""" determinant_ladder.py ===================== One primitive, five arrows: four directions of ascent (size, richness, dimension-of-map, degree) plus the grey value probe. All the visuals from the journey, as callable functions, plus a numeric menu. Spyder quick start: >>> from determinant_ladder import menu >>> menu() Or call any figure directly, e.g.: >>> import determinant_ladder as dl >>> dl.fig_rainbow_ports() Figures are shown on screen AND saved into ./figures/. Dependencies: numpy, sympy, scipy, matplotlib. """ import os import itertools import string from math import factorial import numpy as np import sympy as sp import matplotlib.pyplot as plt from matplotlib import cm FIGDIR = 'figures' def _finish(name): os.makedirs(FIGDIR, exist_ok=True) path = os.path.join(FIGDIR, name) plt.savefig(path, dpi=150, bbox_inches='tight') plt.show() print(f'saved {path}') def _perm_sign(p): s, p = 1, list(p) for i in range(len(p)): while p[i] != i: j = p[i]; p[i], p[j] = p[j], p[i]; s = -s return s # ===================================================================== # DIRECTION 0 — the map of the journey # ===================================================================== def fig_index_map(): """Overview: the five-arrow compass — four directions of ascent, one probe, and the fork where the formula demands its hidden hypothesis (commuting branch continues; the non-commuting stub loses det but keeps the trace).""" from matplotlib.patches import FancyArrowPatch fig, ax = plt.subplots(figsize=(15, 10.5)) origin = np.array([0.0, 0.0]) def arm(angle, length, color, name, stops, side, ls='-', lw=4, stop_fs=9, name_shift=(0, 0), head=True): v = np.array([np.cos(np.radians(angle)), np.sin(np.radians(angle))]) tip = origin + v * length ax.annotate('', xy=tip, xytext=origin, arrowprops=dict(arrowstyle='-|>' if head else '-', lw=lw, color=color, linestyle=ls)) ax.text(*(tip + v * 0.10 + np.array(name_shift)), name, color=color, fontsize=12, fontweight='bold', ha='center', va='center') perp = np.array([-v[1], v[0]]) * side for i, s in enumerate(stops): t = length * (0.30 + 0.60 * i / max(len(stops) - 1, 1)) p = origin + v * t ax.plot(*p, 'o', ms=5, color=color) ha = 'left' if perp[0] > 0.15 else ('right' if perp[0] < -0.15 else 'center') va = 'top' if perp[1] < -0.15 else ('bottom' if perp[1] > 0.15 else 'center') ax.text(*(p + perp * 0.05), s, fontsize=stop_fs, color=color, ha=ha, va=va) return v, tip # four full directions of ascent + the grey value probe v_size, _ = arm(15, 1.00, '#1f77b4', 'DIRECTION 1\nSIZE of the matrix (n)', ['n=2: ad-bc, one diagonal', 'n=4: diagonals break', 'n~4.27: drawability inverts', 'n=9: 181,440 paths'], side=-1, name_shift=(0, -0.06)) v_dim, _ = arm(60, 0.92, '#d62728', 'DIRECTION 3\nDIMENSION of the map\n(not independent)', ['2D: Keller 1939 - still OPEN', '3D: FALSE - Alpoge + Fable 2026', 'n>3: false by padding'], side=-1) v_rich, tip_rich = arm(105, 0.96, '#2ca02c', 'DIRECTION 2\nRICHNESS of the entries', ['numbers', 'variables: pure formal symbols', 'functions: symbols over a domain\ndet becomes a FIELD', 'operators (QFT determinants)'], side=-1, name_shift=(-0.52, -0.14), head=False) v_deg, _ = arm(150, 1.00, '#9467bd', 'DIRECTION 4\nDEGREE of the map', ['x^-2, x^-1: Laurent territory', 'x^2: no factorization shuffle', 'x^3: shuffle born - BCW target\n(the conjecture boundary)', 'x^7: the Fable map', 'x^100: 2D verified to here'], side=1) arm(262, 0.62, '#888888', 'VALUE - the probe, not a ladder\n(never changes what kind of thing you have)', ['0 -> 9: bigger, never different', 'negatives: parity mirror, det(-I) = (-1)^n', 'the c in det(cI) = c^n'], side=1, ls=(0, (5, 3)), lw=3, stop_fs=8.5) # the commutativity FORK at the top of RICHNESS: matrix-of-matrix # entries split the road — the commuting branch continues, the # non-commuting branch is a short dead end (det gone, trace kept) fork = origin + v_rich * 0.96 branch_tip = fork + v_rich * 0.26 ax.annotate('', xy=branch_tip, xytext=fork, arrowprops=dict(arrowstyle='-|>', lw=3.5, color='#2ca02c')) ax.text(-0.21, 1.12, 'commuting blocks - monogenic C[M]:\n' 'the formula survives, the ladder continues', fontsize=8.5, color='#2ca02c', ha='left', va='bottom') stub_dir = np.array([np.cos(np.radians(148)), np.sin(np.radians(148))]) stub_end = fork + stub_dir * 0.24 ax.plot([fork[0], stub_end[0]], [fork[1], stub_end[1]], '-', lw=2.5, color='black') ax.plot(*stub_end, 's', ms=11, color='black', zorder=7) ax.text(*(stub_end + np.array([-0.03, 0.05])), 'non-commuting blocks: det undefined\ntrace survives (tr(AB) = tr(BA))', fontsize=8.5, color='black', ha='right', va='bottom', fontweight='bold') ax.text(*(stub_end + np.array([-0.03, -0.02])), 'successors: Dieudonné, quasideterminants', fontsize=7.5, color='#999999', ha='right', va='top') # dimension-of-map is size-of-matrix seen through the derivative a_pt = origin + v_size * 0.62 b_pt = origin + v_dim * 0.62 ax.add_patch(FancyArrowPatch(a_pt, b_pt, connectionstyle='arc3,rad=0.28', arrowstyle='<->', mutation_scale=14, color='gray', lw=1.4, ls=(0, (3, 2)))) ax.text(0.73, 0.33, '= size, via the derivative', fontsize=9, color='gray', ha='center', va='center') # the primitive ax.plot(*origin, 'ko', ms=16, zorder=6) ax.text(-0.62, -0.30, 'THE PRIMITIVE\ndet = signed volume\n(one formula, n! shuffles)', fontsize=12, fontweight='bold', ha='center', va='center') ax.set_xlim(-1.55, 1.55); ax.set_ylim(-0.95, 1.30) ax.set_aspect('equal'); ax.axis('off') ax.set_title('Map of the journey: one primitive, five arrows\n' 'four directions of ascent, one grey probe - and the fork ' 'where the formula demands its hidden hypothesis', fontsize=16, pad=10) plt.tight_layout(); _finish('index_map.png') def fig_hidden_hypothesis(): """The hidden-hypothesis principle: a hypothesis can be vacuously satisfied at low rungs and binding above. Commutativity on the richness arrow (free for numbers/variables/functions, binding at matrix-of-matrix); properness on the dimension arrow (free at n = 1, conjecturally forced at n = 2, demonstrably independent from n = 3). A rhyme to investigate, not a theorem.""" fig, ax = plt.subplots(figsize=(15, 8.5)) rows = [ (0.74, '#2ca02c', 'hidden hypothesis:\nCOMMUTATIVITY\n(richness arrow)', [(1.0, 'numbers', 'free', False), (2.0, 'variables', 'free', False), (3.0, 'functions', 'free', False), (4.2, 'matrix-of-matrix', 'BINDING\n(block det formulas fail;\n' 'only the trace survives)', True)]), (0.30, '#d62728', 'hidden hypothesis:\nPROPERNESS\n(dimension arrow)', [(1.0, 'n = 1', 'free\n(Keller forces linear)', False), (2.5, 'n = 2', 'conjecturally forced\n(THE open question)', False), (4.2, 'n = 3 and up', 'INDEPENDENT\n(non-properness is the\n' "counterexample's escape route)", True)]), ] for y, col, hyp, rungs in rows: ax.annotate('', xy=(4.85, y), xytext=(0.6, y), arrowprops=dict(arrowstyle='-|>', lw=3.5, color=col)) ax.text(0.45, y, hyp, ha='right', va='center', fontsize=11, fontweight='bold', color=col) for x, rung, status, binding in rungs: if binding: ax.plot(x, y, 's', ms=13, color='black', zorder=5) else: ax.plot(x, y, 'o', ms=9, color=col, zorder=5) ax.text(x, y - 0.045, rung, ha='center', va='top', fontsize=10.5, fontweight='bold', color=col) ax.text(x, y + 0.04, status, ha='center', va='bottom', fontsize=9.5, color='#d62728' if binding else '#888888', fontweight='bold' if binding else 'normal') ax.text(2.7, 0.52, 'the same silhouette on two arrows: ' 'satisfied for free below, biting above', ha='center', va='center', fontsize=11, color='#9467bd', style='italic') ax.text(2.7, 0.055, 'Keller re-read: "in low dimension, the properness clause comes ' 'free of charge"\n- true at rung one, unknown at rung two, false ' 'from rung three', ha='center', va='center', fontsize=10.5, color='#d62728') ax.set_xlim(-1.1, 5.1); ax.set_ylim(0, 1.02); ax.axis('off') ax.set_title('The hidden-hypothesis principle - a rhyme to investigate, ' 'not a theorem\na condition can be vacuously satisfied at ' 'low rungs and binding above', fontsize=15, pad=12) plt.tight_layout(); _finish('hidden_hypothesis.png') def fig_ladder_teaser(): """Where the ladder goes — a teaser for Paper II. (a) The von Neumann first steps: 0 = the empty set, 1 = the set containing nothing, each number the gathered history of the nothing