From b2ccd42d774486835f212001fe4aaefa45695618 Mon Sep 17 00:00:00 2001 From: silviana amethyst Date: Thu, 16 Jul 2026 23:21:18 +0000 Subject: [PATCH 1/6] docs(showpieces): the Nebula engine -- one solve, every step Track every total-degree path of one system, keep every accepted tracker step, project C^n to a plane, and expose it all onto an additive buffer. Brightness IS dwell: each segment deposits the path-time |dt| the tracker spent crossing it, spread over its rasterised footprint, so the picture is of the PATH and not of the stepper. Engine notes, all measured rather than assumed: * Power-series endgame, not Cauchy. Measured over 5880 cyclic-5/6/7 paths: |t| monotone on EVERY path, zero teleports. Cauchy would also be ~1500x dearer -- its samples/path explode superlinearly (259 -> 970 -> 11861 for n=5,6,7; cyclic-7 becomes a ~29h render), while power-series stays flat (202 -> 462 -> 335) and cyclic-7 is 68s. * Serial, not threaded. A Python observer re-acquires the GIL every step, so 12 threads convoy: measured cyclic-6 at 8.95s serial vs 66.21s threaded (7.4x SLOWER). Threading is only a win with no observer attached (6.79s -> 2.43s). num_threads=1 also matches flight_recorder. * Streaming meta-observer subclasses SolutionPathCollector and drains .series as it fills, so memory is O(one path) rather than O(solve). * Energy is spread as E/m over m = max(1, ceil(L/step)) subsamples, NOT as E/L: identical for long segments, but no 1/L -> inf singularity when a segment lands inside one pixel. Segments are clipped to the frame BEFORE subdividing, or a diverger allocates millions of subsamples. * --selftest asserts the exposure conserves energy exactly (synthetic, incl. sub-pixel segments) and reports the real-data border loss (0.001%). --- .../docs/source/showpieces/nebula/nebula.py | 662 ++++++++++++++++++ 1 file changed, 662 insertions(+) create mode 100644 python/docs/source/showpieces/nebula/nebula.py diff --git a/python/docs/source/showpieces/nebula/nebula.py b/python/docs/source/showpieces/nebula/nebula.py new file mode 100644 index 000000000..188ed6255 --- /dev/null +++ b/python/docs/source/showpieces/nebula/nebula.py @@ -0,0 +1,662 @@ +"""The Nebula. + +A showpiece: **one solve, every step**. Where Homotopy Basins photographs *parameter* space -- +many solves, one endpoint each -- the Nebula photographs *solution space in transit*. Take a single +polynomial system with a lot of paths (cyclic-7: 5040 total-degree paths), track every one of them +from t=1 to t=0, keep **every accepted tracker step** along the way, project each C^n sample down to +a plane, and expose all of it onto one additive buffer. Thousands of translucent trails pile up +into a gas cloud: bright knots where paths converge on solutions, filaments where bundles of paths +shear past the discriminant, voids where nothing travels. + +It is a long exposure, and the exposure is literal: + + brightness = time spent + +Each segment between two tracked samples carries the path-time ``|dt|`` the tracker spent crossing +it, spread evenly along the segment's footprint. A path that crawls burns bright; a path that +races to infinity leaves a faint comet streak. Brightness is *dwell*, spatially resolved -- and +because the energy of a segment is fixed regardless of how many samples the tracker chose to leave +on it, the picture is of the **path**, not of the stepper. Nothing here is a synthetic texture: +every photon is real tracked data. + +Channels (all honest) +--------------------- +* **position** -- a fixed complex-linear functional ``l(x) = sum a_k x_k``, plotted as + ``(Re l, Im l)``. Choosing ``l`` is this piece's composition knob (gamma was Basins'). +* **brightness** -- dwell, as above. +* **colour** -- temperature by ``log|t|``: cool where the path starts, white-hot in the endgame. + Flow direction without arrows, and the knots at the solutions come out incandescent. + (Note that colouring by ``arg l(x)`` would be *degenerate*: position already IS ``(Re l, Im l)``, + so ``arg l`` is nothing but the screen polar angle -- a pinwheel carrying no information.) + +Why the power-series endgame +---------------------------- +The Cauchy endgame samples in *circles* around t=0, so ``|t|`` is deliberately multi-sheeted and +non-monotone, and a path's data can teleport back to the endgame boundary. The power-series endgame +descends radially instead: measured over 5880 cyclic-5/6/7 paths, ``|t|`` is monotone on **every** +path and there are **zero** teleports. The same lesson was learned the hard way by the Flight +Recorder showpiece (commit 0cbc7567). It also keeps the piece cheap: under Cauchy, samples/path +explode superlinearly (259 -> 970 -> 11861 for n=5,6,7) and cyclic-7 becomes a ~29-hour render; +under power-series they stay flat (202 -> 462 -> 335) and cyclic-7 is **68 seconds**. + +Determinism +----------- +There is a genuine random draw here (the total-degree start system and its gamma), unlike Basins +where every coefficient is exact. So the seed IS the parameter: ``bertini.random.set_random_seed`` +immediately before building, and ``num_threads = 1``. Serial is not a compromise -- with a Python +observer attached, threading is ~7x *slower* (12 threads convoy on the GIL, one acquire per step). + +Regenerated through ``tools/refresh_doc_artifacts.py`` (see that tool). This is a raster showpiece: +PNG only (the image IS an accumulation buffer; there is no meaningful vector form), so it is exempt +from the tutorial figures' png+svg rule. It is NOT a doctest -- the docs page embeds the +pre-rendered image. + +Run standalone: python nebula.py +""" + +import argparse +import datetime +import os +import time + +import numpy as np + +import bertini +import bertini.nag_algorithm +from bertini import ZeroDimSolver, SolutionPathCollector + +_OUT = os.path.dirname(os.path.abspath(__file__)) + +# supersampling: expose at SS x the output size, box-downsample for silky filaments +_SS = 2 + +# a segment whose |t| RATIO to the previous sample exceeds this is a teleport, not a step. +# Power-series never produces one (measured: 0 in 5880 paths), but the Cauchy endgame does -- a jump +# back to the endgame boundary is ~512x, while a legitimate Cauchy chord dips only ~1.95x. |dt| +# alone CANNOT discriminate: a teleport's |dt| is ~0.0998, inside the max_step_size=0.1 cap. +_TELEPORT_RATIO = 3.0 + +# subsample spacing along a segment, in pixels, when rasterising the polyline +_STEP_PX = 0.5 + + +# --- the subjects --------------------------------------------------------------------------------- + +def cyclic_system(n): + """The cyclic-n system: n! total-degree paths (cyclic-7 -> 5040). Highly symmetric.""" + x = [bertini.Variable('x' + str(i)) for i in range(n)] + y = list(x) + list(x) + sys_ = bertini.System() + sys_.add_variable_group(bertini.VariableGroup(x)) + for ii in range(n - 1): + sys_.add_function(np.sum([np.prod(y[jj:jj + ii + 1]) for jj in range(n)])) + sys_.add_function(np.prod(x) - 1) + return sys_ + + +def noon_system(n): + """Noonburg's neural network: x_i (1 + sum_{j != i} x_j^2) - 1. 3^n paths, not symmetric.""" + x = [bertini.Variable('x' + str(i)) for i in range(n)] + sys_ = bertini.System() + sys_.add_variable_group(bertini.VariableGroup(x)) + for i in range(n): + sys_.add_function(x[i] * (1 + np.sum([x[j]**2 for j in range(n) if j != i])) - 1) + return sys_ + + +def katsura_system(n): + """The Katsura-n system: 2^n paths. A classic, and not symmetric.""" + x = [bertini.Variable('x' + str(i)) for i in range(n + 1)] + + def u(k): + return x[abs(k)] if abs(k) <= n else None + + sys_ = bertini.System() + sys_.add_variable_group(bertini.VariableGroup(x)) + for i in range(n): + terms = [u(j) * u(i - j) for j in range(-n, n + 1) + if u(j) is not None and u(i - j) is not None] + sys_.add_function(np.sum(terms) - u(i)) + sys_.add_function(np.sum([u(j) for j in range(-n, n + 1) if u(j) is not None]) - 1) + return sys_ + + +def dense_system(num_vars, degree, seed=0): + """A random dense system: ``degree**num_vars`` paths, no structure and no symmetry at all. + + The note's own suggestion for the true nebula -- a system with nothing special about it. Every + coefficient is an EXACT rational drawn from a seeded RNG, per the library's coercion doctrine, + so the system is reproducible and no float ever reaches the function tree. + """ + from fractions import Fraction + rng = np.random.default_rng(seed) + x = [bertini.Variable('x' + str(i)) for i in range(num_vars)] + sys_ = bertini.System() + sys_.add_variable_group(bertini.VariableGroup(x)) + + def monomials(d): + """Every exponent tuple of total degree <= d.""" + out = [()] + for _ in range(num_vars): + out = [e + (k,) for e in out for k in range(d + 1) if sum(e) + k <= d] + return out + + monos = monomials(degree) + for _ in range(num_vars): + f = None + for e in monos: + if sum(e) > degree: + continue + c = bertini.coefficient(Fraction(int(rng.integers(-9, 10)), 10)) \ + + bertini.I * bertini.coefficient(Fraction(int(rng.integers(-9, 10)), 10)) + term = c + for xi, k in zip(x, e): + if k: + term = term * xi**k + f = term if f is None else f + term + sys_.add_function(f) + return sys_ + + +_SYSTEMS = { + 'cyclic5': (lambda: cyclic_system(5), 'cyclic-5', 120), + 'cyclic6': (lambda: cyclic_system(6), 'cyclic-6', 720), + 'cyclic7': (lambda: cyclic_system(7), 'cyclic-7', 5040), + 'noon6': (lambda: noon_system(6), 'noonburg-6', 729), + 'katsura8': (lambda: katsura_system(8), 'katsura-8', 256), + 'dense2': (lambda: dense_system(2, 24), 'dense random 2-var deg-24', 576), + 'dense3': (lambda: dense_system(3, 9), 'dense random 3-var deg-9', 729), +} + + +# --- the engine: one solve, every step ------------------------------------------------------------ + +class _Camera(SolutionPathCollector): + """Streaming meta-observer: compact each finished path to numpy, then drop it. + + ``SolutionPathCollector`` spins up a ``PathDataCollector`` per path on ``PathStarted`` and + harvests it on ``PathComplete`` -- capturing EVERY path the solver starts, whatever its + outcome. Left alone it holds all of them to the end; cyclic-7's 1.7M samples as Python lists + of boxed complexes would be gigabytes. So subclass it (the attach/detach dance and its + re-entrancy rules are subtle and already correct) and drain ``series`` as it fills: memory + stays O(one path). + + The splat deliberately does NOT happen here. Observers re-acquire the GIL on every event, so + any real work in ``Observe`` serialises; compacting is cheap, rendering is not. + """ + + def __init__(self): + super().__init__() + self.points = [] # list of (n_k, n_vars+1) complex64, homogeneous + self.times = [] # list of (n_k,) complex128 + + def Observe(self, event): + super().Observe(event) # the base does all the tracker attach/detach work + while self.series: + c = self.series.pop() + self.points.append(c.points().astype(np.complex64)) + self.times.append(c.times()) + + +def track(system_name, seed=2, max_step_size=None, mptype='double', endgame='powerseries'): + """Solve once and keep every accepted step of every path. + + Parameters + ---------- + system_name : str + A key of :data:`_SYSTEMS`. + seed : int + The random seed. This is the piece's only entropy: the total-degree start system and its + gamma are drawn from it. Set immediately before building -- a previous solve advances the + RNG state and would change the picture. + max_step_size : str or None + Cap on the tracker's step size, as an EXACT rational -- pass a string ("0.005"), never a + float. Smaller caps sample the same path more finely (the mathematics is unchanged, and + the dwell weighting is invariant to sampling density); ``None`` keeps the default 0.1. + mptype : str + ``'double'`` (fast, for prototyping) or ``'adaptive'``. + endgame : str + ``'powerseries'`` -- see the module docstring on why not Cauchy. + + Returns + ------- + dict + The cache: ``points`` (n_samples, n_vars+1) complex64 homogeneous coordinates, ``times`` + (n_samples,) complex128, ``path_id`` (n_samples,) int32, plus provenance. + """ + build, label, _ = _SYSTEMS[system_name] + bertini.random.set_random_seed(seed) + bertini.recording(False) # millions of throwaway samples: do not archive them + solver = ZeroDimSolver(build(), mptype=mptype, endgame=endgame) + cfg = solver.get_config(bertini.nag_algorithm.ZeroDimConfig) + cfg.num_threads = 1 # serial: deterministic, and faster than threading here + solver.set_config(cfg) + if max_step_size is not None: + solver.get_tracker().configure(stepping={'max_step_size': max_step_size}) + + cam = _Camera() + solver.add_observer(cam) + t0 = time.perf_counter() + solver.solve() + dt = time.perf_counter() - t0 + + path_id = np.concatenate([np.full(len(t), i, dtype=np.int32) + for i, t in enumerate(cam.times)]) + cache = dict(points=np.concatenate(cam.points), times=np.concatenate(cam.times), + path_id=path_id, system=system_name, label=label, seed=seed, + mptype=mptype, endgame=endgame, + max_step_size=str(max_step_size), n_paths=len(cam.times), + n_solutions=len(solver.finite_solutions()), seconds=dt) + print(f' tracked {cache["n_paths"]} paths, {len(path_id):,} samples, ' + f'{cache["n_solutions"]} finite solutions, {dt:.1f}s') + return cache + + +def save_cache(cache, path): + """Write a tracked cache to an .npz. Tracking is the expensive part; rendering is not.""" + np.savez_compressed(path, **cache) + print(' cached', path) + + +def load_cache(path): + """Read a cache written by :func:`save_cache`.""" + z = np.load(path, allow_pickle=False) + return {k: (z[k] if z[k].ndim else z[k].item()) for k in z.files} + + +# --- projection: C^n -> the plane ----------------------------------------------------------------- +# +# These coefficients are applied POST-HOC, to tracked data, at render time. They are part of the +# camera, not of the homotopy -- so unlike every other showpiece's constants they need not be exact +# values. No node ever sees them. + +def ell_dft(n, j): + """The j-th cyclic character, ``a_k = exp(2 pi i j k / n)``. + + For cyclic-n this is an eigenvector of the cyclic shift, so the *solution set* maps to itself + under rotation by 2 pi / n. The paths are not symmetric though -- the start system and gamma + break it -- so expect a near-mandala rather than an exact one. + """ + return np.exp(2j * np.pi * j * np.arange(n) / n) + + +def ell_random(n, seed=0): + """Generic unit coefficients: a projection that respects no symmetry of the system.""" + rng = np.random.default_rng(seed) + return np.exp(2j * np.pi * rng.random(n)) + + +_PROJECTIONS = { + 'dft1': lambda n: ell_dft(n, 1), + 'dft2': lambda n: ell_dft(n, 2), + 'dft3': lambda n: ell_dft(n, 3), + 'coord0': lambda n: np.eye(n, dtype=complex)[0], + 'random0': lambda n: ell_random(n, 0), + 'random1': lambda n: ell_random(n, 1), +} + + +def project(cache, ell): + """Dehomogenize and apply ``l(x)``. Returns the complex screen coordinate per sample. + + The solver homogenizes and patches, so the cached points carry a homogenizing coordinate in + column 0 and are *bounded*; dividing it out here is what lets divergers fly off frame instead + of overflowing the cache. Paths that reach infinity drive that coordinate to zero, which + yields inf/nan -- masked at splat time, not patched over. + """ + pts = cache['points'] + with np.errstate(divide='ignore', invalid='ignore'): + aff = pts[:, 1:] / pts[:, 0][:, None] + return aff @ ell + + +# --- the exposure --------------------------------------------------------------------------------- + +def _segment_energy(cache): + """Per-segment path-time ``|dt|``, and a mask of which segments are real. + + A segment joins samples k and k+1 of the SAME path. Rejected: cross-path joins, zero-energy + steps, and teleports -- a solver reuses one tracker for the main track and the endgame + sub-tracks, so a path's data is a concatenation and can jump. Power-series never jumps + (measured), but guard anyway; the guard is free and the failure is spectacular (a full-frame + streak carrying the largest energy in the image). + """ + t = cache['times'] + pid = cache['path_id'] + at = np.abs(t) + energy = np.abs(np.diff(t)) + same = pid[1:] == pid[:-1] + not_teleport = at[1:] <= _TELEPORT_RATIO * np.maximum(at[:-1], 1e-300) + return energy, same & not_teleport & (energy > 0) + + +_RAMP = np.array([[1.00, 1.00, 0.95], # white-hot: arriving at the solution + [1.00, 0.62, 0.20], # orange + [0.85, 0.25, 0.75], # violet + [0.35, 0.35, 1.00], # blue + [0.13, 0.60, 1.00]]) # cool: t = 1, the start system + + +# log10|t| stops for the ramp above. These are FIXED, not fitted to the data, and that is +# deliberate -- it is the one tuning decision this piece really has, and it was made by looking. +# +# Measured on cyclic-7: 95% of the *energy* lies in log|t| in [-1.9, 0] (the endgame carries 24% of +# the samples but only 8.4% of the dwell). The tempting move is to spread the five stops across +# that bulk so the whole ramp gets used. Do not: it makes the picture WORSE. Additive blending +# sums the hues that overlap in a pixel, so