refactor: add common shaders helpers + fix shadow export

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

CLAUDE.md

@@ -0,0 +1,152 @@
+# CLAUDE.md
+
+This file provides guidance to Claude Code (claude.ai/code) when working with code in this repository.
+
+## What this is
+
+A deep-zoom fractal explorer (Rust + wgpu + egui + WGSL). It zooms past the
+~10¹³× limit of plain `f64` using **perturbation theory**: one high-precision
+reference orbit is computed on the CPU (arbitrary precision via `dashu-float`),
+and every pixel is rendered on the GPU as a cheap `f32` delta from it, with
+rebasing to avoid glitches. The `f32` GPU tier reaches roughly 10³⁰×. Runs
+natively (Vulkan/Metal/DX12) and in the browser (WebGPU only — WebGL2 can't do
+storage buffers, which the fragment shader needs for the reference orbit).
+
+## Commands
+
+```sh
+cargo run --release          # native, run (release matters: fractal math is hot)
+cargo test                   # reference-orbit math, share-link round-trip, WGSL validation
+cargo test --test shader_valid   # just the WGSL parse/validate tests (naga, no GPU needed)
+cargo clippy
+cargo fmt                    # rustfmt.toml just pins edition = "2024"
+```
+
+Web build (WebGPU):
+
+```sh
+rustup target add wasm32-unknown-unknown
+cargo install wasm-bindgen-cli --version 0.2.128   # must match the wasm-bindgen crate version
+./build-web.sh                                     # -> ./dist
+python3 -m http.server -d dist 8080
+```
+
+Native CLI flags (`src/cli.rs`, applied in `FractalApp::apply_cli`): `--kind`,
+`--power`, `--julia re,im`, `--share <fragment>`,
+`--view re,im,half_height[,iterations]`, `--de`, `--buddhabrot`,
+`--buddha-palette`, `--export` (+ `--export-path out.png`). `--headless`
+(`src/headless.rs`) skips the window entirely: it builds the same view from
+the other flags, creates its own offscreen wgpu device, and renders straight
+to a PNG (`--width`/`--height`, default 1920×1080) — implies `--export`'s
+save behavior without needing a GPU-backed window/event loop. Not yet
+supported with `--buddhabrot`. Run `mandelbrot --help` for the full list.
+
+There's no GPU in most sandboxes: `cargo check`/`cargo test --test shader_valid`
+are the fast, headless way to validate a change. `cargo test` also runs but
+doesn't touch the GPU — the reference-orbit tests are pure CPU math (see
+below), and `shader_valid` parses/validates WGSL with `naga` statically instead
+of creating a pipeline.
+
+## Architecture
+
+### The perturbation pipeline (the core mechanism, spans several files)
+
+For a pixel at parameter `c = C_ref + dc`, its orbit is written as
+`y_n = X_n + e_n`, where `X_n` is the (shared, high-precision) reference orbit
+and `e_n` is a small `f32` delta. Whenever `|y_n| < |e_n|` (or the reference
+runs out), rebase: `e ← y_n − X_0`, restart the reference index at 0. This is
+what makes deep zoom cheap — one expensive high-precision orbit, then every
+pixel is a handful of `f32` complex multiplies.
+
+- `src/view.rs` — `ViewState`; center is arbitrary-precision `FBig` (`Big`
+  type alias), pixel scale stays `f64` (still in-range at 10³⁰×). Precision
+  (bits) scales with zoom depth (`precision_for`).
+- `src/fractal/reference.rs` — `FractalKind` enum (Mandelbrot, Burning Ship,
+  Tricorn, Multibrot, Celtic, Perpendicular, Buffalo, Phoenix, Lambda) and
+  `compute_reference`/`compute_set_reference`: iterate the chosen formula at
+  high precision on the CPU, emitting `Z_n` as `f32` pairs — that's the
+  reference orbit the GPU perturbs from.
+- `src/shaders/*.wgsl` — none of these are standalone WGSL modules; WGSL has
+  no `#include`, so each is compiled by concatenating plain-text fragments
+  with `concat!`/`include_str!` at the `create_shader_module` call site (see
+  `renderer.rs`, `buddhabrot.rs`, and `tests/shader_valid.rs`, which must
+  concatenate the same pieces to validate what actually gets built).
+  `common.wgsl` (fullscreen-triangle vertex helper, `cmul`/`cpow`, `KIND_*`
+  constants) is prepended to every shader. `iterate_uniforms.wgsl` (the
+  perturbation-pipeline `Uniforms` struct + `palette()`) is additionally
+  prepended to `mandelbrot.wgsl` and `colorize.wgsl`, which share that layout.
+  Because there's no namespacing, a definition must live in exactly one file
+  among those concatenated together for a given shader — don't redefine a
+  `common.wgsl`/`iterate_uniforms.wgsl` symbol locally.
+- `src/shaders/mandelbrot.wgsl` — the perturbation fragment shader.
+  `advance_delta(z, e)` is the per-kind delta step (`z` = reference point,
+  `e` = current delta); the caller adds `step_add` (= `dc`) afterward — this
+  relies on `c` being additive in every current kind's formula (a kind where
+  it isn't, e.g. a rational map with `c` in a denominator, would need its own
+  step function that consumes `dc` internally instead, plus extra per-step
+  reference data since the orbit point alone wouldn't be enough to recover an
+  exact delta). `fprime(z)` is the derivative used for distance-estimation
+  (DE) shading; exact for holomorphic kinds, an approximation (`~2Z`) for the
+  abs-based ones. A `KIND_*` constant (from `common.wgsl`) must match the
+  matching `FractalKind` variant's discriminant exactly.
+- `src/fractal/renderer.rs` — `FractalRenderer` (wgpu pipelines, uniform +
+  storage buffers, bind groups), `Uniforms` (repr(C) layout that must match
+  the WGSL `Uniforms` struct field-for-field, including padding), and
+  `FractalCallback` (the `egui_wgpu::CallbackTrait` impl: `prepare()` uploads
+  changed buffers and decides whether to re-run the iterate pass, the cheap
+  colourise pass, or just blit the cached texture). Also `ExportRender`, a
+  self-contained tiled renderer used for PNG export off the UI thread.
+- `src/worker.rs` — native background thread for reference-orbit computation
+  (coalesces bursts of requests so a fast drag doesn't compute every
+  intermediate view). The wasm32 build computes inline instead (see the
+  `#[cfg(target_arch = "wasm32")]` branch in `app.rs::ensure_reference`) —
+  **any signature change to `compute_reference`/`compute_set_reference` or
+  `RefRequest`/`RefResult` must be applied to both call sites.**
+- `src/app.rs` — `FractalApp` (the egui app + all UI). Key methods:
+  `should_request`/`ensure_reference` (decide when the reference is stale and
+  dispatch/collect it), `make_uniforms` (assemble the per-frame `Uniforms`),
+  `tick_animations` (drives the "morph c/p/λ" and auto-zoom animations),
+  `default_view_for` (per-kind starting view). `KINDS`, `JULIA_PRESETS`, and
+  `SET_PRESETS` are sized as `[T; FractalKind::<last variant> as usize + 1]` —
+  adding a new `FractalKind` means bumping all three (and adding an empty
+  `&[]` slot to the two preset arrays if the kind has none).
+- `src/fractal/share.rs` — `ShareState`: encodes the full view (mode, kind,
+  full-precision decimal center, zoom, iterations, per-kind constants,
+  coloring) as a `#`-fragment URL for bookmarking/sharing deep-zoom locations.
+
+### Adding a new `FractalKind`
+
+Touches, in order: `reference.rs` (enum variant + CPU iteration formula, and a
+test comparing against a naive `f64` iteration), `common.wgsl` (matching
+`KIND_*` const), `mandelbrot.wgsl` (matching `advance_delta`/`fprime` arms),
+`buddhabrot.wgsl` (matching arm in `advance()`, if the kind makes sense as a
+Buddhabrot), `renderer.rs`
+`Uniforms` (only if the kind needs a new per-kind constant, e.g. Phoenix's
+`phoenix_p`), `share.rs` (encode/decode string tag), `app.rs` (`KINDS` label,
+`JULIA_PRESETS`/`SET_PRESETS` slot, `default_view_for` entry, and optionally a
+UI control for its constant + an animation toggle, following the
+Phoenix/Lambda pattern). If `c` doesn't enter the formula additively (e.g. a
+rational map with `c` in a denominator), the `advance_delta`/`step_add` split
+doesn't work — that needs its own step function plus extra per-step reference
+data uploaded in a second GPU buffer alongside the orbit.
+
+### Buddhabrot is a separate pipeline
+
+`src/fractal/buddhabrot.rs` + `src/shaders/buddhabrot.wgsl` implement the
+Monte-Carlo orbit-density histogram. It does **not** use the perturbation/
+reference-orbit machinery: a Buddhabrot sample's orbit scatters across the
+whole image rather than staying in one pixel, so it's plain `f32` iteration
+from the live view (no deep zoom) via a compute pass that accumulates into a
+histogram buffer, tone-mapped by a fragment pass every frame. Its own
+`KIND_*` iteration formulas in `advance()` must be kept in sync with
+`reference.rs` by hand (there's no shared code path).
+
+### Two-pass render + caching (`renderer.rs`)
+
+The interactive path splits iteration (expensive, perturbation) from
+colourising (cheap, palette remap) into separate offscreen textures, so
+palette/color-scale/offset tweaks skip re-iteration entirely (`geom_differs`
+vs `color_differs` in `renderer.rs` decide which pass reruns). While the user
+is actively panning/zooming, the app renders downscaled with AA off
+(`INTERACT_DOWNSCALE`) and snaps back to full resolution once input settles
+(`INTERACT_SETTLE`).

Cargo.toml

@@ -8,14 +8,16 @@ bytemuck = { version = "1.25.2", features = ["derive"] }
 dashu-float = "0.6.0"
 eframe = { version = "0.36.2", default-features = false, features = ["wgpu", "default_fonts", "x11", "wayland", "accesskit"] }
 egui = "0.36.2"
-futures-channel = { version = "0.3.34", default-features = false, features = ["alloc", "std"] }
 log = "0.4.34"
 png = "0.18.1"
 
 [target.'cfg(not(target_arch = "wasm32"))'.dependencies]
 env_logger = "0.11.11"
+clap = { version = "4.5.51", features = ["derive"] }
+pollster = "1.0.1"
 
 [target.'cfg(target_arch = "wasm32")'.dependencies]
+futures-channel = { version = "0.3.34", default-features = false, features = ["alloc", "std"] }
 console_error_panic_hook = "0.1.7"
 console_log = "1.1.0"
 js-sys = "0.3.105"
@@ -26,6 +28,8 @@ web-sys = { version = "0.3.105", features = ["Window", "Location", "Url", "UrlSe
 # Release: optimize hard (fractal math is hot).
 [profile.release]
 opt-level = 3
+# codegen-units = 1
+debug = true
 
 # Dev: keep our own crate debuggable, but optimize dependencies (dashu, wgpu,
 # egui) so the explorer is actually interactive during development.

build-web.sh

@@ -20,6 +20,7 @@ wasm-bindgen \
     target/wasm32-unknown-unknown/release/mandelbrot.wasm
 
 cp index.html "$OUT/index.html"
+cp favicon.ico "$OUT/favicon.ico"
 
 echo "==> done: $OUT/ (index.html, mandelbrot.js, mandelbrot_bg.wasm)"
 echo "    serve:  python3 -m http.server -d $OUT 8080"

src/cli.rs

@@ -0,0 +1,105 @@
+// Native command-line arguments. Currently mirrors the old `MANDEL_*` debug
+// env vars one-for-one; this is the foundation a future headless (no-window,
+// render-to-file) mode will build on.
+
+use clap::{Parser, ValueEnum};
+
+use crate::fractal::FractalKind;
+
+#[derive(Parser, Debug, Default)]
+#[command(name = "mandelbrot", about = "Deep-zoom fractal explorer", version)]
+pub struct Cli {
+    /// Fractal formula to render.
+    #[arg(long, value_enum)]
+    pub kind: Option<KindArg>,
+
+    /// Exponent for the Multibrot kind (z -> z^power + c), clamped to [2, 8].
+    #[arg(long)]
+    pub power: Option<u32>,
+
+    /// Complex exponent for the Complex Multibrot kind (z -> z^power + c).
+    #[arg(long, value_name = "RE,IM")]
+    pub complex_power: Option<String>,
+
+    /// Start in Julia mode with this seed constant.
+    #[arg(long, value_name = "RE,IM")]
+    pub julia: Option<String>,
+
+    /// Restore a view from a share-link fragment (the part after '#').
+    #[arg(long, value_name = "FRAGMENT")]
+    pub share: Option<String>,
+
+    /// Jump to a view on startup.
+    #[arg(long, value_name = "RE,IM,HALF_HEIGHT[,ITERATIONS]")]
+    pub view: Option<String>,
+
+    /// Enable distance-estimation shading.
+    #[arg(long)]
+    pub de: bool,
+
+    /// Switch to the Buddhabrot renderer.
+    #[arg(long)]
+    pub buddhabrot: bool,
+
+    /// Buddhabrot tonemap palette index.
+    #[arg(long, value_name = "INDEX")]
+    pub buddha_palette: Option<u32>,
+
+    /// Render a PNG export on startup.
+    #[arg(long)]
+    pub export: bool,
+
+    /// Output path for --export/--headless (default: fractal-<timestamp>.png).
+    #[arg(long, value_name = "PATH")]
+    pub export_path: Option<String>,
+
+    /// Run without opening a window: render the current view to a PNG and
+    /// exit. Combine with --kind/--julia/--share/--view etc. to pick what to
+    /// render. Not yet supported with --buddhabrot.
+    #[arg(long)]
+    pub headless: bool,
+
+    /// Output image width in pixels (--headless only).
+    #[arg(long, value_name = "PX", default_value_t = 1920)]
+    pub width: u32,
+
+    /// Output image height in pixels (--headless only).
+    #[arg(long, value_name = "PX", default_value_t = 1080)]
+    pub height: u32,
+}
+
+#[derive(Copy, Clone, Debug, ValueEnum)]
+pub enum KindArg {
+    Mandelbrot,
+    #[value(alias = "ship")]
+    BurningShip,
+    #[value(alias = "mandelbar")]
+    Tricorn,
+    #[value(alias = "multi")]
+    Multibrot,
+    Celtic,
+    #[value(alias = "perp")]
+    Perpendicular,
+    Buffalo,
+    Phoenix,
+    Lambda,
+    #[value(alias = "cmulti")]
+    ComplexMultibrot,
+}
+
+impl From<KindArg> for FractalKind {
+    fn from(k: KindArg) -> Self {
+        match k {
+            KindArg::Mandelbrot => FractalKind::Mandelbrot,
+            KindArg::BurningShip => FractalKind::BurningShip,
+            KindArg::Tricorn => FractalKind::Tricorn,
+            KindArg::Multibrot => FractalKind::Multibrot,
+            KindArg::Celtic => FractalKind::Celtic,
+            KindArg::Perpendicular => FractalKind::Perpendicular,
+            KindArg::Buffalo => FractalKind::Buffalo,
+            KindArg::Phoenix => FractalKind::Phoenix,
+            KindArg::Lambda => FractalKind::Lambda,
+            KindArg::ComplexMultibrot => FractalKind::ComplexMultibrot,
+        }
+    }
+}