before it. (b) The ladder bending back onto its own first rung: enrichment applied to enrichment — categories, then enriched categories. Deliberately spare.""" from matplotlib.patches import Circle, FancyArrowPatch fig, axes = plt.subplots(1, 2, figsize=(13, 5.2)) # --- a: numbers out of nothing ------------------------------------ ax = axes[0] def draw_vn(k, cx, cy, r): ax.add_patch(Circle((cx, cy), r, fill=False, ec='#1f77b4', lw=1.6, alpha=0.9)) if k > 0: step = 2 * r * 0.72 / k x0 = cx - (k - 1) * step / 2 for j in range(k): draw_vn(j, x0 + j * step, cy, r * 0.62 / max(k * 0.85, 1)) labels = ['0 = ∅', '1 = {∅}', '2 = {∅, {∅}}', '3'] for k, (x, r) in enumerate([(0.10, 0.045), (0.34, 0.075), (0.62, 0.105), (0.90, 0.13)]): draw_vn(k, x, 0.56, r) ax.text(x, 0.30, labels[k], ha='center', fontsize=12, color='#1f77b4') if k < 3: ax.annotate('', xy=(x + 0.115 + r * 0.3, 0.56), xytext=(x + r + 0.015, 0.56), arrowprops=dict(arrowstyle='-|>', lw=1.4, color='#999999')) ax.text(0.5, 0.10, 'numbers as the gathered history\nof the nothing ' 'before them', ha='center', fontsize=10.5, style='italic', color='#555555') ax.set_xlim(0, 1.05); ax.set_ylim(0, 1) ax.set_aspect('equal'); ax.axis('off') # --- b: the ladder bends back ------------------------------------- ax = axes[1] for xr in (0.38, 0.52): ax.plot([xr, xr], [0.15, 0.75], color='#2ca02c', lw=2.5) for yr in (0.24, 0.40, 0.56, 0.72): ax.plot([0.38, 0.52], [yr, yr], color='#2ca02c', lw=2) ax.add_patch(FancyArrowPatch((0.52, 0.74), (0.545, 0.20), connectionstyle='arc3,rad=-0.9', arrowstyle='-|>', mutation_scale=16, color='#9467bd', lw=2.2)) ax.text(0.80, 0.47, 'enrichment applied to enrichment:\ncategories, ' 'then enriched categories', ha='left', va='center', fontsize=10.5, color='#9467bd') ax.text(0.45, 0.05, 'the ladder bending back onto its own first rung', ha='center', fontsize=10.5, style='italic', color='#555555') ax.set_xlim(0, 1.55); ax.set_ylim(0, 1); ax.axis('off') fig.suptitle('Where the ladder goes - a teaser', fontsize=15) plt.tight_layout(rect=[0, 0, 1, 0.92]); _finish('ladder_teaser.png') # ===================================================================== # DIRECTION 1 — SIZE of the matrix # ===================================================================== def fig_formulas(): """Reference card: det formulas at n = 1, 2, 3 with letter entries.""" fig, axes = plt.subplots(3, 1, figsize=(10, 10.5)) def draw_matrix(ax, entries, x0, y0, cell=0.9): n = len(entries) for i in range(n): for j in range(n): ax.text(x0 + j*cell + cell/2, y0 - i*cell - cell/2, entries[i][j], ha='center', va='center', fontsize=22) h = n * cell for xb, d in [(x0, 1), (x0 + n*cell, -1)]: ax.plot([xb + d*0.12, xb, xb, xb + d*0.12], [y0, y0, y0 - h, y0 - h], color='black', lw=2.2) ax = axes[0] draw_matrix(ax, [['a']], 0.4, 1.4) ax.text(2.2, 0.95, r'$\det = a$', fontsize=24, va='center') ax.text(2.2, 0.35, 'the number itself - a signed length scale', fontsize=11, va='center', color='gray') ax.text(0.05, 1.75, 'n = 1', fontsize=14, fontweight='bold') ax.set_xlim(0, 10); ax.set_ylim(0, 2); ax.axis('off') ax = axes[1] draw_matrix(ax, [['a', 'b'], ['c', 'd']], 0.4, 2.3) ax.text(3.0, 1.4, r'$\det = ad - bc$', fontsize=24, va='center') ax.text(3.05, 0.75, 'main diagonal minus anti-diagonal\n' '2 terms = 2! shuffles of columns, signed by parity', fontsize=11, va='center', color='gray') ax.text(0.05, 2.65, 'n = 2', fontsize=14, fontweight='bold') ax.set_xlim(0, 10); ax.set_ylim(0, 3); ax.axis('off') ax = axes[2] draw_matrix(ax, [['a', 'b', 'c'], ['d', 'e', 'f'], ['g', 'h', 'i']], 0.4, 3.2) ax.text(4.0, 2.55, r'$\det = a(ei - fh) - b(di - fg) + c(dh - eg)$', fontsize=19, va='center') ax.text(4.0, 1.85, r'$= aei + bfg + cdh - ceg - bdi - afh$', fontsize=17, va='center', color='#444444') ax.text(4.0, 1.15, 'expand along the top row: each letter times the 2x2 det left after\n' 'crossing out its row and column, alternating signs + - +\n' '6 terms = 3! shuffles: three even (+), three odd (-)', fontsize=11, va='center', color='gray') ax.text(0.05, 3.55, 'n = 3', fontsize=14, fontweight='bold') ax.set_xlim(0, 10.6); ax.set_ylim(0.4, 3.9); ax.axis('off') fig.suptitle('The determinant: three formulas, one pattern\n' 'n! signed shuffle-terms - each n uses the (n-1) formula inside itself', fontsize=14, y=0.99) plt.tight_layout(rect=[0, 0, 1, 0.94]); _finish('det_formulas.png') def fig_geometry(): """det = signed length / area / volume, plus the simple-sequence patterns.""" from mpl_toolkits.mplot3d.art3d import Poly3DCollection fig = plt.figure(figsize=(16, 9.5)) ax = fig.add_subplot(2, 3, 1) a = 2.0 ax.annotate('', xy=(1, 0.6), xytext=(0, 0.6), arrowprops=dict(arrowstyle='-|>', lw=3, color='gray')) ax.annotate('', xy=(a, 0.3), xytext=(0, 0.3), arrowprops=dict(arrowstyle='-|>', lw=3, color='#d62728')) ax.text(0.5, 0.68, 'unit interval, length 1', ha='center') ax.text(a/2, 0.18, f'[{a}] . interval -> length {a}', ha='center', color='#d62728') ax.annotate('', xy=(-1.5, 0.05), xytext=(0, 0.05), arrowprops=dict(arrowstyle='-|>', lw=3, color='#1f77b4')) ax.text(-0.75, -0.07, 'det = -1.5: stretch AND flip', ha='center', color='#1f77b4') ax.set_xlim(-2, 2.6); ax.set_ylim(-0.2, 0.85); ax.axis('off') ax.set_title('n = 1: det[a] = a\nsigned LENGTH scale') ax = fig.add_subplot(2, 3, 2) Mx = np.array([[2, 1], [0.5, 1.5]]) sq = np.array([[0, 0], [1, 0], [1, 1], [0, 1]]) par = sq @ Mx.T ax.fill(sq[:, 0], sq[:, 1], color='lightgray', alpha=0.8, label='unit square, area 1') ax.fill(par[:, 0], par[:, 1], color='#d62728', alpha=0.45, label=f'image: area = det = {np.linalg.det(Mx):.2f}') for col, cc in zip(Mx.T, ['#d62728', '#8c564b']): ax.annotate('', xy=col, xytext=(0, 0), arrowprops=dict(arrowstyle='-|>', lw=2.5, color=cc)) ax.set_aspect('equal'); ax.legend(loc='upper left', fontsize=9) ax.set_title('n = 2: det = ad - bc\nsigned AREA of parallelogram') ax = fig.add_subplot(2, 3, 3, projection='3d') M3 = np.array([[1.5, 0.4, 0.3], [0.2, 1.2, 0.5], [0.1, 0.3, 1.4]]) verts = np.array([[x, y, z] for x in (0, 1) for y in (0, 1) for z in (0, 1)]) tv = verts @ M3.T faces = [[0,1,3,2],[4,5,7,6],[0,1,5,4],[2,3,7,6],[0,2,6,4],[1,3,7,5]] ax.add_collection3d(Poly3DCollection([verts[i] for i in faces], facecolor='lightgray', edgecolor='k', alpha=0.15)) ax.add_collection3d(Poly3DCollection([tv[i] for i in faces], facecolor='#d62728', edgecolor='k', alpha=0.35)) ax.set_title(f'n = 3: det = signed VOLUME\nhere det = {np.linalg.det(M3):.2f}') ax.set_xlim(0, 2.3); ax.set_ylim(0, 2.1); ax.set_zlim(0, 2.3) ax = fig.add_subplot(2, 3, 4) cs = np.linspace(-2, 2, 400) for nn, col in [(1, '#1f77b4'), (2, '#2ca02c'), (3, '#d62728')]: ax.plot(cs, cs**nn, lw=2.5, color=col, label=f'n = {nn}: det(cI) = c^{nn}') ax.axhline(0, color='gray', lw=0.7); ax.axvline(0, color='gray', lw=0.7) ax.legend(fontsize=9); ax.set_xlabel('c'); ax.set_ylabel('det') ax.set_title('Dimension enters as an exponent\ndet(-I) = (-1)^n - parity!') ax = fig.add_subplot(2, 3, 5) width = 0.25 cs5 = [-2, -1, 0, 1, 2] for i, (nn, col) in enumerate([(1, '#1f77b4'), (2, '#2ca02c'), (3, '#d62728')]): vals = [np.linalg.det(np.full((nn, nn), float(c))) for c in cs5] ax.bar(np.arange(5) + (i-1)*width, vals, width, color=col, label=f'n = {nn}') ax.set_xticks(range(5), cs5); ax.set_xlabel('c (every entry equals c)') ax.set_ylabel('det'); ax.legend() ax.set_title('All-equal matrices: det = 0 for n >= 2\nsameness = dependence = COLLAPSE') ax = fig.add_subplot(2, 3, 6) ks = np.array([-2, -1, 0, 1, 2]) for nn, col in [(1, '#1f77b4'), (2, '#2ca02c'), (3, '#d62728')]: ax.plot(ks, 10.0**(ks*nn), 'o-', lw=2.5, color=col, label=f'n = {nn}: det = 10^({nn}k)') ax.set_yscale('log'); ax.set_xlabel('k (matrix = 10^k * I)') ax.set_ylabel('det (log scale)'); ax.legend(fontsize=9) ax.set_title('Powers of ten: slopes 1, 2, 3\nvolume scaling compounds per dimension') plt.tight_layout(); _finish('det_journey.png') def fig_permutations(): """Shuffled identities, all 0/1 matrices, and the non-square trap.""" from scipy.spatial import ConvexHull from collections import Counter fig = plt.figure(figsize=(16, 9.5)) for idx, p in enumerate(itertools.permutations(range(3))): ax = plt.subplot2grid((4, 6), (0 if idx < 3 else 1, idx % 3), fig=fig) P = np.zeros((3, 3)); P[range(3), p] = 1 d = int(round(np.linalg.det(P))) ax.imshow(P, cmap='Greys', vmin=0, vmax=1) ax.set_xticks([]); ax.set_yticks([]) ax.set_title(f'det = {d:+d}', fontsize=11, color='#2ca02c' if d > 0 else '#d62728') for spine in ax.spines.values(): spine.set_color('#2ca02c' if d > 0 else '#d62728'); spine.set_linewidth(2) axl = plt.subplot2grid((4, 6), (0, 3), colspan=3, rowspan=2, fig=fig) axl.axis('off') axl.text(0.02, 0.75, 'A: shuffled identities (all 3x3 permutation matrices)', fontsize=12, fontweight='bold') axl.text(0.02, 0.45, 'det = sign of the shuffle: +1 even swaps, -1 odd.