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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Tied-Basis Subspace Physics — Water</title>
<style>
 :root{
 --void:#0B0E1A; --panel:#111527; --line:#232A45;
 --cyan:#7EE8FA; --violet:#C77DFF; --magenta:#FF7EB6;
 --txt:#E8ECF7; --mut:#8A94B8;
 }
 *{box-sizing:border-box;margin:0}
 html,body{height:100%}
 body{background:var(--void);color:var(--txt);
 font:14px/1.5 "IBM Plex Mono","SFMono-Regular",Consolas,monospace;
 display:flex;flex-direction:column;overflow:hidden}
 header{display:flex;align-items:baseline;gap:16px;padding:14px 20px 10px;border-bottom:1px solid var(--line)}
 h1{font:600 17px/1 "Space Grotesk","Avenir Next",system-ui,sans-serif;letter-spacing:.02em}
 h1 em{font-style:normal;background:linear-gradient(90deg,var(--cyan),var(--magenta));
 -webkit-background-clip:text;background-clip:text;color:transparent}
 header .sub{color:var(--mut);font-size:11px}
 main{flex:1;display:flex;min-height:0}
 #stage{flex:1;position:relative;min-width:0}
 canvas{display:block;width:100%;height:100%}
 #cv{cursor:crosshair}
 #hint{position:absolute;left:16px;bottom:12px;color:var(--mut);font-size:11px;pointer-events:none}
 aside{width:250px;border-left:1px solid var(--line);background:var(--panel);
 display:flex;flex-direction:column;overflow-y:auto}
 section{padding:14px 16px;border-bottom:1px solid var(--line)}
 .lbl{font-size:10px;letter-spacing:.14em;color:var(--mut);text-transform:uppercase;margin-bottom:8px}
 .seg{display:flex;gap:6px;flex-wrap:wrap}
 button{background:transparent;border:1px solid var(--line);color:var(--txt);
 font:inherit;font-size:12px;padding:5px 10px;border-radius:3px;cursor:pointer}
 button:hover{border-color:var(--cyan)}
 button:focus-visible{outline:2px solid var(--cyan);outline-offset:1px}
 button.on{border-color:var(--cyan);color:var(--cyan);background:rgba(126,232,250,.08)}
 #tower{width:100%;height:210px;display:block}
 .stat{display:flex;justify-content:space-between;font-size:12px;padding:2px 0}
 .stat b{font-weight:600;color:var(--cyan)}
 .note{font-size:11px;color:var(--mut);margin-top:8px}
 @media (max-width:640px){ aside{width:150px} #tower{height:140px} }
</style>
</head>
<body>
<header>
 <h1>Tied-basis subspace tower <em>· live</em></h1>
 <span class="sub">same skeleton, two constraints: strain (tied basis) for solids · density for water</span>
</header>
<main>
 <div id="stage"><canvas id="cv"></canvas>
 <div id="hint">release the dam · drag near tower to grab · drag from empty space to throw a ball</div>
