feat(game): official first-move tile draw + admin step-by-step replay

Decide who moves first by the official rule: each seated player draws one
tile, the one closest to "A" leads (a blank beats every letter), ties
re-drawing until a single leader remains. Each draw uses honest per-draw
crypto/rand entropy (not the deterministic bag seed), so the recorded draw —
not a seed — is the only account of the outcome. The leader takes seat 0, so
the engine and journal replay are unchanged.

The draw is recorded with the game (game_setup_draws, migration 00013) for
future tournaments, designed as a discrete "player N draws a tile" step.
Friend/AI games draw at creation. Auto-match draws when the game opens,
against a synthetic uuid.Nil opponent whose draw rows (NULL account_id) are
back-filled to the real opponent on join — so the opener's seat is fixed up
front and the existing open-game pre-move is preserved with no reseating.

Admin /_gm/games/:id gains the recorded draw list and a simple step-by-step
board replay (game.ReplayTimeline + a vanilla-JS stepper): a board with
A-O/1-15 headers and highlighted premium squares, placed letters with their
tile value as a subscript, rack panels around the board (seat 0 top, 1
bottom, 2 left, 3 right) with the current player highlighted, and a per-move
log with the tiles drawn and the bag remainder.

Docs: ARCHITECTURE §6/§9, FUNCTIONAL (+_ru), PRERELEASE (FM row), design spec.

