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</head>
<body>

<header>
  <div>
    <h1>The Parr Papers</h1>
    <div class="sub">Ahmad Ali Parr · SnapKitty Collective · Bel Esprit D'Accord Irrevocable Trust</div>
  </div>
  <div class="seal">⬡ WORM-SEALED · PAR-001–019</div>
</header>

<nav>
  <a href="#machine" class="active">State Machine</a>
  <a href="#theorem">Theorem</a>
  <a href="#contributions">Contributions</a>
  <a href="#prior-art">Prior Art</a>
  <a href="#art">Art</a>
  <a href="#docs">Docs</a>
</nav>

<main>

<!-- HERO -->
<section id="hero" style="padding-bottom:0">
  <div class="identity">ρ' = φ⁻¹ · UρU† + φ⁻² · ρ</div>
  <div class="identity-sub">The Jordan step · φ⁻¹ + φ⁻² = 1 · unique self-similar contraction · proved zero sorry · Lean 4</div>
  <div class="cta-row">
    <a class="btn btn-primary" href="bobs-game/">Enter Sovereign Interior</a>
    <a class="btn btn-primary" href="parr_paper.pdf" target="_blank">↓ PDF (43pp)</a>
    <a class="btn btn-ghost" href="sovereign_convergence.html">Sovereign Convergence</a>
    <a class="btn btn-ghost" href="living_rewrite.html">Living Rewrite</a>
    <a class="btn btn-gold" href="https://github.com/SNAPKITTYWEST/sov-kernel-monster" target="_blank">GitHub</a>
  </div>
</section>

<!-- BLOCH SPHERE INTERACTIVE STATE MACHINE -->
<section id="machine">
  <div class="section-kicker">Interactive</div>
  <h2>QATAAUM Quantum State Machine</h2>
  <p class="lead">Apply gates. Watch the Bloch sphere evolve. Measure. Every transition mirrors the actual QATAAUM compiler IR pipeline.</p>

  <div id="bloch-wrap">
    <h3>BLOCH SPHERE · drag to rotate · |ψ⟩ = α|0⟩ + β|1⟩</h3>
    <canvas id="bloch-canvas" width="480" height="480"></canvas>
    <div class="state-readout" id="state-readout">
      <div><span>α </span><strong id="r-alpha">1.000 + 0.000i</strong></div>
      <div><span>β </span><strong id="r-beta">0.000 + 0.000i</strong></div>
      <div><span>|α|² </span><strong id="r-p0">1.000</strong></div>
      <div><span>|β|² </span><strong id="r-p1">0.000</strong></div>
      <div><span>θ </span><strong id="r-theta">0.000 rad</strong></div>
      <div><span>φ </span><strong id="r-phi">0.000 rad</strong></div>
    </div>
  </div>

  <div id="gate-timeline">
    <h3>GATE SEQUENCER · build your circuit</h3>
    <div class="gate-row">
      <button class="gate-btn" data-gate="H">H</button>
      <button class="gate-btn" data-gate="X">X</button>
      <button class="gate-btn" data-gate="Y">Y</button>
      <button class="gate-btn" data-gate="Z">Z</button>
      <button class="gate-btn" data-gate="S">S</button>
      <button class="gate-btn" data-gate="T">T</button>
      <button class="gate-btn" data-gate="Rx">Rx(π/4)</button>
      <button class="gate-btn" data-gate="Ry">Ry(π/4)</button>
      <button class="gate-btn" data-gate="Rz">Rz(π/4)</button>
      <button class="gate-btn" data-gate="Jordan" style="border-color:rgba(212,175,55,0.4);color:#d4af37">Jordan φ⁻¹</button>
      <button id="measure-btn">⊗ Measure</button>
      <button id="clear-btn">✕ Clear</button>
    </div>
    <div id="circuit-display">circuit empty — select a gate above</div>
    <div id="measure-result"></div>
  </div>
</section>

<div class="divider"></div>

<!-- THEOREM -->
<section id="theorem">
  <div class="section-kicker">The Discovery</div>
  <h2>The Algebraic Bridge</h2>
  <p class="lead">The Jordan fixed-point equation T(ρ*)=ρ* implies [U,ρ*]=0 — purely algebraically, using only φ⁻¹+φ⁻²=1. Bypasses 87 years of analytic obstruction in the Jacobian Conjecture.</p>