src/fractal/buddhabrot.rs

@@ -0,0 +1,391 @@
+//! Buddhabrot / Nebulabrot rendering: a Monte-Carlo orbit-density histogram,
+//! accumulated progressively across frames by a compute pass and tone-mapped
+//! to colour by a fragment pass. See `shaders/buddhabrot.wgsl` for the "why"
+//! this is a separate pipeline from the escape-time perturbation renderer.
+
+use eframe::egui_wgpu::{self, wgpu};
+
+/// Random samples dispatched per accumulating frame. Chosen so a frame stays
+/// interactive on a modest GPU even when most samples run the full `b_cap`
+/// (e.g. the view sits entirely inside the set, so nothing escapes).
+const SAMPLES_PER_DISPATCH: u32 = 150_000;
+const WORKGROUP_SIZE: u32 = 64;
+
+/// GPU-side parameters for both the accumulate (compute) and tonemap
+/// (fragment) passes. Layout must match `Uniforms` in `buddhabrot.wgsl`.
+#[repr(C)]
+#[derive(Copy, Clone, PartialEq, bytemuck::Pod, bytemuck::Zeroable)]
+pub struct BuddhabrotUniforms {
+    pub center: [f32; 2],
+    pub half_height: f32,
+    pub aspect: f32,
+    pub phoenix_p: [f32; 2],
+    pub lambda_l: [f32; 2],
+    pub bailout_sq: f32,
+    /// Iteration formula (`FractalKind::shader_id`); `KIND_LAMBDA` samples z0
+    /// instead of c (see the shader's doc comment).
+    pub kind: u32,
+    /// Exponent for the Multibrot kind.
+    pub power: u32,
+    /// Nested escape-iteration caps (r_cap <= g_cap <= b_cap) that bucket an
+    /// orbit's points into the R/G/B histogram planes.
+    pub r_cap: u32,
+    pub g_cap: u32,
+    pub b_cap: u32,
+    /// RNG nonce, bumped every dispatch so each frame samples fresh points.
+    pub seed: u32,
+    pub samples_this_dispatch: u32,
+    /// Tonemap brightness multiplier (user-controlled).
+    pub exposure: f32,
+    pub width: u32,
+    pub height: u32,
+    /// Running total of samples accumulated into the current histogram
+    /// (across all dispatches since the last reset); normalizes brightness.
+    pub total_samples: f32,
+    /// Tonemap colour style: 0 = classic (R/G/B = raw caps), 1 = nebula
+    /// (yellow core, blue halo), 2 = grayscale. Display-only, like `exposure`
+    /// — excluded from `ContentKey` so changing it doesn't reset accumulation.
+    pub palette: u32,
+    /// Padding so `complex_power` (a vec2, 8-byte aligned in the shader)
+    /// starts on an 8-byte boundary.
+    pub _pad0: u32,
+    /// Complex exponent for the Complex Multibrot kind; ignored by other kinds.
+    pub complex_power: [f32; 2],
+}
+
+/// The subset of `BuddhabrotUniforms` that determines the *content* of the
+/// histogram (as opposed to `exposure`, a display-only rescale). A change in
+/// any of these invalidates the accumulated histogram.
+#[derive(Copy, Clone, PartialEq)]
+struct ContentKey {
+    center: [f32; 2],
+    half_height: f32,
+    aspect: f32,
+    phoenix_p: [f32; 2],
+    lambda_l: [f32; 2],
+    bailout_sq: f32,
+    kind: u32,
+    power: u32,
+    complex_power: [f32; 2],
+    r_cap: u32,
+    g_cap: u32,
+    b_cap: u32,
+}
+
+impl From<&BuddhabrotUniforms> for ContentKey {
+    fn from(u: &BuddhabrotUniforms) -> Self {
+        Self {
+            center: u.center,
+            half_height: u.half_height,
+            aspect: u.aspect,
+            phoenix_p: u.phoenix_p,
+            lambda_l: u.lambda_l,
+            bailout_sq: u.bailout_sq,
+            kind: u.kind,
+            power: u.power,
+            complex_power: u.complex_power,
+            r_cap: u.r_cap,
+            g_cap: u.g_cap,
+            b_cap: u.b_cap,
+        }
+    }
+}
+
+/// The histogram buffer and its two bind groups, sized to the widget.
+struct Histogram {
+    buffer: wgpu::Buffer,
+    compute_bind_group: wgpu::BindGroup,
+    tonemap_bind_group: wgpu::BindGroup,
+    width: u32,
+    height: u32,
+}
+
+pub struct BuddhabrotRenderer {
+    compute_pipeline: wgpu::ComputePipeline,
+    compute_bind_group_layout: wgpu::BindGroupLayout,
+    tonemap_pipeline: wgpu::RenderPipeline,
+    tonemap_bind_group_layout: wgpu::BindGroupLayout,
+    uniform_buffer: wgpu::Buffer,
+    histogram: Option<Histogram>,
+    /// What the current histogram's content was last accumulated for; a
+    /// mismatch clears the histogram and restarts accumulation.
+    last_content: Option<ContentKey>,
+    /// Running sample count since the last reset (mirrors what was written
+    /// into `total_samples`, since the callback doesn't own that state).
+    total_samples: f32,
+    seed: u32,
+}
+
+impl BuddhabrotRenderer {
+    pub fn new(device: &wgpu::Device, target_format: wgpu::TextureFormat) -> Self {
+        let shader = device.create_shader_module(wgpu::ShaderModuleDescriptor {
+            label: Some("buddhabrot"),
+            source: wgpu::ShaderSource::Wgsl(
+                concat!(
+                    include_str!("../shaders/common.wgsl"),
+                    include_str!("../shaders/buddhabrot.wgsl"),
+                )
+                .into(),
+            ),
+        });
+
+        let uniform_buffer = device.create_buffer(&wgpu::BufferDescriptor {
+            label: Some("buddhabrot uniforms"),
+            size: std::mem::size_of::<BuddhabrotUniforms>() as u64,
+            usage: wgpu::BufferUsages::UNIFORM | wgpu::BufferUsages::COPY_DST,
+            mapped_at_creation: false,
+        });
+
+        let compute_bind_group_layout =
+            device.create_bind_group_layout(&wgpu::BindGroupLayoutDescriptor {
+                label: Some("buddhabrot compute bind group layout"),
+                entries: &[
+                    wgpu::BindGroupLayoutEntry {
+                        binding: 0,
+                        visibility: wgpu::ShaderStages::COMPUTE,
+                        ty: wgpu::BindingType::Buffer {
+                            ty: wgpu::BufferBindingType::Uniform,
+                            has_dynamic_offset: false,
+                            min_binding_size: None,
+                        },
+                        count: None,
+                    },
+                    wgpu::BindGroupLayoutEntry {
+                        binding: 1,
+                        visibility: wgpu::ShaderStages::COMPUTE,
+                        ty: wgpu::BindingType::Buffer {
+                            ty: wgpu::BufferBindingType::Storage { read_only: false },
+                            has_dynamic_offset: false,
+                            min_binding_size: None,
+                        },
+                        count: None,
+                    },
+                ],
+            });
+        let compute_pipeline_layout =
+            device.create_pipeline_layout(&wgpu::PipelineLayoutDescriptor {
+                label: Some("buddhabrot compute pipeline layout"),
+                bind_group_layouts: &[Some(&compute_bind_group_layout)],
+                immediate_size: 0,
+            });
+        let compute_pipeline = device.create_compute_pipeline(&wgpu::ComputePipelineDescriptor {
+            label: Some("buddhabrot compute pipeline"),
+            layout: Some(&compute_pipeline_layout),
+            module: &shader,
+            entry_point: Some("cs_main"),
+            compilation_options: Default::default(),
+            cache: None,
+        });
+
+        let tonemap_bind_group_layout =
+            device.create_bind_group_layout(&wgpu::BindGroupLayoutDescriptor {
+                label: Some("buddhabrot tonemap bind group layout"),
+                entries: &[
+                    wgpu::BindGroupLayoutEntry {
+                        binding: 0,
+                        visibility: wgpu::ShaderStages::FRAGMENT,
+                        ty: wgpu::BindingType::Buffer {
+                            ty: wgpu::BufferBindingType::Uniform,
+                            has_dynamic_offset: false,
+                            min_binding_size: None,
+                        },
+                        count: None,
+                    },
+                    wgpu::BindGroupLayoutEntry {
+                        binding: 2,
+                        visibility: wgpu::ShaderStages::FRAGMENT,
+                        ty: wgpu::BindingType::Buffer {
+                            ty: wgpu::BufferBindingType::Storage { read_only: true },
+                            has_dynamic_offset: false,
+                            min_binding_size: None,
+                        },
+                        count: None,
+                    },
+                ],
+            });
+        let tonemap_pipeline_layout =
+            device.create_pipeline_layout(&wgpu::PipelineLayoutDescriptor {
+                label: Some("buddhabrot tonemap pipeline layout"),
+                bind_group_layouts: &[Some(&tonemap_bind_group_layout)],
+                immediate_size: 0,
+            });
+        let tonemap_pipeline = device.create_render_pipeline(&wgpu::RenderPipelineDescriptor {
+            label: Some("buddhabrot tonemap pipeline"),
+            layout: Some(&tonemap_pipeline_layout),
+            vertex: wgpu::VertexState {
+                module: &shader,
+                entry_point: Some("vs_main"),
+                buffers: &[],
+                compilation_options: Default::default(),
+            },
+            fragment: Some(wgpu::FragmentState {
+                module: &shader,
+                entry_point: Some("fs_tonemap"),
+                targets: &[Some(wgpu::ColorTargetState {
+                    format: target_format,
+                    blend: None,
+                    write_mask: wgpu::ColorWrites::ALL,
+                })],
+                compilation_options: Default::default(),
+            }),
+            primitive: wgpu::PrimitiveState::default(),
+            depth_stencil: None,
+            multisample: wgpu::MultisampleState::default(),
+            multiview_mask: None,
+            cache: None,
+        });
+
+        Self {
+            compute_pipeline,
+            compute_bind_group_layout,
+            tonemap_pipeline,
+            tonemap_bind_group_layout,
+            uniform_buffer,
+            histogram: None,
+            last_content: None,
+            total_samples: 0.0,
+            seed: 0,
+        }
+    }
+
+    /// Ensure the histogram buffer exists at `width`×`height`, recreating (and
+    /// resetting accumulation) on a size change.
+    fn ensure_histogram(&mut self, device: &wgpu::Device, width: u32, height: u32) {
+        if let Some(h) = &self.histogram
+            && h.width == width
+            && h.height == height
+        {
+            return;
+        }
+
+        let plane = (width as u64) * (height as u64);
+        let buffer = device.create_buffer(&wgpu::BufferDescriptor {
+            label: Some("buddhabrot histogram"),
+            size: plane * 3 * std::mem::size_of::<u32>() as u64,
+            usage: wgpu::BufferUsages::STORAGE | wgpu::BufferUsages::COPY_DST,
+            mapped_at_creation: false,
+        });
+
+        let compute_bind_group = device.create_bind_group(&wgpu::BindGroupDescriptor {
+            label: Some("buddhabrot compute bind group"),
+            layout: &self.compute_bind_group_layout,
+            entries: &[
+                wgpu::BindGroupEntry {
+                    binding: 0,
+                    resource: self.uniform_buffer.as_entire_binding(),
+                },
+                wgpu::BindGroupEntry {
+                    binding: 1,
+                    resource: buffer.as_entire_binding(),
+                },
+            ],
+        });
+        let tonemap_bind_group = device.create_bind_group(&wgpu::BindGroupDescriptor {
+            label: Some("buddhabrot tonemap bind group"),
+            layout: &self.tonemap_bind_group_layout,
+            entries: &[
+                wgpu::BindGroupEntry {
+                    binding: 0,
+                    resource: self.uniform_buffer.as_entire_binding(),
+                },
+                wgpu::BindGroupEntry {
+                    binding: 2,
+                    resource: buffer.as_entire_binding(),
+                },
+            ],
+        });
+
+        self.histogram = Some(Histogram {
+            buffer,
+            compute_bind_group,
+            tonemap_bind_group,
+            width,
+            height,
+        });
+        // New (zero-initialized) buffer: accumulation starts fresh.
+        self.last_content = None;
+        self.total_samples = 0.0;
+    }
+}
+
+/// Per-frame paint callback. `accumulate` controls whether a new batch of
+/// samples is dispatched this frame (a content change always forces one
+/// dispatch regardless, so a parameter/view change is never left blank).
+pub struct BuddhabrotCallback {
+    pub uniforms: BuddhabrotUniforms,
+    pub accumulate: bool,
+    /// Widget size in physical pixels — the histogram resolution.
+    pub size_px: [u32; 2],
+}
+
+impl egui_wgpu::CallbackTrait for BuddhabrotCallback {
+    fn prepare(
+        &self,
+        device: &wgpu::Device,
+        queue: &wgpu::Queue,
+        _screen_descriptor: &egui_wgpu::ScreenDescriptor,
+        egui_encoder: &mut wgpu::CommandEncoder,
+        resources: &mut egui_wgpu::CallbackResources,
+    ) -> Vec<wgpu::CommandBuffer> {
+        let Some(renderer) = resources.get_mut::<BuddhabrotRenderer>() else {
+            return Vec::new();
+        };
+
+        let width = self.size_px[0].max(1);
+        let height = self.size_px[1].max(1);
+        renderer.ensure_histogram(device, width, height);
+
+        let content = ContentKey::from(&self.uniforms);
+        let content_changed = renderer.last_content != Some(content);
+        let should_dispatch = content_changed || self.accumulate;
+
+        if let Some(histogram) = &renderer.histogram {
+            if content_changed {
+                egui_encoder.clear_buffer(&histogram.buffer, 0, None);
+                renderer.total_samples = 0.0;
+                renderer.last_content = Some(content);
+            }
+
+            let mut uniforms = self.uniforms;
+            uniforms.width = width;
+            uniforms.height = height;
+            if should_dispatch {
+                renderer.seed = renderer.seed.wrapping_add(1);
+                renderer.total_samples += SAMPLES_PER_DISPATCH as f32;
+                uniforms.seed = renderer.seed;
+                uniforms.samples_this_dispatch = SAMPLES_PER_DISPATCH;
+            } else {
+                uniforms.samples_this_dispatch = 0;
+            }
+            uniforms.total_samples = renderer.total_samples;
+            queue.write_buffer(&renderer.uniform_buffer, 0, bytemuck::bytes_of(&uniforms));
+
+            if should_dispatch {
+                let mut pass = egui_encoder.begin_compute_pass(&wgpu::ComputePassDescriptor {
+                    label: Some("buddhabrot accumulate pass"),
+                    timestamp_writes: None,
+                });
+                pass.set_pipeline(&renderer.compute_pipeline);
+                pass.set_bind_group(0, &histogram.compute_bind_group, &[]);
+                let workgroups = SAMPLES_PER_DISPATCH.div_ceil(WORKGROUP_SIZE);
+                pass.dispatch_workgroups(workgroups, 1, 1);
+            }
+        }
+        Vec::new()
+    }
+
+    fn paint(
+        &self,
+        _info: egui::PaintCallbackInfo,
+        render_pass: &mut wgpu::RenderPass<'static>,
+        resources: &egui_wgpu::CallbackResources,
+    ) {
+        if let Some(renderer) = resources.get::<BuddhabrotRenderer>()
+            && let Some(histogram) = &renderer.histogram
+        {
+            render_pass.set_pipeline(&renderer.tonemap_pipeline);
+            render_pass.set_bind_group(0, &histogram.tonemap_bind_group, &[]);
+            render_pass.draw(0..3, 0..1);
+        }
+    }
+}

src/fractal/mod.rs

@@ -1,13 +1,16 @@
 //! GPU fractal rendering: wgpu pipeline, uniforms, reference orbit, and the
 //! egui paint callback.
 
+pub mod buddhabrot;
 pub mod reference;
 pub mod renderer;
 pub mod share;
 
+pub use buddhabrot::{BuddhabrotCallback, BuddhabrotRenderer, BuddhabrotUniforms};
 pub use reference::{FractalKind, compute_reference, compute_set_reference};
-pub use renderer::{
-    ExportRender, FractalCallback, FractalRenderer, MAX_REF_POINTS, Uniforms,
-    encode_png_with_progress,
-};
+#[cfg(target_arch = "wasm32")]
+pub use renderer::encode_png_with_progress;
+#[cfg(not(target_arch = "wasm32"))]
+pub use renderer::export_to_png_blocking;
+pub use renderer::{ExportRender, FractalCallback, FractalRenderer, MAX_REF_POINTS, Uniforms};
 pub use share::ShareState;

src/fractal/reference.rs

@@ -10,30 +10,53 @@
 //! * Mandelbrot-set: `z0 = 0`, `c = view center` (the c-plane point per pixel).
 //! * Julia-set:      `z0 = view center`, `c = fractal constant` (fixed per view).
 
-use crate::view::Big;
+use crate::view::{Big, big_from_f64};
 
 /// The iteration formula. Must be kept in sync with `advance_delta` and the
 /// `KIND_*` constants in the shader.
+#[repr(u8)]
 #[derive(Clone, Copy, PartialEq, Eq, Debug)]
 pub enum FractalKind {
     /// `z -> z^2 + c`.
-    Mandelbrot,
+    Mandelbrot = 0,
     /// `z -> (|Re z| + i|Im z|)^2 + c`.
-    BurningShip,
+    BurningShip = 1,
     /// `z -> conj(z)^2 + c` (the Mandelbar).
-    Tricorn,
+    Tricorn = 2,
     /// `z -> z^power + c` (power >= 2).
-    Multibrot,
+    Multibrot = 3,
+    /// `z -> |Re(z^2)| + i·Im(z^2) + c` (abs on the real output of the square).
+    Celtic = 4,
+    /// `z -> (x^2 - y^2) - 2·x·|y|·i + c` (abs on the imaginary input).
+    Perpendicular = 5,
+    /// `z -> |Re(z^2)| - |Im(z^2)|·i + c` (abs on both outputs).
+    Buffalo = 6,
+    /// `z -> z^2 + c + p·z_{n-1}` (two-term recurrence; `p` is `phoenix_p`).
+    Phoenix = 7,
+    /// `z -> lambda·z(1 - z)` (logistic map).
+    Lambda = 8,
+    /// `z -> z^power + c`, where `power` is a complex constant (the
+    /// `complex_power` argument), via the principal branch `z^p = exp(p·ln z)`.
+    ComplexMultibrot = 9,
 }
 
 impl FractalKind {
-    /// Integer id matching the shader's `KIND_*` constants.
-    pub fn shader_id(self) -> u32 {
+    pub fn description(&self) -> &str {
         match self {
-            FractalKind::Mandelbrot => 0,
-            FractalKind::BurningShip => 1,
-            FractalKind::Tricorn => 2,
-            FractalKind::Multibrot => 3,
+            FractalKind::Mandelbrot => {
+                "The Mandelbrot set is the most famous fractal set, obtained with the simplest escape-time formula. This set represents all Julia fractals: each points of the Mandelbrot set is related to a specific Julia fractal."
+            }
+            FractalKind::BurningShip => {
+                "A variation of the famous Mandelbrot set, using absolute values on the real and imaginary part of each iterations."
+            }
+            FractalKind::Tricorn => "The Tricorn set is obtained using the same formula as the Mandelbrot set, taking the complex conjugate of the previous iteration.",
+            FractalKind::Multibrot => "Multibrot use the same formula as the Mandelbrot set, with a bigger exposant.",
+            FractalKind::Celtic => "",
+            FractalKind::Perpendicular => "",
+            FractalKind::Buffalo => "",
+            FractalKind::Phoenix => "",
+            FractalKind::Lambda => "",
+            FractalKind::ComplexMultibrot => "Like Multibrot, but the exponent itself is a complex number instead of a plain integer, via z^p = exp(p·ln z).",
         }
     }
 }
@@ -46,6 +69,7 @@ const REFERENCE_ESCAPE_SQ: f64 = 1.0e10;
 /// Compute the reference orbit `Z_0..Z_{len-1}` where `Z_0 = z0` and
 /// `Z_{n+1} = f(Z_n, c)` for the given `kind` (and `power`, for Multibrot), up
 /// to `max_iter` steps at `precision` bits. Each entry is `[re, im]` in f32.
+#[allow(clippy::too_many_arguments)]
 pub fn compute_reference(
     z0_re: &Big,
     z0_im: &Big,
@@ -55,12 +79,27 @@ pub fn compute_reference(
     precision: usize,
     kind: FractalKind,
     power: u32,
+    phoenix_p: (f64, f64),
+    lambda_l: (f64, f64),
+    complex_power: (f64, f64),
 ) -> Vec<[f32; 2]> {
     let cr = c_re.clone().with_precision(precision).value();
     let ci = c_im.clone().with_precision(precision).value();
 
     let mut zr = z0_re.clone().with_precision(precision).value();
     let mut zi = z0_im.clone().with_precision(precision).value();
+    // Previous iterate, for the Phoenix two-term recurrence (Y_{-1} = 0).
+    let mut zr_prev = big_zero(precision);
+    let mut zi_prev = big_zero(precision);
+    // Phoenix distortion constant `p` (a small fixed complex number).
+    let pr = big_from_f64(phoenix_p.0, precision);
+    let pi = big_from_f64(phoenix_p.1, precision);
+    // Lambda distortion constant `l` (a small fixed complex number).
+    let lr = big_from_f64(lambda_l.0, precision);
+    let li = big_from_f64(lambda_l.1, precision);
+    // Complex Multibrot exponent (a fixed complex number).
+    let cpow_re = big_from_f64(complex_power.0, precision);
+    let cpow_im = big_from_f64(complex_power.1, precision);
 
     let mut points: Vec<[f32; 2]> = Vec::with_capacity(max_iter as usize + 1);
 
