diff --git a/docs/ARCHITECTURE.md b/docs/ARCHITECTURE.md
index ba9f4b6b..4ffb3469 100644
--- a/docs/ARCHITECTURE.md
+++ b/docs/ARCHITECTURE.md
@@ -34,8 +34,12 @@ effects), [DATA.md](DATA.md) (generated data), [ASSETS.md](ASSETS.md) (art and a
   against the spec (`server/match/fields.js` `validateClientResult`), runs the battles no connected human owns (bots, a
   dropped player, an authority that missed its deadline) and can re-simulate results (`SP_VERIFY`). `SP_COMBAT=server`
   is the older mode in which the server runs every battle and streams snapshots. DESIGN §14.
-- **Determinism** is what makes this work: fixed 1/30 s ticks, a seeded PRNG (`server/sim/rng.js`), no wall clock and
-  no `Math.random` in the sim, so the same spec and data give the same battle on every machine, bit for bit. The golden
+- **Determinism** is what makes this work: fixed 1/30 s ticks, a seeded PRNG (`server/sim/rng.js`), no wall clock,
+  no `Math.random` and only correctly rounded arithmetic in the sim, so the same spec and data give the same battle on
+  every machine, bit for bit. ECMA-262 leaves `Math.hypot`, `sin`, `cos`, `atan2`, `pow` (and `**`) implementation-
+  approximated and the engines differ in their last bits (V8, SpiderMonkey and JavaScriptCore each compute `hypot`
+  differently; Chrome's and Node's `sin` differ), so the sim takes them from `server/sim/detmath.js`, built only from
+  + − × ÷ and `Math.sqrt`; ESLint and `test/sim/detmath.test.js` refuse the Math ones in `server/sim/`. The golden
   results (§5) rely on it.
 
 ## 2. The WebSocket protocol
diff --git a/docs/SIM.md b/docs/SIM.md
index 45d84c7b..08a53a3f 100644
--- a/docs/SIM.md
+++ b/docs/SIM.md
@@ -3,7 +3,9 @@
 Audience: **content authors** (kits, bonds, garrisons, items, bands, enemies, bosses, devices, choices) and the
 **match owner** who drives `Battle`. The normative contract is DESIGN.md §5 ([design/engine.md](design/engine.md));
 this file documents the concrete implementation, every hook and helper, the SkillSpec schema with worked examples,
-the profession defaults and the test harness. Everything here is deterministic: the only randomness is `battle.rng()`.
+the profession defaults and the test harness. Everything here is deterministic: the only randomness is `battle.rng()`,
+and the only floating-point functions are correctly rounded ones (`detmath.js` below), so a battle gives the same bits in
+every browser and on the server.
 
 ```
 server/sim/
@@ -13,6 +15,8 @@ server/sim/
                    tiles (relocation, bodies, tactical points), displacement, economy (DP, layers, coins), events
   constants.js     TICK, MOVE_SCALE, ATTACK_PAUSE, element numbers, tuning knobs
   rng.js           mulberry32 PRNG (+ int/range/chance/pick/shuffle/weighted)
+  detmath.js       hypot / powi / sin / cos / atan2 with the same bits on every engine — the sim never calls the
+                   implementation-approximated Math functions (hypot, sin, cos, atan2, pow, ** …; lint + test enforce it)
   grid.js          stage grid, tile semantics, 8-dir A* (no corner cutting), obstacles
   units.js         Unit model + stat aggregation
   buffs.js         Buff model, mod keys, status catalogue
@@ -851,7 +855,7 @@ instance) and skip `'counter'` / `'reflect'` damage. When the guard trips, the l
 | `push(enemy, force, {from, dir, fixed, fixedAngle, inward, effect})`, `pull(enemy, force, {to, center, stop})`, `pullToFront(enemy, unit, force)`, `forceLevel(enemy, force)`, `pushDistance(enemy, force, {effect})` | the official 位移 (PRTS 游戏数据基础 §重量公式 / 推与拉; user playtest #6 item 14): 受力等级 = 力度 (微小力 −1, 小力 0, 中力 1, 较大力 2, 大力 3 …) − current 重量等级 (massLevel, 失重 counts). Push distance per level (`constants.js PUSH_TILES`, PRTS 推与拉's 弹道 column): ≤ −3 → 0, −2 → 0.12, −1 → 0.44, 0 → 1.7, 1 → 2.14, 2 → 2.96, ≥ 3 → 3.53 tiles; `effect` = a 特效 push (`PUSH_TILES_EFFECT`: −2 → 0.085, −1 → 0.374, 0 → 1.562, 1 → 1.987, 2 → 2.773, ≥ 3 → 3.331 — 见行者 S1 / S2, `PUSH_EFFECT_SKILLS`; every other pusher uses the 弹道 column [ASSUMED]); radial (away from `from`; the client's buff template `knockback[relative]`: 莫斯提马 S3, 山 S3, 琳琅诗怀雅 S3 — also PRTS 备注 "推开效果为径向推动") unless `dir` (directional; template `knockback[dir]` = Knockback {`_useSourceDirection` true, `_decreaseForceLevelWhenNotInDirection` 2}: 推击手, 野鬃 S2 "往攻击方向" (charpack char_496_wildmn: buff `wildmn_s_2[force]`) — > 45° off or < 0.25 tile ⇒ radial and level −2; `fixed` waives both (圣聆初雪 S1 朝部署方向, `KnockBackWithCharacterDirection`), `fixedAngle` only the angle (见行者 S2)); `inward` = a push towards `from` (薄绿 S2), stopping at the 急停 radius. Pull: level ≥ 0 → to `to` / the 急停 radius around `center`, −1 → 35 % of the way, −2 → 0.03, ≤ −3 → 0; `pullToFront` aims at the 拉力起点 0.5 tile ahead of the unit with the 急停 radius 0.6708 around it (never moves an enemy the unit itself blocks — nor does a pull towards an ally's centre, e.g. the 流形 S3 pulse) A 静态刚体 (data `staticBody`: every air unit of the mode except “炎佑”, plus 昆图斯 — PRTS 特殊机制 "可以进入失衡状态…但物理层面上无法产生任何速度或移动", "与单位的行动方式无关") moves 0 from every source while the skills that reach it still hit it (薄绿 S2, 锏 S3, the 钩索师 …; player report after 0.1.0) |
 | `displace(enemy, {x, y}, tiles)` | the raw mover behind push / pull: along passable tiles (by `motion`: FLY in the rect, else ground-passable), no weight rule; nothing for 失衡免疫 (`noDisplace`: 近地悬浮 incl. 喷气人's 飞行模式, 浮空, the 胄 parts, 守墓石像's statue and flight), 静态刚体 (`def.staticBody`) and leaders (`_displaceable`); unblocks + re-paths |
 | `after(seconds, fn, {owner})`, `every(seconds, fn, {owner, immediate})` | return `{cancel()}`; fn(battle, sched) |
-| `fx(kind, params)`, `rng()` (+ `rng.int/range/chance/pick/shuffle/weighted`) | never use Math.random |
+| `fx(kind, params)`, `rng()` (+ `rng.int/range/chance/pick/shuffle/weighted`) | never use Math.random; for distances, angles and powers use `hypot`, `sin`, `cos`, `atan2`, `powi` from `server/sim/detmath.js`, never `Math.hypot` / `sin` / `cos` / `atan2` / `pow` / `**` (their last bits differ between browsers) |
 | `forceAttack(unit, targets?, { noAmmo })` | an immediate attack with the current profile (hooks, attack SP, a running ammo skill's bullet); `noAmmo: true` = spends no bullet and emits no `ammoUsed` (圣约送葬人 extra attack). Returns true when it attacked |
 | `findTacticalPoint(unit)`, `groundPathTiles()` | tactical point (战术点, tactician 援军 / talent tokens): a free (`isReservedTile`) walkable, standable (`canStand`: never the 深水区) tile of the initial range **on an enemy ground path first** (`groundPathTiles`: grid paths of every non-FLY route, cached per grid version), then nearest (Chebyshev, rows break ties), then the lowest tile key |
 | `effectiveProfile(unit)`, `reduceElement(unit, amount, el?)` | |
diff --git a/eslint.config.js b/eslint.config.js
index 43a85fe6..3073ce80 100644
--- a/eslint.config.js
+++ b/eslint.config.js
@@ -39,6 +39,10 @@ const rules = {
   'preserve-caught-error': 'warn',
 };
 
+// ECMA-262 §21.3.2: the Math functions whose results are "implementation-approximated".
+const APPROXIMATED_MATH = ['acos', 'acosh', 'asin', 'asinh', 'atan', 'atanh', 'atan2', 'cbrt', 'cos', 'cosh', 'exp',
+  'expm1', 'hypot', 'log', 'log1p', 'log10', 'log2', 'pow', 'sin', 'sinh', 'tan', 'tanh'];
+
 const nodeFiles = [
   'server/**/*.js',
   'shared/**/*.js',
@@ -79,6 +83,20 @@ export default [
     },
     rules,
   },
+  {
+    // The sim runs in every player's browser and on the server, and must give the same bits everywhere. ECMA-262
+    // leaves these Math functions (and **) implementation-approximated: engines differ in the last bits.
+    files: ['server/sim/**/*.js'],
+    rules: {
+      'no-restricted-properties': ['error', ...APPROXIMATED_MATH.map((property) => ({
+        object: 'Math', property, message: 'implementation-approximated (engines differ in the last bits): use server/sim/detmath.js',
+      }))],
+      'no-restricted-syntax': ['error',
+        { selector: "BinaryExpression[operator='**']", message: 'implementation-approximated: use powi from server/sim/detmath.js' },
+        { selector: "AssignmentExpression[operator='**=']", message: 'implementation-approximated: use powi from server/sim/detmath.js' },
+      ],
+    },
+  },
   {
     // Node test runner, plus browser tests whose page.evaluate callbacks use DOM globals.
     // PIXI is the page global those render tests read inside evaluate().
diff --git a/jsconfig.json b/jsconfig.json
index a8f02f2c..aa30a313 100644
--- a/jsconfig.json
+++ b/jsconfig.json
@@ -18,6 +18,7 @@
   "include": [
     "shared/**/*.js",
     "server/sim/rng.js",
+    "server/sim/detmath.js",
     "server/sim/dir.js",
     "server/sim/constants.js",
     "types/**/*.js"
diff --git a/server/sim/ai.js b/server/sim/ai.js
index c0fb8750..c9a9f0eb 100644
--- a/server/sim/ai.js
+++ b/server/sim/ai.js
@@ -38,6 +38,7 @@ import { sortEnemyTargets, sortAllyTargets, canTargetEnemy, canTargetAlly, tileK
 import { reduceElement } from './damage.js';
 import { straightClear } from './grid.js';
 import { moveFeared, endFear } from './fear.js';
+import { hypot, powi } from './detmath.js';
 
 // ---------------------------------------------------------------------------------------------------------------
 // profiles
@@ -325,13 +326,13 @@ export function resolveHit(b, u, prof, target, info, x, y) {
       let best = null, bd = Infinity;
       for (const e of b.enemiesInRadius(prev.x, prev.y, prof.chain.radius || CHAIN_RADIUS)) {
         if (hit.has(e.id) || !canTargetEnemy(u, e, prof)) continue;
-        const d = Math.hypot(e.x - prev.x, e.y - prev.y);
+        const d = hypot(e.x - prev.x, e.y - prev.y);
         if (d < bd - 1e-9 || (Math.abs(d - bd) <= 1e-9 && best && e.spawnSeq < best.spawnSeq)) { bd = d; best = e; }
       }
       if (!best) break;
       hit.add(best.id);
       b._ev(['atk', prev.id, best.id, 'chain']);
-      const d = b.dealDamage(u, best, { amount: atk * scale * skillMul * Math.pow(1 - (prof.chain.falloff ?? 0.15), k), type: baseType, isAttack: true, isSkill: info.isSkill, tags: ['chain'], attackId });
+      const d = b.dealDamage(u, best, { amount: atk * scale * skillMul * powi(1 - (prof.chain.falloff ?? 0.15), k), type: baseType, isAttack: true, isSkill: info.isSkill, tags: ['chain'], attackId });
       dealtTotal += d;
       if (prof.chain.sluggish && best.alive) b.applyStatus(best, 'sluggish', { duration: prof.chain.sluggish, source: u });
       if (each) each(best, d, 'chain');
@@ -389,7 +390,7 @@ function doHeal(b, u, prof, t) {
       if (!best) break;
       seen.add(best.id);
       b._ev(['atk', prev.id, best.id, 'chainHeal']);
-      b.heal(u, best, amount * Math.pow(1 - (h.falloff ?? 0.25), k));
+      b.heal(u, best, amount * powi(1 - (h.falloff ?? 0.25), k));
       prev = best;
     }
   }
@@ -475,7 +476,7 @@ function planLeg(b, e, leg) {
   R.version = b.grid.version;
   // suffix lengths for remaining-distance queries
   const suf = new Float64Array(pts.length);
-  for (let i = pts.length - 2; i >= 0; i--) suf[i] = suf[i + 1] + Math.hypot(pts[i + 1].x - pts[i].x, pts[i + 1].y - pts[i].y);
+  for (let i = pts.length - 2; i >= 0; i--) suf[i] = suf[i + 1] + hypot(pts[i + 1].x - pts[i].x, pts[i + 1].y - pts[i].y);
   R.suffix = suf;
 }
 
@@ -497,7 +498,7 @@ function tailLength(b, e, idx) {
       let len = 0;
       if (L.t === 'move') {
         const [sr, sc] = starts[i];
-        len = Math.hypot(L.r - sr, L.c - sc);
+        len = hypot(L.r - sr, L.c - sc);
         if (e.motion !== 'FLY' && b.grid.inBounds(sr, sc)) {
           const f = legField(b, L.r, L.c, sr * COLS + sc);
           const fl = f ? b.grid.fieldLength(f, sr * COLS + sc) : Infinity;
@@ -519,9 +520,9 @@ export function remainingDistance(b, e) {
   const leg = R.legs[R.legIdx];
   if (leg && leg.t === 'move' && R.pts && R.ptIdx < R.pts.length) {
     const p = R.pts[R.ptIdx];
-    d += Math.hypot(p.x - e.x, p.y - e.y) + R.suffix[R.ptIdx];
+    d += hypot(p.x - e.x, p.y - e.y) + R.suffix[R.ptIdx];
   } else if (leg && leg.t === 'move') {
-    d += Math.hypot(leg.c - e.x, leg.r - e.y);
+    d += hypot(leg.c - e.x, leg.r - e.y);
   }
   return d;
 }
@@ -614,7 +615,7 @@ function moveAttracted(b, e, dt) {
   let moved = false;
   while (dist > 1e-9 && A.i < A.pts.length) {
     const p = A.pts[A.i];
-    const dx = p.x - e.x, dy = p.y - e.y, d = Math.hypot(dx, dy);
+    const dx = p.x - e.x, dy = p.y - e.y, d = hypot(dx, dy);
     if (d <= dist) { e.x = p.x; e.y = p.y; dist -= d; A.i++; } else { e.x += (dx / d) * dist; e.y += (dy / d) * dist; dist = 0; }
     moved = true;
   }
@@ -667,7 +668,7 @@ function advanceRoute(b, e, dt, R, standing = false) {
       if (R.ptIdx >= R.pts.length) break;
       const p = R.pts[R.ptIdx];
       const dx = p.x - e.x, dy = p.y - e.y;
-      const d = Math.hypot(dx, dy);
+      const d = hypot(dx, dy);
       if (d <= dist) {
         e.x = p.x; e.y = p.y;
         dist -= d;
@@ -848,7 +849,7 @@ function enemyAttack(b, e, prevCd) {
       if (!tt || !tt.alive || !e.alive && !rangedShot) return;
       b.dealDamage(e, tt, { amount: e.s.atk * (e.profile?.atkScale ?? 1), type, isAttack: true, attackId, isProjectile });
     };
-    if (rangedShot && Math.hypot(t.x - e.x, t.y - e.y) > 0.75) {
+    if (rangedShot && hypot(t.x - e.x, t.y - e.y) > 0.75) {
       b.addProjectile({ from: e, target: t, speed: PROJECTILE_SPEEDS.enemy, visual: 'enemy', source: e, onHit: (c) => hit(c.target, true) });
     } else hit(t);
   }
diff --git a/server/sim/battle/displacement.js b/server/sim/battle/displacement.js
index 88d6d790..12a2cfc7 100644
--- a/server/sim/battle/displacement.js
+++ b/server/sim/battle/displacement.js
@@ -4,6 +4,7 @@
 
 import { COLS, PUSH_TILES, PUSH_TILES_EFFECT, PULL_WEAK_SHARE, PULL_CRAWL, PULL_ORIGIN, PULL_STOP_RADIUS, PUSH_DIRECTIONAL_MIN_DIST } from '../constants.js';
 import { fin } from './util.js';
+import { hypot } from '../detmath.js';
 
 /**
  * Official push distance (tiles) of a 受力等级 (constants.js PUSH_TILES: ≤ −3 → 0, ≥ 3 → the 3 value); `effect` = a 特效
@@ -42,10 +43,10 @@ export class BattleDisplacement {
     if (!this._displaceable(e)) return 0;
     let level = this.forceLevel(e, force);
     const fx0 = fin(from?.x, e.x), fy0 = fin(from?.y, e.y);
-    const vx = e.x - fx0, vy = e.y - fy0, d = Math.hypot(vx, vy);
+    const vx = e.x - fx0, vy = e.y - fy0, d = hypot(vx, vy);
     let ux = 0, uy = 0;
     const dirX = dir ? fin(dir.x, 0) : 0, dirY = dir ? fin(dir.y, 0) : 0;
-    const dl = Math.hypot(dirX, dirY);
+    const dl = hypot(dirX, dirY);
     if (dl > 0) {
       ux = dirX / dl; uy = dirY / dl;
       if (from && !fixed && (d < PUSH_DIRECTIONAL_MIN_DIST || (!fixedAngle && vx * ux + vy * uy < d * Math.SQRT1_2))) {
@@ -71,7 +72,7 @@ export class BattleDisplacement {
     // an enemy the puller itself blocks already stands in front of it (at contact) [ASSUMED: no pull, no unblocking]
     if (center && center.side === 'ally' && e.blockedBy === center) return 0;
     const tx = fin(to.x, e.x), ty = fin(to.y, e.y);
-    const dx = tx - e.x, dy = ty - e.y, d0 = Math.hypot(dx, dy);
+    const dx = tx - e.x, dy = ty - e.y, d0 = hypot(dx, dy);
     if (!(d0 > 1e-6)) return 0;
     const ux = dx / d0, uy = dy / d0;
     // travel until inside the stop circle around `center` (smaller root of |e + t·u − c| = stop), else up to `to`
@@ -122,7 +123,7 @@ export class BattleDisplacement {
   displace(e, dir, distance) {
     if (!this._displaceable(e) || !dir) return 0;
     const dxv = fin(dir.x, 0), dyv = fin(dir.y, 0);
-    const len = Math.hypot(dxv, dyv);
+    const len = hypot(dxv, dyv);
     if (!(len > 0)) return 0;
     const eff = Math.min(fin(distance, 0), 2 * COLS);
     if (!(eff > 0)) return 0;
diff --git a/server/sim/body.js b/server/sim/body.js
index 52191434..6e65f3d0 100644
--- a/server/sim/body.js
+++ b/server/sim/body.js
@@ -24,6 +24,7 @@
 // `hitRect`).
 
 import { ROWS, COLS } from './constants.js';
+import { hypot } from './detmath.js';
 
 /**
  * Validated hit area of a data record (`{ w, h, dx, dy }`, tiles) or null for a regular (point) unit.
@@ -83,10 +84,10 @@ export function bodyInKeys(u, keys) {
 /** Distance (tiles) from point (x, y) to the unit's body: its position, or its rectangle (0 inside). */
 export function bodyDist(u, x, y) {
   const R = hitRect(u);
-  if (!R) return Math.hypot(u.x - x, u.y - y);
+  if (!R) return hypot(u.x - x, u.y - y);
   const dx = x < R.x0 ? R.x0 - x : x > R.x1 ? x - R.x1 : 0;
   const dy = y < R.y0 ? R.y0 - y : y > R.y1 ? y - R.y1 : 0;
-  return Math.hypot(dx, dy);
+  return hypot(dx, dy);
 }
 
 /** Is the unit's body within `r` tiles of point (x, y)? (the same ≤ r + 1e-9 test as Battle.enemiesInRadius) */
diff --git a/server/sim/buffs.js b/server/sim/buffs.js
index 86bca4b7..08744909 100644
--- a/server/sim/buffs.js
+++ b/server/sim/buffs.js
@@ -8,6 +8,7 @@
 // Additive mod keys scale with stacks (value × stacks); *Mul keys multiply (value ^ stacks).
