Before rendering, confirm that python3 "$EDU_SKILL_ROOT/scripts/precheck.py" dist was already run in Step 5 and printed ALL CHECKS PASSED. If you skipped it or are unsure, run it now — it takes seconds. Do NOT render with failing prechecks.

npx hyperframes lint and npx hyperframes validate pass with zero errors (run from dist/). Video rendered to MP4 via npx hyperframes render inside dist/. The output .mp4 file must exist inside dist/ and be reported to the user. Preview-only is NOT sufficient — rendering is mandatory. After rendering: check_render_overlap.py passes AND you have extracted+viewed a frame from every box/connector/label/chart scene and confirmed no clipped/covered text, no line-into-box, no misaligned boxes — fixing coordinates and re-rendering until clean. AND python3 "$EDU_SKILL_ROOT/scripts/postcheck.py" dist prints ALL POST-RENDER CHECKS PASSED (every line/curve the scene should draw is actually painted in output.mp4 — 线/曲线没漏画).

Design System

Read design-system.md before writing ANY HTML. It defines the "Aurora Scholar" light-themed visual identity. Use its exact color tokens, font specs, and component styles. Do not invent colors.

Problem Type Reference

Non-Negotiable Rules

1. Caption text from original script (字幕必须用原始脚本文本). Video captions/subtitles MUST use text from captions.json (which contains the original narration script text with timestamps measured from TTS output). The timestamps are 100% accurate because they are measured from each sentence's actual TTS audio duration during sentence-by-sentence synthesis. The OpenCC t2s conversion is no longer needed — transcript text comes directly from the original script.
2. KaTeX for layout-dependent math; plain HTML/Chinese for simple symbols (复杂公式用KaTeX，简单符号用HTML或中文). Use KaTeX only when the expression needs math layout features (fractions, roots, summations, matrices). For simple comparisons and standalone symbols (≤, ≥, °, ×), prefer HTML entities or Chinese text directly — e.g., write <span>OP' ≤ 1</span> not katex.render("OP' \\leq 1", ...) — this avoids the fragile JS backslash-escaping pipeline entirely (see Rule #25). Never display raw LaTeX source code. When you DO use KaTeX in JS strings, ALWAYS double-escape LaTeX backslashes — write "\\\\dfrac{1}{2}" not "\\dfrac{1}{2}". The \\d escape is silent: JS turns \\d → d, so \\dfrac becomes dfrac and KaTeX renders italic text "dfrac12" instead of a fraction. This is the single most common rendering bug. After building all compositions, run python3 "$EDU_SKILL_ROOT/scripts/check_katex_escaping.py" dist and fix every reported line BEFORE proceeding to Step 6.
3. Chinese pacing. Narration at 3.5-4.0 characters/second. Leave 0.5-1.0s pauses between steps.
4. Three layers per scene. Background treatment (wave texture + aurora mesh orbs) + content layer + accent elements. No flat single-layer scenes. See design-system.md "Background Treatment" for the exact CSS pattern and aurora palette guide.
5. Solid opaque panels — frosted glass is FORBIDDEN (禁止毛玻璃，完全避免遮挡). All content panels use the solid panel style from design-system.md — OPAQUE background:#ffffff (or white alpha ≥ 0.92), depth from borders + layered box-shadow + inset top highlight. NEVER use backdrop-filter / -webkit-backdrop-filter, and NEVER a see-through translucent panel background — a blurred/see-through panel washes out and OCCLUDES the problem text / diagram / formulas behind it (this is a hard defect). Enforced by scripts/check_no_glass.py (in precheck.py): render is blocked if any glass/translucent panel is found.
6. SVG geometry. Geometric figures use SVG path drawing animation, never static images.
7. Deterministic. No Math.random(), Date.now(), or async timeline construction.
8. Delegate. Use the hyperframes skill for composition rules, hyperframes-cli for CLI commands. TTS uses DashScope Qwen-TTS via the official SDK (dashscope.MultiModalConversation, model qwen3-tts-flash, HTTP) — no self-hosted node, no URL wiring; needs DASHSCOPE_API_KEY. Synthesize sentences concurrently (thread pool) for speed, and take per-sentence timestamps from each returned clip's measured duration (see step-3-voice-generation.md). This skill defines the math-tutorial domain logic only.
9. Dark text on light backgrounds (浅色背景深色文字). This is a light-theme design system. All text — KaTeX equations, SVG labels, Chinese body text, HTML table cells — MUST use dark colors (#0f172a or darker). Every composition MUST include the full "Mandatory Global Color Reset" block from design-system.md: root selector with color: #0f172a, .katex, .katex * { color: #0f172a; }, .katex-mathml { display: none !important; }, and table, th, td { color: #0f172a; }. Never use #fff, #f8fafc, #e8ecf4, or any light color for text on the light background.
10. Pre-built assets first. Before writing custom HTML/CSS for any visual object (car, train, candle, battery, lens, etc.), check assets/ASSET_CATALOG.md. If a matching component exists, copy its CSS and HTML verbatim. If NO match exists, follow the "Quality Fallback Template" in ASSET_CATALOG.md — every custom object must have ≥3 gradient layers, inset shadows, ground shadow, glow halo, and no CSS @keyframes. Compare against the candle component for quality level.
11. No emoji. Never use emoji characters (🔧⚙️🔵🔴🟢⭕📐✅❌⚡💡 etc.) anywhere in visible text — titles, labels, captions, formula notes, phase titles, badge text, or any string rendered in the video. Headless Chromium has no emoji font installed; all emoji render as □ (tofu boxes) in the final video. Use plain Chinese text or SVG/CSS shapes instead.
12. Math symbols: simple via HTML entities, complex via KaTeX — NO Chinese inside KaTeX (简单符号用HTML实体，复杂公式用KaTeX，KaTeX禁止包含中文). Common math comparison and operator symbols are safe as HTML entities in Noto Sans SC / Inter: ≤ (≤), ≥ (≥), ≠ (≠), ° (°), × (×), ÷ (÷), ± (±), ² (²), ³ (³) — use these directly in HTML text instead of KaTeX when no layout structure is needed (see Rule #25). Greek letters (α, β, γ, δ, ε, θ, λ, μ, π, φ, ω) are safe as Unicode characters in HTML text — Inter-Variable.woff2 contains all Greek glyphs (Noto Sans SC does NOT). The font-family stack "Noto Sans SC", Inter, sans-serif will fall back to Inter for Greek. NEVER write the English word ("alpha", "beta", "theta") — always use the Unicode character (α, β, θ). Do NOT insert other specialized Unicode math symbols (∠, △, ⊥, ∥, √, ∞, ₁, ₂, etc.) directly into HTML text — use KaTeX for these in formulas, or Chinese equivalents in non-formula context ("角ABC" not "∠ABC", "三角形" not "△"). For inline subscripts/superscripts in labels, use HTML <sub>/<sup> tags (e.g., F<sub>1</sub> not F₁). The only safe non-ASCII characters in plain text are standard CJK Unified Ideographs (U+4E00–U+9FFF), common CJK punctuation, the HTML-entity math symbols listed above, and Greek letters (U+0370–U+03FF). CRITICAL: Never put Chinese text inside KaTeX \text{...} or any KaTeX command. KaTeX's math fonts (KaTeX_Main, KaTeX_Math, etc.) do NOT contain CJK glyphs — any Chinese character inside \text{} renders as □ tofu boxes. When a formula needs to be mixed with Chinese text, break it into separate HTML elements:

    <!-- WRONG — Chinese in \text{} renders as □□□ -->
    <span id="eq" data-tex="n \text{ 还是 } V"></span>
    
    <!-- CORRECT — Chinese in HTML, math in KaTeX -->
    <span>判断操作对 </span><span id="eq-n" data-tex="n"></span><span> 或 </span><span id="eq-v" data-tex="V"></span><span> 的影响</span>
    

13. KaTeX CSS must be inlined, fonts self-hosted (字幕和公式的CSS必须内联，字体离线). The HyperFrames compiler processes CDN <link rel="stylesheet"> tags by extracting ONLY @font-face rules and discarding all other CSS. This silently breaks KaTeX layout (fractions, subscripts, spacing render as flat plain text). Never use <link rel="stylesheet" href="...katex.min.css">. Instead: copy the shipped assets/katex/katex.min.css into dist/katex/, rewrite its font URLs to the local self-hosted path (url(./katex/fonts/), and paste the full CSS as <style id="katex-inline-css"> in every composition and in index.html. Load katex.min.js from the local dist/katex/ copy, not a CDN. See step-5 for the exact procedure.
14. Circuit schematic physics accuracy (电路原理图物理正确性). When drawing circuit schematic diagrams (电路图) in SVG: (a) battery symbol — 长线 = 正极(+), 短线 = 负极(-), never reverse the labels; (b) current direction — conventional current flows from battery + terminal through external circuit to - terminal, all arrows must form a consistent closed loop; (c) ammeter — must be wired in series, current enters + terminal; (d) voltmeter — must be wired in parallel with dashed branch wires (stroke-dasharray), + terminal toward higher potential; (e) wire segmentation (导线分段) — wires must STOP at component terminals, never draw a continuous wire that passes through components (components are NOT decorative overlays on wires); (f) switch must break the circuit (开关必须断路) — incoming wire ends at switch pivot, outgoing wire starts at contact terminal, switch orientation must match wire direction (vertical wire → vertical switch), no zero-length line segments; (g) layout distribution (布局分布) — distribute series components across multiple sides of the rectangular loop, do NOT stack all components on one side; an empty side of the loop may be JUST a wire (导线) — never add/duplicate a component only to "fill" or "balance" a side; (h) no duplicate wires — each wire gap between components is drawn exactly once; (h2) component inventory correctness (元件清单正确性) — 每个真实元件恰好出现一次 — the set of components must match the problem's actual circuit; single-instance instruments (变阻器/滑动变阻器, 电流表, 电压表, 开关, 电源) must each appear EXACTLY ONCE — never invent or duplicate a component (drawing two 变阻器 in series is a physics error). Only true multiples in the problem (e.g. 两个灯泡 L₁/L₂, 两个电阻 R₁/R₂) may repeat. check_circuit_inventory.py gates this; (h3) circuit loop MUST be closed / 电源两端必须都接线 (回路闭合) — the main series loop is a single closed path: trace it from the 电源 + terminal through every component back to the 电源 - terminal — every gap must have a wire, and the power source's BOTH terminals must each connect to a wire. No unwired component terminal, no dangling wire stub (悬空导线端点 — a wire end that lands in empty space instead of on a component terminal or another wire at a corner). In physical wiring diagrams whose wire coordinates live in a JS array (var wires=[["w1",x1,y1,x2,y2],…]), it is easy to forget the segment that closes the left side back up to the source — do not. check_circuit_closed.py gates this; (i) physical wiring diagram layout (实物连接图布局) — "连接电表" and "连接主回路" operation scenes must use a rectangular loop layout (NOT a flat horizontal line) with ammeter IN the series loop (on the bottom return wire) and voltmeter on a visually separate dashed parallel branch; wire routing stays compact (≤100px margin beyond components); use Circuit Wiring Operation (C12) template from math-components.md, NOT Chemistry Operation Flow (C9); see circuit-schematic-guide.md Section 12; (j) voltmeter connection wires must not protrude (电压表连接导线禁止突出) — the sch-voltmeter template has NO built-in wire stubs; terminal endpoints are at the circle edge (±30 from center, not ±50); all dashed connection wires must be drawn as separate <line> elements from the main circuit T-junction point to the voltmeter circle edge — wires must start at the junction and end at the circle edge, never extending past either point; (k) voltmeter branch junctions must have dots and clean routing (电压表分支点必须有圆点且路由干净) — every T-junction where a voltmeter dashed wire meets the main circuit wire must have a filled junction dot (<circle r="4-5">); the dashed wire endpoint coordinates must be exactly ON the main wire (no offset); position the voltmeter so its terminals align with junction points to allow straight dashed lines; if L-shaped routing is unavoidable, use a single <path> with stroke-linejoin="round" — NEVER two separate <line> elements (they create gaps/protrusions at corners). Use sch-* templates from assets/ASSET_CATALOG.md and read references/circuit-schematic-guide.md for the full pre-flight checklist.
15. Caption formulas use KaTeX inline rendering (字幕公式必须用KaTeX渲染). Video captions must display proper math notation — never show spoken-form Chinese like "G M m除以R的平方". When assembling caption <div> elements in index.html, replace all spoken-form math expressions from captions.json with <span class="cm">LaTeX code</span> elements. The root index.html must load KaTeX JS and include a script that renders all .cm spans with katex.render() in inline mode. Use \tfrac for fractions (compact, fits single-line captions). KaTeX angle pitfall: for degrees use ^\circ, for arcminutes use a plain ASCII ' (prime), for arcseconds use ''. NEVER use \' or \" in KaTeX math — these are text-mode accent commands that cause the entire formula to render as red error text. See step-6 for the full template, common replacements table, KaTeX pitfalls, and examples.