\n' 'Volume never changes (|det| = 1); only ORIENTATION flips.\n' 'Green = rotations, red = mirror reflections. Half of each.', fontsize=10.5) for col, nn in enumerate((2, 3)): ax = plt.subplot2grid((4, 6), (2, col*2), colspan=2, rowspan=2, fig=fig) dets = Counter() for bits in itertools.product([0, 1], repeat=nn*nn): dets[int(round(np.linalg.det(np.array(bits, float).reshape(nn, nn))))] += 1 keys = sorted(dets) colors = ['#d62728' if k == 0 else '#1f77b4' for k in keys] ax.bar([str(k) for k in keys], [dets[k] for k in keys], color=colors) for i, k in enumerate(keys): ax.text(i, dets[k], str(dets[k]), ha='center', va='bottom', fontsize=9) ax.set_title(f'B: all {2**(nn*nn)} binary {nn}x{nn} matrices\n' 'det = 0 (collapse) in red', fontsize=11) ax.set_xlabel('det value'); ax.set_ylabel('count') ax = plt.subplot2grid((4, 6), (2, 4), colspan=2, rowspan=2, fig=fig) A = np.array([[1.0, 0.0, 1.0], [0.0, 1.0, 1.0]]) verts3 = np.array(list(itertools.product([0, 1], repeat=3))) shadow = verts3 @ A.T hull = ConvexHull(shadow); poly = shadow[hull.vertices] ax.fill(poly[:, 0], poly[:, 1], color='#9467bd', alpha=0.4) ax.plot(shadow[:, 0], shadow[:, 1], 'ko', ms=5) ax.set_title('C: 2x3 matrix - det UNDEFINED\n3D cube crushed to flat zonogon;\n' f'sqrt(det(AA^T)) = {np.sqrt(np.linalg.det(A @ A.T)):.3f}', fontsize=11) ax.set_aspect('equal') plt.tight_layout(); _finish('perm_journey.png') def fig_sarrus(ns=(2, 3, 4)): """Positive determinant terms as colored paths; Sarrus breaks at n = 4.""" fig, axes = plt.subplots(1, len(ns), figsize=(7.3*len(ns), 8.5)) if len(ns) == 1: axes = [axes] cmaps = plt.cm.tab20(np.linspace(0, 1, 20)) for ax, n in zip(axes, ns): L = string.ascii_lowercase M = [[L[i*n + j] for j in range(n)] for i in range(n)] terms = [p for p in itertools.permutations(range(n)) if _perm_sign(p) == 1] for i in range(n): for j in range(n): ax.add_patch(plt.Rectangle((j, n-1-i), 1, 1, fill=False, edgecolor='lightgray', lw=1)) ax.text(j + 0.5, n-1-i + 0.5, M[i][j], ha='center', va='center', fontsize=30 - 3*n, fontweight='bold', zorder=5) labels = [] for k, p in enumerate(terms): pts = np.array([[p[i] + 0.5, n-1-i + 0.5] for i in range(n)]) off = (k - len(terms)/2) * 0.045 po = pts + np.array([off, off]) col = cmaps[k % 20] ax.plot(po[:, 0], po[:, 1], '-', lw=3, color=col, alpha=0.85) ax.annotate('', xy=po[-1], xytext=po[-2], arrowprops=dict(arrowstyle='-|>', lw=3, color=col)) labels.append(''.join(M[i][p[i]] for i in range(n))) ax.set_xlim(-0.6, n + 0.6); ax.set_ylim(-1.6, n + 0.4) ax.set_aspect('equal'); ax.axis('off') ax.set_title(f'n = {n}: {len(terms)} positive terms', fontsize=15) ax.text(n/2, -0.55, ' + '.join(labels), ha='center', fontsize=15 if n < 4 else 10.5, wrap=True) if n == 4: ax.text(n/2, -1.25, 'Sarrus BREAKS here: 8 wrap-diagonals, 12 terms needed', ha='center', fontsize=11, color='#d62728') fig.suptitle('Positive determinant terms as paths - the diagonal picture is\n' 'a low-dimensional accident', fontsize=16, y=1.0) plt.tight_layout(rect=[0, 0, 1, 0.93]); _finish('sarrus_lines.png') def _ports(paths, n, R=0.33): node_users = {} for lid, p, pts in paths: for i in range(n): prev_d = pts[i] - pts[i-1] if i > 0 else None next_d = pts[i+1] - pts[i] if i < n-1 else None if prev_d is None: t = next_d elif next_d is None: t = prev_d else: t = prev_d/np.linalg.norm(prev_d) + next_d/np.linalg.norm(next_d) if np.linalg.norm(t) < 1e-9: t = prev_d node_users.setdefault((i, p[i]), []).append( (lid, np.arctan2(t[1], t[0]), t/np.linalg.norm(t))) offsets = {} for node, users in node_users.items(): users_sorted = sorted(users, key=lambda u: (round(u[1], 3), u[0])) m = len(users_sorted) i_row, j_col = node center = np.array([j_col + 0.5, n - 1 - i_row + 0.5]) for k, (lid, ang, t) in enumerate(users_sorted): perp = np.array([-t[1], t[0]]) slot = (k - (m - 1) / 2) / max(m - 1, 1) * 2 * (R * 0.85) offsets[(node, lid)] = center + perp * slot return node_users, offsets def fig_rainbow_ports(n=5, keep_columns=None, show_counts=False): """n x n positive paths, rainbow families by start column, arc-port fanning. keep_columns: e.g. (0, 2) to show only the a- and c-families. show_counts: print each letter's docking count under its circle (paper Figure 6 — 'per-node docking counts').""" L = string.ascii_lowercase M = [[L[i*n + j] for j in range(n)] for i in range(n)] pos = [p for p in itertools.permutations(range(n)) if _perm_sign(p) == 1] if keep_columns is not None: pos = [p for p in pos if p[0] in keep_columns] paths = [] for lid, p in enumerate(pos): pts = [np.array([p[i] + 0.5, n - 1 - i + 0.5]) for i in range(n)] paths.append((lid, p, pts)) R = 0.33 node_users, offsets = _ports(paths, n, R) all_cmaps = [cm.Reds, cm.Oranges, cm.Greens, cm.Blues, cm.Purples, cm.Greys, cm.YlOrBr, cm.BuGn, cm.PuRd] cols_present = sorted({p[0] for _, p, _ in paths}) fam_cmap = {c: all_cmaps[c % len(all_cmaps)] for c in cols_present} counters = {c: 0 for c in cols_present} fam_size = {c: sum(1 for _, p, _ in paths if p[0] == c) for c in cols_present} fig, ax = plt.subplots(figsize=(3*n + 1, 3*n + 2)) for i in range(n): for j in range(n): ax.add_patch(plt.Rectangle((j, n-1-i), 1, 1, fill=False, edgecolor='#dddddd', lw=1)) ax.add_patch(plt.Circle((j + 0.5, n-1-i + 0.5), R, fill=False, edgecolor='#eeeeee', lw=0.8, zorder=1)) for lid, p, pts in paths: c = p[0]; k = counters[c]; counters[c] += 1 col = fam_cmap[c](0.45 + 0.5 * k / max(fam_size[c] - 1, 1)) W = np.array([offsets[((i, p[i]), lid)] for i in range(n)]) ax.plot(W[:, 0], W[:, 1], '-', lw=2.0, color=col, alpha=0.85, zorder=2) ax.annotate('', xy=W[-1], xytext=W[-2], arrowprops=dict(arrowstyle='-|>', lw=2.0, color=col, alpha=0.9), zorder=3) for i in range(n): for j in range(n): ax.text(j + 0.5, n-1-i + 0.5, M[i][j], ha='center', va='center', fontsize=25, fontweight='bold', zorder=6, bbox=dict(boxstyle='circle,pad=0.12', fc='white', ec='none', alpha=0.65)) for c in cols_present: ax.text(c + 0.5, n + 0.22, f'start {M[0][c]}', ha='center', fontsize=13, color=fam_cmap[c](0.8), fontweight='bold') if show_counts: for i in range(n): for j in range(n): cnt = len(node_users.get((i, j), [])) ax.text(j + 0.5, n-1-i + 0.5 - R - 0.05, str(cnt), ha='center', va='top', fontsize=11, fontweight='bold', color='#444444' if cnt else '#bbbbbb', zorder=6) ax.set_xlim(-0.4, n + 0.4); ax.set_ylim(-0.5, n + 0.85) ax.set_aspect('equal'); ax.axis('off') ax.set_title(f'n = {n}: {len(paths)} positive paths with arc-port fanning', fontsize=16, pad=12) plt.tight_layout() suffix = '' if keep_columns is None else '_' + ''.join(M[0][c] for c in keep_columns) _finish(f'rainbow{n}{suffix}.png') def fig_growth(): """Letters n^2 vs positive paths n!/2 for n = 1..9, linear and log.""" ns = np.arange(1, 10) letters = ns**2 paths = np.array([max(factorial(int(n)) // 2, 1) for n in ns]) fig, axes = plt.subplots(1, 2, figsize=(14, 6)) ax = axes[0] ax.plot(ns, letters, 'o-', lw=2.5, color='#1f77b4', label='letters = n^2') ax.plot(ns, paths, 's-', lw=2.5, color='#d62728', label='positive paths = n!/2') ax.set_xlabel('n'); ax.set_ylabel('count') ax.set_title('Linear: paths vanish then explode') ax.legend(); ax.grid(alpha=0.3) ax = axes[1] ax.semilogy(ns, letters, 'o-', lw=2.5, color='#1f77b4', label='letters = n^2') ax.semilogy(ns, paths, 's-', lw=2.5, color='#d62728', label='paths = n!/2') ax.semilogy(ns, paths/letters, '^--', lw=2, color='#9467bd', label='paths per letter') ax.set_xlabel('n'); ax.set_ylabel('count (log)') ax.set_title('Log: polynomial straightens, factorial keeps bending') ax.legend(); ax.grid(alpha=0.3, which='both') plt.tight_layout(); _finish('growth.png') def fig_crossing(): """Exact crossings of x^2 and x!