Challenges/Stadium Exit Gate Bottleneck/Run

Run Detail

completedclaude-opus-5.5

30bad

Every model's answer from this round

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Provider

openrouter

Challenge Version

v1.0

Turns

Duration

Batched — not timed

Total Tokens

55,520

Input Tokens

738

Output Tokens

54,782

Reasoning Tokens

32,207

Reasoning Effort

default

Est. Cost

$0.5493

Started

10/26/2026, 2:59:43 PM

OR Provider

Anthropic

Finish Reason

stop

Generation ID

gen-batch-1793026782-752dd7088d49466641f7

Turns (1)

Turn #1

738 in54,782 out32,207 reasoning55,520 total

user

You write crowd simulations where the behavior emerges from rules each person follows.

Build me a single HTML page: a section of a football stadium emptying through one exit gate.

The build

Top-down view. At least 200 people start in the stands and head for a single gate in a wall.
Every person runs the same rules: move toward the gate, keep some personal space, and push physically against whoever and whatever is in the way. Nobody overlaps anybody.
Urgency control from 0.0 to 1.0, scaling desired speed and contact force.
Gate width control in yards.
A toggle that places a single column a short distance in front of the gate.
On-screen metrics: outflow rate in people per minute measured from the people actually leaving, and a timer for how long the section takes to empty.
A reset control that refills the stands.
Loading the page as #urgency=0.8&gate=1.2&column=1 starts simulation at load with those settings: urgency 0.0 to 1.0, gate width in yards, column 0 or 1. Any values work.
The art is yours.

House rules

One file. Everything inline — no CDN, no external stylesheet, font, script or image. data: URIs are fine.
No libraries. No physics engine — write your own.
No network at runtime.
No localStorage, sessionStorage, cookies or indexedDB — sandboxed frame, they throw.
Return the file as exactly one fenced html block.

Answer in exactly these sections

The prediction

Before the file, two quantitative predictions: whether increasing urgency from 0.3 to 0.9 increases or decreases evacuation time at gate=1.2 and column=0, with the predicted difference in seconds; and whether enabling the column (column=1) increases or decreases evacuation time at urgency=0.9 and gate=1.2, with the predicted difference in seconds.

The page

The single fenced html block.

What it actually does

Where your crowd behaves like a real one and where it does not. Name anything you special-cased: a nudge that breaks a jam, a speed cap at the gate, an allowance for overlap. If your page never jams at the gate, say so.

assistant

The prediction

These numbers were written before running anything. They are not measured.

Urgency 0.3 → 0.9, at gate=1.2, column=0: evacuation time increases, by about +40 s. My estimate is roughly 125 s at urgency 0.3 and roughly 165 s at urgency 0.9, for 220 people through a 1.10 m opening. The reason is the "faster-is-slower" effect. At high urgency, people press into each other at the opening. The sideways friction between bodies forms arches that stall for a few seconds and then burst. At urgency 0.3 people keep a gap, rarely touch at the gate, and flow steadily.

Column on, at urgency=0.9, gate=1.2: evacuation time decreases, by about −20 s (roughly 165 s → 145 s). The column takes the crowd's push off the people standing in the opening, so arches there form less often. I'm less confident in this one than in the first. The column also adds a second bottleneck. In real human trials the effect is weaker and less consistent than in simulations.

The page has a "Check predictions" button. It runs these three setups headless, three seeds each, and reports the measured differences. Trust that over the numbers above.

The page

<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<title>Section 114 · Gate C — stand evacuation</title>
<style>
:root{--bg:#0e1217;--panel:#161b22;--ink:#e6e9ee;--dim:#8b95a3;--acc:#ffcf3f;--line:#28303a}
*{box-sizing:border-box}
html,body{margin:0;height:100%;background:var(--bg);color:var(--ink);font:13px/1.35 system-ui,-apple-system,"Segoe UI",Roboto,sans-serif}
#app{display:flex;height:100vh}
#stage{flex:1;position:relative;min-width:0}
#view{position:absolute;inset:0;width:100%;height:100%;display:block}
#panel{width:330px;flex:none;background:var(--panel);border-left:1px solid var(--line);overflow-y:auto;padding:14px 16px}
h1{font-size:15px;margin:0 0 2px;letter-spacing:.05em;text-transform:uppercase}
h2{font-size:12px;margin:16px 0 4px;letter-spacing:.06em;text-transform:uppercase;color:var(--dim)}
.sub{color:var(--dim);font-size:12px;margin-bottom:10px}
.ctl{margin:10px 0}
.ctl label{display:flex;justify-content:space-between;color:var(--dim);font-size:12px;margin-bottom:4px}
.ctl label b{color:var(--ink);font-weight:600}
input[type=range]{width:100%;accent-color:var(--acc)}
.row{display:flex;gap:8px;align-items:center;flex-wrap:wrap}
button,select,input[type=number]{background:#222a34;color:var(--ink);border:1px solid #343d49;border-radius:6px;padding:6px 10px;font:inherit}
button{cursor:pointer}
button:hover{border-color:var(--acc)}
button.primary{background:var(--acc);color:#1a1a1a;border-color:var(--acc);font-weight:600}
button:disabled{opacity:.5;cursor:default}
input[type=number]{width:84px}
.metrics{display:grid;grid-template-columns:1fr auto;gap:4px 10px;margin:12px 0;padding:10px;background:#10141a;border-radius:8px;border:1px solid var(--line)}
.metrics div:nth-child(odd){color:var(--dim)}
.metrics div:nth-child(even){text-align:right;font-variant-numeric:tabular-nums;font-weight:600}
.metrics .big{font-size:20px}
#chart{width:100%;height:120px;display:block;background:#10141a;border-radius:8px;border:1px solid var(--line)}
.hash{font-family:ui-monospace,Menlo,Consolas,monospace;font-size:11px;color:var(--dim);word-break:break-all;margin-top:4px}
table{width:100%;border-collapse:collapse;font-size:12px;margin-top:6px}
td,th{padding:3px 4px;border-bottom:1px solid var(--line);text-align:left;font-variant-numeric:tabular-nums}
th{color:var(--dim);font-weight:500}
.legend{display:flex;align-items:center;gap:6px;font-size:11px;color:var(--dim);margin-top:4px}
.legend .bar{flex:1;height:8px;border-radius:4px;background:linear-gradient(90deg,#3a7bd5,#f5c542,#e8402f,#ff2bd6)}
.note{font-size:11px;color:var(--dim)}
@media (max-width:820px){#app{flex-direction:column;height:auto}#stage{height:72vh}#panel{width:auto;border-left:0}}
</style>
</head>
<body>
<div id="app">
  <div id="stage"><canvas id="view"></canvas></div>
  <aside id="panel">
    <h1>Section 114 · Gate C</h1>
    <div class="sub">Social-force crowd, one exit. Every spectator runs the same rules.</div>

