Mesh Synchronization

Build "Mesh Synchronization"—a one-page quantum dance topology app for Lovable FREE tier (~5 credits, ONE build message).

**CONSTRAINTS:**
- No auth, no database, no Lovable Cloud. Static only.
- One workspace page + small "About the quantum" footer strip.
- Ship in ONE message.

**REAL QUANTUM (build-time only):**
1. pip install guppylang selene-sim in sandbox.
2. Create quantum/kernel.py: a real .py file with a @guppy Quantum Topological Data Analysis kernel that computes Betti numbers (b0, b1, b2) from Vietoris-Rips filtration on 32-point ensemble synchronization data across 8 filtration radii. Use selene_sim.build, Quest, run_shots APIs. Kernel must be ≤8 qubits, 256 shots.
3. Create quantum/run.py: import kernel, compile, run on Selene emulator, write results to src/data/quantum-results.json as array of {input, output} where output = {b0, b1, b2}. Execute ONCE at build; commit real results.
4. Do NOT call Python at runtime (Cloudflare Workers).

**FRONTEND (TanStack Start, route /):**
- Import src/data/quantum-results.json statically.
- Layout: use shadcn Card (left: filtration-radius Slider + Badge showing active record), center (Tabs for "Topology" + "Trace"), right (3 Badges: b0 clusters, b1 loops, b2 voids).
- "Topology" tab: 2D scatter plot of 32 points, colored by cluster membership (b0).
- "Trace" tab: Card showing raw {b0, b1, b2} + collapsible code block printing quantum/kernel.py source inline (prove it's real QTDA).
- Dark workspace, one accent color (deep purple or teal for dance mood), one typeface (Inter or Geist).
- Micro-interaction: Slider change → fade + scale transition on center plot.

**BUILD ORDER (prevent scope creep):**
1. Write quantum/kernel.py (Guppy QTDA kernel, real file).
2. Write quantum/run.py (compile, run, save JSON).
3. Execute quantum/run.py; commit src/data/quantum-results.json.
4. Create TanStack Start page, import JSON, build Slider + Tabs + Badges.
5. Add Trace tab with inline kernel source.
6. Style dark + accent color + micro-interactions.
7. Deploy.

**Deliverable:** Every number on screen traces to real Selene output from quantum/kernel.py.

--- QUANTINUUM APPENDIX (self-contained reference — paste verbatim) ---
[1] INSTALL (Lovable Linux sandbox):
  python -m pip install --no-cache-dir guppylang selene-sim

[2] IMPORTS (use exactly these — no other quantum libs):
  # quantum/kernel.py and quantum/run.py top
  from guppylang import guppy
  from guppylang.std.builtins import result
  from guppylang.std.quantum import qubit, h, cx, rx, ry, rz, measure, discard, t as tgate, tdg
  from guppylang.std.angles import angle, pi
  from selene_sim import build, Quest
  import math, json, sys, tempfile, importlib.util, uuid
  from pathlib import Path

[3] HARD RULES (violating any breaks the build):
  - @guppy reads source via inspect.getsource → kernels MUST live in a real .py file on disk. No exec(), no REPL strings, no inline templates.
  - Allowed gate set ONLY: h, rx, ry, rz, cx, tgate, tdg. There is NO native ccx/toffoli, cswap, cphase, or crz — decompose using the snippets in [7].
  - Qubit ownership: a qubit passed to a function is moved. You MUST measure() or discard() every qubit exactly once; never reuse after measure.
  - Angle hygiene before baking a float into generated source:
        theta = ((theta + math.pi) % (2.0 * math.pi)) - math.pi
    and write it with repr: f"... {theta!r} ..."  (str(float) can truncate).

[4] SELENE SHOT LOOP (canonical):
  compiled = my_kernel.compile()
  runner   = build(compiled)
  shots = []
  for shot in runner.run_shots(Quest(), n_qubits=N, n_shots=S):
      shots.append({str(lbl): int(v) for lbl, v in shot})
  # N = MAX number of qubits simultaneously LIVE in the kernel.
  # measure(q) releases the slot, so one ancilla reused across k windows still counts as 1.

