"""Reproducible exact tracial examples. No operator theorem is inferred from numerics."""
from pathlib import Path
import json
import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
plt.rcParams["svg.hashsalt"] = "oa-flow-tracial-density-20261009-v1"
from matplotlib.patches import FancyBboxPatch

OUT = Path(__file__).resolve().parent / "assets"
OUT.mkdir(exist_ok=True)
plt.rcParams.update({"font.family": "DejaVu Sans", "font.size": 12,
                     "mathtext.fontset": "dejavusans", "svg.fonttype": "none"})
navy, blue, teal, orange = "#17324d", "#2166ac", "#087f8c", "#bc5726"
fig = plt.figure(figsize=(14, 9), facecolor="#f7fafc")
fig.text(.055, .95, "Tracial densities retain every finite and infinite value",
         fontsize=23, weight="bold", color=navy)
fig.text(.055, .912, "A given faithful normal semifinite trace • arbitrary Hilbert spaces • TD-1–8",
         fontsize=12.5, color=navy)
flow = fig.add_axes([.045, .705, .91, .17]); flow.set_axis_off()
boxes = [
    (r"$\xi\in H_\tau$", "trace GNS vector"),
    (r"$T_\xi\Lambda(x)=r(x)\xi$", "close the actual graph"),
    (r"$h_\xi=T_\xi T_\xi^*$", "affiliated density"),
    (r"$\Phi_{h_\xi}(a)=\langle a\xi,\xi\rangle$", "every positive a"),
]
for j, (formula, label) in enumerate(boxes):
    x = .012 + j*.249
    flow.add_patch(FancyBboxPatch((x, .19), .226, .65,
                   boxstyle="round,pad=0.008,rounding_size=0.04",
                   facecolor="white", edgecolor="#bad0df", linewidth=1.5))
    flow.text(x+.113, .60, formula, ha="center", va="center", fontsize=14, color=blue)
    flow.text(x+.113, .34, label, ha="center", fontsize=10.5, color=navy)
    if j < 3:
        flow.annotate("", xy=(x+.245, .52), xytext=(x+.229, .52),
                      arrowprops=dict(arrowstyle="->", color=navy, lw=1.5))
flow.text(.01, .025, "TD-3: bounded spectral cutoffs lie in the finite trace ideal; their vectors converge to ξ.",
          fontsize=11.3, color=navy)

ax1 = fig.add_axes([.13, .355, .32, .255], facecolor="white")
T = np.array([[1., 1.], [0., 1.]])
h = T @ T.T
lm, lp = (3-np.sqrt(5))/2, (3+np.sqrt(5))/2
sm, sp = np.sqrt(lm), np.sqrt(lp)
assert np.allclose(np.linalg.eigvalsh(T.T@T), [lm, lp])
assert np.allclose(h, [[2, 1], [1, 1]]) and np.trace(h) == 3
for left, right, mass in [(0, sm, 0), (sm, sp, lm), (sp, 2.25, 3)]:
    ax1.plot([left, right], [mass, mass], color=blue, lw=3)
    ax1.plot(left, mass, "o", color=blue, ms=7, zorder=5)
for x, y in [(sm, 0), (sp, lm)]:
    ax1.plot(x, y, "o", mfc="white", mec=blue, mew=2, ms=7, zorder=6)
ax1.axvline(sm, color="#ccd8e1", ls=":", zorder=0)
ax1.axvline(sp, color="#ccd8e1", ls=":", zorder=0)
ax1.set(xlim=(-.03, 2.25), ylim=(-.25, 3.45),
        xlabel=r"singular-value cutoff $R$",
        ylabel=r"$\operatorname{Tr}(T e_R T^*)$")
ax1.set_xticks([0, sm, sp, 2.25], ["0", r"$s_-$", r"$s_+$", "2.25"])
ax1.set_yticks([0, lm, 3], ["0", r"$(3-\sqrt{5})/2$", "3"])
ax1.grid(axis="y", alpha=.2)
ax1.set_title("Finite cutoffs recover the whole vector", loc="left",
              fontsize=14, weight="bold", color=navy, pad=19)
fig.text(.075, .252, r"$T=(1,1;\ 0,1),\quad h=(2,1;\ 1,1),\quad \operatorname{Tr}(h)=3$",
         fontsize=12, color=navy)
fig.text(.075, .214, r"$e_R=1_{[0,R]}(|T|),\quad s_\pm=(\sqrt{5}\pm1)/2$",
         fontsize=12, color=navy)
fig.text(.075, .17, "Exact M₂ example • TD15–TD17; caption TDF1–2",
         fontsize=10.6, color=blue)

ax2 = fig.add_axes([.595, .355, .34, .255], facecolor="white")
N = np.arange(1, 13)
ax2.plot(N, N, "o-", color=orange, lw=2.4, ms=5)
ax2.set(xlim=(.5, 12.5), ylim=(0, 13),
        xlabel=r"finite spectral cutoff $N$",
        ylabel=r"$\langle c^*H_Nc\,e_1,e_1\rangle=N$")
ax2.set_xticks([1, 3, 6, 9, 12]); ax2.set_yticks([0, 3, 6, 9, 12])
ax2.grid(alpha=.2)
ax2.set_title("A bounded sandwich can lose a dense domain", loc="left",
              fontsize=14, weight="bold", color=navy, pad=19)
fig.text(.575, .252, r"$He_n=4^n e_n,\quad v_n=2^{-n},\quad c\xi=\xi_1v$",
         fontsize=12, color=navy)
fig.text(.575, .214, r"$D(q_{c^*Hc})=e_1^\perp,\quad q=0\ \mathrm{there};\quad q=\infty\ \mathrm{outside}$",
         fontsize=12, color=orange)
fig.text(.575, .17, "Infinite-value part P₁ • TD26; caption TDF3–5",
         fontsize=10.6, color=orange)

fig.text(.055, .091, "Full domains decide semifiniteness: a density form is allowed to have a nondense finite domain.",
         fontsize=13, weight="bold", color=navy)
fig.text(.055, .053, "Exact proofs: TRACIAL_DENSITY_PROOF.md, TD-3/6/7. Free human context: Hiai (2020), pp.49–50.",
         fontsize=10.7, color=navy)
for name in ["png", "svg"]:
    fig.savefig(OUT / f"tracial-density.{name}", dpi=180, facecolor=fig.get_facecolor(), **({"metadata": {"Date": None}} if name == "svg" else {}))
plt.close(fig)

# Finite sections verify the displayed identity exactly up to floating-point arithmetic.
for n in [1, 2, 3, 6, 12]:
    v = 2.**(-np.arange(1, n+1))
    H = np.diag(4.**np.arange(1, n+1))
    assert np.isclose(v @ H @ v, n)
(OUT / "tracial-density-numerics.json").write_text(json.dumps({
    "scope": "Finite numerical checks of exact caption formulas; not a theorem proof",
    "matrix": T.tolist(), "density": h.tolist(),
    "eigenvalues": [float(lm), float(lp)], "singular_values": [float(sm), float(sp)],
    "trace": 3, "cutoffs": N.tolist(), "sandwich_values": N.tolist(),
    "rank_one_norm_squared_exact": "1/3",
    "infinite_value_domain_exact": "e_1 perpendicular"
}, indent=2)+"\n", encoding="utf-8")