    <div class="ctl">
      <label>Urgency <b id="urgLbl">0.50</b></label>
      <input id="urg" type="range" min="0" max="1" step="0.01" value="0.5">
      <div class="note" id="urgInfo"></div>
    </div>
    <div class="ctl">
      <label>Gate width (yards) <b id="gateLbl">1.20 yd</b></label>
      <div class="row"><input id="gateR" type="range" min="0.5" max="4" step="0.05" value="1.2" style="flex:1"><input id="gateN" type="number" step="0.05" min="0.1" value="1.2"></div>
    </div>
    <div class="ctl row">
      <label style="display:flex;gap:6px;align-items:center;color:var(--ink);margin:0"><input id="col" type="checkbox"> Column in front of gate</label>
    </div>
    <div class="ctl row">
      <button id="reset" class="primary">Reset / refill stands</button>
      <button id="pause">Pause</button>
      <select id="speed"><option value="1">1×</option><option value="2">2×</option><option value="4">4×</option><option value="8">8×</option></select>
    </div>
    <div class="ctl row">
      <label style="margin:0;color:var(--dim)">View</label>
      <select id="viewMode"><option value="kits">Kits + crush halo</option><option value="pressure">Contact pressure</option></select>
    </div>
    <div class="legend"><span>0</span><span class="bar"></span><span>3 kN/m</span></div>
    <div class="note">Gate width and column restart the run. Urgency applies live.</div>
    <div class="hash" id="hashTxt"></div>

    <div class="metrics">
      <div>Clock</div><div class="big" id="mT">0.0 s</div>
      <div>Section empty after</div><div class="big" id="mEvac">—</div>
      <div>Still inside</div><div id="mIn">—</div>
      <div>Through the gate</div><div id="mOut">—</div>
      <div>Outflow, last 10 s</div><div id="mRate">—</div>
      <div>Outflow, mean (first→last exit)</div><div id="mMean">—</div>
      <div>Longest gap between exits</div><div id="mGap">—</div>
      <div>Stalls ≥ 2 s</div><div id="mStall">—</div>
      <div>Max body compression now / ever</div><div id="mPen">—</div>
      <div>Overlap-cap corrections</div><div id="mProj">—</div>
    </div>
    <canvas id="chart"></canvas>
    <div class="note">Bars: people/min per 5 s bin. Line: cumulative exits.</div>

    <h2>Check the predictions</h2>
    <div class="note">Runs headless: gate 1.2 yd; (A) urgency 0.3, (B) urgency 0.9, (C) urgency 0.9 + column. 3 seeds each.</div>
    <div class="row" style="margin-top:6px"><button id="expBtn">Check predictions</button><span class="note" id="expProg"></span></div>
    <div id="expOut"></div>
  </aside>
</div>
<script>
(function(){
'use strict';
// ---------- world constants (metres, seconds, kg, newtons) ----------
const YD=0.9144;
const W=24, WALL_Y=16, WALL_T=0.5, OUT_Y=WALL_Y+WALL_T, H=21;
const STAND_Y0=0.5, STAND_Y1=10.1, ROW=0.8, AISLES=[6,12,18];
const DT=0.004, MASS=80, TAU=0.5;
const A_SOC=2000, A_WALL=2000, B_WALL=0.08, K_BODY=1.2e5, KAPPA0=2.4e5, LAMBDA=0.4;
const CUT=1.1, CUT2=CUT*CUT, WALL_CUT=0.9;
const MAX_PEN=0.03, VMAX=8;
const COL_R=0.35, COL_DIST=1.4;
const CS=0.1, GC=1.1;
const T_LIMIT=1200;
function urgencyParams(u){
  return { v0:0.6+3.4*u, B:0.08+0.08*(1-u), kappa:KAPPA0*(0.5+0.5*u) };
}
function mulberry32(a){return function(){a|=0;a=a+0x6D2B79F5|0;let t=Math.imul(a^a>>>15,1|a);t=t+Math.imul(t^t>>>7,61|t)^t;return((t^t>>>14)>>>0)/4294967296;};}
const KITS=['#1f5fd1','#1f5fd1','#1f5fd1','#f1f1ee','#173f96','#1f5fd1','#e9e4d4','#2b2f36','#c8372d','#3d8a4b','#d9a62e','#7a4bb0','#1f5fd1','#f1f1ee'];
const SKIN=['#f1c9a5','#d9a47a','#b07a52','#7d5236','#5a3a26','#e8b994'];

