# -*- coding: utf-8 -*- """ Visualisation for Appendices E & F of the IT3 paper. Self-checking: (Z*alpha)^2, the bicone radius and the anchor coordinates are COMPUTED here (not hard-coded), so a successful render == consistency with Tables E/F. Panels that carry no exact number (E.3) are explicitly drawn as schematic (sign/magnitude = ab-initio), matching the text. Changelog vs previous version: * E.2 inset title moved INSIDE the inset (transAxes) so it no longer overprints the panel title; inset lowered to clear the block row. * E.2 explicit "saturated floor S5 (mirror S6)" label added (left gap). * E.4 dead variable `labels` removed. * F.3 central plaque nudged to the 3-circle centroid (0.52). Output: IT3_Appendix_E.png , IT3_Appendix_F.png """ import math import numpy as np import matplotlib.pyplot as plt from matplotlib.patches import Rectangle, Circle # ---- palette locked to the Topological Matrix (Fig.1) ---- COL = dict(s="#E8A317", f="#17BECF", d="#E6005C", p="#9B59B6") BLUE, RED = "#4C8CBF", "#C0504D" ALPHA = 1.0/137.035999084 # ---- floor capacities (mirror-symmetric) and bicone silhouette ---- C = [2, 8, 8, 18, 18, 32, 32, 18, 18, 8, 8, 2] assert C == C[::-1], "capacity sequence must be a palindrome" bicone_r = [math.sqrt(c/2.0) for c in C] # discrete bicone radius CLASSICAL = list(range(6)); RELATIV = list(range(6, 12)) # ---- Diophantine Vacuum Operator: block partition depends on capacity ---- def part(cap): if cap == 32: return [("s", 2), ("f", 16), ("d", 26), ("p", 32)] # S5,S6 if cap == 18: return [("s", 2), ("d", 12), ("p", 18)] # S3,S4,S7,S8 if cap == 8: return [("s", 2), ("p", 8)] # S1,S2,S9,S10 return [("s", 2)] # S0,S11 def block_of(mu, cap): prev = 0 for b, e in part(cap): if mu <= e: return b, mu - prev prev = e return ("?", 0) _STARTS = [0] for c in C: _STARTS.append(_STARTS[-1] + c) def coords(Z): for n in range(12): if _STARTS[n] < Z <= _STARTS[n+1]: mu = Z - _STARTS[n]; cap = C[n]; b, p = block_of(mu, cap) return n, mu, cap, b, p raise ValueError(Z) def is_closed(b, p): # relativistic subshell closure return (b == "s" and p == 2) or (b == "d" and p == 10) or \ (b == "p" and p in (2, 6)) or (b == "f" and p == 14) # ===================================================================== # FIGURE E (2 x 2) # ===================================================================== figE, ax = plt.subplots(2, 2, figsize=(14, 10)) figE.suptitle("Appendix E - Corollaries the standard table lists but cannot derive", fontsize=15, fontweight="bold") # --- E.1 : palindrome + discrete bicone silhouette --- a = ax[0, 0] x = np.arange(12) cols = [BLUE if n in CLASSICAL else RED for n in x] a.bar(x, C, color=cols, edgecolor="k", lw=0.6, zorder=2) for lo, hi in [(1, 2), (3, 4), (5, 6), (7, 8), (9, 10)]: # pair plateaux a.axvspan(lo-0.5, hi+0.5, color="0.90", zorder=0) a2 = a.twinx() a2.plot(x, bicone_r, "o-", color=COL["s"], lw=2, ms=7, label=r"discrete bicone radius $\sqrt{C/2}$", zorder=3) a.scatter([0, 11], [C[0], C[11]], marker="*", s=220, color="gold", edgecolor="k", zorder=4) a.text(0, C[0] + 1.5, "Core\n$S_0$", ha="center", va="bottom", fontsize=9, fontweight="bold") a.text(11, C[11] + 1.5, "End\n$S_{11}$", ha="center", va="bottom", fontsize=9, fontweight="bold") a.set_xticks(x); a.set_xticklabels([f"$S_{{{n}}}$" for n in x], fontsize=9) a.set_ylabel("floor capacity $C(S_n)$") a2.set_ylabel(r"$\sqrt{C/2}$ (bicone radius)") a.set_title("E.1 Palindrome of capacities & bicone silhouette", fontweight="bold") a.set_ylim(0, max(C) + 6); a2.set_ylim(0, max(bicone_r) + 0.5) a2.legend(loc="upper right", fontsize=8); a.grid(axis="y", ls=":", alpha=.4) # --- E.2 : f-block is INTERIOR to d (schematic) + contraction profiles --- a = ax[0, 1] order = part(32) # s,f,d,p on the saturated floor left = 0; ranges = {} for b, e in order: a.add_patch(Rectangle((left, 0.30), e-left, 0.30, color=COL[b], ec="k", lw=0.8)) a.text((left+e)/2, 0.45, f"{b}\n$\\mu$ {left+1}..