 
 import { COLD_ASPD, COLD_FREEZE_DURATION, FREEZE_RES_DOWN, RESIST_DEFAULT, RESIST_PALSY_DECAY } from './constants.js';
+import { powi } from './detmath.js';
 
 /** 抵抗: "麻痹等状态每5秒流失1层" — tick of the resist buff. */
 function resistPalsyDecay({ battle, unit }) {
@@ -200,12 +201,12 @@ export function aggregateMods(buffs) {
         const v = m[k];
         if (typeof v !== 'number' || !Number.isFinite(v)) continue;
         if (k === 'shield') continue;
-        if (k.endsWith('Mul')) mul[k] = (mul[k] ?? 1) * (st === 1 ? v : Math.pow(v, st));
+        if (k.endsWith('Mul')) mul[k] = (mul[k] ?? 1) * (st === 1 ? v : powi(v, st));
         else if ((k === 'dodgePhys' || k === 'dodgeArts') && v > 0) {
           const p = Math.min(1, v);
           const d = dodge[k];
-          if (!d) dodge[k] = { n: st, p, miss: Math.pow(1 - p, st) };
-          else { d.n += st; d.miss *= Math.pow(1 - p, st); }
+          if (!d) dodge[k] = { n: st, p, miss: powi(1 - p, st) };
+          else { d.n += st; d.miss *= powi(1 - p, st); }
         } else {
           add[k] = (add[k] ?? 0) + v * st;
           if (k === 'rangeExtend' && b.persist && b.duration === Infinity) permRangeExtend += v * st;
diff --git a/server/sim/content/bosses.js b/server/sim/content/bosses.js
index 56eecc4d..eb3cd4be 100644
--- a/server/sim/content/bosses.js
+++ b/server/sim/content/bosses.js
@@ -75,6 +75,7 @@ import {
   ensureInstalled, abOf, attach, T, elem, hurt, targetsNear, allTargets, byPriority, areaAllies, areaAlliesInTiles, fieldAllies,
   remainingRoute, stayRoute, stepToward, setHits, hitCount, lpLoss, blinkForward, canCast, absorbArts, nthOf,
 } from './enemies.js';
+import { hypot, powi } from '../detmath.js';
 
 // ---------------------------------------------------------------------------------------------------------------
 // constants
@@ -209,7 +210,7 @@ function rawRouteOf(b, e, tpl) {
 function toGoal(b, x, y, motion = 'WALK') {
   const ends = b.grid.specialTiles('end');
   let best = ends[0] ?? [Math.round(y), Math.round(x)], bd = Infinity;
-  for (const p of ends) { const d = Math.hypot(p[0] - y, p[1] - x); if (d < bd) { bd = d; best = p; } }
+  for (const p of ends) { const d = hypot(p[0] - y, p[1] - x); if (d < bd) { bd = d; best = p; } }
   return { motion, start: [y, x], end: best, checkpoints: [] };
 }
 
@@ -318,7 +319,7 @@ function mirror(b, e, tpl) {
 const opsOnly = (list) => list.filter((u) => u.kind === 'op');
 const nearestOf = (b, e, pred) => {
   let best = null, bd = Infinity;
-  for (const o of b.enemies) { if (!o.alive || o === e || !pred(o)) continue; const d = Math.hypot(o.x - e.x, o.y - e.y); if (d < bd) { bd = d; best = o; } }
+  for (const o of b.enemies) { if (!o.alive || o === e || !pred(o)) continue; const d = hypot(o.x - e.x, o.y - e.y); if (d < bd) { bd = d; best = o; } }
   return best;
 };
 /** Apply a floor to an enemy whose data attack speed is 0 (skill-driven leaders: 铳/管/弦). */
@@ -370,7 +371,7 @@ function stunBlast(b, src, by, r, c, stun, dot, dur, kind) {
 function segDist(px, py, ax, ay, bx, by) {
   const dx = bx - ax, dy = by - ay, L = dx * dx + dy * dy;
   const t = L > 0 ? Math.max(0, Math.min(1, ((px - ax) * dx + (py - ay) * dy) / L)) : 0;
-  return Math.hypot(px - (ax + t * dx), py - (ay + t * dy));
+  return hypot(px - (ax + t * dx), py - (ay + t * dy));
 }
 
 // ---------------------------------------------------------------------------------------------------------------
@@ -708,7 +709,7 @@ function kitSpring(ab, e) {
         const d = e2.mem.ab.dash;
         if (!d) return;
         const g = d.gun && d.gun.alive ? d.gun : gun(b);
-        const arrived = !g || stepToward(e2, g.x, g.y, e2.s.moveSpeed * d.mul * MOVE_SCALE * dt) || Math.hypot(g.x - e2.x, g.y - e2.y) < 1;
+        const arrived = !g || stepToward(e2, g.x, g.y, e2.s.moveSpeed * d.mul * MOVE_SCALE * dt) || hypot(g.x - e2.x, g.y - e2.y) < 1;
         // 追逐模式 "对进入自身0.35半径范围内的我方单位（包括飞行单位）造成一次攻击力100%的物理普通伤害" (until 0.1.3: radius 0.5)
         for (const u of areaAllies(b, e2, e2.x, e2.y, CHARGE_RADIUS)) if (!d.hit.has(u)) { d.hit.add(u); hurt(b, e2, u, e2.s.atk, 'phys'); }
         if (arrived || b.time >= d.until) { e2.mem.ab.dash = null; b.removeBuff(e2, 'boss:dash'); if (e2.route) e2.route.pts = null; }
@@ -730,9 +731,9 @@ function kitSpring(ab, e) {
           for (let k = 0; k < times && t; k++) {
             hit.add(t);
             b.fx('beam', { x: e2.x, y: e2.y, from: e2.id, to: t.id, kind: 'springBullet' });
-            elem(b, e2, t, 'erosion', e2.s.atk * ratio * Math.pow(SPRING_BOUNCE_FALLOFF, k));
+            elem(b, e2, t, 'erosion', e2.s.atk * ratio * powi(SPRING_BOUNCE_FALLOFF, k));
             const prev = t;
-            t = areaAllies(b, e2, prev.x, prev.y, SPRING_BOUNCE_RANGE).filter((u) => !hit.has(u)).sort((p, q) => Math.hypot(p.x - prev.x, p.y - prev.y) - Math.hypot(q.x - prev.x, q.y - prev.y) || aggroCmp(p, q))[0];
+            t = areaAllies(b, e2, prev.x, prev.y, SPRING_BOUNCE_RANGE).filter((u) => !hit.has(u)).sort((p, q) => hypot(p.x - prev.x, p.y - prev.y) - hypot(q.x - prev.x, q.y - prev.y) || aggroCmp(p, q))[0];
           }
         } else { // 十连击
           b.fx('beam', { x: e2.x, y: e2.y, from: e2.id, to: t0.id, kind: 'springCombo' });
diff --git a/server/sim/content/devices.js b/server/sim/content/devices.js
index ebd0c344..84b57608 100644
--- a/server/sim/content/devices.js
+++ b/server/sim/content/devices.js
@@ -37,6 +37,7 @@ import { COLS, ROWS } from '../constants.js';
 import { performAttack } from '../ai.js';
 import { sortEnemyTargets } from '../targeting.js';
 import { DIR_VEC, normDir, oppositeDir } from '../dir.js';
+import { hypot } from '../detmath.js';
 
 const num = (v, d = 0) => (typeof v === 'number' && Number.isFinite(v) ? v : (typeof v === 'string' && v.trim() !== '' && Number.isFinite(+v) ? +v : d));
 /** ASPD blackboard values: a fraction (|v| < 1, e.g. −0.6 / −0.05) is ×100 ASPD, otherwise flat (+20). */
@@ -192,7 +193,7 @@ function deviceTile(battle, r0, c0) {
       const exact = r === r0 && c === c0;
       if (!exact && battle.grid.tile(r, c).build !== 'NONE') continue;
       const walk = !exact && battle.grid.groundPassable(r, c, true) ? 1 : 0;
-      const s = [exact ? 0 : 1, walk, Math.hypot(r - r0, c - c0), r * COLS + c];
+      const s = [exact ? 0 : 1, walk, hypot(r - r0, c - c0), r * COLS + c];
       let less = !bs;
       if (!less) for (let i = 0; i < s.length; i++) { if (s[i] < bs[i] - 1e-9) { less = true; break; } if (s[i] > bs[i] + 1e-9) break; }
       if (less) { best = [r, c]; bs = s; }
@@ -455,7 +456,7 @@ function tickAirflowEnemy(battle, st, e) {
     const r = Math.round(e.y), c = Math.round(e.x);
     const f = r >= 0 && r < ROWS && c >= 0 && c < COLS ? st.flow.get(r * COLS + c) : null;
     const vx = e.x - px, vy = e.y - py;
-    const v = Math.hypot(vx, vy);
+    const v = hypot(vx, vy);
     if (f && v > 1e-6) {
       const rel = airflowRelation((vx * f.fx + vy * f.fy) / v);
       if (rel !== 'vertical') mul = 1 + num(f.bb[`blower_s_enemy[${rel}].move_speed`], 0);
diff --git a/server/sim/content/enemies.js b/server/sim/content/enemies.js
index ed568ca6..1ce33f4d 100644
--- a/server/sim/content/enemies.js
+++ b/server/sim/content/enemies.js
@@ -116,6 +116,7 @@ import { REFLECTION_KITS } from './enemies/reflection.js';
 import { FLY_KITS } from './enemies/fly.js';
 import { SPECIAL_KITS } from './enemies/special.js';
 import { LEADER_KITS } from './enemies/leaders.js';
+import { hypot } from '../detmath.js';
 
 export {
   EROSION, EROSION_BURST, HUSK_REBIRTH, nthOf, abOf, attach, T, silenced, canCast, elem, hurt, alliesInTiles,
@@ -312,7 +313,7 @@ function chaliceShare(b, c) {
   for (const o of b.enemies) {
     if (!o.alive || o === t || o.defId !== 'enemy_1430_lrrook' || !o.mem.ab) continue;
     const a = o.mem.ab.list[0];
-    if (!a || !(a.share > 0) || Math.hypot(o.x - t.x, o.y - t.y) > a.r) continue;
+    if (!a || !(a.share > 0) || hypot(o.x - t.x, o.y - t.y) > a.r) continue;
     const part = c.dmg.amount * a.share;
     c.dmg.amount -= part;
     // a 无来源 burst's share stays 无来源, credited like the burst (damage.js)
diff --git a/server/sim/content/enemies/archetypes.js b/server/sim/content/enemies/archetypes.js
index 028487d0..19185942 100644
--- a/server/sim/content/enemies/archetypes.js
+++ b/server/sim/content/enemies/archetypes.js
@@ -8,6 +8,7 @@ import {
   HUSK_REBIRTH, BOOM_RADIUS, POLLUTION_INTERVAL, STEALTH_RESTORE_BY_KEY, DUCK_STEALTH_RESTORE, nthOf, stOf, safe, num, T, elem, hurt,
   areaAllies, areaAlliesInTiles, zone, spawnChildren, stepToward, setHits, hitCount, setForm, absorbArts, auraBuff,
 } from './helpers.js';
+import { hypot } from '../../detmath.js';
 
 // ---------------------------------------------------------------------------------------------------------------
 // archetypes
@@ -105,7 +106,7 @@ function pathKeysAhead(e) {
   if (!R || !Array.isArray(R.pts)) return keys;
   let x = e.x, y = e.y;
   for (let i = R.ptIdx ?? 0; i < R.pts.length; i++) {
-    const p = R.pts[i], n = Math.max(1, Math.ceil(Math.hypot(p.x - x, p.y - y) * 4));
+    const p = R.pts[i], n = Math.max(1, Math.ceil(hypot(p.x - x, p.y - y) * 4));
     for (let k = 1; k <= n; k++) keys.add(Math.round(y + ((p.y - y) * k) / n) * COLS + Math.round(x + ((p.x - x) * k) / n));
     x = p.x; y = p.y;
   }
@@ -405,7 +406,7 @@ function unbalanced(onMove) {
       a.px = e.x; a.py = e.y; a.hid = e.hidden;
       if (px == null || hid || e.hidden) return;
       const own = e.s.moveSpeed * MOVE_SCALE * dt * 1.5 + 1e-3;
-      const extra = Math.hypot(e.x - px, e.y - py) - own;
+      const extra = hypot(e.x - px, e.y - py) - own;
       a.lx = px; a.ly = py;                                          // where the move started (direction for onMove)
       if (extra > 0.05) onMove(b, e, a, extra);
     },
@@ -605,7 +606,7 @@ export function blinkForward(b, e, dist) {
     let i = R.ptIdx;
     while (left > 1e-9 && i < R.pts.length) {
       const p = R.pts[i];
-      const d = Math.hypot(p.x - e.x, p.y - e.y);
+      const d = hypot(p.x - e.x, p.y - e.y);
       if (d <= left) { e.x = p.x; e.y = p.y; left -= d; i++; } else { stepToward(e, p.x, p.y, left); left = 0; }
     }
   } else if (R && R.legs && R.legs[R.legIdx] && R.legs[R.legIdx].r != null) {
diff --git a/server/sim/content/enemies/fly.js b/server/sim/content/enemies/fly.js
index f90e917c..03d9ab6b 100644
--- a/server/sim/content/enemies/fly.js
+++ b/server/sim/content/enemies/fly.js
@@ -7,6 +7,7 @@ import {
   stepToward, setForm, expose,
 } from './helpers.js';
 import { enemyAura, allyAura, selfFear, skill, kitSelfFear } from './archetypes.js';
+import { hypot } from '../../detmath.js';
 
 // ---------------------------------------------------------------------------------------------------------------
 // constants (numbers that exist nowhere in the data)
@@ -62,7 +63,7 @@ function kitBlackCloud(ab, e) {
   const prey = (b, e2) => b.enemiesInRadius(e2.x, e2.y, grabR).filter((o) => isPrey(b, e2, o));
   return [
     skill(k, (b, e2) => {
-      const list = prey(b, e2).sort((p, q) => Math.hypot(p.x - e2.x, p.y - e2.y) - Math.hypot(q.x - e2.x, q.y - e2.y)).slice(0, GRAB_MAX_PREY);
+      const list = prey(b, e2).sort((p, q) => hypot(p.x - e2.x, p.y - e2.y) - hypot(q.x - e2.x, q.y - e2.y)).slice(0, GRAB_MAX_PREY);
       if (!list.length) return;
       b.addBuff(e2, { key: 'ab:grabbing', duration: dur, flags: { bind: true, noMove: true } });   // 技能持续4秒，期间持有束缚
       b.fx('beam', { x: e2.x, y: e2.y, from: e2.id, to: list[0].id, kind: 'devour' });
@@ -250,12 +251,12 @@ export const FLY_KITS = Object.freeze({
     tick(b, e, a, dt) {
       if (a.t && !a.t.alive) { a.t = null; b.removeBuff(e, 'ab:dive'); }
       if (!a.t) {
-        const l = targetsNear(b, e, e.base.rangeRadius || 1).sort((p, q) => Math.hypot(p.x - e.x, p.y - e.y) - Math.hypot(q.x - e.x, q.y - e.y));
+        const l = targetsNear(b, e, e.base.rangeRadius || 1).sort((p, q) => hypot(p.x - e.x, p.y - e.y) - hypot(q.x - e.x, q.y - e.y));
         if (!l.length) return;
         a.t = l[0];
         b.addBuff(e, { key: 'ab:dive', flags: { noMove: true } });
       }
-      if (!stepToward(e, a.t.x, a.t.y, Math.max(e.s.moveSpeed, 0.5) * MOVE_SCALE * dt) && Math.hypot(a.t.x - e.x, a.t.y - e.y) > 0.3) return;
+      if (!stepToward(e, a.t.x, a.t.y, Math.max(e.s.moveSpeed, 0.5) * MOVE_SCALE * dt) && hypot(a.t.x - e.x, a.t.y - e.y) > 0.3) return;
       hurt(b, e, a.t, e.s.atk, 'phys');
       b.fx('explode', { x: e.x, y: e.y, r: 0.5, kind: 'seed' });
       b.kill(e, null);
diff --git a/server/sim/content/enemies/helpers.js b/server/sim/content/enemies/helpers.js
index e23f3ebd..700f0e6d 100644
--- a/server/sim/content/enemies/helpers.js
+++ b/server/sim/content/enemies/helpers.js
@@ -6,6 +6,7 @@
 import { ELEMENT } from '../../constants.js';
 import { canTargetAlly, sortAllyTargets, areaSelectable, auraSelectable } from '../../targeting.js';
 import { mitigate } from '../../damage.js';
+import { hypot } from '../../detmath.js';
 
 // ---------------------------------------------------------------------------------------------------------------
 // constants (numbers that exist nowhere in the data)
@@ -295,7 +296,7 @@ export function spawnChildren(b, parent, key, n, opts = {}) {
 /** Move a unit straight toward (tx, ty) by `dist` tiles. Returns true on arrival. */
 export function stepToward(u, tx, ty, dist) {
   const dx = tx - u.x, dy = ty - u.y;
-  const d = Math.hypot(dx, dy);
+  const d = hypot(dx, dy);
   if (d <= dist + 1e-9) { u.x = tx; u.y = ty; return true; }
   u.x += (dx / d) * dist;
   u.y += (dy / d) * dist;
diff --git a/server/sim/content/enemies/invisible.js b/server/sim/content/enemies/invisible.js
index 7a9f242e..9e5fc7b5 100644
--- a/server/sim/content/enemies/invisible.js
+++ b/server/sim/content/enemies/invisible.js
@@ -4,6 +4,7 @@
 import { canTargetAlly, enemyStealthed } from '../../targeting.js';
 import { T, elem, hurt, targetsNear, byPriority, auraBuff } from './helpers.js';
 import { stealth, onHitStatus, skill, kitStealth } from './archetypes.js';
+import { hypot } from '../../detmath.js';
 
 // ---------------------------------------------------------------------------------------------------------------
 // constants (numbers that exist nowhere in the data)
@@ -20,8 +21,8 @@ const CROSS_REACH = 6, CROSS_SKILL = 2.5, CROSS_CHARGE = 1.4;
 
 function kitCrossbow(ab) {
   const s = ab.sk.CrossAttack;
-  const aligned = (b, e) => b.allies().filter((u) => canTargetAlly(e, u, true) && (Math.abs(u.y - e.y) < 0.5 || Math.abs(u.x - e.x) < 0.5) && Math.hypot(u.x - e.x, u.y - e.y) <= CROSS_REACH);
-  const nearest = (b, e, l) => l.sort((p, q) => Math.hypot(p.x - e.x, p.y - e.y) - Math.hypot(q.x - e.x, q.y - e.y))[0];
+  const aligned = (b, e) => b.allies().filter((u) => canTargetAlly(e, u, true) && (Math.abs(u.y - e.y) < 0.5 || Math.abs(u.x - e.x) < 0.5) && hypot(u.x - e.x, u.y - e.y) <= CROSS_REACH);
+  const nearest = (b, e, l) => l.sort((p, q) => hypot(p.x - e.x, p.y - e.y) - hypot(q.x - e.x, q.y - e.y))[0];
   // PRTS 重弩突袭者: 天赋 "隐匿（被阻挡，主动攻击期间均可解除）"; 直击 "蓄力1.4s后向目标方向发射1支弩箭，对击中的首个目标造成攻击力100%
   // 的法术伤害与5s晕眩 ※技能持续2.5s": revealed for the whole skill, the bolt leaves CROSS_CHARGE s in — at the nearest unit still
   // in line in the aimed direction; it stands meanwhile [ASSUMED], and a stun / silence or its death before the release
@@ -108,7 +109,7 @@ function kitShadowBlade(ab) {
     tick(b, e) {
       const partner = b.enemiesInRadius(e.x, e.y, 3).find((o) => /enemy_1174_duholy/.test(o.defId));
       const r = partner && partner.mem.ab ? (T(partner.mem.ab, 'traitAbility.range_radius') ?? 1.1) : 1.1;
-      if (partner && Math.hypot(partner.x - e.x, partner.y - e.y) <= r + 1e-9 && e.base.bat > 0) auraBuff(b, e, 'ab:shadowSync', 0.5, { batPct: bat / e.base.bat });
+      if (partner && hypot(partner.x - e.x, partner.y - e.y) <= r + 1e-9 && e.base.bat > 0) auraBuff(b, e, 'ab:shadowSync', 0.5, { batPct: bat / e.base.bat });
     },
   }];
 }
diff --git a/server/sim/content/enemies/leaders.js b/server/sim/content/enemies/leaders.js
index 441794de..3a4fddb8 100644
--- a/server/sim/content/enemies/leaders.js
+++ b/server/sim/content/enemies/leaders.js
@@ -12,6 +12,7 @@ import {
   unblockable, maxTargets, onHitStatus, splashAttack, pathKeysAhead, noAirTargets, resist, nthAttackStatus, lowHpBuff,
   freeAllPrisoners, reborn, frontGuard, faceCrowd, unbalanced, artsBarrier, skill, blinkForward,
 } from './archetypes.js';
+import { hypot, powi } from '../../detmath.js';
 
 // ---------------------------------------------------------------------------------------------------------------
 // constants (numbers that exist nowhere in the data)
@@ -312,7 +313,7 @@ function kitLeaderMisc(key, ab, e) {
             const nx = areaAllies(b, e2, prev.x, prev.y, jr).find((u) => !hit.has(u));
             if (!nx) break;
             hit.add(nx);
-            hurt(b, e2, nx, e2.s.atk * Math.pow(fall, k), 'arts');
+            hurt(b, e2, nx, e2.s.atk * powi(fall, k), 'arts');
             if (nx.alive && cold > 0) b.applyStatus(nx, 'cold', { duration: cold, source: e2 });
             prev = nx;
           }
@@ -323,7 +324,7 @@ function kitLeaderMisc(key, ab, e) {
       skill(cb, (b, e2) => {
         const bb = cb.bb, jump = bb.projectile_range ?? jr, max = bb['chain.max_target'] ?? 3;
         const near = (x, y, r, seen) => b.enemiesInRadius(x, y, r).filter((o) => o !== e2 && !seen.has(o))
-          .sort((p, q) => Math.hypot(p.x - x, p.y - y) - Math.hypot(q.x - x, q.y - y) || p.spawnSeq - q.spawnSeq)[0];
+          .sort((p, q) => hypot(p.x - x, p.y - y) - hypot(q.x - x, q.y - y) || p.spawnSeq - q.spawnSeq)[0];
         const seen = new Set();
         let prev = e2, cur = near(e2.x, e2.y, e2.base.rangeRadius || 3.5, seen);