// ---------- simulation ----------
class Sim{
  constructor(cfg){
    this.cfg=Object.assign({},cfg);
    this.u=cfg.u;
    this.buildGeometry();
    this.buildField();
    this.spawn();
  }
  buildGeometry(){
    const g=Math.min(W-0.4,Math.max(0.05,this.cfg.gateYd*YD));
    this.gateW=g; this.gateL=W/2-g/2; this.gateR=W/2+g/2;
    const R=[[-3,-3,W+3,0],[-3,-3,0,OUT_Y],[W,-3,W+3,OUT_Y],[-3,WALL_Y,this.gateL,OUT_Y],[this.gateR,WALL_Y,W+3,OUT_Y]];
    this.rects=R;
    this.rectArr=new Float64Array(R.length*4);
    R.forEach((q,k)=>{for(let m=0;m<4;m++)this.rectArr[4*k+m]=q[m];});
    this.col=this.cfg.column?{x:W/2,y:WALL_Y-COL_DIST,r:COL_R}:null;
  }
  solidAt(x,y,m){
    for(const q of this.rects){if(x>q[0]-m&&x<q[2]+m&&y>q[1]-m&&y<q[3]+m)return true;}
    if(this.col){const dx=x-this.col.x,dy=y-this.col.y;if(dx*dx+dy*dy<(this.col.r+m)*(this.col.r+m))return true;}
    return false;
  }
  buildField(){
    const nx=Math.ceil(W/CS), ny=Math.ceil(H/CS), N=nx*ny;
    this.fnx=nx; this.fny=ny;
    const blk=new Uint8Array(N), dist=new Float32Array(N).fill(Infinity);
    for(let j=0;j<ny;j++)for(let i=0;i<nx;i++){if(this.solidAt((i+.5)*CS,(j+.5)*CS,0.05))blk[j*nx+i]=1;}
    const hk=[],hv=[];
    const push=(k,v)=>{hk.push(k);hv.push(v);let i=hk.length-1;while(i>0){const p=(i-1)>>1;if(hk[p]<=hk[i])break;let t=hk[p];hk[p]=hk[i];hk[i]=t;t=hv[p];hv[p]=hv[i];hv[i]=t;i=p;}};
    let popK=0;
    const pop=()=>{const v=hv[0];popK=hk[0];const lk=hk.pop(),lv=hv.pop();if(hk.length){hk[0]=lk;hv[0]=lv;let i=0;const L=hk.length;for(;;){const l=2*i+1,r=l+1;let m=i;if(l<L&&hk[l]<hk[m])m=l;if(r<L&&hk[r]<hk[m])m=r;if(m===i)break;let t=hk[m];hk[m]=hk[i];hk[i]=t;t=hv[m];hv[m]=hv[i];hv[i]=t;i=m;}}return v;};
    for(let j=0;j<ny;j++){if((j+.5)*CS>H-0.6){for(let i=0;i<nx;i++){const k=j*nx+i;if(!blk[k]){dist[k]=0;push(0,k);}}}}
    const D2=CS*Math.SQRT2;
    while(hk.length){
      const k=pop(),d=popK; if(d>dist[k])continue;
      const i=k%nx,j=(k/nx)|0;
      for(let dj=-1;dj<=1;dj++)for(let di=-1;di<=1;di++){
        if(!di&&!dj)continue;
        const ii=i+di,jj=j+dj; if(ii<0||jj<0||ii>=nx||jj>=ny)continue;
        const kk=jj*nx+ii; if(blk[kk])continue;
        if(di&&dj&&(blk[j*nx+ii]||blk[jj*nx+i]))continue;
        const nd=d+(di&&dj?D2:CS);
        if(nd<dist[kk]){dist[kk]=nd;push(nd,kk);}
      }
    }
    const dX=new Float32Array(N), dY=new Float32Array(N);
    const val=(i,j,f0)=>{if(i<0||j<0||i>=nx||j>=ny)return f0+CS;const k=j*nx+i;return (blk[k]||!isFinite(dist[k]))?f0+CS:dist[k];};
    for(let j=0;j<ny;j++)for(let i=0;i<nx;i++){
      const k=j*nx+i; if(blk[k]||!isFinite(dist[k]))continue;
      const f0=dist[k];
      let gx=(val(i+1,j,f0)-val(i-1,j,f0)), gy=(val(i,j+1,f0)-val(i,j-1,f0));
      let L=Math.hypot(gx,gy);
      if(L<1e-6){
        const cy=(j+.5)*CS,cx=(i+.5)*CS;
        if(cy>OUT_Y){gx=0;gy=-1;}else{gx=-(W/2-cx);gy=-(OUT_Y-cy);}
        L=Math.hypot(gx,gy)||1;
      }
      dX[k]=-gx/L; dY[k]=-gy/L;
    }
    this.blk=blk; this.dist=dist; this.dirX=dX; this.dirY=dY;
  }
  dirAt(x,y){
    const nx=this.fnx,ny=this.fny;
    const gx=x/CS-0.5, gy=y/CS-0.5;
    const i0=Math.floor(gx), j0=Math.floor(gy), fx=gx-i0, fy=gy-j0;
    let sx=0,sy=0;
    for(let b=0;b<2;b++){const j=j0+b;if(j<0||j>=ny)continue;const wy=b?fy:1-fy;
      for(let a=0;a<2;a++){const i=i0+a;if(i<0||i>=nx)continue;const w=(a?fx:1-fx)*wy;const k=j*nx+i;sx+=w*this.dirX[k];sy+=w*this.dirY[k];}}
    let L=Math.hypot(sx,sy);
    if(L<1e-6){sx=W/2-x;sy=OUT_Y+1-y;L=Math.hypot(sx,sy)||1;}
    this._dx=sx/L; this._dy=sy/L;
  }
  spawn(){
    const want=this.cfg.n, rng=mulberry32((this.cfg.seed|0)*9973+17);
    const X=[],Y=[],RR=[];
    let ymax=STAND_Y1;
    while(X.length<want){
      let ok=false;
      for(let tries=0;tries<3000&&!ok;tries++){
        const r=0.25+0.06*rng();
        const x=0.4+r+rng()*(W-0.8-2*r), y=STAND_Y0+r+rng()*(ymax-STAND_Y0-2*r);
        let clash=false;
        for(let k=0;k<X.length;k++){const dx=x-X[k],dy=y-Y[k],m=r+RR[k]+0.06;if(dx*dx+dy*dy<m*m){clash=true;break;}}
        if(!clash){X.push(x);Y.push(y);RR.push(r);ok=true;}
      }
      if(!ok){ if(ymax>=WALL_Y-2.5)break; ymax=Math.min(WALL_Y-2.5,ymax+0.5); }
    }
    const n=X.length; this.n=n;
    const F=()=>new Float64Array(n);
    this.x=Float64Array.from(X); this.y=Float64Array.from(Y); this.r=Float64Array.from(RR);
    this.vx=F();this.vy=F();this.fx=F();this.fy=F();this.ex=F();this.ey=F();this.press=F();
    this.v0f=F();this.delay=F();this.hd=F();
    this.kit=new Array(n); this.skin=new Array(n);
    for(let i=0;i<n;i++){
      this.v0f[i]=0.9+0.2*rng(); this.delay[i]=2*rng(); this.hd[i]=Math.PI/2;
      this.kit[i]=KITS[(rng()*KITS.length)|0]; this.skin[i]=SKIN[(rng()*SKIN.length)|0];
    }
    this.alive=new Uint8Array(n).fill(1); this.exited=new Uint8Array(n);
    this.cellOf=new Int32Array(n); this.next=new Int32Array(n);
    this.gnx=Math.ceil(W/GC)+1; this.gny=Math.ceil(H/GC)+1;
    this.head=new Int32Array(this.gnx*this.gny);
    this.t=0; this.exitTimes=[]; this.exitedCount=0; this.aliveCount=n;
    this.tEvac=null; this.maxPenNow=0; this.maxPenEver=0; this.projCount=0; this.capCount=0;
    this.longestGap=0; this.stalls=0;
  }
  done(){return this.exitedCount===this.n||this.t>=T_LIMIT;}
  finished(){return this.aliveCount===0||this.t>=T_LIMIT;}
  step(){
    const n=this.n,x=this.x,y=this.y,vx=this.vx,vy=this.vy,fx=this.fx,fy=this.fy,r=this.r,
      alive=this.alive,ex=this.ex,ey=this.ey,press=this.press,head=this.head,next=this.next,cellOf=this.cellOf;