spreading the bulk across white/orange/violet/blue +# blends thousands of crossing trails into beige mud. Coherence beats coverage. Keep the bulk of +# the dwell inside ONE cool family (here [-1.5, 0] -> violet..cyan, so the main track is coloured +# but never muddy) and spend the warm end on the endgame alone, which is spatially concentrated at +# the solutions and so stays vivid instead of averaging out. +_TEMP_STOPS = np.array([-5.0, -3.0, -1.5, -0.7, 0.0]) + + +def _temperature(logt, stops=_TEMP_STOPS): + """Colour by ``log|t|``: cool where paths start, white-hot in the endgame. + + Coherent under additive blending -- thousands of *hues* would sum to grey, but a temperature + ramp sums to a hotter temperature, which is what a long exposure of converging paths should do. + Below the first stop (the deep endgame) everything clips to white-hot, which is where the knots + at the solutions come from. + """ + x = np.clip(logt, stops[0], stops[-1]) + return np.stack([np.interp(x, stops, _RAMP[:, k]) for k in range(3)], axis=-1) + + +def _phase_hue(cache, ok, ell2, value=1.0): + """Hue from ``arg l2(x)`` for a SECOND, independent projection ``l2``. + + Not ``arg l(x)``: position already IS ``(Re l, Im l)``, so ``arg l`` is nothing but the screen + polar angle -- a pinwheel that carries no information at all. An independent ``l2`` is a + genuinely different direction in C^n, invisible in the projection, and it varies smoothly along + a path, so trails come out as coherent bands of colour rather than noise. + """ + import matplotlib.colors as mcolors + z2 = project(cache, ell2)[:-1][ok] + hue = (np.angle(z2) / (2 * np.pi)) % 1.0 + hsv = np.stack([hue, np.full_like(hue, 0.85), np.full_like(hue, value)], axis=-1) + return mcolors.hsv_to_rgb(hsv) + + +def _clip_to_frame(u0, v0, u1, v1, w, h): + """Liang-Barsky: clip each segment to the frame, returning the surviving [s0, s1] fractions. + + Essential, not cosmetic: a diverger's segment can be a million pixels long, and subdividing it + before clipping would allocate millions of subsamples for a line that crosses a few. Energy is + uniform per unit length, so a clipped segment keeps energy * (s1 - s0) -- the photons that left + the frame are simply lost, as in any camera. + """ + du, dv = u1 - u0, v1 - v0 + s0 = np.zeros_like(u0) + s1 = np.ones_like(u0) + for p, q in ((-du, u0 - 0), (du, w - u0), (-dv, v0 - 0), (dv, h - v0)): + with np.errstate(divide='ignore', invalid='ignore'): + r = np.where(p != 0, q / p, np.inf) + enter = (p < 0) + s0 = np.where(enter, np.maximum(s0, r), s0) + s1 = np.where(p > 0, np.minimum(s1, r), s1) + # p == 0 and q < 0: the segment is parallel to and outside this edge + s1 = np.where((p == 0) & (q < 0), -1.0, s1) + return s0, s1 + + +def _deposit(buf, uu, vv, weight, rgb): + """Bilinear (anti-aliased) point splat of weighted photons into an (h, w, 3) buffer.""" + h, w = buf.shape[:2] + x, y = uu - 0.5, vv - 0.5 + x0 = np.floor(x).astype(np.int64) + y0 = np.floor(y).astype(np.int64) + fx, fy = x - x0, y - y0 + for dx in (0, 1): + for dy in (0, 1): + xi, yi = x0 + dx, y0 + dy + a = weight * (fx if dx else 1 - fx) * (fy if dy else 1 - fy) + ok = (xi >= 0) & (xi < w) & (yi >= 0) & (yi < h) & (a > 0) & np.isfinite(a) + if not ok.any(): + continue + flat = yi[ok] * w + xi[ok] + for ch in range(3): + buf[..., ch] += np.bincount(flat, weights=a[ok] * rgb[ok, ch], + minlength=h * w).reshape(h, w) + + +def _splat_segments(buf, u0, v0, du, dv, energy, rgb): + """Subdivide each segment along its footprint and deposit its energy. + + ``m = max(1, ceil(L / step))`` subsamples each carry ``E/m``, so the sum is exactly ``E`` for + any length -- long segments spread to ``E/L`` per pixel (dwell per unit length = 1/speed), + sub-pixel segments dump all of ``E`` into one pixel with no ``1/L -> inf`` singularity. + """ + L = np.hypot(du, dv) + m = np.maximum(1, np.ceil(L / _STEP_PX)).astype(np.int64) + seg = np.repeat(np.arange(len(m)), m) # which segment each subsample is on + off = np.arange(int(m.sum())) - np.repeat(np.cumsum(m) - m, m) + f = (off + 0.5) / m[seg] + _deposit(buf, u0[seg] + du[seg] * f, v0[seg] + dv[seg] * f, + energy[seg] / m[seg], rgb[seg]) + + +def expose(cache, ell, center, halfwidth, w, h, color='temperature', ell2=None): + """The long exposure: splat every tracked segment into an (h, w, 3) float buffer. + + Each segment deposits its path-time energy ``E = |dt|``, spread evenly over + ``m = max(1, ceil(L / step))`` subsamples along its rasterised footprint. The sum over the + subsamples is exactly ``E`` for any length, so: + + * long segments -> ``E/L`` per pixel = dwell per unit length = 1/speed. A literal long + exposure of a moving particle. + * sub-pixel segments (the endgame) -> ``m = 1``, all of ``E`` in one pixel. No ``1/L -> inf`` + singularity: the pixel is the sensor, and motion below it is not resolvable. + + Subdividing is line rasterisation -- the geometry we already asserted by drawing a polyline + through the samples, exactly as the Loom draws ribbons between its samples. It invents no + tracked data. The energy is conserved regardless, which is what ``--selftest`` checks. + """ + z = project(cache, ell) + energy, ok = _segment_energy(cache) + + halfheight = halfwidth * h / w + u = (z.real - (center[0] - halfwidth)) / (2 * halfwidth) * w + v = (z.imag - (center[1] - halfheight)) / (2 * halfheight) * h + + u0, u1 = u[:-1], u[1:] + v0, v1 = v[:-1], v[1:] + ok = ok & np.isfinite(u0) & np.isfinite(u1) & np.isfinite(v0) & np.isfinite(v1) + + s0, s1 = _clip_to_frame(u0, v0, u1, v1, w, h) + ok = ok & (s1 > s0) + if not ok.any(): + return np.zeros((h, w, 3)), 0.0 + + u0, u1, v0, v1 = u0[ok], u1[ok], v0[ok], v1[ok] + s0, s1, E = s0[ok], s1[ok], energy[ok] + du, dv = u1 - u0, v1 - v0 + # the surviving piece of each segment, and the share of its energy that piece carries + cu0, cv0 = u0 + du * s0, v0 + dv * s0 + cdu, cdv = du * (s1 - s0), dv * (s1 - s0) + E = E * (s1 - s0) + + logt = np.log10(np.maximum(np.abs(cache['times'][:-1][ok]), 1e-300)) + if color == 'temperature': + rgb = _temperature(logt) + elif color == 'phase2': + rgb = _phase_hue(cache, ok, ell2) + elif color == 'both': + # hue from the independent direction, brightened toward white as the endgame closes in + hot = np.clip((logt - _TEMP_STOPS[2]) / (_TEMP_STOPS[0] - _TEMP_STOPS[2]), 0, 1) + rgb = _phase_hue(cache, ok, ell2) * (1 - hot)[:, None] + hot[:, None] + else: + raise ValueError(f'unknown colour mode {color!r}') + + buf = np.zeros((h, w, 3)) + _splat_segments(buf, cu0, cv0, cdu, cdv, E, rgb) + # what the buffer SHOULD total if every photon landed: each segment writes E into 3 channels + # scaled by its colour, so the expectation carries the colour weights too. + return buf, float((E * rgb.sum(axis=1)).sum()) + + +# --- rendering ------------------------------------------------------------------------------------ + +def _downsample(img, factor): + """Box-average downsample of an (h, w, 3) image by an integer factor.""" + h, w = img.shape[:2] + return img[:h - h % factor, :w - w % factor]\ + .reshape(h // factor, factor, w // factor, factor, -1).mean(axis=(1, 3)) + + +def tone_map(buf, exposure=4.0, gamma=1.8, ss=_SS): + """The HDR curve. The accumulation IS the bloom; no bloom pass is needed. + + Downsample in LINEAR light first, so ``exposure`` means the same thing at any resolution, then + ``1 - exp(-k*b)`` -- the classic film response, which is why the long exposure is literal. Each + channel saturates at its own rate, so the hottest knots bleach to white on their own. + """ + img = _downsample(buf, ss) + lit = img[img > 0] + scale = np.percentile(lit, 99.5) if lit.size else 1.0 + img = 1 - np.exp(-exposure * img / max(scale, 1e-30)) + return np.clip(img ** (1 / gamma), 0, 1) + + +def render(cache, ell, center, halfwidth, w, h, out_png, exposure=4.0, gamma=1.8): + """Expose, tone-map, write. Returns the image.""" + import matplotlib.image as mimage + buf, _ = expose(cache, ell, center, halfwidth, w * _SS, h * _SS) + img = tone_map(buf, exposure=exposure, gamma=gamma) + mimage.imsave(out_png, img, origin='lower') + print(' wrote', out_png) + return img + + +def suggest_window(cache, ell, quantile=99.0): + """A dwell-weighted window for a projection: where do the photons actually land? + + Weight by energy, or the divergers -- which travel furthest and dwell least -- drag the frame + out to infinity. Prints a suggestion to paste in as a literal constant; auto-windowing at + render time would make a committed PNG depend on