16. Self-hosted offline fonts — Chinese MUST NOT be tofu (中文字体必须离线内嵌). The font files are shipped with this skill under assets/fonts/ and MUST be copied into dist/assets/fonts/ (see step-5 Prerequisites). Every sub-composition file AND the root index.html MUST embed them via an inline @font-face <style> block (the exact block is in step-6 for the root and assets/k12-scholar-font-template.css for compositions). Do NOT use a Google Fonts / CDN <link> and do NOT rely on the compiler's built-in embedding or system fonts (PingFang SC): the render sandbox is air-gapped, so a CDN font that fails to load makes ALL Chinese render as garbled boxes (乱码). Use a CJK-first stack everywhere — font-family: "Noto Sans SC", Inter, sans-serif — and never Inter, sans-serif alone. This applies no matter HOW font-family is set — SVG <text> attr, inline style=, AND CSS class rules. The #1 miss is a small label/eyebrow/badge class (e.g. .block-number{font-family:Inter,sans-serif}) that the design intends for a Latin number like "01" but the model fills with Chinese (对称性/题目所给) → those 3–4 characters render as 豆腐块 while the rest of the card is fine. Any class that could ever hold Chinese must include "Noto Sans SC". Enforced by scripts/check_cjk_font.py (now also scans class-based font-family, not just SVG/inline).
17. No CDN/external URLs at render time (渲染时禁止任何CDN/外部URL). Every <script src>, <link href>, and CSS url() in compositions and index.html MUST use local relative paths. The AP render sandbox is air-gapped with no internet access — CDN URLs that load successfully on a local machine will silently fail on AP, breaking GSAP timelines (no animations), KaTeX rendering (raw LaTeX in captions), and font loading (tofu boxes). Specifically: (a) GSAP — load from ./gsap/gsap.min.js (shipped in assets/gsap/, copied to dist/gsap/ AND dist/compositions/gsap/ in step-5); (b) KaTeX JS — load from ./katex/katex.min.js; (c) KaTeX CSS — inline as <style>, fonts from ./katex/fonts/; (d) body fonts — inline @font-face from ./assets/fonts/. Never use cdn.jsdelivr.net, cdnjs.cloudflare.com, unpkg.com, or any other external host in a src or href attribute. CRITICAL: sub-compositions in dist/compositions/ resolve ./ relative to their own directory — the assets MUST be mirrored into dist/compositions/ (gsap/, katex/, assets/) or all ./ paths silently 404 and GSAP/KaTeX/fonts fail to load (blank scenes, no animations, tofu). Enforced by scripts/check_asset_mirror.py in precheck.py.
18. Defensive rendering — no CSS-hidden content (防御性渲染 — CSS禁止隐藏内容). The #1 cause of blank videos is CSS opacity: 0 on .scene-content or content elements, combined with a JS error that prevents GSAP from revealing them. Rules: (a) Never set opacity: 0, visibility: hidden, or display: none on any content element in CSS — GSAP handles the initial hidden state dynamically via autoAlpha: 0 in fromTo(); (b) Wrap the entire GSAP timeline construction in a try-catch block — if any tween fails, the timeline still registers and content stays visible; (c) Each katex.render() call must have its own try-catch — one bad LaTeX string must not crash the entire composition; (d) Always pass throwOnError: false to katex.render(). See step-5 "Defensive Script Pattern" for the complete code template.
19. Reproducibility — stable structure every run (稳定复现 — 结构固定). The same problem must yield the same well-formed video every time; the biggest source of run-to-run quality swing is the BUILD step deviating from the plan. Rules: (a) Fixed output path — scaffold and render in exactly ./dist at the workspace root; never nest it (math-tutorial-output/dist etc.) — the renderer and gates assume ./dist. (b) One composition per storyboard scene — build exactly the scenes listed in STORYBOARD.md, one concept/ray/step per scene; never merge several planned scenes into one file, never skip a planned scene. index.html must wire ≥ the planned scene count. (c) Every scene self-contained & complete — each composition draws its full content (e.g. an optics scene MUST include the horizontal principal axis through O with 2F/F/O/F′/2F′ marked) and registers window.__timelines[...]. (d) Determinism — no Math.random() / Date.now() / async timeline construction (see rule #7). (e) These are enforced deterministically by $EDU_SKILL_ROOT/scripts/check_scene_coverage.py and $EDU_SKILL_ROOT/scripts/check_no_hidden_content.py inside $EDU_SKILL_ROOT/scripts/precheck.py — precheck.py dist must print ALL CHECKS PASSED before rendering.
20. Root MUST reference scenes via data-composition-src (根合成引用名固定 — 否则全白). In dist/index.html, every scene clip MUST use the EXACT attribute data-composition-src="compositions/scene-X.html" — never data-src, src, or a missing src. HyperFrames only loads/compiles an external scene when it sees data-composition-src; any other name means no scene loads and the whole video renders white (only the caption bar shows). Also: each clip's data-composition-id must match that scene's registered window.__timelines["<id>"], and scenes are laid out directly inside the root composition div (do NOT wrap them in an extra data-composition-id="mt-root" container). Enforced by scripts/check_root_compositions.py in precheck.py.
21. 几何图形坐标必须数学精确 (Geometry SVG coordinates must be mathematically exact). When drawing geometric figures in SVG (Geometry Canvas scenes), all vertex coordinates MUST be computed from the problem's mathematical constraints (perpendicularity, parallelism, similarity ratios, length ratios, etc.) — never placed by visual estimation. Mandatory steps: (a) solve the problem's equations to get exact proportional relationships between all segments before writing any SVG; (b) choose a base unit in the SVG viewBox, then compute every point's coordinates from the ratios; (c) verify perpendicularity — if two lines are perpendicular, compute the dot product of their direction vectors and confirm it equals 0; (d) verify length ratios — compute Euclidean distances and confirm they match the problem's given ratios; (e) write a <!-- GEOMETRY VERIFICATION --> block above the SVG with a POINTS: line declaring all coordinates and ASSERT lines for every problem constraint (parallel, perpendicular, midpoint, on_line, on_segment, intersection, ratio, collinear) — the pre-render gate scripts/check_geometry_verification.py dist mathematically verifies every assertion against the declared coordinates and exits 0; (f) right angle marks (直角符号) must be drawn as small L-shaped squares (not V-shapes or arcs), with edges computed from the actual line direction unit vectors — see step-5-build-components.md "Geometry Coordinate Accuracy" for the verification block format, assertion types, and construction algorithms.
22. 角度弧线必须用叉积确定扫描方向 (Angle arcs must use cross-product to determine sweep direction). When marking a non-right angle at vertex V with arms pointing to adjacent vertices P1 and P2 in SVG, the arc MUST sweep through the interior of the angle being marked. The sweep direction is determined by the cross product of the two arm vectors — never by visual estimation or guesswork. Algorithm:

    // 1. Direction vectors from vertex V to each arm endpoint
    dx1 = P1.x - V.x;  dy1 = P1.y - V.y;
    dx2 = P2.x - V.x;  dy2 = P2.y - V.y;
    
    // 2. Cross product (in SVG screen coordinates, y-down)
    cross = dx1 * dy2 - dy1 * dx2;
    
    // 3. Arc start/end points on each arm at radius distance
    angle1 = Math.atan2(dy1, dx1);
    angle2 = Math.atan2(dy2, dx2);
    startX = V.x + R * Math.cos(angle1);
    startY = V.y + R * Math.sin(angle1);
    endX   = V.x + R * Math.cos(angle2);
    endY   = V.y + R * Math.sin(angle2);
    
    // 4. Sweep flag: cross > 0 → interior is CW → sweep=1
    //                cross < 0 → interior is CCW → sweep=0
    sweepFlag = cross > 0 ? 1 : 0;
    
    // 5. SVG path (large-arc-flag = 0 for angles < 180°)
    d = `M ${startX},${startY} A ${R},${R} 0 0,${sweepFlag} ${endX},${endY}`;
    

    Worked example — rhombus with ∠A = 60°, ∠B = 120°:

    // Vertices: A(200,400) B(400,180) C(600,400) D(400,620), R=30
    
    // ∠A = 60° — arms AB and AD
    dx1=200, dy1=-220; dx2=200, dy2=220;
    cross = 200*220 - (-220)*200 = 88000 > 0 → sweep=1
    // Arc sweeps CW through interior (rightward between AB and AD) ✅
    
    // ∠B = 120° — arms BA and BC
    dx1=-200, dy1=220; dx2=200, dy2=220;
    cross = (-200)*220 - 220*200 = -88000 < 0 → sweep=0
    // Arc sweeps CCW through interior (downward between BA and BC) ✅
    

    Common mistakes: (a) Hardcoding sweep-flag=1 or sweep-flag=0 without computing cross product — causes arcs to appear on the exterior (reflex) side of the angle; (b) Drawing the arc between the wrong pair of arms (e.g., marking ∠ABC but using arms BA and BD); (c) Using a right-angle L-shaped mark for non-90° angles; (d) Forgetting that SVG y-axis points down, which reverses the visual rotation direction. Always verify: visually, the arc must curve TOWARD the interior of the polygon, not away from it.
23. 坐标分隔符只用逗号，禁止分号 (Coordinate separators: comma only, never semicolon). When writing coordinate pairs, point notation, or ordered tuples in KaTeX (e.g., vertex coordinates, intersection points), use only a comma as the separator between components — never a semicolon, and never both. Add a KaTeX thin space \, after the comma for readability. JS string escaping pitfall: in a JavaScript string passed to katex.render(), write \\, (double backslash) so KaTeX receives \,. Writing \; in a JS string does NOT produce a KaTeX thick space — JS treats the unknown escape \; as a literal semicolon character ;, so KaTeX renders a visible ,; (comma followed by semicolon). This is a confirmed rendering bug. Correct examples:

    // In JS strings (double-escaped for JS → KaTeX):
    katex.render("(1,\\, b-1)", el);          // ✅ renders (1, b−1)
    katex.render("(x_0,\\, y_0)", el);        // ✅ renders (x₀, y₀)
    
    // WRONG:
    katex.render("(1,\; b-1)", el);           // ❌ renders (1,; b−1) — JS turns \; into literal ;
    katex.render("(1,\\; b-1)", el);          // ❌ renders (1, b−1) with excessive space
    
    // In HTML (e.g., <span class="cm"> for caption KaTeX):
    <span class="cm">(1,\, b-1)</span>        // ✅
    <span class="cm">(1,\; b-1)</span>        // ❌ overly wide space, inconsistent
    

    General rule: coordinates are (a,\, b) in LaTeX, (a,\\, b) in JS strings. No semicolons anywhere in coordinate notation — this is Chinese math convention (中国数学用逗号分隔坐标，不用分号).
24. Sub-composition files must be HTML fragments, NOT full documents (子合成文件必须是HTML片段，不能是完整HTML文档). Every file in dist/compositions/*.html must be an HTML fragment — a root <div data-composition-id="..."> with inline <style> and <script> blocks. NEVER wrap sub-compositions in <!doctype html>, <html>, <head>, or <body> tags. HyperFrames loads sub-compositions by injecting their content into the parent document; a full HTML document wrapper prevents the renderer from properly executing GSAP scripts inside the sub-composition, causing all content elements to remain invisible (blank white panels with only background texture showing — audio and captions play normally, making the bug deceptive). Only the root index.html is a full HTML document. Enforced by scripts/check_composition_format.py in precheck.py.
25. 简单符号优先用中文或HTML实体，避免不必要的KaTeX (Prefer Chinese/HTML entities over KaTeX for simple math symbols). KaTeX rendering in JS strings requires double-escaped backslashes (\\leq not \leq); a single-backslash error is silent — JS eats \l→l, \a→a, \c→c, producing garbled text like "OP'leq1" or "angleMPNleq90^circ" instead of proper symbols. To avoid this fragile pipeline, prefer plain HTML or Chinese text whenever the expression does NOT need math layout features (fractions, roots, matrices, complex sub/superscripts). Decision table:

    Expression type	Display goal	Method	Code
    Simple comparison	OP' ≤ 1	HTML entity	<span>OP' ≤ 1</span> or OP' &le; 1
    Angle expression	角MPN ≤ 90°	Chinese + entity	<span>角MPN ≤ 90°</span>
    Degree symbol	90°	HTML entity	90&deg; or 90°
    Multiplication	3 × 4	HTML entity	3 &times; 4
    Simple inequality	x ≥ 5	HTML entity	<span>x ≥ 5</span>
    Fraction layout	½	KaTeX	katex.render("\\\\frac{1}{2}", ...)
    Complex formula	quadratic	KaTeX	katex.render("\\\\frac{-b\\\\pm\\\\sqrt{b^2-4ac}}{2a}", ...)
    Root expression	√(a²+b²)	KaTeX	katex.render("\\\\sqrt{a^2+b^2}", ...)
    Subscript variable	x₁, y₂	HTML <sub>	x<sub>1</sub>, y<sub>2</sub>
    Greek letter	α, β, θ	Unicode char	<span>α</span> — NEVER write "alpha"
    Safe HTML entities: ≤ (&le;), ≥ (&ge;), ≠ (&ne;), ° (&deg;), × (&times;), ÷ (&divide;), ± (&plusmn;), ² (&sup2;), ³ (&sup3;).
    Safe Unicode Greek letters (in Inter font): α, β, γ, δ, ε, θ, λ, μ, π, φ, ω — use directly, never write English names.
    Chinese replacements for geometric symbols: "角" (∠), "三角形" (△), "垂直于" (⊥), "平行于" (∥).
    Rule of thumb: if the expression is a simple comparison, a standalone symbol, or a Chinese-friendly geometric term, use HTML/Chinese. If it has fractions, roots, summations, or nested structure, use KaTeX.