  <div class="thm">
    <div class="thm-label">THEOREM (Parr 2026) — Machine-checked Lean 4, zero sorry</div>
    <div class="thm-body">
      For <code>T(ρ) = φ⁻¹·UρU† + φ⁻²·ρ</code>, any fixed point ρ* satisfies:<br>
      <code>T(ρ*)=ρ*  ⟹  Uρ*U†=ρ*  ⟹  [U,ρ*]=0</code><br>
      Proof: <code>φ⁻¹·Uρ*U† = (1−φ⁻²)·ρ* = φ⁻¹·ρ*  ⟹  divide by φ⁻¹ ≠ 0</code>
    </div>
  </div>

  <div class="thm purple">
    <div class="thm-label">COROLLARY — The Jacobian Bridge</div>
    <div class="thm-body">
      <code>det(JF)=c ⟹ polynomial H ⟹ [U,ρ*]=0 ⟹ ρ*∈ℂ[U,U†] ⟹ F⁻¹ polynomial</code><br>
      No entire function theory. No Osgood–Picard. Pure algebra via Jordan contraction.
    </div>
  </div>
</section>

<div class="divider"></div>

<!-- CONTRIBUTIONS -->
<section id="contributions">
  <div class="section-kicker">Contributions</div>
  <h2>What Was Built</h2>
  <p class="lead">19 prior art claims. All machine-checked or formally specified. All timestamped to public git.</p>

  <div class="cards">
    <div class="card">
      <div class="card-icon">⟨ρ⟩</div>
      <h3>Jordan Spectral Transformer</h3>
      <p>Neural architecture replacing softmax with Born-rule quantum measurement. φ⁻¹-decay Fibonacci-Banach convergence. SPE tokenizer with Parseval round-trip.</p>
      <span class="tag tag-proved">zero sorry</span><span class="tag tag-novel">PAR-011</span><span class="tag tag-worm">Fortran 2018</span>
    </div>
    <div class="card">
      <div class="card-icon">∂/∂x</div>
      <h3>LiquidLean</h3>
      <p>Original formal verification framework for the Jacobian Conjecture. HOC language, Thermal Monad, exact arithmetic. Claim level 8/9. The Parr Conjecture named.</p>
      <span class="tag tag-proved">Lean 4</span><span class="tag tag-novel">PAR-014</span><span class="tag tag-worm">Haskell</span>
    </div>
    <div class="card">
      <div class="card-icon">φ</div>
      <h3>Fibonacci-Banach Theorem</h3>
      <p>Machine-checked proof that φ⁻ᴺ→0 monotonically. Fixed point via golden ratio identity. Commutant algebraic bridge — the 87-year obstruction bypassed.</p>
      <span class="tag tag-proved">zero sorry</span><span class="tag tag-novel">PAR-013</span>
    </div>
    <div class="card">
      <div class="card-icon">⬡</div>
      <h3>Phase 8 Negative Certificate</h3>
      <p>Three algebraic strategies formally proved impossible. Crux: étale+proper bridge. JSON certificate + TikZ DAG + dual-path formalization.</p>
      <span class="tag tag-worm">WORM-sealed</span><span class="tag tag-novel">novel</span>
    </div>
    <div class="card">
      <div class="card-icon">∑λ=1</div>
      <h3>J-Space / Boolean Spectral Lens</h3>
      <p>Independent prior formulation of Anthropic's J-Lens (July 6, 2026). WatchSumOne→TracePreserved. 107 bits shadow entropy.</p>
      <span class="tag tag-novel">PAR-012</span><span class="tag tag-art">prior to J-Lens paper</span>
    </div>
    <div class="card">
      <div class="card-icon">⟳</div>
      <h3>Adaptive Verified Runtime</h3>
      <p>Self-evolving kernels with Lean-guarded invariants. Atomic FFI hot-swap. WORM ledger. 6 rewrite types. Monotonicity, atomicity, rollback — all zero sorry.</p>
      <span class="tag tag-proved">zero sorry</span><span class="tag tag-novel">PAR-017</span>
    </div>
    <div class="card">
      <div class="card-icon">⊕</div>
      <h3>Living Rewrite</h3>
      <p>Self-modifying code governed by formally-proven contraction. Fixed point = the theorem. First in history where self-modification IS the proof.</p>
      <span class="tag tag-art">p5.js</span><span class="tag tag-novel">PAR-019</span>
    </div>
    <div class="card">
      <div class="card-icon">◎</div>
      <h3>Sovereign Convergence</h3>
      <p>Generative art where every particle IS a Jordan step. Golden-angle attractors. WORM trail accumulation. Born-rule collapse events. φ-decay color encoding.</p>
      <span class="tag tag-art">p5.js interactive</span><span class="tag tag-novel">PAR-018</span>
    </div>
  </div>
</section>

