Lands the Vale prose linter as a non-blocking docs CI step. Builds on the
scaffold from DOCS-180 (#25466):

- .vale.ini at the repo root configures Google + curated write-good
  + cherry-picked alex rules. Disables Google.Spacing (false positives
  on codersdk type names in the auto-generated API reference),
  Google.EmDash (conflicts with our em-dash ban), Google.Latin (i.e.
  and e.g. are fine), write-good.Passive and write-good.E-Prime
  (judgment-heavy).
- mise.toml pins Vale 3.7.1 via aqua.
- Makefile adds build/vale-$VERSION install, docs/.style/.vale-synced
  sentinel that runs 'vale sync' once per .vale.ini change, and a
  lint/prose target wrapped in '|| true' for v1 non-blocking severity.
- .github/workflows/docs-ci.yaml adds a 'prose' step that lints only
  the changed Markdown files under docs/, with continue-on-error: true
  and a cache for the synced styles and binary.
- .gitignore excludes the synced upstream styles and the sentinel.
- .markdownlint-cli2.jsonc ignores the synced styles so local
  markdownlint runs do not lint upstream READMEs.
- docs/.style/README.md and style-guide.md document how to run Vale
  locally and what the active rule set is.

Severity policy (v1): every rule lands at 'warning'. CI is non-blocking
through continue-on-error: true. A rule promotes to 'error' only when
(a) it is objectively correct and (b) the existing-content violation
count reaches zero. Judgment rules stay at 'suggestion'.

Local run on the full docs corpus produces 0 blocking failures, 391
errors, 5202 warnings, 7682 suggestions across 458 files in ~20s.

Refs DOCS-40.

… filter

The 'docs/**.md' glob in tj-actions/changed-files skips dot-prefixed
directories by default, so the changed-md-docs filter silently dropped
docs/.style/README.md and docs/.style/style-guide.md. The Vale prose
step never fired on PR #25467 as a result.

Drop the second changed-files step and post-filter the changed-md
output in shell. grep '^docs/' keeps only docs paths; grep -v
'^docs/.style/styles/' excludes the synced upstream packages. The
early exit handles PRs that only touched non-docs markdown.

Refs DOCS-40.

Vale's release archive places 'vale' at the archive root with no
leading ./ (unlike typos), so 'tar -xzf - ./vale' matched nothing and
produced 'tar: ./vale: Not found in archive' on CI. Switch to 'vale'.

The local Makefile invocation worked before because build/vale-3.7.1
was already present from the spike artifacts. CI hit the cold path
and exposed the bug.

Resolves the 12 findings from review id 4314529409 on PR #25467.

DEREM-1 (P2): fix `./build/vale-*/vale` glob in docs/.style/README.md;
  the binary is the file itself, not a directory.
DEREM-2 (P3): drop the inaccurate `make lint` claim in style-guide.md;
  point readers at README's Running Vale section instead.
DEREM-3 (P2): split the prose step into `Prepare Vale styles` (no
  continue-on-error) and `prose` (no continue-on-error; --no-exit), so
  the job fails on sync failures but lints non-blocking.
DEREM-4 (P2): replace `|| true` with Vale's native --no-exit in the
  Makefile and rewrite the severity-policy comment to match the
  measured exit-code semantics (Vale exits non-zero only on error
  alerts, regardless of MinAlertLevel).
DEREM-5 (P2): add `Coder` to BasedOnStyles so the empty starter style
  is loaded and ready to receive rule files.
DEREM-6 (P3): replace hardcoded per-package paths in .gitignore and the
  workflow cache step with `styles/*` plus a `!styles/Coder` negation,
  reducing the places future packages need to be listed.
DEREM-7 (P3): already addressed by c3e9e2f (prose filter matches
  docs/.style paths) and 6054d9e (tar layout).
DEREM-8 (P3): remove the duplicate Running Vale block in style-guide.md
  and cross-reference README.md as the single source.
DEREM-9 (P3): add .github/vale-problem-matcher.json and wire the prose
  step with `::add-matcher::`/`--output=line`/`::remove-matcher::` so
  alerts surface as inline PR annotations.
DEREM-10 (P4): trim the Active rule set section in style-guide.md to a
  policy summary plus a pointer to .vale.ini.
DEREM-11 (Nit): rephrase the Makefile sync-sentinel comment to describe
  Make's behavior instead of suggesting `touch`.
DEREM-12 (Nit): swap "(Vale spike, 2026-05-18)" for the more neutral
  "(measured 2026-05-18)" in the Google.Spacing rationale.

Two-layer defense on the cache step protects the hand-authored
Coder rules: the negation excludes styles/Coder from the cached paths,
and the cache key hashes styles/Coder/** so any rule change invalidates
the cache even if the negation behavior ever regresses
(actions/toolkit#713, actions/cache#494).

Verification:
- make lint/actions: clean (zizmor, actionlint).
- make fmt/markdown: no changes.
- make lint/markdown: 0 errors across 463 files.
- make lint/prose: exit 0; baseline 391 errors / 5201 warnings / 7673
  suggestions across 458 files (consistent with the pre-review run).

Three follow-ups from review id 4320840969.

DEREM-13 (P3): drop the redundant `grep -v '^docs/\.style/styles/'`
filter from the prose step. Synced upstream packages are gitignored, so
tj-actions/changed-files never lists them; the filter only ever rejects
tracked Coder/**.md files, which is the opposite of what we want. The
inline comment now documents why no second filter exists.

DEREM-14 (P3): fix the README's --no-exit explanation. The DEREM-4 fix
chain reached .vale.ini, the Makefile, and style-guide.md but missed
docs/.style/README.md. The README now matches the rest: --no-exit
suppresses the exit from the baseline error count produced by
un-overridden Google error-level rules, not from warnings/suggestions
(which never trigger non-zero exit regardless of --no-exit).

DEREM-15 (Nit): same model fix in the docs-ci.yaml prose-step comment.

Verification: make lint/actions, make fmt/markdown, make lint/markdown,
make lint/prose all clean. Vale baseline unchanged (391 errors / 5201
warnings / ~7673 suggestions across 458 files).

DEREM-16 (Nit) from review id 4321386259. `./build/vale-* docs/` is fine
when only one vale-X.Y.Z binary exists in build/, but if a developer
bumps the version in mise.toml without running `make clean`, multiple
binaries coexist and the glob expands to multiple positional args,
breaking the command. Low-risk path because three conditions have to
align (version bump + no clean + manual invocation), but worth a sentence
so a developer who hits it knows to `make clean`.

Adds a warning-level Vale rule that flags headings beginning with an
-ing word (typically a gerund or present participle, like 'Installing'
or 'Configuring').

The rule uses extends: existence with scope: heading and a regex
anchored to the start of the heading text, since Vale's sequence rule
type (which would let us condition on the POS tag VBG) is documented
as sentence-scoped and does not honor scope: heading. Google and
Microsoft style packages both use existence + scope: heading + regex
for all their heading-targeted rules; this rule follows the same
pattern.

A small exceptions list covers words that end in 'ing' but are not
gerunds (Bring, String, Spring, King, Ring, Sting, Sing, Thing, Wing).
Concept-noun gerunds (Logging, Networking, Monitoring, Troubleshooting)
are intentionally NOT in the exceptions list: those headings often
read better as imperatives or full nouns and writers should see the
warning and decide.

Adds a 'Headings' section to docs/.style/style-guide.md with the
'Gerund headings' subsection explaining the rule, examples, and how to
silence individual instances.

Updates docs/.style/styles/Coder/README.md to list DOCS-191.

Closes DOCS-191.