,

backend/src/game-server.ts

@@ -301,6 +301,7 @@ export class GameServer implements GameEvents {
 
   private handlePhysics() {
     const frameStart = process.hrtime.bigint();
+    const frameStartMs = performance.now();
     const delta = Number(frameStart - this.lastPhysicsBase) / 1e9;
     this.lastPhysicsBase = frameStart;
 
@@ -348,8 +349,13 @@ export class GameServer implements GameEvents {
       this.players.step(scaledDelta);
     }
 
+    // Physics has only advanced to the start of this frame less whatever is still in the
+    // accumulator. Timing anything off the send time instead would put up to one physics step
+    // of noise on every interpolated position and on the client's replay anchor.
+    const simulatedThroughMs = frameStartMs - this.physicsAccumulator * 1000;
+
     const propertyUpdates = new Map<GameObject, PropertyUpdatesForObject | undefined>();
-    this.players.stepCommunication(delta, (object) => {
+    this.players.stepCommunication(delta, simulatedThroughMs, (object) => {
       if (!propertyUpdates.has(object)) {
         propertyUpdates.set(object, object.getPropertyUpdates());
       }

backend/src/players/player-container.ts

@@ -82,10 +82,11 @@ export class PlayerContainer {
 
   public stepCommunication(
     deltaTimeInSeconds: number,
+    simulatedThroughMs: number,
     propertyUpdatesOf: (object: GameObject) => PropertyUpdatesForObject | undefined,
   ) {
     this._players.forEach((p) =>
-      p.stepCommunications(deltaTimeInSeconds, propertyUpdatesOf),
+      p.stepCommunications(deltaTimeInSeconds, simulatedThroughMs, propertyUpdatesOf),
     );
   }
 

backend/src/players/player.ts

@@ -160,6 +160,7 @@ export class Player extends PlayerBase {
 
   public stepCommunications(
     deltaTime: number,
+    simulatedThroughMs: number,
     propertyUpdatesOf: (object: GameObject) => PropertyUpdatesForObject | undefined,
   ) {
     const remoteCalls: Array<RemoteCallsForObject> = [];
@@ -180,15 +181,20 @@ export class Player extends PlayerBase {
       this.socket.emit(TransportEvents.Ping, this.pendingPingNonce);
     }
 
+    // Subtracted, not zeroed: zeroing would stretch every interval to the next physics frame.
     if ((this.timeSinceLastMessage += deltaTime) > settings.updateMessageInterval) {
-      this.timeSinceLastMessage = 0;
+      this.timeSinceLastMessage = Math.min(
+        this.timeSinceLastMessage - settings.updateMessageInterval,
+        settings.updateMessageInterval,
+      );
       this.queueAnnouncement();
-      this.queueSnapshot(propertyUpdatesOf);
+      this.queueSnapshot(simulatedThroughMs, propertyUpdatesOf);
       this.sendQueuedCommandsToClient();
     }
   }
 
   private queueSnapshot(
+    simulatedThroughMs: number,
     propertyUpdatesOf: (object: GameObject) => PropertyUpdatesForObject | undefined,
   ) {
     const { topLeft, size } = calculateViewArea(this.center, this.aspectRatio, 1.2);
@@ -227,7 +233,7 @@ export class Player extends PlayerBase {
       }
     }
     this.queueCommandSend(
-      new PropertyUpdatesForObjects(propertyUpdates, performance.now() / 1000),
+      new PropertyUpdatesForObjects(propertyUpdates, simulatedThroughMs / 1000),
     );
 
     if (this.character) {
@@ -236,8 +242,10 @@ export class Player extends PlayerBase {
           this.lastInputClientTimeMs,
           this.character.movementSnapshot,
           this.lastLeapClientTimeMs,
+          // Aged against the instant the pose is from, not the send time: the difference is
+          // the un-simulated remainder, and it would land straight in the client's replay.
           this.lastInputReceiptMs > 0
-            ? Math.max(0, performance.now() - this.lastInputReceiptMs)
+            ? Math.max(0, simulatedThroughMs - this.lastInputReceiptMs)
             : 0,
         ),
       );

frontend/src/main.scss

@@ -127,7 +127,6 @@ body {
 
     .player-tag {
       border-radius: 1000px;
-      transition: transform 200ms;
 