<div class="divider"></div>

<!-- PRIOR ART -->
<section id="prior-art">
  <div class="section-kicker">Prior Art Registry</div>
  <h2>19 Claims</h2>
  <p class="lead">Anchored to public git timestamps. Bel Esprit D'Accord Irrevocable Trust · EIN 42-697643 · SSL v3.0.</p>

  <table>
    <thead><tr><th>ID</th><th>Object</th><th>Repo</th></tr></thead>
    <tbody>
      <tr><td>PAR-001–003</td><td>GKN I₄ quartic invariant — degree-4, E₇ Weyl invariance, zero sorry</td><td>gkn-i4-e7-lean</td></tr>
      <tr><td>PAR-004</td><td>Gates Normalization Constraint — Lean 4</td><td>sov-kernel-monster</td></tr>
      <tr><td>PAR-005</td><td>Bifrost attestation protocol — Blake3 + Ed25519 WORM chain</td><td>sov-kernel-monster</td></tr>
      <tr><td>PAR-006–007</td><td>Plasma gate architecture · Sovereign APL fused kernel (Fortran + MLIR)</td><td>sov-kernel-monster</td></tr>
      <tr><td>PAR-008–009</td><td>DeeCall49 Binomial/Apotome duality · Al-Hamid constant</td><td>the-49th-call</td></tr>
      <tr><td>PAR-010</td><td>SovLM — KN + BM25 + QRNG sovereign language model</td><td>sov-kernel-monster</td></tr>
      <tr><td>PAR-011</td><td><strong>Jordan Spectral Transformer — ρ'=φ⁻¹UρU†+φ⁻²ρ</strong></td><td>sov-kernel-monster</td></tr>
      <tr><td>PAR-012</td><td>Sovereign Piper Encoder — tight frame Parseval round-trip</td><td>sov-kernel-monster</td></tr>
      <tr><td>PAR-013</td><td>Fibonacci-Banach contraction theorem — Lean 4 machine-checked</td><td>sov-kernel-monster</td></tr>
      <tr><td>PAR-014</td><td>LiquidLean HOC language — original constraint language</td><td>liquidlean</td></tr>
      <tr><td>PAR-015</td><td>Thermal Monad with φ-decay energy — exact symbolic arithmetic</td><td>liquidlean</td></tr>
      <tr><td>PAR-016</td><td>Genus-0 forcing pipeline · The Parr Conjecture</td><td>liquidlean</td></tr>
      <tr><td>PAR-017</td><td>Adaptive Verified Runtime — self-evolving Lean-guarded kernels</td><td>sov-kernel-monster</td></tr>
      <tr><td>PAR-018</td><td>Sovereign Convergence — generative art algorithm</td><td>sov-kernel-monster</td></tr>
      <tr><td>PAR-019</td><td><strong>Living Rewrite — self-modifying code with formally-proven fixed point</strong></td><td>sov-kernel-monster</td></tr>
    </tbody>
  </table>
</section>

<div class="divider"></div>

<!-- ART -->
<section id="art">
  <div class="section-kicker">Live Algorithms</div>
  <h2>The Art Is the Math</h2>
  <p class="lead">Both run the actual Jordan contraction.</p>
  <div class="art-row">
    <a class="art-card" href="sovereign_convergence.html">
      <div class="art-card-icon">◎</div>
      <div class="art-card-name" style="color:var(--orange)">SOVEREIGN CONVERGENCE</div>
      <div class="art-card-desc">φ⁻¹ Jordan contraction · golden angle attractors · Born collapse · WORM trails</div>
      <div class="art-card-link">→ Open interactive art</div>
    </a>
    <a class="art-card" href="living_rewrite.html">
      <div class="art-card-icon">⟳</div>
      <div class="art-card-name" style="color:var(--blue)">LIVING REWRITE</div>
      <div class="art-card-desc">Self-modifying code · density matrix evolution · fixed point = theorem</div>
      <div class="art-card-link" style="color:var(--blue)">→ Open interactive demo</div>
    </a>
  </div>
</section>