    const gnx=this.gnx,gny=this.gny;
    const P=urgencyParams(this.u),B=P.B,kap=P.kappa,v0=P.v0;
    const t=this.t;
    head.fill(-1);
    for(let i=0;i<n;i++){
      if(!alive[i])continue;
      let cx=(x[i]/GC)|0,cy=(y[i]/GC)|0;
      if(cx<0)cx=0;else if(cx>=gnx)cx=gnx-1;
      if(cy<0)cy=0;else if(cy>=gny)cy=gny-1;
      const c=cy*gnx+cx; cellOf[i]=c; next[i]=head[c]; head[c]=i;
      if(t<this.delay[i]){ex[i]=0;ey[i]=0;fx[i]=-MASS*vx[i]/TAU;fy[i]=-MASS*vy[i]/TAU;}
      else{
        this.dirAt(x[i],y[i]);
        const dx=this._dx,dy=this._dy; ex[i]=dx; ey[i]=dy;
        const vd=v0*this.v0f[i];
        fx[i]=MASS*(vd*dx-vx[i])/TAU; fy[i]=MASS*(vd*dy-vy[i])/TAU;
      }
      press[i]=0;
    }
    const RA=this.rectArr,NR=RA.length/4,col=this.col;
    let maxPen=0;
    for(let i=0;i<n;i++){
      if(!alive[i])continue;
      const c=cellOf[i],cx=c%gnx,cy=(c/gnx)|0;
      const xi=x[i],yi=y[i],ri=r[i],vxi=vx[i],vyi=vy[i],exi=ex[i],eyi=ey[i];
      let fxi=0,fyi=0,pi=0;
      for(let oy=-1;oy<=1;oy++){const yy=cy+oy;if(yy<0||yy>=gny)continue;
        for(let ox=-1;ox<=1;ox++){const xx=cx+ox;if(xx<0||xx>=gnx)continue;
          for(let j=head[yy*gnx+xx];j!==-1;j=next[j]){
            if(j<=i)continue;
            const dx=xi-x[j],dy=yi-y[j],d2=dx*dx+dy*dy;
            if(d2>CUT2)continue;
            const d=Math.sqrt(d2)||1e-6,nx=dx/d,ny=dy/d,rij=ri+r[j],gap=rij-d;
            let s=A_SOC*(Math.exp(Math.min(gap,0.5)/B)-Math.exp((rij-CUT)/B)); if(s<0)s=0;
            const ci=-(nx*exi+ny*eyi), wi=LAMBDA+(1-LAMBDA)*0.5*(1+ci);
            const cj=nx*ex[j]+ny*ey[j], wj=LAMBDA+(1-LAMBDA)*0.5*(1+cj);
            fxi+=s*wi*nx; fyi+=s*wi*ny; fx[j]-=s*wj*nx; fy[j]-=s*wj*ny;
            if(gap>0){
              const tx=-ny,ty=nx,dvt=(vx[j]-vxi)*tx+(vy[j]-vyi)*ty;
              const fn=K_BODY*gap,ft=kap*gap*dvt;
              const Fx=fn*nx+ft*tx,Fy=fn*ny+ft*ty;
              fxi+=Fx;fyi+=Fy;fx[j]-=Fx;fy[j]-=Fy;
              pi+=fn;press[j]+=fn;
              if(gap>maxPen)maxPen=gap;
            }
          }
        }
      }
      for(let q=0;q<NR;q++){
        const x0=RA[4*q],y0=RA[4*q+1],x1=RA[4*q+2],y1=RA[4*q+3];
        const px=xi<x0?x0:(xi>x1?x1:xi),py=yi<y0?y0:(yi>y1?y1:yi);
        const dx=xi-px,dy=yi-py,d2=dx*dx+dy*dy;
        if(d2>WALL_CUT*WALL_CUT)continue;
        let d,nx,ny;
        if(d2<1e-12){
          const dl=xi-x0,dr=x1-xi,du=yi-y0,dd=y1-yi;let m=dl;nx=-1;ny=0;
          if(dr<m){m=dr;nx=1;ny=0;} if(du<m){m=du;nx=0;ny=-1;} if(dd<m){m=dd;nx=0;ny=1;}
          d=-m;
        }else{d=Math.sqrt(d2);nx=dx/d;ny=dy/d;}
        const gap=ri-d, s=A_WALL*Math.exp(Math.min(gap,0.5)/B_WALL);
        fxi+=s*nx;fyi+=s*ny;
        if(gap>0){const tx=-ny,ty=nx,vt=vxi*tx+vyi*ty,fn=K_BODY*gap,ft=-kap*gap*vt;
          fxi+=fn*nx+ft*tx;fyi+=fn*ny+ft*ty;pi+=fn;if(gap>maxPen)maxPen=gap;}
      }
      if(col){
        const dx=xi-col.x,dy=yi-col.y,dc=Math.sqrt(dx*dx+dy*dy)||1e-6,d=dc-col.r;
        if(d<WALL_CUT){
          const nx=dx/dc,ny=dy/dc,gap=ri-d,s=A_WALL*Math.exp(Math.min(gap,0.5)/B_WALL);
          fxi+=s*nx;fyi+=s*ny;
          if(gap>0){const tx=-ny,ty=nx,vt=vxi*tx+vyi*ty,fn=K_BODY*gap,ft=-kap*gap*vt;
            fxi+=fn*nx+ft*tx;fyi+=fn*ny+ft*ty;pi+=fn;if(gap>maxPen)maxPen=gap;}
        }
      }
      fx[i]+=fxi;fy[i]+=fyi;press[i]+=pi;
    }
    // integrate (semi-implicit Euler)
    const k=DT/MASS;
    for(let i=0;i<n;i++){
      if(!alive[i])continue;
      let a=vx[i]+fx[i]*k,b=vy[i]+fy[i]*k;
      const sp=a*a+b*b;
      if(sp>VMAX*VMAX){const m=VMAX/Math.sqrt(sp);a*=m;b*=m;this.capCount++;}
      vx[i]=a;vy[i]=b;
      x[i]+=a*DT;y[i]+=b*DT;
      if(x[i]<0.05)x[i]=0.05;else if(x[i]>W-0.05)x[i]=W-0.05;
    }
    // hard overlap cap: bodies may compress up to MAX_PEN, never more
    for(let i=0;i<n;i++){
      if(!alive[i])continue;
      const c=cellOf[i],cx=c%gnx,cy=(c/gnx)|0;
      for(let oy=-1;oy<=1;oy++){const yy=cy+oy;if(yy<0||yy>=gny)continue;
        for(let ox=-1;ox<=1;ox++){const xx=cx+ox;if(xx<0||xx>=gnx)continue;
          for(let j=head[yy*gnx+xx];j!==-1;j=next[j]){
            if(j<=i)continue;
            const dx=x[i]-x[j],dy=y[i]-y[j],d2=dx*dx+dy*dy,rij=r[i]+r[j]-MAX_PEN;
            if(d2>=rij*rij)continue;
            const d=Math.sqrt(d2)||1e-6,corr=(rij-d)*0.5,nx=dx/d,ny=dy/d;
            x[i]+=nx*corr;y[i]+=ny*corr;x[j]-=nx*corr;y[j]-=ny*corr;this.projCount++;
          }
        }
      }
      for(let q=0;q<NR;q++){
        const x0=RA[4*q],y0=RA[4*q+1],x1=RA[4*q+2],y1=RA[4*q+3],xi=x[i],yi=y[i];
        const px=xi<x0?x0:(xi>x1?x1:xi),py=yi<y0?y0:(yi>y1?y1:yi);
        const dx=xi-px,dy=yi-py,d2=dx*dx+dy*dy,lim=r[i]-MAX_PEN;
        if(d2>=lim*lim||d2<1e-12)continue;
        const d=Math.sqrt(d2),c2=lim-d;x[i]+=dx/d*c2;y[i]+=dy/d*c2;this.projCount++;
      }
      if(col){
        const dx=x[i]-col.x,dy=y[i]-col.y,dc=Math.sqrt(dx*dx+dy*dy)||1e-6,lim=r[i]-MAX_PEN+col.r;
        if(dc<lim){const c2=lim-dc;x[i]+=dx/dc*c2;y[i]+=dy/dc*c2;this.projCount++;}
      }
    }
    this.t=t+DT;
    for(let i=0;i<n;i++){
      if(!alive[i])continue;
      if(!this.exited[i]&&y[i]>OUT_Y){
        this.exited[i]=1;this.exitedCount++;
        const et=this.t, L=this.exitTimes.length;
        if(L){const g=et-this.exitTimes[L-1];if(g>this.longestGap)this.longestGap=g;if(g>=2)this.stalls++;}
        this.exitTimes.push(et);
        if(this.exitedCount===this.n)this.tEvac=et;
      }
      if(y[i]>H-0.7){alive[i]=0;this.aliveCount--;}
    }
    this.maxPenNow=maxPen; if(maxPen>this.maxPenEver)this.maxPenEver=maxPen;
  }
}