       &.blue {
         color: $bright-blue;
@@ -255,7 +254,6 @@ body {
 
     // Off-screen pointer to the keystone "Heart", tinted by its current owner.
     .keystone-arrow {
-      transition: transform 150ms;
       opacity: 0.9;
 
       @include square($large-icon);

frontend/src/scripts/game.ts

@@ -347,6 +347,9 @@ export class Game extends CommandReceiver {
   }
 
   private draw() {
+    // Before any style writes this frame, so the read cannot force a synchronous layout.
+    this.minimap.measure();
+
     if (this.lastGameState) {
       this.scoreboard.update(this.lastGameState, this.gameObjects.localPlayer?.team);
     }

frontend/src/scripts/helper/prediction/local-character-predictor.ts

@@ -5,8 +5,7 @@ import {
   CharacterMovementState,
   PhysicsBody,
   settings,
-  clamp,
-  easeVec2,
+  followVec2,
   stepCharacterMovement,
   applyLeapImpulse,
   tickPlanetDetachment,
@@ -24,10 +23,12 @@ let frameTimeMs = 0;
 export const setFrameTimeMs = (timeMs: number): void => {
   frameTimeMs = timeMs;
 };
-export const clientTimeMs = (): number => Math.round(frameTimeMs);
+export const clientTimeMs = (): number => frameTimeMs;
 
 const maxReplayMs = 400;
 
+// Only a correction has to fade; the character's own motion is carried straight through, so this
+// buys smoothing without costing responsiveness.
 const smoothSeconds = 0.06;
 
 const snapDistance = 250;
@@ -58,6 +59,7 @@ export class LocalCharacterPredictor {
   private renderHead = new Circle(vec2.create(), headRadius);
   private renderLeftFoot = new Circle(vec2.create(), feetRadius);
   private renderRightFoot = new Circle(vec2.create(), feetRadius);
+  private previousTargets = [vec2.create(), vec2.create(), vec2.create()];
   private hasRender = false;
 
   public get head(): Circle {
@@ -152,7 +154,7 @@ export class LocalCharacterPredictor {
       this.snapRenderTo(predicted);
       this.hasRender = true;
     } else {
-      this.easeRenderTo(predicted, frameSeconds);
+      this.followRenderTo(predicted, frameSeconds);
     }
     return true;
   }
@@ -161,10 +163,10 @@ export class LocalCharacterPredictor {
     const auth = this.authoritative!;
     const movement = this.movement!;
     const now = clientTimeMs();
-    const startMs = clamp(this.replayAnchorMs!, now - maxReplayMs, now);
+    const startMs = Math.max(this.replayAnchorMs!, now - maxReplayMs);
     const windowMs = Math.max(0, now - startMs);
     const steps = Math.floor(windowMs / stepMs);
-    const remainderSeconds = (windowMs - steps * stepMs) / 1000;
+    const partialStep = (windowMs - steps * stepMs) / stepMs;
 
     const state: CharacterMovementState = {
       head: makeBody(auth.head.center, auth.head.radius, upwards),
@@ -191,9 +193,10 @@ export class LocalCharacterPredictor {
     let t = startMs;
     for (let i = 0; i < steps; i++) {
       const input = this.inputHistory.directionAt(t);
+      // Planets first: the server's container holds them ahead of every character.
+      this.world.advance(stepSeconds);
       tickPlanetDetachment(state, stepSeconds);
       stepCharacterMovement(state, this.world, input, stepSeconds);
-      this.world.advance(stepSeconds);
 
       for (const leapMs of this.leapHistory) {
         if (
@@ -213,21 +216,32 @@ export class LocalCharacterPredictor {
       t += stepMs;
     }
 