26. SVG 坐标轴箭头必须用 orient="auto" + 朝右三角形 (Axis arrows: always use a single right-pointing marker with orient="auto"). When drawing coordinate axes in SVG, define ONE arrowhead marker with its polygon pointing in the +x direction of the marker's local coordinate system, and set orient="auto". This same marker works for ALL axis directions (x-axis, y-axis, or any angle) because orient="auto" rotates the marker to match the stroke direction at the endpoint. NEVER define a separate "upward" marker with a polygon pre-rotated to point up — orient="auto" will rotate it an additional -90°, making the arrowhead point LEFT instead of UP. Correct pattern:

    <defs>
      <!-- ONE marker for ALL axes — polygon points right (+x), orient="auto" rotates it -->
      <marker id="axis-arrow" markerWidth="10" markerHeight="7" refX="10" refY="3.5" orient="auto">
        <polygon points="0 0, 10 3.5, 0 7" fill="#0f172a"/>
      </marker>
    </defs>
    <!-- x-axis: line goes right → arrow points right ✅ -->
    <line x1="50" y1="400" x2="570" y2="400" marker-end="url(#axis-arrow)"/>
    <!-- y-axis: line goes up → orient="auto" rotates arrow to point up ✅ -->
    <line x1="300" y1="470" x2="300" y2="30" marker-end="url(#axis-arrow)"/>
    

    Common mistake (y-axis arrow points LEFT):

    <!-- ❌ WRONG — polygon already points up, orient="auto" rotates it -90° more → points LEFT -->
    <marker id="arrowU" orient="auto"><polygon points="0 10, 3.5 0, 7 10"/></marker>
    