// ---------- config / hash ----------
function parseHash(){
  const p=new URLSearchParams((location.hash||'').replace(/^#/,''));
  const num=(k,d)=>{const v=parseFloat(p.get(k));return isFinite(v)?v:d;};
  const u=Math.min(1,Math.max(0,num('urgency',0.5)));
  const g=Math.min(30,Math.max(0.1,num('gate',1.2)));
  const cs=p.get('column');
  let col=false;
  if(cs!==null){const v=parseFloat(cs);col=cs==='true'||(isFinite(v)&&v!==0);}
  const n=Math.round(Math.min(500,Math.max(200,num('n',220))));
  const seed=Math.round(num('seed',1));
  return {u,gateYd:g,column:col,n,seed};
}

// ---------- UI ----------
const $=id=>document.getElementById(id);
const view=$('view'),ctx=view.getContext('2d');
const chart=$('chart'),cctx=chart.getContext('2d');
let cfg=parseHash();
let sim=new Sim(cfg);
let running=true,speed=1,acc=0,last=performance.now(),bgKey='',bg=null;
let L={s:1,ox:0,oy:0,cw:1,ch:1,dpr:1};
let viewMode='kits';

function syncUI(){
  $('urg').value=cfg.u; $('urgLbl').textContent=cfg.u.toFixed(2);
  const P=urgencyParams(cfg.u);
  $('urgInfo').textContent=`desired speed ≈ ${P.v0.toFixed(2)} m/s · max push ≈ ${(MASS*P.v0/TAU|0)} N · personal-space range ${(P.B*100).toFixed(1)} cm · sliding friction ×${(P.kappa/KAPPA0).toFixed(2)}`;
  $('gateR').value=Math.min(4,Math.max(0.5,cfg.gateYd)); $('gateN').value=+cfg.gateYd.toFixed(3);
  $('gateLbl').textContent=`${cfg.gateYd.toFixed(2)} yd = ${(Math.min(W-0.4,cfg.gateYd*YD)).toFixed(2)} m`;
  $('col').checked=cfg.column;
  $('hashTxt').textContent=`#urgency=${+cfg.u.toFixed(2)}&gate=${+cfg.gateYd.toFixed(3)}&column=${cfg.column?1:0}`;
}
function restart(){sim=new Sim(cfg);acc=0;bgKey='';running=true;$('pause').textContent='Pause';}
$('urg').addEventListener('input',e=>{cfg.u=+e.target.value;sim.u=cfg.u;syncUI();});
$('gateR').addEventListener('input',e=>{cfg.gateYd=+e.target.value;syncUI();restart();});
$('gateN').addEventListener('change',e=>{const v=parseFloat(e.target.value);if(isFinite(v)&&v>0){cfg.gateYd=Math.min(30,v);}syncUI();restart();});
$('col').addEventListener('change',e=>{cfg.column=e.target.checked;syncUI();restart();});
$('reset').addEventListener('click',()=>restart());
$('pause').addEventListener('click',()=>{running=!running;$('pause').textContent=running?'Pause':'Resume';});
$('speed').addEventListener('change',e=>{speed=+e.target.value;});
$('viewMode').addEventListener('change',e=>{viewMode=e.target.value;});
window.addEventListener('hashchange',()=>{cfg=parseHash();syncUI();restart();});

// ---------- layout / art ----------
const VX0=-0.6,VX1=W+0.6,VY0=-2.4,VY1=H+0.1;
function layout(){
  const rect=view.getBoundingClientRect(),dpr=Math.min(2,window.devicePixelRatio||1);
  const cw=Math.max(50,rect.width),ch=Math.max(50,rect.height);
  view.width=Math.round(cw*dpr);view.height=Math.round(ch*dpr);
  const s=Math.min(cw/(VX1-VX0),ch/(VY1-VY0));
  L={s,dpr,cw,ch,ox:(cw-(VX1-VX0)*s)/2-VX0*s,oy:(ch-(VY1-VY0)*s)/2-VY0*s};
  const cr=chart.getBoundingClientRect();chart.width=Math.round(cr.width*dpr);chart.height=Math.round(cr.height*dpr);
  bgKey='';
}
window.addEventListener('resize',layout);

