"""Original CC0 typed diagram for actual anchored etale descent, DS.1--DS.23.""" from pathlib import Path import os import argparse import json HERE = Path(__file__).resolve().parent os.environ["MPLCONFIGDIR"] = str(HERE / "runtime-cache") import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt from matplotlib.font_manager import FontProperties from matplotlib.patches import FancyBboxPatch REG = FontProperties(fname=str(HERE / "fonts" / "DejaVuSans.ttf")) BOLD = FontProperties(fname=str(HERE / "fonts" / "DejaVuSans-Bold.ttf")) INK, BLUE, GREEN, PALE = "#17354b", "#165f92", "#08795e", "#edf5f9" def txt(ax, x, y, value, size=15, bold=False, **kw): return ax.text(x, y, value, fontsize=size, color=kw.pop("color", INK), fontproperties=BOLD if bold else REG, **kw) def arrow(ax, start, end, color=BLUE): ax.annotate("", xy=end, xytext=start, arrowprops={"arrowstyle": "->", "lw": 2.2, "color": color}) def box(ax, x, y, width, height, value, size=15): ax.add_patch(FancyBboxPatch((x-width/2, y-height/2), width, height, boxstyle="round,pad=.009", facecolor=PALE, edgecolor=BLUE, linewidth=1.6)) txt(ax, x, y, value, size, True, ha="center", va="center") def panel(fig, rect): ax = fig.add_axes(rect) ax.set_xlim(0, 1) ax.set_ylim(0, 1) ax.axis("off") return ax def render(out): out.mkdir(parents=True, exist_ok=True) matplotlib.rcParams.update({"svg.hashsalt": "actual-anchored-etale-descent-DS-20261006", "svg.fonttype": "path"}) fig = plt.figure(figsize=(18, 12), facecolor="white") fig.text(.04, .956, "Descent retains the actual arrows, coefficient modules and products", fontproperties=BOLD, fontsize=23, color=INK) fig.text(.04, .923, "Hausdorff second-countable LCH étale G ⇉ T; all original coefficients and cycles are C₀(T)-linear.", fontproperties=REG, fontsize=14, color=INK) a = panel(fig, [.045, .565, .425, .33]) txt(a, 0, .94, "DS.1 / DS.13 Range-fibre tensor sum", 18, True) pos = {"x": (.15, .37), "y": (.82, .37), "z": (.48, .70)} for label, point in pos.items(): a.scatter([point[0]], [point[1]], s=55, color=INK) txt(a, point[0], point[1]-.09, label, 16, True, ha="center") arrow(a, (.19, .42), (.43, .66)) arrow(a, (.78, .42), (.53, .66)) arrow(a, (.20, .37), (.77, .37), GREEN) txt(a, .14, .64, "g : x → z", 14) txt(a, .65, .64, "h : y → z", 14) txt(a, .48, .30, "k = h⁻¹g : x → y", 14, True, ha="center") txt(a, .01, .13, "r(h) = r(g) = z; ξ(h) ∈ E₁,z; η(k) ∈ E₂,y.", 14) txt(a, .01, .025, "U₂,h transports η(k) into E₂,z before tensor balancing.", 13) b = panel(fig, [.535, .565, .425, .33]) txt(b, 0, .94, "DS.6–DS.8 Reduced source-column matrix", 18, True) box(b, .14, .64, .22, .16, "source x", 14) box(b, .67, .77, .26, .14, "g : x → z", 14) box(b, .67, .48, .26, .14, "l : x → y", 14) arrow(b, (.27, .67), (.52, .77)) arrow(b, (.27, .61), (.52, .48)) txt(b, .49, .62, "gl⁻¹ : y → z", 14, True) txt(b, 0, .29, "entry (g,l) = πₓ(αg⁻¹(a(gl⁻¹))) : Eₓ → Eₓ", 15, True) txt(b, 0, .15, "The PN regular coefficient is represented on Eₓ.", 14) txt(b, 0, .025, "‖Λπ,r(a)‖ ≤ ‖λA(a)‖; finite matrices cover degenerate πₓ.", 13) fig.text(.06, .525, "Wτ(ξ ⊗ η)(g) = Σr(h)=r(g) ξ(h) ⊗ U₂,h η(h⁻¹g)", fontproperties=BOLD, fontsize=18, color=GREEN) fig.text(.06, .497, "Exact inner-product identity gives an even onto unitary for τ = full or reduced (DS.13–DS.14).", fontproperties=REG, fontsize=13, color=INK) c = panel(fig, [.045, .115, .425, .325]) txt(c, 0, .96, "DS.3 / DS.17 Compact extension and positivity", 17, True) box(c, .22, .76, .37, .14, "K_B(E)", 