/2 via the Gamma function.""" from scipy.special import gamma from scipy.optimize import brentq f = lambda x: gamma(x + 1) / 2 - x**2 x1 = brentq(f, 0.5, 2.0); x2 = brentq(f, 3.5, 5.0) print(f'crossings: x = {x1:.6f} and x = {x2:.6f}') xs = np.linspace(0.5, 6, 500) fig, ax = plt.subplots(figsize=(10, 6.5)) ax.plot(xs, xs**2, lw=2.5, color='#1f77b4', label='letters: x^2') ax.plot(xs, gamma(xs + 1)/2, lw=2.5, color='#d62728', label='paths: Gamma(x+1)/2') ns = np.arange(1, 7) ax.plot(ns, ns**2, 'o', ms=8, color='#1f77b4') ax.plot(ns, [max(factorial(n)//2, 1) for n in ns], 's', ms=8, color='#d62728') for x, lab in [(x1, f'x = {x1:.3f}'), (x2, f'inversion\nx = {x2:.3f}')]: ax.axvline(x, color='gray', ls=':', lw=1.5) ax.plot(x, x**2, 'k*', ms=16, zorder=5) ax.annotate(lab, (x, x**2), textcoords='offset points', xytext=(10, -28), fontsize=11, fontweight='bold') ax.fill_between(xs, xs**2, gamma(xs+1)/2, where=xs**2 > gamma(xs+1)/2, color='#1f77b4', alpha=0.12, label='letters rule (drawable)') ax.fill_between(xs, xs**2, gamma(xs+1)/2, where=gamma(xs+1)/2 > xs**2, color='#d62728', alpha=0.12, label='paths rule (storm)') ax.set_ylim(0, 60); ax.set_xlabel('n (continuous)'); ax.set_ylabel('count') ax.set_title('The two crossings: entries vs terms\n' 'n = 4 is the last gasp of the drawable world (16 vs 12)') ax.legend(loc='upper left'); ax.grid(alpha=0.3) plt.tight_layout(); _finish('crossing.png') # ===================================================================== # DIRECTION 2 — RICHNESS of the entries # ===================================================================== def fig_det_field(): """The function rung: det becomes a field (Whitney cusp map).""" x, y = sp.symbols('x y') F1, F2 = x*y - x**3, y J = sp.Matrix([F1, F2]).jacobian([x, y]) detJ = sp.expand(J.det()) print('F =', (F1, F2), ' det DF =', detJ) f_det = sp.lambdify((x, y), detJ, 'numpy') f_map = sp.lambdify((x, y), [F1, F2], 'numpy') lim = 1.4 g = np.linspace(-lim, lim, 600) X, Y = np.meshgrid(g, g) D = f_det(X, Y) fig, axes = plt.subplots(1, 2, figsize=(15, 7)) ax = axes[0] vmax = np.percentile(np.abs(D), 98) im = ax.pcolormesh(X, Y, D, cmap='RdBu_r', vmin=-vmax, vmax=vmax, shading='auto') csz = ax.contour(X, Y, D, levels=[0], colors='black', linewidths=2.5) ax.set_title('det DF = y - 3x^2, a FIELD over the plane\n' 'red +: preserves, blue -: mirrors, black: collapse (Jacobi alarm)') ax.set_xlabel('x'); ax.set_ylabel('y'); ax.set_aspect('equal') fig.colorbar(im, ax=ax, label='det DF(x, y)') ax = axes[1] t = np.linspace(-lim, lim, 500) for gv in np.linspace(-lim, lim, 27): for (gx, gy) in [(np.full_like(t, gv), t), (t, np.full_like(t, gv))]: u, v = f_map(gx, gy) d = f_det(gx, gy) ax.scatter(u, v, c=np.where(d >= 0, '#d62728', '#1f77b4'), s=0.4, alpha=0.45) try: zero_paths = csz.get_paths() # matplotlib >= 3.8 except AttributeError: zero_paths = [p for coll in csz.collections for p in coll.get_paths()] for path in zero_paths: vp = path.vertices u, v = f_map(vp[:, 0], vp[:, 1]) ax.plot(u, v, 'k-', lw=2.8, zorder=5) ax.set_title('Image of the grid: creased along the fold\n' 'in THIS world, merging happens only by crossing det = 0') ax.set_xlabel('F1'); ax.set_ylabel('F2'); ax.set_aspect('equal') ax.set_xlim(-1.2, 1.2); ax.set_ylim(-1.5, 1.5) plt.tight_layout(); _finish('det_field.png') def fig_block_break(seed=3): """The commutativity cliff: block-determinant formulas vs the truth. det(AD - BC) for a 2x2 block matrix is valid only when the blocks commute — the shuffle formula silently assumed ad = da. The commuting example draws its blocks from the MONOGENIC (singly generated) commutative subalgebra C[M]: polynomials in one fixed matrix always commute; two independent generators generically do not. Meanwhile the TRACE — the additive invariant — walks past the cliff: tr(AB) = tr(BA) holds for non-commuting matrices too (cyclicity).""" rng = np.random.default_rng(seed) I2 = np.eye(2) M = np.array([[1.0, 2.0], [3.0, 4.0]]) # monogenic example: every block lies in C[M], polynomials in ONE matrix com = dict(A=M, B=M @ M - I2, C=2*M + I2, D=M @ M - 3*M) # generic example: independent integer blocks — no reason to commute gen = dict(zip('ABCD', (rng.integers(-3, 4, (2, 2)).astype(float) for _ in range(4)))) cases = [('blocks from the MONOGENIC algebra C[M]\n' '(polynomials in one matrix - always commute)', com), ('generic blocks, two independent generators\n' '(commutativity generically destroyed)', gen)] labels = ['true det of the 4x4', 'det(AD - BC)', 'det(AD - CB)'] colors = ['#1f77b4', '#2ca02c', '#9467bd'] fig, axes = plt.subplots(1, 2, figsize=(13, 6.5)) for ax, (title, blk) in zip(axes, cases): A, B, C, D = blk['A'], blk['B'], blk['C'], blk['D'] big = np.block([[A, B], [C, D]]) vals = [np.linalg.det(big), np.linalg.det(A @ D - B @ C), np.linalg.det(A @ D - C @ B)] vals = [round(v) for v in vals] print(f'{title.splitlines()[0]}: true={vals[0]}, ' f'det(AD-BC)={vals[1]}, det(AD-CB)={vals[2]}') bars = ax.bar(labels, vals, color=colors, width=0.62) pad = 0.02 * max(abs(x) for x in vals) for bar, v in zip(bars, vals): up = v >= 0 ax.text(bar.get_x() + bar.get_width()/2, bar.get_height() + (pad if up else -pad), f'{v}', ha='center', va='bottom' if up else 'top', fontsize=13, fontweight='bold') agree = len(set(vals)) == 1 ax.set_title(title + '\n' + ('all three AGREE' if agree else 'all three DISAGREE'), fontsize=11.5, color='#2ca02c' if agree else '#d62728') ax.axhline(0, color='gray', lw=0.8) ax.tick_params(axis='x', labelsize=9) ax.grid(axis='y', alpha=0.3) # the additive invariant survives the cliff: cyclicity needs no ad = da A, B = gen['A'], gen['B'] tAB, tBA = np.trace(A @ B), np.trace(B @ A) print(f'trace survives the cliff: tr(AB) = {tAB:g}, tr(BA) = {tBA:g} ' f'(equal, though AB != BA) - while the det formulas disagree') axes[1].text(0.5, -0.16, f'yet tr(AB) = tr(BA) = {tAB:g} for these same blocks: ' 'the ADDITIVE invariant survives the cliff', transform=axes[1].transAxes, ha='center', fontsize=10.5, color='#1f77b4') fig.suptitle('Matrix-of-matrix entries: the shuffle formula assumed the ' 'letters commute\nover a non-commutative ring "the" ' 'determinant stops being well-defined - but the trace ' 'survives', fontsize=13.5) plt.tight_layout(rect=[0, 0.04, 1, 0.90]); _finish('block_break.png') def fig_two_vacuums(samples=200, seed=42): """Two operations, two nothings: 0 is the additive identity (empty sum), 1 is the multiplicative identity (empty product) — neither more fundamental. det lives in the multiplicative world (det AB = det A det B, det of the 0x0 matrix = 1); its additive twin is the trace. exp/log bridge the worlds, and det(exp A) = exp(tr A) is the capstone.""" from scipy.linalg import expm from matplotlib.patches import FancyBboxPatch E0 = np.zeros((0, 0)) print(f'det of the 0x0 matrix = {np.linalg.det(E0):g} (empty product); ' f'trace = {np.trace(E0):g} (empty sum)') fig, axes = plt.subplots(1, 3, figsize=(19, 6.5)) # --- panel a: the two worlds -------------------------------------- ax = axes[0] cols = [ (0.19, '#1f77b4', 'ADDITIVE\nWORLD', ['identity: 0', 'empty SUM = 0', 'TRACE:\ntr(A+B) = tr A + tr B', 'tr(0x0 matrix) = 0', 'swap COUNT k (mod 2)']), (0.81, '#d62728', 'MULTIPLICATIVE\nWORLD', ['identity: 1', 'empty PRODUCT = 1', 'DETERMINANT:\ndet(AB) = det A det B', 'det(0x0 matrix) = 1', 'sign (-1)^k']), ] for xc, col, name, rows in cols: ax.add_patch(FancyBboxPatch((xc - 0.17, 0.12), 0.34, 0.72, boxstyle='round,pad=0.02', fc='none', ec=col, lw=2)) ax.text(xc, 0.93, name, ha='center', va='center', fontsize=11.5, fontweight='bold', color=col) for i, r in enumerate(rows): ax.text(xc, 0.76 - 0.145 * i, r, ha='center', va='center', fontsize=9.5, color=col) for y, lab, x0, x1 in [(0.58, 'exp: sums -> products', 0.385, 0.615), (0.40, 'log: products -> sums', 0.615, 0.385)]: ax.annotate('', xy=(x1, y), xytext=(x0, y), arrowprops=dict(arrowstyle='-|>', lw=2.2, color='#9467bd')) ax.text(0.5, y + 0.035, lab, ha='center', fontsize=8, color='#9467bd') ax.text(0.5, 0.035, 'parity bridge: additive swap-count k\n' 'becomes the multiplicative sign (-1)^k', ha='center', fontsize=9, color='#9467bd') ax.set_xlim(0, 1); ax.set_ylim(0, 1); ax.axis('off') ax.set_title('Two operations, two nothings -\nneither origin is more ' 'fundamental', fontsize=12) # --- panel b: the capstone identity ------------------------------- ax = axes[1] rng = np.random.default_rng(seed) trs, logdets, bad = [], [], 0 for _ in range(samples): n = int(rng.integers(2, 6)) A = rng.uniform(-1, 1, (n, n)) / n sign, logabs = np.linalg.slogdet(expm(A)) if sign != 1: bad += 1 trs.append(np.trace(A)); logdets.append(logabs) trs = np.array(trs); logdets = np.array(logdets) if bad: print(f'WARNING: {bad} samples had det(exp A) with sign != +1') disc = np.abs(logdets - trs).max() lo, hi = trs.min() - 0.1, trs.max() + 0.1 ax.plot([lo, hi], [lo, hi], color='#2ca02c', lw=2, label='y = x') ax.scatter(trs, logdets, s=22, color='#9467bd', alpha=0.7, label=f'{samples} random A, n = 2..5') ax.set_xlabel('tr A (additive invariant)') ax.set_ylabel('log det(exp A) (multiplicative invariant, via slogdet)') ax.legend(fontsize=9) ax.set_title('The capstone: det(exp A) = exp(tr A)\n' f'max |log det(e^A) - tr A| = {disc:.1e}', fontsize=12) ax.grid(alpha=0.3) # --- panel c: absorption vs transparency -------------------------- ax = axes[2] x = np.linspace(-3, 3, 200) ax.plot(x, 0 * x, lw=3, color='#d62728', label='x * 0 = 0 - additive vacuum ABSORBS (collapse)') ax.plot(x, 1 * x, lw=3, color='#2ca02c', label='x * 1 = x - multiplicative vacuum is transparent') ax.axhline(0, color='gray', lw=0.6); ax.axvline(0, color='gray', lw=0.6) ax.set_xlabel('x'); ax.set_ylabel('product') ax.legend(fontsize=9, loc='upper left') ax.set_title('Collapse, re-read: one dependent row is the\n' 'additive vacuum invading the multiplicative world', fontsize=12) ax.grid(alpha=0.3) fig.suptitle('The two vacuums: 0 (empty sum) and 1 (empty product) - ' 'trace and determinant as the two shadows of one matrix', fontsize=14) plt.tight_layout(rect=[0, 0, 1, 0.93]); _finish('two_vacuums.png') def fig_two_nothings_finale(samples=50, seed=42): """The two-vacuums layer completed. (a) The hospitality table: each nothing transparent at home, mirror OPPOSITES abroad — zero annihilates the multiplicative world (x*0 = 0), one generates the additive world (1, 1+1, 1+1+1, ... builds the number line). (b) Each nothing's locus in the matrix world — tr A = 0 and det A = 1, i.e. sl_n and SL_n — carried exactly onto each other by exp, since exp(0) = 1; verified numerically. (c) When the two nothings touch, 0 = 1 forces every x = x*1 = x*0 = 0: the zero ring, the one-point arithmetic.""" from scipy.linalg import expm from matplotlib.patches import FancyBboxPatch # panel-b check first: traceless exponentiates to volume-preserving rng = np.random.default_rng(seed) devs = [] for _ in range(samples): n = int(rng.integers(2, 6)) A = rng.uniform(-1, 1, (n, n)) A -= (np.trace(A) / n) * np.eye(n) # project the trace out sign, logabs = np.linalg.slogdet(expm(A)) devs.append(abs(sign * np.exp(logabs) - 1.0)) maxdev = max(devs) print(f'{samples} random traceless matrices (n = 2..5): ' f'max |det(exp A) - 1| = {maxdev:.2e} - traceless exponentiates ' 'to volume-preserving, exactly as exp(0) = 1 demands') if maxdev > 1e-10: print('WARNING: deviation above machine-precision expectations') fig, axes = plt.subplots(1, 3, figsize=(19, 6.8)) # --- a: the hospitality table ------------------------------------- ax = axes[0] ax.text(0.40, 0.90, 'AT HOME', ha='center', fontsize=12, fontweight='bold', color='#555555') ax.text(0.78, 0.90, 'ABROAD', ha='center', fontsize=12, fontweight='bold', color='#555555') ax.text(0.02, 0.62, 'ZERO\nthe additive\nnothing', ha='left', va='center', fontsize=11, fontweight='bold', color='#1f77b4') ax.text(0.02, 0.22, 'ONE\nthe multiplicative\nnothing', ha='left', va='center', fontsize=11, fontweight='bold', color='#d62728') ax.text(0.40, 0.62, 'transparent:\nx + 0 = x', ha='center', va='center', fontsize=11, color='#777777') ax.text(0.40, 0.22, 'transparent:\nx · 1 = x', ha='center', va='center', fontsize=11, color='#777777') ax.text(0.78, 0.70, 'ANNIHILATES: x · 0 = 0', ha='center', fontsize=11, fontweight='bold', color='#d62728') for xs_ in (0.66, 0.72, 0.78, 0.84, 0.90): ax.annotate('', xy=(0.78, 0.545), xytext=(xs_, 0.64), arrowprops=dict(arrowstyle='->', lw=1.1, color='#d62728', alpha=0.7)) ax.plot(xs_, 0.64, 'o', ms=5, color='#d62728') ax.plot(0.78, 0.535, 'o', ms=9, color='#d62728') ax.text(0.78, 0.30, 'GENERATES: 1, 1+1, 1+1+1, ...', ha='center', fontsize=11, fontweight='bold', color='#2ca02c') ax.plot(0.62, 0.20, 'o', ms=9, color='#2ca02c') for k, xs_ in enumerate((0.70, 0.77, 0.84, 0.91)): ax.annotate('', xy=(xs_, 0.20), xytext=(xs_ - 0.055, 0.20), arrowprops=dict(arrowstyle='->', lw=1.1, color='#2ca02c', alpha=0.8)) ax.plot(xs_, 0.20, 'o', ms=5 + k, color='#2ca02c') ax.text(0.78, 0.11, 'builds the whole number line', ha='center', fontsize=9, color='#2ca02c') ax.plot([0.24, 0.24], [0.05, 0.82], color='#cccccc', lw=1) ax.plot([0.58, 0.58], [0.05, 0.82], color='#cccccc', lw=1) ax.plot([0.14, 0.97], [0.44, 0.44], color='#cccccc', lw=1) ax.set_xlim(0, 1); ax.set_ylim(0, 1); ax.axis('off') ax.set_title('The hospitality table: transparent at home,\n' 'mirror OPPOSITES abroad - destruction and genesis', fontsize=12) # --- b: the two loci and the bridge ------------------------------- ax = axes[1] for xc, col, main, sub in [ (0.24, '#1f77b4', 'tr A = 0', "the additive nothing's locus\ntraceless matrices - " '$\\mathfrak{sl}_n$'), (0.76, '#d62728', 'det A = 1', "the multiplicative nothing's locus\nvolume-preserving - " '$SL_n$')]: ax.add_patch(FancyBboxPatch((xc - 0.185, 0.38), 0.37, 0.30, boxstyle='round,pad=0.02', fc='white', ec=col, lw=2.5)) ax.text(xc, 0.60, main, ha='center', va='center', fontsize=15, fontweight='bold', color=col) ax.text(xc, 0.47, sub, ha='center', va='center', fontsize=9.5, color=col) ax.annotate('', xy=(0.555, 0.53), xytext=(0.445, 0.53), arrowprops=dict(arrowstyle='-|>', lw=3, color='#9467bd')) ax.text(0.5, 0.585, 'exp', ha='center', fontsize=13, fontweight='bold', color='#9467bd') ax.text(0.5, 0.30, 'det(exp A) = exp(tr A), and exp(0) = 1:\n' 'traceless exponentiates EXACTLY to volume-preserving', ha='center', fontsize=10.5, color='#9467bd') ax.text(0.5, 0.12, f'numerical check: {samples} random traceless A, ' f'n = 2..5\nmax |det(exp A) - 1| = {maxdev:.1e}', ha='center', fontsize=9.5, color='#555555') ax.text(0.5, 0.85, 'the expedition aimed here with a sentence -\n' 'Lie theory was already standing, with a name on the door', ha='center', fontsize=9.5, color='#555555', style='italic') ax.set_xlim(0, 1); ax.set_ylim(0, 1); ax.axis('off') ax.set_title("Each nothing casts a locus - 'the trace of each\n" "nothing's position' - and the capstone bridges them", fontsize=12) # --- c: the coincidence 0 = 1 ------------------------------------- ax = axes[2] ax.plot(0.30, 0.78, 'o', ms=13, color='#1f77b4') ax.text(0.30, 0.87, '0', ha='center', fontsize=15, fontweight='bold', color='#1f77b4') ax.plot(0.70, 0.78, 'o', ms=13, color='#d62728') ax.text(0.70, 0.87, '1', ha='center', fontsize=15, fontweight='bold', color='#d62728') for x0 in (0.30, 0.70): ax.annotate('', xy=(0.5, 0.45), xytext=(x0, 0.75), arrowprops=dict(arrowstyle='-|>', lw=2, color='black')) ax.text(0.5, 0.66, 'suppose 0 = 1', ha='center', fontsize=12, fontweight='bold') ax.plot(0.5, 0.40, 'o', ms=16, color='black') ax.text(0.5, 0.325, 'then for every x: x = x·1 = x·0 = 0', ha='center', fontsize=11) ax.text(0.5, 0.245, 'the ZERO RING: the unique ring with one element\n' 'every x forced to zero - the whole arithmetic is one point\n' '(so in every nontrivial system, 0 and 1 are provably distinct)', ha='center', va='top', fontsize=10.5) ax.text(0.5, 0.05, 'the only place the two buildings meet\nis a world ' 'with a single point in it', ha='center', va='bottom', fontsize=10, style='italic', color='#555555') ax.set_xlim(0, 1); ax.set_ylim(0, 1); ax.axis('off') ax.set_title('The singularity - when the two nothings touch:\n' 'formally, the ZERO RING (one element, all of it)', fontsize=12) fig.suptitle('The two nothings, complete: transparent at home, opposite ' 'abroad - and one point where they meet', fontsize=14) plt.tight_layout(rect=[0, 0, 1, 0.92]); _finish('two_nothings_finale.png') # ===================================================================== # DIRECTION 3 — DIMENSION of the map (the Jacobian conjecture) # ===================================================================== def _fable_exprs(Z): u = 1 + Z[0]*Z[1] return [u**3*Z[2] + Z[1]**2*u*(4 + 3*Z[0]*Z[1]), Z[1] + 