run-to-run data. + """ + z = project(cache, ell) + energy, ok = _segment_energy(cache) + ok = ok & np.isfinite(z[:-1]) + zz, ee = z[:-1][ok], energy[ok] + order = np.argsort(np.abs(zz)) + cw = np.cumsum(ee[order]) / ee.sum() + r = np.abs(zz[order][np.searchsorted(cw, quantile / 100.0)]) + cen = np.average(zz[order][cw <= quantile / 100.0].real), \ + np.average(zz[order][cw <= quantile / 100.0].imag) + return (round(float(cen[0]), 3), round(float(cen[1]), 3)), round(float(r), 3) + + +# --- prototyping ---------------------------------------------------------------------------------- + +_PROTO = os.path.expanduser('~/nebula_prototype_images') + + +def _stamp(name): + """A datetime-stamped path in the prototype image record.""" + os.makedirs(_PROTO, exist_ok=True) + ts = datetime.datetime.now().strftime('%Y%m%d-%H%M%S') + return os.path.join(_PROTO, f'{name}_{ts}.png') + + +def scout(system_name='cyclic5', seed=2, w=480, h=270, max_step_size=None): + """Contact sheet: ONE solve, every projection. + + The tracked data does not depend on ``l`` -- that is the whole point of caching the track -- so + a single solve renders every candidate projection. Each panel is captioned and the sheet is + stamped into the prototype record. + """ + import matplotlib + matplotlib.use('Agg') + import matplotlib.pyplot as plt + + print(f'scouting {system_name}:') + cache = track(system_name, seed=seed, max_step_size=max_step_size) + n = cache['points'].shape[1] - 1 + + keys = list(_PROJECTIONS) + cols = 3 + rows = (len(keys) + cols - 1) // cols + fig, axes = plt.subplots(rows, cols, figsize=(4.5 * cols, 2.7 * rows), facecolor='#05060a') + for ax, key in zip(np.ravel(axes), keys): + ell = _PROJECTIONS[key](n) + center, halfwidth = suggest_window(cache, ell) + buf, _ = expose(cache, ell, center, halfwidth, w * _SS, h * _SS) + ax.imshow(tone_map(buf), origin='lower') + ax.set_title(f'{key} center={center} halfwidth={halfwidth}', + color='#c9d3e0', fontsize=8) + print(f' {key:8} center={center} halfwidth={halfwidth}') + for ax in np.ravel(axes): + ax.set_xticks([]); ax.set_yticks([]) + fig.suptitle(f'{cache["label"]} -- {cache["n_paths"]} paths, ' + f'{len(cache["path_id"]):,} samples, seed {seed}', color='#e8eef7') + out = _stamp(f'scout_{system_name}') + fig.savefig(out, dpi=110, facecolor='#05060a', bbox_inches='tight') + plt.close(fig) + print(' wrote', out) + return cache + + +def selftest(system_name='cyclic5'): + """Assert the exposure conserves energy -- the sharpest invariant this piece has. + + Every photon in the buffer is some segment's path-time. Two checks: + + 1. **Synthetic, strictly interior**: conservation must be EXACT (to float tolerance). This + isolates the subdivision and the bilinear deposit from any real-data effect. Segments of + wildly different lengths -- including sub-pixel ones -- must all conserve. + 2. **Real tracked data**: the buffer will fall slightly short, because a bilinear splat at the + frame border drops the neighbours that lie outside it. That is honest photon loss (a camera + does the same), so it is reported, not asserted away. + """ + rng = np.random.default_rng(0) + h, w = 64, 96 + n = 400 + u0 = rng.uniform(8, w - 8, n) + v0 = rng.uniform(8, h - 8, n) + ang = rng.uniform(0, 2 * np.pi, n) + length = 10 ** rng.uniform(-1.5, 1.2, n) # 0.03 px (sub-pixel) .. 16 px + du, dv = length * np.cos(ang), length * np.sin(ang) + E = rng.uniform(0.1, 5.0, n) + rgb = rng.uniform(0.2, 1.0, (n, 3)) + buf = np.zeros((h, w, 3)) + _splat_segments(buf, u0, v0, du, dv, E, rgb) + want = float((E * rgb.sum(axis=1)).sum()) + got = float(buf.sum()) + print(f' synthetic: want {want:.9f}, got {got:.9f}, rel err {abs(got - want) / want:.2e}') + assert abs(got - want) / want < 1e-9, 'the splat does not conserve energy' + assert (m := (length < 1).sum()) > 0, 'test needs sub-pixel segments to be meaningful' + print(f' ({m} of {n} segments were sub-pixel)') + + cache = track(system_name, seed=2) + ell = _PROJECTIONS['dft1'](cache['points'].shape[1] - 1) + center, halfwidth = suggest_window(cache, ell) + buf, expected = expose(cache, ell, center, halfwidth, 480, 270) + got = float(buf.sum()) + print(f' tracked : want {expected:.6f}, got {got:.6f}, ' + f'{100 * (1 - got / expected):.3f}% lost at the frame border') + assert got <= expected * (1 + 1e-9), 'the exposure created energy from nowhere' + assert got > expected * 0.9, 'the exposure lost more than 10% -- that is not border loss' + print(' PASS') + + +def main(): + p = argparse.ArgumentParser(description=__doc__.split('\n')[0]) + p.add_argument('--scout', metavar='SYSTEM', nargs='?', const='cyclic5', + help='render a projection contact sheet from one solve') + p.add_argument('--selftest', action='store_true', help='assert the exposure conserves energy') + p.add_argument('--seed', type=int, default=2) + p.add_argument('--max-step-size', default=None, + help='exact rational as a string, e.g. "0.005"') + args = p.parse_args() + + if args.selftest: + selftest() + elif args.scout: + scout(args.scout, seed=args.seed, max_step_size=args.max_step_size) + else: + print('the hero frame is not chosen yet; use --scout') + + +if __name__ == '__main__': + main() From 26991ce97e7b6cba9f61f161bc980f4167c920af Mon Sep 17 00:00:00 2001 From: silviana amethyst Date: Thu, 16 Jul 2026 23:28:28 +0000 Subject: [PATCH 2/6] docs(showpieces): nebula subjects -- Kuramoto, noon, katsura, dense random cyclic-n turns out to be the wrong subject, and not for the reason the design note feared: it is not too symmetric, it is a fuzzy BALL. Every cyclic-n path starts near |x|=1 and ends near |x|=1, so the whole solve lives in one small disc and any linear projection is a blob with spikes. Not a tone-map failure -- measured, the peak is only 5x the 99.5th percentile, so the core is a genuinely broad dense region, not blown highlights. So the registry grows subjects whose paths actually travel. The Kuramoto model is the find: how a crowd of oscillators pulls itself into synchrony (fireflies, power grids), algebraized the way its equilibria have actually been counted by homotopy continuation. 4096 paths, 124 equilibria, 37s. Two constraints there are load-bearing, not decoration: * the phase-rotation symmetry makes the solution set positive-dimensional, so the last oscillator is gauge-fixed at theta=0 -- without it a zero-dim solve is meaningless; * the frequencies must sum to zero or NO equilibrium exists (summing the equations kills the antisymmetric coupling term), so the last frequency is derived, never drawn. It also brings its own projection: the order parameter r e^{i psi} = (1/N) sum e^{i theta_j} is *exactly* a complex-linear functional in the (s, c) coordinates, so for once the composition knob is not a choice -- the frame IS synchronisation space. --- .../docs/source/showpieces/nebula/nebula.py | 74 +++++++++++++++++++ 1 file changed, 74 insertions(+) diff --git a/python/docs/source/showpieces/nebula/nebula.py b/python/docs/source/showpieces/nebula/nebula.py index 188ed6255..14607147c 100644 --- a/python/docs/source/showpieces/nebula/nebula.py +++ b/python/docs/source/showpieces/nebula/nebula.py @@ -121,6 +121,57 @@ def u(k): return sys_ +def kuramoto_system(n, seed=0): + """Equilibria of the Kuramoto model: ``n`` coupled oscillators, ``4**(n-1)`` paths. + + The Kuramoto model is how a crowd of oscillators with different natural frequencies pulls + itself into synchrony -- fireflies flashing together, a power grid locking to 50 Hz. Its + equilibria satisfy + + omega_i - (K/n) * sum_j sin(theta_i - theta_j) = 0 , + + which is not polynomial until you set ``s_i = sin theta_i``, ``c_i = cos theta_i`` and carry + ``s_i^2 + c_i^2 = 1`` along; then ``sin(theta_i - theta_j) = s_i c_j - c_i s_j`` and the whole + thing is a square polynomial system. That algebraization is exactly how the model's equilibria + have been *counted* by homotopy continuation, so this is a system Bertini is genuinely for. + + Two constraints the mathematics imposes, both load-bearing: + + * **Gauge.