from manim import * import numpy as np config.frame_width = 16 config.frame_height = 9 config.pixel_width = 1920 config.pixel_height = 1080 config.background_color = "#070B18" SAFE_X = config.frame_width / 2 - 0.35 SAFE_Y = config.frame_height / 2 - 0.35 SAFE_Y_BOTTOM = config.frame_height / 2 - 0.7 TEXT_MAX_W = config.frame_width - 1.4 BLUEC = "#3FA9F5" REDC = "#E5484D" GREYC = "#8A93A6" GOLD = "#FFC65C" GREENC = "#37D67A" INK = "#E8EEF8" X0, X1 = -3.0, 3.0 Y0, Y1 = -2.5, 2.5 def bumps(x, y, centers): z = 0.0 for (cx, cy, a, s) in centers: z += a * np.exp(-((x - cx) ** 2 + (y - cy) ** 2) / (2 * s ** 2)) return z B_C = [(-1.2, 1.2, 0.75, 1.0), (1.6, 1.4, 0.8, 0.9), (0.0, -1.8, 0.6, 1.1), (0.4, 0.6, 0.7, 1.0)] R_C = [(2.4, 1.6, 0.6, 1.0), (-2.4, 0.4, 0.55, 1.0), (0.8, -2.0, 0.5, 1.2)] def f_blue(x, y): return 0.35 + bumps(x, y, B_C) def f_red(x, y): return 0.20 + bumps(x, y, R_C) def required_zoom_to_fit(points, phi, theta, gamma=0.0, focal_distance=20.0, safe_half_width=None, safe_half_height=None): if safe_half_width is None: safe_half_width = config.frame_width / 2 - 1.4 if safe_half_height is None: safe_half_height = config.frame_height / 2 - 1.2 def rot_z(a): c, s = np.cos(a), np.sin(a) return np.array([[c, -s, 0], [s, c, 0], [0, 0, 1]]) def rot_x(a): c, s = np.cos(a), np.sin(a) return np.array([[1, 0, 0], [0, c, -s], [0, s, c]]) rot = rot_z(gamma) @ rot_x(-phi) @ rot_z(-theta - PI / 2) worst = 0.0 for p in points: x, y, z = rot @ np.array(p, dtype=float) denom = focal_distance - z factor = 1e6 if denom < 0 else focal_distance / denom worst = max(worst, abs(x * factor) / safe_half_width, abs(y * factor) / safe_half_height) return None if worst <= 1.0 else 1.0 / worst class SpatiotemporalBeliefFields(ThreeDScene): # ---------- fixed-frame helpers ---------- def caption(self, s, size=26): t = Text(s, font_size=size, color=INK) if t.width > TEXT_MAX_W: t.scale_to_fit_width(TEXT_MAX_W) t.to_edge(UP, buff=0.35) return t def caption2(self, l1, l2, size=25): t = VGroup(Text(l1, font_size=size, color=INK), Text(l2, font_size=size, color=INK)).arrange(DOWN, buff=0.16) if t.width > TEXT_MAX_W: t.scale_to_fit_width(TEXT_MAX_W) t.to_edge(UP, buff=0.3) return t def swap_caption(self, new, run_time=0.7): self.add_fixed_in_frame_mobjects(new) if self.cap is not None: self.play(FadeOut(self.cap, shift=UP * 0.2), FadeIn(new, shift=UP * 0.2), run_time=run_time) else: self.play(FadeIn(new), run_time=run_time) self.cap = new def swap_formula(self, new, run_time=0.9): new.to_edge(DOWN, buff=0.8) if new.height > 0.95: new.scale_to_fit_height(0.95) new.to_edge(DOWN, buff=0.8) if new.width > TEXT_MAX_W: new.scale_to_fit_width(TEXT_MAX_W) self.add_fixed_in_frame_mobjects(new) if self.fml is not None: self.play(FadeOut(self.fml, shift=DOWN * 0.15), FadeIn(new, shift=DOWN * 