    Why: orient="auto" aligns the marker's local +x axis with the line's tangent direction. For a line from (300,470) to (300,30), the tangent is (0,−1) = upward, so the local +x rotates −90°. A right-pointing triangle (+x direction) correctly becomes upward. An already-upward triangle (−y direction) gets rotated to point left.
27. 受力箭头/坐标轴禁止加辉光滤镜 (NEVER apply a glow/blur filter to an axis-aligned <line>). A perfectly horizontal or vertical <line> — every force arrow (重力 G↓ / 支持力 N↑ / 摩擦力 f / 拉力), coordinate axis, horizontal light ray, or wire — has a degenerate bounding box (zero height when horizontal, zero width when vertical). SVG filters default to filterUnits="objectBoundingBox" with a region of -10%…120% of the bbox, so a zero dimension makes the filter region zero-area → the line AND its marker-end arrowhead render completely invisible. The <text> label carries no filter, so it still shows — producing the deceptive symptom "受力示意图里只有力的字(G/N/v)，没有力的箭头线条." This bug hides on slanted rays (a diagonal line has a nonzero-area bbox and survives), so it only surfaces on vertical/horizontal force arrows. Fix: give force arrows a solid stroke (width 4–6) + marker-end and NO filter — a force arrow reads perfectly without a glow. If a glow on straight strokes is truly needed, make the filter absolute so it ignores the bbox: <filter id="fGlow" filterUnits="userSpaceOnUse" x="0" y="0" width="<viewBoxW>" height="<viewBoxH>">…</filter>. Glow filters remain safe on circles/particles/ions/text (non-degenerate bbox). Enforced by scripts/check_svg_filter_bbox.py in precheck.py.
28. SVG 标签禁止相互重叠 (SVG <text> labels must NOT overlap each other). SVG <text> is positioned by hand (absolute x/y in viewBox units), so labels crammed around a shared point or along a short line collide and the text mashes together (文字重叠). The classic failure is a pulley/lever diagram that places the fulcrum label 支点 O and the two arm labels 动力臂 r / 阻力臂 r on the SAME horizontal diameter through the circle center — three centered labels in ~90px of room overlap into "动力臂 支点O 阻力臂 r". This class of overlap is invisible to the other layout gates (check_scene_overflow.py ignores SVG user coordinates; check_scene_layout.py only checks the DOM wrapper). Rules: (a) never stack multiple labels on a shared point or a short segment — give each label its own side with a clear gap of ≥ ~1 label-height (≥ the label's font-size in viewBox units) between neighbouring boxes; (b) for a center/fulcrum + two arm labels, put the center label ABOVE the line and the arm labels BELOW it (or vice-versa) so their vertical bands don't collide; (c) anchor arm/segment labels text-anchor="end" (left side) / "start" (right side) and push them outward past the shape edge rather than toward the crowded center; (d) offset a label perpendicular to the line/arm it names, not along it; (e) when two point labels sit on the same axis (e.g. A(-3,0) and E(-5,0)), stagger their y or shorten to bare letters so their boxes clear. Do NOT let a point label sit on top of an axis tick number — move the tick or drop the redundant one. Enforced by scripts/check_svg_label_overlap.py in precheck.py (conservative: flags only substantial 2-D overlap, accounts for <g transform="translate"> and GSAP-repositioned elements).
29. 受力/矢量箭头必须正确 (SVG force/vector arrows — fixed-size head, full-length line). Hand-rolled SVG arrows (<line marker-end>) fail in three severe ways at once, as seen in a 蹦床受力 scene (giant sideways arrowheads with no shaft): (a) 箭头太大 — a <marker> with no markerUnits defaults to markerUnits="strokeWidth", so the arrowhead is multiplied by the (thick, 6–8) force stroke → a ~72–96px triangle. Fix: <marker markerUnits="userSpaceOnUse" markerWidth="14" markerHeight="14" refX=… refY=… orient="auto"> sized in absolute px. (b) 方向不对 + (c) 线没画出来 — drawing the <line> collapsed to a point (x1==x2 && y1==y2) and growing it via GSAP attr:{y2}: a zero-length line has no shaft, and orient="auto" on it is undefined so the head points sideways; if the attr tween doesn't seek at render, the shaft never appears. Fix: draw the line at its full, correct length (so orient is right and the shaft always exists) and reveal it with stroke-dashoffset draw-on — never animate a marker line's length from zero. BEST: reuse the prebuilt CSS force-arrow component (assets/components/mechanics/force-arrow.html) — fixed head, rotate for direction, --arrow-length for magnitude. Enforced by scripts/check_svg_arrow.py in precheck.py.
30. 图形标签禁止压在图形上 (SVG labels must NOT overlap the drawing — 文字不压线不压点). A point/vertex/curve letter (A/B/O/F/M…) must never sit ON a line, axis, symmetry line, curve, or vertex dot — it must be offset OUTWARD, away from the figure, by ~18–24 units. Classic failure: a point M on the symmetry axis gets its label placed straddling that axis's x, so the dashed axis line runs straight through the "M". Rules: (a) place a small dot at the point and the letter ~20u beside it, on the side away from the shape centre (never at the exact point coordinate); (b) a point on the x-axis → label ~22u below/above the axis, not on it; the origin O → offset into a quadrant (x−14, y+20); (c) a label near a vertical/symmetry line → anchor end and push left (or start/right) so the stroke clears the glyphs; (d) a curve label sits in the empty area beside the curve. Nudging a label a little to clear the figure is expected and encouraged. This is distinct from label-vs-label overlap (Rule #28) — here the collision is label-vs-geometry. Enforced by scripts/check_svg_label_on_figure.py in precheck.py.