 </div>
 <aside>
 <section>
 <div class="lbl">Actions</div>
 <div class="seg">
 <button id="bWater">Release water</button>
 <button id="bBall">Throw ball</button>
 <button id="bClear">Clear balls</button>
 <button id="bReset">Reset all</button>
 </div>
 </section>
 <section>
 <div class="lbl">Tied strain basis k</div>
 <div class="seg" id="kSeg"></div>
 <div class="note">tower and balls share the same k-dim strain basis</div>
 </section>
 <section>
 <div class="lbl">Tower subspace r</div>
 <div class="seg" id="rSeg"></div>
 <div class="note">the tower's entire motion lives in r coordinates — shown live below</div>
 <canvas id="tower"></canvas>
 </section>
 <section>
 <div class="lbl">Telemetry</div>
 <div class="stat"><span>physics</span><b id="stMs">–</b></div>
 <div class="stat"><span>tower state</span><b id="stDof">–</b></div>
 <div class="stat"><span>balls</span><b id="stBalls">0</b></div>
 <div class="stat"><span>water</span><b id="stWater">0</b></div>
 <div class="stat"><span>max strain</span><b id="stStrain">–</b></div>
 </section>
 </aside>
</main>
<script>
const D = 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const NV=D.NV, NT=D.NT, dt=D.dt, gamma=D.wsub/(1+D.wsub);
const V0=D.V0.flat(), Tm=D.T.flat(), Bm=D.Bm.flat(), G=D.G.flat();
const wEl=D.wEl, mass=D.mass;
const Phi=D.Phi.map(row=>Float64Array.from(row));
const AV0=D.AV0.flat();
let k=2, r=8;
let x=Float64Array.from(V0), v=new Float64Array(2*NV);
const s=new Float64Array(2*NV), xn=new Float64Array(2*NV),
 rhs=new Float64Array(2*NV), strain=new Float64Array(NT);
const extA=new Float64Array(2*NV); // contact accel on tower
let qGlobal=new Float64Array(r);
function Ucols(){const U=[];for(let c=0;c<k;c++)U.push([D.U[0][c],D.U[1][c],D.U[2][c],D.U[3][c]]);return U}
let U=Ucols();
// ---- projectile template ----
const P=D.proj, PN=P.NV, PT=P.NT;
const PV0=P.V0.flat(), PTm=P.T.flat(), PBm=P.Bm.flat(), PG=P.G.flat();
const PwEl=P.wEl, Pmass=P.mass, PAInv=P.AInv;
const balls=[]; const MAXB=6;
function spawnBall(px,py,vx,vy){
 const b={x:new Float64Array(2*PN), v:new Float64Array(2*PN),
 s:new Float64Array(2*PN), xn:new Float64Array(2*PN),
 rhs:new Float64Array(2*PN), ext:new Float64Array(2*PN),
 strain:new Float64Array(PT)};
 for(let i=0;i<PN;i++){ b.x[2*i]=PV0[2*i]+px; b.x[2*i+1]=PV0[2*i+1]+py;
 b.v[2*i]=vx; b.v[2*i+1]=vy; }
 balls.push(b); if(balls.length>MAXB) balls.shift();
}
// generic tied local step + scatter (shared by tower & balls -- same physics)
function localScatter(nT,Tarr,Barr,Garr,wArr,xw,rh,strOut){