-    if (remainderSeconds > 0) {
-      tickPlanetDetachment(state, remainderSeconds);
+    // A shorter final step would not integrate like the server's fixed one, so take a whole
+    // step and read the pose part-way along it instead.
+    if (partialStep > 0) {
+      const from = [state.head, state.leftFoot, state.rightFoot].map((b) =>
+        vec2.clone(b.center),
+      );
+      this.world.advance(stepSeconds);
+      tickPlanetDetachment(state, stepSeconds);
       stepCharacterMovement(
         state,
         this.world,
-        this.inputHistory.directionAt(now),
-        remainderSeconds,
+        this.inputHistory.directionAt(t),
+        stepSeconds,
+      );
+      [state.head, state.leftFoot, state.rightFoot].forEach((body, i) =>
+        vec2.lerp(body.center, from[i], body.center, partialStep),
       );
-      this.world.advance(remainderSeconds);
     }
 
     return state;
   }
 
   private snapRenderTo(state: CharacterMovementState): void {
+    this.previousTargets = [state.head, state.leftFoot, state.rightFoot].map((b) =>
+      vec2.clone(b.center),
+    );
     this.renderHead = new Circle(vec2.clone(state.head.center), state.head.radius);
     this.renderLeftFoot = new Circle(
       vec2.clone(state.leftFoot.center),
@@ -239,14 +253,26 @@ export class LocalCharacterPredictor {
     );
   }
 
-  private easeRenderTo(state: CharacterMovementState, frameSeconds: number): void {
-    this.easePart(this.renderHead, state.head, frameSeconds);
-    this.easePart(this.renderLeftFoot, state.leftFoot, frameSeconds);
-    this.easePart(this.renderRightFoot, state.rightFoot, frameSeconds);
+  private followRenderTo(state: CharacterMovementState, frameSeconds: number): void {
+    this.followPart(this.renderHead, state.head, 0, frameSeconds);
+    this.followPart(this.renderLeftFoot, state.leftFoot, 1, frameSeconds);
+    this.followPart(this.renderRightFoot, state.rightFoot, 2, frameSeconds);
   }
 
-  private easePart(render: Circle, target: PhysicsBody, frameSeconds: number): void {
-    easeVec2(render.center, target.center, frameSeconds, smoothSeconds, snapDistance);
+  private followPart(
+    render: Circle,
+    target: PhysicsBody,
+    index: number,
+    frameSeconds: number,
+  ): void {
+    followVec2(
+      render.center,
+      target.center,
+      this.previousTargets[index],
+      frameSeconds,
+      smoothSeconds,
+      snapDistance,
+    );
     render.radius = target.radius;
   }
 }

frontend/src/scripts/minimap.ts

@@ -8,6 +8,7 @@ export class Minimap {
   private readonly colors: Record<CharacterTeam, string>;
   private readonly smoothed = new Map<Id, vec2>();
   private bufferSize = 0;
+  private size = 0;
 
   constructor() {
     this.element.className = 'minimap';
@@ -23,8 +24,14 @@ export class Minimap {
     };
   }
 
+  // Reading clientWidth after this frame's style writes would force a synchronous layout of
+  // the whole overlay inside the animation frame; the size only changes when the window does.
+  public measure() {
+    this.size = this.element.clientWidth;
+  }
+
   public update(localPosition: vec2 | undefined, players: Array<MinimapPlayer>) {
-    const size = this.element.clientWidth;
+    const size = this.size;
     if (size === 0) {
       return;
     }

frontend/src/scripts/objects/game-object-container.ts

@@ -119,12 +119,13 @@ export class GameObjectContainer extends CommandReceiver {
   }
 
   public render(renderer: Renderer, overlay: HTMLElement, shouldChangeLayout: boolean) {
+    // First: everything below projects world coordinates through this frame's view area.
+    this.camera.draw(renderer);
     this.objects.forEach((o) => {
       if (this.isVisible(o.id)) {
         o.render(renderer, overlay, shouldChangeLayout);
       }
     });
-    this.camera.draw(renderer);
   }
 
   private isVisible(id: Id): boolean {

frontend/src/scripts/objects/types/camera.ts

@@ -1,6 +1,6 @@
 import { vec2 } from 'gl-matrix';
 import { Renderer } from 'sdf-2d';
-import { calculateViewArea, easeVec2, mixRgb, settings } from 'shared';
+import { calculateViewArea, followVec2, mixRgb, settings } from 'shared';
 import { Game } from '../../game';
 import { ScreenShake } from '../../screen-shake';
 