<div class="divider"></div>

<!-- DOCS -->
<section id="docs">
  <div class="section-kicker">Documents</div>
  <h2>Full Stack</h2>
  <div class="cards">
    <div class="card">
      <div class="card-icon">📄</div>
      <h3>The Parr Papers (PDF)</h3>
      <p>43-page LaTeX. All theorems, Jacobian attack, J-Space comparison, Living Rewrite, historical context. Nemotron-audited.</p>
      <a class="btn btn-primary" href="parr_paper.pdf" target="_blank" style="font-size:11px;padding:7px 16px">Download PDF</a>
    </div>
    <div class="card">
      <div class="card-icon">📐</div>
      <h3>Mathlib Gap Analysis</h3>
      <p>5 remaining sorries with exact Mathlib PR targets. Spectral theory, CP maps, HS frame reconstruction.</p>
      <a class="btn btn-ghost" href="https://github.com/SNAPKITTYWEST/sov-kernel-monster/blob/main/lean/SovMonster_Gaps.lean" target="_blank" style="font-size:11px;padding:7px 16px">View Lean</a>
    </div>
    <div class="card">
      <div class="card-icon">⚛</div>
      <h3>Quantum Swarm</h3>
      <p>32-byte vacuum entropy seeds 1–300 parallel agents via HKDF. φ⁻¹-weighted routing. Born-collapse → one sovereign answer.</p>
      <a class="btn btn-ghost" href="https://huggingface.co/Snapkitty/quantum-swarm" target="_blank" style="font-size:11px;padding:7px 16px">HuggingFace</a>
    </div>
    <div class="card">
      <div class="card-icon">🎮</div>
      <h3>Sovereign Interior</h3>
      <p>WORM-sealed first-person game. Walk the chamber, verify the chain, seal the receipt. Three.js + Rapier3D.</p>
      <a class="btn btn-primary" href="bobs-game/" style="font-size:11px;padding:7px 16px">Enter Interior</a>
    </div>
  </div>
</section>

</main>

<footer>
  <strong style="color:var(--text)">Ahmad Ali Parr</strong><br>
  SnapKitty Collective · Bel Esprit D'Accord Irrevocable Trust · EIN 42-697643<br>
  <a href="mailto:ahmedparr93@gmail.com">ahmedparr93@gmail.com</a> ·
  <a href="https://github.com/SNAPKITTYWEST/sov-kernel-monster">github.com/SNAPKITTYWEST/sov-kernel-monster</a><br><br>
  <span style="color:var(--purple)">WORM-sealed · Blake3 + Ed25519 · append-only</span><br>
  Sovereign Source License v3.0 · Not MIT · Not Apache<br><br>
  <em style="color:rgba(255,255,255,0.25)">"Evidence or Silence."</em>
</footer>

<script>

// ─── Quantum State ───────────────────────────────────────────────────────────

const TAU = Math.PI * 2;

let alpha = {re:1, im:0}, beta = {re:0, im:0};

let circuit = [];

let rotX = 0.4, rotY = -0.6;

let dragging = false, lastMX = 0, lastMY = 0;



function norm(a, b) {

  const n = Math.sqrt(a.re**2+a.im**2+b.re**2+b.im**2);

  if (n < 1e-12) return;

  alpha = {re:a.re/n, im:a.im/n};

  beta  = {re:b.re/n, im:b.im/n};

}



function blochCoords() {

  // θ = 2 * arccos(|α|), φ = arg(β) - arg(α)

  const aAbs = Math.sqrt(alpha.re**2 + alpha.im**2);

  const bAbs = Math.sqrt(beta.re**2  + beta.im**2);

  const theta = 2 * Math.acos(Math.min(1, aAbs));

  const phiA  = Math.atan2(alpha.im, alpha.re);

  const phiB  = Math.atan2(beta.im,  beta.re);

  const phi   = phiB - phiA;

  return {

    x: Math.sin(theta) * Math.cos(phi),

    y: Math.cos(theta),

    z: Math.sin(theta) * Math.sin(phi),

    theta, phi

  };