 for(let t=0;t<nT;t++){
 const a=Tarr[3*t],bb=Tarr[3*t+1],c=Tarr[3*t+2];
 const d0x=xw[2*bb]-xw[2*a], d0y=xw[2*bb+1]-xw[2*a+1];
 const d1x=xw[2*c]-xw[2*a], d1y=xw[2*c+1]-xw[2*a+1];
 const B0=Barr[4*t],B1=Barr[4*t+1],B2=Barr[4*t+2],B3=Barr[4*t+3];
 const F0=d0x*B0+d1x*B2, F1=d0x*B1+d1x*B3,
 F2=d0y*B0+d1y*B2, F3=d0y*B1+d1y*B3;
 let rx=F0+F3, ry=F1-F2; const nn=Math.hypot(rx,ry)||1e-12; rx/=nn; ry/=nn;
 const R0=rx,R1=ry,R2=-ry,R3=rx;
 const p0=R0*F0+R2*F2-1, p1=R0*F1+R2*F3, p2=R1*F0+R3*F2, p3=R1*F1+R3*F3-1;
 let q0=0,q1=0,q2=0,q3=0;
 for(let c2=0;c2<k;c2++){ const u=U[c2];
 const z=u[0]*p0+u[1]*p1+u[2]*p2+u[3]*p3;
 q0+=u[0]*z; q1+=u[1]*z; q2+=u[2]*z; q3+=u[3]*z; }
 const pn=Math.hypot(q0,q1,q2,q3);
 if(pn>0.5){ const sc2=0.5/pn; q0*=sc2;q1*=sc2;q2*=sc2;q3*=sc2; } // strain limiting (inversion guard)
 if(strOut) strOut[t]=Math.min(pn,0.5);
 const s0=1+gamma*q0, s1=gamma*q1, s2=gamma*q2, s3=1+gamma*q3;
 const P0=R0*s0+R1*s2, P1=R0*s1+R1*s3, P2=R2*s0+R3*s2, P3=R2*s1+R3*s3;
 const wt=wArr[t]*(1+D.wsub);
 const g0=Garr[6*t],g1=Garr[6*t+1],g2=Garr[6*t+2],g3=Garr[6*t+3],g4=Garr[6*t+4],g5=Garr[6*t+5];
 rh[2*a]+=wt*(g0*P0+g3*P1); rh[2*a+1]+=wt*(g0*P2+g3*P3);
 rh[2*bb]+=wt*(g1*P0+g4*P1); rh[2*bb+1]+=wt*(g1*P2+g4*P3);
 rh[2*c]+=wt*(g2*P0+g5*P1); rh[2*c+1]+=wt*(g2*P2+g5*P3);
 }
}
// ---- contacts: acceleration-based penalty (mass-independent gain, stable explicit) ----
const KA=2500, CA=25, RC=0.17;
function contacts(){
 extA.fill(0);
 for(const b of balls) b.ext.fill(0);
 for(let bi=0;bi<balls.length;bi++){
 const b=balls[bi];
 for(let i=0;i<PN;i++){
 const px=b.x[2*i], py=b.x[2*i+1];
 for(let j=0;j<NV;j++){
 const dx=px-x[2*j], dy=py-x[2*j+1];
 if(Math.abs(dx)>RC||Math.abs(dy)>RC) continue;
 const d=Math.hypot(dx,dy); if(d>=RC||d<1e-9) continue;
 const nx=dx/d, ny=dy/d, pen=RC-d;
 const vn=(b.v[2*i]-v[2*j])*nx+(b.v[2*i+1]-v[2*j+1])*ny;
 const a=Math.max(KA*pen-CA*vn,0);
 b.ext[2*i]+=a*nx; b.ext[2*i+1]+=a*ny;
 extA[2*j]-=a*nx*0.5; extA[2*j+1]-=a*ny*0.5;
 }
 if(py<0.0){
 const a=KA*(-py)-CA*b.v[2*i+1];
 b.ext[2*i+1]+=Math.max(a,0);
 b.ext[2*i]-=8*b.v[2*i]; // floor friction
 }
 }
 for(let bj=bi+1;bj<balls.length;bj++){
 const b2=balls[bj];
 for(let i=0;i<PN;i++) for(let j2=0;j2<PN;j2++){
 const dx=b.x[2*i]-b2.x[2*j2], dy=b.x[2*i+1]-b2.x[2*j2+1];
 if(Math.abs(dx)>RC||Math.abs(dy)>RC) continue;
 const d=Math.hypot(dx,dy); if(d>=RC||d<1e-9) continue;
 const nx=dx/d, ny=dy/d;
 const vn=(b.v[2*i]-b2.v[2*j2])*nx+(b.v[2*i+1]-b2.v[2*j2+1])*ny;
 const a=Math.max(KA*(RC-d)-CA*vn,0)*0.5;
