diff --git a/LOTUS_FEATURES.md b/LOTUS_FEATURES.md
index d6a4c2649..095c32b60 100644
--- a/LOTUS_FEATURES.md
+++ b/LOTUS_FEATURES.md
@@ -706,8 +706,9 @@ KaTeX-rendered math in messages, two paths:
 
 - **Spec path (CS-API §11.5):** `<span/div data-mx-maths="…">` in `formatted_body` renders the attribute's LaTeX (block for div, inline for span); on render failure the element's child fallback content shows instead
 - **Plain-text path:** `$…$` (inline) and `$$…$$` (block) with conservative rules — escape-aware (`\$`), currency-guarded (`$5 and $10` stays text), never inside `code`/`pre`
+- **Outgoing interop:** on send, the composer converts `$…$`/`$$…$$` to spec `<span/div data-mx-maths>` HTML in `formatted_body` (extracted before markdown so LaTeX isn't mangled; off inside code), so math renders on Element and every other client — not just Lotus. The plain `body` keeps literal `$…$` as the fallback
 - KaTeX + its CSS load lazily on first math encountered — zero cost to the main bundle
-- Files: `src/app/utils/mathParse.ts` (+14 tests), `components/math/KaTeX.tsx`, `plugins/react-custom-html-parser.tsx`
+- Files: `src/app/utils/mathParse.ts` (+14 tests), `components/math/KaTeX.tsx`, `plugins/react-custom-html-parser.tsx` (render), `components/editor/output.ts` (+ `output.test.ts`, outgoing)
 
 ### Image / Video Captions
 
diff --git a/LOTUS_TODO.md b/LOTUS_TODO.md
index c244dbb9d..c20d51171 100644
--- a/LOTUS_TODO.md
+++ b/LOTUS_TODO.md
@@ -153,9 +153,15 @@ After Phases A–C the client spec is ~complete. What's left, flagged by **what
 
 A minimal audio editor for soundboard clips and voice content. Scope: (1) **trim/clip** an audio file to a chosen start/end (waveform scrubber, in/out handles); (2) **upload a video file → strip and discard the video track, keep only the audio** (extract audio, then the source video is dropped — never uploaded/stored); (3) minimal edits only (trim, maybe gain/normalize, fade in/out) — not a full DAW. Likely Web Audio API (`AudioContext.decodeAudioData` → trim `AudioBuffer` → re-encode) + `MediaRecorder`/an encoder for output; video demux via a `<video>`+`MediaElementSource` capture or ffmpeg.wasm (weigh bundle cost). Feeds the soundboard uploader (`utils/soundboardClips.ts`, `SoundboardPackEditor`) and attachments. Design under TDS + native-cinny law. Big build — plan a dedicated session; evaluate ffmpeg.wasm size/CSP (wasm) before committing.
 
-### [ ] P4-4 · Math / LaTeX Rendering (LOW PRIORITY)
+### [x] P4-4 · Math / LaTeX Rendering — DONE
 
-Render `$…$` / `$$…$$` via KaTeX; graceful fallback to raw text. **Sanitizer must be patched** — `src/app/utils/sanitize.ts` (sanitize-html, `disallowedTagsMode:'discard'`) strips all MathML: add `<math><mi><mo><mn><mrow><mfrac><msqrt><mroot><msub><msup><msubsup><munder><mover><mtable><mtr><mtd>…` + `annotation` to `permittedHtmlTags`, and `xmlns`/`display`/`mathvariant` to `permittedTagToAttributes`. Parser: split text nodes on `/(\$\$.*?\$\$|\$.*?\$)/g` in `react-custom-html-parser.tsx` → `<KaTeX>`. Lazy-import `katex/dist/katex.min.css` only when a math block renders. Verify KaTeX bundle-size impact.
+Rendering shipped (KaTeX, `$…$`/`$$…$$` + spec `data-mx-maths`, lazy-loaded,
+`<pre>/<code>`-guarded) — see LOTUS_FEATURES.md. **Outgoing cross-client interop
+added (2026-07):** the composer now emits spec `data-mx-maths` HTML on send
+(`editor/output.ts`, reusing `splitMathSegments`), so math a Lotus user types
+renders on Element and every other client, not just Lotus. Deferred: multi-line
+block `$$…$$` (spans editor paragraph nodes) still renders on Lotus via the
+plain-body `$…$` path only.
 
 ### [~] P5-20 · Quick Reply from Browser Notification (partial)