion — pure
+- `[6][next]` integer ops (negatives, abs, sign rules) — pure
+
+**Đại số (lớp 7-9)**
+- `[7][next]` polynomial: add/subtract/scale (1 var) — pure
+- `[7][next]` polynomial multiply (foil + general) — pure
+- `[8][next]` factoring: common factor, grouping, identities (a²-b², a²±2ab+b², a³±b³) — pure
+- `[8][later]` factoring: AC method for trinomials — pure
+- `[7][next]` linear equation 1 var: solve + step trace — pure
+- `[8][later]` linear system 2 var: substitution + elimination + step trace — pure
+- `[8][later]` inequality 1 var (with sign-flip on negative mult) — pure
+- `[9][later]` quadratic: discriminant, roots, Viète — pure
+- `[9][later]` rational expression: simplify, restrict domain — pure
+- `[9][later]` square root: simplify √n, conjugate denom — pure
+- `[7][later]` linear function: y = ax + b, plot, slope/intercept — pure + render
+- `[9][later]` quadratic function: y = ax² + bx + c, vertex, axis — pure + render
+
+**Hình học (lớp 6-9, beyond current 3 lessons)**
+- `[6][next]` polygon area/perimeter (rect, tri, parallelogram, trapezoid) — pure
+- `[7][later]` Pythagoras + converse — pure (already in vec via `len`/`dist`)
+- `[7][later]` triangle inequality predicate — pure
+- `[8][later]` transformations: translate, rotate, reflect, dilate (2D matrix or pointwise) — pure
+- `[8][later]` similarity: SAS/SSS/AA — pure (mathmax has SSS-position-strict; permutation-invariant variants are real lessons later)
+- `[9][later]` circle: chord/secant/tangent, arc length, inscribed quadrilateral — pure (extend `circle.js`)
+- `[9][later]` trig ratios (sin/cos/tan in right triangle) — pure
+- `[9][YAGNI]` 3D solids (cube/cylinder/cone/sphere V+SA) — pure but render layer is heavy; defer
+- `[8][later]` coordinate geometry: distance, midpoint, slope between points — pure
+- `[7][YAGNI]` compass+straightedge construction primitives — interaction layer is its own project; defer
+
+**Thống kê & Xác suất (lớp 7-9)**
+- `[7][later]` mean/median/mode/range — pure
+- `[7][later]` frequency table + bin — pure
+- `[8][later]` variance / std dev — pure
+- `[9][later]` basic probability (favorable/total, independent events) — pure
+- `[9][YAGNI]` combinatorics (nCr, nPr) for advanced lessons — defer
+
+### 1.2 Lesson interaction logic
+
+- `[now]` drag w/ pointer + keyboard + ARIA — exists (`use:draggable`)
+- `[now]` projector (snap to circle, line) — exists (used by inscribed angle)
+- `[next]` slider (one-knob parameter, e.g. similarity ratio k) — trivial; native `<input type="range">` w/ Tailwind
+- `[next]` snap-to-grid (integer or fraction lattice) — extend draggable action w/ `snap` opt
+- `[next]` snap-to-angle (15°/30°/45° increments) — variant of projector
+- `[next]` toggle/checkbox to show/hide auxiliary lines — Svelte `$state` only
+- `[next]` multi-choice tap (1-correct, multi-correct) — interaction primitive
+- `[next]` drag-bin categorize (sort items into buckets) — interaction primitive
+- `[next]` drag-reorder (sequence proof steps, sort numbers) — interaction primitive
+- `[next]` number input + validation — bare `<input type="number">` w/ derived feedback
+- `[later]` expression input (e.g. `2x+3`) + parser — needs grammar; large; only when first algebra lesson lands
+- `[later]` fraction input (numerator/denominator pair) — small variant of number input
+- `[later]` step-state machine (assertion → justification → next step) — for proof lessons
+- `[later]` undo/redo stack — generic over `$state` snapshots
+- `[later]` restart/reset to initial config — trivial; per-lesson save initial `$state`
+- `[later]` progressive hint reveal — `$state` index + `<details>` or button-driven
+- `[later]` animation playback (apply transformation step-by-step) — WAAPI or Svelte transitions
+- `[later]` plot point on coordinate plane (click to place) — needs canvas grid render
+- `[later]` draw line/ray (drag two points, render) — extension of draggable
+- `[YAGNI]` freehand sketch — overkill; pedagogy doesn't need it
+
+### 1.3 Pedagogy / scoring logic
+
+- `[now]` sandbox mode — current lessons are pure exploration; no goal
+- `[next]` goal-state detection (predicate: "is this a right triangle?", "did you simplify fully?") — emerges from lesson; pure