 b.ext[2*i]+=a*nx; b.ext[2*i+1]+=a*ny;
 b2.ext[2*j2]-=a*nx; b2.ext[2*j2+1]-=a*ny;
 }
 }
 }
}
// ============== WATER: Position-Based Fluids (density constraint) ==============
const DTW=1/60;
const WH=0.11, RHO0=1.0, WSP=WH*0.55, WMASS=RHO0*WSP*WSP, WEPS=100.0, WITER=2;
const POLY6=4/(Math.PI*Math.pow(WH,8)), SPIKY=-30/(Math.PI*Math.pow(WH,5));
const XMIN=-4.4,XMAX=4.4,YMIN=0.02,YMAX=7.0;
let wN=0;
const WCAP=900;
const wx=new Float64Array(WCAP), wy=new Float64Array(WCAP),
 wvx=new Float64Array(WCAP), wvy=new Float64Array(WCAP),
 wpx=new Float64Array(WCAP), wpy=new Float64Array(WCAP),
 wl=new Float64Array(WCAP), wrho=new Float64Array(WCAP),
 wax=new Float64Array(WCAP), way=new Float64Array(WCAP);
function releaseWater(){
 wN=0;
 for(let gy=0; gy<22 && wN<WCAP; gy++)
 for(let gx=0; gx<16 && wN<WCAP; gx++){
 wx[wN]=-4.25+gx*WSP+((gy%2)?WSP*0.5:0);
 wy[wN]=2.0+gy*WSP;
 wvx[wN]=0; wvy[wN]=0; wN++;
 }
}
// spatial hash (rebuilt per substep)
const HGRID=new Map(); const TGRID=new Map();
function hkey(cx,cy){ return cx*100003+cy; }
function buildGrid(){
 HGRID.clear();
 for(let i=0;i<wN;i++){
 const kk=hkey(Math.floor(wpx[i]/WH),Math.floor(wpy[i]/WH));
 let l=HGRID.get(kk); if(!l){l=[];HGRID.set(kk,l);} l.push(i);
 }
}
function forEachNb(px,py,fn){
 const cx=Math.floor(px/WH), cy=Math.floor(py/WH);
 for(let ox=-1;ox<=1;ox++) for(let oy=-1;oy<=1;oy++){
 const l=HGRID.get(hkey(cx+ox,cy+oy)); if(!l) continue;
 for(const j of l) fn(j);
 }
}
let wPhase=0;
function waterStep(){
 if(wN===0) return;
 wPhase^=1; if(wPhase) return; // fluid runs at 60Hz
 for(let i=0;i<wN;i++){
 wvy[i]+= -9.8*DTW + way[i]*DTW; wvx[i]+= wax[i]*DTW;
 wpx[i]=wx[i]+DTW*wvx[i]; wpy[i]=wy[i]+DTW*wvy[i];
 }
 for(let it=0; it<WITER; it++){
 buildGrid();
 for(let i=0;i<wN;i++){
 let rho=0, gx=0, gy=0, g2=0;
 const px=wpx[i], py=wpy[i];
 forEachNb(px,py,(j)=>{
 const dx=px-wpx[j], dy=py-wpy[j], d2=dx*dx+dy*dy;
 if(d2>=WH*WH) return;
 const q=WH*WH-d2; rho+=WMASS*POLY6*q*q*q;
 const d=Math.sqrt(d2);
 if(d>1e-9){
 const c=SPIKY*(WH-d)*(WH-d)/d*WMASS/RHO0;
 const ggx=c*dx, ggy=c*dy;
 gx+=ggx; gy+=ggy; g2+=ggx*ggx+ggy*ggy;
 }
 });
 wrho[i]=rho;
 const C=rho/RHO0-1;
 wl[i]= C>0 ? -C/(g2+gx*gx+gy*gy+WEPS) : 0;
 }
 for(let i=0;i<wN;i++){
 let cx=0, cy=0;
 const px=wpx[i], py=wpy[i], li=wl[i];
 forEachNb(px,py,(j)=>{
 if(j===i) return;
 const dx=px-wpx[j], dy=py-wpy[j], d2=dx*dx+dy*dy;
 if(d2>=WH*WH) return;
 const d=Math.sqrt(d2); if(d<=1e-9) return;
 const c=SPIKY*(WH-d)*(WH-d)/d*WMASS/RHO0*(li+wl[j]);
 cx+=c*dx; cy+=c*dy;
 });
