Path params starting with `$` are rewritten to "undefined" on page load

Which project does this relate to?

Router

Describe the bug

A path param value starting with $ is destroyed on full page load: the router rewrites the URL to /files/undefined via history.replaceState and the param value becomes the string "undefined".

With pathParamsAllowedCharacters: ['$'], the router itself generates such URLs: a <Link to="/files/$filePath" params={{ filePath: '$EXAMPLE_CODE/file.abap' }}> navigates to /files/$EXAMPLE_CODE%2Ffile.abap and works fine client-side, but reloading that URL mangles it.

Claude's root cause analysis: on mount, <Transitioner> round-trips the current location through buildLocation({ to: router.latestLocation.pathname, ... }). buildLocation treats to as a route template, so interpolatePath interprets the concrete segment $EXAMPLE_CODE%2Ffile.abap as a param placeholder named EXAMPLE_CODE%2Ffile.abap.
No such param exists, so it interpolates "undefined" (encodeParam(key, params, decoder) ?? "undefined" in router-core's path.ts). The rebuilt publicHref differs from the current one, so the broken location is committed with replace: true. A second symptom of the same ambiguity: the rebuilt to containing $ doesn't exactly match any route template, so buildLocation sets destRoutes = [] and skips all search middlewares for that navigation.

Complete minimal reproducer

https://stackblitz.com/edit/github-arh22bkl?file=src%2Fmain.tsx

Steps to Reproduce the Bug

1. npm install && npm run dev
2. Open http://localhost:5173/ and click the link — the URL becomes /files/$EXAMPLE_CODE%2Ffile.abap and the view renders the param correctly
3. Reload the page
4. The URL is replaced with /files/undefined and useParams() returns filePath: "undefined"

Expected behavior

As a user, I expected a page reload of a router-generated URL to render the same route with the same param value, but the URL is rewritten to /files/undefined and the param value is lost.

Screenshots or Videos

No response

Platform

• Router / Start Version: 1.169.2
• OS: macOS 26.5.1
• Browser: Chrome
• Browser Version: 149
• Bundler: vite
• Bundler Version: 8.0.10 (reproduction uses vite 7 as I ran into some napi crashes in Stackblitz)

Additional context

Workaround: percent-encode $ as %24 in the pathname (e.g. via history.replaceState) before the router parses the location.

Start in router-core's path.ts, especially interpolatePath and the encodeParam fallback, then trace how Transitioner calls buildLocation on mount. Use the linked StackBlitz reproducer to verify the reload case, and consider regression coverage that confirms a router-generated path containing a leading $ keeps its URL and parameter value after reload.