{e}", ha="center", va="center", color="white", fontsize=9, fontweight="bold") ranges[b] = (left, e); left = e fc, dc = ranges["f"], ranges["d"] a.annotate("", xy=((dc[0]+dc[1])/2, 0.22), xytext=((fc[0]+fc[1])/2, 0.22), arrowprops=dict(arrowstyle="->", color="k", lw=1.5)) a.text(16, 0.12, "f-cavity is topologically INSIDE d => no transverse growth", ha="center", va="top", fontsize=9, style="italic") # NEW: explicit floor label in the free left-top gap (clears the inset at x>=0.60) a.text(8, 0.66, "saturated floor $S_5$ (mirror $S_6$)", ha="center", va="bottom", fontsize=8.5, color="0.25", fontweight="bold") # inset: lowered (y 0.62..0.92) and title drawn INSIDE via transAxes (no overflow) ins = a.inset_axes([0.60, 0.62, 0.38, 0.30]) t = np.arange(1, 15) lanth = 1.0 - 0.045*t actin = 1.0 - 0.045*t - 0.18*np.exp(-((t-2.5)**2)/3.0) ins.plot(t, lanth, "-", color=BLUE, lw=1.6, label="Ln (monotone)") ins.plot(t, actin, "--", color=RED, lw=1.6, label="An (early-enhanced)") ins.set_xticks([]); ins.set_yticks([]) ins.text(0.5, 0.97, "contraction (schematic)", transform=ins.transAxes, ha="center", va="top", fontsize=7.5) ins.legend(fontsize=6.5, loc="lower left") a.set_xlim(0, 32); a.set_ylim(0, 1.1); a.axis("off") a.set_title("E.2 Lanthanide/actinide contraction = geometry (direction only)", fontweight="bold", fontsize=10) # --- E.3 : Og magnetic NON-ADDITIVITY (schematic, sign-free) --- a = ax[1, 0] xg = np.array([1, 2, 3, 4, 5]); yg = np.array([1, 3, 6, 10, 15]) a.plot(xg, yg, "o-", color="0.3", lw=1.8, label="He..Xe (monotone $\\propto\\langle r^2\\rangle$)") a.plot([5, 6], [15, 21], ":", color="0.3", lw=1.8, label="group extrapolation") a.scatter([6], [21], marker="x", s=80, color="0.3") a.scatter([6, 6], [9, 30], s=90, color=[COL["d"], COL["p"]], edgecolor="k", zorder=4) a.annotate("", xy=(6, 30), xytext=(6, 9), arrowprops=dict(arrowstyle="<->", color=RED, lw=2)) a.text(6.15, 19.5, "non-additivity\ngap (split $7p$ shell)", color=RED, fontsize=9, fontweight="bold", va="center") a.text(6.15, 9, "$7p_{1/2}$ contracted", fontsize=8, color=COL["d"], va="center") a.text(6.15, 30, "$7p_{3/2}$ + Van Vleck", fontsize=8, color=COL["p"], va="center") a.set_xticks([1, 2, 3, 4, 5, 6]) a.set_xticklabels(["He", "Ne", "Ar", "Kr", "Xe", "Og"]) a.set_xlim(0.5, 7.8) a.set_ylabel(r"relative $|\chi|$ (schematic a.u.)") a.set_title("E.3 Magnetic non-additivity of Og (sign = ab-initio)", fontweight="bold") a.legend(fontsize=8, loc="upper left"); a.grid(ls=":", alpha=.4) # --- E.4 : group-12 (Z*alpha)^2 and the phase sequence --- a = ax[1, 1] Z12 = [30, 48, 80, 112]; nm12 = ["Zn", "Cd", "Hg", "Cn"] phase = ["(solid)", "(solid)", "(liquid)", "(gas)"] za2 = [(Z*ALPHA)**2 for Z in Z12] barc = [BLUE, BLUE, RED, RED] bars = a.bar(nm12, za2, color=barc, edgecolor="k", lw=0.7) a.axhline(0.20, ls="--", color="k", lw=1.3) a.text(3.45, 0.215, "bond-quenching threshold\n(phenomenological)", ha="right", va="bottom", fontsize=8) for b, v in zip(bars, za2): a.text(b.get_x()+b.get_width()/2, v+0.015, f"{v:.3f}", ha="center", fontsize=9) a.set_xticks(range(4)) a.set_xticklabels([f"{n}\n{p}" for n, p in zip(nm12, phase)], fontweight="bold", fontsize=10) # phase fused into tick a.set_ylabel(r"$(Z\alpha)^2$ (computed)") a.set_title(r"E.4 Why Hg is the unique liquid metal -> Zn,Cd(s)->Hg(l)->Cn(g)", fontweight="bold", fontsize=10) a.set_ylim(0, 0.85); a.grid(axis="y", ls=":", alpha=.4) figE.tight_layout(rect=[0, 0, 1, 0.96]) figE.savefig("IT3_Appendix_E.png", dpi=200, bbox_inches="tight") # ===================================================================== # FIGURE