 wpx[i]=Math.min(Math.max(px+cx,XMIN),XMAX);
 wpy[i]=Math.min(Math.max(py+cy,YMIN),YMAX);
 }
 // position-level pushout from tower vertices (seals the lattice)
 TGRID.clear();
 for(let j=0;j<NV;j++){
 const kk=hkey(Math.floor(x[2*j]/WH),Math.floor(x[2*j+1]/WH));
 let l=TGRID.get(kk); if(!l){l=[];TGRID.set(kk,l);} l.push(j);
 }
 const RP=0.11;
 for(let i=0;i<wN;i++){
 const cx2=Math.floor(wpx[i]/WH), cy2=Math.floor(wpy[i]/WH);
 for(let ox=-2;ox<=2;ox++) for(let oy=-2;oy<=2;oy++){
 const l=TGRID.get(hkey(cx2+ox,cy2+oy)); if(!l) continue;
 for(const j of l){
 const dx=wpx[i]-x[2*j], dy=wpy[i]-x[2*j+1];
 const d=Math.hypot(dx,dy);
 if(d<RP&&d>1e-9){ const push=0.35*(RP-d)/d;
 wpx[i]+=dx*push; wpy[i]+=dy*push; }
 }
 }
 }
 }
 for(let i=0;i<wN;i++){
 let vx2=(wpx[i]-wx[i])/DTW*0.998, vy2=(wpy[i]-wy[i])/DTW*0.998;
 if(vx2>15)vx2=15; if(vx2<-15)vx2=-15; if(vy2>15)vy2=15; if(vy2<-15)vy2=-15;
 wvx[i]=vx2; wvy[i]=vy2; wx[i]=wpx[i]; wy[i]=wpy[i];
 }
 // XSPH viscosity
 buildGrid();
 for(let i=0;i<wN;i++){
 let dvx=0,dvy=0;
 forEachNb(wx[i],wy[i],(j)=>{
 const dx=wx[i]-wx[j], dy=wy[i]-wy[j], d2=dx*dx+dy*dy;
 if(d2>=WH*WH) return;
 const q=WH*WH-d2; const w2=POLY6*q*q*q*WMASS/RHO0*0.05;
 dvx+=w2*(wvx[j]-wvx[i]); dvy+=w2*(wvy[j]-wvy[i]);
 });
 wvx[i]+=dvx; wvy[i]+=dvy;
 }
}
// water<->solid coupling: accel penalty via water grid (uses wpx from LAST substep grid state)
function waterSolidContacts(){
 if(wN===0) return;
 for(let i=0;i<wN;i++){ wax[i]=0; way[i]=0; }
 // grid over CURRENT positions
 HGRID.clear();
 for(let i=0;i<wN;i++){
 const kk=hkey(Math.floor(wx[i]/WH),Math.floor(wy[i]/WH));
 let l=HGRID.get(kk); if(!l){l=[];HGRID.set(kk,l);} l.push(i);
 }
 const RCW=0.12, KAW=800, CAW=35;
 // tower vertices query nearby particles
 for(let j=0;j<NV;j++){
 const tx=x[2*j], ty=x[2*j+1];
 forEachNb(tx,ty,(i)=>{
 const dx=wx[i]-tx, dy=wy[i]-ty, d=Math.hypot(dx,dy);
 if(d>=RCW||d<1e-9) return;
 const nx=dx/d, ny=dy/d;
 const vn=(wvx[i]-v[2*j])*nx+(wvy[i]-v[2*j+1])*ny;
 const a=Math.max(KAW*(RCW-d)-CAW*vn,0);
 wax[i]+=a*nx; way[i]+=a*ny;
 extA[2*j]-=a*nx*0.25; extA[2*j+1]-=a*ny*0.25;
 });
 }
 // ball vertices
 for(const b of balls){
 for(let k2=0;k2<PN;k2++){
 const tx=b.x[2*k2], ty=b.x[2*k2+1];
 forEachNb(tx,ty,(i)=>{
 const dx=wx[i]-tx, dy=wy[i]-ty, d=Math.hypot(dx,dy);
 if(d>=RCW||d<1e-9) return;
 const nx=dx/d, ny=dy/d;
 const vn=(wvx[i]-b.v[2*k2])*nx+(wvy[i]-b.v[2*k2+1])*ny;
 const a=Math.max(KAW*(RCW-d)-CAW*vn,0);
 wax[i]+=a*nx; way[i]+=a*ny;
 b.ext[2*k2]-=a*nx*0.6; b.ext[2*k2+1]-=a*ny*0.6;
 });
 }
 }
}