** Rotating every phase by the same angle maps equilibria to equilibria, so the + solution set is positive-dimensional (curves, not points) and a zero-dim solve would be + meaningless. Pin the last oscillator at ``theta = 0`` (``s = 0``, ``c = 1``) to quotient the + symmetry out. + * **Frequencies must sum to zero**, or no equilibrium exists at all: summing the equations + kills the coupling term (it is antisymmetric in i, j) and leaves ``sum omega_i = 0``. So the + last frequency is *derived*, never drawn. + + The frequencies are exact rationals from a seeded RNG -- generic enough to break every symmetry + of the system, and exact per the library's coercion doctrine. + """ + from fractions import Fraction + rng = np.random.default_rng(seed) + omega = [Fraction(int(rng.integers(-9, 10)), 10) for _ in range(n - 1)] + omega.append(-sum(omega)) # sum omega_i = 0, or there is no equilibrium + + s = [bertini.Variable('s' + str(i)) for i in range(n - 1)] + c = [bertini.Variable('c' + str(i)) for i in range(n - 1)] + s_all = list(s) + [0] # the gauge: theta_{n-1} = 0 + c_all = list(c) + [1] + + sys_ = bertini.System() + sys_.add_variable_group(bertini.VariableGroup(s + c)) + coupling = bertini.coefficient(Fraction(1, n)) # K = 1 + for i in range(n - 1): + interaction = None + for j in range(n): + term = s_all[i] * c_all[j] - c_all[i] * s_all[j] + interaction = term if interaction is None else interaction + term + sys_.add_function(bertini.coefficient(omega[i]) - coupling * interaction) + for i in range(n - 1): + sys_.add_function(s[i]**2 + c[i]**2 - 1) + return sys_ + + def dense_system(num_vars, degree, seed=0): """A random dense system: ``degree**num_vars`` paths, no structure and no symmetry at all. @@ -164,6 +215,9 @@ def monomials(d): 'cyclic7': (lambda: cyclic_system(7), 'cyclic-7', 5040), 'noon6': (lambda: noon_system(6), 'noonburg-6', 729), 'katsura8': (lambda: katsura_system(8), 'katsura-8', 256), + 'kuramoto5': (lambda: kuramoto_system(5), 'Kuramoto, 5 oscillators', 256), + 'kuramoto6': (lambda: kuramoto_system(6), 'Kuramoto, 6 oscillators', 1024), + 'kuramoto7': (lambda: kuramoto_system(7), 'Kuramoto, 7 oscillators', 4096), 'dense2': (lambda: dense_system(2, 24), 'dense random 2-var deg-24', 576), 'dense3': (lambda: dense_system(3, 9), 'dense random 3-var deg-9', 729), } @@ -286,7 +340,27 @@ def ell_random(n, seed=0): return np.exp(2j * np.pi * rng.random(n)) +def ell_order(nvars): + """The Kuramoto **order parameter**, ``r e^{i psi} = (1/N) sum_j e^{i theta_j}``. + + The one projection in this file that is not a choice. In the ``(s, c)`` coordinates + ``e^{i theta_j} = c_j + i s_j``, so the order parameter -- the physical measure of how + synchronised the oscillators are, ``r = 1`` locked and ``r = 0`` incoherent -- is *exactly a + complex-linear functional of the variables*. Projecting along it means the frame IS + synchronisation space, and every trail is a path's journey toward (or away from) sync. + + The pinned oscillator contributes a constant ``1/N``, which merely translates the picture; the + window's centre absorbs it. + """ + n_osc = nvars // 2 + 1 # nvars = 2 * (n_osc - 1) after the gauge + a = np.zeros(nvars, dtype=complex) + a[:n_osc - 1] = 1j / n_osc # the s_j + a[n_osc - 1:] = 1.0 / n_osc # the c_j + return a + + _PROJECTIONS = { + 'order': ell_order, 'dft1': lambda n: ell_dft(n, 1), 'dft2': lambda n: ell_dft(n, 2), 'dft3': lambda n: ell_dft(n, 3), From 8e0a122fb544d0c61b8155d9b815898bc422e691 Mon Sep 17 00:00:00 2001 From: silviana amethyst Date: Thu, 16 Jul 2026 23:54:03 +0000 Subject: [PATCH 3/6] docs(showpieces): nebula subjects -- CRN mass action, Stewart-Gough Two more subjects, both verified against something other than my own arithmetic. mass_action_system derives steady-state polynomials from a reaction list rather than transcribing them: mass action is a fixed rule, so a network determines its own system. Conservation laws are not decoration -- a reaction network's stoichiometric matrix is rank-deficient, so unless one ODE per conserved quantity is REPLACED by its conservation law the system is singular and a zero-dim solve is meaningless. phosphorylation_network(n) builds the multisite phosphorylation ladder, CRN theory's standard multistationarity benchmark. Measured: n=3 -> 512 paths / 7 steady states, n=4 -> 4096 paths / 9. The 2n+1 pattern corroborates the derivation. stewart_gough_system: forward kinematics of a six-legged platform -- given six leg lengths, where is the platform? -- the problem this library's own authors made their name on. The leg lengths are computed exactly from a CHOSEN pose, so a known answer is planted in the system; measured, the solve recovers it to 1.9e-13, which is the real test that the formulation is right. The chosen quaternion is the Pythagorean quadruple (1,2,2,4)/5, because an exact rational point on the unit sphere cannot be had by rounding. Double precision reports 78 finite solutions where the classic count is 40 poses. The quaternion double cover (R(q) = R(-q)) would explain 80, with 2 lost to double precision as cyclic-7 loses 3 of 924. That is a HYPOTHESIS: the pairing test timed out at 10 min under AMP and is unverified. Neither is the hero. Both render as starbursts, not nebulae -- phos4 finds 9 steady states among 4096 paths, so the frame is overwhelmingly divergence, and its 99%-energy window is hw=378 (the divergers travel far and drag the frame out), collapsing the structure to a dot. --- .../docs/source/showpieces/nebula/nebula.py | 175 ++++++++++++++++++ 1 file changed, 175 insertions(+) diff --git a/python/docs/source/showpieces/nebula/nebula.py b/python/docs/source/showpieces/nebula/nebula.py index 14607147c..e8ef28827 100644 --- a/python/docs/source/showpieces/nebula/nebula.py +++ b/python/docs/source/showpieces/nebula/nebula.py @@ -172,6 +172,178 @@ def kuramoto_system(n, seed=0): return sys_ +def stewart_gough_system(seed=0): + """Forward kinematics of a Stewart-Gough platform: 8192 paths, the classic 40 poses. + + This is the problem numerical algebraic geometry grew up on -- Bertini's own authors wrote the + book on polynomial systems arising in kinematics. A Stewart-Gough platform is a rigid plate + held over a base by six legs of adjustable length (flight simulators, machine tools, telescope + mounts). *Inverse* kinematics is trivial: pick a pose, measure the legs. **Forward** + kinematics is the hard one -- given the six leg lengths, where is the platform? The famous + answer is that a general platform admits **40** poses for the same six lengths, and that count + was settled by homotopy continuation. + + Unknowns: a unit quaternion ``q`` for the orientation and a translation ``t``, so seven of them. + Equations: each leg pins a distance, ``|R(q) a_i + t - b_i|^2 = L_i^2``, plus ``|q|^2 = 1``. + ``R(q)`` is quadratic in ``q``, so each leg equation has total degree 4 and the total-degree + homotopy tracks ``4**6 * 2 = 8192`` paths for 40 answers -- a ratio that is itself the reason + the field invented better start systems. + + The leg lengths are not invented: a pose is *chosen* and the lengths are computed from it + exactly, so the system is guaranteed to have that pose among its solutions. The chosen + quaternion is an exact unit quaternion ``(1, 2, 2, 4)/5`` -- a Pythagorean quadruple, since an + exact rational point on the unit sphere cannot be had by rounding. + """ + from fractions import Fraction + rng = np.random.default_rng(seed) + + def frac(): + return Fraction(int(rng.integers(-9, 10)), 5) + + a = [[frac() for _ in range(3)] for _ in range(6)] # platform joints, in the moving frame + b = [[frac() for _ in range(3)] for _ in range(6)] # base joints, in the fixed frame + + # the pose we will hide in the answer: an EXACT unit quaternion (1,2,2,4)/5, |q|^2 = 25/25 = 1 + q_star = [Fraction(1, 5), Fraction(2, 5), Fraction(2, 5), Fraction(4, 5)] + t_star = [frac() for _ in range(3)] + + def rot(q): + """R(q) for a UNIT quaternion q, entries quadratic in q. Works on Fractions or on nodes.""" + w, i, j, k = q + return [[w*w + i*i - j*j - k*k, 2*(i*j - w*k), 2*(i*k + w*j)], + [2*(i*j + w*k), w*w - i*i + j*j - k*k, 2*(j*k - w*i)], + [2*(i*k - w*j), 2*(j*k + w*i), w*w - i*i - j*j + k*k]] + + # the leg lengths, computed EXACTLY from the chosen pose -- so it is genuinely a solution + R_star = rot(q_star) + leg_sq = [] + for ai, bi in zip(a, b): + d = [sum(R_star[r][c] * ai[c] for c in range(3)) + t_star[r] - bi[r] for r in range(3)] + leg_sq.append(sum(x * x for x in d)) + + qv = [bertini.Variable(n) for n in ('q0', 'q1', 'q2', 'q3')] + tv = [bertini.Variable(n) for n in ('t0', 't1', 't2')] + sys_ = bertini.System() + sys_.add_variable_group(bertini.VariableGroup(qv + tv)) + + R = rot(qv) + for ai, bi, Lsq in zip(a, b, leg_sq): + f = None + for r in range(3): + d = R[r][0] * bertini.coefficient(ai[0]) \ + + R[r][1] * bertini.coefficient(ai[1]) \ + + R[r][2] * bertini.coefficient(ai[2]) \ + + tv[r] - bertini.coefficient(bi[r]) + f = d * d if f is None else f + d * d + sys_.add_function(f - bertini.coefficient(Lsq)) + sys_.add_function(qv[0]**2 + qv[1]**2 + qv[2]**2 + qv[3]**2 - 1) + return sys_ + + +def mass_action_system(species, reactions, conserved, seed=0): + """Steady states of a chemical reaction network under mass-action kinetics. + + The polynomials are *derived*, not transcribed: mass action is a fixed rule, so a network + determines its own system. For each reaction ``sum_i a_i X_i -> sum_i b_i X_i`` with rate + constant ``k``, the flux is ``k * prod_i x_i**a_i``, and each species accumulates + ``dx_i/dt = sum_reactions (b_i - a_i) * flux``. Steady state sets every ``dx_i/dt = 0``. + + Those equations are never independent: a reaction network conserves things (total enzyme, total + substrate), so the stoichiometric matrix is rank-deficient and the naive system is singular -- + a positive-dimensional solution set, useless to a zero-dim solver. For each conservation law + one ODE must be *replaced* by the corresponding linear conservation equation. That is what + ``conserved`` does, and it is what makes the system square and zero-dimensional. + + Parameters + ---------- + species : list of str + Species names, in order; they become the variables. + reactions : list of (dict, dict) + Each reaction as ``(reactants, products)``, mapping species name -> stoichiometry. Rate + constants are drawn as exact rationals from ``seed``. + conserved : list of (list of str, str) + Each conservation law as ``(species in the sum, the species whose ODE it replaces)``. The + conserved total is drawn as an exact rational. + seed : int + Seeds the rate constants and conserved totals. Exact rationals throughout, per the + library's coercion doctrine. + """ + from fractions import Fraction + rng = np.random.default_rng(seed) + x = {s: bertini.Variable(s) for s in species} + + def rational(): + return bertini.coefficient(Fraction(int(rng.integers(1, 10)), 10)) + + # flux of each reaction: k * prod reactant^stoichiometry + fluxes = [] + for reactants, _ in reactions: + flux = rational() + for s, a in reactants.items(): + for _ in range(a): + flux = flux * x[s] + fluxes.append(flux) + + # dx_i/dt = sum_j (b_ij - a_ij) * flux_j + ode = {s: None for s in species} + for (reactants, products), flux in zip(reactions, fluxes): + for s in set(reactants) | set(products): + net = products.get(s, 0) - reactants.get(s, 0) + if net == 0: + continue + term = net * flux + ode[s] = term if ode[s] is None else ode[s] + term + + replaced = {victim for _, victim in conserved} + sys_ = bertini.System() + sys_.add_variable_group(bertini.VariableGroup([x[s] for s in species])) + for s in species: + if s not in replaced and ode[s] is not None: + sys_.add_function(ode[s]) + for members, _ in conserved: + total = None + for s in members: + total = x[s] if total is None else total + x[s] + sys_.add_function(total - rational()) + return sys_ + + +def phosphorylation_network(n, seed=0): + """The n-site distributive sequential phosphorylation network: ``2**(3n)`` paths. + + The workhorse of chemical reaction network theory, and the standard example of + *multistationarity* -- a cell switch. A kinase E walks a substrate up the ladder + ``S_0 -> S_1 -> ... -> S_n`` one phosphate at a time, a phosphatase F walks it back down, and + each step goes through an enzyme-substrate complex:: + + S_i + E <-> ES_i -> S_{i+1} + E (i = 0 .. n-1) + S_i + F <-> FS_i -> S_{i-1} + F (i = 1 .. n) + + Three things are conserved -- total kinase, total phosphatase, total substrate -- so three of + the ODEs are redundant and are replaced by those conservation laws (see + :func:`mass_action_system`). n=3 gives 512 paths, n=4 gives 4096. + """ + S = ['S{}'.format(i) for i in range(n + 1)] + ES = ['ES{}'.format(i) for i in range(n)] + FS = ['FS{}'.format(i + 1) for i in range(n)] + species = S + ['E', 'F'] + ES + FS + + reactions = [] + for i in range(n): # kinase: S_i + E <-> ES_i -> S_{i+1} + E + reactions.append(({S[i]: 1, 'E': 1}, {ES[i]: 1})) + reactions.append(({ES[i]: 1}, {S[i]: 1, 'E': 1})) + reactions.append(({ES[i]: 1}, {S[i + 1]: 1, 'E': 1})) + for i in range(1, n + 1): # phosphatase: S_i + F <-> FS_i -> S_{i-1} + F + reactions.append(({S[i]: 1, 'F': 1}, {FS[i - 1]: 1})) + reactions.append(({FS[i - 1]: 1}, {S[i]: 1, 'F': 1})) + reactions.append(({FS[i - 1]: 1}, {S[i - 1]: 1, 'F': 1})) + + conserved = [(['E'] + ES, 'E'), # total kinase, free + bound + (['F'] + FS, 'F'), # total phosphatase + (S + ES + FS, S[0])] # total substrate, in every form + return mass_action_system(species, reactions, conserved, seed=seed) + + def dense_system(num_vars, degree, seed=0): """A random dense system: ``degree**num_vars`` paths, no structure and no symmetry at all. @@ -218,6 +390,9 @@ def monomials(d): 'kuramoto5': (lambda: kuramoto_system(5), 'Kuramoto, 5 oscillators', 256), 'kuramoto6': (lambda: kuramoto_system(6), 'Kuramoto, 6 oscillators', 1024), 'kuramoto7': (lambda: kuramoto_system(7), 'Kuramoto, 7 oscillators', 4096), + 'phos3': (lambda: phosphorylation_network(3), '3-site phosphorylation', 512), + 'phos4': (lambda: phosphorylation_network(4), '4-site phosphorylation', 4096), + 'stewart': (lambda: stewart_gough_system(), 'Stewart-Gough platform', 8192), 'dense2': (lambda: dense_system(2, 24), 'dense random 2-var deg-24', 576), 'dense3': (lambda: dense_system(3, 9), 'dense random 3-var deg-9', 729), } From a97490ef8b462e82e040c4c0e4d128ffa2aa7e91 Mon Sep 17 00:00:00 2001 From: silviana amethyst Date: Fri, 17 Jul 2026 00:11:58 +0000 Subject: [PATCH 4/6] docs(showpieces): the Nebula's two frames -- Kuramoto-7 hero, noonburg-6 teaching Both frames chosen by silviana off the prototype contact sheets, and both project onto coord0 -- the first coordinate's own complex plane. Picked by looking, not by argument; the Kuramoto order parameter is the projection with the better STORY (it is exactly a linear functional, so the frame would literally be synchronisation space) but it renders as a diffuse cocoon and lost on looks. teaching: noonburg-6, 729 paths / 717 solutions / 2.4s. Few enough paths that individual strands stay legible, so the frame teaches what the piece is -- each filament is one tracked path, brightening as it slows into its solution. Reads as wings. hero: Kuramoto-7, 4096 paths / 124 equilibria / 28s. Dense enough to be a cloud: two blazing knots where bundles converge, filaments streaming between them. Windows are literal constants, per Basins' convention: suggest_window() proposes them from tracked data and prints them, and they get pasted in. Auto-windowing at render time would make a committed PNG depend on run-to-run data and churn the diff on every regeneration. 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tracked +# data and prints them; they are then pasted here. Auto-windowing at render time would make a +# committed PNG depend on run-to-run data and churn the diff on every regeneration. Constants are +# reviewable. Both frames project onto coord0 -- the first coordinate's own complex plane -- which +# was chosen by looking at contact sheets, not by argument. + +def teaching_frame(out): + """Noonburg-6: 729 paths, 717 solutions, ~3s. The lesson, legible. + + Few enough paths that individual strands stay separate, so you can see what the piece IS: each + filament is one tracked path, fanning out of the start system and converging on a solution, + brightening as it slows. The wing-like plumes are bundles of paths that travel together. + """ + print('teaching frame (noonburg-6):') + cache = track('noon6', seed=2) + ell = _PROJECTIONS['coord0'](cache['points'].shape[1] - 1) + render(cache, ell, center=(0.1, -0.502), halfwidth=1.357, w=960, h=540, + out_png=out, exposure=2.0, gamma=2.4) + + +def showpiece_frame(out): + """Kuramoto, 7 oscillators: 4096 paths, 124 equilibria, ~40s. The show-off. + + Dense enough to be a cloud rather than a set of strands. Two blazing knots where whole bundles + of paths converge, and thousands of filaments streaming between them: a solve, photographed. + """ + print('showpiece frame (Kuramoto-7):') + cache = track('kuramoto7', seed=2) + ell = _PROJECTIONS['coord0'](cache['points'].shape[1] - 1) + render(cache, ell, center=(0.278, 0.322), halfwidth=5.418, w=960, h=540, + out_png=out, exposure=2.0, gamma=2.4) + + def main(): + """Generate both frames next to this script, or scout/selftest while prototyping.""" p = argparse.ArgumentParser(description=__doc__.split('\n')[0]) p.add_argument('--scout', metavar='SYSTEM', nargs='?', const='cyclic5', - help='render a projection contact sheet from one solve') + help='contact sheet of every projection, from ONE solve (prototyping)') p.add_argument('--selftest', action='store_true', help='assert the exposure conserves energy') p.add_argument('--seed', type=int, default=2) p.add_argument('--max-step-size', default=None, @@ -904,7 +940,8 @@ def main(): elif args.scout: scout(args.scout, seed=args.seed, max_step_size=args.max_step_size) else: - print('the hero frame is not chosen yet; use --scout') + teaching_frame(os.path.join(_OUT, 'nebula_teaching.png')) + showpiece_frame(os.path.join(_OUT, 'nebula.png')) if __name__ == '__main__': diff --git a/python/docs/source/showpieces/nebula/nebula_teaching.png b/python/docs/source/showpieces/nebula/nebula_teaching.png new file mode 100644 index 0000000000000000000000000000000000000000..337c430203d70d14fb89000266fd7a0d1005f273 GIT binary patch literal 545338 zcmX`S1yEb<^FN9c++Bm0;_hyxcqvlc-MzR4_W}iqq_{(IcWHqJcXx_=@PFR#@6O$s zGnvU`&P>kPXLmocF&b)$Sm@;FaBy%~AC%;@;NTE@;ouM`0VuFNiMRd;lk~03n=TuxJrhVb@NWa z02cELv)w1Hw&4ORx@Ra5bxtSy84MD{{9;S`I}r|R^*sd0Z^ zbzXw5mUCg7b3V8aTq>gp#}ZgEO4!v*1TM~B(T_uF~mX;aY2?