@@ -97,8 +136,49 @@ pub fn compute_reference(
                 let (pr, pi) = complex_pow(&zr, &zi, power.max(2), precision);
                 (pr + &cr, pi + &ci)
             }
+            FractalKind::Celtic => {
+                // |Re(z^2)| + i·Im(z^2): abs the real output of the square.
+                let re = big_abs(&zr.sqr() - &zi.sqr()) + &cr;
+                let im = ((&zr * &zi) << 1) + &ci;
+                (re, im)
+            }
+            FractalKind::Perpendicular => {
+                // (x^2 - y^2) - 2·x·|y| i: abs the imaginary input.
+                let re = &zr.sqr() - &zi.sqr() + &cr;
+                let im = &ci - ((&zr * &big_abs(zi.clone())) << 1);
+                (re, im)
+            }
+            FractalKind::Buffalo => {
+                // |Re(z^2)| - |Im(z^2)| i: abs both outputs.
+                let re = big_abs(&zr.sqr() - &zi.sqr()) + &cr;
+                let im = &ci - big_abs((&zr * &zi) << 1);
+                (re, im)
+            }
+            FractalKind::Phoenix => {
+                // z^2 + c + p·z_{n-1}.
+                let re2 = &zr.sqr() - &zi.sqr();
+                let im2 = (&zr * &zi) << 1;
+                let pzr = &pr * &zr_prev - &pi * &zi_prev;
+                let pzi = &pr * &zi_prev + &pi * &zr_prev;
+                (re2 + &cr + pzr, im2 + &ci + pzi)
+            }
+            FractalKind::Lambda => {
+                // λ·z(1 - z): logistic map.
+                let re2 = 1 - &zr;
+                let im2 = -&zi;
+                let lzr = &lr * &zr - &li * &zi;
+                let lzi = &lr * &zi + &li * &zr;
+                (&lzr * &re2 - &lzi * &im2, re2 * lzi + lzr * im2)
+            }
+            FractalKind::ComplexMultibrot => {
+                let (pr, pi) = complex_pow_complex(&zr, &zi, &cpow_re, &cpow_im, precision);
+                (pr + &cr, pi + &ci)
+            }
         };
 
+        // Shift the previous iterate (only the Phoenix arm reads it).
+        zr_prev = zr;
+        zi_prev = zi;
         zr = new_zr.with_precision(precision).value();
         zi = new_zi.with_precision(precision).value();
     }
@@ -130,8 +210,35 @@ fn complex_pow(zr: &Big, zi: &Big, power: u32, precision: usize) -> (Big, Big) {
     (rr, ri)
 }
 
+/// `true` if `x` is (numerically) zero. The f64 check is exact for a true
+/// zero; only matters here to special-case `ln(0)`.
+fn is_big_zero(x: &Big) -> bool {
+    x.to_f64().value() == 0.0
+}
+
+/// `(zr + i zi)^(pr + i pi)` for a complex exponent, via the principal branch
+/// `z^p = exp(p·ln z)` where `ln z = ln|z| + i·arg(z)`. Used by
+/// `ComplexMultibrot`; must be kept in sync with the shader's `cpow`.
+/// `z = 0` is special-cased to `0` (the formula's `ln(0)` would otherwise
+/// panic; this is the correct limit for the `Re(p) > 0` region the UI
+/// exposes).
+fn complex_pow_complex(zr: &Big, zi: &Big, pr: &Big, pi: &Big, precision: usize) -> (Big, Big) {
+    if is_big_zero(zr) && is_big_zero(zi) {
+        return (big_zero(precision), big_zero(precision));
+    }
+    let r2 = &zr.sqr() + &zi.sqr();
+    let ln_r = r2.ln() >> 1; // 0.5 * ln(r2) = ln(sqrt(r2)); exact halving.
+    let theta = zi.atan2(zr);
+    let exp_re = (pr * &ln_r - pi * &theta).with_precision(precision).value();
+    let exp_im = (pr * &theta + pi * &ln_r).with_precision(precision).value();
+    let mag = exp_re.exp();
+    let (sin_a, cos_a) = exp_im.sin_cos();
+    (&mag * &cos_a, &mag * &sin_a)
+}
+
 /// Convenience: parameter-plane ("Mandelbrot-set") reference (`z0 = 0`,
 /// `c = center`) for any `kind`.
+#[allow(clippy::too_many_arguments)]
 pub fn compute_set_reference(
     center_re: &Big,
     center_im: &Big,
@@ -139,10 +246,23 @@ pub fn compute_set_reference(
     precision: usize,
     kind: FractalKind,
     power: u32,
+    phoenix_p: (f64, f64),
+    lambda_l: (f64, f64),
+    complex_power: (f64, f64),
 ) -> Vec<[f32; 2]> {
     let zero = big_zero(precision);
     compute_reference(
-        &zero, &zero, center_re, center_im, max_iter, precision, kind, power,
+        &zero,
+        &zero,
+        center_re,
+        center_im,
+        max_iter,
+        precision,
+        kind,
+        power,
+        phoenix_p,
+        lambda_l,
+        complex_power,
     )
 }
 
@@ -156,7 +276,17 @@ mod tests {
     fn reference_matches_naive_f64() {
         let cr = Big::try_from(-0.75_f64).unwrap();
         let ci = Big::try_from(0.1_f64).unwrap();
-        let points = compute_set_reference(&cr, &ci, 60, 200, FractalKind::Mandelbrot, 2);
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Mandelbrot,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
 
         // Independent naive f64 orbit.
         let (c_re, c_im) = (-0.75_f64, 0.1_f64);
@@ -166,8 +296,14 @@ mod tests {
             // significant figures.
             let tol_re = 1e-4 * (1.0 + zr.abs());
             let tol_im = 1e-4 * (1.0 + zi.abs());
-            assert!((point[0] as f64 - zr).abs() < tol_re, "re mismatch: {point:?} vs {zr}");
-            assert!((point[1] as f64 - zi).abs() < tol_im, "im mismatch: {point:?} vs {zi}");
+            assert!(
+                (point[0] as f64 - zr).abs() < tol_re,
+                "re mismatch: {point:?} vs {zr}"
+            );
+            assert!(
+                (point[1] as f64 - zi).abs() < tol_im,
+                "im mismatch: {point:?} vs {zi}"
+            );
             let nzr = zr * zr - zi * zi + c_re;
             let nzi = 2.0 * zr * zi + c_im;
             zr = nzr;
@@ -180,7 +316,17 @@ mod tests {
     fn interior_orbit_runs_full_length() {
         let cr = Big::try_from(-0.2_f64).unwrap();
         let ci = Big::try_from(0.0_f64).unwrap();
-        let points = compute_set_reference(&cr, &ci, 500, 120, FractalKind::Mandelbrot, 2);
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            500,
+            120,
+            FractalKind::Mandelbrot,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
         assert_eq!(points.len(), 501, "interior orbit should not escape");
     }
 
@@ -189,7 +335,17 @@ mod tests {
     fn burning_ship_reference_matches_naive_f64() {
         let cr = Big::try_from(-1.75_f64).unwrap();
         let ci = Big::try_from(-0.03_f64).unwrap();
-        let points = compute_set_reference(&cr, &ci, 60, 200, FractalKind::BurningShip, 2);
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::BurningShip,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
 
         let (c_re, c_im) = (-1.75_f64, -0.03_f64);
         let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
@@ -209,7 +365,17 @@ mod tests {
     fn multibrot3_reference_matches_naive_f64() {
         let cr = Big::try_from(0.3_f64).unwrap();
         let ci = Big::try_from(0.2_f64).unwrap();
-        let points = compute_set_reference(&cr, &ci, 60, 200, FractalKind::Multibrot, 3);
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Multibrot,
+            3,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
 
         let (c_re, c_im) = (0.3_f64, 0.2_f64);
         let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
@@ -242,6 +408,9 @@ mod tests {
             200,
             FractalKind::Mandelbrot,
             2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (mut zr, mut zi) = (0.15_f64, -0.1_f64);
@@ -256,4 +425,180 @@ mod tests {
             zi = nzi;
         }
     }
+
+    /// Celtic reference matches a naive f64 iteration: real = |x^2 - y^2| + cr.
+    #[test]
+    fn celtic_reference_matches_naive_f64() {
+        let cr = Big::try_from(-0.6_f64).unwrap();
+        let ci = Big::try_from(0.4_f64).unwrap();
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Celtic,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
+
+        let (c_re, c_im) = (-0.6_f64, 0.4_f64);
+        let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
+        for point in &points {
+            let tol = 1e-4 * (1.0 + zr.abs().max(zi.abs()));
+            assert!((point[0] as f64 - zr).abs() < tol, "re: {point:?} vs {zr}");
+            assert!((point[1] as f64 - zi).abs() < tol, "im: {point:?} vs {zi}");
+            let nzr = (zr * zr - zi * zi).abs() + c_re;
+            let nzi = 2.0 * zr * zi + c_im;
+            zr = nzr;
+            zi = nzi;
+        }
+    }
+
+    /// Perpendicular reference matches a naive f64 iteration:
+    /// real = x^2 - y^2 + cr, imag = -2·x·|y| + ci.
+    #[test]
+    fn perpendicular_reference_matches_naive_f64() {
+        let cr = Big::try_from(-0.7_f64).unwrap();
+        let ci = Big::try_from(-0.2_f64).unwrap();
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Perpendicular,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
+
+        let (c_re, c_im) = (-0.7_f64, -0.2_f64);
+        let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
+        for point in &points {
+            let tol = 1e-4 * (1.0 + zr.abs().max(zi.abs()));
+            assert!((point[0] as f64 - zr).abs() < tol, "re: {point:?} vs {zr}");
+            assert!((point[1] as f64 - zi).abs() < tol, "im: {point:?} vs {zi}");
+            let nzr = zr * zr - zi * zi + c_re;
+            let nzi = -2.0 * zr * zi.abs() + c_im;
+            zr = nzr;
+            zi = nzi;
+        }
+    }
+
+    /// Buffalo reference matches a naive f64 iteration:
+    /// real = |x^2 - y^2| + cr, imag = -|2·x·y| + ci.
+    #[test]
+    fn buffalo_reference_matches_naive_f64() {
+        let cr = Big::try_from(-1.2_f64).unwrap();
+        let ci = Big::try_from(-0.35_f64).unwrap();
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Buffalo,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
+
+        let (c_re, c_im) = (-1.2_f64, -0.35_f64);
+        let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
+        for point in &points {
+            let tol = 1e-4 * (1.0 + zr.abs().max(zi.abs()));
+            assert!((point[0] as f64 - zr).abs() < tol, "re: {point:?} vs {zr}");
+            assert!((point[1] as f64 - zi).abs() < tol, "im: {point:?} vs {zi}");
+            let nzr = (zr * zr - zi * zi).abs() + c_re;
+            let nzi = -(2.0 * zr * zi).abs() + c_im;
+            zr = nzr;
+            zi = nzi;
+        }
+    }
+
+    /// Phoenix reference matches a naive f64 two-term iteration
+    /// `z_{n+1} = z_n^2 + c + p·z_{n-1}` (z_0 = 0, z_{-1} = 0).
+    #[test]
+    fn phoenix_reference_matches_naive_f64() {
+        let cr = Big::try_from(0.5667_f64).unwrap();
+        let ci = Big::try_from(0.0_f64).unwrap();
+        let p = (-0.5_f64, 0.0_f64);
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Phoenix,
+            2,
+            p,
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
+
+        let (c_re, c_im) = (0.5667_f64, 0.0_f64);
+        let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
+        let (mut pr, mut pi) = (0.0_f64, 0.0_f64); // previous iterate
+        for point in &points {
+            let tol = 1e-4 * (1.0 + zr.abs().max(zi.abs()));
+            assert!((point[0] as f64 - zr).abs() < tol, "re: {point:?} vs {zr}");
+            assert!((point[1] as f64 - zi).abs() < tol, "im: {point:?} vs {zi}");
+            // p·z_{n-1} = (p.0 + i p.1)(pr + i pi).
+            let pzr = p.0 * pr - p.1 * pi;
+            let pzi = p.0 * pi + p.1 * pr;
+            let nzr = zr * zr - zi * zi + c_re + pzr;
+            let nzi = 2.0 * zr * zi + c_im + pzi;
+            pr = zr;
+            pi = zi;
+            zr = nzr;
+            zi = nzi;
+        }
+    }
+
+    /// Complex Multibrot (power 2.5 + 0.3i) reference matches a naive f64
+    /// iteration of `z^p = exp(p·ln z)`.
+    #[test]
+    fn complex_multibrot_reference_matches_naive_f64() {
+        let cr = Big::try_from(0.1_f64).unwrap();
+        let ci = Big::try_from(-0.2_f64).unwrap();
+        let power = (2.5_f64, 0.3_f64);
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::ComplexMultibrot,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            power,
+        );
+
+        // Naive f64 complex power via z^p = exp(p * ln z), ln z = ln|z| + i*arg(z).
+        fn naive_cpow(zr: f64, zi: f64, pr: f64, pi: f64) -> (f64, f64) {
+            if zr == 0.0 && zi == 0.0 {
+                return (0.0, 0.0);
+            }
+            let ln_r = 0.5 * (zr * zr + zi * zi).ln();
+            let theta = zi.atan2(zr);
+            let exp_re = pr * ln_r - pi * theta;
+            let exp_im = pr * theta + pi * ln_r;
+            let mag = exp_re.exp();
+            (mag * exp_im.cos(), mag * exp_im.sin())
+        }
+
+        let (c_re, c_im) = (0.1_f64, -0.2_f64);
+        let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
+        for point in &points {
+            let tol = 1e-4 * (1.0 + zr.abs().max(zi.abs()));
+            assert!((point[0] as f64 - zr).abs() < tol, "re: {point:?} vs {zr}");
+            assert!((point[1] as f64 - zi).abs() < tol, "im: {point:?} vs {zi}");
+            let (pr, pi) = naive_cpow(zr, zi, power.0, power.1);
+            let nzr = pr + c_re;
+            let nzi = pi + c_im;
+            zr = nzr;
+            zi = nzi;
+        }
+    }
 }

src/fractal/renderer.rs

@@ -13,10 +13,46 @@ use std::sync::Arc;
 
 use eframe::egui_wgpu::{self, wgpu};
 
+use crate::lights::{Light, MAX_LIGHT_COUNT};
+
 /// Maximum reference-orbit length (points) the storage buffer can hold. Also
 /// bounds the iteration count. 128k points * 8 bytes = 1 MiB.
 pub const MAX_REF_POINTS: usize = 1 << 17;
 