let drag=-1, mouse=[0,0], throwing=false, throwStart=[0,0];
function step(){
 contacts();
 waterSolidContacts();
 // ---- tower (reduced solve) ----
 for(let i=0;i<NV;i++){
 let ax=extA[2*i], ay=-9.8+extA[2*i+1];
 if(drag===i){ ax+=120*(mouse[0]-x[2*i]); ay+=120*(mouse[1]-x[2*i+1]); }
 s[2*i]=x[2*i]+dt*v[2*i]+dt*dt*ax;
 s[2*i+1]=x[2*i+1]+dt*v[2*i+1]+dt*dt*ay;
 }
 xn.set(s);
 for(let it=0;it<5;it++){
 for(let i=0;i<NV;i++){ const m=mass[i]/(dt*dt);
 rhs[2*i]=m*s[2*i]-AV0[2*i]; rhs[2*i+1]=m*s[2*i+1]-AV0[2*i+1]; }
 localScatter(NT,Tm,Bm,G,wEl,xn,rhs,it===4?strain:null);
 const bq=new Float64Array(r);
 for(let c2=0;c2<r;c2++){ let acc=0; for(let i=0;i<2*NV;i++) acc+=Phi[i][c2]*rhs[i]; bq[c2]=acc; }
 const AI=D.ArInv[r]; qGlobal=new Float64Array(r);
 for(let i2=0;i2<r;i2++){ let acc=0; for(let j2=0;j2<r;j2++) acc+=AI[i2][j2]*bq[j2]; qGlobal[i2]=acc; }
 for(let i=0;i<2*NV;i++){ let acc=V0[i]; for(let c2=0;c2<r;c2++) acc+=Phi[i][c2]*qGlobal[c2]; xn[i]=acc; }
 }
 for(let i=0;i<2*NV;i++){ v[i]=(xn[i]-x[i])/dt*0.999; x[i]=xn[i]; }
 // ---- balls (full tiny dense solve, same local step) ----
 for(const b of balls){
 for(let i=0;i<PN;i++){
 b.s[2*i]=b.x[2*i]+dt*b.v[2*i]+dt*dt*b.ext[2*i];
 b.s[2*i+1]=b.x[2*i+1]+dt*b.v[2*i+1]+dt*dt*(-9.8+b.ext[2*i+1]);
 }
 b.xn.set(b.s);
 for(let it=0;it<5;it++){
 for(let i=0;i<PN;i++){ const m=Pmass[i]/(dt*dt);
 b.rhs[2*i]=m*b.s[2*i]; b.rhs[2*i+1]=m*b.s[2*i+1]; }
 localScatter(PT,PTm,PBm,PG,PwEl,b.xn,b.rhs,it===4?b.strain:null);
 for(let i=0;i<PN;i++){ let ax=0,ay=0;
 for(let j2=0;j2<PN;j2++){ const w2=PAInv[i][j2]; ax+=w2*b.rhs[2*j2]; ay+=w2*b.rhs[2*j2+1]; }
 b.xn[2*i]=ax; b.xn[2*i+1]=ay; }
 }
 for(let i=0;i<2*PN;i++){ let vv=(b.xn[i]-b.x[i])/dt*0.999;
 if(vv>15)vv=15; if(vv<-15)vv=-15;
 b.v[i]=vv; b.x[i]=b.xn[i]; }
 }
 waterStep();
}
// ---------- rendering ----------
const cv=document.getElementById('cv'), ctx=cv.getContext('2d');
const tw=document.getElementById('tower'), tctx=tw.getContext('2d');
function fit(cnv){ const d=window.devicePixelRatio||1;
 cnv.width=cnv.clientWidth*d; cnv.height=cnv.clientHeight*d;
 cnv.getContext('2d').setTransform(d,0,0,d,0,0); }
function world2px(){ const w=cv.clientWidth,h=cv.clientHeight;
 const sc=Math.min(w/9.6,h/(D.H+3.2));
 return {sc, ox:w/2, oy:h-30}; }
function lerp3(t){
 const c=[126,232,250], m=[199,125,255], p=[255,126,182];
 const a=t<.5?c:m, bb=t<.5?m:p, u=(t<.5?t:t-.5)*2;
 return `rgb(${a.map((v2,i)=>Math.round(v2+(bb[i]-v2)*u)).join(',')})`;
}
function drawMesh(nT,Tarr,xw,strArr,sref,X,Y){
 for(let t=0;t<nT;t++){