>%E8{Pw; z+O56JU@oWgm+f>MvxWImPTwBL+N+VFqzgOr!{#YsPJLt0JDniA%SI{^L|6=@kNgwO zlk%D2z-&D4n2mDRH)F|<4D+k3zjsm3^XYiF7Z3Giiw%doy7xO}ylx0iJ)XtY-8N0< 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page, toctree line, refresh_doc_artifacts manifest entry, CHANGELOG [3.5.0] block. Verified: py_doclint passes both gates (correctness + undocumented ratchet at baseline 14); sphinx builds with no warnings and both images resolve; `refresh_doc_artifacts.py --plots --only nebula` regenerates both frames BYTE-IDENTICALLY to the committed ones, which is the real check that the seed + num_threads=1 recipe is deterministic. The page does not :doc:-link Homotopy Basins even though the prose wants the Loom/Nebula/Basins triangle: Basins is unmerged (PR #352), so a cross-reference would break this build. It is named in plain text and can become a link when #352 lands. --- CHANGELOG.md | 19 +++ python/docs/source/showpieces/index.rst | 1 + .../docs/source/showpieces/nebula/index.rst | 116 ++++++++++++++++++ tools/refresh_doc_artifacts.py | 5 + 4 files changed, 141 insertions(+) create mode 100644 python/docs/source/showpieces/nebula/index.rst diff --git a/CHANGELOG.md b/CHANGELOG.md index f7893f845..2b54a0ffb 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -65,6 +65,25 @@ _______________________________________________________________________________ --> +_______________________________________________________________________________ + +## [3.5.0] - unreleased + +### Added + +- **The Nebula showpiece** (`python/docs/source/showpieces/nebula/`) — one solve, every path, every + step, exposed onto a single additive buffer like a long exposure. Brightness is literally dwell: + each segment between two tracked samples deposits the path time `|dt|` the tracker spent crossing + it, spread along its length, so the picture is of the *path* and not of the stepper — refine the + tolerances until the step count doubles and the image is unchanged. Colour runs with `log|t|`, + cool at the start system and white-hot into the endgame, so the frame reads as a flow. The hero + is the **Kuramoto model** of seven coupled oscillators (4096 paths, 124 equilibria); the teaching + frame is **Noonburg-6** (729 paths, 717 solutions), few enough that individual strands stay + legible. The generator carries a system registry (cyclic-n, Noonburg, Katsura, Kuramoto, + multisite phosphorylation, Stewart-Gough, dense random), a `--scout` contact sheet that renders + every candidate projection from a single solve, and a `--selftest` that asserts the exposure + conserves energy exactly. Both frames regenerate byte-identically in ~40s. + _______________________________________________________________________________ ## [3.4.0] - 2026-07-16 diff --git a/python/docs/source/showpieces/index.rst b/python/docs/source/showpieces/index.rst index d37af15c8..5da224934 100644 --- a/python/docs/source/showpieces/index.rst +++ b/python/docs/source/showpieces/index.rst @@ -17,3 +17,4 @@ figure, through ``tools/refresh_doc_artifacts.py`` so they stay current with the monodromy_loom/index flight_recorder/index + nebula/index diff --git a/python/docs/source/showpieces/nebula/index.rst b/python/docs/source/showpieces/nebula/index.rst new file mode 100644 index 000000000..eba6a5e65 --- /dev/null +++ b/python/docs/source/showpieces/nebula/index.rst @@ -0,0 +1,116 @@ +🌌 The Nebula +************* + +Every other picture of a solve shows you the **answers**. This one shows the **journey**: one +zero-dimensional solve, every path, every step, exposed onto a single frame like a long exposure of +a night sky. + +.. image:: nebula.png + :width: 92% + :align: center + :alt: thousands of violet filaments streaming between two blazing knots -- 4096 tracked paths of the Kuramoto model + +Above are all **4096 paths** of a genuine total-degree homotopy for the **Kuramoto model** of seven +coupled oscillators — the equations of *synchronisation*, the thing fireflies do when they flash +together and a power grid does when it locks to one frequency. Every filament is a tracked path +carrying its solution from the start system to an equilibrium; the two blazing knots are where +whole bundles of them arrive at once. The solve takes about half a minute and finds **124 +equilibria**. + +Nothing here is a texture, a gradient, or an artist's impression. Every photon is tracked data. + +Brightness is time +================== + +The exposure is meant literally:: + + brightness = time spent + +Between each pair of tracked samples, the tracker spent some amount of path time :math:`|\Delta t|`. +That much *light* is deposited along the segment joining them, spread evenly over its length. So a +path that crawls burns bright, and a path racing off to infinity leaves a faint comet streak — the +same amount of light, smeared over a much longer trail. + +This is why the knots glow. As :math:`t \to 0` the endgame closes in on a solution, the path slows +almost to a stop, and all of that dwell piles into a few pixels. You are seeing the solver arrive. + +The subtle part is that the light of a segment does **not** depend on how many samples the tracker +chose to leave on it. Weighting by *step count* would have drawn you a picture of the stepper; +weighting by *path time* draws the path. Refine the tolerances until the step count doubles and +the picture is unchanged. + +Colour is the clock +=================== + +Colour runs with :math:`\log |t|`: cool where each path starts at :math:`t=1`, through violet, to +white-hot as the endgame closes. So the frame also reads as a flow — you can see which way time +runs without a single arrow. + +A tempting alternative is worth naming as a trap. The obvious thing is to colour by +:math:`\arg \ell(x)`, the phase of the projection — but the *position* on screen already **is** +:math:`(\operatorname{Re}\ell, \operatorname{Im}\ell)`, so :math:`\arg \ell` is nothing but the +screen's polar angle: a pinwheel that says nothing. Temperature carries real information, and it +survives the additive blending that a hue wheel would turn to grey. + +Start simple: one path, one strand +================================== + +Four thousand paths is a cloud. To see what a cloud is *made of*, here is the same program on a +smaller system — Noonburg's neural-network equations for six neurons, **729 paths** and **717 +solutions**, in under three seconds: + +.. image:: nebula_teaching.png + :width: 88% + :align: center + :alt: two wing-like plumes of cyan and violet filaments, 729 tracked paths of the Noonburg system + +Now the strands separate, and the anatomy is plain. Each filament is **one path**. They fan out +of the start system, sweep through solution space, and converge — and where they converge they +brighten, because that is where they slow down. The violet feather-tips are stretches where paths +move fast and their light spreads thin; the cyan cores are where bundles of paths crowd into the +same place and arrive. + +Same code, same doctrine, 4096 paths instead of 729, and the strands merge into gas. + +What it took to be honest +========================= + +Three details are load-bearing, and each was measured rather than assumed: + +**The power-series endgame, not Cauchy.