0.15), run_time=run_time) else: self.play(Write(new), run_time=run_time) self.fml = new def swap_label(self, new, run_time=0.6): if new.width > 8.5: new.scale_to_fit_width(8.5) new.move_to(np.array([-SAFE_X + new.width / 2, -2.05, 0])) self.add_fixed_in_frame_mobjects(new) if self.lbl is not None: self.play(FadeOut(self.lbl), FadeIn(new), run_time=run_time) else: self.play(FadeIn(new), run_time=run_time) self.lbl = new # ---------- scene ---------- def construct(self): self.cap = None self.fml = None self.lbl = None axes = ThreeDAxes( x_range=[X0, X1, 1], y_range=[Y0, Y1, 1], z_range=[0, 2.4, 1], x_length=6, y_length=5, z_length=2.4, axis_config={"stroke_width": 2, "color": "#4A5570", "include_ticks": False}, ) P = axes.c2p # ---- hexagonal cell index floor ---- hexes = VGroup() R = 0.42 dx = 1.5 * R dy = np.sqrt(3) * R i = 0 cx = X0 while cx <= X1 + 1e-6: offset = 0.0 if i % 2 == 0 else dy / 2 cy = Y0 + offset while cy <= Y1 + 1e-6: verts = [P(cx + R * np.cos(a), cy + R * np.sin(a), 0) for a in np.linspace(0, TAU, 7)[:-1]] if X0 - 0.01 <= cx <= X1 + 0.01 and Y0 - 0.01 <= cy <= Y1 + 0.01: h = Polygon(*verts, stroke_color="#2E3A57", stroke_width=1.2) h.set_fill("#101A33", opacity=0.35) hexes.add(h) cy += dy cx += dx i += 1 border = VMobject(stroke_color="#5C6B8F", stroke_width=2.5) border.set_points_as_corners([P(X0, Y0, 0), P(X1, Y0, 0), P(X1, Y1, 0), P(X0, Y1, 0), P(X0, Y0, 0)]) slab_top = 2.4 pillars = VGroup(*[ Line(P(x, y, 0), P(x, y, slab_top), stroke_width=1.2, color="#22304C") for x, y in [(X0, Y0), (X1, Y0), (X1, Y1), (X0, Y1)] ]) corner_pts = [P(x, y, z) for x in (X0, X1) for y in (Y0, Y1) for z in (0, slab_top)] phi, theta = 65 * DEGREES, -45 * DEGREES zoom = required_zoom_to_fit(corner_pts, phi, theta) self.set_camera_orientation(phi=phi, theta=theta, zoom=zoom or 1.0) title = Text("Spatiotemporal belief fields: from cell to route to certified plan", font_size=30, color="#DCE6F7") if title.width > TEXT_MAX_W: title.scale_to_fit_width(TEXT_MAX_W) title.to_edge(UP, buff=0.35) self.add_fixed_in_frame_mobjects(title) self.play(FadeIn(title), run_time=1.2) self.play(LaggedStart(*[FadeIn(h) for h in hexes], lag_ratio=0.004), Create(border), run_time=2.6) self.play(Create(pillars), FadeIn(axes), run_time=1.2) xlab = Text("stores / geography", font_size=22, color="#9FB2D6") xlab.move_to(P(0, Y0 - 0.7, 0)) ylab = Text("time (day, hour)", font_size=22, color="#9FB2D6") ylab.move_to(P(X1 + 0.95, 0, 0)) self.add_fixed_orientation_mobjects(xlab, ylab) self.play(FadeIn(xlab), FadeIn(ylab), run_time=1.0) self.play(FadeOut(title), run_time=0.6) self.swap_caption(self.caption("Each cell carries a feature tensor, partitioned by source and cadence.")) self.swap_label(MathTex(r"\text{cell index }\mathcal{C}\text{ over the operational volume }\Omega", font_size=28, color="#9FB2D6")) self.swap_formula(MathTex( r"U_c=\{\,U^{\mathrm{endo}}_c,\ U^{\mathrm{exo},b}_c,\ U^{\mathrm{exo},x}_c,\ W_c\,\}", font_size=38, color=INK)) self.begin_ambient_camera_rotation(rate=0.04) self.wait(3.2) # ================= Scene 2 ================= def mk_surface(f, color): s = Surface( lambda u, v: P(u, v, f(u, v)), u_range=[X0, X1], v_range=[Y0, Y1], resolution=(26, 24), fill_opacity=0.55, checkerboard_colors=False, stroke_width=0.7, ) s.set_fill(color, opacity=0.55) s.set_stroke(interpolate_color(ManimColor(color), WHITE, 0.45), width=0.7, opacity=0.55) return s blue = mk_surface(f_blue, BLUEC) red = mk_surface(f_red, REDC) site_xy = [(-2.4, -1.9), (-1.5, -0.7), (-0.4, 0.4), (0.7, -1.2), (1.5, 1.0), (2.4, -0.3), (0.2, 1.9), (-2.0, 1.4)] heights = [0.9, 1.4, 0.7, 1.1, 1.6, 0.8, 1.2, 1.0] stems = VGroup() caps = VGroup() for (sx, sy), h in zip(site_xy, heights): stems.add(Line(P(sx, sy, 0), P(sx, sy, h), stroke_width=6, color=GREYC)) caps.add(Dot3D(P(sx, sy, h), radius=0.075, color="#D5DCE8")) self.swap_caption(self.caption("Fields are exogenous. Inventory is a stem, not a wave.")) self.swap_label(MathTex(r"U^{\mathrm{endo}}_c:\ \text{inventory position } s\ \ (\text{discrete; only tasks move it})", font_size=26, color=GREYC)) self.swap_formula(MathTex(r"s\in\mathbb{Z}_{\ge 0}^{N}", font_size=38, color=GREYC)) self.play(LaggedStart(*[GrowFromEdge(s, IN) for s in stems], lag_ratio=0.12), run_time=2.2) self.play(FadeIn(caps), run_time=0.6) self.swap_label(MathTex(r"U^{\mathrm{exo},b}_c:\ \text{belief-driven, functions of }(\mu_b,\Sigma_b),\ \text{intra-hour}", font_size=25, color=BLUEC)) self.play(Create(blue), run_time=2.8) self.swap_label(MathTex(r"U^{\mathrm{exo},x}_c:\ \text{external feeds (traffic, weather, road class)}", font_size=25, color=REDC)) self.play(Create(red), run_time=2.4) self.wait(0.8) rng = np.random.default_rng(7) pxy = [(rng.uniform(X0 + 0.4, X1 - 0.4), rng.uniform(Y0 + 0.4, Y1 - 0.4)) for _ in range(30)] dots = VGroup(*[Dot3D(P(px, py, f_blue(px, py) + 0.35), radius=0.055, color="#BFF0FF") for px, py in pxy]) self.swap_caption(self.caption("The ensemble is a belief over exogenous parameters.")) self.swap_label(MathTex(r"\{(\theta^{(i)},w^{(i)})\}_{i=1}^{P}", font_size=34, color="#BFF0FF")) self.swap_formula(MathTex( r"\theta^{(i)}_t\sim p(\theta_t\mid\theta^{(i)}_{t-1})\ (\text{predict}),\qquad " r"w^{(i)}_t\propto w^{(i)}_{t-1}\,p(o_t\mid\theta^{(i)}_t)\ (\text{update})", font_size=32, color=INK)) self.play(LaggedStart(*[FadeIn(d, scale=0.4) for d in dots], lag_ratio=0.03), run_time=2.4) self.wait(1.4) # censored read ox, oy = site_xy[1] visit = Sphere(radius=0.13, resolution=(12, 12)).set_color(GOLD) visit.move_to(P(ox, oy, heights[1])) self.swap_caption(self.caption2( "Observing sales = v with stock v is the tail event D \u2265 v,", "not a demand count of v. The likelihood uses the tail event.")) self.swap_formula(MathTex(r"p(o_t\mid\theta)=\Pr\!