3*Z[0]*u**2*Z[2] + 3*Z[0]*Z[1]**2*(4 + 3*Z[0]*Z[1]), 2*Z[0] - 3*Z[0]**2*Z[1] - Z[0]**3*Z[2]] def fable_verify(): """Symbolic check: det DF = -2 and the triple collision.""" Z = sp.symbols('z1 z2 z3') F = sp.Matrix(_fable_exprs(Z)) detJ = sp.expand(F.jacobian(Z).det()) print('det DF =', detJ) for p in [(0, 0, sp.Rational(-1, 4)), (1, sp.Rational(-3, 2), sp.Rational(13, 2)), (-1, sp.Rational(3, 2), sp.Rational(13, 2))]: print(f'F{p} =', list(F.subs(dict(zip(Z, p))).T)) def fig_fable_wells(s_max=4.5, t_max=4.0, n=800, name='jacobian_wells.png'): """Three wells: the collision plane of the 2026 counterexample. s_max, t_max set the half-width of the plane window (zoom out by passing larger values), n the grid resolution per axis.""" def F(z1, z2, z3): u = 1 + z1*z2 return (u**3*z3 + z2**2*u*(4 + 3*z1*z2), z2 + 3*z1*u**2*z3 + 3*z1*z2**2*(4 + 3*z1*z2), 2*z1 - 3*z1**2*z2 - z1**3*z3) target = np.array([-0.25, 0.0, 0.0]) P = [np.array([0.0, 0.0, -0.25]), np.array([1.0, -1.5, 6.5]), np.array([-1.0, 1.5, 6.5])] origin = sum(P) / 3 e1 = P[1] - P[0]; e1 /= np.linalg.norm(e1) v = P[2] - P[0]; e2 = v - (v @ e1) * e1; e2 /= np.linalg.norm(e2) s = np.linspace(-s_max, s_max, n); t = np.linspace(-t_max, t_max, n) S, T = np.meshgrid(s, t) X = origin[0] + S*e1[0] + T*e2[0] Y = origin[1] + S*e1[1] + T*e2[1] Zc = origin[2] + S*e1[2] + T*e2[2] F1, F2, F3 = F(X, Y, Zc) dist = np.sqrt((F1-target[0])**2 + (F2-target[1])**2 + (F3-target[2])**2) logd = np.log10(dist + 1e-12) fig, ax = plt.subplots(figsize=(9, 7)) im = ax.pcolormesh(S, T, logd, cmap='inferno', shading='auto', vmin=-2.5, vmax=np.percentile(logd, 99)) for Pt in P: d = Pt - origin ax.plot(d @ e1, d @ e2, 'o', ms=11, mfc='none', mec='cyan', mew=2.2) ax.set_title('The 2026 counterexample: three points, one image\n' 'det DF = -2 everywhere - merging with ZERO local collapse') ax.set_xlabel('plane coordinate s'); ax.set_ylabel('plane coordinate t') fig.colorbar(im, ax=ax, label='log10 distance to target') plt.tight_layout(); _finish(name) def _collide_search(f, trials, box, seed): """Core collision hunter: random targets, least-squares probes for a DIFFERENT preimage. An instrument — its finds are evidence, not proof.""" from scipy.optimize import least_squares rng = np.random.default_rng(seed) found = [] for _ in range(trials): xx = rng.uniform(-box, box, 3) fx = np.asarray(f(*xx), float) y0 = rng.uniform(-box, box, 3) try: sol = least_squares(lambda y: np.asarray(f(*y), float) - fx, y0, method='lm', max_nfev=3000) except Exception: continue yy = sol.x if (np.linalg.norm(np.asarray(f(*yy), float) - fx) < 1e-8 and np.linalg.norm(xx - yy) > 1e-2): found.append((xx, yy)) return found def fable_collide(trials=150, box=3.0, seed=7): """Numerically hunt for collision pairs of the Fable map.""" Z = sp.symbols('z1 z2 z3') f = sp.lambdify(Z, _fable_exprs(Z), 'numpy') found = _collide_search(f, trials, box, seed) print(f'{len(found)} collision pairs found (of {trials} trials)') for xx, yy in found[:5]: print(' x =', np.round(xx, 3), ' y =', np.round(yy, 3)) return found def _fable_np(z1, z2, z3): """Vectorised numpy evaluation of the Fable map.""" u = 1 + z1*z2 return (u**3*z3 + z2**2*u*(4 + 3*z1*z2), z2 + 3*z1*u**2*z3 + 3*z1*z2**2*(4 + 3*z1*z2), 2*z1 - 3*z1**2*z2 - z1**3*z3) def _shear_polys(which): """The tame shears used as disguises: z2-shift p(z1), z3-shift q(z1, z2). All triangular with unit diagonal — Jacobian det 1.""" if which == 1: return (lambda a: a**2), (lambda a, b: 2*a*b) if which == 2: return (lambda a: 1 - a - a**3), (lambda a, b: b*(3 + a**2)) return (lambda a: a**3 - 2*a), (lambda a, b: b*(1 + 2*a)) def _disguised_np(which=1): """The Fable map conjugated by a tame unit-Jacobian shear: G = P^-1 o F o P. G is again a polynomial Keller map, and by the chain rule det DG = det DF = -2 EXACTLY (each shear factor has Jacobian determinant 1). We evaluate the composition directly — expanding it symbolically is possible but explosively large, and the numbers are identical either way.""" p, q = _shear_polys(which) fwd = lambda z1, z2, z3: (z1, z2 + p(z1), z3 + q(z1, z2)) inv = lambda w1, w2, w3: (w1, w2 - p(w1), w3 - q(w1, w2 - p(w1))) def G(z1, z2, z3): return inv(*_fable_np(*fwd(z1, z2, z3))) return G def _check_shear_inverts(which, rng): """Verify P^-1(P(z)) = z at random points. (A finite-difference det of the CONJUGATED map is numerically hopeless — the chain-rule factors are huge and cancel to -2 only in exact arithmetic — so we check the piece that could actually be wrong: that the shear really inverts.)""" p, q = _shear_polys(which) z = rng.uniform(-3, 3, (5, 3)) w = np.column_stack([z[:, 0], z[:, 1] + p(z[:, 0]), z[:, 2] + q(z[:, 0], z[:, 1])]) back = np.column_stack([w[:, 0], w[:, 1] - p(w[:, 0]), w[:, 2] - q(w[:, 0], w[:, 1] - p(w[:, 0]))]) return float(np.abs(back - z).max()) def fig_escape_analysis(trials=250, seed=11): """The full escape analysis (paper Figure: three panels). Left: the escape scale of collision pairs is crushed by a mere change of coordinates — raw fiber norm is NOT conjugation-invariant. Middle: separation vs escape — no merges near the origin. Right: a properness probe — where images of huge spheres land in target space. Bright ridges are the escape valve made visible. All evidence, never proof.""" maps = [('original F', _fable_np), ('disguise $P_1^{-1} \\circ F \\circ P_1$', _disguised_np(1)), ('disguise $P_2^{-1} \\circ F \\circ P_2$', _disguised_np(2)), ('disguise $P_3^{-1} \\circ F \\circ P_3$', _disguised_np(3))] # det DG = -2 exactly, by the chain rule (unit-Jacobian shears); the # piece that could actually be wrong is the shear inversion — check it rng0 = np.random.default_rng(seed) for k in (1, 2, 3): err = _check_shear_inverts(k, rng0) print(f'disguise {k}: max |P^-1(P(z)) - z| = {err:.2e} at random ' 'points; det DG = -2 everywhere by the chain rule') fig, axes = plt.subplots(1, 3, figsize=(19, 6.2)) ax = axes[0] all_pairs = {} for i, (name, f) in enumerate(maps): pairs = _collide_search(f, trials, 3.0, seed) all_pairs[name] = pairs if not pairs: print(f'{name}: no pairs found in {trials} trials') ax.text(i, 1.0, 'no pairs\nfound', ha='center', va='center', fontsize=10, color='gray') continue esc = np.array([max(np.linalg.norm(x), np.linalg.norm(y)) for x, y in pairs]) print(f'{name}: {len(pairs)} pairs, escape scale ' f'median {np.median(esc):.4g}, max {esc.max():.4g}') jit = np.random.default_rng(seed + i).uniform(-0.14, 0.14, len(esc)) ax.scatter(np.full(len(esc), i) + jit, esc, s=30, alpha=0.75, color=['#d62728', '#1f77b4', '#9467bd', '#2ca02c'][i]) ax.plot([i - 0.22, i + 0.22], [np.median(esc)]*2, 'k-', lw=2.5) ax.set_yscale('log') ax.set_xticks(range(len(maps))) ax.set_xticklabels([m[0] for m in maps], fontsize=8) ax.set_ylabel('escape scale max(|x|, |y|) of found pairs') ax.set_title('Same map, new coordinates: the escape scale\n' 'is crushed - raw fiber norm is not the invariant', fontsize=11) ax.grid(alpha=0.3, which='both', axis='y') ax = axes[1] pairs = all_pairs['original F'] esc = np.array([max(np.linalg.norm(x), np.linalg.norm(y)) for x, y in pairs]) sep = np.array([np.linalg.norm(x - y) for x, y in pairs]) ax.scatter(sep, esc, s=35, color='#d62728', alpha=0.8) ax.set_xscale('log'); ax.set_yscale('log') ax.set_xlabel('separation |x - y|'); ax.set_ylabel('escape max(|x|, |y|)') ax.set_title('Borrowing room at infinity:\nno merges happen near the origin', fontsize=11) ax.grid(alpha=0.3, which='both') ax = axes[2] rng = np.random.default_rng(seed) W = 30.0 edges = np.linspace(-W, W, 241) H = np.zeros((240, 240)) for R in (25, 50, 100, 200): pts = rng.normal(size=(600_000, 3)) pts *= R / np.linalg.norm(pts, axis=1, keepdims=True) F1, F2, F3 = _fable_np(pts[:, 0], pts[:, 1], pts[:, 2]) m = (np.abs(F1) < W) & (np.abs(F2) < W) & (np.abs(F3) < W) h, _, _ = np.histogram2d(F1[m], F2[m], bins=(edges, edges)) H += h im = ax.pcolormesh(edges, edges, np.log10(H.T + 1), cmap='inferno', shading='auto') ax.set_xlabel('F1'); ax.set_ylabel('F2') ax.set_title('Properness probe: where images of spheres\n' '|z| = 25..200 still land at finite distance', fontsize=11) fig.colorbar(im, ax=ax, label='log10(1 + hits)') fig.suptitle('The escape analysis: non-properness is the valve - and the ' 'right invariant is still an open hunt', fontsize=14) plt.tight_layout(rect=[0, 