@@ -9,12 +9,20 @@ const snapDistance = 1500;
 
 export class Camera {
   public readonly center: vec2 = vec2.create();
+  private readonly previousTarget = vec2.create();
   private aspectRatio?: number;
 
   constructor(private readonly game: Game) {}
 
   public follow(target: vec2, deltaTimeInSeconds: number) {
-    easeVec2(this.center, target, deltaTimeInSeconds, followSeconds, snapDistance);
+    followVec2(
+      this.center,
+      target,
+      this.previousTarget,
+      deltaTimeInSeconds,
+      followSeconds,
+      snapDistance,
+    );
   }
 
   public draw(renderer: Renderer) {

shared/src/helper/ease.ts

@@ -13,3 +13,22 @@ export const easeVec2 = (
   vec2.distance(out, target) > snapDistance
     ? vec2.copy(out, target)
     : vec2.lerp(out, out, target, smoothing(deltaTimeInSeconds, seconds));
+
+// Follows a *moving* target. `easeVec2` alone lags one by an amount that depends on the frame
+// duration, so an uneven frame — or a dropped one — changes the lag and the follower appears to
+// speed up or stall. Carrying the target's own displacement first leaves only the residual to
+// decay: a target at a steady speed is tracked exactly, whatever the frame times do, and a
+// correction still fades out over `seconds`.
+export const followVec2 = (
+  out: vec2,
+  target: vec2,
+  previousTarget: vec2,
+  deltaTimeInSeconds: number,
+  seconds: number,
+  snapDistance: number,
+): vec2 => {
+  out[0] += target[0] - previousTarget[0];
+  out[1] += target[1] - previousTarget[1];
+  vec2.copy(previousTarget, target);
+  return easeVec2(out, target, deltaTimeInSeconds, seconds, snapDistance);
+};

shared/src/physics/character-movement.ts

@@ -2,6 +2,7 @@ import { vec2 } from 'gl-matrix';
 import { settings } from '../settings';
 import { GroundSurface, PhysicsBody } from './sdf';
 import { interpolateAngles } from './interpolate-angles';
+import { smoothing } from '../helper/ease';
 
 export const headRadius = 50;
 export const feetRadius = 20;
@@ -69,11 +70,20 @@ export const characterCenter = (
   return vec2.scale(center, center, 1 / 3);
 };
 
-const setDirection = (state: CharacterMovementState, direction: vec2) => {
+// Tuned so one physics step still turns the same 20% it always did, but a replay that splits
+// or merges steps now lands on the same facing instead of one that depends on the step count.
+const directionSmoothingSeconds =
+  settings.targetPhysicsDeltaTimeInSeconds / Math.log(1 / (1 - 0.2));
+
+const setDirection = (
+  state: CharacterMovementState,
+  direction: vec2,
+  deltaTimeInSeconds: number,
+) => {
   state.direction = interpolateAngles(
     state.direction,
     Math.atan2(direction[1], direction[0]) + Math.PI / 2,
-    0.2,
+    smoothing(deltaTimeInSeconds, directionSmoothingSeconds),
   );
 };
 
@@ -255,7 +265,7 @@ export const stepCharacterMovement = (
   vec2.zero(state.rightFoot.velocity);
 
   const movementForce = vec2.scale(
-    inputDirection,
+    vec2.create(),
     inputDirection,
     settings.maxAcceleration,
   );
@@ -272,7 +282,11 @@ export const stepCharacterMovement = (
     applyForce(state.rightFoot, rightFootGravity, deltaTimeInSeconds);
 
     const sumForce = vec2.subtract(vec2.create(), leftFootGravity, movementForce);
-    setDirection(state, vec2.length(sumForce) === 0 ? vec2.fromValues(0, -1) : sumForce);
+    setDirection(
+      state,
+      vec2.length(sumForce) === 0 ? vec2.fromValues(0, -1) : sumForce,
+      deltaTimeInSeconds,
+    );
   } else {
     carryWithRotatingPlanet(state, deltaTimeInSeconds);
 
