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Step 1 of #proj-mobile:#60 (Computers rack overview, web parity, 1.24): the web's toolbar logic is ported into compose/shared as pure, tested code. The UI does not change. The toolbar, list body and strings are steps 2 and 3. There is no changeset because nothing user-visible ships in this PR; the UI PR will add it.

Web source of truth: botiverse/slock origin/staging @ fabcd51b4. This is newer than the spec's 194e5d3; see the spec gaps below. Paths are relative to packages/web/src/.

What's added

• RaftComputersRackFilters.kt (next to RaftSettingsComputersModel.kt): the pure functions, the toolbar state and RaftComputersRackToolbarStore.
• data/ComputersRackPreferences.kt: the ComputersRackGroupBy and ComputersRackSort* types, plus KeyValueComputersRackPreferenceRepository over SharedPreferenceStore. It stores Group (computers:rackGroup) and Sort (computers:rackSort, the web's JSON shape). View is delegated to the existing KeyValueComputersRackViewPreferenceRepository (computers:rackView).
• hostProvider (nullable) is carried through ComputerDto → ComputerSummary → SettingsComputerSnapshot and the durable cache. The field name was checked against packages/server/src/services/machineReadModel.ts:163 (hostProvider: isHostProvider(machine.hostProvider) ? machine.hostProvider : null), packages/server/src/db/schema.ts:6481 (host_provider) and packages/shared/src/index.ts:2038-2056 (HOST_PROVIDER_LABELS, isHostProvider).

Rules → web reference

Rule	Web	Kotlin
Search: trimmed, case-insensitive, over name, hostname, owner display name or name (only while ownerUserId is set), and owner.name	features/computer/computersRackFilter.ts:55-73	raftFilterRackComputers
Status: All / Online / Offline (anything not online) / Needs attention (upgrade waiting; a Computer that is not online; online with low disk)	computersRackFilter.ts:43-47,74-76, utils/computerUpgradeIndicator.ts getComputerAttentionStatus	raftRackNeedsAttention
Runtime: any chosen runtime; Other = a runtime with no usage provider; options = providers present (catalog order), then Other; a choice whose computers have gone filters nothing; picking prunes it and keeps the options' order	computersRackFilter.ts:22-40,49-53,77-80, features/computer/ComputersRackToolbar.tsx:118-156	raftRackRuntimeFilterOptions, raftRackEffectiveRuntimeFilter, toggleRuntime
Group options: none, prefix, provider	store/computersRackFilterPreference.ts:22	ComputersRackGroupBy
Name prefix: longest shared word prefix (split on - _ .), case-insensitive, lone groups fold up or go to Other, title = most common spelling	computersRackFilter.ts:85-169	raftRackNamePrefixGroups
Provider: HOST_PROVIDER_LABELS; unknown or none goes to Personal computers	computersRackFilter.ts:187-197	RAFT_HOST_PROVIDER_LABELS
Group order: fallback last; under a numeric sort, by best member; otherwise A to Z	computersRackFilter.ts:206-239, features/computer/ComputersRackList.tsx:124-137	raftGroupRackComputers, raftRackGroupRank
Sort keys: Name, remaining:<provider> for each provider present, Disk, Memory; default Name A to Z; a new key starts at its first direction (name asc, numbers desc); no value goes last both ways; ties keep the list order; a stored runtime sort whose runtime is gone falls back to the default	features/computer/computersRackSort.ts:19-121, features/computer/computersRackToolbarModel.ts:68-85, ComputersRackToolbar.tsx:58-61,196-248	raftSortRackComputers, raftRackEffectiveSort, raftRackSortMenuSections, raftRackSortDirectionCopy
Remaining: the tightest window of the tightest account; rate_limited = 0; reauth / unsupported / error = no value; limit_reached with no ratio = fully used	computersRackSort.ts:49-75	raftRackRemaining
Offline section at the end; none under Status: Offline; open by default when nothing else is online	ComputersRackList.tsx:102-123	raftComputersRackLayout
Clear all resets search, Status and Runtime, never Group, Sort or View; it shows while any of those three is active	store/computersRackFilterPreference.ts:55-57,66-68,81-83	clearedFilters, RaftComputersRackFilters.active
Search, Status and Runtime are session-only and reset on another server; Group, Sort and View are stored; an unreadable stored value reads as the default	store/computersRackFilterPreference.ts:35-51,71-78, store/computersRackSortPreference.ts:30-47, store/computersRackViewPreference.ts	RaftComputersRackToolbarStore.onServer, KeyValueComputersRackPreferenceRepository

Tests (35 new, all commonTest)

• RaftComputersRackFiltersTest (30) mirrors features/computer/tests/computersRackFilter.behavior.test.ts (all 14 tests: same fleets, same name-prefix table, same 20-seed order-independence shuffle, same remaining cases), plus the row-order expectations of computersRackSort.behavior.test.tsx and computersRackToolbar.behavior.test.tsx: search, Status, Runtime, Group, every sort key in both directions, provider groups under a remaining sort, a stale runtime choice, Clear all, server change, persistence, and the sort sheet's key/direction steps. The cases are tables, not copied tests.
• ComputersRackPreferencesTest (4): the Group and Sort round trips (every key × direction), unreadable stored values, and View sharing the existing key.
• ComputersApiTest.hostProviderMapsFromTheMachineReadModelAndIsNullWhenAbsent (1).
• Existing suites rerun and green: ComputersApiTest, LocalPreferencesTest, RaftComputersRackModelTest, SharedComputersCacheFirstTest.

Spec vs web differences (followed the web)

1. Name-prefix grouping changed after the spec's commit. The spec (B4) says "strips -N/_N, a group needs 2+ members". Staging (#14042, 77ba471b5) groups by the longest shared word prefix instead: it splits on - _ ., compares case-insensitively, and folds lone groups up (for example, artin-hostinger-root pulls artin-fluence-* into artin). The spec's line range computersRackFilter.ts:85-166 is now :85-239.
2. Search also matches owner.name when the Computer has no owner. Web filterRackMachines adds machine.owner?.name even when ownerUserId is null. Only the display name requires an owner.
3. "Needs attention" for offline machines applies only to Computers (isComputer). An offline bare daemon is not counted. A machine whose status is unknown counts as offline.
4. Picking the current sort key again changes nothing in the Sort select. The desktop ⋯ menu instead resets the direction. The phone sheet follows the select.
5. Collation. The web uses Intl.Collator(locale, {numeric, sensitivity: "base"}). Common Kotlin has no collator, so raftRackCompareText approximates it: numeric runs, case and common Latin accents ignored, punctuation < digits < letters. There is no locale tailoring, so zh pinyin order for Han names falls back to code point order.
6. Naming and test files. The spec's B8 store name is SharedComputersRackToolbarStore; the code uses RaftComputersRackToolbarStore. The spec points to the toolbar and sort behaviour tests, but most of the logic tests are in computersRackFilter.behavior.test.ts, which this PR mirrors in full.
7. Web store lifetime is the page/tab; the mobile store is meant to be one per app session. Its owner is wired in step 2.

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