0, 1, 0.92]); _finish('escape_analysis.png') def fig_escape(trials=150): """Escape-scale scatter: merging partners sit far out (non-properness).""" pairs = fable_collide(trials=trials) esc = np.array([max(np.linalg.norm(x), np.linalg.norm(y)) for x, y in pairs]) sep = np.array([np.linalg.norm(x - y) for x, y in pairs]) fig, ax = plt.subplots(figsize=(8, 6.5)) ax.scatter(sep, esc, s=35, color='#d62728', alpha=0.8) ax.set_xscale('log'); ax.set_yscale('log') ax.set_xlabel('separation |x - y|'); ax.set_ylabel('escape max(|x|, |y|)') ax.set_title('Borrowing room at infinity:\n' 'no merges happen near the origin - partners live far out') ax.grid(alpha=0.3, which='both') plt.tight_layout(); _finish('escape.png') def fig_information_ladder(): """The +C lens: the ladder map F -> DF -> det DF, with what each step discards. Differentiation forgets exactly one additive constant (DF = D(F + c)); the determinant forgets vastly more. Registered under Direction 3 because its punchline is the conjecture itself: does the Keller condition climb back up? From dimension three, it does not.""" from matplotlib.patches import FancyBboxPatch # the +C fact, verified symbolically rather than asserted x, y, c1, c2 = sp.symbols('x y c1 c2') F = sp.Matrix([x*y - x**3, y + x**2]) Fc = F + sp.Matrix([c1, c2]) diff = sp.simplify(F.jacobian([x, y]) - Fc.jacobian([x, y])) print('sympy check: D(F + c) - DF =', diff.tolist(), 'for F =', list(F.T), 'and symbolic constants c = (c1, c2)') fig, ax = plt.subplots(figsize=(12, 9)) boxes = [ (0.82, '#1f77b4', 'the MAP F : C^n -> C^n', 'all the information there is'), (0.50, '#2ca02c', 'the JACOBIAN MATRIX FIELD DF', 'a matrix at every point'), (0.18, '#9467bd', 'the DETERMINANT FIELD det DF', 'one number at every point'), ] for yc, col, main, sub in boxes: ax.add_patch(FancyBboxPatch((0.24, yc - 0.075), 0.50, 0.15, boxstyle='round,pad=0.015', fc='white', ec=col, lw=2.5)) ax.text(0.49, yc + 0.022, main, ha='center', va='center', fontsize=13, color=col, fontweight='bold') ax.text(0.49, yc - 0.038, sub, ha='center', va='center', fontsize=9.5, color=col) for y0, y1, disc in [ (0.735, 0.585, 'discarded: exactly ONE additive\nconstant, ' 'DF = D(F + c) -\ndifferentiation\'s "+C",\na single global datum'), (0.415, 0.265, 'discarded: VASTLY more -\nmany matrix fields\n' 'share one determinant field')]: ax.annotate('', xy=(0.40, y1), xytext=(0.40, y0), arrowprops=dict(arrowstyle='-|>', lw=3, color='#d62728')) ax.text(0.36, (y0 + y1) / 2, disc, ha='right', va='center', fontsize=9.5, color='#d62728') for y0, y1, lab, col in [ (0.585, 0.735, 'climb back: integrate,\nrecover F up to +c', '#2ca02c'), (0.265, 0.415, 'climb back? the Keller condition\n' 'det DF = const != 0 - the conjecture\nasked whether this is enough', '#555555')]: ax.annotate('', xy=(0.58, y1), xytext=(0.58, y0), arrowprops=dict(arrowstyle='-|>', lw=2, color=col, linestyle=(0, (4, 3)))) ax.text(0.62, (y0 + y1) / 2, lab, ha='left', va='center', fontsize=9.5, color=col) ax.text(0.49, 0.02, 'the Jacobian conjecture asks whether the Keller ' 'condition is strong enough to climb back up -\n' 'from dimension three, it is not', ha='center', va='bottom', fontsize=11, color='black') ax.set_xlim(0, 1); ax.set_ylim(0, 1); ax.axis('off') ax.set_title('The information ladder: each step down forgets\n' 'differentiation forgets +C; the determinant forgets ' 'almost everything', fontsize=14, pad=10) plt.tight_layout(); _finish('information_ladder.png') # ===================================================================== # DIRECTION 4 — DEGREE of the map # ===================================================================== def fig_degree_axis(): """Milestone diagram of the degree axis: Laurent territory, the shuffle born at degree 3, the Fable map at 7, the 2D verification frontier at 100.""" from matplotlib.patches import FancyArrowPatch fig, ax = plt.subplots(figsize=(15, 6.5)) xs = {-2: 0.0, -1: 1.0, 0: 2.0, 1: 3.0, 2: 4.2, 3: 5.6, 7: 7.6, 100: 9.8} names = {-2: r'$x^{-2}$', -1: r'$x^{-1}$', 0: r'$1$', 1: r'$x$', 2: r'$x^{2}$', 3: r'$x^{3}$', 7: r'$x^{7}$', 100: r'$x^{100}$'} ax.annotate('', xy=(10.6, 0), xytext=(-0.8, 0), arrowprops=dict(arrowstyle='-|>', lw=3, color='#9467bd')) for bx in (6.6, 8.7): # the axis is NOT to scale — mark the jumps ax.plot([bx-0.07, bx+0.07], [-0.09, 0.09], color='white', lw=7, zorder=3) ax.plot([bx-0.12, bx+0.02], [-0.1, 0.1], color='#9467bd', lw=2, zorder=4) ax.plot([bx-0.02, bx+0.12], [-0.1, 0.1], color='#9467bd', lw=2, zorder=4) ax.axvspan(-0.8, 2.5, color='gray', alpha=0.13, zorder=0) ax.text(0.85, -0.62, 'LAURENT TERRITORY\nnegative powers - already met in ' "Tao's\nparametrization: Laurent in a,\npolynomial in b and c", ha='center', fontsize=9.5, color='#555555') for d, x in xs.items(): ax.plot(x, 0, 'o', ms=7, color='#9467bd', zorder=5) ax.text(x, -0.17, names[d], ha='center', va='top', fontsize=14) marks = [ (2, 0.40, '#1f77b4', 'o', 'degree 2: NO shuffle\na quadratic splits one ' 'way once\nscalings are fixed - nothing to permute'), (3, 0.82, '#9467bd', 'D', 'degree 3: the shuffle is BORN\nthree roots ' 'can be permuted - the true\nboundary behind the Jacobian conjecture'), (7, 0.40, '#d62728', '*', 'degree 7: the Fable map\n(the 2026 ' 'counterexample\nin three variables)'), (100, 0.82, '#1f77b4', 's', 'degree 100: the 2D conjecture\nis verified ' 'up to here - and still open'), ] for d, ty, col, mk, txt in marks: x = xs[d] ax.plot(x, 0, mk, ms=17 if mk == '*' else 11, color=col, zorder=6) ax.plot([x, x], [0.10, ty - 0.04], color=col, lw=1, ls=':') ax.text(x, ty, txt, ha='center', va='bottom', fontsize=9.5, color=col) ax.add_patch(FancyArrowPatch((9.8, -0.5), (xs[3] + 0.08, -0.1), connectionstyle='arc3,rad=-0.25', arrowstyle='-|>', mutation_scale=16, color='#2ca02c', lw=2)) ax.text(7.9, -0.72, 'Bass-Connell-Wright: EVERY Keller map, any degree,\n' 'reduces to degree 3 (paying with more variables)', ha='center', fontsize=9.5, color='#2ca02c') # honesty marker: 7 is where THIS construction landed, not a threshold ax.annotate('', xy=(xs[7] - 0.12, 0.20), xytext=(xs[3] + 0.12, 0.20), arrowprops=dict(arrowstyle='<->', lw=1.4, color='#555555')) ax.text((xs[3] + xs[7]) / 2, 0.24, 'minimal counterexample degree in C^3: OPEN, boxed 3..7\n' '(a cubic one exists - but in C^19, via the BCW push)', ha='center', va='bottom', fontsize=8.5, color='#555555') ax.set_xlim(-1.1, 11.0); ax.set_ylim(-1.05, 1.15); ax.axis('off') ax.set_title('DIRECTION 4 - DEGREE: the axis that was secretly running the show\n' 'no shuffle at degree 2, shuffle born at degree 3, counterexample ' 'at degree 7, 2D verified to degree 100 (axis not to scale)', fontsize=14, pad=12) plt.tight_layout(); _finish('degree_axis.png') def fig_cross_compare(): """The same letters-vs-paths pattern probed along two directions: SIZE of the matrix, and DEGREE of the map (at n = 2 and n = 3). Crossings, quotient trend-breaks, and the degeneracy ladder on each axis. Heuristic counting only: the Fable map exists DESPITE 1329 > 360 — crossings locate rarity, not impossibility.""" from scipy.special import gamma, comb from scipy.optimize import brentq def C(a, b): # continuous binomial via Gamma return gamma(a + 1) / (gamma(b + 1) * gamma(a - b + 1)) # ---- exact crossings ------------------------------------------------ x_size = brentq(lambda x: gamma(x + 1)/2 - x**2, 3.5, 5.0) f2 = lambda d: (C(2*d, 2) - 1) - 2*C(d + 2, 2) # n=2: constraints - freedom f3 = lambda d: (C(3*d, 3) - 1) - 3*C(d + 3, 3) x_d2 = brentq(f2, 3.0, 6.0) x_d3 = brentq(f3, 2.0, 3.0) print(f'SIZE crossing: n = {x_size:.4f} ' '(paths n!