 const a=Tarr[3*t],bb=Tarr[3*t+1],c=Tarr[3*t+2], u=Math.min(strArr[t]/sref,1);
 ctx.strokeStyle=lerp3(u); ctx.globalAlpha=0.35+0.65*u; ctx.lineWidth=1;
 ctx.beginPath(); ctx.moveTo(X(xw,a),Y(xw,a)); ctx.lineTo(X(xw,bb),Y(xw,bb));
 ctx.lineTo(X(xw,c),Y(xw,c)); ctx.closePath(); ctx.stroke();
 }
 ctx.globalAlpha=1;
}
function draw(msPhys){
 const {sc,ox,oy}=world2px();
 ctx.clearRect(0,0,cv.clientWidth,cv.clientHeight);
 const X=(xw,i)=>ox+xw[2*i]*sc, Y=(xw,i)=>oy-xw[2*i+1]*sc;
 ctx.strokeStyle="#232A45"; ctx.lineWidth=2;
 ctx.beginPath(); ctx.moveTo(0,oy+1); ctx.lineTo(cv.clientWidth,oy+1); ctx.stroke();
 let smax=0;
 for(let t=0;t<NT;t++) smax=Math.max(smax,strain[t]);
 for(const b of balls) for(let t=0;t<PT;t++) smax=Math.max(smax,b.strain[t]);
 const sref=Math.max(smax,0.02);
 drawMesh(NT,Tm,x,strain,sref,X,Y);
 for(const b of balls) drawMesh(PT,PTm,b.x,b.strain,sref,X,Y);
 for(let i=0;i<wN;i++){
 const sp2=Math.hypot(wvx[i],wvy[i]);
 ctx.fillStyle=lerp3(Math.min(sp2/6,1));
 ctx.globalAlpha=0.85;
 const px2=ox+wx[i]*sc, py2=oy-wy[i]*sc;
 ctx.fillRect(px2-1.6,py2-1.6,3.2,3.2);
 }
 ctx.globalAlpha=1;
 for(const f of D.fixed){ ctx.fillStyle="#FF7EB6"; ctx.fillRect(X(x,f)-2.5,Y(x,f)-2.5,5,5); }
 if(drag>=0){ ctx.strokeStyle="#7EE8FA"; ctx.setLineDash([4,4]);
 ctx.beginPath(); ctx.moveTo(X(x,drag),Y(x,drag));
 ctx.lineTo(ox+mouse[0]*sc,oy-mouse[1]*sc); ctx.stroke(); ctx.setLineDash([]); }
 if(throwing){ ctx.strokeStyle="#C77DFF"; ctx.setLineDash([3,5]);
 ctx.beginPath(); ctx.moveTo(ox+throwStart[0]*sc,oy-throwStart[1]*sc);
 ctx.lineTo(ox+mouse[0]*sc,oy-mouse[1]*sc); ctx.stroke(); ctx.setLineDash([]);
 ctx.fillStyle="#C77DFF";
 ctx.beginPath(); ctx.arc(ox+throwStart[0]*sc,oy-throwStart[1]*sc,P.rad*sc,0,7); ctx.fill();
 ctx.globalAlpha=1; }
 document.getElementById('stMs').textContent=msPhys.toFixed(2)+" ms/frame";
 document.getElementById('stDof').textContent=r+" numbers";
 document.getElementById('stBalls').textContent=balls.length;
 document.getElementById('stWater').textContent=wN;
 document.getElementById('stStrain').textContent=(100*smax).toFixed(1)+" %";
 // latent tower
 const Wp=tw.clientWidth,Hp=tw.clientHeight;
 tctx.clearRect(0,0,Wp,Hp);
 const lay=Math.min(r,24), lh=(Hp-24)/lay, pw=Wp*0.4;
 let qm=0.02; for(let i=0;i<r;i++) qm=Math.max(qm,Math.abs(qGlobal[i]));
 for(let i=0;i<lay;i++){
 const q=qGlobal[i]/qm, cy2=Hp-14-i*lh, cx=Wp/2+q*Wp*0.22, t=i/Math.max(lay-1,1);
 tctx.strokeStyle=lerp3(t); tctx.globalAlpha=0.35+0.6*Math.min(Math.abs(q),1);
 tctx.beginPath();
 tctx.moveTo(cx-pw/2,cy2); tctx.lineTo(cx-pw/2+10,cy2-lh*0.55);