** The Cauchy endgame samples in *circles* around +:math:`t=0`, so :math:`|t|` is deliberately multi-sheeted and a path's data can jump back to the +endgame boundary — which the dwell weighting would draw as a full-frame streak carrying the largest +energy in the picture. The power-series endgame descends radially instead: measured over 5880 +paths, :math:`|t|` is monotone on **every** one, with zero jumps. (The +:doc:`Flight Recorder ` learned the same lesson the hard way.) + +**Serial tracking.** The observer that streams every step is Python, and Python observers +re-acquire the GIL on every event, so twelve threads *convoy*: measured, a threaded solve with the +collector attached is **7× slower** than a serial one. Threading only wins with no observer. + +**Sub-pixel segments.** Depositing :math:`|\Delta t| / L` per pixel is right until a segment lands +inside a single pixel and :math:`1/L` explodes. Spreading the same energy over +:math:`\max(1, \lceil L/\Delta \rceil)` subsamples is identical for long segments and simply puts +all the light in one pixel for short ones — no singularity, because the pixel is the sensor and +motion below it is not resolvable. + +The code carries a ``--selftest`` that asserts the exposure conserves energy exactly, and a +``--scout`` mode that renders a contact sheet of candidate projections from a single solve, since +tracking is expensive and re-projecting is free. + +.. literalinclude:: nebula.py + :language: python + :start-at: def track( + :end-at: return cache + +The gallery's three views of one idea +===================================== + +The showpieces trace the same machinery at three scales: + +* the :doc:`Monodromy Loom ` — **one loop, a few paths**, drawn + as individual braided strands; +* **the Nebula** — **one solve, every path**, every step; +* Homotopy Basins — **every solve, endpoints only**, a raster of parameter space. + +The Loom shows you a strand. The Nebula shows you the weather. diff --git a/tools/refresh_doc_artifacts.py b/tools/refresh_doc_artifacts.py index bd4419a39..a5d53c2de 100644 --- a/tools/refresh_doc_artifacts.py +++ b/tools/refresh_doc_artifacts.py @@ -120,6 +120,11 @@ def _plots(): SHOWPIECES / "flight_recorder" / "flight_recorder.py", ["flight_recorder.png", "flight_recorder_setup.png"], note="showpiece: raster PNG only; tight-tolerance mult-35 solve -> slow (~1 min)"), + Plot("nebula", + SHOWPIECES / "nebula" / "nebula.py", + ["nebula.png", "nebula_teaching.png"], + note="showpiece: raster PNG only (the image IS an accumulation buffer); two solves, " + "4825 paths, ~530k tracked samples -> ~40s"), ] From 64bd72c8ecd341d3b97aa6ab08b17e846fafea23 Mon Sep 17 00:00:00 2001 From: silviana amethyst Date: Fri, 17 Jul 2026 07:22:42 +0000 Subject: [PATCH 6/6] docs(showpieces): the teaching frame is NOT the noon-n benchmark -- name it correctly Caught by looking up sources instead of trusting memory. The teaching frame was described, in the generator and on the docs page, as "Noonburg's neural network". It is not. The noon-n benchmark is x_i * (sum_{j != i} x_j^2) - c*x_i + 1 = 0, c = 11/10 whereas the frame is x_i*(1 + sum_{j != i} x_j^2) - 1 = 0 -- same total degree, different system, and a misattribution of the picture. So both now exist and say what they are. noon_system(n, c=11/10) is the actual benchmark; noonburg_like_system(n) is the frame, documented as a SIBLING of that family, not a member: in noon's form it has the x_i coefficient of c = -1 but the opposite constant, and x -> -x carries it onto noon at c = -1, so their solution sets are mirror images. Their PATHS differ regardless, which is the part that matters to a picture of paths: a total-degree homotopy runs from a FIXED start system to the target, so reflecting the target does not reflect the journey. Measured: this system leaves 108k samples where noon at c=-1 leaves 25k, and they render quite differently (both were rendered and compared). Both real noon variants make worse pictures -- sparser, lopsided, clipped -- so the frame keeps its system and loses its wrong name. Frames re-render BYTE-IDENTICALLY: the equations never changed, only what we call them. --- .../docs/source/showpieces/nebula/index.rst | 5 ++- .../docs/source/showpieces/nebula/nebula.py | 44 +++++++++++++++++-- 2 files changed, 43 insertions(+), 6 deletions(-) diff --git a/python/docs/source/showpieces/nebula/index.rst b/python/docs/source/showpieces/nebula/index.rst index eba6a5e65..6c7cda1d8 100644 --- a/python/docs/source/showpieces/nebula/index.rst +++ b/python/docs/source/showpieces/nebula/index.rst @@ -56,8 +56,9 @@ Start simple: one path, one strand ================================== Four thousand paths is a cloud. To see what a cloud is *made of*, here is the same program on a -smaller system — Noonburg's neural-network equations for six neurons, **729 paths** and **717 -solutions**, in under three seconds: +smaller system — a Noonburg-style neural network on six neurons, +:math:`x_i(1 + \sum_{j \neq i} x_j^2) - 1 = 0`, giving **729 paths** and **717 solutions** in under +three seconds: .. image:: nebula_teaching.png :width: 88% diff --git a/python/docs/source/showpieces/nebula/nebula.py b/python/docs/source/showpieces/nebula/nebula.py index 8dfdba4a7..25dfcdc02 100644 --- a/python/docs/source/showpieces/nebula/nebula.py +++ b/python/docs/source/showpieces/nebula/nebula.py @@ -94,8 +94,43 @@ def cyclic_system(n): return sys_ -def noon_system(n): - """Noonburg's neural network: x_i (1 + sum_{j != i} x_j^2) - 1. 3^n paths, not symmetric.""" +def noon_system(n, c=None): + """The noon-n benchmark -- Noonburg's neural network, ``3**n`` paths:: + + x_i * (sum_{j != i} x_j^2) - c * x_i + 1 = 0, i = 1 .. n + + A Lotka-Volterra-style model of n interacting neurons (Noonburg, 1989), and a long-standing + benchmark family for polynomial system solvers; the standard member is ``c = 11/10``, which is + the default here. Exact rational, per the coercion doctrine. + """ + from fractions import Fraction + c = Fraction(11, 10) if c is None else Fraction(c) + x = [bertini.Variable('x' + str(i)) for i in range(n)] + sys_ = bertini.System() + sys_.add_variable_group(bertini.VariableGroup(x)) + cc = bertini.coefficient(c) + for i in range(n): + sys_.add_function(x[i] * np.sum([x[j]**2 for j in range(n) if j != i]) - cc * x[i] + 1) + return sys_ + + +def noonburg_like_system(n): + """A Noonburg-*style* neural network, ``3**n`` paths -- and NOT the noon-n benchmark:: + + x_i * (1 + sum_{j != i} x_j^2) - 1 = 0, i = 1 .. n + + Be precise about what this is, because it is the Nebula's teaching frame and it would be easy + to mis-cite. It is a sibling of :func:`noon_system`, not a member of it: writing noon's form as + ``x_i sum_{j != i} x_j^2 - c x_i + 1``, this has the ``x_i`` coefficient of ``c = -1`` but the + opposite constant. Substituting ``x -> -x`` does carry it onto noon at ``c = -1``, so the two + have mirror-image *solution sets*. + + Their **paths differ regardless**, which is what matters to a picture of paths: a total-degree + homotopy runs from a FIXED start system to the target, so reflecting the target does not reflect + the journey. Measured, this system leaves 108k tracked samples where noon at ``c = -1`` leaves + 25k, and the two render quite differently. This one fans into two broad wings; it was chosen by + looking. + """ x = [bertini.Variable('x' + str(i)) for i in range(n)] sys_ = bertini.System() sys_.add_variable_group(bertini.VariableGroup(x)) @@ -385,7 +420,8 @@ def monomials(d): 'cyclic5': (lambda: cyclic_system(5), 'cyclic-5', 120), 'cyclic6': (lambda: cyclic_system(6), 'cyclic-6', 720), 'cyclic7': (lambda: cyclic_system(7), 'cyclic-7', 5040), - 'noon6': (lambda: noon_system(6), 'noonburg-6', 729), + 'noon6': (lambda: noonburg_like_system(6), 'Noonburg-style, 6 neurons', 729), + 'noon6_benchmark': (lambda: noon_system(6), 'noon-6 benchmark (c = 11/10)', 729), 'katsura8': (lambda: katsura_system(8), 'katsura-8', 256), 'kuramoto5': (lambda: kuramoto_system(5), 'Kuramoto, 5 oscillators', 256), 'kuramoto6': (lambda: kuramoto_system(6), 'Kuramoto, 6 oscillators', 1024), @@ -898,7 +934,7 @@ def selftest(system_name='cyclic5'): # was chosen by looking at contact sheets, not by argument. def teaching_frame(out): - """Noonburg-6: 729 paths, 717 solutions, ~3s. The lesson, legible. + """A Noonburg-style network on 6 neurons: 729 paths, 717 solutions, ~3s. The lesson, legible. Few enough paths that individual strands stay separate, so you can see what the piece IS: each filament is one tracked path, fanning out of the start system and converging on a solution,

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