/2 overtake letters n^2)') print(f'DEGREE crossing (n=2): d = {x_d2:.4f} ' f'(= 2 + sqrt(7) = {2 + np.sqrt(7):.4f})') print(f'DEGREE crossing (n=3): d = {x_d3:.4f}') print(f'sanity, Tao numbers at n=3, d=7: freedom {3*int(comb(10, 3))}, ' f'constraints {int(comb(21, 3)) - 1}') fig, axes = plt.subplots(2, 3, figsize=(19, 10), gridspec_kw={'height_ratios': [2.4, 1]}) def panel(ax, axq, xs, free, cons, xstar, title, xlabel, marks, free_lab, cons_lab): ax.semilogy(xs, free, lw=2.5, color='#1f77b4', label=free_lab) ax.semilogy(xs, cons, lw=2.5, color='#d62728', label=cons_lab) ax.axvline(xstar, color='gray', ls=':', lw=1.5) ax.plot(xstar, np.interp(xstar, xs, free), 'k*', ms=16, zorder=5) ax.annotate(f'crossing\n{xstar:.3f}', (xstar, np.interp(xstar, xs, free)), textcoords='offset points', xytext=(8, -34), fontweight='bold') for xm, lab, col in marks: ax.axvline(xm, color=col, ls='--', lw=1.2, alpha=0.7) ax.text(xm, 0.98, lab, transform=ax.get_xaxis_transform(), fontsize=8.5, color=col, ha='center', va='top') ax.set_title(title, fontsize=11) ax.set_ylabel('count (log)') ax.legend(fontsize=9, loc='lower right') ax.grid(alpha=0.3, which='both') # quotient panel: the trend-break axq.plot(xs, cons / free, lw=2.2, color='#9467bd') axq.axhline(1, color='gray', lw=1) axq.axvline(xstar, color='gray', ls=':', lw=1.5) axq.set_yscale('log') axq.set_xlabel(xlabel); axq.set_ylabel('quotient') axq.set_title('constraints / freedom - crosses 1 at the break', fontsize=10) axq.grid(alpha=0.3, which='both') # Panel A: SIZE direction xs = np.linspace(0.05, 7, 500) panel(axes[0, 0], axes[1, 0], xs, xs**2, gamma(xs + 1)/2, x_size, 'SIZE direction: letters $n^2$ vs positive paths $n!/2$\n' 'vacuum det(0x0) = 1; point n = 1: letters = paths = 1', 'n (continuous)', [(0, 'vacuum\ndet=1', '#2ca02c'), (1, 'point\nn=1', '#2ca02c')], 'letters $n^2$', 'paths $n!/2$') # Panel B: DEGREE direction at n = 2 ds = np.linspace(1.0, 8, 500) panel(axes[0, 1], axes[1, 1], ds, 2*C(ds + 2, 2), C(2*ds, 2) - 1, x_d2, 'DEGREE direction, n = 2: freedom vs Keller constraints\n' 'd=0: det$\\equiv$0 forbidden; d=1: auto-constant; open to d=100', 'degree d', [(1, 'linear:\nauto-Keller', '#2ca02c')], 'freedom $2\\binom{d+2}{2}$', 'constraints $\\binom{2d}{2}-1$') # Panel C: DEGREE direction at n = 3 ds3 = np.linspace(1.0, 8, 500) panel(axes[0, 2], axes[1, 2], ds3, 3*C(ds3 + 3, 3), C(3*ds3, 3) - 1, x_d3, 'DEGREE direction, n = 3: freedom vs constraints\n' 'crossing between Wang (d=2 true) and shuffle birth (d=3)', 'degree d', [(2, 'Wang:\nd=2 TRUE', '#2ca02c'), (3, 'shuffle born\nBCW target', '#d62728'), (7, 'Fable map\n360 vs 1329', 'black')], 'freedom $3\\binom{d+3}{3}$', 'constraints $\\binom{3d}{3}-1$') fig.suptitle('One pattern, probed along two directions: where counting ' 'flips from underdetermined to overdetermined\n' '(heuristic counting only - the Fable map exists DESPITE ' '1329 > 360; crossings locate rarity, not impossibility)', fontsize=13, y=1.00) plt.tight_layout(rect=[0, 0, 1, 0.94]); _finish('cross_compare.png') def fig_dstar_curve(): """The counting flip d*(n) as one curve: where Keller constraints overtake coefficient freedom, as a function of dimension n. Asymptotes to the degenerate floor d = 1; crosses Wang's line d = 2 at n ~ 3.36 — the built-in guardrail (counting calls quadratics overdetermined where Wang proves them safe forever). The 4.273 size crossing is off-family.""" from scipy.special import gamma from scipy.optimize import brentq def C(a, b): return gamma(a + 1) / (gamma(b + 1) * gamma(a - b + 1)) def dstar(n): f = lambda d: (C(n*d, n) - 1) - n*C(d + n, n) return brentq(f, 1.0001, 12) ns = np.arange(2, 15) ds = np.array([dstar(n) for n in ns]) for n, d in zip(ns, ds): print(f'n={n:2d} d* = {d:.4f}') fig, ax = plt.subplots(figsize=(10.5, 6.5)) nn = np.linspace(2, 14, 300) dd = [dstar(x) for x in nn] ax.plot(nn, dd, lw=2.5, color='#9467bd', label='d*(n): counting flip (constraints overtake freedom)') ax.plot(ns, ds, 'o', ms=7, color='#9467bd') ax.axhline(1, color='gray', ls='--', lw=1.5) ax.text(13.9, 0.62, 'd = 1: linear maps, auto-Keller\n' '(the degenerate floor - asymptote)', ha='right', va='bottom', fontsize=9, color='gray') ax.axhline(2, color='#2ca02c', ls=':', lw=1.5) ax.text(13.9, 2.06, "d = 2: Wang's theorem - conjecture TRUE here " 'in every n', ha='right', fontsize=9, color='#2ca02c') ax.annotate(f'n=2: d* = 2+√7 ≈ {ds[0]:.3f}', (2, ds[0]), textcoords='offset points', xytext=(12, 8), fontsize=10, fontweight='bold') ax.annotate(f'n=3: d* ≈ {ds[1]:.3f}\n(the bracket: between Wang ' 'and the shuffle)', (3, ds[1]), textcoords='offset points', xytext=(14, 6), fontsize=10, fontweight='bold') ncross = brentq(lambda x: dstar(x) - 2, 2.5, 5) print(f'd*(n) crosses the Wang line d = 2 at n = {ncross:.4f}') ax.plot(ncross, 2, 'k*', ms=15, zorder=5) ax.annotate(f'curve crosses the Wang line at n ≈ {ncross:.3f}:\n' 'beyond this, counting calls even quadratics\n' 'overdetermined - yet Wang PROVES d=2 safe forever.\n' 'Counting locates rarity, not truth.', (ncross, 2), textcoords='offset points', xytext=(95, 62), fontsize=9, arrowprops=dict(arrowstyle='->', lw=0.9, color='gray')) ax.scatter([4.273], [4.6], marker='*', s=180, color='#1f77b4', zorder=5) ax.annotate('the SIZE crossing 4.273 - an isolated point from a\n' 'DIFFERENT family (drawability), NOT on this curve', (4.273, 4.6), textcoords='offset points', xytext=(14, -22), fontsize=9, color='#1f77b4') ax.set_xlabel('dimension n'); ax.set_ylabel('degree d') ax.set_title('The collapsing threshold as its own object: d*(n) ' 'asymptotes to the degenerate floor\nas dimension grows, ' 'the underdetermined window is squeezed onto the linear rung') ax.set_ylim(0.5, 5.2); ax.legend(loc='upper right', fontsize=9) ax.grid(alpha=0.3) plt.tight_layout(); _finish('dstar_asymptote.png') # ===================================================================== # MENU # ===================================================================== _CATALOG = { '0': ('Map of the journey (index figure)', [ ('1', 'The five-arrow compass (probe, size, richness, degree, map dim)', fig_index_map), ('2', 'Hidden hypotheses: commutativity & properness (rhyme, not theorem)', fig_hidden_hypothesis), ('3', 'Where the ladder goes - the Paper II teaser', fig_ladder_teaser)]), '1': ('Direction 1 - SIZE of the matrix', [ ('1', 'Formula reference card (n = 1, 2, 3)', fig_formulas), ('2', 'Geometry + simple sequences (length/area/volume)', fig_geometry), ('3', 'Permutation matrices, binary matrices, non-square trap', fig_permutations), ('4', 'Sarrus paths n = 2, 3, 4 (diagonals break)', fig_sarrus), ('5', 'Rainbow n = 5, all 60 paths, port fanning', fig_rainbow_ports), ('6', 'Rainbow n = 5, a- and c-families with docking counts', lambda: fig_rainbow_ports(5, keep_columns=(0, 2), show_counts=True)), ('7', 'Growth: letters vs paths, n = 1..9', fig_growth), ('8', 'Exact crossings via Gamma (x = 0.672, 4.273)', fig_crossing)]), '2': ('Direction 2 - RICHNESS of the entries', [ ('1', 'det as a FIELD: Whitney cusp, folds on det = 0', fig_det_field), ('2', 'Matrix-of-matrix: the commutativity fork (det needs it, trace does not)', fig_block_break), ('3', 'The two vacuums: 0 vs 1, trace vs det, det(exp A) = exp(tr A)', fig_two_vacuums), ('4', 'The two nothings, complete: hospitality, loci, zero ring', fig_two_nothings_finale)]), '3': ('Direction 3 - DIMENSION of the map (Jacobian conjecture)', [ ('1', 'Verify the 2026 counterexample symbolically', fable_verify), ('2', 'Three wells on the collision plane', fig_fable_wells), ('3', 'Hunt collision pairs numerically', fable_collide), ('4', 'Escape scatter (non-properness signature)', fig_escape), ('5', 'Full escape analysis: disguises + properness probe', fig_escape_analysis), ('6', 'The information ladder: F -> DF -> det DF (the +C lens)', fig_information_ladder)]), '4': ('Direction 4 - DEGREE of the map', [ ('1', 'The degree axis: Laurent to x^100, shuffle born at 3', fig_degree_axis), ('2', 'Freedom vs constraints: crossings along SIZE and DEGREE', fig_cross_compare), ('3', 'The curve d*(n): counting flip vs dimension, Wang guardrail', fig_dstar_curve)]), } def menu(): """Interactive numeric menu. Type numbers, q to quit, b to go back.""" while True: print('\n=== THE DETERMINANT LADDER ===') for key, (name, _) in sorted(_CATALOG.items()): print(f' {key}: {name}') choice = input('direction (q quits): ').strip().lower() if choice in ('q', 'quit', 'exit'): return if choice not in _CATALOG: print('unknown option'); continue name, figs = _CATALOG[choice] while True: print(f'\n--- {name} ---') for k, label, _ in figs: print(f' {k}: {label}') sub = input('figure (b back, q quits): ').strip().lower() if sub in ('q', 'quit'): return if sub in ('b', 'back'): break match = [fn for k, lbl, fn in figs if k == sub] if not match: print('unknown option'); continue try: match[0]() except Exception as exc: print(f'error while drawing: {exc}') if __name__ == '__main__': menu()