\big[D\ge v\mid\theta\big]", font_size=36, color=GOLD)) self.play(FadeIn(visit, scale=0.5), stems[1].animate.set_color(GOLD), run_time=1.0) anims = [] for i, (px, py) in enumerate(pxy): if (px - ox) ** 2 + (py - oy) ** 2 < 2.2: anims.append(dots[i].animate.move_to(P(px, py, f_blue(px, py) + 0.95)).set_color(GOLD)) self.play(*anims, run_time=2.0) self.wait(1.6) # ================= Scene 3 ================= self.stop_ambient_camera_rotation() route_xy = [(-2.8, -2.3)] + site_xy[:1] + [site_xy[1], site_xy[2], site_xy[4], site_xy[6]] pts = [P(x, y, f_blue(x, y) + 0.45) for x, y in route_xy] gamma_path = VMobject(stroke_color=WHITE, stroke_width=7) gamma_path.set_points_smoothly(pts) glow = gamma_path.copy().set_stroke(WHITE, width=16, opacity=0.18) home = Sphere(radius=0.14, resolution=(12, 12)).set_color("#FFE28A") home.move_to(pts[0]) self.swap_caption(self.caption2( "The relational signal: a kernel-weighted path integral of the", "feature field along the planned route \u03b3\u2096,\u1d04.")) self.swap_label(MathTex(r"\gamma_{k,c}:\ \text{agent }k\text{ from }c_k\text{ to }c", font_size=28, color=WHITE)) self.play(FadeIn(home, scale=0.5), FadeOut(visit), run_time=0.6) self.play(Create(gamma_path), Create(glow), run_time=3.0) self.swap_formula(MathTex( r"\alpha_{k,c,m}=\int_{\gamma_{k,c}} W_m(\tau)\,U\big(x(\tau)\big)\,d\tau,\qquad " r"W_m(\tau)=\begin{pmatrix}\cos(\omega_m\tau)\\ \sin(\omega_m\tau)\end{pmatrix}", font_size=32, color=INK)) self.wait(2.4) # profile strip + amplitude bars (fixed frame, upper-left band) taus = np.linspace(0, 1, 120) prof = [] for t in taus: k = t * (len(route_xy) - 1) i0 = int(min(k, len(route_xy) - 2)) fr = k - i0 xg = route_xy[i0][0] * (1 - fr) + route_xy[i0 + 1][0] * fr yg = route_xy[i0][1] * (1 - fr) + route_xy[i0 + 1][1] * fr prof.append(f_blue(xg, yg) + f_red(xg, yg)) prof = np.array(prof) prof_n = (prof - prof.min()) / (prof.max() - prof.min() + 1e-9) pax = Axes(x_range=[0, 1, 0.5], y_range=[0, 1.1, 0.5], x_length=3.6, y_length=1.2, axis_config={"include_tip": False, "include_ticks": False, "stroke_width": 2, "color": "#7C89A8"}) pcurve = VMobject(stroke_color=GOLD, stroke_width=3.5) pcurve.set_points_smoothly([pax.c2p(t, v) for t, v in zip(taus[::4], prof_n[::4])]) plab = MathTex(r"U(x(\tau))\ \text{along}\ \gamma", font_size=24, color="#9FB2D6") plab.next_to(pax, DOWN, buff=0.14) amps = [] for m, w in enumerate([2 * PI, 4 * PI, 6 * PI], start=1): a = np.trapezoid(prof_n * np.cos(w * taus), taus) b = np.trapezoid(prof_n * np.sin(w * taus), taus) amps.append(np.hypot(a, b)) amps = np.array(amps) / (max(amps) + 1e-9) bars = VGroup() for m, a in enumerate(amps): bar = Rectangle(width=0.34, height=max(0.12, a * 