[5] DRIVER PATTERN — sweep a kernel over many inputs (closures do NOT work):
  ROOT = Path(__file__).resolve().parent.parent
  if str(ROOT) not in sys.path: sys.path.insert(0, str(ROOT))

  def run_one(params: dict, shots: int = 256):
      # Bake params as literals into a fresh .py file that imports your kernel helpers.
      src = (
          "from quantum.kernel import guppy, my_helper\n"
          "@guppy\n"
          "def program() -> None:\n"
          f"    my_helper({params['a']!r}, {params['b']!r})\n"
      )
      tmp = Path(tempfile.gettempdir()) / "qprogs"; tmp.mkdir(exist_ok=True)
      name = f"prog_{uuid.uuid4().hex[:8]}"
      path = tmp / f"{name}.py"; path.write_text(src)
      spec = importlib.util.spec_from_file_location(name, path)
      mod  = importlib.util.module_from_spec(spec)
      sys.modules[name] = mod                       # register BEFORE exec_module
      spec.loader.exec_module(mod)
      runner = build(mod.program.compile())
      out = []
      for shot in runner.run_shots(Quest(), n_qubits=5, n_shots=shots):
          out.append({str(l): int(v) for l, v in shot})
      return out

[6] PER-QUBIT INTEGER DECODE (host-side):
  # kernel emits: for j in range(n): result(f"x{j}", measure(q[j]))
  def decode(rec, n):
      x = 0
      for j in range(n): x |= (rec.get(f"x{j}", 0) & 1) << j
      return x

[7] DECOMPOSITION LIBRARY (copy verbatim into quantum/kernel.py):

  # ---- Toffoli (CCX) from H, CX, T, Tdg — 6-T standard decomposition ----
  @guppy
  def toffoli(c1: qubit, c2: qubit, tgt: qubit) -> None:
      h(tgt)
      cx(c2, tgt); tdg(tgt)
      cx(c1, tgt); tgate(tgt)
      cx(c2, tgt); tdg(tgt)
      cx(c1, tgt); tgate(c2); tgate(tgt)
      h(tgt)
      cx(c1, c2); tgate(c1); tdg(c2)
      cx(c1, c2)

  # ---- CSWAP (Fredkin) from CX + Toffoli ----
  @guppy
  def cswap(c: qubit, a: qubit, b: qubit) -> None:
      cx(b, a)
      toffoli(c, a, b)
      cx(b, a)

  # ---- Controlled phase exp(i*theta) on |11> from rz + cx ----
  @guppy
  def cphase(c: qubit, d: qubit, theta: float) -> None:
      rz(d, angle(theta / 2.0))
      cx(c, d)
      rz(d, angle(-theta / 2.0))
      cx(c, d)

  # ---- Amplitude-encoded feature state (3 floats in [0,1] → 2-qubit state) ----
  @guppy
  def prep_features(q0: qubit, q1: qubit, a: float, b: float, c: float) -> None:
      ry(q0, angle(a))
      ry(q1, angle(b))
      cx(q0, q1)
      rz(q1, angle(c))

  # ---- SWAP test kernel; HOST inverts:  F = clamp(2*P(anc=0) - 1, 0, 1) ----
  @guppy
  def swap_test(ai: float, bi: float, ci: float,
                aj: float, bj: float, cj: float) -> None:
      anc = qubit()
      pi0 = qubit(); pi1 = qubit()
      pj0 = qubit(); pj1 = qubit()
      prep_features(pi0, pi1, ai, bi, ci)
      prep_features(pj0, pj1, aj, bj, cj)
      h(anc)
      cswap(anc, pi0, pj0)
      cswap(anc, pi1, pj1)
      h(anc)
      result("anc", measure(anc))
      discard(pi0); discard(pi1); discard(pj0); discard(pj1)
  # n_qubits = 5 for swap_test above.