         while (cur && seen.size < max) {
@@ -357,7 +358,7 @@ function kitLeaderMisc(key, ab, e) {
 function segDist(px, py, ax, ay, bx, by) {
   const dx = bx - ax, dy = by - ay, L = dx * dx + dy * dy;
   const t = L > 0 ? Math.max(0, Math.min(1, ((px - ax) * dx + (py - ay) * dy) / L)) : 0;
-  return Math.hypot(px - (ax + t * dx), py - (ay + t * dy));
+  return hypot(px - (ax + t * dx), py - (ay + t * dy));
 }
 
 /** A second normal hit on the same target ("二连击"): same damage type, no further on-hit procs. */
@@ -454,7 +455,7 @@ function kitWarden(ab) {
 function kitXi(ab) {
   const cross = ab.sk.CrossAttack, sb = ab.sk.ShieldBurst, sb2 = ab.sk.ShieldBurstReborn;
   const P = { form2: false };
-  const byDist = (b, e) => allTargets(b, e).sort((p, q) => Math.hypot(p.x - e.x, p.y - e.y) - Math.hypot(q.x - e.x, q.y - e.y) || aggroCmp(p, q));
+  const byDist = (b, e) => allTargets(b, e).sort((p, q) => hypot(p.x - e.x, p.y - e.y) - hypot(q.x - e.x, q.y - e.y) || aggroCmp(p, q));
   const crossAt = (b, e, t) => {
     const r0 = t.tileR, c0 = t.tileC;
     b.fx('telegraph', { x: c0, y: r0, r: XI_CROSS_REACH, kind: 'xiCross', tiles: 'plus', id: e.id });
@@ -783,7 +784,7 @@ export const LEADER_KITS = Object.freeze({
     if (st > 0) b.applyStatus(e, 'stun', { duration: st, source: null });
   })],
   enemy_10138_xdsnow: (ab) => [unbalanced((b, e, a) => {             // 雪孩子 · pushed / pulled into high ground ⇒ hitWall.value damage
-    const dx = e.x - a.lx, dy = e.y - a.ly, d = Math.hypot(dx, dy);
+    const dx = e.x - a.lx, dy = e.y - a.ly, d = hypot(dx, dy);
     if (!(d > 0)) return;
     const r = Math.round(e.y + (dy / d) * 0.6), c = Math.round(e.x + (dx / d) * 0.6);
     if (b.grid.isLow(r, c) && b.grid.groundPassable(r, c)) return;   // stopped by nothing: no collision
diff --git a/server/sim/content/enemies/special.js b/server/sim/content/enemies/special.js
index e14efd6c..0a64f969 100644
--- a/server/sim/content/enemies/special.js
+++ b/server/sim/content/enemies/special.js
@@ -12,6 +12,7 @@ import {
   unblockable, runWhenHit, blockWeight, taunt, maxTargets, ep, nthAttackPower, lowHpBuff, deathBoom, setFloat, float,
   reborn, frontGuard, faceMove, unbalanced, skill, kitEp, kitPrisoner, kitStun3,
 } from './archetypes.js';
+import { hypot } from '../../detmath.js';
 
 // ---------------------------------------------------------------------------------------------------------------
 // constants (numbers that exist nowhere in the data)
@@ -245,7 +246,7 @@ export const SPECIAL_KITS = Object.freeze({
   enemy_1183_mlasrt: (ab) => [ep('erosion', T(ab, 'EpDamage.attack@ep_damage_ratio') ?? 0), nthAttackPower(nthOf(ab.sk.PowerAttack), (ab.sk.PowerAttack && ab.sk.PowerAttack.bb.atk_scale) || 1)], // 无胄盟清扫小队 · erosion; every 4th attack ×1.5
   enemy_1273_stmgun_2: (ab) => {                                     // 高准度伦蒂尼姆城防自行炮 · locks the highest max-HP unit in range, bombards the highest HP% on its 9 tiles
     // the allies in its range (the engine's ranged reach: radius + the ally collider)
-    const inRange = (b, e) => allTargets(b, e).filter((a) => Math.hypot(a.x - e.x, a.y - e.y) <= (e.base.rangeRadius || 0) + ALLY_COLLIDER_RADIUS + 1e-9);
+    const inRange = (b, e) => allTargets(b, e).filter((a) => hypot(a.x - e.x, a.y - e.y) <= (e.base.rangeRadius || 0) + ALLY_COLLIDER_RADIUS + 1e-9);
     return [skill(ab.sk.Cannon, (b, e) => {
       const lock = inRange(b, e).sort((p, q) => q.s.maxHp - p.s.maxHp)[0];
       if (!lock) return;
@@ -279,7 +280,7 @@ export const SPECIAL_KITS = Object.freeze({
     spawn(b, e) { watchDeaths(b, e); },
     otherDeath(c, b, e) {
       const u = c.unit;
-      if (!u || u === e || c.reason !== 'killed' || Math.hypot(u.x - e.x, u.y - e.y) > (T(ab, 'Attack.range_radius') ?? TORTURER_RADIUS) + 1e-9) return;
+      if (!u || u === e || c.reason !== 'killed' || hypot(u.x - e.x, u.y - e.y) > (T(ab, 'Attack.range_radius') ?? TORTURER_RADIUS) + 1e-9) return;
       b.heal(e, e, e.s.maxHp * (T(ab, 'Attack.hp_ratio') ?? 0), { self: true });
       b.addBuff(e, { key: 'ab:torture', refresh: 'stack', stacks: 1, maxStacks: T(ab, 'Attack.max_stack_cnt') ?? 1, persist: true, mods: { atkPct: T(ab, 'Attack.atk') ?? 0 } });
     },
diff --git a/server/sim/content/items/battle.js b/server/sim/content/items/battle.js
index e88da4e8..9cda94b7 100644
--- a/server/sim/content/items/battle.js
+++ b/server/sim/content/items/battle.js
@@ -47,6 +47,7 @@ import {
   alliesAround, passiveBuff, fxOn, battleStore, contentInfo, itemsOf, directMods,
 } from '../support/index.js';
 import { mitigate, hasHp, periodicDamage, isHpLoss } from '../../damage.js';
+import { hypot } from '../../detmath.js';
 
 // =====================================================================================================================
 // data helpers
@@ -465,7 +466,7 @@ const BY_ITEM = {
     const r = num(p.radius, 3), sc = num(p.damage_scale, 1);
     S.on('hit', (c) => {
       if (c.source !== u || !c.target || c.target.side !== 'enemy' || c.dmg.type === 'element') return;
-      if (Math.hypot(c.target.x - u.x, c.target.y - u.y) >= r - 1e-6) c.dmg.mul *= sc;
+      if (hypot(c.target.x - u.x, c.target.y - u.y) >= r - 1e-6) c.dmg.mul *= sc;
     });
   },
   // 炎国短刀: each skill activation +atk (≤ atk_buff_cnt stacks) — one 直接乘算 bonus of atk × stacks
diff --git a/server/sim/content/kits/ops/chess_char_2_15-akkord.js b/server/sim/content/kits/ops/chess_char_2_15-akkord.js
index c78f3997..174c143b 100644
--- a/server/sim/content/kits/ops/chess_char_2_15-akkord.js
+++ b/server/sim/content/kits/ops/chess_char_2_15-akkord.js
@@ -2,6 +2,7 @@
 // Conventions of the tier-2 kits: ../shared/tier2.js; kit contract and rules: ../README.md.
 
 import { num, talentBb, traitBb, onHitBy, alliesInGridOf, toggleBuff } from '../shared/tier1.js';
+import { hypot } from '../../../detmath.js';
 
 export default {
   // ---------------------------------------------------------------------------------------------------------------
@@ -33,7 +34,7 @@ export default {
           const mn = num(tb.min_dist), mx = num(tb.max_dist, 4), ds = num(tb.damage_scale);
           onHitBy(battle, unit, ({ target, dmg }) => {
             if (!dmg.isAttack) return;
-            const d = Math.hypot(target.x - unit.x, target.y - unit.y);
+            const d = hypot(target.x - unit.x, target.y - unit.y);
             dmg.amount *= 1 + ds * (mx > mn ? Math.max(0, Math.min(1, (d - mn) / (mx - mn))) : 1);
           });
         }
diff --git a/server/sim/content/kits/ops/chess_char_3_04-swire2.js b/server/sim/content/kits/ops/chess_char_3_04-swire2.js
index 5004c3b5..0b137b44 100644
--- a/server/sim/content/kits/ops/chess_char_3_04-swire2.js
+++ b/server/sim/content/kits/ops/chess_char_3_04-swire2.js
@@ -7,6 +7,7 @@ import {
   num, defOf, talentBb, moduleTalentBb, selectedId, altSkills, alive, gridKeys, fx, copyGrid, textNum, freeTile,
   groundTile, enemiesOn,
 } from '../shared/tier3.js';
+import { powi } from '../../../detmath.js';
 
 /** 琳琅诗怀雅 S3's coin range (PRTS 备注 "前方范围2-4"; range_table "2-4", facing right). */
 const SWIRE2_COIN_GRID = Object.freeze([[1, 1], [0, 0], [0, 1], [0, 2], [-1, 1]]);
@@ -267,7 +268,7 @@ export default {
           battle.on('fatal', (ctx) => {
             if (ctx.unit !== unit || ctx.prevented) return;
             const n = unit.mem.saveCount ?? 0;
-            const cost = Math.abs(num(t1.cost, -5)) * Math.pow(num(t1.cost_multi, 2), n);
+            const cost = Math.abs(num(t1.cost, -5)) * powi(num(t1.cost_multi, 2), n);
             const pl = battle.getPlayer(unit.ownerId);
             if (!pl || pl.dp + 1e-9 < cost) return;
             battle.addDp(unit.ownerId, -cost);
diff --git a/server/sim/content/kits/ops/chess_char_3_07-forcer.js b/server/sim/content/kits/ops/chess_char_3_07-forcer.js
index a3183016..4b84b175 100644
--- a/server/sim/content/kits/ops/chess_char_3_07-forcer.js
+++ b/server/sim/content/kits/ops/chess_char_3_07-forcer.js
@@ -3,6 +3,7 @@
 
 import { PUSH_DIRECTIONAL_MIN_DIST } from '../../../constants.js';
 import { num, defOf, talentBb, traitBb, onTiles, gridKeys, fx, copyGrid } from '../shared/tier3.js';
+import { hypot } from '../../../detmath.js';
 
 /** Two enemy bodies touch within this distance (tiles) — 见行者 collision stun. */
 const COLLIDE = 0.6;
@@ -33,7 +34,7 @@ export default {
             // (only the angle: a target nearer than 0.25 tile still turns radial at 受力等级 −2) — by the official
             // 力度 − 重量 distance of a 特效 push (PRTS 推与拉 names 见行者's skills 特效类; Battle.push / pushDistance);
             // stopped short of it ⇒ it hit a wall
-            const near = Math.hypot(e.x - unit.x, e.y - unit.y) < PUSH_DIRECTIONAL_MIN_DIST;
+            const near = hypot(e.x - unit.x, e.y - unit.y) < PUSH_DIRECTIONAL_MIN_DIST;
             const expect = battle.pushDistance(e, near ? force - 2 : force, { effect: true });
             const moved = battle.push(e, force, { from: unit, dir: { x: unit.fwd[1], y: unit.fwd[0] }, fixedAngle: true, effect: true });
             const wall = expect > 0 && moved + 0.05 < expect;
diff --git a/server/sim/content/kits/ops/chess_char_4_01-rmixer.js b/server/sim/content/kits/ops/chess_char_4_01-rmixer.js
index 21564fc2..03e2d796 100644
--- a/server/sim/content/kits/ops/chess_char_4_01-rmixer.js
+++ b/server/sim/content/kits/ops/chess_char_4_01-rmixer.js
@@ -7,6 +7,7 @@ import {
   AURA, num, tbb, grid, nationOf, enemiesOnRange, targetsInRange, whileDeployed, reveal, skillActive, isSel, alt,
   instantKind, withDefaults,
 } from '../shared/tier4.js';
+import { hypot } from '../../../detmath.js';
 
 /** Is module `id` the active one? (elites; `'none'` / normal chess ⇒ false) */
 const moduleIs = (def, id) => !!(def?.raw?.module?.active && def.raw.module.id === id);
@@ -39,7 +40,7 @@ export default withDefaults({
               const n = num(bb.charge, 1);
               const cand = battle.alliesInRadius(unit.x, unit.y, 1.5, unit.ownerId)
                 .filter((a) => a !== unit && a.kind === 'op' && nationOf(a) === 'laterano' && a.skill && a.skill.active && a.skill.kind === 'ammo')
-                .sort((a, b) => Math.hypot(a.x - unit.x, a.y - unit.y) - Math.hypot(b.x - unit.x, b.y - unit.y) || a.id - b.id);
+                .sort((a, b) => hypot(a.x - unit.x, a.y - unit.y) - hypot(b.x - unit.x, b.y - unit.y) || a.id - b.id);
               if (!cand[0] || !(n > 0)) return;
               cand[0].skill.addAmmo(n);
               battle.fx('reload', { x: cand[0].x, y: cand[0].y, id: cand[0].id, n });
diff --git a/server/sim/content/kits/ops/chess_char_4_04-ines.js b/server/sim/content/kits/ops/chess_char_4_04-ines.js
index 1570baa0..f2055310 100644
--- a/server/sim/content/kits/ops/chess_char_4_04-ines.js
+++ b/server/sim/content/kits/ops/chess_char_4_04-ines.js
@@ -49,7 +49,7 @@ export default withDefaults({
             unit.mem.sentry = null;
             unit.mem.inesSentryAt = null;
             const ax = from.x, ay = from.y, bx = unit.x, by = unit.y;
-            const L2 = (bx - ax) ** 2 + (by - ay) ** 2;
+            const L2 = (bx - ax) * (bx - ax) + (by - ay) * (by - ay);
             const w = num(bb.projectile_range, 1.4);
             const hits = [];
             for (const e of battle.aliveEnemies()) {
diff --git a/server/sim/content/kits/ops/chess_char_4_05-beewax.js b/server/sim/content/kits/ops/chess_char_4_05-beewax.js
index f5552993..07d63a93 100644
--- a/server/sim/content/kits/ops/chess_char_4_05-beewax.js
+++ b/server/sim/content/kits/ops/chess_char_4_05-beewax.js
@@ -5,6 +5,7 @@ import { COLS } from '../../../constants.js';
 import {
   num, tbb, grid, targetsInRange, targetsInGrid, toggleBuff, skillActive, withDefaults,
 } from '../shared/tier4.js';
+import { hypot } from '../../../detmath.js';
 
 /**
  * Free tile for a summon/device: in the rect, standable, not reserved (Battle.isReservedTile: empty, no knocked-out
@@ -39,7 +40,7 @@ export default withDefaults({
           for (const k of unit.rangeKeys || []) {
             const r = (k / COLS) | 0, c = k % COLS;
             if (!freeTile(battle, r, c)) continue;
-            const d = Math.hypot(c - ref.x, r - ref.y) + (battle.grid.groundPassable(r, c) ? 0 : 5);
+            const d = hypot(c - ref.x, r - ref.y) + (battle.grid.groundPassable(r, c) ? 0 : 5);
             if (d < bd - 1e-9) { bd = d; best = [r, c]; }
           }
           if (!best) return;
diff --git a/server/sim/content/kits/ops/chess_char_4_10-aroma.js b/server/sim/content/kits/ops/chess_char_4_10-aroma.js
index 54006970..30eb73fa 100644
--- a/server/sim/content/kits/ops/chess_char_4_10-aroma.js
+++ b/server/sim/content/kits/ops/chess_char_4_10-aroma.js
@@ -3,6 +3,7 @@
 
 import { bodyInKeys } from '../../../body.js';
 import { num, tbb, keySet, skillActive, isSel, alt, instantKind, withDefaults } from '../shared/tier4.js';
+import { hypot } from '../../../detmath.js';
 
 /** 阿罗玛's 非首次标记 (talent 起泡性能测试): one per enemy, shared by every 阿罗玛 (PRTS 备注). */
 const AROMA_MARK = 'aroma:bubbled';
@@ -71,7 +72,7 @@ export default withDefaults({
           const lo = num(tb.min_dist, 0), hi = num(tb.max_dist, 4);
           battle.on('hit', (c) => {
             if (c.source !== unit || !c.dmg.isAttack) return;
-            const d = Math.hypot(c.target.x - unit.x, c.target.y - unit.y);
+            const d = hypot(c.target.x - unit.x, c.target.y - unit.y);
             c.dmg.mul *= 1 + ds * Math.max(0, Math.min(1, (d - lo) / Math.max(1e-6, hi - lo)));
           }, { owner: unit });
         }
diff --git a/server/sim/content/kits/ops/chess_char_4_12-glady.js b/server/sim/content/kits/ops/chess_char_4_12-glady.js
index 53b58e7e..98e7700c 100644
--- a/server/sim/content/kits/ops/chess_char_4_12-glady.js
+++ b/server/sim/content/kits/ops/chess_char_4_12-glady.js
@@ -6,6 +6,7 @@ import {
   AURA, num, tbb, moduleBb, grid, batFlat, enemyHasTag, targetsInRange, whileDeployed, pulse, alt, instantKind,
   pullToFront, withDefaults,
 } from '../shared/tier4.js';
+import { hypot } from '../../../detmath.js';
 
 const ABYSSAL = new Set(['char_143_ghost', 'char_263_skadi', 'char_474_glady', 'char_4145_ulpia', 'char_1023_ghost2']);
 const isAbyssal = (u) => u?.def?.raw?.groupId === 'abyssal' || ABYSSAL.has(u?.def?.charId ?? u?.def?.raw?.charId);
@@ -28,7 +29,7 @@ export default withDefaults({
     };
     // module HOK-Y: "向自身拖拽较远的敌人时力度提升一个等级"
     const farR = num(mb['skill@range_radius'], num(mb['attack@range_radius'], 0)), farF = num(mb['skill@delta_force'], num(mb['attack@delta_force'], 0));
-    const selfForce = (unit, e, f) => (farR > 0 && farF > 0 && Math.hypot(e.x - unit.x, e.y - unit.y) > farR + 1e-9 ? f + farF : f);
+    const selfForce = (unit, e, f) => (farR > 0 && farF > 0 && hypot(e.x - unit.x, e.y - unit.y) > farR + 1e-9 ? f + farF : f);
     const pullSelf = (battle, unit, e, f) => { if (e && e.alive) dragDmg(battle, unit, e, pullToFront(battle, unit, e, selfForce(unit, e, f))); };
     return {
       skills: alt(def, {
diff --git a/server/sim/content/kits/ops/chess_char_4_20-fartth.js b/server/sim/content/kits/ops/chess_char_4_20-fartth.js
index bdccdae5..83691ad5 100644
--- a/server/sim/content/kits/ops/chess_char_4_20-fartth.js
+++ b/server/sim/content/kits/ops/chess_char_4_20-fartth.js
@@ -6,6 +6,7 @@ import { bodyInKeys } from '../../../body.js';
 import { absoluteRangeKeys, sortEnemyTargets } from '../../../targeting.js';
 import { isHpLoss } from '../../../damage.js';
 import { num, tbb, whileDeployed, toggleBuff, skillActive, isSel, alt, withDefaults } from '../shared/tier4.js';
+import { hypot } from '../../../detmath.js';
 
 const LINE = Object.freeze(Array.from({ length: COLS }, (_, i) => Object.freeze([0, i])));
 const tileKey = (u) => Math.round(u.y) * COLS + Math.round(u.x);
@@ -83,7 +84,7 @@ export default withDefaults({
         battle.on('hit', (c) => {
           if (c.source !== unit || !c.dmg.isAttack) return;
           if (S3 && skillActive(unit) && unit.baseRangeKeys && !bodyInKeys(c.target, unit.baseRangeKeys)) c.dmg.mul *= num(bb.damage_scale, 1.25);
-          if (ds > 0) { const d = Math.hypot(c.target.x - unit.x, c.target.y - unit.y); c.dmg.mul *= 1 + ds * Math.max(0, Math.min(1, (d - lo) / Math.max(1e-6, hi - lo))); }
+          if (ds > 0) { const d = hypot(c.target.x - unit.x, c.target.y - unit.y); c.dmg.mul *= 1 + ds * Math.max(0, Math.min(1, (d - lo) / Math.max(1e-6, hi - lo))); }
         }, { owner: unit });
       },
     };
diff --git a/server/sim/content/kits/ops/chess_char_5_09-cetsyr.js b/server/sim/content/kits/ops/chess_char_5_09-cetsyr.js
index a165bc29..09052789 100644
--- a/server/sim/content/kits/ops/chess_char_5_09-cetsyr.js
+++ b/server/sim/content/kits/ops/chess_char_5_09-cetsyr.js
@@ -4,6 +4,7 @@
 import { COLS } from '../../../constants.js';
 import { rotateOffset } from '../../../dir.js';
 import { AURA_IV, AURA_DUR, num, talent, traitBb, skillGrid, isOp, leaderOf, whileOn } from '../shared/tier5.js';
+import { hypot, sin, cos } from '../../../detmath.js';
 
 /** 魔王 T1 微尘: collision radius of one mote (PRTS 备注 0.4); orbit radius / speed come from the talent bb. */
 const MOTE_HIT_RADIUS = 0.4;
@@ -82,11 +83,11 @@ export default {
               const ang = spd * t + (2 * Math.PI * k) / cnt;
               // the orbit turns with his direction: angle 0 = straight ahead, π/2 = his left hand
               const [fr, fc] = unit.fwd, [lr, lc] = rotateOffset(1, 0, unit.dir);
-              const mx = unit.x + orbit * (fc * Math.cos(ang) + lc * Math.sin(ang)), my = unit.y + orbit * (fr * Math.cos(ang) + lr * Math.sin(ang));
+              const mx = unit.x + orbit * (fc * cos(ang) + lc * sin(ang)), my = unit.y + orbit * (fr * cos(ang) + lr * sin(ang));
               let hit = null, hd = Infinity;
               for (const a of cands) {
