"""Original groupoid regular interface illustration. Expression CC0-1.0.""" from pathlib import Path import argparse import hashlib import json import os import tempfile HERE = Path(__file__).resolve().parent os.environ.setdefault("MPLCONFIGDIR", str(Path(tempfile.gettempdir()) / "ncg-groupoid-figure-cache")) import matplotlib matplotlib.use("Agg") from matplotlib import pyplot as plt from matplotlib.font_manager import FontProperties from matplotlib.patches import FancyArrowPatch, Rectangle, Circle ap = argparse.ArgumentParser() ap.add_argument("--output-dir", type=Path, default=HERE) out = ap.parse_args().output_dir.resolve() out.mkdir(parents=True, exist_ok=True) REG = FontProperties(fname=str(HERE / "fonts/DejaVuSans.ttf")) BOLD = FontProperties(fname=str(HERE / "fonts/DejaVuSans-Bold.ttf")) plt.rcParams.update({"svg.fonttype": "path", "svg.hashsalt": "groupoid-regular-interface-20261006"}) INK, BLUE, RED, GREEN = "#243b53", "#2467a4", "#b4394a", "#217459" fig, axs = plt.subplots(2, 2, figsize=(17, 11)) fig.patch.set_facecolor("white") for a in axs.flat: a.set_xlim(0, 1); a.set_ylim(0, 1); a.axis("off") def text(a, x, y, s, size=11, bold=False, color=INK, ha="center"): a.text(x, y, s, fontsize=size, fontproperties=BOLD if bold else REG, color=color, ha=ha, va="center", transform=a.transAxes, linespacing=1.35, zorder=10) def arrow(a, p, q, color=BLUE, curve=0, width=2): a.add_patch(FancyArrowPatch(p, q, arrowstyle="-|>", mutation_scale=16, linewidth=width, color=color, transform=a.transAxes, connectionstyle="arc3,rad=" + str(curve))) a = axs[0, 0] text(a, .5, .96, "A. A nonzero corner is not the balanced quotient", 13, True) labels = ["11", "12", "21", "22"] for j, label in enumerate(labels): text(a, .31 + .13*j, .82, label, 12, True) text(a, .165, .69 - .115*j, label, 12, True) for i in range(4): for j in range(4): x, y = .26 + .13*j, .638 - .115*i inside = i in (0, 3) and j in (0, 3) a.add_patch(Rectangle((x, y), .12, .105, facecolor="#e4f4eb" if inside else "#eef1f5", edgecolor=GREEN if inside else "#cad3dc", transform=a.transAxes)) matrix_labels = {(0, 0): "Δ(e₁₁)", (0, 3): "Δ(e₁₂)", (3, 0): "Δ(e₂₁)", (3, 3): "Δ(e₂₂)"} text(a, x + .06, y + .052, matrix_labels.get((i, j), "·"), 9, color=GREEN if inside else "#9aabb9") text(a, .54, .22, "Green 2 × 2 corner on span{11,22}: P M₄ P ≅ M₂", 11, True, GREEN) text(a, .5, .125, "Balancing generator in all M₄: diag(0,1,−1,0)", 11, True, RED) text(a, .5, .058, "Eₐ,₁₂ generator E₁₂,ᵦ = Eₐ,ᵦ ⇒ quotient = 0", 10, color=RED) a = axs[0, 1] text(a, .5, .96, "B. Transport changes the coefficient's unit fibre", 13, True) nodes = {"z": (.17, .38), "y": (.81, .38), "x": (.49, .76)} for label, pos in nodes.items(): a.add_patch(Circle(pos, .035, facecolor="#eaf2fb", edgecolor=BLUE, linewidth=2, transform=a.transAxes)) text(a, *pos, label, 13, True) arrow(a, (.20, .41), (.46, .724)) arrow(a, (.78, .41), (.52, .724)) arrow(a, (.21, .38), (.765, .38)) text(a, .245, .625, "g : z → x", 11, color=BLUE) text(a, .72, .625, "γ : y → x", 11, color=BLUE) text(a, .49, .32, "γ⁻¹g : z → y", 11, True, BLUE) text(a, .49, .23, "V_z → V_y → V_x → V_z", 13, True) text(a, .49, .15, "U_(γ⁻¹g) U_γ U_g⁻¹ = identity on V_z", 10) text(a, .49, .058, "Jξ(g) = U_g⁻¹ξ(g) · one unit measure · reduced regular norm", 10, True, GREEN) a = axs[1, 0] text(a, .5, .96, "C. The actual smooth graph leaves a whole source line", 13, True) text(a, .5, .865, "G = (ℝ_t × ℝ_r × S¹_n) / ℤ · n-circle and C₁ suppressed", 11) left, right, bottom, top = .11, .94, .25, .72 def rmap(r): return left + (right - left)*(r + 1.5)/5 def tmap(t): return bottom + (top - bottom)*t arrow(a, (left-.015, bottom), (right+.025, bottom), INK, width=1.3) arrow(a, (left, bottom-.015), (left, top+.04), INK, width=1.3) text(a, .965, .245, "r", 12, True) text(a, .08, .77, "t", 12, True) for t, label in [(0, "0"), (.25, "1/4"), (.75, "3/4"), (1, "1")]: a.plot([left, right], [tmap(t), tmap(t)], color="#c6d1dd", linestyle=":" if t not in (0, 1) else "-", linewidth=1, transform=a.transAxes) text(a, .07, tmap(t), label, 9) for j in [-1, 0, 1, 2, 3]: a.add_patch(Rectangle((rmap(j-1/8), tmap(.25)), (right-left)*(.25)/5, (top-bottom)*.5, facecolor="#c8e5d5", edgecolor=GREEN, linewidth=1.4, transform=a.transAxes)) text(a, rmap(j), .195, str(j), 10) text(a, .53, .64, "Support boxes: t ∈ (1/4,3/4), r ∈ (j−1/8,j+1/8)", 9, color=GREEN) text(a, .53, .115, "Fixed k averages t only: |ξ⟩⟨ξ|_t ⊗ identity on r,n,C₁", 10, True) text(a, .53, .037, "Source r is unbounded. Deck phase e^(−ijπ/5) is retained.", 10) a = axs[1, 1] text(a, .5, .96, "D. One fixed localized resolvent preserves every ψⱼ", 13, True) text(a, .5, .86, "D = A(−i∂_n), A² = 1 · ψⱼ = ξ(t) ζ(r−j) u(n)", 11) xs = [.2, .34, .48, .62, .76] for i, x in enumerate(xs): a.plot([x, x], [.43, .72], color=BLUE, linewidth=4, transform=a.transAxes) a.plot([x], [.72], "o", color=BLUE, markersize=6, transform=a.transAxes) text(a, x, .385, "j=" + str(i-1), 9) text(a, .08, .72, "1", 11, True) text(a, .08, .43, "0", 10) text(a, .5, .79, "Input norm = image norm = 1", 12, True, BLUE) text(a, .885, .55, "…", 22, True, BLUE) text(a, .5, .30, "Π(k)(1+D²)⁻¹ ψⱼ = ψⱼ", 14, True) a.add_patch(Rectangle((.1, .105), .8, .12, facecolor="#fff0f1", edgecolor=RED, transform=a.transAxes)) text(a, .5, .165, "Distinct images are orthogonal: distance √2", 12, True, RED) text(a, .5, .047, "Finite sample of a proved infinite sequence; no compact subsequence.", 10) fig.subplots_adjust(left=.02, right=.985, bottom=.09, top=.92, wspace=.14, hspace=.12) fig.text(.5, .985, "Regular groupoid absorption: exact norm, distinct index and compactness tests", ha="center", va="top", fontproperties=BOLD, fontsize=18, color=INK) fig.text(.5, .045, "Figure 11M.1. Proposition 11M.1; Theorems 11M.2–11M.3. " "Full proofs: (GR.1)–(GR.18). Canonical doubled unit-line index is zero: (GR.19)–(GR.21).", ha="center", fontproperties=REG, fontsize=10, color=INK) notice = (HERE / "FONT-NOTICE.txt").read_text(encoding="utf-8") description = ("Original mathematical illustration CC0-1.0. " "Complete font notice:\n" + notice) png, svg = out / "groupoid-regular-interface.png", out / "groupoid-regular-interface.svg" fig.savefig(png, dpi=160, metadata={"Description": description, "Software": "Matplotlib " + matplotlib.__version__}) fig.savefig(svg, metadata={"Date": None, "Description": description}) svg.write_text(svg.read_text(encoding="utf-8"),encoding="utf-8",newline="\n") plt.close(fig) data = { "proof_locators": ["GR.1", "GR.2", "GR.3", "GR.4", "GR.1–GR.21"], "A": {"tensor_basis": ["11", "12", "21", "22"], "balance_diagonal": [0, 1, -1, 0], "corner_basis": ["11", "22"], "balanced_quotient": "zero algebra", "corner": "M2"}, "B": {"gamma": "y->x", "g": "z->x", "gamma_inverse_g": "z->y", "transport_chain": ["Vz->Vy", "Vy->Vx", "Vx->Vz"], "one_unit_measure": True, "isotropy_labels_retained": True}, "C": {"range_strip": [0, 1], "source_line": "R", "normal_circle": "R/2piZ", "alpha": "2pi(sqrt(2)-1)", "beta": "pi/5", "full_inverse_phase": "exp(-ij*pi/5)", "inverse_determinant": "exp(-2ij*pi/5)", "t_support_box": ["1/4", "3/4"], "r_radius": "1/8", "displayed_source_centres": [-1, 0, 1, 2, 3], "boxes_are_support_bounds_not_function_profiles": True, "normal_and_clifford_factors_suppressed": True}, "D": {"sample_centres": [-1, 0, 1, 2, 3], "input_norms": [1]*5, "output_norms": [1]*5, "pairwise_image_distance": "sqrt(2)", "infinite_statement_requires_written_proof": True}, "matplotlib": matplotlib.__version__, "font_notice_embedded": True, "outputs": [{"path": p.name, "bytes": p.stat().st_size, "sha256": hashlib.sha256(p.read_bytes()).hexdigest().upper()} for p in [png, svg]] } (out / "FIGURE-DATA.json").write_text(json.dumps(data, indent=2) + "\n", encoding="utf-8")