+- `[next]` correct/wrong feedback (visual + ARIA-live announcement) — primitive
+- `[next]` partial credit / "almost" feedback — judgment call per lesson
+- `[later]` exercise variants (seeded random problem generator per lesson) — pure
+- `[later]` mastery flag (3-correct-in-a-row) — localStorage key per lesson
+- `[later]` hint reveal w/ optional cost (star deduction) — needs scoring vocabulary
+- `[later]` streak counter — localStorage per user
+- `[later]` worked-example fading (fully-shown → partial → solo) — pedagogical pattern; high effort
+- `[later]` self-check button vs auto-grade — per-lesson UX choice
+- `[later]` quiz primitive: MC, free-response, drag-match, ordering — composes interaction primitives
+- `[YAGNI]` spaced repetition (SM-2 etc) — wrong tool for middle-school topic mastery
+- `[YAGNI]` adaptive difficulty (next-problem easier/harder by perf) — overkill; static pacing is fine
+- `[YAGNI]` leaderboards / social — out of scope for self-paced learning
+
+### 1.4 App-level logic
+
+- `[now]` lesson registry + per-lesson copy module — exists
+- `[next]` registry: add `prerequisites: []`, `tags: []`, `estimatedMinutes` fields — schema bump
+- `[next]` next/prev lesson nav (registry-driven, ordered by topic+grade) — only after lesson 4
+- `[next]` per-lesson `<svelte:head>` SEO (title, desc, canonical, OG) — pulled from copy module
+- `[next]` topic landing page enhancements (group by grade, show progress) — extends `/hinh-hoc/`
+- `[later]` grade-filter UI on topic landing — tabs or chips
+- `[later]` search/filter by tag, slug — client-side over registry (no server)
+- `[later]` localStorage persistence: completed lessons, last-state per lesson, prefs — pure JS module
+- `[later]` deep-link state: encode lesson config in URL hash (e.g. `#a=60,80&b=180,80`) — for sharing problems
+- `[later]` sitemap.xml generated at build — SvelteKit `+server.js` or static
+- `[later]` robots.txt — trivial static
+- `[later]` OG images per lesson — needs render pipeline (satori/playwright); defer to lesson 6+
+- `[later]` i18n: add EN parallel to copy.vi.js — only when EN audience signal exists
+- `[later]` dark mode — Tailwind `dark:` + prefers-color-scheme; cheap but needs design pass
+- `[later]` reduced motion — already gated in port spec; honor when animations land
+- `[later]` print stylesheet (worksheet export) — `@media print` rules
+- `[later]` PDF export of lesson — needs render pipeline; defer
+- `[later]` privacy-respecting analytics (Plausible / Umami) — one snippet; cheap; pick later
+- `[YAGNI]` user accounts / auth — static-site killer; don't add
+- `[YAGNI]` server-side anything (Postgres, API) — kills GitHub Pages deploy
+- `[YAGNI]` PWA / service worker / offline — premature optimization
+- `[YAGNI]` Sentry / error reporting — overkill for static educational site
+- `[YAGNI]` content CMS (Sanity/Contentful) — copy modules are the CMS
+
+---
+
+## 2 · Strategy comparison
+
+| Strategy | Pros | Cons | Verdict |
+|---|---|---|---|
+| **A. Topic-first** (finish hình học → số học → đại số) | Deep math-engine reuse per topic; topic-coherent UX | App-level features lag; users see uneven progress across topics | Reject — uneven curriculum hurts perceived completeness |
+| **B. Infra-first** (build scoring + persistence + i18n + registry v2 before lessons) | Each new lesson cheap once infra exists | YAGNI violation; infra built without consumers always rots; 3 lessons doesn't justify it | Reject — recipe for over-engineering |
+| **C. Lesson-driven, infra emerges** *(recommended)* | Pure YAGNI; every line has a consumer; matches how the geom-engine actually evolved | Some rework when 3rd consumer reveals shared shape | **Adopt** — proven by the port itself |
+
+**Why C wins**: the existing geom-engine (vec / triangle / circle / ticks) emerged exactly this way — it's pure and small because every function has 1+ caller. Doing infra-first now would invert that virtue.
+
+---
+
+## 3 · Decomposition into sub-projects
+
+Each sub-project = its own brainstorm → plan → implement cycle. Order is **suggested**, not enforced; pick what unblocks the most lessons.