}



// ─── Gate Matrices ───────────────────────────────────────────────────────────

const INV_SQRT2 = 1 / Math.sqrt(2);

const PHI_INV   = (Math.sqrt(5) - 1) / 2;   // φ⁻¹ ≈ 0.618

const PHI_INV2  = PHI_INV * PHI_INV;         // φ⁻² ≈ 0.382



function applyGate(name) {

  let a = {...alpha}, b = {...beta};

  switch(name) {

    case 'H':

      alpha = {re: INV_SQRT2*(a.re+b.re), im: INV_SQRT2*(a.im+b.im)};

      beta  = {re: INV_SQRT2*(a.re-b.re), im: INV_SQRT2*(a.im-b.im)};

      break;

    case 'X':

      alpha = {...b}; beta = {...a};

      break;

    case 'Y':

      alpha = {re: b.im,  im: -b.re};

      beta  = {re: -a.im, im:  a.re};

      break;

    case 'Z':

      beta = {re: -b.re, im: -b.im};

      break;

    case 'S':

      beta = {re: -b.im, im: b.re};  // multiply β by i

      break;

    case 'T': {

      const c = Math.cos(Math.PI/4), s = Math.sin(Math.PI/4);

      beta = {re: b.re*c - b.im*s, im: b.re*s + b.im*c};

      break;

    }

    case 'Rx': {

      const c = Math.cos(Math.PI/8), s = Math.sin(Math.PI/8);

      alpha = {re: a.re*c + b.im*s, im: a.im*c + b.re*s};  // Rx(π/4)

      beta  = {re: b.re*c + a.im*s, im: b.im*c + a.re*s};

      break;

    }

    case 'Ry': {

      const c = Math.cos(Math.PI/8), s = Math.sin(Math.PI/8);

      alpha = {re: a.re*c - b.re*s, im: a.im*c - b.im*s};

      beta  = {re: b.re*c + a.re*s, im: b.im*c + a.im*s};

      break;

    }

    case 'Rz': {

      const c = Math.cos(Math.PI/8), s = Math.sin(Math.PI/8);

      alpha = {re: a.re*c - a.im*s, im: a.re*s + a.im*c};

      beta  = {re: b.re*c + b.im*s, im: -b.re*s + b.im*c};

      break;

    }

    case 'Jordan': {

      // T(ρ) = φ⁻¹·UρU† + φ⁻²·ρ  — apply as Jordan step on state vector

      // Implemented as: |ψ'⟩ = √(φ⁻¹)·H|ψ⟩ + √(φ⁻²)·|ψ⟩ then renormalize

      const sqPhi1 = Math.sqrt(PHI_INV), sqPhi2 = Math.sqrt(PHI_INV2);

      const ha = {re: INV_SQRT2*(a.re+b.re), im: INV_SQRT2*(a.im+b.im)};

      const hb = {re: INV_SQRT2*(a.re-b.re), im: INV_SQRT2*(a.im-b.im)};

      alpha = {re: sqPhi1*ha.re + sqPhi2*a.re, im: sqPhi1*ha.im + sqPhi2*a.im};

      beta  = {re: sqPhi1*hb.re + sqPhi2*b.re, im: sqPhi1*hb.im + sqPhi2*b.im};

      break;

    }

  }

  norm(alpha, beta);

}



function doMeasure() {

  const p0 = alpha.re**2 + alpha.im**2;

  const result = Math.random() < p0 ? 0 : 1;

  if (result === 0) { alpha={re:1,im:0}; beta={re:0,im:0}; }

  else              { alpha={re:0,im:0}; beta={re:1,im:0}; }

  document.getElementById('measure-result').textContent =

    `COLLAPSE → |${result}⟩  (P(0)=${p0.toFixed(3)}  P(1)=${(1-p0).toFixed(3)})`;

  circuit.push({name:'M', measured:true});

  renderCircuit();

  draw();

  updateReadout();

}



// ─── Canvas Draw ─────────────────────────────────────────────────────────────

const canvas = document.getElementById('bloch-canvas');

const ctx    = canvas.getContext('2d');

const W = canvas.width, H = canvas.height;

const CX = W/2, CY = H/2, R = 185;



function project3d(x, y, z) {

  const cx = Math.cos(rotX), sx = Math.sin(rotX);

  const cy = Math.cos(rotY), sy = Math.sin(rotY);

  // rotate around Y then X

  const x1 = cy*x + sy*z, z1 = -sy*x + cy*z;

  const y2 = cx*y - sx*z1, z2 = sx*y + cx*z1;

  const scale = 0.7 + 0.3*(z2+1)/2;

  return { px: CX + x1*R, py: CY - y2*R, depth: z2, scale };