 tctx.lineTo(cx+pw/2+10,cy2-lh*0.55); tctx.lineTo(cx+pw/2,cy2);
 tctx.closePath(); tctx.stroke();
 tctx.fillStyle=tctx.strokeStyle;
 for(const dx of [-pw/2,pw/2]) tctx.fillRect(cx+dx-1.2,cy2-1.2,2.4,2.4);
 }
 tctx.globalAlpha=1;
}
// ---------- interaction ----------
function evtWorld(e){ const rect=cv.getBoundingClientRect(), {sc,ox,oy}=world2px();
 const px=(e.touches?e.touches[0].clientX:e.clientX)-rect.left,
 py=(e.touches?e.touches[0].clientY:e.clientY)-rect.top;
 return [(px-ox)/sc,(oy-py)/sc]; }
function down(e){ const [wx,wy]=evtWorld(e); mouse=[wx,wy];
 let best=-1,bd=1e9;
 for(let i=0;i<NV;i++){ const d=(x[2*i]-wx)**2+(x[2*i+1]-wy)**2; if(d<bd){bd=d;best=i} }
 if(bd<0.09){ drag=best; }
 else { throwing=true; throwStart=[wx,wy]; }
 e.preventDefault(); }
function move(e){ if(drag>=0||throwing){ mouse=evtWorld(e); e.preventDefault(); } }
function up(){
 if(throwing){
 let vx=(mouse[0]-throwStart[0])*4, vy=(mouse[1]-throwStart[1])*4;
 const sp=Math.hypot(vx,vy);
 if(sp>8){ vx*=8/sp; vy*=8/sp; }
 spawnBall(throwStart[0],Math.max(throwStart[1],P.rad+0.02),vx,vy);
 throwing=false;
 }
 drag=-1; }
cv.addEventListener('mousedown',down); addEventListener('mousemove',move); addEventListener('mouseup',up);
cv.addEventListener('touchstart',down,{passive:false});
addEventListener('touchmove',move,{passive:false}); addEventListener('touchend',up);
document.getElementById('bWater').onclick=()=>releaseWater();
document.getElementById('bBall').onclick=()=>spawnBall(-3.2,3.2,6.5,1.0);
document.getElementById('bClear').onclick=()=>{balls.length=0};
document.getElementById('bReset').onclick=()=>{x=Float64Array.from(V0);v.fill(0);balls.length=0;wN=0;};
function seg(id,opts,cur,cb){ const el=document.getElementById(id);
 el.innerHTML=''; for(const o of opts){ const b=document.createElement('button');
 b.textContent=o; if(o===cur)b.classList.add('on');
 b.onclick=()=>{cb(o); seg(id,opts,o,cb)}; el.appendChild(b);} }
seg('kSeg',[1,2,3],k,o=>{k=o;U=Ucols();});
seg('rSeg',D.rOptions,r,o=>{r=o;});
// ---------- main loop ----------
let acc=0,last=performance.now(),msAvg=0,lastW=0,lastH=0;
function loop(now){
 if(cv.clientWidth!==lastW||cv.clientHeight!==lastH){fit(cv);fit(tw);lastW=cv.clientWidth;lastH=cv.clientHeight;}
 acc+=Math.min(now-last,50); last=now;
 const t0=performance.now();
 while(acc>=1000/120){ step(); acc-=1000/120; }
 msAvg=msAvg*0.9+(performance.now()-t0)*0.1;
 draw(msAvg);
 requestAnimationFrame(loop);
}
requestAnimationFrame(loop);
</script>
</body>
</html>