[8] CLASSICAL CROSS-CHECK (NumPy reference — commit alongside quantum result):
  import numpy as np
  I = np.eye(2); X = np.array([[0,1],[1,0]])
  def RY(t): c,s = math.cos(t/2), math.sin(t/2); return np.array([[c,-s],[s,c]])
  def RZ(t): return np.array([[np.exp(-1j*t/2),0],[0,np.exp(1j*t/2)]])
  CX = np.array([[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]])
  def prep_psi(a,b,c):
      s = np.kron(RY(a)@np.array([1,0]), RY(b)@np.array([1,0]))
      s = CX @ s
      s = np.kron(I, RZ(c)) @ s
      return s
  def classical_fidelity(p, q):
      return float(abs(np.vdot(prep_psi(*p), prep_psi(*q)))**2)
  # Commit BOTH quantum and classical values per record:
  #   {"input": [...], "quantum": 0.873, "classical": 0.881, "shots": 256}

[9] FRONTEND HANDOFF (TanStack Start):
  # quantum/run.py last step:
  Path("src/data/quantum-results.json").write_text(json.dumps({
      "records": records,                          # list of {input, quantum, classical, ...}
      "circuit": {"n_qubits": 5, "shots": 256},
      "kernel_source": Path("quantum/kernel.py").read_text(),  # show on /code tab
  }, indent=2))
  # In React:  import data from "@/data/quantum-results.json"  (resolveJsonModule:true).
  # Render data.records; render data.kernel_source in a <pre> inside the Quantum trace tab.

[10] BUILD ORDER (do not deviate — keeps you under 5 credits):
  1. Write quantum/kernel.py with the helpers from [7] you actually need.
  2. Write quantum/run.py: driver pattern from [5], Selene loop from [4], cross-check from [8], JSON write from [9].
  3. Execute once:  python quantum/run.py   (commit src/data/quantum-results.json).
  4. Build the React route: static JSON import, shadcn Card + Tabs + Slider + Badge wiring.
  5. Deploy. No Python at runtime; every on-screen number traces to committed Selene output.

[11] OUTPUT CONTRACT — selene_run v1 (write this exact shape to src/data/quantum-results.json):
  {
    "schemaVersion": 1,
    "experiment": "<idea-slug>",
    "title": "<idea title>",
    "description": "<one plain-English sentence — what a non-quantum dev sees>",
    "kernel": { "snippet": "<first 40 lines of quantum/kernel.py>", "qubits": <int>, "shotsPerRow": <int> },
    "verdict": { "text": "<one-sentence headline of the result>", "good": true },
    "metrics": [ { "name": "<label>", "value": <number>, "unit": "<%|bits|Hz|...>", "good": true } ],
    "series":  [ { "id": "<id>", "kind": "histogram"|"bar"|"line", "title": "...",
                   "xLabel": "...", "yLabel": "...", "yKeys": ["count"],
                   "points": [ { "label": "<x>", "values": { "count": <num> } } ] } ]
  }
  Reason: one schema = one renderer = a polished demo without bespoke React per idea.
  All metric / series VALUES must come from real Selene shot data committed in step 3 — no mocks.

[12] RENDERER — create src/components/SeleneRunView.tsx with EXACTLY this body (no deps beyond react + tailwind):

