feedback7 R23-TICK: an attack interval or a time-SP cost of a whole number of frames takes exactly that many — 1 s is 30 frames, not 31 (PR #402 by @Cloudnyco, ported onto 0.2.2); golden: every family moves (roster, bonds, fields, matches, standins, diy — regenerated in the next commit)

Seen (PR #402, measured again on 0.2.2): TICK = 1/30 is not a binary fraction. An attack cooldown counted down from 1 s
stops at 2.1e-16 after thirty steps and the fire check `atkCd > 0` waited a 31st frame; 300 time gains of 1/30 SP add up
to 9.999999999999975 and `sp >= cost` charged a 10-SP skill a frame late. Over the 265 golden battles 30.85 % of the
consecutive attacks on a whole-frame interval came a frame late (control, other intervals: 0.09 %), and 4 157
end-of-tick states sat within 1e-9 under a cost. The engine already took timers within 1e-9 as done (skill timeLeft,
buffs, every() / after(), enemy abilities, nextCastAt, the 失衡 end); these two did not.

Official: PRTS 作战机制/sandbox 「帧对齐机制对攻速的影响」 — one frame = 1/30 s; since 2019-12-24 a non-integer frame count
is rounded (四舍五入); an exact 1 s is 30 frames under any rule.

Now: ai.js attackCountdown(cd, dt) (left within 1e-9 of 0 is 0) for the operators' cooldown (the canAct guard kept) and
the enemies' (not while stunned or hidden; the 0.2.2 失衡 swing cut untouched) — an enemy's whole-frame wind-up is exact
too (0.4 s: 12 frames, 13 before). The same countdown for the two strict `cd -= dt; if (cd > 0)` attack clocks of the
same shape: the 双眼皮 turrets (content/devices.js; BAT 4 s: 121 → 120 frames) and 寒檀's S2 icicles (0.5 s: 16 → 15).
SkillRuntime.gainSp takes SP within 1e-9 of the cost as full for every charge (a tiny negative remainder → 0);
_normalize compares the same way. Left as they are: a non-integer count still ends on the frame that crosses 0
(⌈30 × interval⌉ — the official 四舍五入 is an older, separate difference); the count-up content clocks that restart from
0 (bosses.js self-timers, kit auras — PR #402's list); attack / hurt SP (+1 on an integer cost) unchanged.

Tests: test/sim/tick-tolerance.test.js (new, 7: the PR's four adapted — residue, attackCountdown over bat 0.1–10 s ×
14 ASPD = ⌈30 × interval⌉, 1 s operator every 30 / 3 s ranged enemy every 90, time SP 4/10/16/20/30/50 at exactly 30 × N
gains for one and two charges — plus the 1.2 s / 1.25 s control (36 / 38), the turret at 0 layers (120) and 寒檀's
icicles (15); the four engine tests fail on the parent: 31 / 91, cost 4 at 121, 121, 16). The PR's seven test edits,
re-run: the same 8 assertions failed on the new timing and each edit passes on the parent and on the fix —
feedback2-surtr-stealth (count once no arrow is in flight), kits_alt_t2 拉普兰德 S1 (dummy ATK 300 → 250: 120 × 15 = her
1791 HP by 59.5 s), kits_alt_t6 (medic smoke at half HP; 余 S1 up to the first 燃烧 burst), op_jesca2 (the T6 shield
returns a tick after the cast), op_lmlee (hits before the burst), op_logos (count after the clear), op_narant (the first
attack thrown after the new enemies). node --test (with the golden of the next commit): 5703 tests, 5701 pass, 0 fail,
2 skipped; eslint 0 errors (135 warnings, the parent's 135); typecheck, check:imports, smoke ok.

feedback7 R23-TICK: golden for whole-frame attack intervals and SP costs (PR #402) — 281 of 283 scenarios move: roster 49/49, bonds 46/46, fields 22/22, matches 18/18, standins 10/10, diy 136/138; solo-NORMAL-1 now loses its Final Assault (below)

npm run golden:update on the previous commit. Every family moves because both sides attack on time (more enemy kinds
than operator forms sit on whole-frame intervals: 3 s, 4 s, 1 s …) and time SP charges a frame sooner. Measured over
the 265 battles: whole-frame consecutive attacks a frame late 30.85 % → 0.04 % (control 0.09 % → 0.13 %; 1 s gaps at
30 frames 256 → 3 776 of ~3 800, 3 s at 90 frames 0 → 6 657 of 6 657), SP float-short states 4 157 → 0. Moved battles'
totals: kills 10 647 → 10 625, leaks 12 398 → 12 445, deaths 2 913 → 2 944. Unchanged: diy-040, diy-069.