function paintBg(){
  bg=document.createElement('canvas');bg.width=view.width;bg.height=view.height;
  const g=bg.getContext('2d'),s=L.s,d=L.dpr;
  g.fillStyle='#0b0e12';g.fillRect(0,0,bg.width,bg.height);
  g.setTransform(s*d,0,0,s*d,L.ox*d,L.oy*d);
  const px=1/s;
  // pitch
  g.fillStyle='#2f7d3a';g.fillRect(VX0,VY0,VX1-VX0,-0.25-VY0);
  for(let x=VX0;x<VX1;x+=2){g.fillStyle='rgba(255,255,255,0.05)';g.fillRect(x,VY0,1,-0.25-VY0);}
  g.fillStyle='#f4f4f0';g.fillRect(VX0,-1.0,VX1-VX0,0.08);
  // ad boards
  const ads=['#c8372d','#1d1d1d','#f2c230','#1f5fd1'];
  for(let i=0,x=VX0;x<VX1;i++,x+=3){g.fillStyle=ads[i%4];g.fillRect(x,-0.32,3,0.3);}
  g.fillStyle='rgba(255,255,255,0.25)';g.fillRect(VX0,-0.04,VX1-VX0,0.04);
  // stands: terrace bands
  g.fillStyle='#4a515c';g.fillRect(0,0,W,STAND_Y1+0.3);
  let rowIdx=0;
  for(let yy=STAND_Y0;yy+ROW<=STAND_Y1+0.01;yy+=ROW,rowIdx++){
    g.fillStyle=rowIdx%2?'#555d69':'#5b6370';g.fillRect(0,yy,W,ROW);
    g.fillStyle='rgba(255,255,255,0.10)';g.fillRect(0,yy,W,0.03);
    g.fillStyle='rgba(0,0,0,0.25)';g.fillRect(0,yy+ROW-0.05,W,0.05);
    const seatCol=(rowIdx%4===3)?'#e0b23a':'#1f4fa8',back=(rowIdx%4===3)?'#a8821f':'#163a7c';
    for(let sx=0.3;sx+0.42<W-0.2;sx+=0.5){
      let inA=false;for(const a of AISLES){if(Math.abs(sx+0.2-a)<0.75)inA=true;}
      if(inA)continue;
      g.fillStyle=seatCol;g.fillRect(sx+0.03,yy+0.28,0.36,0.3);
      g.fillStyle=back;g.fillRect(sx,yy+ROW-0.22,0.42,0.12);
    }
  }
  for(const a of AISLES){
    g.fillStyle='#6c7480';g.fillRect(a-0.5,0,1.0,STAND_Y1+0.3);
    g.fillStyle='rgba(0,0,0,0.22)';
    for(let yy=0.2;yy<STAND_Y1+0.3;yy+=0.4)g.fillRect(a-0.5,yy,1.0,0.025);
    g.fillStyle='#f2c230';g.fillRect(a-0.5,0,0.04,STAND_Y1+0.3);g.fillRect(a+0.46,0,0.04,STAND_Y1+0.3);
  }
  // concourse
  g.fillStyle='#343a43';g.fillRect(0,STAND_Y1+0.3,W,WALL_Y-STAND_Y1-0.3);
  g.fillStyle='rgba(242,194,48,0.85)';g.fillRect(0,STAND_Y1+0.3,W,0.06);
  g.strokeStyle='rgba(255,255,255,0.04)';g.lineWidth=px;
  for(let x=0;x<=W;x+=1.5){g.beginPath();g.moveTo(x,STAND_Y1+0.36);g.lineTo(x,WALL_Y);g.stroke();}
  // painted chevrons following the route field
  g.strokeStyle='rgba(242,194,48,0.22)';g.lineWidth=0.08;g.lineCap='round';g.lineJoin='round';
  for(let yy=STAND_Y1+1.2;yy<WALL_Y-0.6;yy+=1.6)for(let xx=1.5;xx<W-1;xx+=2.2){
    if(sim.solidAt(xx,yy,0.6))continue;
    sim.dirAt(xx,yy);const dx=sim._dx,dy=sim._dy,nx=-dy,ny=dx;
    g.beginPath();g.moveTo(xx-dx*0.25+nx*0.3,yy-dy*0.25+ny*0.3);g.lineTo(xx+dx*0.15,yy+dy*0.15);g.lineTo(xx-dx*0.25-nx*0.3,yy-dy*0.25-ny*0.3);g.stroke();
  }
  // outside plaza
  g.fillStyle='#6a5f55';g.fillRect(VX0,OUT_Y,VX1-VX0,H-0.6-OUT_Y);
  g.strokeStyle='rgba(0,0,0,0.18)';g.lineWidth=px;
  let br=0;
  for(let yy=OUT_Y;yy<H-0.6;yy+=0.3,br++){g.beginPath();g.moveTo(VX0,yy);g.lineTo(VX1,yy);g.stroke();
    for(let xx=VX0+(br%2?0.3:0);xx<VX1;xx+=0.6){g.beginPath();g.moveTo(xx,yy);g.lineTo(xx,yy+0.3);g.stroke();}}
  g.fillStyle='#9b9389';g.fillRect(VX0,H-0.6,VX1-VX0,0.12);
  g.fillStyle='#24272c';g.fillRect(VX0,H-0.48,VX1-VX0,VY1-H+0.48);
  g.fillStyle='#d9d4c4';for(let x=VX0;x<VX1;x+=1.6)g.fillRect(x,H-0.12,0.8,0.05);
  // threshold hazard stripes in the opening
  g.save();g.beginPath();g.rect(sim.gateL,WALL_Y,sim.gateW,WALL_T);g.clip();
  g.fillStyle='#e0b23a';g.fillRect(sim.gateL,WALL_Y,sim.gateW,WALL_T);
  g.fillStyle='#1a1a1a';
  for(let x=sim.gateL-1;x<sim.gateR+1;x+=0.25){g.beginPath();g.moveTo(x,WALL_Y);g.lineTo(x+0.12,WALL_Y);g.lineTo(x+0.12+WALL_T,OUT_Y);g.lineTo(x+WALL_T,OUT_Y);g.closePath();g.fill();}
  g.restore();
  // walls
  const wallFill=(x0,x1)=>{
    if(x1<=x0)return;
    g.fillStyle='#8d8f93';g.fillRect(x0,WALL_Y,x1-x0,WALL_T);
    g.fillStyle='rgba(0,0,0,0.25)';g.fillRect(x0,OUT_Y-0.07,x1-x0,0.07);
    g.fillStyle='rgba(255,255,255,0.18)';g.fillRect(x0,WALL_Y,x1-x0,0.05);
    g.strokeStyle='rgba(0,0,0,0.18)';g.lineWidth=px;
    for(let x=x0+0.9;x<x1;x+=1.8){g.beginPath();g.moveTo(x,WALL_Y);g.lineTo(x,OUT_Y);g.stroke();}
  };
  wallFill(VX0,sim.gateL);wallFill(sim.gateR,VX1);
  g.fillStyle='#8d8f93';g.fillRect(VX0,-0.02,0.6,OUT_Y);g.fillRect(W,-0.02,0.6,OUT_Y);
  // gate posts + folded leaves
  g.fillStyle='#c9ced6';
  g.fillRect(sim.gateL-0.18,WALL_Y-0.05,0.18,WALL_T+0.1);g.fillRect(sim.gateR,WALL_Y-0.05,0.18,WALL_T+0.1);
  g.strokeStyle='#d8dde4';g.lineWidth=0.05;
  const leaf=Math.min(1.4,sim.gateW/2);
  g.beginPath();g.moveTo(sim.gateL,OUT_Y+0.05);g.lineTo(sim.gateL-leaf,OUT_Y+0.05);g.moveTo(sim.gateR,OUT_Y+0.05);g.lineTo(sim.gateR+leaf,OUT_Y+0.05);g.stroke();
  // column
  if(sim.col){const c=sim.col;
    g.fillStyle='rgba(0,0,0,0.35)';g.beginPath();g.arc(c.x+0.08,c.y+0.1,c.r,0,7);g.fill();
    g.fillStyle='#a9abb0';g.beginPath();g.arc(c.x,c.y,c.r,0,7);g.fill();
    g.strokeStyle='rgba(0,0,0,0.3)';g.lineWidth=0.03;g.beginPath();g.arc(c.x,c.y,c.r*0.65,0,7);g.stroke();
    g.fillStyle='rgba(255,255,255,0.3)';g.beginPath();g.arc(c.x-0.1,c.y-0.1,c.r*0.35,0,7);g.fill();}
  // labels in screen space
  g.setTransform(d,0,0,d,0,0);
  const S=(x,y)=>[L.ox+x*s,L.oy+y*s];
  g.textAlign='center';g.textBaseline='middle';
  g.font=`700 ${Math.max(9,0.34*s)|0}px system-ui,sans-serif`;g.fillStyle='#22262c';
  const lx=sim.gateL-2.4>0.5?sim.gateL-2.4:sim.gateR+2.4;
  let [a,b]=S(lx,WALL_Y+WALL_T/2);g.fillText('GATE C  ▸',a,b);
  g.font=`700 ${Math.max(9,0.45*s)|0}px system-ui,sans-serif`;g.fillStyle='rgba(255,255,255,0.10)';
  [a,b]=S(W/2,(STAND_Y1+WALL_Y)/2-0.6);g.fillText('CONCOURSE',a,b);
  [a,b]=S(3,-1.6);g.font=`600 ${Math.max(9,0.38*s)|0}px system-ui,sans-serif`;g.fillStyle='rgba(255,255,255,0.55)';g.fillText('PITCH',a,b);
  [a,b]=S(W-3,(OUT_Y+H-0.6)/2);g.fillStyle='rgba(255,255,255,0.35)';g.fillText('OUTSIDE',a,b);
  // scale bar
  const [sx0,sy0]=S(W-5.5,H-0.3);g.fillStyle='#e6e9ee';g.fillRect(sx0,sy0,5*s,2);
  g.font=`${Math.max(9,0.3*s)|0}px system-ui,sans-serif`;g.textAlign='left';g.fillText('5 m ≈ 5.5 yd',sx0,sy0-8);
  bgKey=key();
}
function key(){return `${view.width}x${view.height}|${sim.gateW}|${!!sim.col}`;}
function heat(p){ // p in N/m
  const t=Math.min(1,Math.max(0,p/3000));
  const stops=[[58,123,213],[245,197,66],[232,64,47],[255,43,214]];
  const f=t*3,i=Math.min(2,Math.floor(f)),u=f-i,A=stops[i],B=stops[i+1];
  return `rgb(${A[0]+(B[0]-A[0])*u|0},${A[1]+(B[1]-A[1])*u|0},${A[2]+(B[2]-A[2])*u|0})`;
}
function draw(){
  if(bgKey!==key())paintBg();
  const d=L.dpr,s=L.s;
  ctx.setTransform(1,0,0,1,0,0);ctx.drawImage(bg,0,0);
  ctx.setTransform(s*d,0,0,s*d,L.ox*d,L.oy*d);
  const n=sim.n,x=sim.x,y=sim.y,r=sim.r,al=sim.alive;
  ctx.fillStyle='rgba(0,0,0,0.28)';
  for(let i=0;i<n;i++){if(!al[i])continue;ctx.beginPath();ctx.arc(x[i]+0.05,y[i]+0.07,r[i],0,6.2832);ctx.fill();}
  for(let i=0;i<n;i++){
    if(!al[i])continue;
    const sp=Math.hypot(sim.vx[i],sim.vy[i]);
    let target=sim.hd[i];
    if(sp>0.15)target=Math.atan2(sim.vy[i],sim.vx[i]);else if(sim.ex[i]||sim.ey[i])target=Math.atan2(sim.ey[i],sim.ex[i]);
    let dA=target-sim.hd[i];while(dA>Math.PI)dA-=2*Math.PI;while(dA<-Math.PI)dA+=2*Math.PI;
    sim.hd[i]+=dA*0.2;
    const h=sim.hd[i],ri=r[i],p=sim.press[i]/(2*Math.PI*ri);
    ctx.fillStyle=viewMode==='pressure'?heat(p):sim.kit[i];
    ctx.beginPath();ctx.ellipse(x[i],y[i],ri,ri*0.92,h+Math.PI/2,0,6.2832);ctx.fill();
    ctx.lineWidth=0.025;ctx.strokeStyle='rgba(0,0,0,0.55)';ctx.stroke();
    if(viewMode==='kits'&&p>350){
      ctx.lineWidth=0.06;ctx.strokeStyle=heat(p);ctx.globalAlpha=Math.min(1,0.35+p/2500);
      ctx.beginPath();ctx.arc(x[i],y[i],ri+0.03,0,6.2832);ctx.stroke();ctx.globalAlpha=1;
    }
    ctx.fillStyle='#2a1d14';ctx.beginPath();ctx.arc(x[i]+Math.cos(h)*ri*0.12,y[i]+Math.sin(h)*ri*0.12,ri*0.46,0,6.2832);ctx.fill();
    ctx.fillStyle=sim.skin[i];ctx.beginPath();ctx.arc(x[i]+Math.cos(h)*ri*0.24,y[i]+Math.sin(h)*ri*0.24,ri*0.3,0,6.2832);ctx.fill();
  }
  // HUD
  ctx.setTransform(d,0,0,d,0,0);
  ctx.textAlign='left';ctx.textBaseline='top';
  ctx.fillStyle='rgba(10,12,16,0.7)';ctx.fillRect(10,10,236,54);
  ctx.fillStyle='#ffcf3f';ctx.font='700 22px ui-monospace,Menlo,Consolas,monospace';
  ctx.fillText('T+ '+fmtClock(sim.tEvac!=null?sim.tEvac:sim.t),18,15);
  ctx.fillStyle='#cfd5dd';ctx.font='12px system-ui,sans-serif';
  ctx.fillText(`${sim.n-sim.exitedCount} inside · ${sim.exitedCount} out · urgency ${sim.u.toFixed(2)}`,18,44);
  if(sim.tEvac!=null||sim.t>=T_LIMIT){
    const msg=sim.tEvac!=null?`Section empty in ${sim.tEvac.toFixed(1)} s`:`Not empty after ${T_LIMIT} s`;
    ctx.font='700 20px system-ui,sans-serif';const w=ctx.measureText(msg).width+30;
    ctx.fillStyle='rgba(10,12,16,0.82)';ctx.fillRect(L.cw/2-w/2,L.ch*0.38,w,40);
    ctx.fillStyle='#ffcf3f';ctx.textAlign='center';ctx.fillText(msg,L.cw/2,L.ch*0.38+10);
  }
  const minD=2*0.25;
  if(sim.gateW<minD){ctx.textAlign='center';ctx.fillStyle='#ff7a6a';ctx.font='600 14px system-ui,sans-serif';
    ctx.fillText(`Gate (${sim.gateW.toFixed(2)} m) is narrower than the slimmest body (${minD.toFixed(2)} m): nobody can leave.`,L.cw/2,L.ch-30);}
}
function fmtClock(t){const m=Math.floor(t/60),s=t-m*60;return `${String(m).padStart(2,'0')}:${s.toFixed(1).padStart(4,'0')}`;}