1.15), fill_color=GOLD, fill_opacity=0.85, stroke_width=0) lab = MathTex(rf"\alpha_{{{m+1}}}", font_size=24, color=GOLD) col = VGroup(bar, lab).arrange(DOWN, buff=0.12) bars.add(col) bars.arrange(RIGHT, buff=0.32, aligned_edge=DOWN) strip = VGroup(VGroup(pax, pcurve, plab), bars).arrange(RIGHT, buff=0.6, aligned_edge=DOWN) box = RoundedRectangle(corner_radius=0.12, width=strip.width + 0.5, height=strip.height + 0.45, stroke_color="#3B4A6B", stroke_width=2, fill_color="#0C1226", fill_opacity=0.92).move_to(strip) panel3 = VGroup(box, strip).to_corner(UL, buff=0.75).shift(DOWN * 0.9) world = VGroup(axes, hexes, border, pillars, stems, caps, blue, red, dots, gamma_path, glow, home, xlab, ylab) self.play(world.animate.scale(0.62, about_point=ORIGIN).shift(RIGHT * 3.1), run_time=1.4) self.add_fixed_in_frame_mobjects(panel3) self.play(FadeOut(self.lbl), FadeIn(box), Create(pax), run_time=0.9) self.lbl = None self.play(Create(pcurve), FadeIn(plab), run_time=1.4) self.play(LaggedStart(*[GrowFromEdge(b, DOWN) for b in bars], lag_ratio=0.25), run_time=1.5) self.wait(1.6) self.swap_caption(self.caption2( "Decisions read the shadow-price field \u03b2, the gradient of a", "concave quadratic surrogate \u2014 monotone-substitutable by construction.")) self.swap_formula(MathTex( r"A(b,\alpha)=A_0+\sum_{i=1}^{d}(\mu_b)_i A_i+\sum_{m=1}^{|\alpha|}\alpha_m A^{\alpha}_m", font_size=34, color=INK)) self.wait(2.2) self.swap_formula(MathTex( r"V(s,b,\alpha)=c(b)+g(b)^{\top}s+\tfrac{1}{2}s^{\top}A(b,\alpha)s,\qquad " r"\beta=\nabla_s V=g(b)+A(b,\alpha)s", font_size=31, color=INK)) self.wait(2.6) self.swap_formula(MathTex( r"A_{kl}=A_{lk}\le 0,\qquad A_{kk}-\sum_{l\neq k}A_{kl}\le 0\quad(\text{class }\mathcal{A})", font_size=33, color=INK)) self.wait(2.4) # greedy task: one stem grows by delta self.play(FadeOut(panel3), run_time=0.6) self.swap_caption(self.caption2( "A task executes only if its exact finite gain is positive;", "the shadow-price update is O(N), no matrix solve.")) def W(p): return p * 0.62 + RIGHT * 3.1 tall = Line(W(P(*site_xy[4], 0)), W(P(site_xy[4][0], site_xy[4][1], heights[4] + 0.7)), stroke_width=6, color=GREENC) self.play(Transform(stems[4], tall), caps[4].animate.move_to(W(P(site_xy[4][0], site_xy[4][1], heights[4] + 0.7))), run_time=1.4) self.swap_formula(MathTex( r"G_\delta(d;s)=V(s+\delta d)-V(s)=\delta\,\beta(s)^{\top}d+\tfrac{1}{2}\delta^{2}d^{\top}Ad", font_size=33, color=GREENC)) self.wait(2.6) # ================= Scene 4 ================= panel_ax = Axes(x_range=[0, 1.05, 0.5], y_range=[0, 1.15, 0.5], x_length=3.2, y_length=1.8, axis_config={"include_tip": False, "stroke_width": 2, "color": "#7C89A8", "include_ticks": False}) curve = panel_ax.plot(lambda x: 0.95 * np.tanh(2.6 * x), x_range=[0, 0.86], color=GOLD, stroke_width=4) wall = Line(panel_ax.c2p(0.88, 0), panel_ax.c2p(0.88, 