End results that flip (one frame cascades; the causes as measured with the matches' battles instrumented):
- solo-NORMAL-1: won → lost in round 14 (passed 14 → 13, team LP 8 → 0): the Final Assault on 卢西恩 (pool 390 000) runs
  as before for two minutes (his 0.4 s wind-up a frame shorter), then the team's damage on the pool trails — 884 left
  when the overtime drain (from 300 game s) takes the last of the 17 LP at 334 s; before, it emptied at 318 s, 8 LP left.
- solo-HARD-1: LP 10 → 12 — round 8 lets one more through (26 → 25; rounds 11–12 even it out) and the Final Assault ends
  at 300.6 s, before the overtime drain (304.8 s and 2 LP before).
- solo-NORMAL-2: LP 9 → 25 — round 5 lets one more through (26 → 25, a coin less), the bot's shop path parts from round
  6 and its later board (突袭, 232 layers at round 13 against 94) clears rounds 12–13 with no leak (7 + 18 before).
- coop2-NORMAL-3: team LP 63 → 68 — ai_1's round-13 battle leaks 1 instead of 6 (its round-12 leaker now goes to 联防).
- coop2-ABYSS-10-boosted: team LP 583 → 630 — the Final Assault ends at 606.6 s (639.6) and the Hidden Core at 335.9 s
  (397.8): 47 LP less of overtime drain.
- coop2-NORMAL-14-standins: team LP 41 → 40 — ai_0's round-13 battle lets one 潜伏者 through (23 → 22).
- roster-016: cleared at 220.3 s → time-out at 230 s (主角阵营角色 leaks instead of falling; kills 101 → 105, leaks 26 → 22).
- roster-041: time-out → cleared at 189.0 s, perfect (the last 粉碎攻坚组长 falls inside the 190 s).
- bond-suntShip-high: cleared at 164.2 s → time-out at 190 s (kills 11 → 13, deaths 4 → 2).
- standin-04: perfect → 10 leaks (deaths 8 → 10; 萨卡兹大剑组长 / 萨卡兹穿刺手 / 澪 get through).
- diy-005 / diy-008 (艾丽妮 / 赫拉格): cleared at 225.3 s → time-out at 230 s (a last 宿主重装士兵 / 深池方阵指挥官 walking).
- diy-127 (白铁 / 酒神): time-out → cleared at 225.8 s (deaths 7 → 13, leaks 57 → 72).
- diy-131 (死芒): perfect → 1 leak (a 掠海漂移体).
The other matches keep their ends (solo-FUNNY-1/2, solo-HARD-2, solo-ABYSS-1/2, coop2-HARD-4, coop3-ABYSS-5,
coop4-FUNNY-6, coop4-ABYSS-7, coop2-NORMAL-8-serverrun, solo-HARD-9-boosted, coop2-NORMAL-70-diy). PR #402's own
0.2.1 table (coop2-NORMAL-14-standins loss → win, solo-HARD-2 / solo-ABYSS-2 earlier exits) was measured before §27 and
does not repeat here.

server/sim/ai.js

@@ -75,8 +75,19 @@ export function effectiveProfile(u) {
 // ---------------------------------------------------------------------------------------------------------------
 // ally attack loop
 
+/**
+ * One tick off an attack cooldown. What is left within 1e-9 of 0 is 0 (the engine's timer tolerance, as skills.js
+ * `timeLeft`, buff intervals and `every()`): 1 s counted down in thirty steps of 1/30 leaves 2.1e-16 in floating point,
+ * which held every attack whose interval is a whole number of ticks — 1 s, 3 s, 4 s … — one tick longer (PR #402; PRTS
+ * 作战机制 帧对齐: 1 s is 30 frames). A non-integer count still ends on the tick that crosses 0 (docs/SIM.md §2).
+ */
+export function attackCountdown(cd, dt) {
+  const left = cd - dt;
+  return left > 1e-9 ? left : 0;
+}
+
 export function updateAlly(b, u, dt) {
-  if (u.atkCd > 0 && u.canAct) u.atkCd = Math.max(0, u.atkCd - dt);
+  if (u.atkCd > 0 && u.canAct) u.atkCd = attackCountdown(u.atkCd, dt);
   if (u.blocking.length) enforceBlockCapacity(b, u);
   if (!u.canAct || !u.profile) return;
   // a rangeExtend change (buff added / expired) rebuilds the range — also for units that never attack (auras)
@@ -542,7 +553,7 @@ export function updateEnemy(b, e, dt) {
   if (e.s.flags.levitate && e.unbalanceUntil > b.time) e.unbalanceUntil = -Infinity;
   const unbalanced = b.time < e.unbalanceUntil - 1e-9;   // (the 1e-9: a float-summed end such as 35/30 s keeps its frame)
   const prevCd = e.atkCd;
-  if (e.atkCd > 0 && !stunned && !e.hidden) e.atkCd = Math.max(0, e.atkCd - dt);
+  if (e.atkCd > 0 && !stunned && !e.hidden) e.atkCd = attackCountdown(e.atkCd, dt);
   // a stun / freeze / sleep / 浮空 / 失衡 — or leaving the field — takes the enemy out of its attack: a swing short of its
   // damage frame does not land, and the attack starts again from its wind-up afterwards (enemyAttack)
   if (e.swing && (stunned || e.hidden || unbalanced)) e.swing = false;