31. 关系图/流程图节点必须打组、箭头不压框、图形内白字必显 (SVG node-and-connector diagrams). When a scene draws a food chain / feedback loop / flow diagram from SVG boxes + arrows + in-shape labels, three failures ruin it (seen in a 生态平衡 case: white 负反馈 invisible, arrows poking into the 方框). Rules: (a) a label inside a shape MUST be in the SAME <g> as that shape — <g><circle/><text/></g>, never independent siblings; otherwise a GSAP scale/pulse/back.out overshoot or svgOrigin residue moves the shape but not the text, so the label slides off the shape onto the panel and disappears (the "白色文字没显示" bug). Animate/position the group, not the shape alone. (b) white SVG <text> is allowed ONLY inside a dark shape that contains it (white-on-light is invisible; #ffffff text is otherwise forbidden — see Rule on light-theme colors), and it must be grouped with that shape per (a); size the shape so the text fits with margin. (c) arrow/connector endpoints stop ≥ ~10u before the target box edge (account for markerWidth) — never end a marker-end line/path inside or flush against a <rect> node, or the arrowhead hides under / pokes into the 方框. (d) edge labels ("取食"/"捕食") go in the gap between boxes with clearance, not against a box; space boxes so connector + label + clearances fit. Enforced by scripts/check_svg_node_graph.py in precheck.py.
32. SVG 高度必须有界，禁止 width:100%;height:auto 撑爆场景 (SVG height must be bounded — never let width drive an unbounded height). A content <svg> styled width:100%; height:auto sizes its width to the panel (~1380px in a full-width panel) and then, because height:auto, sets its height from the viewBox aspect ratio — a near-square viewBox like 0 0 1000 760 becomes ~1050px tall, far past the ~860px usable height (1080 − top padding − 180px caption safe zone). The SVG overflows the frame and the scene looks like it "suddenly got bigger" (场景突然变大) than the earlier text scenes. This is invisible to the literal-px overflow gate (there is no px > frame). Rules: (a) never write width:100%; height:auto on a content SVG with a tall/near-square viewBox (and equally never width:100%; aspect-ratio:1 — a CSS aspect-ratio drives the height from the width just the same); (b) bound the height — either add max-height:<usable>px (e.g. max-height:760px) so the SVG letterboxes, or size it by height (height:100% inside a bounded flex/grid box) and let preserveAspectRatio="xMidYMid meet" shrink the width to fit; (c) alternatively choose a viewBox aspect ratio that matches the available box (wide/landscape, ratio ≲ 0.5) so width:100% keeps the height under ~820px. The origin of the "后面的渲染场景突然变大" bug was exactly a 图象验证 (graph) scene with #svg{width:100%;height:auto} on a 1000×760 viewBox. Enforced deterministically by scripts/check_svg_height_bound.py (browserless) and measured at runtime by scripts/check_scene_fit.py, both in precheck.py.
33. 禁止对带定位 transform 属性的 SVG 元素补间 x/y/rotation/scale (never GSAP-tween a transform prop on a transform-positioned SVG element — 否则物体飞出画面). An SVG element placed with a hard-coded transform attribute, e.g. <g transform="translate(390,364) rotate(-30)"> (a block on an incline, a rotated arrow, a positioned symbol), must NOT be the target of a GSAP tween that animates a transform shorthand (x, y, rotation, rotationX/Y/Z, scale, scaleX/Y, xPercent, yPercent, skewX/Y). GSAP parses the element's existing transform into its own model, then the tween overwrites it — e.g. fromTo("g[transform]", {y:10}, {y:0}) drives translateY from its real 364 down to 0, so the element jumps ~364px off its position and flies out of frame (物体飞出画面). The classic failure is a broad selector like g[transform] or a bare g/rect selector that happens to hit a positioned group. Rules: (a) on a transform-positioned SVG element, animate ONLY opacity/autoAlpha (never x/y/rotation/scale); (b) to also slide/scale it, NEST — keep the positioning transform on a static outer <g transform=...> and put the fade/slide tween on an inner <g> (or wrapper); (c) if you truly must animate its transform, use attr:{transform:'translate(...) rotate(...)'} with the full transform written out at both ends, never a bare x/y shorthand; (d) never target positioned groups with a broad g[transform] / bare-tag selector. Animating x/y/scale on elements without a transform attribute (plain circles, HTML panels positioned by CSS) is fine. (e) 对 SVG 元素做 scale/rotation 缩放/旋转时，绝不要用 px 单位的 transformOrigin（如 transformOrigin:"470px 270px"）——在 SVG 上它按元素包围盒本地坐标换算，对一个子元素坐标远离 SVG 原点的 <g> 缩放会附带一个很大的平移量，把元素甩飞/错位（即使该 <g> 没有 transform 属性、纯靠子元素绝对坐标定位，也会飞——这正是"变阻器缩短动画把整个 R₁ 甩到画面外"的真实事故）。改用 GSAP 专给 SVG 的 svgOrigin:"x y"（用户坐标系，如 svgOrigin:"470 270"），或把元素套进 <g transform="translate(...)"> 再对内层缩放。transformOrigin 用百分比（如 "50% 50%"）在 SVG 上是安全的。 Enforced by scripts/check_svg_transform_anim.py in precheck.py.