1.12), color=REDC, stroke_width=6) curve_lab = Text("curvature in A (soft saturation)", font_size=18, color=GOLD) curve_lab.next_to(panel_ax, DOWN, buff=0.16) wall_lab = MathTex(r"W_c=1", font_size=26, color=REDC) wall_lab.next_to(panel_ax, UP, buff=0.16) pbody = VGroup(panel_ax, curve, wall, curve_lab, wall_lab) prect = RoundedRectangle(corner_radius=0.12, width=pbody.width + 0.5, height=pbody.height + 0.4, stroke_color="#3B4A6B", stroke_width=2, fill_color="#0C1226", fill_opacity=0.92).move_to(pbody) panel = VGroup(prect, pbody).to_corner(UL, buff=0.75).shift(DOWN * 0.9) self.add_fixed_in_frame_mobjects(panel) self.swap_caption(self.caption("Curvature for soft saturation, walls for hard cliffs.")) self.swap_formula(MathTex(r"A'=\Pi A \Pi", font_size=40, color=INK)) self.play(FadeIn(panel), run_time=1.4) self.wait(1.6) self.swap_caption(self.caption("Masking wall cells preserves the class \U0001d4d0 \u2014 no refit needed.")) self.wait(1.8) wx = 1.5 window = Surface(lambda u, v: P(wx, u, v), u_range=[Y0, Y1], v_range=[0, 2.2], resolution=(10, 10), fill_opacity=0.28, checkerboard_colors=False, stroke_width=0.6) window.set_fill(REDC, opacity=0.28) window.set_stroke(REDC, width=0.8, opacity=0.5) alt_xy = [(-2.8, -2.3), (-1.6, -1.2), (-0.2, -0.2), (1.5, 0.4), (2.4, 1.6)] alt_base = VMobject(stroke_width=5) alt_base.set_points_smoothly([P(x, y, f_blue(x, y) + 0.45) for x, y in alt_xy]) gamma_p = DashedVMobject(alt_base, num_dashes=60).set_stroke(GREYC, width=5) for m in (window, gamma_p): m.scale(0.62, about_point=ORIGIN).shift(RIGHT * 3.1) self.swap_caption(self.caption("A time-window closure is a wall in the field.")) self.play(FadeIn(window), Create(gamma_p), run_time=1.8) self.play(gamma_p.animate.set_stroke(REDC, width=6), run_time=0.9) excl = Text("excluded, reason recorded: time-window", font_size=24, color=REDC) excl.move_to(np.array([-SAFE_X + excl.width / 2, -2.05, 0])) self.add_fixed_in_frame_mobjects(excl) self.lbl = excl self.play(FadeIn(excl, scale=1.1), run_time=0.8) self.wait(1.6) self.swap_caption(self.caption2( "Per-decision read: operator weights combine per-objective shadow prices.", "Walls are never part of the weights.")) self.swap_formula(MathTex(r"\beta_{\mathrm{wt},j}=\sum_o w_o\,\beta_{o,j}", font_size=40, color=GOLD)) self.wait(2.4) # two-layer diagram self.play(FadeOut(panel), FadeOut(excl), FadeOut(window), FadeOut(gamma_p), FadeOut(self.fml), run_time=0.9) self.lbl = None self.fml = None lower = VGroup( Text("weighted read inside a plan", font_size=23, color=GOLD), MathTex(r"\beta_{\mathrm{wt},j}=\sum_o w_o\,\beta_{o,j}", font_size=30, color=GOLD), ).arrange(DOWN, buff=0.2) lrect = RoundedRectangle(corner_radius=0.12, width=lower.width + 0.6, height=lower.height + 0.4, stroke_color=GOLD, stroke_width=2, fill_color="#141021", fill_opacity=0.94).move_to(lower) layer_low = VGroup(lrect, lower) ladder = VGroup( Text("1. feasible", font_size=22, color=GREENC), Text("2. avoidable cost", font_size=22, color=INK), Text("3. physical burden", font_size=22, color=INK), Text("4. total time", font_size=22, color=INK), ).arrange(RIGHT, buff=0.45) upper = VGroup( Text("plan-acceptance layer (production)", font_size=23, color=GREENC), ladder, Text("lexicographic at the plan level; hard violations are not traded against cost.", font_size=19, color="#9FB2D6"), ).arrange(DOWN, buff=0.2) urect = RoundedRectangle(corner_radius=0.12, width=upper.width + 0.6, height=upper.height + 0.4, stroke_color=GREENC, stroke_width=2, fill_color="#0C1226", fill_opacity=0.94).move_to(upper) layer_up = VGroup(urect, upper) stack = VGroup(layer_up, layer_low).arrange(DOWN, buff=0.5) connector = Arrow(layer_low.get_top(), layer_up.get_bottom(), buff=0.06, stroke_width=3, color="#9FB2D6", max_tip_length_to_length_ratio=0.25) block = VGroup(stack, connector) if block.width > TEXT_MAX_W: block.scale_to_fit_width(TEXT_MAX_W) block.move_to(ORIGIN).shift(DOWN * 0.4) self.add_fixed_in_frame_mobjects(block) self.swap_caption(self.caption("Two layers: weighted read within a plan, lexicographic gate between plans.")) self.play(FadeIn(layer_low), run_time=0.9) self.play(GrowArrow(connector), run_time=0.6) self.play(FadeIn(layer_up), run_time=1.0) self.wait(2.6) accepted = Text("ACCEPTED", font_size=28, color=GREENC, weight=BOLD) accepted.move_to(np.array([-SAFE_X + accepted.width / 2, -2.05, 0])) self.add_fixed_in_frame_mobjects(accepted) self.play(gamma_path.animate.set_stroke("#B9FFD6", width=8), FadeIn(accepted, scale=1.1), run_time=1.1) self.wait(1.4) self.play( FadeOut(block), FadeOut(accepted), FadeOut(blue), FadeOut(red), FadeOut(dots), FadeOut(gamma_path), FadeOut(glow), FadeOut(stems), FadeOut(caps), FadeOut(home), FadeOut(axes), FadeOut(xlab), FadeOut(ylab), FadeOut(hexes), FadeOut(border), FadeOut(pillars), run_time=1.4, ) # ================= Scene 5 ================= def mk_slab(dz, color, seed): g = VGroup() for i in range(7): x = X0 + i * (X1 - X0) / 6 g.add(Line(P(x, Y0, 0), P(x, Y1, 0), stroke_width=1.0, color=color)) for j in range(6): y = Y0 + j * (Y1 - Y0) / 5 g.add(Line(P(X0, y, 0), P(X1, y, 0), stroke_width=1.0, color=color)) b = VMobject(stroke_color=color, stroke_width=2.2) b.set_points_as_corners([P(X0, Y0, 0), P(X1, Y0, 0), P(X1, Y1, 0), P(X0, Y1, 0), P(X0, Y0, 0)]) g.add(b) r = np.random.default_rng(seed) xs = np.sort(r.uniform(X0 + 0.4, X1 - 0.4, 6)) ys = np.sort(r.uniform(Y0 + 0.4, Y1 - 0.4, 6)) path = VMobject(stroke_color=WHITE, stroke_width=5) path.set_points_smoothly([P(x, y, 0.12) for x, y in zip(xs, ys)]) g.add(path) g.shift(OUT * dz) return g, path s1, p1 = mk_slab(0.0, "#3E4C6E", 1) s2, p2 = mk_slab(1.5, "#3E4C6E", 2) s3, p3 = mk_slab(3.0, "#3E4C6E", 3) fleet = VGroup(s1, s2, s3) pts_fleet = [c + OUT * dz for dz in (0, 1.5, 3.0) for c in [P(x, y, 0) for x in (X0, X1) for y in (Y0, Y1)]] z2 = required_zoom_to_fit(pts_fleet, 62 * DEGREES, -50 * DEGREES) self.move_camera(phi=62 * DEGREES, theta=-50 * DEGREES, zoom=z2 or 1.0, run_time=1.5) self.swap_caption(self.caption2( "In the asset-allocation regime the loop selects certified fragments;", "A\u2096\u2097 \u2264 0 makes the selection submodular.")) self.swap_label(MathTex(r"\text{feasibility-certified fragments }\gamma_f", font_size=28, color=WHITE)) self.play(FadeIn(s1), run_time=0.8) self.play(FadeIn(s2), run_time=0.8) self.play(FadeIn(s3), run_time=0.8) self.swap_formula(MathTex( r"V_{\mathrm{frag}}(x,b,\alpha)=g(b)^{\top}x+\tfrac{1}{2}x^{\top}A(b,\alpha)x,\qquad " r"x\in\{0,1\}^{F}", font_size=33, color=INK)) self.wait(2.6) self.play(s3.animate.set_color(GREYC).set_opacity(0.25).shift(IN * 3.0), run_time=1.6) self.play(p1.animate.set_stroke(width=7), p2.animate.set_stroke(width=7), run_time=1.2) self.wait(1.6) self.play(FadeOut(fleet), FadeOut(self.cap), FadeOut(self.fml), FadeOut(self.lbl), run_time=1.2) self.cap = None self.fml = None self.lbl = None # ================= Scene 6: Provenance ================= head = Text("Provenance", font_size=34, color=GREENC) items = VGroup(*[ Text(t, font_size=22, color=INK) for t in [ "cells read, and their features", "wall state", "basis matrices and operator weights in effect", "world-model batch backing U\u1d49\u02e3\u1d52,\u1d47", "external feeds backing U\u1d49\u02e3\u1d52,\u02e3", "posterior spread at cells read", ] ]).arrange(DOWN, aligned_edge=LEFT, buff=0.22) rec = VGroup(head, items).arrange(DOWN, aligned_edge=LEFT, buff=0.32) rrect = RoundedRectangle(corner_radius=0.15, width=rec.width + 0.8, height=rec.height + 0.6, stroke_color=GREENC, stroke_width=2.5, fill_color="#0C1226", fill_opacity=0.94).move_to(rec) record = VGroup(rrect, rec).move_to(ORIGIN).shift(UP * 0.5) self.add_fixed_in_frame_mobjects(record) self.play(FadeIn(rrect), FadeIn(head), run_time=0.9) self.play(LaggedStart(*[FadeIn(m, shift=RIGHT * 0.2) for m in items], lag_ratio=0.18), run_time=2.2) final = VGroup( Text("Every decision records the state it was computed against.", font_size=26, color="#DCE6F7"), Text("Replayable under different weights, candidates, or counterfactual walls.", font_size=26, color="#DCE6F7"), ).arrange(DOWN, buff=0.18) if final.width > TEXT_MAX_W: final.scale_to_fit_width(TEXT_MAX_W) final.next_to(record, DOWN, buff=0.5) if final.get_bottom()[1] < -SAFE_Y_BOTTOM: final.move_to(np.array([0, -SAFE_Y_BOTTOM + final.height / 2, 0])) self.add_fixed_in_frame_mobjects(final) self.play(FadeIn(final, shift=UP * 0.2), run_time=1.2) self.wait(2.5)

Concave Surrogate Optimization