build: unify the gate so root make check covers the backend (closes #16)

Root `make check` only ever ran the frontend, so the "main is always
green" policy was satisfied vacuously: the Go backend could be entirely
broken and the root gate stayed green.

- The backend moves onto scripts-to-rule-them-all. Its test, lint, fmt,
  fmt-check, build, run and clean implementations now live in
  `backend/script/`, and `backend/Makefile` is thin shims. The backend
  is its own project (own module, README, LICENSE, linter config,
  Dockerfile stage), and `Dockerfile.backend` only copies `backend/`
  into its builder, so its scripts have to live under `backend/`.
- The root `script/test`, `script/lint`, `script/fmt` and
  `script/fmt-check` now run the frontend step and then the matching
  `backend/script/*` step, so `script/check` — and therefore the
  pre-commit hook — gates both halves. The frontend-only steps moved
  into `script/frontend-*` so nothing is duplicated.
- `script/frontend-check` is the frontend half of the gate, exposed as
  the `check-frontend` target, for the frontend Dockerfile: its build
  stage is a node image with no Go toolchain. The backend half is gated
  by `Dockerfile.backend`, and `script/cibuild` builds both images, so
  the two Dockerfiles together still gate the whole repo. The
  `check-backend` target is the mirror of it.
- `script/cibuild` builds both images through one `build_image` helper,
  and the Gitea workflow's only build step is `script/cibuild`; the raw
  `docker build -f Dockerfile.backend .` is gone from the workflow.
  `script/docker` likewise builds and tags both images.
- `backend/Makefile`'s `hooks` target is removed. It wrote the same
  `.git/hooks/pre-commit` as `script/install-precommit`, so the two
  clobbered each other and the developer silently ended up gating on
  only one half of the repo. `script/install-precommit` is now the only
  installer, and the hook it writes runs the repo-wide `script/check`.
- `backend/Makefile`'s `docker` target is removed too: the backend image
  builds from the repo root with a root-level Dockerfile, so it belongs
  to the root `script/docker` and `script/cibuild` rather than to a
  backend script that would have to reach outside `backend/`.
- `backend/script/lint` verifies that `.golangci.yml` still matches its
  pinned sha256 before running the linter. Offline hash comparison, no
  network.

READMEs at the root and in `backend/` document every script, and
`TODO.md` records the change.

Dockerfile

@@ -5,10 +5,14 @@ COPY package.json yarn.lock ./
 RUN yarn install --frozen-lockfile
 RUN apk add --no-cache git make
 COPY . .
-# make check runs script/check (test + lint + fmt-check); its test step
-# is the production yarn build, so this both produces dist/ and gates the
-# image on lint/fmt-check/test regressions, not merely a broken build.
-RUN make check
+# make check-frontend runs script/frontend-check (test + lint +
+# fmt-check for the frontend); its test step is the production yarn
+# build, so this both produces dist/ and gates the image on
+# lint/fmt-check/test regressions, not merely a broken build. It is the
+# frontend half of `make check` rather than all of it because this stage
+# is a node image with no Go toolchain; the backend half is gated by
+# Dockerfile.backend, and script/cibuild builds both images.
+RUN make check-frontend
 
 # nginx:stable-alpine as of 2026-02-22
 FROM nginx@sha256:15e96e59aa3b0aada3a121296e3bce117721f42d88f5f64217ef4b18f458c6ab