                 if (a.findBuff('cetsyr:mote')) continue;
-                const d = Math.hypot(a.x - mx, a.y - my);
+                const d = hypot(a.x - mx, a.y - my);
                 if (d <= MOTE_HIT_RADIUS + 1e-9 && d < hd) { hd = d; hit = a; }
               }
               if (!hit) continue;
diff --git a/server/sim/content/kits/ops/chess_char_5_15-thorn2.js b/server/sim/content/kits/ops/chess_char_5_15-thorn2.js
index b0312bcd..5de800cd 100644
--- a/server/sim/content/kits/ops/chess_char_5_15-thorn2.js
+++ b/server/sim/content/kits/ops/chess_char_5_15-thorn2.js
@@ -8,6 +8,7 @@ import {
   AURA_IV, AURA_DUR, RING1, num, on, talent, skillGrid, maxCharges, mods, dist, isOp, selectedId, lazySkills,
   instantKind, whileOn, permBuff,
 } from '../shared/tier5.js';
+import { hypot } from '../../../detmath.js';
 
 /** Straight-road length through each tile (max of the horizontal and vertical runs of ground-passable tiles). */
 const ROAD_CACHE = new WeakMap();
@@ -134,7 +135,7 @@ export default {
           }
           const extra = battle.alliesInGrid(unit).some((a) => a !== unit && isOp(a)) ? Math.min(num(t0.projectile_extend), num(t0.projectile_extend_max, Infinity)) : 0;
           // it drifts away from her deployment tile ("移动方向始终为远离棘刺部署位置中心的方向")
-          const dx = t.x - unit.x, dy = t.y - unit.y, len = Math.hypot(dx, dy) || 1;
+          const dx = t.x - unit.x, dy = t.y - unit.y, len = hypot(dx, dy) || 1;
           // `key`: the fx re-sent every second as the unit drifts and grows is this one zone — the client updates it in
           // place instead of stacking a new layer each second (render/fx/zones.js; community report of 2026-10-06)
           const key = `thorn2:${unit.id}:${unit.mem.zoneSeq = (unit.mem.zoneSeq ?? 0) + 1}`;
diff --git a/server/sim/content/kits/ops/chess_char_6_02-sbell2.js b/server/sim/content/kits/ops/chess_char_6_02-sbell2.js
index 2bc0d456..b5c3f9a4 100644
--- a/server/sim/content/kits/ops/chess_char_6_02-sbell2.js
+++ b/server/sim/content/kits/ops/chess_char_6_02-sbell2.js
@@ -4,6 +4,7 @@
 import { COLS } from '../../../constants.js';
 import { bodyInKeys } from '../../../body.js';
 import { num, tbb, moduleBb, live, keyOf, enemiesIn, isTok, instantKind, AROUND8, aura } from '../shared/tier6.js';
+import { hypot } from '../../../detmath.js';
 
 // ------------------------------------------------------------------------------------------------------------------
 // 圣聆初雪 chess_char_6_02 (阵法术师) — S3 群山俯首; 无垠的雪景; 圣山的祝福; module 千分之一的心
@@ -217,7 +218,7 @@ function lureGoal(battle, unit, e) {
   for (const [dr, dc] of AROUND8) {
     const r = unit.tileR + dr, c = unit.tileC + dc;
     if (!battle.grid.groundPassable(r, c)) continue;
-    const d = Math.hypot(e.x - c, e.y - r);
+    const d = hypot(e.x - c, e.y - r);
     if (d < bd - 1e-9) { bd = d; goal = [r, c]; }
   }
   return goal;
diff --git a/server/sim/content/kits/ops/chess_char_6_05-pasngr.js b/server/sim/content/kits/ops/chess_char_6_05-pasngr.js
index 5b3aa393..e0733833 100644
--- a/server/sim/content/kits/ops/chess_char_6_05-pasngr.js
+++ b/server/sim/content/kits/ops/chess_char_6_05-pasngr.js
@@ -5,6 +5,7 @@ import { absoluteRangeKeys, canTargetEnemy } from '../../../targeting.js';
 import { CHAIN_RADIUS } from '../../../constants.js';
 import { bodyDist, bodyInKeys } from '../../../body.js';
 import { num, bv, tbb, live, ANY, enemiesIn, batOf, N4, aura } from '../shared/tier6.js';
+import { powi } from '../../../detmath.js';
 
 // ------------------------------------------------------------------------------------------------------------------
 // 异客 chess_char_6_05 (链术师) — S3 辉煌裂片; 机理分析; 孤卒
@@ -25,7 +26,7 @@ function pasngr(bb, chess, def) {
     for (let i = 0; i < count && prev; i++) {
       hit.add(prev.id);
       battle.fx('lightning', { x: prev.x, y: prev.y, id: prev.id, src: unit.id });
-      battle.dealDamage(unit, prev, { amount: unit.s.atk * scale * Math.pow(1 - num(ch.falloff, 0.15), i), type: 'arts', isSkill: true, isAttack: true, tags: ['skill', 'storm'] });
+      battle.dealDamage(unit, prev, { amount: unit.s.atk * scale * powi(1 - num(ch.falloff, 0.15), i), type: 'arts', isSkill: true, isAttack: true, tags: ['skill', 'storm'] });
       if (slug > 0 && prev.alive) battle.applyStatus(prev, 'sluggish', { duration: slug, source: unit });
       let best = null, bd = Infinity;
       for (const x of battle.foesInRadius(prev.x, prev.y, ch.radius || CHAIN_RADIUS)) {
diff --git a/server/sim/content/kits/ops/chess_char_6_18-whitw2.js b/server/sim/content/kits/ops/chess_char_6_18-whitw2.js
index 057f6309..08efbb74 100644
--- a/server/sim/content/kits/ops/chess_char_6_18-whitw2.js
+++ b/server/sim/content/kits/ops/chess_char_6_18-whitw2.js
@@ -5,6 +5,7 @@ import { sortEnemyTargets, canTargetEnemy } from '../../../targeting.js';
 import { rotateOffset } from '../../../dir.js';
 import { bodyDist } from '../../../body.js';
 import { num, tbb, hasBond, ANY, selectedSkill, skillGridOf, WHOLE_FIELD, bstate, aura } from '../shared/tier6.js';
+import { hypot, sin, cos, atan2 } from '../../../detmath.js';
 
 // ------------------------------------------------------------------------------------------------------------------
 // 荒芜拉普兰德 chess_char_6_18 (驭械术师) — S3 终幕·浩劫; 头狼; 叙拉古的荣幸
@@ -39,10 +40,10 @@ function whitw2(bb, chess, def) {
   // ① `k` drones leave her evenly spread, the first along her facing (row 0 is the bottom row: angles in the (col, row)
   // plane, counter-clockwise)
   const releaseDrones = (unit, k) => {
-    const a0 = Math.atan2(unit.fwd[0], unit.fwd[1]);
+    const a0 = atan2(unit.fwd[0], unit.fwd[1]);
     for (let i = 0; i < k; i++) {
       const a = a0 + (2 * Math.PI * i) / k;
-      unit.mem.drones.push({ x: unit.x, y: unit.y, hx: Math.cos(a), hy: Math.sin(a), v: WHITW2_SPREAD.v0, age: 0, phase: 'spread', t: null, cd: 0, rampId: null, ramp: 0, orbit: null });
+      unit.mem.drones.push({ x: unit.x, y: unit.y, hx: cos(a), hy: sin(a), v: WHITW2_SPREAD.v0, age: 0, phase: 'spread', t: null, cd: 0, rampId: null, ramp: 0, orbit: null });
     }
   };
   // speed v → v + acc·dt (capped); the distance covered at the mean of the two (exact under constant acceleration)
@@ -56,7 +57,7 @@ function whitw2(bb, chess, def) {
     let best = null, bd = Infinity, bh = Infinity;
     for (const e of battle.enemies) {
       if (!ok(e)) continue;
-      const de = Math.hypot(e.x - d.x, e.y - d.y), dh = Math.hypot(e.x - unit.x, e.y - unit.y);
+      const de = hypot(e.x - d.x, e.y - d.y), dh = hypot(e.x - unit.x, e.y - unit.y);
       if (de < bd - 1e-9 || (de <= bd + 1e-9 && dh < bh - 1e-9)) { best = e; bd = de; bh = dh; }
     }
     return best;
@@ -65,9 +66,9 @@ function whitw2(bb, chess, def) {
   const circle = (d, dt) => {
     const { r, v } = WHITW2_ORBIT;
     if (!d.orbit) d.orbit = { cx: d.x - d.hy * r, cy: d.y + d.hx * r };
-    const a = Math.atan2(d.y - d.orbit.cy, d.x - d.orbit.cx) + (v / r) * dt;
-    d.x = d.orbit.cx + r * Math.cos(a); d.y = d.orbit.cy + r * Math.sin(a);
-    d.hx = -Math.sin(a); d.hy = Math.cos(a);
+    const a = atan2(d.y - d.orbit.cy, d.x - d.orbit.cx) + (v / r) * dt;
+    d.x = d.orbit.cx + r * cos(a); d.y = d.orbit.cy + r * sin(a);
+    d.hx = -sin(a); d.hy = cos(a);
   };
   // one tick of one drone: spread → (pick) → chase → on the target, attacking like a normal drone. [ASSUMED] its attack
   // clock runs all the time (one attack per interval of hers at most, whatever it chased in between) and its first hit
@@ -103,7 +104,7 @@ function whitw2(bb, chess, def) {
     if (d.phase === 'chase') {
       const s = accelerate(d, WHITW2_CHASE, dt);
       if (bodyDist(t, d.x, d.y) > s + 1e-9) {
-        const dx = t.x - d.x, dy = t.y - d.y, L = Math.hypot(dx, dy);
+        const dx = t.x - d.x, dy = t.y - d.y, L = hypot(dx, dy);
         if (L > 1e-9) { d.hx = dx / L; d.hy = dy / L; }
         d.x += d.hx * s; d.y += d.hy * s;
         return;
diff --git a/server/sim/content/kits/ops/op-aglna2.js b/server/sim/content/kits/ops/op-aglna2.js
index 1ec707e7..d1534370 100644
--- a/server/sim/content/kits/ops/op-aglna2.js
+++ b/server/sim/content/kits/ops/op-aglna2.js
@@ -63,6 +63,7 @@ import { bodyInKeys } from '../../../body.js';
 import { hasHp } from '../../../damage.js';
 import { RESIST_STATUSES } from '../../../buffs.js';
 import { TICK } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_aglna2_1';
 const S2 = 'skchr_aglna2_2';
@@ -342,7 +343,7 @@ export default {
           if (!st || !skillOn(unit, S3) || !up(unit) || st.flight || !unit.canAct || unit.blocking.length) return;
           const tgt = moveTarget(battle, unit);
           if (!tgt) return;
-          const dur = Math.hypot(tgt.r - unit.tileR, tgt.c - unit.tileC) / MOVE_SPEED;
+          const dur = hypot(tgt.r - unit.tileR, tgt.c - unit.tileC) / MOVE_SPEED;
           const act = unit.skill.activations;
           st.flight = tgt;
           battle.releaseBlocked(unit);
diff --git a/server/sim/content/kits/ops/op-cqbw.js b/server/sim/content/kits/ops/op-cqbw.js
index c3f73a88..1753b9a9 100644
--- a/server/sim/content/kits/ops/op-cqbw.js
+++ b/server/sim/content/kits/ops/op-cqbw.js
@@ -62,6 +62,7 @@ import { sortEnemyTargets, canTargetEnemy } from '../../../targeting.js';
 import { bodyInKeys } from '../../../body.js';
 import { hasHp, isHpLoss } from '../../../damage.js';
 import { COLS, PROJECTILE_SPEEDS } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_cqbw_1';
 const S2 = 'skchr_cqbw_2';
@@ -133,7 +134,7 @@ function mineTile(battle, unit) {
   let best = [], bd = Infinity;
   for (const t of tiles) {
     let d = Infinity;
-    for (const e of foes) d = Math.min(d, Math.hypot(e.x - t[1], e.y - t[0]));
+    for (const e of foes) d = Math.min(d, hypot(e.x - t[1], e.y - t[0]));
     if (d < bd - 1e-9) { bd = d; best = [t]; } else if (Math.abs(d - bd) <= 1e-9) best.push(t);
   }
   return best.length === 1 ? best[0] : battle.rng.pick(best);
diff --git a/server/sim/content/kits/ops/op-ebnhlz.js b/server/sim/content/kits/ops/op-ebnhlz.js
index 6bbe699a..6ff9a79f 100644
--- a/server/sim/content/kits/ops/op-ebnhlz.js
+++ b/server/sim/content/kits/ops/op-ebnhlz.js
@@ -67,6 +67,7 @@ import { num, talentBb, moduleBb, traitBb, skillRec, batMod, up, toggleBuff } fr
 import { canTargetEnemy, sortEnemyTargets } from '../../../targeting.js';
 import { hasHp } from '../../../damage.js';
 import { COLS } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_ebnhlz_1';
 const S2 = 'skchr_ebnhlz_2';
@@ -116,7 +117,7 @@ function remnantTiles(battle, unit) {
   const rank = new Map();
   foes.forEach((e, i) => { const k = Math.round(e.y) * COLS + Math.round(e.x); if (!rank.has(k)) rank.set(k, i); });
   battle.rng.shuffle(tiles);   // the random order of the last group, and of equal distances [ASSUMED]
-  const dist = (r, c) => { let d = Infinity; for (const e of foes) d = Math.min(d, Math.hypot(e.x - c, e.y - r)); return d; };
+  const dist = (r, c) => { let d = Infinity; for (const e of foes) d = Math.min(d, hypot(e.x - c, e.y - r)); return d; };
   const keyed = tiles.map(([r, c], i) => ({ r, c, i, on: rank.get(r * COLS + c) ?? Infinity, d: dist(r, c) }));
   keyed.sort((a, b) => a.on - b.on || a.d - b.d || a.i - b.i);
   return keyed.map((x) => [x.r, x.c]);
diff --git a/server/sim/content/kits/ops/op-haak.js b/server/sim/content/kits/ops/op-haak.js
index b4901146..921d923f 100644
--- a/server/sim/content/kits/ops/op-haak.js
+++ b/server/sim/content/kits/ops/op-haak.js
@@ -50,6 +50,7 @@
 import { num, talentBb, moduleBb, moduleOn, skillRec, statBuff, toggleBuff, up } from '../shared/tier1.js';
 import { toLocal } from '../../../dir.js';
 import { COLS } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_haak_1';
 const S2 = 'skchr_haak_2';
@@ -80,7 +81,7 @@ export function frontAlly(battle, unit, heal = false) {
     if (heal && (a.s.flags.noHeal || a.s.flags.healFree || a.profile?.noHeal)) continue;
     const dr = a.tileR - unit.tileR, dc = a.tileC - unit.tileC;
     const [lr, lc] = toLocal(dr, dc, unit.dir);
-    const k = [lr === 0 && lc > 0 ? 0 : 1, Math.hypot(dr, dc), -(a.s.taunt || 0), -(a.aggroSeq || a.deploySeq || 0), a.id];
+    const k = [lr === 0 && lc > 0 ? 0 : 1, hypot(dr, dc), -(a.s.taunt || 0), -(a.aggroSeq || a.deploySeq || 0), a.id];
     let less = !bk;
     for (let i = 0; !less && i < k.length; i++) {
       if (k[i] === bk[i]) continue;
diff --git a/server/sim/content/kits/ops/op-haruka.js b/server/sim/content/kits/ops/op-haruka.js
index 1457251d..eec091ea 100644
--- a/server/sim/content/kits/ops/op-haruka.js
+++ b/server/sim/content/kits/ops/op-haruka.js
@@ -50,6 +50,7 @@
 import { num, up, traitBb, talentBb, skillRec, batMod, giveSp, protectMods } from '../shared/tier1.js';
 import { absoluteRangeKeys } from '../../../targeting.js';
 import { COLS } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_haruka_1';
 const S2 = 'skchr_haruka_2';
@@ -217,7 +218,7 @@ export default {
             if (c.source !== unit || c.opts?.regen || !skillOn(unit, S2) || !(c.amount > 0) || !(scale > 0) || !n) return;
             const t = c.target;
             const foes = battle.foesInRadius(t.x, t.y, r);
-            foes.sort((x, y) => Math.hypot(x.x - t.x, x.y - t.y) - Math.hypot(y.x - t.x, y.y - t.y) || x.spawnSeq - y.spawnSeq);
+            foes.sort((x, y) => hypot(x.x - t.x, x.y - t.y) - hypot(y.x - t.x, y.y - t.y) || x.spawnSeq - y.spawnSeq);
             const amount = c.amount * scale;
             for (const e of foes.slice(0, n)) battle.dealDamage(unit, e, { amount, type: 'arts', isSkill: true, tags: ['skill', 'haruka:s2'] });
             if (foes.length) battle.fx('aoe', { x: t.x, y: t.y, radius: r, id: unit.id, skill: 'haruka:s2' });
diff --git a/server/sim/content/kits/ops/op-hsgma2.js b/server/sim/content/kits/ops/op-hsgma2.js
index 0aaeb458..dcd1c29c 100644
--- a/server/sim/content/kits/ops/op-hsgma2.js
+++ b/server/sim/content/kits/ops/op-hsgma2.js
@@ -57,6 +57,7 @@ import { canTargetEnemy } from '../../../targeting.js';
 import { bodyInRadius } from '../../../body.js';
 import { hasHp } from '../../../damage.js';
 import { COLS } from '../../../constants.js';
+import { sin, cos, atan2 } from '../../../detmath.js';
 
 const S1 = 'skchr_hsgma2_1';
 const S2 = 'skchr_hsgma2_2';
@@ -98,7 +99,7 @@ export default {
     /** S2: the shield's centre after `turned` degrees (row 0 is the bottom row: counter-clockwise = a growing angle). */
     const shieldAt = (unit, sh) => {
       const a = sh.a0 + (sh.turned * Math.PI) / 180;
-      return { x: unit.x + SHIELD_START * Math.cos(a), y: unit.y + SHIELD_START * Math.sin(a) };
+      return { x: unit.x + SHIELD_START * cos(a), y: unit.y + SHIELD_START * sin(a) };
     };
     const touching = (battle, unit, p) => battle.enemies.filter((e) => canTargetEnemy(unit, e, GROUND) && bodyInRadius(e, p.x, p.y, SHIELD_RADIUS));
     /** S2: one shield pulse — 停顿, the arts hit, and her heal of heal_ratio × the HP it removed (ignoring her 禁疗). */
@@ -128,7 +129,7 @@ export default {
             if (!('hits' in s2Attack)) return;
             plainS2(); // the throw was this attack: the shield flies from now on
             const [fr, fc] = unit.fwd;
-            unit.mem.hsShield = { a0: Math.atan2(fr, fc), turned: 0, acc: 0 };
+            unit.mem.hsShield = { a0: atan2(fr, fc), turned: 0, acc: 0 };
             battle.fx('zone', { x: unit.x, y: unit.y, radius: SHIELD_START + SHIELD_RADIUS, dur: SHIELD_TURN / SHIELD_SPEED, id: unit.id, skill: 'hsgma2:shield' });
             shieldPulse(battle, unit, shieldAt(unit, unit.mem.hsShield));
           },
diff --git a/server/sim/content/kits/ops/op-ifrit.js b/server/sim/content/kits/ops/op-ifrit.js
index 2b6b92df..65457df2 100644
--- a/server/sim/content/kits/ops/op-ifrit.js
+++ b/server/sim/content/kits/ops/op-ifrit.js
@@ -52,6 +52,7 @@
 import { num, talentBb, moduleBb, traitBb, skillRec, up, giveSp } from '../shared/tier1.js';
 import { bodyInKeys } from '../../../body.js';
 import { hasHp } from '../../../damage.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_ifrit_1';
 const S2 = 'skchr_ifrit_2';
@@ -197,7 +198,7 @@ export default {
           battle.on('hit', (c) => {
             const e = c.target;
             if (c.source !== unit || !e || e.side !== 'enemy' || c.dmg.type === 'element' || !up(unit)) return;
-            const d = Math.hypot(e.x - unit.x, e.y - unit.y);
+            const d = hypot(e.x - unit.x, e.y - unit.y);
             c.dmg.mul *= 1 + ds * (hi > lo ? Math.max(0, Math.min(1, (d - lo) / (hi - lo))) : 1);
           }, { owner: unit });
         }
diff --git a/server/sim/content/kits/ops/op-jesca2.js b/server/sim/content/kits/ops/op-jesca2.js
index 8fe618bb..41f552b9 100644
--- a/server/sim/content/kits/ops/op-jesca2.js
+++ b/server/sim/content/kits/ops/op-jesca2.js
@@ -58,6 +58,7 @@
 import { num, talentBb, traitBb, moduleOn, skillRec, batMod, up, giveSp, installReveal } from '../shared/tier1.js';
 import { offsetTile, dirFromDelta, dirVec } from '../../../dir.js';
 import { isHpLoss } from '../../../damage.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_jesca2_1';
 const S2 = 'skchr_jesca2_2';
@@ -182,7 +183,7 @@ function fireBomb(battle, unit, b3) {
   };
   (unit.mem.jescaShells ??= []).push(bomb);
   unit.atkCd = unit.s.interval;   // the shell is her attack: the timer restarts [ASSUMED]
-  battle.fx('bombardShell', { x: bomb.tx, y: bomb.ty, id: unit.id, r: bomb.radius, t: Math.hypot(bomb.tx - bomb.x, bomb.ty - bomb.y) / BOMB_SPEED });
+  battle.fx('bombardShell', { x: bomb.tx, y: bomb.ty, id: unit.id, r: bomb.radius, t: hypot(bomb.tx - bomb.x, bomb.ty - bomb.y) / BOMB_SPEED });
 }
 
 /** Move the shells one tick; a shell touching an enemy or reaching its end point explodes. */
@@ -192,7 +193,7 @@ function flyShells(battle, unit, dt) {
   const keep = [];
   for (const s of list) {