+/// Format of the intermediate iteration-data texture holding, per pixel,
+/// `(ci, DE factor, interior fraction)`. 32-bit float keeps the smooth iteration
+/// count precise at deep zoom. Color-renderable and read with nearest sampling
+/// (iteration data must never be linearly filtered across escape boundaries), so
+/// no `float32-filterable` feature is needed.
+const DATA_FORMAT: wgpu::TextureFormat = wgpu::TextureFormat::Rgba32Float;
+
+/// True when the two uniforms differ in any field the iteration pass depends on
+/// (i.e. anything except the palette / colour scale / offset).
+fn geom_differs(a: &Uniforms, b: &Uniforms) -> bool {
+    a.span != b.span
+        || a.max_iter != b.max_iter
+        || a.ref_len != b.ref_len
+        || a.bailout_sq != b.bailout_sq
+        || a.is_julia != b.is_julia
+        || a.aa_level != b.aa_level
+        || a.kind != b.kind
+        || a.power != b.power
+        || a.complex_power != b.complex_power
+        || a.dc_offset != b.dc_offset
+        || a.phoenix_p != b.phoenix_p
+        || a.de_coloring != b.de_coloring
+}
+
+/// True when the two uniforms differ in a colour-only field (remappable by the
+/// cheap colourise pass without re-iterating).
+fn color_differs(a: &Uniforms, b: &Uniforms) -> bool {
+    a.color_offset != b.color_offset
+        || a.color_scale != b.color_scale
+        || a.palette_id != b.palette_id
+        || a.shadow_palette_id != b.shadow_palette_id
+        || a.shadow != b.shadow
+}
+
 /// GPU-side view + coloring parameters. Layout must match `Uniforms` in the
 /// WGSL shader; total size is a multiple of 16 bytes for uniform-buffer rules.
 #[repr(C)]
@@ -33,65 +69,109 @@ pub struct Uniforms {
     /// 0 = Mandelbrot, 1 = Julia.
     pub is_julia: u32,
     pub palette_id: u32,
+    pub shadow_palette_id: u32,
     /// Supersampling factor per axis: 1 = off, 2 = 2×2 (4 samples).
     pub aa_level: u32,
     /// Iteration formula (`FractalKind::shader_id`).
     pub kind: u32,
     /// Exponent for the Multibrot kind.
     pub power: u32,
+    pub _pad: [u32; 1],
     /// Complex offset of the view center from the reference center, so a stale
     /// or reused reference (computed at a slightly different center) still maps
     /// correctly. Added to every pixel's per-pixel offset.
     pub dc_offset: [f32; 2],
+    /// Distortion constant `p` for the Phoenix map (`z^2 + c + p·z_{n-1}`);
+    /// ignored by other kinds. Kept next to `dc_offset` so both `vec2`s land on
+    /// 8-byte boundaries, matching the shader's layout.
+    pub phoenix_p: [f32; 2],
+    /// Distortion constant `l` for the Lambda map (`l·z(1 - z)`);
+    /// ignored by other kinds.
+    pub lambda_l: [f32; 2],
+    /// Complex exponent for the Complex Multibrot kind (`z^power + c`);
+    /// ignored by other kinds.
+    pub complex_power: [f32; 2],
     /// 0 = escape-time coloring, 1 = distance-estimation shading.
     pub de_coloring: u32,
-    /// Padding to a 16-byte multiple (uniform buffer requirement).
-    pub _pad: u32,
+    // 0 = classic colors, 1 = shadows
+    pub shadow: u32,
 }
 
-/// Offscreen texture the fractal is rendered into, plus the bind group used to
-/// blit it. Recreated whenever the widget's pixel size changes.
+/// Offscreen textures for the two-pass render, recreated whenever the widget's
+/// pixel size changes:
+/// * `data_view` — the iteration pass's output (see [`DATA_FORMAT`]).
+/// * `color_view` — the colourise pass's output; the blit source.
+///   plus the bind groups that read them.
 struct CacheTarget {
-    view: wgpu::TextureView,
+    data_view: wgpu::TextureView,
+    color_view: wgpu::TextureView,
+    /// Colourise pass input: uniforms + the data texture.
+    colorize_bind_group: wgpu::BindGroup,
+    /// Blit pass input: the colour texture + sampler.
     blit_bind_group: wgpu::BindGroup,
     width: u32,
     height: u32,
 }
 
-/// State the cache texture was last rendered with. If the next frame's inputs
-/// match this, the cache is still valid and the fractal shader is skipped.
-struct RenderedState {
+/// What the iteration-data texture was last computed with. If the next frame's
+/// geometry inputs match, iteration is skipped and only colour may be redone.
+struct IterState {
     uniforms: Uniforms,
     generation: u64,
     width: u32,
     height: u32,
 }
 
+/// What the colour texture was last computed with. If the next frame's colour
+/// inputs (and size) match and iteration did not re-run, colourise is skipped.
+struct ColorState {
+    uniforms: Uniforms,
+    width: u32,
+    height: u32,
+}
+
 pub struct FractalRenderer {
-    pipeline: wgpu::RenderPipeline,
+    /// Iteration pass: perturbation iterate → data texture (`fs_data`).
+    iterate_pipeline: wgpu::RenderPipeline,
+    /// Combined iterate + colour in one pass (`fs_color`), used only by export.
+    export_pipeline: wgpu::RenderPipeline,
     bind_group_layout: wgpu::BindGroupLayout,
     uniform_buffer: wgpu::Buffer,
     ref_buffer: wgpu::Buffer,
+    lights_buffer: wgpu::Buffer,
     bind_group: wgpu::BindGroup,
     target_format: wgpu::TextureFormat,
     /// Generation of the reference orbit currently uploaded to `ref_buffer`.
     uploaded_generation: u64,
 
-    /// Blit pipeline + resources that copy the cache texture to egui's surface.
+    /// Colourise pass: data texture → colour texture (palette mapping).
+    colorize_pipeline: wgpu::RenderPipeline,
+    colorize_bind_group_layout: wgpu::BindGroupLayout,
+
+    /// Blit pipeline + resources that copy the colour texture to egui's surface.
     blit_pipeline: wgpu::RenderPipeline,
     blit_bind_group_layout: wgpu::BindGroupLayout,
     blit_sampler: wgpu::Sampler,
-    /// The offscreen cache; `None` until the first frame sizes it.
+    /// The offscreen textures; `None` until the first frame sizes them.
     cache: Option<CacheTarget>,
-    /// What the cache currently holds; `None` forces a re-render.
-    rendered: Option<RenderedState>,
+    /// What the data texture holds; `None` forces re-iteration.
+    iterated: Option<IterState>,
+    /// What the colour texture holds; `None` forces a recolour.
+    colored: Option<ColorState>,
 }
 
 impl FractalRenderer {
     pub fn new(device: &wgpu::Device, target_format: wgpu::TextureFormat) -> Self {
         let shader = device.create_shader_module(wgpu::ShaderModuleDescriptor {
             label: Some("mandelbrot"),
-            source: wgpu::ShaderSource::Wgsl(include_str!("../shaders/mandelbrot.wgsl").into()),
+            source: wgpu::ShaderSource::Wgsl(
+                concat!(
+                    include_str!("../shaders/common.wgsl"),
+                    include_str!("../shaders/iterate_uniforms.wgsl"),
+                    include_str!("../shaders/mandelbrot.wgsl"),
+                )
+                .into(),
+            ),
         });
 
         let uniform_buffer = device.create_buffer(&wgpu::BufferDescriptor {
@@ -108,6 +188,13 @@ impl FractalRenderer {
             mapped_at_creation: false,
         });
 
+        let lights_buffer = device.create_buffer(&wgpu::BufferDescriptor {
+            label: Some("lights parameters"),
+            size: (MAX_LIGHT_COUNT * std::mem::size_of::<Light>()) as u64,
+            usage: wgpu::BufferUsages::UNIFORM | wgpu::BufferUsages::COPY_DST,
+            mapped_at_creation: false,
+        });
+
         let bind_group_layout = device.create_bind_group_layout(&wgpu::BindGroupLayoutDescriptor {
             label: Some("fractal bind group layout"),
             entries: &[
@@ -131,6 +218,19 @@ impl FractalRenderer {
                     },
                     count: None,
                 },
+                // Only read by the export pipeline's shadow branch (`fs_color`
+                // with the custom-lights palette); the iterate pipeline
+                // (`fs_data`) ignores it, but both pipelines share this layout.
+                wgpu::BindGroupLayoutEntry {
+                    binding: 2,
+                    visibility: wgpu::ShaderStages::FRAGMENT,
+                    ty: wgpu::BindingType::Buffer {
+                        ty: wgpu::BufferBindingType::Uniform,
+                        has_dynamic_offset: false,
+                        min_binding_size: None,
+                    },
+                    count: None,
+                },
             ],
         });
 
@@ -146,6 +246,10 @@ impl FractalRenderer {
                     binding: 1,
                     resource: ref_buffer.as_entire_binding(),
                 },
+                wgpu::BindGroupEntry {
+                    binding: 2,
+                    resource: lights_buffer.as_entire_binding(),
+                },
             ],
         });
 
@@ -155,8 +259,9 @@ impl FractalRenderer {
             immediate_size: 0,
         });
 
-        let pipeline = device.create_render_pipeline(&wgpu::RenderPipelineDescriptor {
-            label: Some("fractal pipeline"),
+        // Iteration pass: perturbation iterate → data texture (color-independent).
+        let iterate_pipeline = device.create_render_pipeline(&wgpu::RenderPipelineDescriptor {
+            label: Some("fractal iterate pipeline"),
             layout: Some(&pipeline_layout),
             vertex: wgpu::VertexState {
                 module: &shader,
@@ -166,6 +271,114 @@ impl FractalRenderer {
             },
             fragment: Some(wgpu::FragmentState {
                 module: &shader,
+                entry_point: Some("fs_data"),
+                targets: &[Some(wgpu::ColorTargetState {
+                    format: DATA_FORMAT,
+                    blend: None,
+                    write_mask: wgpu::ColorWrites::ALL,
+                })],
+                compilation_options: Default::default(),
+            }),
+            primitive: wgpu::PrimitiveState::default(),
+            depth_stencil: None,
+            multisample: wgpu::MultisampleState::default(),
+            multiview_mask: None,
+            cache: None,
+        });
+
+        // Combined iterate + colour in one pass — for PNG export only.
+        let export_pipeline = device.create_render_pipeline(&wgpu::RenderPipelineDescriptor {
+            label: Some("fractal export pipeline"),
+            layout: Some(&pipeline_layout),
+            vertex: wgpu::VertexState {
+                module: &shader,
+                entry_point: Some("vs_main"),
+                buffers: &[],
+                compilation_options: Default::default(),
+            },
+            fragment: Some(wgpu::FragmentState {
+                module: &shader,
+                entry_point: Some("fs_color"),
+                targets: &[Some(wgpu::ColorTargetState {
+                    format: target_format,
+                    blend: None,
+                    write_mask: wgpu::ColorWrites::ALL,
+                })],
+                compilation_options: Default::default(),
+            }),
+            primitive: wgpu::PrimitiveState::default(),
+            depth_stencil: None,
+            multisample: wgpu::MultisampleState::default(),
+            multiview_mask: None,
+            cache: None,
+        });
+
+        // Colourise pass: data texture + colour uniforms → colour texture.
+        let colorize_shader = device.create_shader_module(wgpu::ShaderModuleDescriptor {
+            label: Some("colorize"),
+            source: wgpu::ShaderSource::Wgsl(
+                concat!(
+                    include_str!("../shaders/common.wgsl"),
+                    include_str!("../shaders/iterate_uniforms.wgsl"),
+                    include_str!("../shaders/colorize.wgsl"),
+                )
+                .into(),
+            ),
+        });
+        let colorize_bind_group_layout =
+            device.create_bind_group_layout(&wgpu::BindGroupLayoutDescriptor {
+                label: Some("colorize bind group layout"),
+                entries: &[
+                    wgpu::BindGroupLayoutEntry {
+                        binding: 0,
+                        visibility: wgpu::ShaderStages::FRAGMENT,
+                        ty: wgpu::BindingType::Buffer {
+                            ty: wgpu::BufferBindingType::Uniform,
+                            has_dynamic_offset: false,
+                            min_binding_size: None,
+                        },
+                        count: None,
+                    },
+                    wgpu::BindGroupLayoutEntry {
+                        binding: 1,
+                        visibility: wgpu::ShaderStages::FRAGMENT,
+                        ty: wgpu::BindingType::Texture {
+                            // Nearest only: iteration data must not be filtered.
+                            sample_type: wgpu::TextureSampleType::Float { filterable: false },
+                            view_dimension: wgpu::TextureViewDimension::D2,
+                            multisampled: false,
+                        },
+                        count: None,
+                    },
+                    wgpu::BindGroupLayoutEntry {
+                        binding: 2,
+                        visibility: wgpu::ShaderStages::FRAGMENT,
+                        ty: wgpu::BindingType::Buffer {
+                            ty: wgpu::BufferBindingType::Uniform,
+                            has_dynamic_offset: false,
+                            min_binding_size: None,
+                        },
+                        count: None,
+                    },
+                ],
+            });
+        let colorize_pipeline_layout =
+            device.create_pipeline_layout(&wgpu::PipelineLayoutDescriptor {
+                label: Some("colorize pipeline layout"),
+                bind_group_layouts: &[Some(&colorize_bind_group_layout)],
+                immediate_size: 0,
+            });
+        let colorize_pipeline = device.create_render_pipeline(&wgpu::RenderPipelineDescriptor {
+            label: Some("colorize pipeline"),
+            layout: Some(&colorize_pipeline_layout),
+            vertex: wgpu::VertexState {
+                module: &colorize_shader,
+                entry_point: Some("vs_main"),
+                buffers: &[],
+                compilation_options: Default::default(),
+            },
+            fragment: Some(wgpu::FragmentState {
+                module: &colorize_shader,
                 entry_point: Some("fs_main"),
                 targets: &[Some(wgpu::ColorTargetState {
                     format: target_format,
@@ -184,7 +397,13 @@ impl FractalRenderer {
         // Blit pipeline: samples the cache texture onto egui's surface.
         let blit_shader = device.create_shader_module(wgpu::ShaderModuleDescriptor {
             label: Some("blit"),
-            source: wgpu::ShaderSource::Wgsl(include_str!("../shaders/blit.wgsl").into()),
+            source: wgpu::ShaderSource::Wgsl(
+                concat!(
+                    include_str!("../shaders/common.wgsl"),
+                    include_str!("../shaders/blit.wgsl"),
+                )
+                .into(),
+            ),
         });
 
         let blit_bind_group_layout =
@@ -250,18 +469,23 @@ impl FractalRenderer {
         });
 
         Self {
-            pipeline,
+            iterate_pipeline,
+            export_pipeline,
             bind_group_layout,
             uniform_buffer,
             ref_buffer,
+            lights_buffer,
             bind_group,
             target_format,
             uploaded_generation: u64::MAX,
+            colorize_pipeline,
+            colorize_bind_group_layout,
             blit_pipeline,
             blit_bind_group_layout,
             blit_sampler,
             cache: None,
-            rendered: None,
+            iterated: None,
+            colored: None,
         }
     }
 
@@ -275,13 +499,29 @@ impl FractalRenderer {
             return;
         }
 
-        let texture = device.create_texture(&wgpu::TextureDescriptor {
-            label: Some("fractal cache"),
-            size: wgpu::Extent3d {
+        let extent = wgpu::Extent3d {
             width,
             height,
             depth_or_array_layers: 1,
-            },
+        };
+
+        // Iteration-data texture (color-independent escape data).
+        let data_texture = device.create_texture(&wgpu::TextureDescriptor {
+            label: Some("fractal data"),
+            size: extent,
+            mip_level_count: 1,
+            sample_count: 1,
+            dimension: wgpu::TextureDimension::D2,
+            format: DATA_FORMAT,
+            usage: wgpu::TextureUsages::RENDER_ATTACHMENT | wgpu::TextureUsages::TEXTURE_BINDING,
+            view_formats: &[],
+        });
+        let data_view = data_texture.create_view(&wgpu::TextureViewDescriptor::default());
+
+        // Colour texture (colourise output; blit source).
+        let color_texture = device.create_texture(&wgpu::TextureDescriptor {
+            label: Some("fractal color cache"),
+            size: extent,
             mip_level_count: 1,
             sample_count: 1,
             dimension: wgpu::TextureDimension::D2,
@@ -289,7 +529,26 @@ impl FractalRenderer {
             usage: wgpu::TextureUsages::RENDER_ATTACHMENT | wgpu::TextureUsages::TEXTURE_BINDING,
             view_formats: &[],
         });
-        let view = texture.create_view(&wgpu::TextureViewDescriptor::default());
+        let color_view = color_texture.create_view(&wgpu::TextureViewDescriptor::default());
+
+        let colorize_bind_group = device.create_bind_group(&wgpu::BindGroupDescriptor {
+            label: Some("colorize bind group"),
+            layout: &self.colorize_bind_group_layout,
+            entries: &[
+                wgpu::BindGroupEntry {
+                    binding: 0,
+                    resource: self.uniform_buffer.as_entire_binding(),
+                },
+                wgpu::BindGroupEntry {
+                    binding: 1,
+                    resource: wgpu::BindingResource::TextureView(&data_view),
+                },
+                wgpu::BindGroupEntry {
+                    binding: 2,
+                    resource: self.lights_buffer.as_entire_binding(),
+                },
+            ],
+        });
 
         let blit_bind_group = device.create_bind_group(&wgpu::BindGroupDescriptor {
             label: Some("blit bind group"),
@@ -297,7 +556,7 @@ impl FractalRenderer {
             entries: &[
                 wgpu::BindGroupEntry {
                     binding: 0,
-                    resource: wgpu::BindingResource::TextureView(&view),
+                    resource: wgpu::BindingResource::TextureView(&color_view),
                 },
                 wgpu::BindGroupEntry {
                     binding: 1,
@@ -307,21 +566,30 @@ impl FractalRenderer {
         });
 
         self.cache = Some(CacheTarget {
-            view,
+            data_view,
+            color_view,
+            colorize_bind_group,
             blit_bind_group,
             width,
             height,
         });
-        // New texture → old render is gone.
-        self.rendered = None;
+        // New textures → old renders are gone.
+        self.iterated = None;
+        self.colored = None;
     }
 
     /// Handles needed to build a standalone [`ExportRender`] off the UI thread:
     /// the (immutable) pipeline and its bind-group layout, plus the target
     /// format. Cloned so the caller can drop the render-state lock before use.
-    pub fn export_handles(&self) -> (wgpu::RenderPipeline, wgpu::BindGroupLayout, wgpu::TextureFormat) {
+    pub fn export_handles(
+        &self,
+    ) -> (
+        wgpu::RenderPipeline,
+        wgpu::BindGroupLayout,
+        wgpu::TextureFormat,
+    ) {
         (
-            self.pipeline.clone(),
+            self.export_pipeline.clone(),
             self.bind_group_layout.clone(),
             self.target_format,
         )
@@ -361,6 +629,7 @@ impl ExportRender {
         height: u32,
         uniforms: Uniforms,
         reference: &[[f32; 2]],
+        lights: &[Light],
     ) -> Self {
         let uniform_buffer = device.create_buffer(&wgpu::BufferDescriptor {
             label: Some("export uniforms"),
@@ -381,6 +650,19 @@ impl ExportRender {
             queue.write_buffer(&ref_buffer, 0, bytemuck::cast_slice(&reference[..count]));
         }
 
