"""Render the exact countable-absorption coordinate diagram. Original figure and renderer: CC0-1.0 to the extent of rights held. Requires Python 3 and Matplotlib. Run this file from any working directory. The formulas prove the bijection; the finite checks below validate diagram data. """ from pathlib import Path import json import shutil import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt from matplotlib import font_manager from matplotlib.patches import Rectangle OUT = Path(__file__).resolve().parent def beta(j, k): assert j >= 0 and k >= 0 return (2**j) * (2*k + 1) - 1 def inverse(n): assert n >= 0 odd = n + 1 j = 0 while odd % 2 == 0: odd //= 2 j += 1 return j, (odd - 1) // 2 def main(): OUT.mkdir(parents=True, exist_ok=True) for n in range(4096): assert beta(*inverse(n)) == n for j in range(10): for k in range(64): assert inverse(beta(j, k)) == (j, k) assert beta(1, 2) == 9 and beta(2, 1) == 11 plt.rcParams.update({ "font.family": "DejaVu Sans", "font.size": 14, "mathtext.fontset": "dejavusans", "svg.fonttype": "none", "savefig.facecolor": "#f8fafc", }) blue, orange = "#1662a2", "#b24e0f" ink, muted = "#14283d", "#475569" fig = plt.figure(figsize=(16, 7.2), facecolor="#f8fafc") fig.text(.045, .935, "Countable absorption: an exact coordinate map", fontsize=25, weight="bold", color=ink) fig.text(.045, .88, r"$\beta(j,k)=2^j(2k+1)-1$ and " r"$U(\delta_k\otimes\delta_j)=\delta_{\beta(j,k)}$", fontsize=21, color=ink) left = fig.add_axes([.045, .22, .49, .55]) left.set_xlim(-.8, 6) left.set_ylim(-.1, 5) left.axis("off") left.text(-.7, 4.85, "FINITE SAMPLE OF THE PARTITION", fontsize=14, weight="bold", color=muted) left.text(-.6, 4.15, r"$j\backslash k$", color=muted, fontsize=17, ha="center", va="center") for k in range(6): left.text(k+.5, 4.15, str(k), fontsize=17, color=muted, ha="center", va="center") for j in range(4): y = 3-j left.text(-.6, y+.45, str(j), fontsize=17, color=muted, ha="center", va="center") for k in range(6): n = beta(j, k) color = blue if n == 11 else orange if n == 9 else ink fill = "#dceefe" if n == 11 else "#ffebd8" if n == 9 else "#e9eef5" left.add_patch(Rectangle((k+.035, y+.035), .93, .83, facecolor=fill, edgecolor="white", lw=1.5)) left.text(k+.5, y+.45, str(n), fontsize=21, color=color, ha="center", va="center", weight="bold" if n in (9, 11) else "normal") left.text(-.65, -.08, "Rows and columns continue; the formula covers every n ≥ 0.", fontsize=12, color=muted, va="top") right = fig.add_axes([.575, .22, .385, .55]) right.set_xlim(0, 1) right.set_ylim(0, 1) right.axis("off") right.text(0, .97, "ONE MATRIX UNIT, TWO COORDINATES", fontsize=14, weight="bold", color=muted) right.text(.18, .82, "SOURCE", ha="center", color=blue, fontsize=12, weight="bold") right.text(.83, .82, "DESTINATION", ha="center", color=orange, fontsize=12, weight="bold") right.text(.18, .68, r"$\delta_1\otimes\delta_2$", ha="center", va="center", fontsize=24, color=blue) right.text(.83, .68, r"$\delta_2\otimes\delta_1$", ha="center", va="center", fontsize=24, color=orange) right.annotate("", xy=(.66, .68), xytext=(.35, .68), arrowprops=dict(arrowstyle="->", lw=2, color=ink)) right.text(.5, .76, r"$E_{2,1}\otimes e_{1,2}$", ha="center", fontsize=19, color=ink) for x in (.18, .83): right.annotate("", xy=(x, .32), xytext=(x, .57), arrowprops=dict(arrowstyle="->", lw=2, color=ink)) right.text(x-.075, .445, r"$U$", fontsize=22, ha="center", color=ink) right.text(.18, .22, r"$\delta_{11}$", ha="center", va="center", fontsize=27, color=blue) right.text(.83, .22, r"$\delta_9$", ha="center", va="center", fontsize=27, color=orange) right.annotate("", xy=(.68, .22), xytext=(.32, .22), arrowprops=dict(arrowstyle="->", lw=2, color=ink)) right.text(.5, .295, r"$E_{9,11}$", ha="center", fontsize=22, color=ink) right.text(.5, .025, r"$\Xi(E_{2,1}\otimes e_{1,2})=E_{9,11}$", ha="center", fontsize=20, color=ink) fig.text(.045, .10, r"Inverse: $\nu(n)=v_2(n+1),\quad " r"\kappa(n)=\left((n+1)/2^{\nu(n)}-1\right)/2,\quad " r"U^*\delta_n=\delta_{\kappa(n)}\otimes\delta_{\nu(n)}$", fontsize=18, color=ink) fig.text(.045, .038, "Exact formulas: CDEC.7.a–g. The model B(ℓ²) is properly infinite and semifinite; rank-one projections are finite.", fontsize=13, color=muted) fig.savefig(OUT / "absorption-map.png", dpi=160) fig.savefig(OUT / "absorption-map.svg") plt.close(fig) data = { "formula": "beta(j,k)=2^j(2k+1)-1", "domain": "j,k in N_0; output in N_0", "inverse": {"nu": "v_2(n+1)", "kappa": "((n+1)/2^nu(n)-1)/2"}, "table": [{"j": j, "values": [beta(j, k) for k in range(6)]} for j in range(4)], "source": {"k": 1, "j": 2, "n": 11}, "destination": {"k": 2, "j": 1, "n": 9}, "matrix_unit": {"input": "E_{2,1} tensor e_{1,2}", "output": "E_{9,11}"}, "map": "Xi(X)=U X U*", "inverse_map": "Xi^{-1}(y)=U* y U; (i,j) entry V_i* y V_j", "proof_locators": ["CDEC.7.a", "CDEC.7.b", "CDEC.7.c", "CDEC.7.d", "CDEC.7.e", "CDEC.7.f", "CDEC.7.g"], "finite_validation": {"inverse_n_checked": 4096, "j_checked": 10, "k_checked": 64}, "finite_validation_is_not_a_proof": True, } (OUT / "data.json").write_text(json.dumps(data, indent=2) + "\n", encoding="utf-8") font = Path(font_manager.findfont("DejaVu Sans")) license_path = font.parent / "LICENSE_DEJAVU" if license_path.is_file(): shutil.copyfile(license_path, OUT / "FONT-LICENSE.txt") print(json.dumps({"rendered": ["absorption-map.png", "absorption-map.svg"], "data": "data.json", "validation": "passed"})) if __name__ == "__main__": main()