  import * as React from "react";
  type Point  = { label: string; values: Record<string, number> };
  type Series = { id: string; kind: "histogram"|"bar"|"line"; title: string;
                  xLabel?: string; yLabel?: string; yKeys: string[]; points: Point[] };
  type Metric = { name: string; value: number; unit?: string; good?: boolean };
  export type SeleneRun = {
    schemaVersion: 1; experiment: string; title: string; description: string;
    kernel: { snippet: string; qubits: number; shotsPerRow: number };
    verdict: { text: string; good: boolean };
    metrics: Metric[]; series: Series[]; notes?: string;
  };
  const fmt = (n: number) => Math.abs(n) >= 100 ? n.toFixed(0) : Math.abs(n) >= 1 ? n.toFixed(2) : n.toFixed(3);
  function Bars({ s }: { s: Series }) {
    const max = Math.max(1, ...s.points.flatMap(p => s.yKeys.map(k => p.values[k] ?? 0)));
    return (
      <div className="space-y-1">
        {s.points.map((p, i) => (
          <div key={i} className="flex items-center gap-2 text-xs">
            <div className="w-20 truncate text-muted-foreground">{p.label}</div>
            <div className="flex-1 h-3 bg-muted rounded-sm overflow-hidden">
              <div className="h-full bg-primary" style={{ width: `${(100*(p.values[s.yKeys[0]]??0))/max}%` }} />
            </div>
            <div className="w-12 text-right tabular-nums">{fmt(p.values[s.yKeys[0]]??0)}</div>
          </div>
        ))}
      </div>
    );
  }
  function Line({ s }: { s: Series }) {
    const W=320, H=120, P=20;
    const ys = s.points.map(p => p.values[s.yKeys[0]] ?? 0);
    const min = Math.min(...ys), max = Math.max(...ys), span = max - min || 1;
    const pts = ys.map((y, i) => {
      const x = P + (i*(W-2*P))/Math.max(1, ys.length-1);
      const yy = H - P - ((y - min)/span)*(H - 2*P);
      return `${x},${yy}`;
    }).join(" ");
    return (
      <svg viewBox={`0 0 ${W} ${H}`} className="w-full h-32">
        <polyline fill="none" stroke="currentColor" strokeWidth="2" points={pts} className="text-primary" />
      </svg>
    );
  }
  export function SeleneRunView({ run }: { run: SeleneRun }) {
    return (
      <div className="space-y-6">
        <header>
          <div className="text-xs uppercase tracking-wider text-muted-foreground">{run.experiment}</div>
          <h2 className="text-2xl font-semibold">{run.title}</h2>
          <p className="text-sm text-muted-foreground">{run.description}</p>
          <div className={`mt-2 inline-block px-3 py-1 rounded-full text-xs ${run.verdict.good?"bg-emerald-500/15 text-emerald-400":"bg-amber-500/15 text-amber-400"}`}>
            {run.verdict.text}
          </div>
        </header>
        <section className="grid grid-cols-2 md:grid-cols-4 gap-3">
          {run.metrics.map((m, i) => (
            <div key={i} className="rounded-lg border border-border p-3">
              <div className="text-[10px] uppercase tracking-wider text-muted-foreground">{m.name}</div>
              <div className="text-xl font-semibold tabular-nums">{fmt(m.value)}<span className="text-xs text-muted-foreground ml-1">{m.unit}</span></div>
            </div>
          ))}
        </section>
        <section className="space-y-6">
          {run.series.map(s => (
            <div key={s.id} className="rounded-lg border border-border p-4">
              <div className="flex items-baseline justify-between mb-3">
                <div className="text-sm font-medium">{s.title}</div>
                <div className="text-[10px] text-muted-foreground">{s.xLabel} / {s.yLabel}</div>
              </div>
              {s.kind === "line" ? <Line s={s} /> : <Bars s={s} />}
            </div>
          ))}
        </section>
        <footer className="text-[11px] text-muted-foreground">
          kernel: {run.kernel.qubits} qubits · {run.kernel.shotsPerRow} shots/row
        </footer>
      </div>
    );
  }

  Then in the route: import data from "@/data/quantum-results.json"; <SeleneRunView run={data as any} />.
  Quantum trace tab: <pre>{data.kernel.snippet}</pre>.

[HOOK] QTDA — Betti numbers from a point cloud.
  Per point: amplitude-encode with prep_features() from [7] on ≤ 8 qubits, measure all → empirical distribution.
  Host-side post-processing:
    - Pairwise distance d_ij = 1 - quantum_fidelity (use swap_test() from [7] across all pairs).
    - For each filtration radius r ∈ {r0..r7}: build graph with edge iff d_ij ≤ r.
    - b0 = #connected components (BFS/union-find).
    - b1 = E - V + b0   (independent cycles for a 1-complex).
    - b2 = 0 unless you fill triangles (optional).
  selene_run mapping:
    metrics: [ {"name":"b0","value":b0_at_r_mid}, {"name":"b1","value":b1_at_r_mid}, {"name":"b2","value":0} ]
    series:  [ {"id":"betti-curve","kind":"line","title":"Betti numbers vs radius",
                "xLabel":"r","yLabel":"count","yKeys":["b0"],
                "points":[{"label":f"{r:.2f}","values":{"b0":b0,"b1":b1}} for r,b0,b1,_ in scan]} ]