     const dx = s.tx - s.x, dy = s.ty - s.y;
-    const d = Math.hypot(dx, dy);
+    const d = hypot(dx, dy);
     const step = BOMB_SPEED * dt;
     let hit = battle.foesInRadius(s.x, s.y, BOMB_HIT_RADIUS).length > 0;
     if (!hit) {
diff --git a/server/sim/content/kits/ops/op-kalts2.js b/server/sim/content/kits/ops/op-kalts2.js
index b8419e75..2041934d 100644
--- a/server/sim/content/kits/ops/op-kalts2.js
+++ b/server/sim/content/kits/ops/op-kalts2.js
@@ -76,6 +76,7 @@
 import { num, talentBb, skillRec, batMod, up } from '../shared/tier1.js';
 import { releaseSkillSummon } from '../../tokens.js';
 import { COLS } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_kalts2_1';
 const S2 = 'skchr_kalts2_2';
@@ -235,7 +236,7 @@ function startTransport(battle, unit, anchor) {
     }, { owner: unit });
   }
   const from = { x: unit.x, y: unit.y }, to = { x: anchor.x, y: anchor.y };
-  const T = Math.hypot(to.x - from.x, to.y - from.y) / TRANSPORT_SPEED;
+  const T = hypot(to.x - from.x, to.y - from.y) / TRANSPORT_SPEED;
   const f = { t0: battle.time, T, from, to, hit: new Set(), anchor, r: anchor.tileR, c: anchor.tileC, dir: anchor.dir };
   unit.mem.k2flight = f;
   battle.releaseBlocked(unit);
diff --git a/server/sim/content/kits/ops/op-mantra.js b/server/sim/content/kits/ops/op-mantra.js
index 893e7396..285a6f58 100644
--- a/server/sim/content/kits/ops/op-mantra.js
+++ b/server/sim/content/kits/ops/op-mantra.js
@@ -46,6 +46,7 @@ import { hasHp } from '../../../damage.js';
 import { canTargetEnemy, enemyStealthed } from '../../../targeting.js';
 import { bodyInKeys } from '../../../body.js';
 import { COLS, PALSY_MAX } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_mantra_1';
 const S2 = 'skchr_mantra_2';
@@ -91,7 +92,7 @@ function nearestFoe(battle, unit, x, y, r, skip) {
   let best = null, bd = Infinity;
   for (const e of battle.enemiesInRadius(x, y, r)) {
     if (skip.has(e) || !canTargetEnemy(unit, e, ANY)) continue;
-    const d = Math.hypot(e.x - x, e.y - y);
+    const d = hypot(e.x - x, e.y - y);
     if (d < bd - 1e-9 || (Math.abs(d - bd) <= 1e-9 && best && e.spawnSeq < best.spawnSeq)) { bd = d; best = e; }
   }
   return best;
diff --git a/server/sim/content/kits/ops/op-mcnist.js b/server/sim/content/kits/ops/op-mcnist.js
index ea118724..31cd8bc2 100644
--- a/server/sim/content/kits/ops/op-mcnist.js
+++ b/server/sim/content/kits/ops/op-mcnist.js
@@ -58,6 +58,7 @@
 import { num, skillRec, batMod, up } from '../shared/tier1.js';
 import { COLS } from '../../../constants.js';
 import { dirVec } from '../../../dir.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_mcnist_1';
 const S2 = 'skchr_mcnist_2';
@@ -197,7 +198,7 @@ function grafKit(owner, { life, t2, b2, b3, aura }) {
 function chargeGraf(battle, mcn, t, b3) {
   if (!up(t)) return;
   const { stop, land } = chargeEnd(battle, t);
-  const dist = Math.hypot(stop[0] - t.tileR, stop[1] - t.tileC);
+  const dist = hypot(stop[0] - t.tileR, stop[1] - t.tileC);
   t.mem.charging = true;
   battle.fx('charge', { x: t.x, y: t.y, id: t.id, tx: stop[1], ty: stop[0] });
   battle.retreat(t, { reason: 'expired', permanent: true });
diff --git a/server/sim/content/kits/ops/op-narant.js b/server/sim/content/kits/ops/op-narant.js
index 0695f09c..45316af9 100644
--- a/server/sim/content/kits/ops/op-narant.js
+++ b/server/sim/content/kits/ops/op-narant.js
@@ -42,6 +42,7 @@
 import { num, talentBb, moduleBb, traitBb, skillRec, statBuff, toggleBuff, onHitBy, enemiesInGrid, cheb, up, giveSp } from '../shared/tier1.js';
 import { canTargetEnemy } from '../../../targeting.js';
 import { isHpLoss } from '../../../damage.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_narant_1';
 const S2 = 'skchr_narant_2';
@@ -138,7 +139,7 @@ export default {
         if (left <= 0 || !ctx.home()) { ctx.comeBack(x, y); return; }
         const cands = battle.foesInRadius(x, y, BOUNCE_RADIUS).filter((e) => e !== on && canTargetEnemy(unit, e, ctx.profile));
         if (!cands.length) { ctx.comeBack(x, y); return; }
-        const d2 = (e) => (e.x - x) ** 2 + (e.y - y) ** 2;
+        const d2 = (e) => (e.x - x) * (e.x - x) + (e.y - y) * (e.y - y);
         cands.sort((a, b) => (hit.has(a) ? 1 : 0) - (hit.has(b) ? 1 : 0) || d2(a) - d2(b) || a.spawnSeq - b.spawnSeq);
         const tgt = cands[0];
         left--;
@@ -158,7 +159,7 @@ export default {
     const dash = (ctx) => {
       const { battle, unit } = ctx;
       let dx = ctx.x - unit.x, dy = ctx.y - unit.y;
-      const len = Math.hypot(dx, dy);
+      const len = hypot(dx, dy);
       if (len > 1e-9) { dx /= len; dy /= len; } else { const [fr, fc] = unit.fwd; dx = fc; dy = fr; }
       const ahead = DASH_SPEED * Math.max(0, num(b2['attack@move_ahead_time'], 0.5));
       battle.addProjectile({ from: { x: ctx.x, y: ctx.y }, to: { x: ctx.x + dx * ahead, y: ctx.y + dy * ahead }, speed: DASH_SPEED, visual: 'boomerang', source: unit, hitDead: true,
diff --git a/server/sim/content/kits/ops/op-phatm2.js b/server/sim/content/kits/ops/op-phatm2.js
index deab16ea..8011d2b1 100644
--- a/server/sim/content/kits/ops/op-phatm2.js
+++ b/server/sim/content/kits/ops/op-phatm2.js
@@ -68,6 +68,7 @@ import { absoluteRangeKeys } from '../../../targeting.js';
 import { bodyInKeys } from '../../../body.js';
 import { hasHp, burstLocked } from '../../../damage.js';
 import { MIN_DAMAGE_RATIO } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_phatm2_1';
 const S2 = 'skchr_phatm2_2';
@@ -134,7 +135,7 @@ function callKit(t, owner, b2) {
         const keys = keysAt(grid, tok);
         const cands = battle.enemiesInKeys([...keys], tok, ANY).filter((e) => !lured.includes(e) && !e.s.flags.attract
           && (e.motion === 'FLY' || !!battle.grid.waypoints(Math.round(e.y), Math.round(e.x), tok.tileR, tok.tileC)));
-        cands.sort((a, b) => (isElite(b) - isElite(a)) || (Math.hypot(a.x - tok.x, a.y - tok.y) - Math.hypot(b.x - tok.x, b.y - tok.y)) || a.spawnSeq - b.spawnSeq);
+        cands.sort((a, b) => (isElite(b) - isElite(a)) || (hypot(a.x - tok.x, a.y - tok.y) - hypot(b.x - tok.x, b.y - tok.y)) || a.spawnSeq - b.spawnSeq);
         for (const e of cands) {
           if (lured.length >= cap) break;
           if (battle.applyStatus(e, 'attract', { duration: Math.max(0.05, endAt - battle.time), source: tok, point: [tok.tileR, tok.tileC] })) {
@@ -146,7 +147,7 @@ function callKit(t, owner, b2) {
       // the first lured enemy within `arrive` of it: it leaves
       battle.on('tick', () => {
         if (!up(tok)) return;
-        if (lured.some((e) => e.alive && mine(e) && Math.hypot(e.x - tok.x, e.y - tok.y) <= arrive + 1e-9)) battle.retreat(tok, { reason: 'expired', permanent: true });
+        if (lured.some((e) => e.alive && mine(e) && hypot(e.x - tok.x, e.y - tok.y) <= arrive + 1e-9)) battle.retreat(tok, { reason: 'expired', permanent: true });
       }, { owner: tok });
       // leaving (any reason): 停顿 + the damage over time on every selectable enemy of its x-1; the 诱导 ends
       battle.on('death', (c) => {
diff --git a/server/sim/content/kits/ops/op-phenxi.js b/server/sim/content/kits/ops/op-phenxi.js
index c0530e75..f06c9805 100644
--- a/server/sim/content/kits/ops/op-phenxi.js
+++ b/server/sim/content/kits/ops/op-phenxi.js
@@ -56,6 +56,7 @@
 import { num, talentBb, traitBb, moduleOn, skillRec, up, statBuff } from '../shared/tier1.js';
 import { toLocal } from '../../../dir.js';
 import { COLS, PROJECTILE_SPEEDS } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_phenxi_1';
 const S2 = 'skchr_phenxi_2';
@@ -136,7 +137,7 @@ function s3Shot(battle, unit, b3) {
   battle.addProjectile({ from: unit, to: { x, y }, speed: S3_SPEED, visual: 'bomb', source: unit, onHit: () => {
     battle.fx('aoe', { x, y, radius: S3_RADIUS, id: unit.id, skill: 'phenxi:s3' });
     for (const e of battle.foesInRadius(x, y, S3_RADIUS, true)) {
-      const sc = Math.hypot(e.x - x, e.y - y) <= core + 1e-9 ? hi : lo;
+      const sc = hypot(e.x - x, e.y - y) <= core + 1e-9 ? hi : lo;
       battle.dealDamage(unit, e, { amount: unit.s.atk * sc * unit.s.atkScaleMul, type: 'phys', isAttack: true, isSkill: true, isSplash: true, tags: ['skill', 'phenxi:s3'] });
     }
   } });
diff --git a/server/sim/content/kits/ops/op-poca.js b/server/sim/content/kits/ops/op-poca.js
index 90c01826..77c5dda9 100644
--- a/server/sim/content/kits/ops/op-poca.js
+++ b/server/sim/content/kits/ops/op-poca.js
@@ -36,6 +36,7 @@
 
 import { num, talentBb, moduleBb, moduleOn, traitBb, skillRec, statBuff, onHitBy, enemiesInGrid, up } from '../shared/tier1.js';
 import { hasHp } from '../../../damage.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skcom_atk_up[3]';
 const S2 = 'skchr_poca_2';
@@ -161,7 +162,7 @@ export default {
         if (ds > 0) {
           onHitBy(battle, unit, ({ target, dmg }) => {
             if (!dmg.isAttack) return;
-            const d = Math.hypot(target.x - unit.x, target.y - unit.y);
+            const d = hypot(target.x - unit.x, target.y - unit.y);
             dmg.mul *= 1 + ds * Math.max(0, Math.min(1, (d - lo) / Math.max(1e-6, hi - lo)));
           });
         }
diff --git a/server/sim/content/kits/ops/op-radian.js b/server/sim/content/kits/ops/op-radian.js
index 02b338ef..d3f7242d 100644
--- a/server/sim/content/kits/ops/op-radian.js
+++ b/server/sim/content/kits/ops/op-radian.js
@@ -64,6 +64,7 @@ import { acquireTargets } from '../../../ai.js';
 import { aggregateMods } from '../../../buffs.js';
 import { dirVec } from '../../../dir.js';
 import { COLS } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_radian_1';
 const S2 = 'skchr_radian_2';
@@ -160,7 +161,7 @@ function flyBullets(battle, t, dt) {
     const hit = () => {
       const near = battle.foesInRadius(b.x, b.y, BULLET_RADIUS).filter((e) => e.alive && !e.isFlying);
       if (!near.length) return false;
-      near.sort((p, q) => Math.hypot(p.x - b.x, p.y - b.y) - Math.hypot(q.x - b.x, q.y - b.y) || p.spawnSeq - q.spawnSeq);
+      near.sort((p, q) => hypot(p.x - b.x, p.y - b.y) - hypot(q.x - b.x, q.y - b.y) || p.spawnSeq - q.spawnSeq);
       battle.dealDamage(t, near[0], { amount: b.atk, type: 'phys', isAttack: true, tags: [TAG_BULLET] });
       return true;
     };
diff --git a/server/sim/content/kits/ops/op-ray.js b/server/sim/content/kits/ops/op-ray.js
index 684d2d33..f32a0aaf 100644
--- a/server/sim/content/kits/ops/op-ray.js
+++ b/server/sim/content/kits/ops/op-ray.js
@@ -63,6 +63,7 @@ import { sortEnemyTargets } from '../../../targeting.js';
 import { bodyInKeys } from '../../../body.js';
 import { PUSH_DIRECTIONAL_MIN_DIST } from '../../../constants.js';
 import { startCountdown } from '../../tokens.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_ray_1';
 const S2 = 'skchr_ray_2';
@@ -159,7 +160,7 @@ export default {
               if (!target.alive) { unit.mem.rayLoadExtra = num(unit.mem.rayLoadExtra) + Math.max(0, Math.floor(num(b1.cnt, 1))); return; }
               const force = num(b1.force, 1);
               const [fr, fc] = unit.fwd;
-              const vx = target.x - unit.x, vy = target.y - unit.y, d = Math.hypot(vx, vy);
+              const vx = target.x - unit.x, vy = target.y - unit.y, d = hypot(vx, vy);
               if (d >= PUSH_DIRECTIONAL_MIN_DIST && vx * fc + vy * fr >= d * Math.SQRT1_2) battle.push(target, force, { from: unit, dir: { x: fc, y: fr }, fixed: true });
               else battle.push(target, force - 1, { from: unit });
             },
diff --git a/server/sim/content/kits/ops/op-sleach.js b/server/sim/content/kits/ops/op-sleach.js
index c8ee9388..c555bf94 100644
--- a/server/sim/content/kits/ops/op-sleach.js
+++ b/server/sim/content/kits/ops/op-sleach.js
@@ -58,6 +58,7 @@ import { num, talentBb, moduleBb, traitBb, moduleOn, skillRec, up } from '../sha
 import { absoluteRangeKeys, sortEnemyTargets } from '../../../targeting.js';
 import { offsetTile } from '../../../dir.js';
 import { COLS } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skcom_assist_cost[3]';
 const S2 = 'skchr_sleach_2';
@@ -150,7 +151,7 @@ function s3Tile(battle, unit, grid) {
   for (const k of keys) {
     const r = (k / COLS) | 0, c = k % COLS;
     if (!ground(r, c) || battle.unitAt(r, c)) continue;
-    const d = Math.hypot(c - top.x, r - top.y);
+    const d = hypot(c - top.x, r - top.y);
     if (d < bd - 1e-9) { bd = d; best = [r, c]; }
   }
   return best ?? [unit.tileR, unit.tileC];
diff --git a/server/sim/content/kits/ops/op-typhon.js b/server/sim/content/kits/ops/op-typhon.js
index 48205bff..0ee71708 100644
--- a/server/sim/content/kits/ops/op-typhon.js
+++ b/server/sim/content/kits/ops/op-typhon.js
@@ -41,6 +41,7 @@
 import { num, talentBb, moduleOn, traitBb, skillRec, batMod, onHitBy, skillBusy, up } from '../shared/tier1.js';
 import { canTargetEnemy, sortEnemyTargets } from '../../../targeting.js';
 import { ROWS, COLS } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skcom_quickattack[3]';
 const S2 = 'skchr_typhon_2';
@@ -204,7 +205,7 @@ export default {
         if (ds > 0) {
           onHitBy(battle, unit, ({ target, dmg }) => {
             if (!dmg.isAttack) return;
-            const d = Math.hypot(target.x - unit.x, target.y - unit.y);
+            const d = hypot(target.x - unit.x, target.y - unit.y);
             dmg.mul *= 1 + ds * Math.max(0, Math.min(1, (d - lo) / Math.max(1e-6, hi - lo)));
           });
         }
diff --git a/server/sim/content/kits/ops/op-veen.js b/server/sim/content/kits/ops/op-veen.js
index 3f7e69c3..6693710b 100644
--- a/server/sim/content/kits/ops/op-veen.js
+++ b/server/sim/content/kits/ops/op-veen.js
@@ -42,6 +42,7 @@
 
 import { num, talentBb, moduleBb, traitBb, skillRec, batMod, toggleBuff } from '../shared/tier1.js';
 import { canTargetEnemy, sortEnemyTargets } from '../../../targeting.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_veen_1';
 const S2 = 'skchr_veen_2';
@@ -115,7 +116,7 @@ export default {
           if (!pool.length) pool = near;
           let next = null, bd = Infinity;
           for (const e of pool) {
-            const d = Math.hypot(e.x - cur.x, e.y - cur.y);
+            const d = hypot(e.x - cur.x, e.y - cur.y);
             if (d < bd - 1e-9 || (Math.abs(d - bd) <= 1e-9 && next && e.spawnSeq < next.spawnSeq)) { bd = d; next = e; }
           }
           if (!next) return;   // nobody within reach: the bounces left are lost
diff --git a/server/sim/content/kits/ops/op-weedy.js b/server/sim/content/kits/ops/op-weedy.js
index b3433bfc..2bc59b8b 100644
--- a/server/sim/content/kits/ops/op-weedy.js
+++ b/server/sim/content/kits/ops/op-weedy.js
@@ -49,6 +49,7 @@ import { num, traitBb, skillRec, up, giveSp } from '../shared/tier1.js';
 import { absoluteRangeKeys } from '../../../targeting.js';
 import { COLS } from '../../../constants.js';
 import { startCountdown } from '../../tokens.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_weedy_1';
 const S2 = 'skchr_weedy_2';
@@ -92,7 +93,7 @@ export function rupture(battle, src, e, { duration, perTile, interval }) {
     return;
   }
   const st = { x: e.x, y: e.y, moved: 0, acc: 0 };
-  const step = (u) => { st.moved += Math.hypot(u.x - st.x, u.y - st.y); st.x = u.x; st.y = u.y; };
+  const step = (u) => { st.moved += hypot(u.x - st.x, u.y - st.y); st.x = u.x; st.y = u.y; };
   const flush = (u) => {
     const m = st.moved;
     st.moved = 0;
@@ -121,7 +122,7 @@ export function liquidNitrogen(battle, { shooter, src, atk, sbb, grid, force, pi
   const keys = absoluteRangeKeys(grid ?? GRID_4_1, shooter.tileR, shooter.tileC, shooter.dir, 0);
   const cands = battle.enemiesInKeys(keys, shooter, { canHitFly: true });
   if (!cands.length) return false;
-  const key = pick === 'nearest' ? (e) => Math.hypot(e.x - shooter.x, e.y - shooter.y) : (e) => battle.remainingDistance(e);
+  const key = pick === 'nearest' ? (e) => hypot(e.x - shooter.x, e.y - shooter.y) : (e) => battle.remainingDistance(e);
   let target = null, best = Infinity;
   for (const e of cands) {
     const v = key(e);
diff --git a/server/sim/content/kits/ops/op-wisdel.js b/server/sim/content/kits/ops/op-wisdel.js
index b9241676..01220201 100644
--- a/server/sim/content/kits/ops/op-wisdel.js
+++ b/server/sim/content/kits/ops/op-wisdel.js
@@ -61,6 +61,7 @@ import { num, talentBb, traitBb, skillRec, up, batMod } from '../shared/tier1.js
 import { absoluteRangeKeys, sortEnemyTargets } from '../../../targeting.js';
 import { hasHp } from '../../../damage.js';
 import { COLS } from '../../../constants.js';
+import { hypot } from '../../../detmath.js';
 
 const S1 = 'skchr_wisdel_1';
 const S2 = 'skchr_wisdel_2';
@@ -138,7 +139,7 @@ function shadowTile(battle, host) {
   for (const k of host.rangeKeys ?? []) {
     const r = Math.floor(k / COLS), c = k % COLS;
     if (!battle.grid.inRect(r, c) || !battle.grid.canStand(r, c, { ranged: true }) || battle.isReservedTile(r, c)) continue;
-    const d = Math.hypot(r - host.tileR, c - host.tileC);
+    const d = hypot(r - host.tileR, c - host.tileC);
     if (d < bd - 1e-9 || (Math.abs(d - bd) <= 1e-9 && (r < best[0] || (r === best[0] && c < best[1])))) { bd = d; best = [r, c]; }
   }
   return best;
diff --git a/server/sim/content/kits/shared/tier5.js b/server/sim/content/kits/shared/tier5.js
index 4227cbb2..14b622ae 100644
--- a/server/sim/content/kits/shared/tier5.js
+++ b/server/sim/content/kits/shared/tier5.js
@@ -38,6 +38,7 @@
 
 import { bodyInKeys } from '../../../body.js';
 import { absoluteRangeKeys, sortEnemyTargets } from '../../../targeting.js';
+import { hypot } from '../../../detmath.js';
 
 // ---- text-only constants (the official blackboards carry no key for these) --------------------------------------
 /** 华法琳 S1 "只当目标生命值不满一半时才会触发"; 塞雷娅 S1 "血量小于等于一半"; 山 module "生命值高于50%时". */
@@ -70,7 +71,7 @@ const maxCharges = (chess, def) => num(def?.skill?.maxCharges, num(chess?.skill?