backend/internal/game/seating.go

@@ -0,0 +1,156 @@
+package game
+
+import (
+	"crypto/rand"
+	"fmt"
+	"math/big"
+
+	"github.com/google/uuid"
+
+	"scrabble/backend/internal/engine"
+)
+
+// maxSeedingRounds caps the first-move draw's tie re-draws. With a real bag a tie
+// breaks with positive probability each round, so this is reached only by a
+// degenerate (e.g. test) entropy source that ties forever; the cap turns that into
+// an error instead of an infinite loop.
+const maxSeedingRounds = 1000
+
+// SetupDraw is one recorded tile draw of the first-move seeding — one row of
+// game_setup_draws (docs/ARCHITECTURE.md §6, §9): the round, the draw order within
+// it, the seated account that drew, and the decoded tile with its rank. It is
+// dictionary-independent: the letter, the blank flag and the numeric rank describe
+// the draw without any alphabet table.
+type SetupDraw struct {
+	Round   int
+	PickNo  int
+	Account uuid.UUID
+	Letter  string
+	Blank   bool
+	Rank    int
+}
+
+// seedingResult is the outcome of the first-move seeding: the winning account, the
+// seated accounts rotated so the winner leads (seat 0), and the full draw log to
+// persist.
+type seedingResult struct {
+	winner uuid.UUID
+	order  []uuid.UUID
+	draws  []SetupDraw
+}
+
+// drawIntn returns a uniformly random integer in [0, n) for n > 0. It is the
+// entropy seam of the first-move seeding: production uses crypto/rand (cryptoIntn),
+// so every draw is honestly random with no single seed; tests inject a
+// deterministic source.
+type drawIntn func(n int) (int, error)
+
+// cryptoIntn draws a uniform integer in [0, n) from crypto/rand — the honest,
+// seedless entropy the first-move draw requires.
+func cryptoIntn(n int) (int, error) {
+	v, err := rand.Int(rand.Reader, big.NewInt(int64(n)))
+	if err != nil {
+		return 0, fmt.Errorf("game: first-move draw entropy: %w", err)
+	}
+	return int(v.Int64()), nil
+}
+
+// seedFirstMove runs the official first-move draw over accounts for variant v,
+// drawing tiles with the entropy source intn. Each round every contender draws one
+// tile (without replacement) from a fresh full bag; the tile closest to "A" wins, a
+// blank beating every letter; contenders tied for the best tile re-draw in the next
+// round until a single leader remains. It returns the leader, the rotation that
+// seats the leader first (preserving the others' seating order), and every draw for
+// the record. It performs no I/O beyond calling intn.
+func seedFirstMove(v engine.Variant, accounts []uuid.UUID, intn drawIntn) (seedingResult, error) {
+	if len(accounts) < 2 {
+		return seedingResult{}, fmt.Errorf("game: first-move seeding needs at least 2 accounts, got %d", len(accounts))
+	}
+	full, err := engine.SetupBag(v)
+	if err != nil {
+		return seedingResult{}, err
+	}
+	contenders := append([]uuid.UUID(nil), accounts...)
+	var draws []SetupDraw
+	for round := 1; ; round++ {
+		if round > maxSeedingRounds {
+			return seedingResult{}, fmt.Errorf("game: first-move seeding unresolved after %d rounds", maxSeedingRounds)
+		}
+		bag := append([]engine.SetupTile(nil), full...)
+		picks := make([]engine.SetupTile, len(contenders))
+		for i, acc := range contenders {
+			tile, rest, err := drawSetupTile(bag, intn)
+			if err != nil {
+				return seedingResult{}, err
+			}
+			bag = rest
+			picks[i] = tile
+			draws = append(draws, SetupDraw{
+				Round: round, PickNo: i, Account: acc,
+				Letter: tile.Letter, Blank: tile.Blank, Rank: tile.Rank,
+			})
+		}
+		winners := bestContenders(contenders, picks)
+		if len(winners) == 1 {
+			return seedingResult{winner: winners[0], order: rotateToFirst(accounts, winners[0]), draws: draws}, nil
+		}
+		contenders = winners
+	}
+}
+
+// drawSetupTile removes one uniformly random tile from bag using the entropy source
+// intn and returns it with the shrunk bag (a fresh slice, leaving bag untouched).
+// It is the per-tile draw primitive — the seam a future manual "player N draws a
+// tile" tournament API will drive, one call per external request.
+func drawSetupTile(bag []engine.SetupTile, intn drawIntn) (engine.SetupTile, []engine.SetupTile, error) {
+	if len(bag) == 0 {
+		return engine.SetupTile{}, nil, fmt.Errorf("game: first-move draw from an empty bag")
+	}
+	i, err := intn(len(bag))
+	if err != nil {
+		return engine.SetupTile{}, nil, err
+	}
+	if i < 0 || i >= len(bag) {
+		return engine.SetupTile{}, nil, fmt.Errorf("game: first-move draw index %d out of range %d", i, len(bag))
+	}
+	tile := bag[i]
+	rest := make([]engine.SetupTile, 0, len(bag)-1)
+	rest = append(rest, bag[:i]...)
+	rest = append(rest, bag[i+1:]...)
+	return tile, rest, nil
+}
+
+// bestContenders returns the contenders whose drawn tile has the lowest (best)
+// rank — the sole winner if one, else the tied set that re-draws. picks is aligned
+// with contenders by index.
+func bestContenders(contenders []uuid.UUID, picks []engine.SetupTile) []uuid.UUID {
+	best := picks[0].Rank
+	for _, p := range picks[1:] {
+		if p.Rank < best {
+			best = p.Rank
+		}
+	}
+	var winners []uuid.UUID
+	for i, p := range picks {
+		if p.Rank == best {
+			winners = append(winners, contenders[i])
+		}
+	}
+	return winners
+}
+
+// rotateToFirst returns accounts rotated cyclically so winner sits first (seat 0),
+// preserving the seating order of the rest (docs/ARCHITECTURE.md §6). winner must be
+// present in accounts.
+func rotateToFirst(accounts []uuid.UUID, winner uuid.UUID) []uuid.UUID {
+	i := 0
+	for ; i < len(accounts); i++ {
+		if accounts[i] == winner {
+			break
+		}
+	}
+	out := make([]uuid.UUID, 0, len(accounts))
+	out = append(out, accounts[i:]...)
+	out = append(out, accounts[:i]...)
+	return out
+}