@@ -310,7 +324,7 @@ export const stepCharacterMovement = (
     if (vec2.length(gravity) <= settings.planetDetachmentForceThreshold) {
       state.currentPlanet = undefined;
     }
-    setDirection(state, gravity);
+    setDirection(state, gravity, deltaTimeInSeconds);
   }
 
   keepPosture(state);

test/render-smoothness.test.ts

@@ -0,0 +1,288 @@
+import { describe, it, expect } from 'vitest';
+import {
+  LocalCharacterPredictor,
+  setFrameTimeMs,
+} from '../frontend/src/scripts/helper/prediction/local-character-predictor';
+import { ClientCharacterWorld } from '../frontend/src/scripts/helper/prediction/client-character-world';
+import {
+  stepCharacterMovement,
+  tickPlanetDetachment,
+  headRadius,
+  feetRadius,
+} from '../shared/src/physics/character-movement';
+import { followVec2 } from '../shared/src/helper/ease';
+import { settings } from '../shared/src/settings';
+
+const stepSeconds = settings.targetPhysicsDeltaTimeInSeconds;
+const snapshotMs = settings.updateMessageInterval * 1000;
+const heartbeatMs = settings.clientSendInterval * 1000;
+const planetRadius = 800;
+
+// The client's frame clock is rebased to its first frame (sdf-2d passes currentTime - startTime),
+// so it shares no origin with the server's clock. Anything that only works when the two happen to
+// agree is not a fix.
+const clientClockOffsetMs = 987_654.321;
+
+const vertices = Array.from({ length: settings.planetEdgeCount }, (_, i) => {
+  const angle = (i / settings.planetEdgeCount) * -Math.PI * 2;
+  return [planetRadius * Math.cos(angle), planetRadius * Math.sin(angle)];
+});
+
+const planetAt = (rotation: number) =>
+  ({ id: 1, vertices, snapshotRotation: rotation, snapshotRotationSpeed: 0 }) as never;
+
+const body = (center: Array<number>, radius: number) => ({
+  center: center.slice(),
+  radius,
+  velocity: [0, 0],
+  lastNormal: [0, 1],
+  restitution: 0,
+});
+
+interface Conditions {
+  fps: number;
+  /** One way. The round trip is twice this. */
+  wireMs: number;
+  /** Client CPU spent between the frame timestamp and the socket emit at the end of the loop. */
+  emitLagMs: number;
+  /** Peak random variation in frame duration, as a fraction of the nominal frame. */
+  frameJitter: number;
+  /** Probability that a frame is missed entirely and the next one is twice as long. */
+  dropRate: number;
+  /** Server work that can delay a socket callback past the packet's arrival. */
+  serverBusyMs: number;
+}
+
+interface Result {
+  /** Peak-to-peak variation in apparent speed, over its mean. Zero means perfectly even motion. */
+  speedSpread: number;
+  /** Peak-to-peak wobble of the character's screen position about its own mean, in world units. */
+  screenWobble: number;
+  /** How far the rendered character trails the true server pose, in world units. */
+  lagUnits: number;
+}
+
+// Runs a real server tick loop, a real wire, and a real render loop against the predictor, and
+// reports how steady the character's *on-screen* travel is. On-screen, not in world space: the
+// camera follows the character, so what the player sees is the residual between the two.
+const measure = ({
+  fps,
+  wireMs,
+  emitLagMs,
+  frameJitter,
+  dropRate,
+  serverBusyMs,
+}: Conditions): Result => {
+  const world = new ClientCharacterWorld();
+  world.sync([planetAt(0)]);
+
+  const server: any = {
+    head: body([0, planetRadius + 90], headRadius),
+    leftFoot: body([-33, planetRadius + 35], feetRadius),
+    rightFoot: body([33, planetRadius + 35], feetRadius),
+    direction: 0,
+    currentPlanet: world.surfaceById(1),
+    secondsSinceOnSurface: 0,
+    bodyVelocity: [0, 0],
+  };
+
+  const input = [1, 0];
+  const predictor = new LocalCharacterPredictor();
+  predictor.setStrength(settings.playerMaxStrength);
+
+  // Deterministic frame-time noise: a real display never delivers a perfectly even cadence.