F (1 x 3) # ===================================================================== figF, ax = plt.subplots(1, 3, figsize=(18, 5.5), gridspec_kw={"width_ratios": [2.3, 1.0, 1.2]}) figF.suptitle("Appendix F - Island of stability: topological proton anchors & survival", fontsize=16, fontweight="bold") # --- F.1 : node belts S6, S7 with anchors and the 126 'hole' --- a = ax[0] ANCHORS = {112: "d10", 114: "p1/2^2", 118: "p6", 120: "8s2", 130: "d10", 136: "p6"} HOLE = {126} def draw_belt(axx, n, y): cap = C[n] for mu in range(1, cap+1): Z = _STARTS[n] + mu b, p = block_of(mu, cap) axx.add_patch(Rectangle((mu-1, y), 1, 0.8, color=COL[b], ec="white", lw=1.0)) if Z in ANCHORS: axx.add_patch(Rectangle((mu-1, y), 1, 0.8, fill=False, ec="white", lw=2.4)) axx.plot(mu-0.5, y+0.4, marker="*", ms=14, color="white", markeredgecolor="k") axx.text(mu-0.5, y-0.15, f"{Z}", ha="center", va="top", fontsize=8, fontweight="bold", color=COL[b]) if Z in HOLE: axx.plot(mu-0.5, y+0.4, marker="x", ms=12, mew=2.4, color="white") axx.text(mu-0.5, y-0.15, f"{Z}\nno e-closure", ha="center", va="top", fontsize=8, fontweight="bold", color="red") axx.text(-0.8, y+0.4, f"$S_{{{n}}}$", ha="right", va="center", fontsize=12, fontweight="bold") axx.text(cap/2, y+0.9, "fill order: " + r" $\rightarrow$ ".join(b for b, _ in part(cap)), ha="center", va="bottom", color="black", fontsize=9, fontweight="bold") draw_belt(a, 6, 1.8) draw_belt(a, 7, 0.3) a.set_xlim(-2.5, 33); a.set_ylim(-0.5, 3.2); a.axis("off") a.set_title("F.1 Closed relativistic subshells (anchors) vs Z=126 (mid-d, no closure)", fontweight="bold", fontsize=11) # --- F.2 : nuclear vs IT3 overlap matrix --- a = ax[1] rows = ["Z=114", "Z=120", "Z=126", "N=184"] M = np.array([[1, 1], [1, 1], [1, 0], [1, 2]]) # 1=yes 0=no 2=n/a cmap = plt.cm.colors.ListedColormap(["#C0504D", "#4CAF50", "#999999"]) a.imshow(M, cmap=cmap, vmin=0, vmax=2, aspect="auto") lab = {1: "YES", 0: "NO", 2: "n/a"} for i in range(4): for j in range(2): a.text(j, i, lab[M[i, j]], ha="center", va="center", color="white", fontweight="bold", fontsize=12) a.set_xticks([0, 1]); a.set_xticklabels(["nuclear\nshell model", "IT$^3$\ne-topology"], fontsize=10) a.set_yticks(range(4)); a.set_yticklabels(rows, fontsize=10) a.set_title("F.2 Overlap (126 & N=184 not mirrored)", fontweight="bold", fontsize=11) # --- F.3 : Venn of the three factors --- a = ax[2] a.set_xlim(0, 1); a.set_ylim(0, 1); a.axis("off") circ = [((0.35, 0.60), "#999999"), ((0.65, 0.60), COL["d"]), ((0.50, 0.35), COL["f"])] for (cx, cy), col in circ: a.add_patch(Circle((cx, cy), 0.26, color=col, alpha=0.30, ec=col, lw=1.6)) a.text(0.25, 0.90, "(i) nuclear shell min\n[not derived]", ha="center", va="center", fontsize=9, fontweight="bold") a.text(0.75, 0.90, "(ii) closed-subshell belt\n[derived]", ha="center", va="center", fontsize=9, fontweight="bold") a.text(0.50, 0.05, "(iii) accelerated $\\alpha$-channel\n[derived]", ha="center", va="center", fontsize=9, fontweight="bold") a.text(0.50, 0.52, "observed\nisland\n(114,120)", ha="center", va="center", # 3-way centroid fontsize=10, fontweight="bold", color="black", bbox=dict(boxstyle="round,pad=0.4", fc="white", ec="k", lw=1.5)) a.set_title("F.3 Island = intersection of three factors", fontweight="bold", fontsize=11) figF.tight_layout(rect=[0, 0, 1, 0.93]) figF.savefig("IT3_Appendix_F.png", dpi=200, bbox_inches="tight") plt.show() # ---- self-check (must match Tables E/F) ---- print("OK anchors:", {Z: coords(Z) for Z in [112,114,118,120,126,130,136]}) print("OK (Za)^2 group12:", [round((Z*ALPHA)**2,4) for Z in Z12]) print("OK palindrome:", C == C[::-1], " bicone_r:", [round(r,3) for r in bicone_r])