 const batPct = (sec, chess) => { const b = num(chess?.stats?.bat, 1) || 1; const v = num(sec) / b; return v ? Math.max(-0.9, v) : 0; };
 /** Drop zero / non-finite entries (a zero mod is noise in the buff list). */
 const mods = (m) => { const o = {}; for (const k of Object.keys(m)) { const v = m[k]; if (typeof v === 'number' && Number.isFinite(v) && v !== 0) o[k] = v; } return o; };
-const dist = (a, b) => Math.hypot(a.x - b.x, a.y - b.y);
+const dist = (a, b) => hypot(a.x - b.x, a.y - b.y);
 const nationOf = (u) => u?.def?.raw?.nationId ?? null;
 const inFaction = (u, bond, nations) => !!u?.def && ((u.def.bonds || []).includes(bond) || nations.includes(nationOf(u)));
 const isAbyssal = (u) => u?.def?.raw?.groupId === 'abyssal' || ABYSSAL.has(u?.def?.charId);
diff --git a/server/sim/content/kits/shared/tier6.js b/server/sim/content/kits/shared/tier6.js
index 5e040169..0f9999b7 100644
--- a/server/sim/content/kits/shared/tier6.js
+++ b/server/sim/content/kits/shared/tier6.js
@@ -111,6 +111,7 @@
 import { absoluteRangeKeys } from '../../../targeting.js';
 import { COLS, ROWS, PULL_STOP_RADIUS } from '../../../constants.js';
 import { hasHp } from '../../../damage.js';
+import { hypot } from '../../../detmath.js';
 
 // ------------------------------------------------------------------------------------------------------------------
 // helpers
@@ -237,7 +238,7 @@ function bestTile(battle, tiles) {
   let best = tiles[0], bd = Infinity;
   for (const t of tiles) {
     let d = Infinity;
-    for (const e of en) d = Math.min(d, Math.hypot(e.x - t[1], e.y - t[0]));
+    for (const e of en) d = Math.min(d, hypot(e.x - t[1], e.y - t[0]));
     if (d < bd - 1e-9) { bd = d; best = t; }
   }
   return best;
diff --git a/server/sim/content/tokens.js b/server/sim/content/tokens.js
index 9322368b..77f8a7c4 100644
--- a/server/sim/content/tokens.js
+++ b/server/sim/content/tokens.js
@@ -92,6 +92,7 @@ import { genericKit } from './generic.js';
 import { bardRegen } from '../professions.js';
 import { normDir, localOrder } from '../dir.js';
 import { SKILL_SUMMON_START_DEPLOY } from '../../../shared/constants.js';
+import { hypot } from '../detmath.js';
 
 const num = (v, d = 0) => (typeof v === 'number' && Number.isFinite(v) ? v : (typeof v === 'string' && v.trim() !== '' && Number.isFinite(+v) ? +v : d));
 const GRID_3X3 = Object.freeze([[1, -1], [1, 0], [1, 1], [0, -1], [0, 0], [0, 1], [-1, -1], [-1, 0], [-1, 1]]);
@@ -451,7 +452,7 @@ export function findSummonTile(battle, owner, placement = 'melee', opts = {}) {
     let cover = 0;
     if (placement === 'enemy' || placement === 'melee') cover = countNear(enemies, r, c, 1);
     else if (placement === 'ally') cover = countNear(injured, r, c, 1);
-    const d = placement === 'ally' ? Math.hypot(c - owner.x, r - owner.y) : Math.hypot(c - focus.x, r - focus.y);
+    const d = placement === 'ally' ? hypot(c - owner.x, r - owner.y) : hypot(c - focus.x, r - focus.y);
     // last: the tile's offset in the owner's facing-RIGHT frame (for a RIGHT owner = the tile key order), so equal
     // candidates resolve the same way whichever direction the owner faces
     const [lr, lc] = localOrder(r - owner.tileR, c - owner.tileC, owner.dir);
@@ -1236,7 +1237,7 @@ function yanyouKit(bb, raw) {
     const keys = [];
     const r0 = Math.floor(u.y - radius), r1 = Math.ceil(u.y + radius), c0 = Math.floor(u.x - radius), c1 = Math.ceil(u.x + radius);
     for (let r = Math.max(0, r0); r <= Math.min(ROWS - 1, r1); r++) {
-      for (let c = Math.max(0, c0); c <= Math.min(COLS - 1, c1); c++) if (Math.hypot(c - u.x, r - u.y) <= radius + 1e-9) keys.push(r * COLS + c);
+      for (let c = Math.max(0, c0); c <= Math.min(COLS - 1, c1); c++) if (hypot(c - u.x, r - u.y) <= radius + 1e-9) keys.push(r * COLS + c);
     }
     u.rangeKeys = keys;
     u.rangeKeySet = new Set(keys);
@@ -1299,7 +1300,7 @@ function yanyouKit(bb, raw) {
           const tgt = lockOf(unit) ?? topAggroEnemy(battle, unit);
           if (tgt) {
             const dx = tgt.x - unit.x, dy = tgt.y + num(unit.mem.hoverDy, 0) - unit.y;
-            const d = Math.hypot(dx, dy);
+            const d = hypot(dx, dy);
             if (d > YANYOU_STOP + 1e-6) {
               const step = Math.min(d - YANYOU_STOP, speed * dt);
               const R = battle.rect;
@@ -1351,7 +1352,7 @@ function airTile(battle, playerId, taken) {
       if (taken.has(r * COLS + c) || held.has(r * COLS + c) || !tileFree(battle, r, c)) continue;
       const t = battle.grid.tile(r, c);
       const cls = t.build === 'NONE' && t.pass !== 'ALL' ? 0 : t.build === 'NONE' ? 1 : 2;
-      const s = [cls, Math.hypot(r - cr, c - cc), r * COLS + c];
+      const s = [cls, hypot(r - cr, c - cc), r * COLS + c];
       let less = !bs;
       if (!less) for (let i = 0; i < 3; i++) { if (s[i] < bs[i] - 1e-9) { less = true; break; } if (s[i] > bs[i] + 1e-9) break; }
       if (less) { best = [r, c]; bs = s; }
diff --git a/server/sim/detmath.js b/server/sim/detmath.js
new file mode 100644
index 00000000..1aed1596
--- /dev/null
+++ b/server/sim/detmath.js
@@ -0,0 +1,252 @@
+// server/sim/detmath.js — engine-independent math for the sim. Every function here gives the same bits on every
+// conforming JavaScript engine.
+//
+// ECMA-262 lets Math.hypot, sin, cos, atan2, pow (and **) return "implementation-approximated" values, and the engines
+// differ in the last bits: V8's hypot is not the one of SpiderMonkey (fdlibm) or JavaScriptCore (sqrt(x*x + y*y));
+// Chrome 154's sin / cos / atan2 are correctly rounded while Node 24–26's are fdlibm's; JavaScriptCore computes an
+// integer power by repeated squaring. + − × ÷ and Math.sqrt are correctly rounded IEEE 754 operations, and the language
+// fuses nothing (no FMA, no extended precision), so code built only from them is reproducible bit for bit. That is
+// what these functions are. The sim must not call the approximated Math functions (eslint.config.js, and
+// test/sim/detmath.test.js scans server/sim/).
+//
+//   hypot(x, y)         √(x² + y²) as Math.sqrt(x*x + y*y), rescaled by exact powers of two outside 2^±500
+//   powi(x, n)          xⁿ for an integer n: square-and-multiply in double-double, one final rounding
+//   sin(x), cos(x)      fdlibm 5.3 (__kernel_sin / __kernel_cos / __ieee754_rem_pio2)
+//   atan2(y, x)         fdlibm (FreeBSD msun e_atan2.c / s_atan.c)
+//
+// sin / cos / atan2 are the algorithms V8 runs in Node, so they return what the server always computed (its golden
+// digests stay); hypot and powi are at least as accurate as V8's. ±∞ / NaN arguments take the exactly specified
+// results of the Math functions.
+
+const F64 = new Float64Array(1);
+const I32 = new Int32Array(F64.buffer);
+const HW = new Uint8Array(new Uint16Array([1]).buffer)[0] === 1 ? 1 : 0; // index of the high word
+const LW = 1 - HW;
+/** @param {number} x */
+const hi = (x) => { F64[0] = x; return I32[HW]; };
+/** @param {number} x */
+const lo = (x) => { F64[0] = x; return I32[LW] >>> 0; };
+/** @param {number} h @param {number} l */
+const fromWords = (h, l) => { I32[HW] = h; I32[LW] = l; return F64[0]; };
+/** exact 2^e for a normal e @param {number} e */
+const pow2 = (e) => fromWords((e + 1023) << 20, 0);
+const P500 = pow2(500), PM500 = pow2(-500), P600 = pow2(600), PM600 = pow2(-600);
+
+/**
+ * √(x² + y²).
+ * @param {number} x
+ * @param {number} y
+ */
+export function hypot(x, y) {
+  // eslint-disable-next-line no-restricted-properties -- ±∞ / NaN only: those results are exact by the spec
+  if (!Number.isFinite(x) || !Number.isFinite(y)) return Math.hypot(x, y);
+  const m = Math.max(Math.abs(x), Math.abs(y));
+  if (m > P500) { const a = x * PM600, b = y * PM600; return Math.sqrt(a * a + b * b) * P600; }
+  if (m < PM500 && m !== 0) { const a = x * P600, b = y * P600; return Math.sqrt(a * a + b * b) * PM600; }
+  return Math.sqrt(x * x + y * y);
+}
+
+// ---- integer powers ----------------------------------------------------------------------------------------------
+
+const SPLIT = 134217729; // 2^27 + 1 (Dekker)
+const ACC = [0, 0];
+const BASE = [0, 0];
+/** (ah + al)·(bh + bl) in double-double, into out (TwoProduct by Dekker's split: no FMA) */
+function mul2(ah, al, bh, bl, out) {
+  const p = ah * bh;
+  let t = SPLIT * ah;
+  const ahh = t - (t - ah), ahl = ah - ahh;
+  t = SPLIT * bh;
+  const bhh = t - (t - bh), bhl = bh - bhh;
+  const e = ((ahh * bhh - p) + ahh * bhl + ahl * bhh) + ahl * bhl + (ah * bl + al * bh);
+  const s = p + e;
+  out[0] = s;
+  out[1] = e - (s - p);
+}
+
+/**
+ * xⁿ for an integer n (the sim's powers are stack counts, chain bounces …). The square-and-multiply runs in
+ * double-double and rounds once, so the result is correctly rounded in practice; a non-integer exponent or a huge
+ * base falls back to Math.pow.