+
+### 3.1 Sub-project: "Lesson 4 + 5" (number theory + algebra debut)
+- Pick 1 số học lesson (recommend: **GCD/LCM via Euclidean game** — drag two numbers; watch ladder reduce; goal: hit (gcd, 0))
+- Pick 1 đại số lesson (recommend: **factor a² - b² visualizer** — drag two square sides; show area = (a+b)(a-b))
+- Triggers: KaTeX (almost certainly), goal-state detection primitive, possibly slider primitive
+- Don't build: scoring system, persistence, hint UI, undo
+- Risk: LOW (both lessons are bounded, single-screen, single-goal)
+
+### 3.2 Sub-project: "Quiz primitives" (only after 2+ lessons need it)
+- MC component, free-response w/ check, drag-match
+- Reuses the same `goal-state` predicate shape from §3.1
+- Schema for quiz items in copy modules (questions, choices, correct, explain)
+- Triggers: per-lesson "Bài tập" tab vs sandbox-only
+- Don't build: scoring back-end, attempt history (defer to §3.3)
+- Risk: LOW-MEDIUM (component design needs 2 real lessons to validate)
+
+### 3.3 Sub-project: "Local progress" (localStorage)
+- Schema: `mathmax/v1/progress` → `{ [lessonSlug]: { completed: bool, lastVisited: ISOString, attempts: number } }`
+- Pure module + Svelte store wrapper
+- Migration story: version key in storage; if missing, treat as fresh
+- Triggers: first lesson w/ "completion" state; landing page shows ✓ on completed lessons
+- Don't build: cross-device sync, accounts, server persistence
+- Risk: LOW (pure browser API)
+
+### 3.4 Sub-project: "Registry v2 + nav" (after 6+ lessons)
+- Add `prerequisites`, `tags`, `estimatedMinutes`, `order` to registry entries
+- Generate prev/next links at build time
+- Topic landing page groups lessons by grade with progress badges
+- Triggers: lesson list becomes too long to scan; users ask "what's next"
+- Don't build: complex search; tags-cloud UI
+- Risk: LOW (just data shape; no new behavior)
+
+### 3.5 Sub-project: "EN locale" (only with audience signal)
+- Parallel `copy.en.js` per lesson + `site.en.js`
+- Locale switcher in header (defaults to `vi`; remembers in localStorage)
+- URL strategy: `?lang=en` query param OR `/en/...` prefix — decide later
+- Triggers: external request, partner school, analytics showing non-VN traffic
+- Don't build: third locale, automatic translation, RTL
+- Risk: LOW once triggered; HIGH effort otherwise (translation cost is the actual blocker)
+
+### 3.6 Sub-project: "Polish" (after 6+ lessons)
+- Sitemap.xml + robots.txt
+- OG images (per-lesson; consider Satori @ build)
+- Dark mode (design pass)
+- Print stylesheet for paper worksheets
+- Analytics (Plausible — privacy-respecting)
+- Triggers: lesson count past 6; SEO becomes worth caring about
+- Risk: LOW (mostly mechanical)
+
+---
+
+## 4 · Cross-cutting recommendations
+
+### Math expression rendering (KaTeX trigger)
+First algebra lesson needs `\frac{a}{b}`, `\sqrt{x}`, `x^2`. Add KaTeX **then**, not before. Pin one place to render: a tiny `<Tex math="..." />` Svelte component wrapping `katex.render`.
+
+### Goal-state detection pattern
+Don't build a "scoring engine." Each lesson exports a `goal(state) → { reached: bool, hint?: string }` pure function. Page renders feedback. Three lessons later, harvest the shape into a shared interface. Don't pre-design.
+
+### Pure-module discipline
+Every math content logic above goes under `src/lib/<topic>-engine/` (e.g. `numtheory-engine/`, `algebra-engine/`). Same shape as `geom-engine/`: pure JS, JSDoc, Vitest, no DOM. Lessons import. Period.
+
+### "Sắp ra mắt" guard
+Don't enable Số học / Đại số landing tiles until **at least one lesson per topic ships**. Empty topics are worse than missing ones.
+
+### Vietnamese-first slugs
+Every new lesson: VN slug (`uoc-chung-lon-nhat`, not `gcd-game`). Confirmed in port spec.
+
+### A11y by default
+Every new interactive primitive: keyboard equivalent + ARIA from day one. The draggable action set the bar; don't regress.
+
+---
+
+## 5 · Open questions
+
+1. **Which two lessons next?** §3.1 recommends GCD-Euclidean and factor-a²-b² but you may have stronger pedagogy preferences (textbook alignment, school partnerships, etc.).
+2. **KaTeX bundle size** — KaTeX is ~300KB gzipped w/ fonts. Acceptable for a math site? (Recommend yes; alternative `MathJax` is bigger.)
+3. **Analytics provider** — Plausible (paid hosted), Umami (self-host), or none? Defer until §3.6.
+4. **Lesson "completion" definition** — does sandbox-only count as complete on first visit, or do we require a goal? Affects §3.3 schema.
+5. **EN locale priority** — do you have any concrete audience signal (collaborators, partner schools, GitHub stars from EN-speakers), or is it purely speculative? If speculative, drop to "never" until signal arrives.
+6. **Worksheet/print export demand** — is this asked-for by teachers, or my speculation? If speculation, drop.
+7. **Curriculum source-of-truth** — are lessons aligned to a specific Vietnamese textbook (Cánh Diều, Kết nối tri thức, Chân trời sáng tạo)? If yes, that constrains lesson selection and ordering.
+
+---
+
+**Status:** DONE
+**Summary:** Survey of all 4-dimension logics (math content × interaction × pedagogy × app), with grade + risk tags. Recommended Strategy C (lesson-driven). Six concrete sub-projects scoped, ordered, and gated on real triggers. Hard YAGNI rules: no SR, no auth, no PWA, no analytics until lesson 6+.
+**Concerns:** Q7 (textbook alignment) is the biggest unknown — may rewrite §3.1 lesson picks. Q1 needs your call before any next planning step.