Golden Examples (Few-Shot References)

Before building compositions for a new problem, read the matching golden example to see the complete pipeline (PROBLEM → ANALYSIS → SCRIPT → STORYBOARD → HTML compositions → index.html) executed at top quality. These are condensed from real high-scoring videos — no binary files, just the essential pipeline artifacts and representative composition code.

⚙️ Two output modalities. Most requests → a narrated MP4 via the Step 0–6 pipeline (use the geometry / process-animation examples above). But when the user explicitly wants an interactive web page (拖动参数、实时联动，不是视频), switch to the interactive golden example above — same Aurora look & offline assets, but the deliverable is a self-contained index.html opened by double-click, not a rendered video.

34. KaTeX 必须在合成加载时【同步】渲染，并按 id/data-composition-id 作用域选择；禁止无 id 的 #合成id 选择器，禁止用 setTimeout 延迟渲染 (render KaTeX synchronously at load, scoped correctly; never a #composition-id selector without a matching id, never defer with setTimeout). 已确认两类"公式空白"坑（画面空白，选项/步骤/公式看不到）：(1) 缺根 id（确定性、最常见） — 根 div 是 <div data-composition-id="mt-x">（data-composition-id 是 data 属性，不是 DOM id），但渲染循环用 document.querySelectorAll("#mt-x [data-tex]")；#mt-x 匹配 0 个 → 不渲染 → 公式全空。修复：给根 div 同时加 id="mt-x"（保留 data-composition-id），或用 [data-composition-id="mt-x"] / document.currentScript.closest("[data-composition-id]") 作 id 无关作用域。由 scripts/check_composition_root_id.py 拦截（q067 弧长公式只显示"="就是此因，加 id 后重渲已修复）。(2) 必须同步渲染，切勿延迟 — HyperFrames 渲染器按 seek GSAP 时间轴的方式确定性抓帧，不保证执行 setTimeout/requestAnimationFrame 等异步回调；因此 KaTeX 要在合成内联 <script> 加载时同步渲染（此时 index.html 已在 <head> 同步载入 katex，全局可用），不要把 katex.render 包进 setTimeout/轮询，也不要依赖 onload/末尾追加脚本（追加在根 </div> 之后的 <script> 注入时会被丢弃、根本不执行）。优先用静态 data-tex 属性 + 同步 querySelectorAll 渲染循环（见 step-5 Rule 4）。已知未解边缘情形：个别场景在 hyperframes validate/单独打开时公式正常、但最终 MP4 里仍空白（该场景的命令式 KaTeX 未被抓帧保留，如 q062 的标注框）——遇到务必以渲染出的 MP4 为准复核，必要时重写该场景标记。

References Index