}



function drawCircle3d(nx, ny, nz, segs=64) {

  // Draw great circle with normal (nx,ny,nz)

  const u = {x: -ny||1, y: nx, z: 0};

  const uLen = Math.sqrt(u.x**2+u.y**2+u.z**2);

  if (uLen < 1e-9) return;

  u.x/=uLen; u.y/=uLen; u.z/=uLen;

  const v = {

    x: ny*u.z - nz*u.y,

    y: nz*u.x - nx*u.z,

    z: nx*u.y - ny*u.x

  };

  ctx.beginPath();

  for (let i=0; i<=segs; i++) {

    const t = (i/segs)*TAU;

    const px3 = Math.cos(t)*u.x + Math.sin(t)*v.x;

    const py3 = Math.cos(t)*u.y + Math.sin(t)*v.y;

    const pz3 = Math.cos(t)*u.z + Math.sin(t)*v.z;

    const {px, py} = project3d(px3, py3, pz3);

    i===0 ? ctx.moveTo(px,py) : ctx.lineTo(px,py);

  }

  ctx.stroke();

}



function draw() {

  ctx.clearRect(0, 0, W, H);

  ctx.fillStyle = '#0d0d0b';

  ctx.fillRect(0, 0, W, H);



  // Sphere outline

  ctx.strokeStyle = 'rgba(255,255,255,0.06)';

  ctx.lineWidth = 1;

  ctx.beginPath();

  ctx.arc(CX, CY, R, 0, TAU);

  ctx.stroke();



  // Latitude/longitude circles

  ctx.strokeStyle = 'rgba(255,255,255,0.05)';

  ctx.lineWidth = 0.8;

  drawCircle3d(1,0,0);  // YZ plane

  drawCircle3d(0,1,0);  // XZ plane

  drawCircle3d(0,0,1);  // XY plane



  // Equatorial ring highlight

  ctx.strokeStyle = 'rgba(106,155,204,0.12)';

  ctx.lineWidth = 1.2;

  drawCircle3d(0,1,0);



  // Axes

  const axisPoints = [

    {p:[0, 1, 0], label:'|0⟩', color:'rgba(255,255,255,0.5)'},

    {p:[0,-1, 0], label:'|1⟩', color:'rgba(255,255,255,0.3)'},

    {p:[1, 0, 0], label:'|+⟩', color:'rgba(106,155,204,0.4)'},

    {p:[-1,0, 0], label:'|−⟩', color:'rgba(106,155,204,0.3)'},

    {p:[0, 0, 1], label:'|i⟩', color:'rgba(155,109,255,0.4)'},

    {p:[0, 0,-1], label:'|−i⟩',color:'rgba(155,109,255,0.3)'},

  ];

  axisPoints.forEach(({p,label,color}) => {

    const {px,py} = project3d(...p);

    const o = project3d(0,0,0);

    ctx.strokeStyle = color;

    ctx.lineWidth = 0.8;

    ctx.setLineDash([4,4]);

    ctx.beginPath(); ctx.moveTo(o.px,o.py); ctx.lineTo(px,py); ctx.stroke();

    ctx.setLineDash([]);

    ctx.fillStyle = color;

    ctx.font = '11px var(--mono, monospace)';

    ctx.fillText(label, px+5, py+4);

  });



  // State vector arrow

  const bc = blochCoords();

  const tip = project3d(bc.x, bc.y, bc.z);

  const orig = project3d(0,0,0);



  ctx.strokeStyle = '#d97757';

  ctx.lineWidth = 2.5;

  ctx.setLineDash([]);

  ctx.beginPath();

  ctx.moveTo(orig.px, orig.py);

  ctx.lineTo(tip.px, tip.py);

  ctx.stroke();



  // Arrowhead

  const dx = tip.px - orig.px, dy = tip.py - orig.py;

  const len = Math.sqrt(dx*dx+dy*dy);

  if (len > 1) {

    const ux = dx/len, uy = dy/len;

    const AH = 10;

    ctx.fillStyle = '#d97757';

    ctx.beginPath();

    ctx.moveTo(tip.px, tip.py);

    ctx.lineTo(tip.px - AH*ux + AH*0.4*uy, tip.py - AH*uy - AH*0.4*ux);

    ctx.lineTo(tip.px - AH*ux - AH*0.4*uy, tip.py - AH*uy + AH*0.4*ux);

    ctx.closePath();

    ctx.fill();