+ * @param {number} x
+ * @param {number} n
+ */
+export function powi(x, n) {
+  // eslint-disable-next-line no-restricted-properties -- outside the integer domain only (never in the sim)
+  if (!Number.isInteger(n) || !Number.isFinite(x) || Math.abs(x) > P500) return Math.pow(x, n);
+  if (n < 0) return 1 / powi(x, -n);
+  ACC[0] = 1; ACC[1] = 0; BASE[0] = x; BASE[1] = 0;
+  let k = n;
+  while (k > 0) {
+    if (k % 2 === 1) mul2(ACC[0], ACC[1], BASE[0], BASE[1], ACC);
+    k = Math.floor(k / 2);
+    if (k > 0) mul2(BASE[0], BASE[1], BASE[0], BASE[1], BASE);
+  }
+  return ACC[0] + ACC[1];
+}
+
+// ---- sin / cos (fdlibm 5.3) --------------------------------------------------------------------------------------
+
+const S1 = -1.66666666666666324348e-01, S2 = 8.33333333332248946124e-03, S3 = -1.98412698298579493134e-04;
+const S4 = 2.75573137070700676789e-06, S5 = -2.50507602534068634195e-08, S6 = 1.58969099521155010221e-10;
+/** sin on [-π/4, π/4]; y is the tail of x, iy = 0 when y is 0 */
+function kSin(x, y, iy) {
+  if ((hi(x) & 0x7fffffff) < 0x3e400000 && Math.trunc(x) === 0) return x;
+  const z = x * x, v = z * x;
+  const r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
+  if (iy === 0) return x + v * (S1 + z * r);
+  return x - ((z * (0.5 * y - v * r) - y) - v * S1);
+}
+
+const C1 = 4.16666666666666019037e-02, C2 = -1.38888888888741095749e-03, C3 = 2.48015872894767294178e-05;
+const C4 = -2.75573143513906633035e-07, C5 = 2.08757232129817482790e-09, C6 = -1.13596475577881948265e-11;
+/** cos on [-π/4, π/4]; y is the tail of x */
+function kCos(x, y) {
+  const ix = hi(x) & 0x7fffffff;
+  if (ix < 0x3e400000 && Math.trunc(x) === 0) return 1;
+  const z = x * x;
+  const r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
+  if (ix < 0x3fd33333) return 1 - (0.5 * z - (z * r - x * y));
+  const qx = ix > 0x3fe90000 ? 0.28125 : fromWords(ix - 0x00200000, 0);
+  return (1 - qx) - ((0.5 * z - qx) - (z * r - x * y));
+}
+
+const NPIO2_HW = [
+  0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A,
+  0x4031475C, 0x4032D97C, 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, 0x403DD85A, 0x403F6A7A,
+  0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
+  0x404858EB, 0x404921FB,
+];
+const INV_PIO2 = 6.36619772367581382433e-01;
+const PIO2_1 = 1.57079632673412561417e+00, PIO2_1T = 6.07710050650619224932e-11;
+const PIO2_2 = 6.07710050630396597660e-11, PIO2_2T = 2.02226624879595063154e-21;
+const PIO2_3 = 2.02226624871116645580e-21, PIO2_3T = 8.47842766036889956997e-32;
+const TWO_PI = 6.283185307179586;
+const Y = [0, 0];
+
+/** x − n·π/2 into Y (head, tail); returns n. |x| ≤ 2^19·π/2: fdlibm's medium path; beyond that (never in the sim)
+ * x is first taken modulo the double nearest 2π (exact %), which stays deterministic though not accurate. */
+function remPio2(x0) {
+  let x = x0;
+  if ((hi(x) & 0x7fffffff) > 0x413921fb) x = x % TWO_PI;
+  const hx = hi(x), ix = hx & 0x7fffffff;
+  if (ix <= 0x3fe921fb) { Y[0] = x; Y[1] = 0; return 0; }
+  if (ix < 0x4002d97c) { // |x| < 3π/4: n = ±1
+    if (hx > 0) {
+      let z = x - PIO2_1;
+      if (ix !== 0x3ff921fb) { Y[0] = z - PIO2_1T; Y[1] = (z - Y[0]) - PIO2_1T; } else { z -= PIO2_2; Y[0] = z - PIO2_2T; Y[1] = (z - Y[0]) - PIO2_2T; }
+      return 1;
+    }
+    let z = x + PIO2_1;
+    if (ix !== 0x3ff921fb) { Y[0] = z + PIO2_1T; Y[1] = (z - Y[0]) + PIO2_1T; } else { z += PIO2_2; Y[0] = z + PIO2_2T; Y[1] = (z - Y[0]) + PIO2_2T; }
+    return -1;
+  }
+  let t = Math.abs(x);
+  const n = Math.trunc(t * INV_PIO2 + 0.5);
+  let r = t - n * PIO2_1, w = n * PIO2_1T;
+  Y[0] = r - w;
+  if (!(n < 32 && ix !== NPIO2_HW[n - 1])) {
+    const j = ix >> 20;
+    if (j - ((hi(Y[0]) >> 20) & 0x7ff) > 16) { // a 2nd step, good to 118 bits
+      t = r; w = n * PIO2_2; r = t - w; w = n * PIO2_2T - ((t - r) - w); Y[0] = r - w;
+      if (j - ((hi(Y[0]) >> 20) & 0x7ff) > 49) { // a 3rd, 151 bits
+        t = r; w = n * PIO2_3; r = t - w; w = n * PIO2_3T - ((t - r) - w); Y[0] = r - w;
+      }
+    }
+  }
+  Y[1] = (r - Y[0]) - w;
+  if (hx < 0) { Y[0] = -Y[0]; Y[1] = -Y[1]; return -n; }
+  return n;
+}
+
+/** @param {number} x radians */
+export function sin(x) {
+  if (!Number.isFinite(x)) return NaN;
+  if ((hi(x) & 0x7fffffff) <= 0x3fe921fb) return kSin(x, 0, 0);
+  switch (remPio2(x) & 3) {
+    case 0: return kSin(Y[0], Y[1], 1);
+    case 1: return kCos(Y[0], Y[1]);
+    case 2: return -kSin(Y[0], Y[1], 1);
+    default: return -kCos(Y[0], Y[1]);
+  }
+}
+
+/** @param {number} x radians */
+export function cos(x) {
+  if (!Number.isFinite(x)) return NaN;
+  if ((hi(x) & 0x7fffffff) <= 0x3fe921fb) return kCos(x, 0);
+  switch (remPio2(x) & 3) {
+    case 0: return kCos(Y[0], Y[1]);
+    case 1: return -kSin(Y[0], Y[1], 1);
+    case 2: return -kCos(Y[0], Y[1]);
+    default: return kSin(Y[0], Y[1], 1);
+  }
+}
+
+// ---- atan2 (fdlibm, FreeBSD msun) --------------------------------------------------------------------------------
+
+const ATAN_HI = [4.63647609000806093515e-01, 7.85398163397448278999e-01, 9.82793723247329054082e-01, 1.57079632679489655800e+00];
+const ATAN_LO = [2.26987774529616870924e-17, 3.06161699786838301793e-17, 1.39033110312309984516e-17, 6.12323399573676603587e-17];
+const AT = [
+  3.33333333333329318027e-01, -1.99999999998764832476e-01, 1.42857142725034663711e-01, -1.11111104054623557880e-01,
+  9.09088713343650656196e-02, -7.69187620504482999495e-02, 6.66107313738753120669e-02, -5.83357013379057348645e-02,
+  4.97687799461593236017e-02, -3.65315727442169155270e-02, 1.62858201153657823623e-02,
+];
+/** @param {number} x0 */
+function atan(x0) {
+  let x = x0;
+  const hx = hi(x), ix = hx & 0x7fffffff;
+  let id;
+  if (ix >= 0x44100000) { // |x| ≥ 2^66
+    if (ix > 0x7ff00000 || (ix === 0x7ff00000 && lo(x) !== 0)) return x + x;
+    return hx > 0 ? ATAN_HI[3] + ATAN_LO[3] : -ATAN_HI[3] - ATAN_LO[3];
+  }
+  if (ix < 0x3fdc0000) { // |x| < 0.4375
+    if (ix < 0x3e200000) return x;
+    id = -1;
+  } else {
+    x = Math.abs(x);
+    if (ix < 0x3ff30000) {
+      if (ix < 0x3fe60000) { id = 0; x = (2 * x - 1) / (2 + x); } else { id = 1; x = (x - 1) / (x + 1); }
+    } else if (ix < 0x40038000) { id = 2; x = (x - 1.5) / (1 + 1.5 * x); } else { id = 3; x = -1 / x; }
+  }
+  const z = x * x, w = z * z;
+  const s1 = z * (AT[0] + w * (AT[2] + w * (AT[4] + w * (AT[6] + w * (AT[8] + w * AT[10])))));
+  const s2 = w * (AT[1] + w * (AT[3] + w * (AT[5] + w * (AT[7] + w * AT[9]))));
+  if (id < 0) return x - x * (s1 + s2);
+  const r = ATAN_HI[id] - ((x * (s1 + s2) - ATAN_LO[id]) - x);
+  return hx < 0 ? -r : r;
+}
+
+const PI_O_4 = 7.8539816339744827900E-01, PI_O_2 = 1.5707963267948965580E+00, PI = 3.1415926535897931160E+00;
+const PI_LO = 1.2246467991473532e-16;
+/**
+ * The angle of (x, y), in (−π, π].
+ * @param {number} y
+ * @param {number} x
+ */
+export function atan2(y, x) {
+  if (Number.isNaN(x) || Number.isNaN(y)) return NaN;
+  const hx = hi(x), ix = hx & 0x7fffffff, lx = lo(x);
+  const hy = hi(y), iy = hy & 0x7fffffff, ly = lo(y);
+  if (((hx - 0x3ff00000) | lx) === 0) return atan(y); // x = 1
+  let m = ((hy >>> 31) & 1) | ((hx >>> 30) & 2); // 2·sign(x) + sign(y)
+  if ((iy | ly) === 0) return m === 2 ? PI : m === 3 ? -PI : y; // y = ±0
+  if ((ix | lx) === 0) return hy < 0 ? -PI_O_2 : PI_O_2; // x = ±0
+  if (ix === 0x7ff00000) { // x = ±∞
+    if (iy === 0x7ff00000) return [PI_O_4, -PI_O_4, 3 * PI_O_4, -3 * PI_O_4][m];
+    return [0, -0, PI, -PI][m];
+  }
+  if (iy === 0x7ff00000) return hy < 0 ? -PI_O_2 : PI_O_2; // y = ±∞
+  const k = (iy - ix) >> 20;
+  let z;
+  if (k > 60) { z = PI_O_2 + 0.5 * PI_LO; m &= 1; } // |y/x| > 2^60
+  else if (hx < 0 && k < -60) z = 0; // |y|/x < −2^60
+  else z = atan(Math.abs(y / x));
+  switch (m) {
+    case 0: return z;
+    case 1: return -z;
+    case 2: return PI - (z - PI_LO);
+    default: return (z - PI_LO) - PI;
+  }
+}
diff --git a/server/sim/fear.js b/server/sim/fear.js
index 19314e9a..0c2b93df 100644
--- a/server/sim/fear.js
+++ b/server/sim/fear.js
@@ -27,6 +27,7 @@
 
 import { MOVE_SCALE, COLS } from './constants.js';
 import { straightClear } from './grid.js';
+import { hypot } from './detmath.js';
 
 /** 半径10格 — the fan's radius (tiles). */
 export const FEAR_RADIUS = 10;
@@ -80,7 +81,7 @@ const passable = (g, fly, r, c) => (fly ? g.flyPassable(r, c) : g.walkable(r, c,
  */
 export function reachableTiles(b, e, st) {
   if (!st || st.self) return [];
-  const dx = st.hx - st.sx, dy = st.hy - st.sy, d = Math.hypot(dx, dy);
+  const dx = st.hx - st.sx, dy = st.hy - st.sy, d = hypot(dx, dy);
   if (!(d > 1e-9)) return [];
   const ux = dx / d, uy = dy / d;
   const g = b.grid, R = g.rect, fly = e.motion === 'FLY';
@@ -90,7 +91,7 @@ export function reachableTiles(b, e, st) {
   const out = [];
   for (let r = R.r0; r <= R.r1; r++) {
     for (let c = R.c0; c <= R.c1; c++) {
-      const vx = c - st.hx, vy = r - st.hy, L = Math.hypot(vx, vy);
+      const vx = c - st.hx, vy = r - st.hy, L = hypot(vx, vy);
       if (L > FEAR_RADIUS + 1e-9) continue;
       if (L > 1e-9 && vx * ux + vy * uy < FEAR_HALF_COS * L - 1e-9) continue;
       if (!passable(g, fly, r, c) || g.tile(r, c).special === 'end') continue;
@@ -219,7 +220,7 @@ export function moveFeared(b, e, dt) {
     for (let guard = 8; dist > 1e-9 && guard > 0; guard--) {
       if (!m.pts || m.i >= m.pts.length) pickCheckpoint(b, e, m);
       const p = m.pts[m.i];
-      const dx = p.x - e.x, dy = p.y - e.y, d = Math.hypot(dx, dy);
+      const dx = p.x - e.x, dy = p.y - e.y, d = hypot(dx, dy);
       if (d <= dist) { e.x = p.x; e.y = p.y; dist -= d; m.i++; } else { e.x += (dx / d) * dist; e.y += (dy / d) * dist; dist = 0; }
       moved = moved || d > 1e-9;
     }
diff --git a/server/sim/grid.js b/server/sim/grid.js
index f416a251..8eb6404d 100644
--- a/server/sim/grid.js
+++ b/server/sim/grid.js
@@ -54,6 +54,7 @@
 // off-centre unit may walk straight to a tile centre (ai.js planLeg, fear.js: else it steps back to its tile centre).
 
 import { ROWS, COLS } from './constants.js';
+import { hypot } from './detmath.js';
 
 /** Move cost of an obstacle-like tile (client `Tile.get_moveCost`: obstacle-like ? 1000 : 1). */
 export const OBSTACLE_COST = 1000;
@@ -376,7 +377,7 @@ export class Grid {
     for (let i = stack.length - 1; i >= 0; i--) {
       const a = stack[i], b = f.next[a];
       const ar = (a / COLS) | 0, ac = a - ar * COLS, br = (b / COLS) | 0, bc = b - br * COLS;
-      f.len[a] = f.len[b] + Math.hypot(br - ar, bc - ac);
+      f.len[a] = f.len[b] + hypot(br - ar, bc - ac);
     }
     return f.len[k];
   }
@@ -420,7 +421,7 @@ export class Grid {
   /** Length (tiles) of a polyline of [r,c] points. */
   static pathLength(pts) {
     let len = 0;
-    for (let i = 1; i < pts.length; i++) len += Math.hypot(pts[i][0] - pts[i - 1][0], pts[i][1] - pts[i - 1][1]);
+    for (let i = 1; i < pts.length; i++) len += hypot(pts[i][0] - pts[i - 1][0], pts[i][1] - pts[i - 1][1]);
     return len;
   }
 }
diff --git a/server/sim/projectiles.js b/server/sim/projectiles.js
index e60733bb..5ed195a9 100644
--- a/server/sim/projectiles.js
+++ b/server/sim/projectiles.js
@@ -5,6 +5,7 @@
 // mid-flight the projectile fizzles, unless `hitDead: true` (then it lands at the last known position).
 
 import { PROJECTILE_SPEED } from './constants.js';
+import { hypot } from './detmath.js';
 
 let seq = 0;
 const fin = (v, d) => (typeof v === 'number' && Number.isFinite(v) ? v : d);
@@ -55,7 +56,7 @@ export class ProjectileSystem {
         else p.target = null;
       }
       const dx = p.tx - p.x, dy = p.ty - p.y;
-      const d = Math.hypot(dx, dy);
+      const d = hypot(dx, dy);
       const step = p.speed * dt;
       if (d <= step || p.age >= p.maxAge) {
         p.x = p.tx; p.y = p.ty;
diff --git a/server/sim/spec.js b/server/sim/spec.js
index 09996ddb..225099a1 100644
--- a/server/sim/spec.js
+++ b/server/sim/spec.js
@@ -320,7 +320,7 @@ export function resultDigest(result) {
 const fnum = (v, d = 0) => (typeof v === 'number' && Number.isFinite(v) ? v : d);
 const cap = (list, n) => (Array.isArray(list) ? list.slice(0, n) : []);
 const isKey = (v) => typeof v === 'string' && v.length > 0 && v.length <= 64 && /^[A-Za-z0-9_\-.:]+$/.test(v);
-const uidOr = (v) => (Number.isInteger(v) && v >= 1 && v <= 2 ** 31 ? v : null);
+const uidOr = (v) => (Number.isInteger(v) && v >= 1 && v <= 0x80000000 ? v : null);
 const keyOr = (v) => (isKey(v) ? v : null);
 
 function compactMods(m) {
diff --git a/test/golden/diy.json b/test/golden/diy.json
index 84a26ecc..c467dac7 100644
--- a/test/golden/diy.json
+++ b/test/golden/diy.json
@@ -1085,26 +1085,26 @@
       "bossHpLeft": null,
       "rngDraws": 0,
       "errors": {"count":0,"kinds":[]},
-      "players": {"p1":{"killed":111,"total":111,"perfect":true,"leaks":0,"leaksUncounted":0,"coins":0,"dmg":1987984,"bossDmg":0,"heal":0,"deaths":0,"layerGains":{}}},
+      "players": {"p1":{"killed":111,"total":111,"perfect":true,"leaks":0,"leaksUncounted":0,"coins":0,"dmg":1988064,"bossDmg":0,"heal":0,"deaths":0,"layerGains":{}}},
       "units": [
         ["p1",1,"op","chess_char_5_diy1_b",0,"uniequip_002_narant",2331,1,10,9,139430,0,0,91,1],
         ["p1",2,"op","chess_char_5_diy2_b",1,"uniequip_002_narant",2331,1,11,5,155980,0,0,88,3],
         ["p1",3,"op","chess_char_5_diy1_b",2,"uniequip_002_narant",2331,1,1,0,740,0,0,5,0],
         ["p1",4,"op","chess_char_5_diy2_b",0,"uniequip_003_narant",2541,1,3,1,12484,0,0,4,1],
-        ["p1",5,"op","chess_char_5_diy1_b",1,"uniequip_003_narant",2541,1,5,16,222860,0,0,87,3],
-        ["p1",6,"op","chess_char_5_diy2_b",2,"uniequip_003_narant",2541,1,7,11,255194,0,0,170,4],
-        ["p1",7,"op","chess_char_6_diy1_a",0,null,2000,1,9,18,305504,0,0,107,1],
-        ["p1",8,"op","chess_char_6_diy2_a",1,null,2000,1,12,23,447750,0,0,177,4],
+        ["p1",5,"op","chess_char_5_diy1_b",1,"uniequip_003_narant",2541,1,5,18,216963,0,0,85,3],
+        ["p1",6,"op","chess_char_5_diy2_b",2,"uniequip_003_narant",2541,1,7,11,262559,0,0,170,4],
+        ["p1",7,"op","chess_char_6_diy1_a",0,null,2000,1,9,19,302382,0,0,106,1],
+        ["p1",8,"op","chess_char_6_diy2_a",1,null,2000,1,12,27,452511,0,0,177,4],
         ["p1",9,"op","chess_char_6_diy1_a",2,null,2000,1,2,0,251,0,0,5,0],
         ["p1",10,"op","chess_char_6_diy2_b",0,"none",2331,1,4,0,12573,0,0,4,1],
-        ["p1",11,"op","chess_char_6_diy1_b",1,"none",2331,1,6,18,198679,0,0,90,3],
-        ["p1",12,"op","chess_char_6_diy2_b",2,"none",2331,1,8,10,236538,0,0,173,4]
+        ["p1",11,"op","chess_char_6_diy1_b",1,"none",2331,1,6,10,197025,0,0,89,3],
+        ["p1",12,"op","chess_char_6_diy2_b",2,"none",2331,1,8,11,235164,0,0,173,4]
       ],
       "enemies": {"enemy_1010_demon":[54,54,0,0,755827],"enemy_1320_wdrrl_2":[18,18,0,0,839808],"enemy_1422_lrsldr_2":[39,39,0,0,400308]},
-      "hooks": [1,123,5886,1001,1001,2296,2296,0,111,111,25,20,0,9735,233,0,111,0,1,111,0,0,0],
-      "events": {"atk":1001,"deploy":12,"die":111,"dmg":2296,"engage":12,"fx":453,"skill":45,"spawn":123,"status":245},
-      "fx": {"aoe":137,"bounce":316},
-      "snaps": "b054f186 5f1791df 62fad2b9 772b925e 3c33e5ff 792df80b b43d3d8f 4283ce62 933a9c03 44991c3e b9347ef2 941827a5 73c09893 9ddf9d2c dbaf0a96 63e46fbe d21c76f5 55bdfa26 bfaf6c80 6968f2d8"
+      "hooks": [1,123,5886,997,997,2292,2292,0,111,111,25,20,0,9744,235,0,111,0,1,111,0,0,0],
+      "events": {"atk":997,"deploy":12,"die":111,"dmg":2292,"engage":12,"fx":451,"skill":45,"spawn":123,"status":251},
+      "fx": {"aoe":137,"bounce":314},
+      "snaps": "b054f186 5f1791df 3d017ddc 6caa5d05 8c321828 7e0d74c4 924d8c48 bc44fa99 448a38b4 f74013c9 0e587d4d 2185ab3b 9482eaf7 c6e6f386 7ce17fa6 af9ad4b3 97bb47c5 500c935b 824afe6d 6e3d1bcb"
     },
     "diy-028": {
       "about": "6_diy1_b=narant/S1/002_narant 6_diy2_b=narant/S2/002_narant 6_diy1_b=narant/S3/002_narant 6_diy2_b=narant/S1/003_narant 6_diy1_b=narant/S2/003_narant 6_diy2_b=narant/S3/003_narant 5_diy1_a=logos/S1 5_diy2_a=logos/S2 5_diy1_a=logos/S3 5_diy2_b=logos/S1 5_diy1_b=logos/S2 5_diy2_b=logos/S3",
diff --git a/test/golden/matches.json b/test/golden/matches.json
index 092f6a2f..a8c1f602 100644
--- a/test/golden/matches.json
+++ b/test/golden/matches.json
@@ -755,7 +755,7 @@
       "boost": null,
       "setup": {"stageId":"act2autochess_m03","bossId":"boss_4","hiddenBossId":"boss_9","factions":["FLY","TIMES","REFLECTION"]},
       "end": {"victory":false,"roundsPassed":11,"hiddenReached":false,"hiddenCleared":false,"reason":"eliminated","teamLp":null,"bossPoolLeft":null,"virtualMs":115700},
-      "rng": {"setup":25,"shop":525,"waves":0,"draft":75,"bots":24735,"meta":10},
+      "rng": {"setup":25,"shop":510,"waves":0,"draft":75,"bots":24735,"meta":8},
       "errors": {"engine":0,"logged":0,"sim":0,"dispatcher":0},
       "rounds": [