+        // Only read by the shadow branch's custom-lights palette; harmless
+        // (zeroed) for every other coloring mode.
+        let lights_buffer = device.create_buffer(&wgpu::BufferDescriptor {
+            label: Some("export lights"),
+            size: (MAX_LIGHT_COUNT * std::mem::size_of::<Light>()) as u64,
+            usage: wgpu::BufferUsages::UNIFORM | wgpu::BufferUsages::COPY_DST,
+            mapped_at_creation: false,
+        });
+        let mut light_bytes = [0u8; size_of::<Light>() * MAX_LIGHT_COUNT];
+        let n = lights.len().min(MAX_LIGHT_COUNT);
+        light_bytes[..n * size_of::<Light>()].copy_from_slice(bytemuck::cast_slice(&lights[..n]));
+        queue.write_buffer(&lights_buffer, 0, &light_bytes);
+
         let bind_group = device.create_bind_group(&wgpu::BindGroupDescriptor {
             label: Some("export bind group"),
             layout: bind_group_layout,
@@ -393,6 +675,10 @@ impl ExportRender {
                     binding: 1,
                     resource: ref_buffer.as_entire_binding(),
                 },
+                wgpu::BindGroupEntry {
+                    binding: 2,
+                    resource: lights_buffer.as_entire_binding(),
+                },
             ],
         });
 
@@ -530,6 +816,59 @@ impl ExportRender {
     }
 }
 
+/// Render `er` tile by tile (blocking on the GPU after each tile so progress
+/// reflects real work), read it back, and encode the result as PNG bytes.
+/// Blocks the calling thread throughout, so it's only for native targets:
+/// the UI export path runs it on a background thread, headless rendering
+/// runs it directly since it has no frame loop to share a thread with.
+#[cfg(not(target_arch = "wasm32"))]
+pub fn export_to_png_blocking(
+    device: &wgpu::Device,
+    queue: &wgpu::Queue,
+    er: &ExportRender,
+    mut on_progress: impl FnMut(&'static str, f32),
+) -> Vec<u8> {
+    // Progress budget: rendering fills [0, RENDER_END], encoding the rest.
+    const RENDER_END: f32 = 0.6;
+
+    for t in 0..er.tiles {
+        er.render_tile(device, queue, t);
+        let _ = device.poll(wgpu::PollType::Wait {
+            submission_index: None,
+            timeout: None,
+        });
+        let done = (t + 1) as f32 / er.tiles as f32;
+        on_progress("Rendering", RENDER_END * done);
+    }
+    er.copy_to_readback(device, queue);
+
+    let (tx, rx) = std::sync::mpsc::channel();
+    er.readback()
+        .slice(..)
+        .map_async(wgpu::MapMode::Read, move |res| {
+            let _ = tx.send(res);
+        });
+    let _ = device.poll(wgpu::PollType::Wait {
+        submission_index: None,
+        timeout: None,
+    });
+    let _ = rx.recv();
+
+    on_progress("Encoding", RENDER_END);
+    let png = {
+        let data = er
+            .readback()
+            .slice(..)
+            .get_mapped_range()
+            .expect("map readback buffer");
+        encode_png_with_progress(&data, er.width, er.height, er.padded_bpr, er.swap_rb, |f| {
+            on_progress("Encoding", RENDER_END + (0.97 - RENDER_END) * f)
+        })
+    };
+    er.readback().unmap();
+    png
+}
+
 /// Convert a padded BGRA/RGBA readback into tightly-packed RGBA8 and encode it
 /// as PNG bytes, reporting progress in `[0, 1]` via `on_progress` as rows are
 /// streamed to the compressor (encoding is the slow, subdividable phase).
@@ -578,10 +917,12 @@ pub fn encode_png_with_progress(
 
 /// A per-frame paint callback. Carries this frame's uniforms plus a reference to
 /// the current reference orbit (cheap `Arc` clone). The orbit is only re-uploaded
-/// to the GPU when its `generation` changes, and the fractal is only re-rendered
-/// into the cache when the uniforms, generation, or `size_px` change.
+/// when its `generation` changes; the expensive iteration pass re-runs only when
+/// a geometry input changes, and colour-only changes re-run just the cheap
+/// colourise pass (see `prepare`).
 pub struct FractalCallback {
     pub uniforms: Uniforms,
+    pub lights: Vec<Light>,
     pub reference: Arc<Vec<[f32; 2]>>,
     pub generation: u64,
     /// Widget size in physical pixels — the cache texture resolution.
@@ -615,28 +956,44 @@ impl egui_wgpu::CallbackTrait for FractalCallback {
             renderer.uploaded_generation = self.generation;
         }
 
-        // Re-render the cache only when what it depends on changed.
-        let dirty = renderer.rendered.as_ref().is_none_or(|r| {
+        // Iteration (expensive) re-runs only when the geometry inputs change;
+        // colourise (cheap) re-runs when it did, or when only a colour changed —
+        // so palette / colour-scale / offset tweaks (e.g. colour cycling) skip
+        // the perturbation entirely.
+        let iter_dirty = renderer.iterated.as_ref().is_none_or(|r| {
             r.generation != self.generation
                 || r.width != width
                 || r.height != height
-                || bytemuck::bytes_of(&r.uniforms) != bytemuck::bytes_of(&self.uniforms)
+                || geom_differs(&r.uniforms, &self.uniforms)
         });
-        if !dirty {
-            return Vec::new();
+        let color_dirty = iter_dirty
+            || renderer.colored.as_ref().is_none_or(|c| {
+                c.width != width || c.height != height || color_differs(&c.uniforms, &self.uniforms)
+            })
+            || true;
+
+        if !color_dirty {
+            return Vec::new(); // cache still valid; paint() just blits it
         }
 
+        // Both passes read the uniform buffer; refresh it once.
         queue.write_buffer(
             &renderer.uniform_buffer,
             0,
             bytemuck::bytes_of(&self.uniforms),
         );
+        let mut bytes = [0; size_of::<Light>() * MAX_LIGHT_COUNT];
+        bytes[..self.lights.len() * size_of::<Light>()]
+            .copy_from_slice(bytemuck::cast_slice(&self.lights));
+        queue.write_buffer(&renderer.lights_buffer, 0, &bytes);
 
         if let Some(cache) = &renderer.cache {
+            if iter_dirty {
+                // Iteration pass: perturbation iterate → data texture.
                 let mut pass = egui_encoder.begin_render_pass(&wgpu::RenderPassDescriptor {
-                label: Some("fractal cache pass"),
+                    label: Some("fractal iterate pass"),
                     color_attachments: &[Some(wgpu::RenderPassColorAttachment {
-                    view: &cache.view,
+                        view: &cache.data_view,
                         depth_slice: None,
                         resolve_target: None,
                         ops: wgpu::Operations {
@@ -649,17 +1006,46 @@ impl egui_wgpu::CallbackTrait for FractalCallback {
                     occlusion_query_set: None,
                     multiview_mask: None,
                 });
-            pass.set_pipeline(&renderer.pipeline);
+                pass.set_pipeline(&renderer.iterate_pipeline);
                 pass.set_bind_group(0, &renderer.bind_group, &[]);
                 pass.draw(0..3, 0..1);
             }
 
-        renderer.rendered = Some(RenderedState {
+            // Colourise pass: data texture → colour texture.
+            let mut pass = egui_encoder.begin_render_pass(&wgpu::RenderPassDescriptor {
+                label: Some("fractal colorize pass"),
+                color_attachments: &[Some(wgpu::RenderPassColorAttachment {
+                    view: &cache.color_view,
+                    depth_slice: None,
+                    resolve_target: None,
+                    ops: wgpu::Operations {
+                        load: wgpu::LoadOp::Clear(wgpu::Color::BLACK),
+                        store: wgpu::StoreOp::Store,
+                    },
+                })],
+                depth_stencil_attachment: None,
+                timestamp_writes: None,
+                occlusion_query_set: None,
+                multiview_mask: None,
+            });
+            pass.set_pipeline(&renderer.colorize_pipeline);
+            pass.set_bind_group(0, &cache.colorize_bind_group, &[]);
+            pass.draw(0..3, 0..1);
+        }
+
+        if iter_dirty {
+            renderer.iterated = Some(IterState {
                 uniforms: self.uniforms,
                 generation: self.generation,
                 width,
                 height,
             });
+        }
+        renderer.colored = Some(ColorState {
+            uniforms: self.uniforms,
+            width,
+            height,
+        });
         Vec::new()
     }
 

src/fractal/share.rs

@@ -21,31 +21,54 @@ pub struct ShareState {
     pub half_height: f64,
     pub iterations: u32,
     pub julia_c: (f64, f64),
+    /// Distortion constant for the Phoenix kind (ignored by others).
+    pub phoenix_p: (f64, f64),
+    /// Distortion constant for the Lambda kind (ignored by others).
+    pub lambda_l: (f64, f64),
+    /// Complex exponent for the Complex Multibrot kind (ignored by others).
+    pub complex_power: (f64, f64),
     pub color_scale: f32,
     pub color_offset: f32,
     /// Palette index (`palette_id` in the shader).
     pub palette: u32,
+    /// Shadow palette index (`shadow_palette_id` in the shader).
+    pub shadow_palette: u32,
 }
 
 impl ShareState {
     pub fn encode(&self) -> String {
         let mut s = String::new();
         s.push_str(if self.julia { "m=j" } else { "m=m" });
-        s.push_str(&format!("&f={}", match self.kind {
+        s.push_str(&format!(
+            "&f={}",
+            match self.kind {
                 FractalKind::Mandelbrot => "mandel",
                 FractalKind::BurningShip => "burning",
                 FractalKind::Multibrot => "multi",
                 FractalKind::Tricorn => "tricorn",
-        }));
+                FractalKind::Celtic => "celtic",
+                FractalKind::Perpendicular => "perp",
+                FractalKind::Buffalo => "buffalo",
+                FractalKind::Phoenix => "phoenix",
+                FractalKind::Lambda => "lambda",
+                FractalKind::ComplexMultibrot => "cmulti",
+            }
+        ));
         s.push_str(&format!("&pw={}", self.power));
         s.push_str(&format!(
             "&re={}&im={}&hh={}&it={}",
             self.center_re, self.center_im, self.half_height, self.iterations
         ));
         s.push_str(&format!("&jr={}&ji={}", self.julia_c.0, self.julia_c.1));
+        s.push_str(&format!("&px={}&py={}", self.phoenix_p.0, self.phoenix_p.1));
+        s.push_str(&format!("&lx={}&ly={}", self.lambda_l.0, self.lambda_l.1));
         s.push_str(&format!(
-            "&cs={}&co={}&pal={}",
-            self.color_scale, self.color_offset, self.palette
+            "&cpr={}&cpi={}",
+            self.complex_power.0, self.complex_power.1
+        ));
+        s.push_str(&format!(
+            "&cs={}&co={}&pal={}&spal={}",
+            self.color_scale, self.color_offset, self.palette, self.shadow_palette
         ));
         s
     }
@@ -68,7 +91,13 @@ impl ShareState {
                     "multi" => FractalKind::Multibrot,
                     "burning" => FractalKind::BurningShip,
                     "tricorn" => FractalKind::Tricorn,
-                    _ => FractalKind::Mandelbrot
+                    "celtic" => FractalKind::Celtic,
+                    "perp" => FractalKind::Perpendicular,
+                    "buffalo" => FractalKind::Buffalo,
+                    "phoenix" => FractalKind::Phoenix,
+                    "lambda" => FractalKind::Lambda,
+                    "cmulti" => FractalKind::ComplexMultibrot,
+                    _ => FractalKind::Mandelbrot,
                 })
                 .unwrap_or(FractalKind::Mandelbrot),
             power: map.get("pw").and_then(|s| s.parse().ok()).unwrap_or(2),
@@ -80,9 +109,22 @@ impl ShareState {
                 map.get("jr").and_then(|s| s.parse().ok()).unwrap_or(-0.8),
                 map.get("ji").and_then(|s| s.parse().ok()).unwrap_or(0.156),
             ),
+            phoenix_p: (
+                map.get("px").and_then(|s| s.parse().ok()).unwrap_or(-0.5),
+                map.get("py").and_then(|s| s.parse().ok()).unwrap_or(0.0),
+            ),
+            lambda_l: (
+                map.get("lx").and_then(|s| s.parse().ok()).unwrap_or(-0.5),
+                map.get("ly").and_then(|s| s.parse().ok()).unwrap_or(0.0),
+            ),
+            complex_power: (
+                map.get("cpr").and_then(|s| s.parse().ok()).unwrap_or(2.0),
+                map.get("cpi").and_then(|s| s.parse().ok()).unwrap_or(0.0),
+            ),
             color_scale: map.get("cs").and_then(|s| s.parse().ok()).unwrap_or(0.02),
             color_offset: map.get("co").and_then(|s| s.parse().ok()).unwrap_or(0.0),
             palette: map.get("pal").and_then(|s| s.parse().ok()).unwrap_or(0),
+            shadow_palette: map.get("spal").and_then(|s| s.parse().ok()).unwrap_or(0),
         })
     }
 }
@@ -95,16 +137,20 @@ mod tests {
     fn round_trip() {
         let s = ShareState {
             julia: true,
-            kind: FractalKind::Multibrot,
+            kind: FractalKind::Phoenix,
             power: 5,
             center_re: "-0.743643887037158704752191506114774".into(),
             center_im: "0.131825904205311970493132056385139".into(),
             half_height: 1.5e-20,
             iterations: 4000,
             julia_c: (-0.123, 0.745),
+            phoenix_p: (-0.5, 0.1),
+            lambda_l: (-0.5, 0.0),
+            complex_power: (2.5, 0.3),
             color_scale: 0.02,
             color_offset: 0.25,
             palette: 3,
+            shadow_palette: 1,
         };
         let d = ShareState::decode(&s.encode()).unwrap();
         assert_eq!(d.julia, s.julia);
@@ -115,7 +161,10 @@ mod tests {
         assert_eq!(d.half_height, s.half_height);
         assert_eq!(d.iterations, s.iterations);
         assert_eq!(d.julia_c, s.julia_c);
+        assert_eq!(d.phoenix_p, s.phoenix_p);
+        assert_eq!(d.complex_power, s.complex_power);
         assert_eq!(d.palette, s.palette);
+        assert_eq!(d.shadow_palette, s.shadow_palette);
     }
 
     #[test]

src/headless.rs

@@ -0,0 +1,83 @@
+// Headless PNG rendering: parse the CLI, build the exact same view/state the
+// windowed app would from it, then render straight to a file. No window, no
+// event loop, no worker-thread debounce (nothing to debounce for a one-shot
+// render); it just creates its own wgpu device, computes the reference orbit
+// once, and renders through the same `ExportRender` path the "Export PNG"
+// button uses.
+
+use eframe::egui_wgpu::wgpu;
+
+use crate::app::{FractalApp, unix_timestamp};
+use crate::cli::Cli;
+use crate::fractal::{ExportRender, FractalRenderer, export_to_png_blocking};
+
+/// Cap on the output image dimension (px), to stay within GPU texture limits.
+const MAX_DIM: u32 = 8192 * 16;
+
+pub fn run(cli: Cli) -> Result<(), String> {
+    if cli.buddhabrot {
+        return Err("headless mode doesn't support --buddhabrot yet".into());
+    }
+
+    let width = cli.width.clamp(16, MAX_DIM);
+    let height = cli.height.clamp(16, MAX_DIM);
+    let export_path = cli
+        .export_path
+        .clone()
+        .unwrap_or_else(|| format!("fractal-{}.png", unix_timestamp()));
+
+    let mut app = FractalApp::default_state();
+    app.apply_cli(cli);
+
+    eprintln!("computing reference orbit…");
+    app.compute_reference_blocking();
+
+    let (device, queue) = pollster::block_on(request_device())?;
+    let format = wgpu::TextureFormat::Rgba8UnormSrgb;
+    let renderer = FractalRenderer::new(&device, format);
+    let (pipeline, bind_group_layout, format) = renderer.export_handles();
+
+    let uniforms = app.make_uniforms(width as f64 / height as f64);
+    let er = ExportRender::new(
+        &device,
+        &queue,
+        pipeline,
+        &bind_group_layout,
+        format,
+        width,
+        height,
+        uniforms,
+        app.reference_points(),
+        app.lights(),
+    );
+
+    eprintln!("rendering {width}×{height}…");
+    let png = export_to_png_blocking(&device, &queue, &er, |phase, fraction| {
+        eprint!("\r{phase} {:>3.0}%", fraction * 100.0);
+    });
+    eprintln!();
+
+    std::fs::write(&export_path, &png).map_err(|e| format!("save failed: {e}"))?;
+    println!("saved {export_path} ({width}×{height})");
+    Ok(())
+}
+
+/// Set up a wgpu device with no surface/window attached, matching the limits
+/// `main::wgpu_options` requests for the windowed app (the fractal fragment
+/// shader needs storage buffers, which downlevel/WebGL-style limits disallow).
+async fn request_device() -> Result<(wgpu::Device, wgpu::Queue), String> {
+    let instance = wgpu::Instance::default();
+    let adapter = instance
+        .request_adapter(&wgpu::RequestAdapterOptions::default())
+        .await
+        .map_err(|e| format!("no compatible GPU adapter: {e}"))?;
+    adapter
+        .request_device(&wgpu::DeviceDescriptor {
+            label: Some("headless fractal device"),
+            required_features: wgpu::Features::empty(),
+            required_limits: adapter.limits(),
+            ..Default::default()
+        })
+        .await
+        .map_err(|e| format!("failed to create device: {e}"))
+}

src/lights.rs

@@ -0,0 +1,55 @@
+use std::f32::consts::PI;
+
+use bytemuck::{Pod, Zeroable};
+use egui::{Color32, Ui};
+
+/// Maximum number of simultaneous lights.
+pub const MAX_LIGHT_COUNT: usize = 16;
+
+#[derive(Clone, Copy, PartialEq, Zeroable, Pod)]
+#[repr(C)]
+pub struct Light {
+    pub azimuth: f32,
+    pub altitude: f32,
+    pub color: Color32,
+    pub _pad: u32,
+}
+
+impl Default for Light {
+    fn default() -> Self {
+        Self {
+            azimuth: PI / 4.,
+            altitude: PI / 4.,
+            color: Color32::WHITE,
+            _pad: 0,
+        }
+    }
+}
+
+impl Light {
+    pub fn widget(&mut self, ui: &mut Ui) -> bool {
+        let formater = |v, _| format!("{}°", ((v as f32 * 180. / PI) as u32));
+        ui.horizontal(|ui| {
+            let del = ui.button("-").clicked();
+            ui.label("color:");
+            ui.color_edit_button_srgba(&mut self.color);
+            ui.label("θ:");
+            ui.add(
+                egui::DragValue::new(&mut self.azimuth)
+                    .range(0.0..=PI * 2.)
+                    .custom_formatter(formater)
+                    .speed(0.02),
+            );
+            ui.label("φ:");
+            ui.add(
+                egui::DragValue::new(&mut self.altitude)
+                    .range(0.0..=PI / 2.)
+                    .custom_formatter(formater)
+                    .speed(0.02),
+            );
+
+            del
+        })
+        .inner
+    }
+}

src/main.rs

@@ -1,3 +1,7 @@
+// Without these, rust fails to infer Send/Sync trait impls
+// Probably caused by the new trait solver
+#![recursion_limit = "256"]
+
 // Fractal Explorer — Rust + wgpu + egui + WGSL deep-zoom Mandelbrot.
 //
 // A single binary drives both native and web (WASM/WebGPU) builds; the two
@@ -6,8 +10,13 @@
 
 mod app;
 mod fractal;
+mod lights;
 mod view;
 
