feat: playable arcade — rooms, identity, client, deployment

Round length is now bounded: the multiplier follows a hyperbolic curve
diverging at 60s, replacing an exponential one where a 275x crash point
produced a two-and-a-half minute round.

Fixes seed reveal, which silently failed every round because pgx cannot
encode a fixed-size byte array as bytea.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

Dockerfile

@@ -0,0 +1,14 @@
+# Build the server as a static binary with the client embedded in it, then ship
+# it on a minimal base. The result is one file with no runtime dependencies.
+FROM golang:1.26-alpine AS build
+WORKDIR /src
+COPY go.mod go.sum ./
+RUN go mod download
+COPY . .
+RUN CGO_ENABLED=0 go build -trimpath -ldflags="-s -w" -o /out/arcade ./cmd/arcade
+
+FROM gcr.io/distroless/static-debian12
+COPY --from=build /out/arcade /arcade
+EXPOSE 8080
+USER nonroot:nonroot
+ENTRYPOINT ["/arcade"]