// ---------- metrics ----------
function updateMetrics(){
  const S=sim,et=S.exitTimes,n=et.length;
  $('mT').textContent=S.t.toFixed(1)+' s';
  $('mEvac').textContent=S.tEvac!=null?S.tEvac.toFixed(1)+' s':(S.t>=T_LIMIT?'not empty':'—');
  $('mIn').textContent=`${S.n-S.exitedCount} / ${S.n}`;
  $('mOut').textContent=String(S.exitedCount);
  if(n){
    let c=0;for(let k=n-1;k>=0&&et[k]>S.t-10;k--)c++;
    $('mRate').textContent=(c*6).toFixed(0)+' /min';
  }else $('mRate').textContent='—';
  $('mMean').textContent=n>=2?((n-1)*60/(et[n-1]-et[0])).toFixed(1)+' /min':'—';
  $('mGap').textContent=n>=2?S.longestGap.toFixed(2)+' s':'—';
  $('mStall').textContent=String(S.stalls);
  $('mPen').textContent=`${(S.maxPenNow*100).toFixed(1)} / ${(S.maxPenEver*100).toFixed(1)} cm`;
  $('mProj').textContent=String(S.projCount)+(S.capCount?` · speed cap ${S.capCount}`:'');
}
function drawChart(){
  const d=L.dpr,w=chart.width/d,h=chart.height/d;
  cctx.setTransform(d,0,0,d,0,0);cctx.clearRect(0,0,w,h);
  const et=sim.exitTimes,T=Math.max(60,Math.ceil((sim.tEvac||sim.t)/30)*30);
  const bins=Math.ceil(T/5),cnt=new Array(bins).fill(0);
  for(const t of et){const b=Math.min(bins-1,Math.floor(t/5));cnt[b]++;}
  let maxR=60;for(const c of cnt)maxR=Math.max(maxR,c*12);
  const pl=26,pr=6,pt=6,pb=16,iw=w-pl-pr,ih=h-pt-pb;
  cctx.fillStyle='#ffcf3f';
  for(let b=0;b<bins;b++){const v=cnt[b]*12,bh=v/maxR*ih;cctx.fillRect(pl+b/bins*iw+0.5,pt+ih-bh,Math.max(1,iw/bins-1),bh);}
  cctx.strokeStyle='#6fb6ff';cctx.lineWidth=1.5;cctx.beginPath();cctx.moveTo(pl,pt+ih);
  et.forEach((t,k)=>cctx.lineTo(pl+t/T*iw,pt+ih-(k+1)/sim.n*ih));
  cctx.stroke();
  cctx.fillStyle='#8b95a3';cctx.font='10px system-ui,sans-serif';cctx.textAlign='left';cctx.textBaseline='top';
  cctx.fillText(`${maxR|0}/min`,2,pt);cctx.fillText('0',2,pt+ih-8);
  cctx.textAlign='right';cctx.fillText(`${T} s`,w-pr,h-12);cctx.textAlign='left';cctx.fillText('0',pl,h-12);
}