+#[cfg(not(target_arch = "wasm32"))]
+mod cli;
+#[cfg(not(target_arch = "wasm32"))]
+mod headless;
 #[cfg(not(target_arch = "wasm32"))]
 mod worker;
 
@@ -25,13 +34,12 @@ fn wgpu_options() -> eframe::egui_wgpu::WgpuConfiguration {
 
     let mut options = eframe::egui_wgpu::WgpuConfiguration::default();
     if let WgpuSetup::CreateNew(setup) = &mut options.wgpu_setup {
-        setup.device_descriptor = std::sync::Arc::new(|adapter: &wgpu::Adapter| {
-            wgpu::DeviceDescriptor {
+        setup.device_descriptor =
+            std::sync::Arc::new(|adapter: &wgpu::Adapter| wgpu::DeviceDescriptor {
                 label: Some("fractal wgpu device"),
                 required_features: wgpu::Features::empty(),
                 required_limits: adapter.limits(),
                 ..Default::default()
-            }
             });
         #[cfg(target_arch = "wasm32")]
         {
@@ -43,11 +51,24 @@ fn wgpu_options() -> eframe::egui_wgpu::WgpuConfiguration {
 
 #[cfg(not(target_arch = "wasm32"))]
 fn main() -> eframe::Result {
+    use clap::Parser as _;
+
     env_logger::builder()
         .filter_level(log::LevelFilter::Info)
         .parse_default_env()
         .init();
 
+    let cli = cli::Cli::parse();
+    if cli.headless {
+        return match headless::run(cli) {
+            Ok(()) => Ok(()),
+            Err(e) => {
+                eprintln!("error: {e}");
+                std::process::exit(1);
+            }
+        };
+    }
+
     let native_options = eframe::NativeOptions {
         renderer: eframe::Renderer::Wgpu,
         wgpu_options: wgpu_options(),

src/shaders/blit.wgsl

@@ -13,12 +13,7 @@ struct VsOut {
 
 @vertex
 fn vs_main(@builtin(vertex_index) idx: u32) -> VsOut {
-    var verts = array<vec2<f32>, 3>(
-        vec2<f32>(-1.0, -1.0),
-        vec2<f32>(3.0, -1.0),
-        vec2<f32>(-1.0, 3.0),
-    );
-    let p = verts[idx];
+    let p = fullscreen_triangle_pos(idx);
     var out: VsOut;
     out.pos = vec4<f32>(p, 0.0, 1.0);
     // Map NDC to texture UV. v is flipped so the cache's top row (rendered at

src/shaders/buddhabrot.wgsl

@@ -0,0 +1,283 @@
+// Buddhabrot / Nebulabrot rendering: a Monte-Carlo density histogram of
+// escaping orbits, accumulated progressively across frames by a compute pass,
+// then tone-mapped to colour by a fragment pass every frame.
+//
+// This does NOT use the deep-zoom perturbation/reference-orbit machinery in
+// mandelbrot.wgsl: Buddhabrot's structure is a global Monte-Carlo property of
+// the whole basin (a random sample's orbit scatters across the *whole* image,
+// not just its own pixel), so the "gather" per-pixel model doesn't apply, and
+// deep zoom isn't meaningful for it the way it is for the escape-time set.
+// Samples are iterated directly in f32 from the current view's bounds.
+//
+// Sampling convention: for KIND_LAMBDA the formula z -> l*z*(1-z) has no `c`
+// term at all (l is a fixed distortion constant, not a per-sample parameter),
+// so the randomly sampled point instead seeds z0 (a "Julia-Buddhabrot" over
+// z0 with l fixed). Every other kind samples c with z0 = 0, matching its
+// ordinary parameter plane.
+//
+// A sample's orbit is only plotted if it escapes within b_cap iterations (the
+// classic Buddhabrot rule: only escaping orbits are drawn). Its points are
+// then splat into up to three histogram channels by cap (r_cap <= g_cap <=
+// b_cap): fast-escaping (common) orbits light all three channels (bright),
+// slow-escaping (rare) orbits only light the b_cap channel — the classic
+// Nebulabrot false-colour split.
+//
+// Two-pass iteration avoids needing a per-thread orbit buffer sized to
+// max_iter: the first pass just finds the escape iteration (if any); the
+// second replays the same orbit from scratch, splatting each point.
+
+struct Uniforms {
+    center: vec2<f32>,
+    half_height: f32,
+    aspect: f32,
+    phoenix_p: vec2<f32>,
+    lambda_l: vec2<f32>,
+    bailout_sq: f32,
+    kind: u32,
+    power: u32,
+    r_cap: u32,
+    g_cap: u32,
+    b_cap: u32,
+    seed: u32,
+    samples_this_dispatch: u32,
+    exposure: f32,
+    width: u32,
+    height: u32,
+    total_samples: f32,
+    // Tonemap colour style: 0 = classic (R/G/B = raw caps), 1 = nebula
+    // (yellow core, blue halo), 2 = grayscale.
+    palette: u32,
+    // Padding so `complex_power` (a vec2, 8-byte aligned) starts on an
+    // 8-byte boundary. NOT vec3<u32> — that type aligns to 16 bytes in WGSL
+    // (unlike Rust's `[u32; 3]`, which aligns to 4), which silently added 32
+    // bytes instead of 16 and mismatched the Rust struct's size (a wgpu
+    // validation error at dispatch time: "size 96 where the shader expects
+    // 112").
+    _pad0: u32,
+    // Complex exponent for the Complex Multibrot kind; unused by other kinds.
+    complex_power: vec2<f32>,
+};
+
+const PALETTE_NEBULA: u32 = 0u;
+const PALETTE_YELLOW: u32 = 1u;
+const PALETTE_GRAYSCALE: u32 = 2u;
+
+@group(0) @binding(0) var<uniform> u: Uniforms;
+// Compute pass: read-write atomic histogram (3 planes of width*height, R/G/B).
+@group(0) @binding(1) var<storage, read_write> histogram: array<atomic<u32>>;
+// Tonemap pass: read-only plain view of the same buffer.
+@group(0) @binding(2) var<storage, read> tm_histogram: array<u32>;
+
+// --- RNG: a small, fast integer hash (WGSL has no native RNG). ---
+fn hash_u32(x: u32) -> u32 {
+    var h = x;
+    h = h ^ (h >> 16u);
+    h = h * 0x7feb352du;
+    h = h ^ (h >> 15u);
+    h = h * 0x846ca68bu;
+    h = h ^ (h >> 16u);
+    return h;
+}
+fn rand01(seed: u32) -> f32 {
+    return f32(hash_u32(seed)) * (1.0 / 4294967295.0);
+}
+
+fn complex_pow(z: vec2<f32>, p: u32) -> vec2<f32> {
+    var r = vec2<f32>(1.0, 0.0);
+    for (var i: u32 = 0u; i < p; i = i + 1u) {
+        r = cmul(r, z);
+    }
+    return r;
+}
+
+// One iteration step z_n -> z_{n+1} for the current kind. `zp` is the
+// previous iterate (z_{n-1}), used only by the Phoenix two-term recurrence.
+// Must match `FractalKind` in reference.rs (the direct, non-perturbative form
+// of the same formulas).
+fn advance(z: vec2<f32>, zp: vec2<f32>, c: vec2<f32>) -> vec2<f32> {
+    if u.kind == KIND_BURNING_SHIP {
+        return vec2<f32>(z.x * z.x - z.y * z.y, 2.0 * abs(z.x * z.y)) + c;
+    } else if u.kind == KIND_TRICORN {
+        return vec2<f32>(z.x * z.x - z.y * z.y, -2.0 * z.x * z.y) + c;
+    } else if u.kind == KIND_MULTIBROT {
+        return complex_pow(z, clamp(u.power, 2u, 8u)) + c;
+    } else if u.kind == KIND_CELTIC {
+        return vec2<f32>(abs(z.x * z.x - z.y * z.y), 2.0 * z.x * z.y) + c;
+    } else if u.kind == KIND_PERPENDICULAR {
+        return vec2<f32>(z.x * z.x - z.y * z.y, -2.0 * z.x * abs(z.y)) + c;
+    } else if u.kind == KIND_BUFFALO {
+        return vec2<f32>(abs(z.x * z.x - z.y * z.y), -abs(2.0 * z.x * z.y)) + c;
+    } else if u.kind == KIND_PHOENIX {
+        let sq = vec2<f32>(z.x * z.x - z.y * z.y, 2.0 * z.x * z.y);
+        return sq + c + cmul(u.phoenix_p, zp);
+    } else if u.kind == KIND_LAMBDA {
+        // l * z * (1 - z); c is unused (see file doc comment above).
+        return cmul(u.lambda_l, cmul(z, vec2<f32>(1.0 - z.x, -z.y)));
+    } else if u.kind == KIND_COMPLEX_MULTIBROT {
+        return cpow(z, u.complex_power) + c;
+    }
+    return vec2<f32>(z.x * z.x - z.y * z.y, 2.0 * z.x * z.y) + c; // Mandelbrot
+}
+
+// Map a complex-plane point to a flat pixel index, or -1 if outside the
+// current viewport (the sampling region and the display region are the same).
+//
+// This must be the exact inverse of how `view.rs::pan_pixels`/`zoom_at_pixel`
+// relate screen pixels to world points (those are the confirmed-correct,
+// user-tested ground truth — NOT the shader-comment-derived convention tried
+// here previously, which was wrong: dragging/zooming treat +y screen exactly
+// like +x, no flip, so screen-down means im *increasing*, not decreasing).
+fn pixel_index(p: vec2<f32>) -> i32 {
+    let half_w = u.half_height * u.aspect;
+    let uu = (p.x - u.center.x) / half_w * 0.5 + 0.5;
+    let vv = 0.5 + (p.y - u.center.y) / u.half_height * 0.5;
+    if uu < 0.0 || uu >= 1.0 || vv < 0.0 || vv >= 1.0 {
+        return -1;
+    }
+    let px = i32(uu * f32(u.width));
+    let py = i32(vv * f32(u.height));
+    return py * i32(u.width) + px;
+}
+
+// Splat one visited orbit point into the R/G/B histogram planes it qualifies
+// for by the orbit's total escape iteration `n` (nested caps: a fast escape
+// lights all three; only a slow, rare one lights just the blue plane).
+fn splat(p: vec2<f32>, n: u32) {
+    let idx = pixel_index(p);
+    if idx < 0 {
+        return;
+    }
+    let plane = i32(u.width) * i32(u.height);
+    if n <= u.b_cap {
+        atomicAdd(&histogram[idx + 2 * plane], 1u);
+    }
+    if n <= u.g_cap {
+        atomicAdd(&histogram[idx + plane], 1u);
+    }
+    if n <= u.r_cap {
+        atomicAdd(&histogram[idx], 1u);
+    }
+}
+
+@compute @workgroup_size(64)
+fn cs_main(@builtin(global_invocation_id) gid: vec3<u32>) {
+    if gid.x >= u.samples_this_dispatch {
+        return;
+    }
+
+    let base = hash_u32(gid.x ^ (u.seed * 0x9e3779b9u));
+    let rx = rand01(base);
+    let ry = rand01(hash_u32(base ^ 0x68bc21ebu));
+    let half_w = u.half_height * u.aspect;
+    let sample = vec2<f32>(
+        u.center.x + (rx * 2.0 - 1.0) * half_w,
+        u.center.y + (ry * 2.0 - 1.0) * u.half_height,
+    );
+
+    var c = sample;
+    var z0 = vec2<f32>(0.0, 0.0);
+    if u.kind == KIND_LAMBDA {
+        c = vec2<f32>(0.0, 0.0); // unused by the Lambda step
+        z0 = sample;
+    }
+
+    // First pass: just find the escape iteration (if any).
+    var zp = vec2<f32>(0.0, 0.0);
+    var z = z0;
+    var n: u32 = 0u;
+    var escaped = false;
+    loop {
+        if dot(z, z) > u.bailout_sq {
+            escaped = true;
+            break;
+        }
+        if n >= u.b_cap {
+            break;
+        }
+        let next = advance(z, zp, c);
+        zp = z;
+        z = next;
+        n = n + 1u;
+    }
+    if !escaped || n == 0u {
+        return;
+    }
+
+    // Second pass: replay the same orbit, splatting each visited point.
+    // z0 itself is not splat: it's the same fixed point (0,0), or the sample
+    // itself for Lambda, for every orbit — plotting it would just spike the
+    // origin instead of showing the orbit's actual shape.
+    zp = vec2<f32>(0.0, 0.0);
+    z = z0;
+    for (var i: u32 = 0u; i < n; i = i + 1u) {
+        let next = advance(z, zp, c);
+        zp = z;
+        z = next;
+        splat(z, n);
+    }
+}
+
+// --- Tonemap: histogram counts -> colour, drawn as a fullscreen triangle. ---
+
+@vertex
+fn vs_main(@builtin(vertex_index) idx: u32) -> @builtin(position) vec4<f32> {
+    return vec4<f32>(fullscreen_triangle_pos(idx), 0.0, 1.0);
+}
+
+@fragment
+fn fs_tonemap(@builtin(position) pos: vec4<f32>) -> @location(0) vec4<f32> {
+    let x = i32(pos.x);
+    let y = i32(pos.y);
+    if x < 0 || y < 0 || x >= i32(u.width) || y >= i32(u.height) {
+        return vec4<f32>(0.0, 0.0, 0.0, 1.0);
+    }
+    let idx = y * i32(u.width) + x;
+    let plane = i32(u.width) * i32(u.height);
+    let r = f32(tm_histogram[idx]);
+    let g = f32(tm_histogram[idx + plane]);
+    let b = f32(tm_histogram[idx + 2 * plane]);
+
+    // Normalize by the *average* density (total samples / pixel count) rather
+    // than total samples alone, so the scale stays sane across widget sizes
+    // and sample-dispatch rates. Buddhabrot density is extremely peaked (the
+    // brightest pixels run tens of times the average), so the compressive
+    // exponential tonemap only needs a small fraction of the average to reach
+    // full brightness at those peaks; 0.05 is a hand-tuned starting point,
+    // the exposure slider covers the rest.
+    let avg_density = max(u.total_samples / f32(u.width * u.height), 1.0e-6);
+    let scale = u.exposure * 0.05 / avg_density;
+    // Per-cap brightness, each already compressed to [0,1]. Nested caps mean
+    // r <= g <= b pointwise (every orbit counted in a smaller cap is also
+    // counted in every larger one), so fb alone is the full escaping-orbit
+    // density and fr picks out just the common, fast-escaping ones.
+    let fr = 1.0 - exp(-r * scale);
+    let fg = 1.0 - exp(-g * scale);
+    let fb = 1.0 - exp(-b * scale);
+
+    var col: vec3<f32>;
+    if u.palette == PALETTE_YELLOW {
+        // fr is *not* a good stand-alone brightness signal: with c sampled
+        // uniformly over the whole viewport, nearly every sample outside the
+        // set escapes within a handful of iterations and splats a couple of
+        // points near itself, so fr is a near-uniform wash across the entire
+        // image (not concentrated near the boundary the way fb is) — adding
+        // it directly (tried first, both raw and gamma-lifted) drags that
+        // wash up to full brightness and floods the background with solid
+        // colour. Instead use it as a *multiplicative* warm (yellow) tint on
+        // top of fb's brightness, so it only shows up where fb is already
+        // bright (i.e. real near-boundary density) and stays near-zero across
+        // the background (fb ≈ 0 there, so warmth * fb ≈ 0 regardless of fr).
+        col = vec3<f32>(
+            fb + fb * fr * 1.3,
+            fb + fb * fr * 0.6,
+            fb,
+        );
+    } else if u.palette == PALETTE_GRAYSCALE {
+        // fb is the full escaping-orbit density (the cumulative superset);
+        // reuse it directly as a single luminance channel.
+        col = vec3<f32>(fb, fb, fb);
+    } else {
+        col = vec3<f32>(fr, fg, fb); // classic: raw per-cap R/G/B
+    }
+    return vec4<f32>(clamp(col, vec3<f32>(0.0), vec3<f32>(1.0)), 1.0);
+}