+  let seed = 12345;
+  const random = () => ((seed = (seed * 1103515245 + 12345) & 0x7fffffff) / 0x7fffffff);
+
+  const nominalFrameMs = 1000 / fps;
+  const toServer: Array<{ atMs: number; clientStampMs: number }> = [];
+  const toClient: Array<{ atMs: number; ackAgeMs: number; clientStampMs: number; pose: any }> = [];
+  const samples: Array<{
+    screen: Array<number>;
+    world: Array<number>;
+    atMs: number;
+    error: number;
+  }> = [];
+
+  // The client's own view of the character, and the camera that chases it.
+  const camera = [0, planetRadius + 90];
+  const cameraPreviousTarget = [0, planetRadius + 90];
+
+  let simMs = 0;
+  let nextSnapshotMs = 0;
+  let nextFrameAtMs = 0;
+  let lastHeartbeatClientMs = -Infinity;
+  let lastInputClientTimeMs = 0;
+  let lastInputReceiptMs = 0;
+  let hasPressed = false;
+
+
+  while (simMs < 8000) {
+    tickPlanetDetachment(server, stepSeconds);
+    stepCharacterMovement(server, world, input.slice() as never, stepSeconds);
+    simMs += stepSeconds * 1000;
+
+    while (toServer.length > 0 && toServer[0].atMs <= simMs) {
+      const packet = toServer.shift()!;
+      if (packet.clientStampMs > lastInputClientTimeMs) {
+        lastInputClientTimeMs = packet.clientStampMs;
+        // Socket callbacks run between physics frames, so this is the true arrival instant,
+        // not the frame boundary — plus however long the frame in progress still had to run.
+        lastInputReceiptMs = packet.atMs + serverBusyMs;
+      }
+    }
+
+    if (simMs >= nextSnapshotMs) {
+      nextSnapshotMs += snapshotMs;
+      toClient.push({
+        atMs: simMs + wireMs,
+        clientStampMs: lastInputClientTimeMs,
+        ackAgeMs: lastInputReceiptMs > 0 ? Math.max(0, simMs - lastInputReceiptMs) : 0,
+        pose: {
+          head: server.head.center.slice(),
+          leftFoot: server.leftFoot.center.slice(),
+          rightFoot: server.rightFoot.center.slice(),
+          movement: {
+            direction: server.direction,
+            bodyVelocity: server.bodyVelocity.slice(),
+            leftFootNormal: server.leftFoot.lastNormal.slice(),
+            rightFootNormal: server.rightFoot.lastNormal.slice(),
+            groundPlanetId: 1,
+            secondsSinceOnSurface: server.secondsSinceOnSurface,
+          },
+        },
+      });
+    }
+
+    while (toClient.length > 0 && toClient[0].atMs <= simMs) {
+      const packet = toClient.shift()!;
+      predictor.acknowledge(
+        packet.clientStampMs,
+        packet.pose.movement,
+        -Infinity,
+        packet.ackAgeMs,
+      );
+      predictor.setAuthoritative(
+        { center: packet.pose.head, radius: headRadius } as never,
+        { center: packet.pose.leftFoot, radius: feetRadius } as never,
+        { center: packet.pose.rightFoot, radius: feetRadius } as never,
+      );
+    }
+
+    while (nextFrameAtMs <= simMs) {
+      const frameClientMs = nextFrameAtMs + clientClockOffsetMs;
+      setFrameTimeMs(frameClientMs);
+
+      // Movement input is edge-triggered, exactly as KeyboardListener sends it: recorded once
+      // when the key goes down, then held by both sides.
+      if (!hasPressed) {
+        hasPressed = true;
+        predictor.recordInput(input as never);
+      }
+
+      const dropped = random() < dropRate;
+      const jitter = 1 + (random() * 2 - 1) * frameJitter;
+      const frameMs = nominalFrameMs * jitter * (dropped ? 2 : 1);
+
+      if (!dropped) {
+        predictor.update([planetAt(0)], frameMs / 1000);
+        const head = predictor.head.center;
+        followVec2(
+          camera as never,
+          head,
+          cameraPreviousTarget as never,
+          frameMs / 1000,
+          0.08,
+          1500,
+        );
+        samples.push({
+          screen: [head[0] - camera[0], head[1] - camera[1]],