// ---------- headless experiment ----------
let exp=null;
$('expBtn').addEventListener('click',()=>{
  if(exp)return;
  const seeds=[1,2,3],confs=[{k:'A',u:0.3,c:false},{k:'B',u:0.9,c:false},{k:'C',u:0.9,c:true}];
  const jobs=[];for(const c of confs)for(const s of seeds)jobs.push({conf:c,seed:s,res:null,sim:null});
  exp={jobs,idx:0};running=false;$('pause').textContent='Resume';$('expBtn').disabled=true;
  setTimeout(expTick,0);
});
function expTick(){
  const E=exp,job=E.jobs[E.idx];
  if(!job.sim)job.sim=new Sim({u:job.conf.u,gateYd:1.2,column:job.conf.c,n:cfg.n,seed:job.seed});
  const t0=performance.now();
  while(performance.now()-t0<35&&!job.sim.done()){for(let k=0;k<100&&!job.sim.done();k++)job.sim.step();}
  $('expProg').textContent=`run ${E.idx+1}/${E.jobs.length} · ${job.conf.k} seed ${job.seed} · t=${job.sim.t.toFixed(0)} s`;
  if(job.sim.done()){job.res=job.sim.tEvac;job.sim=null;E.idx++;renderExp();}
  if(E.idx<E.jobs.length)setTimeout(expTick,0);
  else{$('expProg').textContent='done';$('expBtn').disabled=false;exp=null;}
}
function renderExp(){
  const E=exp;const groups={A:[],B:[],C:[]};
  for(const j of E.jobs)if(j.res!==null||j.sim===null&&E.jobs.indexOf(j)<E.idx)groups[j.conf.k].push(j.res);
  const mean=a=>{const v=a.filter(z=>z!=null);return v.length?v.reduce((p,q)=>p+q,0)/v.length:null;};
  const f=v=>v==null?'—':v.toFixed(1)+' s';
  const lab={A:'A: urgency 0.3',B:'B: urgency 0.9',C:'C: urgency 0.9 + column'};
  let h='<table><tr><th>Setup (gate 1.2 yd)</th><th>runs</th><th>mean</th></tr>';
  for(const k of ['A','B','C'])h+=`<tr><td>${lab[k]}</td><td>${groups[k].map(z=>z==null?'DNF':z.toFixed(0)).join(', ')}</td><td>${f(mean(groups[k]))}</td></tr>`;
  const a=mean(groups.A),b=mean(groups.B),c=mean(groups.C);
  h+='</table><table>';
  h+=`<tr><td>B − A (urgency 0.3→0.9)</td><td>${a!=null&&b!=null?(b-a>=0?'+':'')+(b-a).toFixed(1)+' s':'—'}</td><td class="note">predicted +40 s</td></tr>`;
  h+=`<tr><td>C − B (add column)</td><td>${b!=null&&c!=null?(c-b>=0?'+':'')+(c-b).toFixed(1)+' s':'—'}</td><td class="note">predicted −20 s</td></tr></table>`;
  $('expOut').innerHTML=h;
}

// ---------- main loop ----------
function frame(now){
  const dtR=Math.min(0.1,(now-last)/1000);last=now;
  if(running&&!sim.finished()){
    acc+=dtR*speed;
    const maxSteps=Math.ceil(0.05*speed/DT);let k=0;
    while(acc>=DT&&k<maxSteps){sim.step();acc-=DT;k++;}
    if(k>=maxSteps)acc=0;
  }
  draw();updateMetrics();
  if((frame.c=(frame.c||0)+1)%10===0)drawChart();
  requestAnimationFrame(frame);
}
syncUI();layout();drawChart();
requestAnimationFrame(t=>{last=t;frame(t);});
})();
</script>
</body>
</html>

What it actually does

The rules (Helbing–Farkas–Vicsek social force, written from scratch). Each person is a disc 0.50–0.62 m across, with a mass of 80 kg. Three forces act on everyone:

Drive: they accelerate toward their desired velocity with a 0.5 s time constant.
Personal space: an exponential push from nearby people that is weaker for people behind them, and the same kind of push from walls.
Contact: a stiff body-compression spring plus sliding friction, once bodies touch.

Urgency changes four things:

Desired speed: 0.6 → 4.0 m/s.
The pushing force that drive delivers through contacts: up to about 650 N.
The sliding-friction coefficient: ×0.5 → ×1.0.
The personal-space range: 16 cm → 8 cm. Calm people keep a gap; urgent people accept contact.

Outflow and the empty timer only count real crossings of the gate's outer face. Each is timestamped in simulation time on a fixed 4 ms step.

Where it behaves like a real crowd:

A half-ring forms around the opening.
Density and pressure peak just in front of the gate.
At high urgency people arrive at the opening faster than it can clear them, and outflow comes in bursts. The pressure view shows force chains through the crowd.
The gate's capacity per metre at low urgency is in the right range for a single narrow door.

Where it does not:

Bodies are discs. There's no turning sideways or shoulder-first squeezing through, no stepping gait, no falls, no fallen people becoming obstacles. Real crushes are deadly mostly through falls and compressive asphyxia; nothing here models injury.
Everyone knows the shortest route perfectly. Direction comes from a precomputed distance field to the outside. There's no herding, no following, no groups or families, and no one turning back.
The seats are paint. The stands behave as an open terrace. Real seat rows would funnel everyone into the aisles first, which would be the actual first bottleneck.
Friction is viscous, not static. It resists sliding speed and cannot hold a stationary arch forever. So jams here are transient: they stall and then collapse. My page never jams permanently at the gate unless the gate is narrower than a person. Real crowds can lock solid.
No random noise term. Jams break only through people's differing speeds, start delays and body sizes. Runs are deterministic per seed.

Special cases, named:

Overlap allowance plus hard cap. Bodies may compress up to 3 cm under the contact spring; that compression is what produces contact force. Beyond 3 cm a position correction separates them. So "nobody overlaps" means "no more than 3 cm of body compression, ever." The panel shows the current and maximum compression and how many times the cap fired. If that count climbs, the cap — not the physics — is holding the crowd apart, and the run should be read with suspicion.
Safety speed cap of 8 m/s. This is a guard against numerical blow-up, not a behaviour. Its trigger count is shown if it ever fires.
Reaction delay. Each person waits a random 0–2 s before moving.
Smooth cutoff. The personal-space force is cut off at 1.1 m centre-to-centre and shifted down so it reaches zero smoothly at that distance.
Restarts. Changing gate width or the column restarts the run; urgency changes apply live.
No other interventions. There's no jam-breaking nudge, no speed limit at the gate, and no special rule near the column.

Caveat on my predictions: I haven't run this here. The direction of the urgency effect is the part I'm fairly confident in. The magnitudes, and especially the column result, could easily come out smaller or with the opposite sign. With my parameters the column might mainly add a second narrow passage rather than relieve pressure. Press Check predictions to measure all three setups.

Attachments

anthropic/claude-opus-5.5 — page