src/shaders/colorize.wgsl

@@ -0,0 +1,48 @@
+// Colourise pass: map the iteration pass's per-pixel escape data (from
+// `mandelbrot.wgsl`'s `fs_data`) through the palette. This is the only
+// color-dependent step, so changing the palette / colour scale / offset (e.g.
+// colour cycling) re-runs just this cheap pass — the expensive perturbation
+// iteration in the data texture is reused untouched.
+//
+// The data texture holds, per texel: R = ci (palette parameter), G = DE
+// darkening factor, B = interior fraction (for boundary anti-aliasing). It is
+// the same resolution as this pass's target, so we read it with `textureLoad`
+// at the fragment's integer pixel coordinate (nearest — iteration data must not
+// be linearly filtered across escape boundaries).
+
+@group(0) @binding(0) var<uniform> u: Uniforms;
+@group(0) @binding(1) var data_tex: texture_2d<f32>;
+@group(0) @binding(2) var<uniform> lights: array<Light, 16>;
+
+@vertex
+fn vs_main(@builtin(vertex_index) idx: u32) -> @builtin(position) vec4<f32> {
+    return vec4<f32>(fullscreen_triangle_pos(idx), 0.0, 1.0);
+}
+
+@fragment
+fn fs_main(@builtin(position) pos: vec4<f32>) -> @location(0) vec4<f32> {
+    if u.shadow != 0u {
+        let x = i32(pos.x);
+        let y = i32(pos.y);
+        if textureLoad(data_tex, vec2<i32>(x, y), 0).b != 0. {
+            return vec4<f32>(0.1, 0.1, 0.1, 1.0);
+        } else {
+            let h0 = textureLoad(data_tex, vec2<i32>(x, y), 0).g;
+            let h1 = textureLoad(data_tex, vec2<i32>(x + 1, y), 0).g;
+            let h2 = textureLoad(data_tex, vec2<i32>(x, y + 1), 0).g;
+            let normal = normal_from_heights(h0, h1, h2);
+            return vec4<f32>(shadow_color(normal), 1.0);
+        }
+    } else {
+        let d = textureLoad(data_tex, vec2<i32>(i32(pos.x), i32(pos.y)), 0);
+        let ci = d.r;
+        let de = d.g;
+        let interior_frac = d.b;
+
+        var col = classic_color(ci, de);
+        // Anti-alias the set boundary: fade toward black by the fraction of the
+        // pixel's sub-samples that landed in the interior.
+        col = col * (1.0 - interior_frac);
+        return vec4<f32>(col, 1.0);
+    }
+}

src/shaders/common.wgsl

@@ -0,0 +1,51 @@
+// Shared helpers, concatenated into every shader at build time via
+// `concat!`/`include_str!` (see renderer.rs / buddhabrot.rs). Keep this file
+// free of anything that differs between pipelines (e.g. a `Uniforms` struct —
+// mandelbrot/colorize and buddhabrot each have their own shape) since every
+// shader gets the whole thing spliced in.
+
+// Fullscreen triangle vertex position: one triangle that covers the whole
+// viewport (cheaper than a quad's two), shared by every full-screen vertex
+// shader in this project.
+fn fullscreen_triangle_pos(idx: u32) -> vec2<f32> {
+    var verts = array<vec2<f32>, 3>(
+        vec2<f32>(-1.0, -1.0),
+        vec2<f32>(3.0, -1.0),
+        vec2<f32>(-1.0, 3.0),
+    );
+    return verts[idx];
+}
+
+// Complex multiply.
+fn cmul(a: vec2<f32>, b: vec2<f32>) -> vec2<f32> {
+    return vec2<f32>(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x);
+}
+
+// z^p for a complex exponent p, via the principal branch z^p = exp(p * ln z),
+// ln z = ln|z| + i*arg(z). z = 0 maps to 0 (the correct limit for the
+// Re(p) > 0 region the UI exposes; ln(0) would otherwise be -inf).
+fn cpow(z: vec2<f32>, p: vec2<f32>) -> vec2<f32> {
+    let r2 = dot(z, z);
+    if r2 < 1e-30 {
+        return vec2<f32>(0.0, 0.0);
+    }
+    let ln_r = 0.5 * log(r2);
+    let theta = atan2(z.y, z.x);
+    let mag = exp(p.x * ln_r - p.y * theta);
+    let ang = p.x * theta + p.y * ln_r;
+    return mag * vec2<f32>(cos(ang), sin(ang));
+}
+
+// Iteration formula selector, shared by the perturbation (mandelbrot.wgsl)
+// and direct (buddhabrot.wgsl) iteration paths. Must match `FractalKind` in
+// reference.rs.
+const KIND_MANDELBROT: u32 = 0u;
+const KIND_BURNING_SHIP: u32 = 1u;
+const KIND_TRICORN: u32 = 2u;
+const KIND_MULTIBROT: u32 = 3u;
+const KIND_CELTIC: u32 = 4u;
+const KIND_PERPENDICULAR: u32 = 5u;
+const KIND_BUFFALO: u32 = 6u;
+const KIND_PHOENIX: u32 = 7u;
+const KIND_LAMBDA: u32 = 8u;
+const KIND_COMPLEX_MULTIBROT: u32 = 9u;

src/shaders/iterate_uniforms.wgsl

@@ -0,0 +1,161 @@
+// Shared by mandelbrot.wgsl (writes the per-pixel data texture) and
+// colorize.wgsl (reads it): the iteration pass and the colour remap pass
+// must agree on both the uniform layout and the palette function.
+
+// Must match the Rust `Uniforms` struct in renderer.rs field-for-field,
+// including padding.
+struct Uniforms {
+    span: vec2<f32>,
+    max_iter: u32,
+    ref_len: u32,
+    color_offset: f32,
+    color_scale: f32,
+    bailout_sq: f32,
+    is_julia: u32,
+    palette_id: u32,
+    shadow_palette_id: u32,
+    aa_level: u32,
+    // Iteration formula (see the KIND_* constants in common.wgsl).
+    kind: u32,
+    // Exponent for the Multibrot kind.
+    power: u32,
+    dc_offset: vec2<f32>,
+    // Distortion constant p for the Phoenix map (z^2 + c + p*z_{n-1}); unused
+    // by other kinds. Placed by dc_offset so both vec2s stay 8-byte aligned.
+    phoenix_p: vec2<f32>,
+    // Distortion constant l for the Lambda map (l*z(1 - z_{n-1})); unused
+    // by other kinds.
+    lambda_l: vec2<f32>,
+    // Complex exponent for the Complex Multibrot kind (z^power + c); unused
+    // by other kinds.
+    complex_power: vec2<f32>,
+    // 0 = escape-time coloring, 1 = distance-estimation shading.
+    de_coloring: u32,
+    // 0 = classic colors, 1 = shadows
+    shadow: u32,
+};
+
+// Smooth cyclic palettes (Inigo Quilez cosine palettes), selected by id.
+fn palette(id: u32, t: f32) -> vec3<f32> {
+    if id == 4u {
+        return vec3<f32>(t, t, t); // grayscale
+    }
+    let a = vec3<f32>(0.5, 0.5, 0.5);
+    let b = vec3<f32>(0.5, 0.5, 0.5);
+    var c = vec3<f32>(1.0, 1.0, 1.0);
+    var d = vec3<f32>(0.00, 0.10, 0.20); // 0: amber / blue
+    if id == 1u {
+        d = vec3<f32>(0.00, 0.33, 0.67); // rainbow
+    } else if id == 2u {
+        d = vec3<f32>(0.30, 0.20, 0.20); // warm ember
+    } else if id == 3u {
+        c = vec3<f32>(1.0, 1.0, 0.5);
+        d = vec3<f32>(0.80, 0.90, 0.30); // lime / magenta
+    }
+    return a + b * cos(6.28318530718 * (c * t + d));
+}
+
+// Classic (non-shadow) escape colouring: palette lookup at the smoothed
+// iteration count `ci`, darkened by the distance-estimate factor `de`
+// (sqrt-compressed so the darkening falls off more gently near the
+// boundary). Shared by the colourise pass's classic branch (colorize.wgsl,
+// applied to an already-averaged data texel) and the PNG-export pass
+// (mandelbrot.wgsl's `fs_color`, applied per sub-sample pre-AA) — the two
+// places a fully escaped point is turned into a final pixel colour.
+fn classic_color(ci: f32, de: f32) -> vec3<f32> {
+    let t = fract(ci * u.color_scale + u.color_offset);
+    return palette(u.palette_id, t) * sqrt(de);
+}
+
+// A single directional/point light, set by the UI's light list. `color`'s
+// alpha channel doubles as intensity (see `shadow_color`'s use of
+// `light_color.a`). Each shader that binds a `lights: array<Light, 16>`
+// uniform (colorize.wgsl, mandelbrot.wgsl's export shadow path) uses this
+// same layout.
+struct Light {
+    azimuth: f32,
+    altitude: f32,
+    color: u32,
+    _pad: u32,
+};
+
+fn compute_light(normal: vec3<f32>, light: vec3<f32>) -> vec3<f32> {
+    return vec3<f32>(max(0., dot(normal, normalize(light))));
+}
+
+fn uncharted2tonemap(x: vec3<f32>) -> vec3<f32> {
+    let A = 0.15; // Shoulder strength
+    let B = 0.50; // Linear strength
+    let C = 0.10; // Linear angle
+    let D = 0.20; // Toe strength
+    let E = 0.02; // Toe numerator / shoarder angle/etc.
+    let F = 0.30; // Toe denominator
+
+    return ((x * (A * x + C * B) + D * E) / (x * (A * x + B) + D * F)) - E / F;
+}
+
+fn filmic(color: vec3<f32>, white_point: f32) -> vec3<f32> {
+    let exposure_bias = 2.0;
+    let curr = uncharted2tonemap(color * exposure_bias);
+
+    // Valeur blanche maximale de référence
+    let white_scale = vec3(1.0) / uncharted2tonemap(vec3(white_point));
+    return curr * white_scale;
+}
+
+fn s(color: vec3<f32>, k: f32, c: f32) -> vec3<f32> {
+    return 1. / (1. + exp(-k * (color - c)));
+}
+
+fn contrast(color: vec3<f32>, k: f32, c: f32) -> vec3<f32> {
+    let color_c = s(color, k, c);
+
+    return (color_c - s(vec3<f32>(0), k, c)) / (s(vec3<f32>(1), k, c) - s(vec3<f32>(0), k, c));
+}
+
+// Surface normal from three height samples (`h0` at the pixel, `h1` one pixel
+// to the right, `h2` one pixel down), treating DE as a height field. Only the
+// differences matter, so callers don't need to pass pixel coordinates — a
+// texture-backed caller (colorize.wgsl) and a live-sampled caller
+// (mandelbrot.wgsl's export shadow path) can share this.
+fn normal_from_heights(h0: f32, h1: f32, h2: f32) -> vec3<f32> {
+    let d0 = vec3<f32>(0.0, 0.0, h0);
+    let d1 = vec3<f32>(1.0, 0.0, h1);
+    let d2 = vec3<f32>(0.0, 1.0, h2);
+    return normalize(cross(d1 - d0, d2 - d0));
+}
+
+// Shade a DE-derived surface normal per `u.shadow_palette_id`: 0 = grayscale
+// key light, 1 = red/blue two-tone, 2 = the user's custom `lights` list.
+// Shared by the interactive shadow pass (colorize.wgsl) and the PNG-export
+// shadow path (mandelbrot.wgsl's `fs_color`), which must render identically.
+fn shadow_color(normal: vec3<f32>) -> vec3<f32> {
+    var color: vec3<f32>;
+    if u.shadow_palette_id == 0u {
+        color = compute_light(normal, vec3<f32>(.5, .5, .5)) + vec3<f32>(0.58, 0.85, 1.) * 0.2;
+
+        color = filmic(color, 2.5);
+        color = contrast(color, 4., 0.67);
+    } else if u.shadow_palette_id == 1u {
+        color = compute_light(normal, vec3<f32>(0., .5, .5)) * vec3<f32>(1., 0.5, 0.5) + compute_light(normal, vec3<f32>(0.5, 0., .5)) * vec3<f32>(0.5, 1., 1.);
+
+        color = filmic(color, 4.2);
+    } else {
+        color = vec3<f32>(0);
+        var light_count = 0;
+        for (var i = 0u; i < 16; i++) {
+            let light_color = unpack4x8unorm(lights[i].color);
+            if any(light_color != vec4<f32>(0)) {
+                light_count += 1;
+            }
+
+            color += compute_light(normal, vec3<f32>(
+                cos(lights[i].azimuth) * cos(lights[i].altitude),
+                sin(lights[i].azimuth) * cos(lights[i].altitude),
+                sin(lights[i].altitude))) * light_color.xyz * light_color.a;
+        }
+
+        color = filmic(color, 1. + f32(light_count));
+    }
+    return color;
+}

src/shaders/mandelbrot.wgsl

@@ -12,32 +12,11 @@
 // the reference index to 0 and carry the full value as the new delta (valid
 // because X_0 = 0).
 
-struct Uniforms {
-    span: vec2<f32>,
-    max_iter: u32,
-    ref_len: u32,
-    color_offset: f32,
-    color_scale: f32,
-    bailout_sq: f32,
-    is_julia: u32,
-    palette_id: u32,
-    aa_level: u32,
-    // Iteration formula: 0 Mandelbrot, 1 Burning Ship, 2 Tricorn, 3 Multibrot.
-    kind: u32,
-    // Exponent for the Multibrot kind.
-    power: u32,
-    dc_offset: vec2<f32>,
-    // 0 = escape-time coloring, 1 = distance-estimation shading.
-    de_coloring: u32,
-};
-
-const KIND_MANDELBROT: u32 = 0u;
-const KIND_BURNING_SHIP: u32 = 1u;
-const KIND_TRICORN: u32 = 2u;
-const KIND_MULTIBROT: u32 = 3u;
-
 @group(0) @binding(0) var<uniform> u: Uniforms;
 @group(0) @binding(1) var<storage, read> ref_orbit: array<vec2<f32>>;
+// Only read by `fs_color`'s shadow branch (custom-lights palette); the
+// iteration pass (`fs_data`) never touches it.
+@group(0) @binding(2) var<uniform> lights: array<Light, 16>;
 
 struct VsOut {
     @builtin(position) pos: vec4<f32>,
@@ -47,12 +26,7 @@ struct VsOut {
 
 @vertex
 fn vs_main(@builtin(vertex_index) idx: u32) -> VsOut {
-    var verts = array<vec2<f32>, 3>(
-        vec2<f32>(-1.0, -1.0),
-        vec2<f32>(3.0, -1.0),
-        vec2<f32>(-1.0, 3.0),
-    );
-    let ndc = verts[idx];
+    let ndc = fullscreen_triangle_pos(idx);
     var out: VsOut;
     out.pos = vec4<f32>(ndc, 0.0, 1.0);
     // Flip y so +imaginary points up the screen.
@@ -60,22 +34,23 @@ fn vs_main(@builtin(vertex_index) idx: u32) -> VsOut {
     return out;
 }
 
-// Complex multiply.
-fn cmul(a: vec2<f32>, b: vec2<f32>) -> vec2<f32> {
-    return vec2<f32>(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x);
-}
-
 // Complex conjugate.
 fn conj(a: vec2<f32>) -> vec2<f32> {
     return vec2<f32>(a.x, -a.y);
 }
 
+// Complex division a / b.
+fn cdiv(a: vec2<f32>, b: vec2<f32>) -> vec2<f32> {
+    let d = dot(b, b);
+    return vec2<f32>(a.x * b.x + a.y * b.y, a.y * b.x - a.x * b.y) / d;
+}
+
 // |c + d| - |c|, evaluated exactly (no catastrophic cancellation even when the
 // sum crosses zero). This is what makes the Burning Ship delta correct through
 // the sign flips that happen all along the axes, where the ship's detail lives.
 fn diffabs(c: f32, d: f32) -> f32 {
     let cd = c + d;
-    if (c >= 0.0) {
+    if c >= 0.0 {
         return select(-(2.0 * c + d), d, cd >= 0.0);
     }
     return select(-d, 2.0 * c + d, cd > 0.0);
@@ -109,11 +84,52 @@ fn multibrot_delta(z: vec2<f32>, e: vec2<f32>, p: u32) -> vec2<f32> {
     return acc;
 }
 