         {
@@ -808,27 +808,27 @@
         },
         {
           "round": 9,
-          "ai_0": [1,3,12,0,1,4,8,"47bf424b",157,"deputShip 2/24, indomShip 1/17, kazimierzShip 1/28, siracusaShip 0/24, skillfulShip 0/19, soloShip 1/14, suntShip 1/14, swiftShip 1/60",36,53,17,76],
+          "ai_0": [1,3,12,0,0,4,8,"e73e84c6",170,"deputShip 2/24, indomShip 1/29, kazimierzShip 1/29, siracusaShip 0/24, skillfulShip 0/19, soloShip 1/14, suntShip 1/14, swiftShip 1/60",32,48,16,76],
           "ai_1": [1,25,12,0,0,4,8,"fec52ed0",54,"arcaneShip 1/17, indomShip 1/18, kazimierzShip 0/7, siracusaShip 1/12, soloShip 1/0, suntShip 1/0, swiftShip 0/17, victoriaShip 1/7, visiShip 0/4, yanShip 0/2",19,43,24,76],
           "ai_2": [1,15,12,0,0,4,8,"549a87ca",142,"deputShip 1/12, indomShip 1/12, siracusaShip 1/88, skillfulShip 1/2, soloShip 1/18, suntShip 1/10, swiftShip 0/6, yanShip 0/1",52,59,7,80]
         },
         {
           "round": 10,
-          "ai_0": [0,0,13,0,0,5,0,"811c9dc5",0,"deputShip 0/28, indomShip 0/41, kazimierzShip 0/33, siracusaShip 0/24, skillfulShip 0/19, soloShip 0/18, suntShip 0/18, swiftShip 0/66",8,18,10,88],
+          "ai_0": [0,0,12,0,0,5,0,"811c9dc5",0,"deputShip 0/28, indomShip 0/53, kazimierzShip 0/34, siracusaShip 0/24, skillfulShip 0/19, soloShip 0/18, suntShip 0/18, swiftShip 0/66",8,18,10,86],
           "ai_1": [1,15,12,0,0,5,8,"fd30bf27",65,"arcaneShip 1/17, indomShip 1/24, kazimierzShip 0/7, siracusaShip 1/12, soloShip 1/0, suntShip 1/0, swiftShip 0/17, victoriaShip 1/12, visiShip 0/4, yanShip 0/2",8,18,10,88],
-          "ai_2": [1,10,12,0,0,5,8,"1a4bfd5d",172,"deputShip 1/14, indomShip 1/16, siracusaShip 1/106, skillfulShip 1/4, soloShip 1/20, suntShip 1/12, swiftShip 0/6, yanShip 0/1",13,18,5,91]
+          "ai_2": [1,10,12,0,2,5,8,"1a4bfd5d",172,"deputShip 1/14, indomShip 1/16, siracusaShip 1/106, skillfulShip 1/4, soloShip 1/20, suntShip 1/12, swiftShip 0/6, yanShip 0/1",13,18,5,92]
         },
         {
           "round": 11,
-          "ai_0": [0,0,0,0,0,5,0,"811c9dc5",0,"deputShip 0/28, indomShip 0/41, kazimierzShip 0/33, siracusaShip 0/24, skillfulShip 0/19, soloShip 0/18, suntShip 0/18, swiftShip 0/66",null,null,null,88],
-          "ai_1": [1,5,12,0,0,5,8,"6eebc22c",74,"arcaneShip 2/31, indomShip 1/28, kazimierzShip 0/11, siracusaShip 0/12, soloShip 1/0, suntShip 1/0, swiftShip 0/21, victoriaShip 1/15, visiShip 0/4, yanShip 0/2",5,25,20,101],
-          "ai_2": [1,3,12,0,0,5,8,"a5108c6b",202,"deputShip 1/16, indomShip 1/20, kazimierzShip 0/8, kjeragShip 0/8, siracusaShip 1/130, skillfulShip 0/4, soloShip 1/22, suntShip 2/14, swiftShip 0/14, yanShip 0/1",18,25,7,103]
+          "ai_0": [0,0,0,0,0,5,0,"811c9dc5",0,"deputShip 0/28, indomShip 0/53, kazimierzShip 0/34, siracusaShip 0/24, skillfulShip 0/19, soloShip 0/18, suntShip 0/18, swiftShip 0/66",null,null,null,86],
+          "ai_1": [1,5,12,0,1,5,8,"38b1f273",92,"arcaneShip 2/29, indomShip 1/38, kazimierzShip 0/7, siracusaShip 0/12, soloShip 1/0, suntShip 1/0, swiftShip 0/17, victoriaShip 1/25, visiShip 0/4, yanShip 0/2",7,25,18,99],
+          "ai_2": [1,4,14,0,0,5,8,"a5108c6b",208,"deputShip 1/16, indomShip 1/20, kazimierzShip 0/8, kjeragShip 0/8, siracusaShip 1/136, skillfulShip 0/4, soloShip 1/22, suntShip 2/14, swiftShip 0/14, yanShip 0/1",19,25,6,105]
         },
         {
           "round": 12,
-          "ai_0": [0,0,0,0,0,5,0,"811c9dc5",0,"deputShip 0/28, indomShip 0/41, kazimierzShip 0/33, siracusaShip 0/24, skillfulShip 0/19, soloShip 0/18, suntShip 0/18, swiftShip 0/66",null,null,null,88],
-          "ai_1": [0,0,12,0,0,5,0,"811c9dc5",0,"arcaneShip 0/45, indomShip 0/38, kazimierzShip 0/11, siracusaShip 0/12, swiftShip 0/21, victoriaShip 0/24, visiShip 0/4, yanShip 0/2",5,57,52,116],
-          "ai_2": [0,0,12,0,0,6,0,"811c9dc5",0,"deputShip 0/20, indomShip 0/20, kazimierzShip 0/8, kjeragShip 0/8, siracusaShip 0/136, skillfulShip 0/4, soloShip 0/26, suntShip 0/18, swiftShip 0/14, yanShip 0/1",13,57,44,116]
+          "ai_0": [0,0,0,0,0,5,0,"811c9dc5",0,"deputShip 0/28, indomShip 0/53, kazimierzShip 0/34, siracusaShip 0/24, skillfulShip 0/19, soloShip 0/18, suntShip 0/18, swiftShip 0/66",null,null,null,86],
+          "ai_1": [0,0,13,0,0,6,0,"811c9dc5",0,"arcaneShip 0/35, indomShip 0/48, kazimierzShip 0/7, siracusaShip 0/12, swiftShip 0/17, victoriaShip 0/34, visiShip 0/4, yanShip 0/2",5,57,52,110],
+          "ai_2": [0,0,12,0,0,6,0,"811c9dc5",0,"deputShip 0/20, indomShip 0/20, kazimierzShip 0/8, kjeragShip 0/8, siracusaShip 0/142, skillfulShip 0/4, soloShip 0/26, suntShip 0/18, swiftShip 0/14, yanShip 0/1",14,57,43,118]
         }
       ],
       "players": {
@@ -839,12 +839,12 @@
           "eliminatedRound": 10,
           "lp": 0,
           "bandId": "band_chiave",
-          "title": {"id":"comment_5","name":"万事俱备","picId":"comment_icon_5","text":"装备配发数量最多"},
+          "title": {"id":"comment_6","name":"挥金如土","picId":"comment_icon_6","text":"消耗资金最多"},
           "trophies": 2,
           "reward": 191,
-          "stats": {"activatedLayers":0,"bossDamage":0,"dmgDealt":1727766,"gold":88,"itemsEquipped":9,"kills":225,"leaks":38,"lpLost":30,"merges":4,"perfectRounds":6,"refreshes":13},
+          "stats": {"activatedLayers":0,"bossDamage":0,"dmgDealt":1678337,"gold":86,"itemsEquipped":8,"kills":221,"leaks":37,"lpLost":30,"merges":4,"perfectRounds":6,"refreshes":12},
           "lineup": [],
-          "bonds": ["deputShip 0/28","indomShip 0/41","kazimierzShip 0/33","siracusaShip 0/24","skillfulShip 0/19","soloShip 0/18","suntShip 0/18","swiftShip 0/66"]
+          "bonds": ["deputShip 0/28","indomShip 0/53","kazimierzShip 0/34","siracusaShip 0/24","skillfulShip 0/19","soloShip 0/18","suntShip 0/18","swiftShip 0/66"]
         },
         "ai_1": {
           "alive": false,
@@ -853,12 +853,12 @@
           "eliminatedRound": 12,
           "lp": 0,
           "bandId": "band_sarkazb",
-          "title": {"id":"comment_4","name":"精英云集","picId":"comment_icon_4","text":"晋升精锐次数最多"},
+          "title": {"id":"comment_5","name":"万事俱备","picId":"comment_icon_5","text":"装备配发数量最多"},
           "trophies": 2,
           "reward": 255,
-          "stats": {"activatedLayers":0,"bossDamage":0,"dmgDealt":2389171,"gold":116,"itemsEquipped":19,"kills":220,"leaks":120,"lpLost":50,"merges":7,"perfectRounds":6,"refreshes":12},
+          "stats": {"activatedLayers":0,"bossDamage":0,"dmgDealt":2293866,"gold":110,"itemsEquipped":10,"kills":222,"leaks":118,"lpLost":50,"merges":5,"perfectRounds":6,"refreshes":10},
           "lineup": [],
-          "bonds": ["arcaneShip 0/45","indomShip 0/38","kazimierzShip 0/11","siracusaShip 0/12","swiftShip 0/21","victoriaShip 0/24","visiShip 0/4","yanShip 0/2"]
+          "bonds": ["arcaneShip 0/35","indomShip 0/48","kazimierzShip 0/7","siracusaShip 0/12","swiftShip 0/17","victoriaShip 0/34","visiShip 0/4","yanShip 0/2"]
         },
         "ai_2": {
           "alive": false,
@@ -867,12 +867,12 @@
           "eliminatedRound": 12,
           "lp": 0,
           "bandId": "band_lmlee",
-          "title": {"id":"comment_6","name":"挥金如土","picId":"comment_icon_6","text":"消耗资金最多"},
+          "title": {"id":"comment_4","name":"精英云集","picId":"comment_icon_4","text":"晋升精锐次数最多"},
           "trophies": 2,
           "reward": 255,
-          "stats": {"activatedLayers":0,"bossDamage":0,"dmgDealt":3734992,"gold":116,"itemsEquipped":8,"kills":216,"leaks":100,"lpLost":34,"merges":6,"perfectRounds":2,"refreshes":16},
+          "stats": {"activatedLayers":0,"bossDamage":0,"dmgDealt":3450226,"gold":118,"itemsEquipped":9,"kills":218,"leaks":98,"lpLost":33,"merges":6,"perfectRounds":2,"refreshes":17},
           "lineup": [],
-          "bonds": ["deputShip 0/20","indomShip 0/20","kazimierzShip 0/8","kjeragShip 0/8","siracusaShip 0/136","skillfulShip 0/4","soloShip 0/26","suntShip 0/18","swiftShip 0/14","yanShip 0/1"]
+          "bonds": ["deputShip 0/20","indomShip 0/20","kazimierzShip 0/8","kjeragShip 0/8","siracusaShip 0/142","skillfulShip 0/4","soloShip 0/26","suntShip 0/18","swiftShip 0/14","yanShip 0/1"]
         }
       }
     },
diff --git a/test/sim/detmath.test.js b/test/sim/detmath.test.js
new file mode 100644
index 00000000..02d67c3f
--- /dev/null
+++ b/test/sim/detmath.test.js
@@ -0,0 +1,141 @@
+// server/sim/detmath.js — the sim's engine-independent math (hypot, powi, sin, cos, atan2). The functions use only
+// + − × ÷ and Math.sqrt, so their results are the same bits on every engine; the digests below pin them (the same
+// digests hold in Chrome, Firefox and Safari). Also: the special values of ECMA-262, powi correctly rounded and hypot
+// within 1 ulp against exact BigInt references, and no sim file calls an implementation-approximated Math function.
+import { test } from 'node:test';
+import assert from 'node:assert/strict';
+import { readdirSync, readFileSync } from 'node:fs';
+import { join, dirname, relative } from 'node:path';
+import { fileURLToPath } from 'node:url';
+import { hypot, powi, sin, cos, atan2 } from '../../server/sim/detmath.js';
+
+const ROOT = join(dirname(fileURLToPath(import.meta.url)), '../..');
+
+const lcg = (seed) => { let s = seed >>> 0; return () => { s = (Math.imul(s, 1664525) + 1013904223) >>> 0; return s / 4294967296; }; };
+const WORDS = new Uint32Array(2), F = new Float64Array(WORDS.buffer);
+/** FNV-1a over the IEEE 754 bits of the values */
+const digest = (values) => {
+  let h = 0x811c9dc5;
+  for (const v of values) { F[0] = v; for (const w of WORDS) for (let k = 0; k < 4; k++) { h ^= (w >>> (8 * k)) & 0xff; h = Math.imul(h, 16777619); } }
+  return (h >>> 0).toString(16).padStart(8, '0');
+};
+const N = 20000;
+
+test('the results are pinned bit for bit (engine-independent digests)', () => {
+  const runs = {
+    hypot: () => { const r = lcg(1); return Array.from({ length: N }, () => hypot((r() * 2 - 1) * 20, (r() < 0.3 ? 0 : (r() * 2 - 1) * 20))); },
+    powi: () => { const r = lcg(2); return Array.from({ length: N }, () => powi(r() * 3, Math.floor(r() * 41))); },
+    sin: () => { const r = lcg(3); return Array.from({ length: N }, () => sin((r() * 2 - 1) * 4 * Math.PI)); },
+    cos: () => { const r = lcg(4); return Array.from({ length: N }, () => cos((r() * 2 - 1) * 4 * Math.PI)); },
+    atan2: () => { const r = lcg(5); return Array.from({ length: N }, () => atan2((r() * 2 - 1) * 20, (r() * 2 - 1) * 20)); },
+  };
+  const want = { hypot: '6e3e3a7d', powi: 'f80a3234', sin: '73aedcb1', cos: '51b52432', atan2: 'a0ab93f0' };
+  for (const [name, run] of Object.entries(runs)) assert.equal(digest(run()), want[name], name);
+});
+
+test('values that tell the engines apart', () => {
+  // V8's Math.hypot(3, 2) is 3.6055512754639896; correctly rounded √13 is 3.605551275463989 (also Firefox, Safari)
+  assert.equal(hypot(3, 2), 3.605551275463989);
+  // fdlibm (Node 24–26); Chrome 154's correctly rounded Math.sin / Math.atan2 give 0.7931254945086103 / 0.6023179867387402
+  assert.equal(sin(2.22566898269194), 0.7931254945086104);
+  assert.equal(cos(0.74090688893123), 0.7378567508940902);
+  assert.equal(atan2(0.5098986645868919, 0.74162210852462), 0.6023179867387403);
+  // correctly rounded; Safari's Math.pow squares and multiplies in double and gives 2.593742460100002
+  assert.equal(powi(1.1, 10), 2.5937424601000023);
+  assert.equal(powi(0.85, 3), 0.6141249999999999);
+});
+
+test('special values follow ECMA-262', () => {
+  const same = (a, b, msg) => assert.ok(Object.is(a, b), `${msg}: ${a} vs ${b}`);
+  for (const [x, y, want] of [[Infinity, NaN, Infinity], [NaN, -Infinity, Infinity], [NaN, 1, NaN], [0, 0, 0], [-0, -0, 0], [0, -2, 2], [-3, 0, 3]]) same(hypot(x, y), want, `hypot(${x}, ${y})`);
+  for (const x of [0, -0]) { same(sin(x), x, 'sin ±0'); same(cos(x), 1, 'cos ±0'); }
+  for (const x of [Infinity, -Infinity, NaN]) { same(sin(x), NaN, `sin(${x})`); same(cos(x), NaN, `cos(${x})`); }
+  const P = Math.PI, H = Math.PI / 2, Q = Math.PI / 4;
+  const table = [
+    [0, 1, 0], [-0, 1, -0], [0, -1, P], [-0, -1, -P], [0, 0, 0], [-0, 0, -0], [0, -0, P], [-0, -0, -P],
+    [1, 0, H], [-1, 0, -H], [1, -0, H], [-1, -0, -H],
+    [1, Infinity, 0], [-1, Infinity, -0], [1, -Infinity, P], [-1, -Infinity, -P],
+    [Infinity, 1, H], [-Infinity, 1, -H], [Infinity, Infinity, Q], [-Infinity, Infinity, -Q], [Infinity, -Infinity, 3 * Q], [-Infinity, -Infinity, -3 * Q],
+    [NaN, 1, NaN], [1, NaN, NaN], [1, 1, Q], [-1, -1, -3 * Q],
+  ];
+  for (const [y, x, want] of table) same(atan2(y, x), want, `atan2(${y}, ${x})`);
+  same(powi(2, 0), 1, 'x^0'); same(powi(0, 0), 1, '0^0'); same(powi(0, 3), 0, '0^3'); same(powi(-2, 3), -8, '(-2)^3');
+  same(powi(2, -2), 0.25, 'x^-n'); same(powi(Infinity, 2), Infinity, '∞²'); same(powi(NaN, 0), 1, 'NaN^0');
+});
+
+// exact references: x = M·2^E with M an integer
+const DV = new DataView(new ArrayBuffer(8));
+const decompose = (x) => {
+  DV.setFloat64(0, Math.abs(x));
+  const hiw = DV.getUint32(0), low = DV.getUint32(4), be = (hiw >>> 20) & 0x7ff;
+  const M = (BigInt(hiw & 0xfffff) << 32n) | BigInt(low);
+  return be === 0 ? [M, -1074] : [M | (1n << 52n), be - 1075];
+};
+const bitlen = (n) => n.toString(2).length;
+/** round M·2^e (M > 0) to the nearest double, ties to even; `sticky` = the true value is a little above M·2^e */
+const roundBig = (M, e, sticky = false) => {
+  const L = bitlen(M);
+  if (L <= 53 && !sticky) return Number(M) * 2 ** e;
+  const sh = BigInt(Math.max(0, L - 53));
+  let q = M >> sh;
+  const rem = M & ((1n << sh) - 1n), half = sh > 0n ? 1n << (sh - 1n) : 0n;
+  if (rem > half || (rem === half && (sticky || (q & 1n)))) q++;
+  return Number(q) * 2 ** (e + Number(sh));
+};
+const isqrt = (n) => { if (n < 2n) return n; let x = BigInt(Math.floor(Math.sqrt(Number(n)))); for (;;) { const y = (x + n / x) >> 1n; if (y >= x) { while (x * x > n) x--; while ((x + 1n) * (x + 1n) <= n) x++; return x; } x = y; } };
+const exactHypot = (x, y) => {
+  if (x === 0 || y === 0) return Math.abs(x || y);
+  const [mx, ex] = decompose(x), [my, ey] = decompose(y), E = Math.min(ex, ey);
+  const S = (mx * mx << BigInt(2 * (ex - E))) + (my * my << BigInt(2 * (ey - E)));
+  const k = Math.max(0, Math.ceil((130 - bitlen(S)) / 2));
+  const q = isqrt(S << BigInt(2 * k));
+  return roundBig(q, E - k, q * q !== S << BigInt(2 * k));
+};
+const exactPowi = (x, n) => { if (n === 0) return 1; const [M, E] = decompose(x); return roundBig(M ** BigInt(n), E * n); };
+const ORD = new BigInt64Array(F.buffer);
+const ulps = (a, b) => { F[0] = a; const ia = ORD[0]; F[0] = b; return Number(ia - ORD[0]); };
+
+test('powi is correctly rounded (exact BigInt reference)', () => {
+  for (let b = 1; b <= 300; b++) for (let n = 0; n <= 24; n++) {
+    const x = b / 100;
+    assert.equal(powi(x, n), exactPowi(x, n), `${x}^${n}`);
+  }
+});
+
+test('hypot is within 1 ulp of √(x² + y²), and symmetric', () => {
+  const r = lcg(7);
+  for (let i = 0; i < 3000; i++) {
+    const x = (r() * 2 - 1) * 20, y = (r() * 2 - 1) * 20;
+    assert.ok(Math.abs(ulps(hypot(x, y), exactHypot(x, y))) <= 1, `hypot(${x}, ${y})`);
+    assert.equal(hypot(x, y), hypot(y, x)); assert.equal(hypot(-x, y), hypot(x, y));
+  }
+  for (const s of [1e300, 1e-300]) assert.ok(Math.abs(ulps(hypot(3 * s, 4 * s), 5 * s)) <= 1, `rescaled ${s}`);
+});
+
+test('sin, cos and atan2 stay within 2 ulp of the engine and keep their symmetries', () => {
+  const r = lcg(9);
+  for (let i = 0; i < 20000; i++) {
+    const x = (r() * 2 - 1) * 10;
+    assert.ok(Math.abs(ulps(sin(x), Math.sin(x))) <= 2 && Math.abs(ulps(cos(x), Math.cos(x))) <= 2, `x = ${x}`);
+    assert.ok(Object.is(sin(-x), -sin(x)) && Object.is(cos(-x), cos(x)), `symmetry at ${x}`);
+    const y = (r() * 2 - 1) * 10;
+    assert.ok(Math.abs(ulps(atan2(y, x), Math.atan2(y, x))) <= 2, `atan2(${y}, ${x})`);
+    assert.ok(Object.is(atan2(-y, x), -atan2(y, x)), `atan2 odd in y at (${y}, ${x})`);
+  }
+});
+
+test('the sim never calls an implementation-approximated Math function', () => {
+  const APPROX = /\bMath\s*\.\s*(acos|acosh|asin|asinh|atan|atanh|atan2|cbrt|cos|cosh|exp|expm1|hypot|log|log1p|log10|log2|pow|sin|sinh|tan|tanh)\b/;
+  const files = [];
+  const walk = (dir) => { for (const e of readdirSync(dir, { withFileTypes: true })) { const p = join(dir, e.name); if (e.isDirectory()) walk(p); else if (e.name.endsWith('.js')) files.push(p); } };
+  walk(join(ROOT, 'server/sim'));
+  assert.ok(files.length > 50);
+  const bad = [];
+  for (const f of files) {
+    const rel = relative(ROOT, f).replaceAll('\\', '/');
+    if (rel === 'server/sim/detmath.js') continue;
+    const code = readFileSync(f, 'utf8').replace(/\/\*[\s\S]*?\*\//g, '').replace(/(^|[^:'"`])\/\/.*$/gm, '$1');
+    if (APPROX.test(code)) bad.push(`${rel}: ${code.match(APPROX)[0]}`);
+  }
+  assert.deepEqual(bad, [], 'use server/sim/detmath.js');
+});