16) box(c, .78, .76, .40, .14, "K_Bτ(XE,τ)", 16) arrow(c, (.42, .76), (.56, .76), GREEN) txt(c, .5, .865, "K ↦ K ⊗ 1", 13, True, ha="center") txt(c, .02, .58, "θeb,fc ⊗ 1 = θ(e ⊗ iB b),(f ⊗ iB c)", 14, True) txt(c, .02, .47, "iB(B) lies in Bτ, so both rank-one vectors are actual.", 13) box(c, .22, .29, .37, .13, "L(E) / K(E)", 13) box(c, .78, .29, .40, .13, "L(XE,τ) / K(XE,τ)", 12) arrow(c, (.42, .29), (.56, .29), GREEN) txt(c, .5, .405, "positivity", 12, True, ha="center") txt(c, .02, .10, "z iA(an) → z transfers the product compression to Λ(z).", 12.5) txt(c, .02, .01, "The closed positive cone gives DS.17 on both completions.", 12.5) d = panel(fig, [.535, .115, .425, .325]) txt(d, 0, .96, "DS.20–DS.21 The coefficient quotient square", 17, True) for x, y, label in [(.18, .73, "A_m"), (.79, .73, "B_m"), (.18, .29, "A_r"), (.79, .29, "B_r")]: box(d, x, y, .21, .12, label, 17) arrow(d, (.30, .73), (.66, .73), GREEN) arrow(d, (.30, .29), (.66, .29), GREEN) arrow(d, (.18, .65), (.18, .38)) arrow(d, (.79, .65), (.79, .38)) txt(d, .49, .83, "jm(x)", 14, True, ha="center") txt(d, .49, .16, "jr(x)", 14, True, ha="center") txt(d, .06, .49, "[qA]", 13, True) txt(d, .84, .49, "[qB]", 13, True) txt(d, .485, .49, "commutes in KK(A_m,B_r)", 12.5, True, ha="center") txt(d, 0, .005, "XE,m ⊗qB B_r ≅ XE,r; the operator is the same pointwise F.", 12.5) fig.text(.045, .064, "Typed schematic, not a geometric model of G. Complete proofs: DS.1–DS.23; product existence input: anchored KG.1–KG.4.", fontproperties=REG, fontsize=12, color=INK) fig.text(.045, .043, "Original CC0 diagram. Human-source orientation: Alistair Miller, Functors between Kasparov categories from étale groupoid correspondences, §2.4.", fontproperties=REG, fontsize=10.5, color=INK) notice = (HERE / "FONT-NOTICE.txt").read_text(encoding="utf-8") fig.savefig(out / "anchored-etale-descent.png", dpi=200, metadata={"Software": "Matplotlib; original CC0 typed diagram", "Description": notice}) fig.savefig(out / "anchored-etale-descent.svg", metadata={"Date": None, "Creator": "Original CC0 anchored etale descent diagram", "Description": notice}) plt.close(fig) data = {"proof_locators": ["DS.1", "DS.3", "DS.6", "DS.7", "DS.8", "DS.13", "DS.14", "DS.17", "DS.20", "DS.21", "DS.23"], "standing_scope": "Hausdorff second-countable locally compact etale groupoids; original C0(T) anchors", "range_sum": {"g": "x->z", "h": "y->z", "h^-1g": "x->y", "common_range": "z", "xi_h_fibre": "E1_z", "eta_hinv_g_fibre": "E2_y", "transported_eta_fibre": "E2_z"}, "source_matrix": {"g": "x->z", "l": "x->y", "g l^-1": "y->z", "entry": "pi_x(alpha_g^-1(a(g l^-1)))", "domain": "E_x", "codomain": "E_x"}, "compact_extension": "theta_(e b),(f c) tensor 1 = theta_(e tensor i_B b),(f tensor i_B c)", "quotient_square": "[q_A] tensor j_r(x) = j_m(x) tensor [q_B] in KK(A_m,B_r)", "module_base_change": "X_E,m tensor_(B_m,q_B) B_r is isomorphic to X_E,r", "diagram_coordinates": "Normalized page layout only, no mathematical metric coordinates asserted", "not_inferred": ["reduced exactness", "global norm-continuity of unlocalized F", "coefficient-free quotient inverse", "original-unit Dirac factors"], "product_existence_provider": "ANCHORED-KK-FOUNDATIONS.md KG.1-KG.4"} (out / "anchored-etale-descent-data.json").write_text(json.dumps(data, indent=2) + "\n", encoding="utf-8") if __name__ == "__main__": parser = argparse.ArgumentParser() parser.add_argument("--output-dir", type=Path, default=HERE / "figures") render(parser.parse_args().output_dir.resolve())