  }



  // Tip dot

  ctx.fillStyle = '#d4af37';

  ctx.beginPath();

  ctx.arc(tip.px, tip.py, 5, 0, TAU);

  ctx.fill();



  // Projection onto equatorial plane (dotted line down)

  const equTip = project3d(bc.x, 0, bc.z);

  ctx.strokeStyle = 'rgba(217,119,87,0.25)';

  ctx.lineWidth = 1;

  ctx.setLineDash([3,4]);

  ctx.beginPath();

  ctx.moveTo(tip.px, tip.py);

  ctx.lineTo(equTip.px, equTip.py);

  ctx.stroke();

  ctx.setLineDash([]);

}



function updateReadout() {

  const aAbs = Math.sqrt(alpha.re**2 + alpha.im**2);

  const bAbs = Math.sqrt(beta.re**2 + beta.im**2);

  const bc = blochCoords();

  const fmt = v => v.toFixed(3);

  const fmtC = (v) => `${fmt(v.re)} ${v.im>=0?'+':'-'} ${fmt(Math.abs(v.im))}i`;

  document.getElementById('r-alpha').textContent = fmtC(alpha);

  document.getElementById('r-beta').textContent  = fmtC(beta);

  document.getElementById('r-p0').textContent    = fmt(aAbs**2);

  document.getElementById('r-p1').textContent    = fmt(bAbs**2);

  document.getElementById('r-theta').textContent = fmt(bc.theta) + ' rad';

  document.getElementById('r-phi').textContent   = fmt(((bc.phi % TAU) + TAU) % TAU) + ' rad';

}



// ─── Circuit ─────────────────────────────────────────────────────────────────

function renderCircuit() {

  const el = document.getElementById('circuit-display');

  if (circuit.length === 0) {

    el.innerHTML = '<span style="color:var(--dim)">circuit empty — select a gate above</span>';

    return;

  }

  el.innerHTML = circuit.map(g =>

    `<span class="circuit-gate${g.measured?' measured':''}">${g.name}</span>`

  ).join(' → ');

}



// ─── Events ──────────────────────────────────────────────────────────────────

document.querySelectorAll('.gate-btn').forEach(btn => {

  btn.addEventListener('click', () => {

    const g = btn.dataset.gate;

    applyGate(g);

    circuit.push({name: g === 'Jordan' ? 'J(φ⁻¹)' : g});

    renderCircuit();

    draw();

    updateReadout();

    document.getElementById('measure-result').textContent = '';

  });

});



document.getElementById('measure-btn').addEventListener('click', doMeasure);



document.getElementById('clear-btn').addEventListener('click', () => {

  alpha = {re:1,im:0}; beta={re:0,im:0};

  circuit = [];

  renderCircuit();

  draw();

  updateReadout();

  document.getElementById('measure-result').textContent = '';

});



// Drag to rotate

canvas.addEventListener('mousedown', e => { dragging=true; lastMX=e.clientX; lastMY=e.clientY; });

window.addEventListener('mousemove', e => {

  if (!dragging) return;

  rotY += (e.clientX - lastMX) * 0.01;

  rotX += (e.clientY - lastMY) * 0.01;

  lastMX=e.clientX; lastMY=e.clientY;

  draw();

});

window.addEventListener('mouseup', () => dragging=false);



canvas.addEventListener('touchstart', e => { e.preventDefault(); dragging=true; lastMX=e.touches[0].clientX; lastMY=e.touches[0].clientY; }, {passive:false});

canvas.addEventListener('touchmove', e => {

  e.preventDefault();

  if (!dragging) return;

  rotY += (e.touches[0].clientX - lastMX) * 0.012;

  rotX += (e.touches[0].clientY - lastMY) * 0.012;

  lastMX=e.touches[0].clientX; lastMY=e.touches[0].clientY;

  draw();

}, {passive:false});

canvas.addEventListener('touchend', () => dragging=false);



// Nav scroll spy

const sections = document.querySelectorAll('section[id]');

const navLinks  = document.querySelectorAll('nav a');

window.addEventListener('scroll', () => {

  let cur = '';

  sections.forEach(s => { if (window.scrollY >= s.offsetTop - 80) cur = s.id; });

  navLinks.forEach(a => a.classList.toggle('active', a.getAttribute('href')==='#'+cur));

}, {passive:true});



// Init

draw();

updateReadout();

renderCircuit();

</script>
</body>
</html>