server/sim/content/devices.js

@@ -34,7 +34,7 @@
 // keeps those tiles NONE (grid.js DEPLOY_REFUSED_TILES), so automatic placements never use them [ASSUMED].
 
 import { COLS, ROWS } from '../constants.js';
-import { performAttack } from '../ai.js';
+import { performAttack, attackCountdown } from '../ai.js';
 import { sortEnemyTargets } from '../targeting.js';
 import { DIR_VEC, normDir, oppositeDir } from '../dir.js';
 import { hypot } from '../detmath.js';
@@ -241,7 +241,7 @@ const TURRET_PROFILE = Object.freeze({ attack: 'ranged', dmgType: 'arts', projec
 function tickTurret(battle, u, dt) {
   const T = u.mem.turret;
   if (!u.alive || !T) return;
-  T.cd -= dt;
+  T.cd = attackCountdown(T.cd, dt);   // (the engine's attack countdown: a whole number of ticks takes exactly that many)
   if (T.cd > 0 || u.s.flags.stun) return;
   const L = u.ownerId != null ? topLayers(battle, u.ownerId) : 0;
   const bonus = Math.min(L * T.aspdPer, T.aspdMax);

server/sim/content/kits/ops/chess_char_5_21-sntlla.js

@@ -2,6 +2,7 @@
 // Conventions of the tier-5 kits: ../shared/tier5.js; kit contract and rules: ../README.md.
 
 import { COLS } from '../../../constants.js';
+import { attackCountdown } from '../../../ai.js';
 import { toLocal } from '../../../dir.js';
 import { num, talent, skillGrid, batPct, mods, lazySkills, whileOn } from '../shared/tier5.js';
 
@@ -33,7 +34,7 @@ export default {
         attack: { noAttack: true },
         onStart({ unit }) { unit.mem.iceCd = 0; unit.mem.iceRow = 0; },
         onTick({ battle, unit, dt }) {
-          unit.mem.iceCd -= dt;
+          unit.mem.iceCd = attackCountdown(unit.mem.iceCd, dt);   // (the engine's attack countdown, PR #402)
           if (unit.mem.iceCd > 0 || !unit.canAct || unit.s.flags.disarm) return;
           if (!battle.enemiesInKeys(unit.rangeKeys, unit, { canHitFly: true }).length) return;
           unit.mem.iceCd = unit.s.interval;

server/sim/skills.js

@@ -338,9 +338,11 @@ export class SkillRuntime {
     }
     if (cost <= 0) return 0; // free skills never recharge (see reset)
     this.sp += amt;
-    while (this.sp >= cost && this.charges < this.maxCharges) {
+    // within 1e-9 of the cost is the cost (the timer tolerance of `timeLeft` below; PR #402): 300 time gains of 1/30 SP
+    // add up to 9.999999999999975 in floating point, which held a 10-SP charge one tick
+    while (this.sp > cost - 1e-9 && this.charges < this.maxCharges) {
       this.charges++;
-      if (this.charges < this.maxCharges) this.sp -= cost;
+      if (this.charges < this.maxCharges) this.sp = Math.max(0, this.sp - cost);
       else this.sp = cost;
     }
     if (this.charges >= this.maxCharges) this.sp = cost;
@@ -352,9 +354,9 @@ export class SkillRuntime {
     if (this.noSkill || this.kind === 'passive') return;
     const cost = this.spCost;
     if (cost <= 0) { if (this.sp > 0) { this.sp = 0; this.charges = this.maxCharges; } return; } // became free
-    while (this.sp >= cost && this.charges < this.maxCharges) {
+    while (this.sp > cost - 1e-9 && this.charges < this.maxCharges) {   // (the 1e-9 of gainSp)
       this.charges++;
-      this.sp = this.charges < this.maxCharges ? this.sp - cost : cost;
+      this.sp = this.charges < this.maxCharges ? Math.max(0, this.sp - cost) : cost;
     }
     if (this.sp > cost) this.sp = cost;
   }