+          world: [head[0], head[1]],
+          atMs: frameClientMs,
+          error: Math.hypot(head[0] - server.head.center[0], head[1] - server.head.center[1]),
+        });
+      }
+
+      // Stamped at the top of the frame, emitted at the bottom.
+      if (frameClientMs - lastHeartbeatClientMs >= heartbeatMs - 1) {
+        lastHeartbeatClientMs = frameClientMs;
+        toServer.push({
+          atMs: nextFrameAtMs + emitLagMs + wireMs,
+          clientStampMs: frameClientMs,
+        });
+      }
+
+      nextFrameAtMs += frameMs;
+    }
+  }
+
+  const tail = samples.slice(Math.floor(samples.length / 2));
+
+  // Apparent speed: world travel divided by how long the frame was on screen. A longer frame
+  // should show proportionally more travel; anything else reads as a lurch.
+  const speeds = tail
+    .slice(1)
+    .map(
+      (s, i) =>
+        Math.hypot(s.world[0] - tail[i].world[0], s.world[1] - tail[i].world[1]) /
+        (s.atMs - tail[i].atMs),
+    );
+  const meanSpeed = speeds.reduce((a, b) => a + b, 0) / speeds.length;
+
+  // Screen position: the camera settles to a constant lag, so anything left is visible wobble.
+  const centre = [0, 1].map((axis) => {
+    const values = tail.map((s) => s.screen[axis]);
+    return values.reduce((a, b) => a + b, 0) / values.length;
+  });
+  const wobble = Math.max(
+    ...tail.map((s) => Math.hypot(s.screen[0] - centre[0], s.screen[1] - centre[1])),
+  );
+
+  if ((globalThis as any).__dump) {
+    console.log('--- apparent speed (u/ms) ---');
+    speeds.slice(0, 40).forEach((v, i) => console.log(`${i} ${v.toFixed(4)}`));
+  }
+  return {
+    speedSpread: (Math.max(...speeds) - Math.min(...speeds)) / meanSpeed,
+    screenWobble: wobble,
+    lagUnits: tail.reduce((sum, s) => sum + s.error, 0) / tail.length,
+  };
+};
+
+const CONDITIONS: Array<[string, Conditions]> = [
+  ['localhost, clean frames', { fps: 60, wireMs: 0.2, emitLagMs: 2, frameJitter: 0, dropRate: 0, serverBusyMs: 0 }],
+  ['localhost, 144 Hz', { fps: 144, wireMs: 0.2, emitLagMs: 2, frameJitter: 0, dropRate: 0, serverBusyMs: 0 }],
+  ['localhost, frame jitter', { fps: 60, wireMs: 0.2, emitLagMs: 2, frameJitter: 0.15, dropRate: 0, serverBusyMs: 0 }],
+  ['localhost, 2% dropped', { fps: 60, wireMs: 0.2, emitLagMs: 2, frameJitter: 0.05, dropRate: 0.02, serverBusyMs: 0 }],
+  ['localhost, variable CPU', { fps: 60, wireMs: 0.2, emitLagMs: 8, frameJitter: 0.15, dropRate: 0.02, serverBusyMs: 0 }],
+  ['30 ms link', { fps: 60, wireMs: 15, emitLagMs: 2, frameJitter: 0.05, dropRate: 0, serverBusyMs: 0 }],
+  ['120 ms link', { fps: 60, wireMs: 60, emitLagMs: 2, frameJitter: 0.05, dropRate: 0, serverBusyMs: 0 }],
+];
+
+describe('local character render smoothness', () => {
+  it('keeps on-screen travel even under realistic clocks and frame pacing', () => {
+    ;(globalThis as any).__dump = true
+    measure(CONDITIONS[0][1])
+    ;(globalThis as any).__dump = false
+    const results: Array<[string, Result]> = CONDITIONS.map(([label, c]) => [label, measure(c)]);
+
+    for (const [label, r] of results) {
+      console.log(
+        `${label.padEnd(26)} speed=${(r.speedSpread * 100).toFixed(1).padStart(6)}% ` +
+          `wobble=${r.screenWobble.toFixed(2).padStart(6)}u ` +
+          `lag=${r.lagUnits.toFixed(1).padStart(5)}u `,
+      );
+    }
+
+    for (const [label, r] of results) {
+      // Even travel: what the eye reads as smooth motion.
+      expect(r.speedSpread, label).toBeLessThan(0.05);
+      // No sub-pixel swimming against the camera, which follows the same character.
+      expect(r.screenWobble, label).toBeLessThan(1);
+    }
+
+    // Prediction exists to cancel latency; a follower that lags undoes that on every link.
+    const localhost = results.find(([label]) => label === 'localhost, clean frames')![1];
+    expect(localhost.lagUnits).toBeLessThan(5);
+  });
+});