+// Number of terms kept in `complex_multibrot_delta`'s series. Truncation, not
+// exactness: unlike `multibrot_delta` (a finite binomial sum for an integer
+// power), a complex power has no finite expansion, so this converges rather
+// than terminates. Fine as long as perturbation's usual invariant (|e| << |z|,
+// kept true by rebasing) holds, since each extra term is O(w^k) smaller.
+const COMPLEX_MULTIBROT_TERMS: u32 = 16u;
+
+// Perturbation delta for z -> z^p with a complex p: (Z+e)^p - Z^p.
+//
+// When |e| << |Z| (the common case: it's the whole reason perturbation
+// works), forming Z+e directly would round e away in f32, so instead expand
+// = Z^p * ((1+w)^p - 1), w = e/Z, as a Taylor series in w: (1+w)^p - 1 =
+// sum_{k=1}^N C(p,k) w^k, with the complex binomial coefficient built up
+// incrementally: C(p,k) = C(p,k-1) * (p-(k-1)) / k. Unlike `multibrot_delta`
+// (a finite binomial sum for an integer power), this only *converges* — and
+// only for |w| < 1 — rather than terminating exactly.
+//
+// Right after a rebase (or near a reference point close to zero, where w is
+// singular), e is *not* small relative to Z — that's normal perturbation
+// dynamics, not a deep-zoom edge case — and the series above would diverge.
+// But forming Z+e directly is numerically safe exactly there (e isn't many
+// orders of magnitude smaller than Z), so fall back to a plain subtraction.
+fn complex_multibrot_delta(z: vec2<f32>, e: vec2<f32>, p: vec2<f32>) -> vec2<f32> {
+    // |w|^2 = |e|^2 / |Z|^2; inf or nan (Z ~ 0, or both ~ 0) correctly fails
+    // the `< 0.25` test below and falls through to the direct branch.
+    let w2 = dot(e, e) / dot(z, z);
+    if w2 < 0.25 {
+        let w = cdiv(e, z);
+        var wk = vec2<f32>(1.0, 0.0); // w^0
+        var coef = vec2<f32>(1.0, 0.0); // C(p,0)
+        var acc = vec2<f32>(0.0, 0.0);
+        for (var k: u32 = 1u; k <= COMPLEX_MULTIBROT_TERMS; k = k + 1u) {
+            coef = cdiv(cmul(coef, p - vec2<f32>(f32(k - 1u), 0.0)), vec2<f32>(f32(k), 0.0));
+            wk = cmul(wk, w);
+            acc = acc + cmul(coef, wk);
+        }
+        return cmul(cpow(z, p), acc);
+    }
+    return cpow(z + e, p) - cpow(z, p);
+}
+
 // One perturbation step of the current fractal's delta: e -> f(Z+e) - f(Z),
 // where `z` is the reference orbit value X_m. `step_add` (dc) is added by the
 // caller. Must match `FractalKind` on the CPU side.
 fn advance_delta(z: vec2<f32>, e: vec2<f32>) -> vec2<f32> {
-    if (u.kind == KIND_BURNING_SHIP) {
+    if u.kind == KIND_BURNING_SHIP {
         // (|x| + i|y|)^2 has real part x^2 - y^2 (an ordinary square delta) and
         // imaginary part 2|x y|. The imaginary delta is 2(|x y| - |X Y|); diffabs
         // computes it exactly, even where the product x y changes sign — which the
@@ -122,14 +138,37 @@ fn advance_delta(z: vec2<f32>, e: vec2<f32>) -> vec2<f32> {
         let base = 2.0 * cmul(z, e) + cmul(e, e);
         let dp = z.x * e.y + z.y * e.x + e.x * e.y;
         return vec2<f32>(base.x, 2.0 * diffabs(z.x * z.y, dp));
-    } else if (u.kind == KIND_TRICORN) {
+    } else if u.kind == KIND_TRICORN {
         let cz = conj(z);
         let ce = conj(e);
         return 2.0 * cmul(cz, ce) + cmul(ce, ce);
-    } else if (u.kind == KIND_MULTIBROT) {
+    } else if u.kind == KIND_MULTIBROT {
         return multibrot_delta(z, e, clamp(u.power, 2u, 8u));
+    } else if u.kind == KIND_CELTIC {
+        // z^2 delta split: sq.x = delta of Re(z^2), sq.y = delta of Im(z^2).
+        // Celtic abs the real output, so |Re(z^2)| delta = diffabs(Re(Z^2), sq.x).
+        let sq = 2.0 * cmul(z, e) + cmul(e, e);
+        return vec2<f32>(diffabs(z.x * z.x - z.y * z.y, sq.x), sq.y);
+    } else if u.kind == KIND_BUFFALO {
+        // Abs both outputs: real |Re(z^2)|, imag -|Im(z^2)| (Im(Z^2) = 2 X Y).
+        let sq = 2.0 * cmul(z, e) + cmul(e, e);
+        return vec2<f32>(diffabs(z.x * z.x - z.y * z.y, sq.x),
+                         -diffabs(2.0 * z.x * z.y, sq.y));
+    } else if u.kind == KIND_PERPENDICULAR {
+        // real x^2 - y^2 (ordinary square delta), imag -2 x |y|.
+        // d(-2 x |y|) = -2[ X·(|Y+ey|-|Y|) + ex·|Y+ey| ]; diffabs gives |Y+ey|-|Y|.
+        let sq = 2.0 * cmul(z, e) + cmul(e, e);
+        let da = diffabs(z.y, e.y);        // |Y + ey| - |Y|
+        let abs_yf = abs(z.y) + da;        // |Y + ey|
+        return vec2<f32>(sq.x, -2.0 * (z.x * da + e.x * abs_yf));
+    } else if u.kind == KIND_LAMBDA {
+        // Lambda map: z^{n+1} = λ·z·(1-z). Delta: e = λ·e·(1-2z-e).
+        let one_minus_2z_minus_e = vec2<f32>(1.0 - 2.0 * z.x - e.x, -2.0 * z.y - e.y);
+        return cmul(u.lambda_l, cmul(e, one_minus_2z_minus_e));
+    } else if u.kind == KIND_COMPLEX_MULTIBROT {
+        return complex_multibrot_delta(z, e, u.complex_power);
     }
-    return 2.0 * cmul(z, e) + cmul(e, e); // Mandelbrot
+    return 2.0 * cmul(z, e) + cmul(e, e); // Mandelbrot (and Phoenix square part)
 }
 
 // Derivative f'(Z) of the iteration map at the full value Z, used to propagate
@@ -138,43 +177,38 @@ fn advance_delta(z: vec2<f32>, e: vec2<f32>) -> vec2<f32> {
 // Burning Ship / Tricorn we use |f'| ~ |2Z|, which keeps the DE magnitude close
 // enough to de-speckle filaments.
 fn fprime(z: vec2<f32>) -> vec2<f32> {
-    if (u.kind == KIND_MULTIBROT) {
+    if u.kind == KIND_MULTIBROT {
         let p = clamp(u.power, 2u, 8u);
         var zk = vec2<f32>(1.0, 0.0); // Z^0
         for (var k: u32 = 1u; k < p; k = k + 1u) {
             zk = cmul(zk, z); // -> Z^{p-1}
         }
         return f32(p) * zk;
+    } else if u.kind == KIND_LAMBDA {
+        // Lambda: f'(z) = λ·(1-2z).
+        return cmul(u.lambda_l, vec2<f32>(1.0 - 2.0 * z.x, -2.0 * z.y));
+    } else if u.kind == KIND_COMPLEX_MULTIBROT {
+        // f'(z) = p * z^(p-1).
+        return cmul(u.complex_power, cpow(z, u.complex_power - vec2<f32>(1.0, 0.0)));
     }
     return 2.0 * z;
 }
 
-// Smooth cyclic palettes (Inigo Quilez cosine palettes), selected by id.
-fn palette(id: u32, t: f32) -> vec3<f32> {
-    if (id == 4u) {
-        return vec3<f32>(t, t, t); // grayscale
-    }
-    let a = vec3<f32>(0.5, 0.5, 0.5);
-    let b = vec3<f32>(0.5, 0.5, 0.5);
-    var c = vec3<f32>(1.0, 1.0, 1.0);
-    var d = vec3<f32>(0.00, 0.10, 0.20); // 0: amber / blue
-    if (id == 1u) {
-        d = vec3<f32>(0.00, 0.33, 0.67); // rainbow
-    } else if (id == 2u) {
-        d = vec3<f32>(0.30, 0.20, 0.20); // warm ember
-    } else if (id == 3u) {
-        c = vec3<f32>(1.0, 1.0, 0.5);
-        d = vec3<f32>(0.80, 0.90, 0.30); // lime / magenta
-    }
-    return a + b * cos(6.28318530718 * (c * t + d));
-}
+// Escape data for one sample: `ci` is the (color-independent) palette parameter,
+// `de` the distance-estimate darkening factor in [0,1], `escaped` false for the
+// interior of the set. Splitting iteration from coloring lets a colour change be
+// remapped cheaply (see the colourise pass) without re-iterating.
+struct Sample {
+    ci: f32,
+    de: f32,
+    escaped: bool,
+};
 
-// Perturbation iterate + color a single sample. `offset` is the per-pixel
-// offset in complex units. For Mandelbrot it is the c-plane offset added every
-// step (delta starts at 0); for Julia it is the z-plane offset that seeds the
-// initial delta (c is fixed, so nothing is added per step). Interior pixels
-// return black.
-fn shade(offset: vec2<f32>, px: f32) -> vec3<f32> {
+// Perturbation iterate a single sample. `offset` is the per-pixel offset in
+// complex units. For Mandelbrot it is the c-plane offset added every step (delta
+// starts at 0); for Julia it is the z-plane offset that seeds the initial delta
+// (c is fixed, so nothing is added per step).
+fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
     let z0 = ref_orbit[0]; // reference start (0 for Mandelbrot, center for Julia)
 
     var step_add = offset;
@@ -183,7 +217,11 @@ fn shade(offset: vec2<f32>, px: f32) -> vec3<f32> {
     // (starts at 0, gains +1 each step); for Julia it is d/dz0 (starts at 1).
     var dz = vec2<f32>(0.0, 0.0);
     var dz_seed = vec2<f32>(1.0, 0.0);
-    if (u.is_julia != 0u) {
+    // Previous-iterate state for the Phoenix two-term recurrence (delta of
+    // y_{n-1}, and its derivative for DE). Both start at 0 (y_{-1} = 0).
+    var e_prev = vec2<f32>(0.0, 0.0);
+    var dz_prev = vec2<f32>(0.0, 0.0);
+    if u.is_julia != 0u {
         step_add = vec2<f32>(0.0, 0.0);
         e = offset;
         dz = vec2<f32>(1.0, 0.0);
@@ -200,27 +238,39 @@ fn shade(offset: vec2<f32>, px: f32) -> vec3<f32> {
             z = xm + e;
 
             let z2 = dot(z, z);
-        if (z2 > u.bailout_sq) {
+            if z2 > u.bailout_sq {
                 escaped = true;
                 break;
             }
-        if (n >= u.max_iter) {
+            if n >= u.max_iter {
                 break; // interior
             }
 
         // Propagate the derivative of the full orbit (unaffected by rebasing,
         // which only re-expresses the same value). Only when DE is enabled.
-        if (u.de_coloring != 0u) {
-            dz = cmul(fprime(z), dz) + dz_seed;
+        // Phoenix's two-term map adds p·dz_{n-1} and carries the previous dz.
+            if u.de_coloring != 0u {
+                var dz_new = cmul(fprime(z), dz) + dz_seed;
+                if u.kind == KIND_PHOENIX {
+                    dz_new = dz_new + cmul(u.phoenix_p, dz_prev);
+                    dz_prev = dz;
+                }
+                dz = dz_new;
             }
 
         // Advance the delta by this fractal's formula (+ dc for the set plane).
+        // Phoenix additionally adds p·e_{n-1} and carries the previous delta.
+            let e_old = e;
             e = advance_delta(xm, e) + step_add;
+            if u.kind == KIND_PHOENIX {
+                e = e + cmul(u.phoenix_p, e_prev);
+                e_prev = e_old;
+            }
             m = m + 1u;
             n = n + 1u;
 
         // Keep the reference index valid and the delta small.
-        if (m >= u.ref_len) {
+            if m >= u.ref_len {
             // Reference exhausted: any pixel that followed it this far has
             // effectively escaped (interior pixels rebase before reaching here).
                 z = ref_orbit[u.ref_len - 1u] + e;
@@ -228,16 +278,21 @@ fn shade(offset: vec2<f32>, px: f32) -> vec3<f32> {
                 break;
             }
             let y = ref_orbit[m] + e;
-        if (dot(y, y) < dot(e, e)) {
+            if dot(y, y) < dot(e, e) {
             // Rebase to index 0: carry the full value as the new delta. Valid
             // because y_n = X[0] + (y_n - X[0]); for Mandelbrot X[0]=0.
+            // Phoenix: after rebasing the implied previous reference is Y[-1]=0,
+            // so the previous delta becomes the full previous value y_n (= z).
+                if u.kind == KIND_PHOENIX {
+                    e_prev = z;
+                }
                 e = y - z0;
                 m = 0u;
             }
         }
 
-    if (!escaped) {
-        return vec3<f32>(0.0, 0.0, 0.0); // interior of the set
+    if !escaped {
+        return Sample(0.0, 1.0, false); // interior of the set
     }
 
     let z2 = dot(z, z);
@@ -250,10 +305,9 @@ fn shade(offset: vec2<f32>, px: f32) -> vec3<f32> {
     // sqrt compresses the huge iteration counts of deep zooms so the palette
     // varies smoothly instead of aliasing into speckle.
     let ci = sqrt(max(smooth_i, 0.0));
-    let t = fract(ci * u.color_scale + u.color_offset);
-    var col = palette(u.palette_id, t);
 
-    if (u.de_coloring != 0u) {
+    var de = 1.0;
+    if u.de_coloring != 0u {
         // Exterior distance estimate (complex-plane units): |z|·ln|z| / |dz|.
         // Divided by the pixel footprint it becomes a distance in pixels; we
         // darken toward the boundary (< ~1 px away) so filaments stay crisp
@@ -261,37 +315,104 @@ fn shade(offset: vec2<f32>, px: f32) -> vec3<f32> {
         // boundary simply reads as dark, which is the correct limit.
         let zmag = sqrt(max(z2, 1.0));
         let dzmag = sqrt(max(dot(dz, dz), 1e-20));
-        let de = zmag * log(zmag) / dzmag;
-        let de_px = de / max(px, 1e-30);
-        col = col * clamp(de_px, 0.0, 1.0);
+        let d = zmag * log(zmag) / dzmag;
+        var max_de = 1.;
+        if u.shadow != 0u {
+            max_de = 1000.;
         }
-    return col;
+        de = clamp(d / max(px, 1e-30), 0.0, max_de);
+    }
+    return Sample(ci, de, true);
 }
 
-@fragment
-fn fs_main(in: VsOut) -> @location(0) vec4<f32> {
-    let base = in.centered * u.span + u.dc_offset;
-
-    // Screen-space complex-units-per-pixel. Derivatives must be evaluated in
-    // uniform control flow, so take them here; used to place sub-pixel AA
-    // samples and to convert the distance estimate into pixels.
-    let dx = dpdx(base);
-    let dy = dpdy(base);
-    let px = length(abs(dx) + abs(dy)); // ~ complex units per pixel (footprint)
+// Map a sample's escape data through the palette (+ DE darkening). This is the
+// only color-dependent step, so it can be redone without re-iterating. Interior
+// samples are black.
+fn color_sample(s: Sample) -> vec3<f32> {
+    if !s.escaped {
+        return vec3<f32>(0.0, 0.0, 0.0);
+    }
+    return classic_color(s.ci, s.de);
+}
 
+// Supersampled escape data at one point: average (ci, DE factor) over the
+// AA grid's escaped sub-samples, plus the fraction that landed in the
+// interior. Shared by `fs_data` (writes it straight to the data texture) and
+// `fs_color`'s shadow branch (used both at the pixel and at its two
+// neighbours, to build a DE height field without a texture round-trip).
+fn aggregate_sample(base: vec2<f32>, dx: vec2<f32>, dy: vec2<f32>, px: f32) -> vec3<f32> {
     let aa = max(u.aa_level, 1u);
-    if (aa <= 1u) {
-        return vec4<f32>(shade(base, px), 1.0);
-    }
-
-    var acc = vec3<f32>(0.0, 0.0, 0.0);
     let inv = 1.0 / f32(aa);
+    var ci_sum = 0.0;
+    var de_sum = 0.0;
+    var escaped_n = 0u;
     for (var sy: u32 = 0u; sy < aa; sy = sy + 1u) {
         for (var sx: u32 = 0u; sx < aa; sx = sx + 1u) {
-            // Sample centers evenly spread across the pixel, jitter in (-0.5, 0.5).
             let jx = (f32(sx) + 0.5) * inv - 0.5;
             let jy = (f32(sy) + 0.5) * inv - 0.5;
-            acc = acc + shade(base + jx * dx + jy * dy, px);
+            let s = iterate_sample(base + jx * dx + jy * dy, px);
+            if s.escaped {
+                ci_sum = ci_sum + s.ci;
+                de_sum = de_sum + s.de;
+                escaped_n = escaped_n + 1u;
+            }
+        }
+    }
+    let total = f32(aa * aa);
+    let ci_avg = select(0.0, ci_sum / f32(escaped_n), escaped_n > 0u);
+    let de_avg = select(1.0, de_sum / f32(escaped_n), escaped_n > 0u);
+    let interior_frac = 1.0 - f32(escaped_n) / total;
+    return vec3<f32>(ci_avg, de_avg, interior_frac);
+}
+
+// Iteration pass: write per-pixel escape data (color-independent) so a colour
+// change is remapped by the cheap colourise pass without re-iterating.
+//   R = ci (palette parameter), G = DE factor, B = interior fraction (for AA).
+// AA is grid-supersampled here; the interior fraction lets the colourise pass
+// anti-alias the set boundary (blend toward black) after the fact.
+@fragment
+fn fs_data(in: VsOut) -> @location(0) vec4<f32> {
+    let base = in.centered * u.span + u.dc_offset;
+    let dx = dpdx(base);
+    let dy = dpdy(base);
+    let px = length(abs(dx) + abs(dy));
+
+    return vec4<f32>(aggregate_sample(base, dx, dy, px), 1.0);
+}
+
+// Combined iterate + colour in a single pass, for PNG export (which never needs
+// incremental recolouring). The interactive path uses fs_data + the colourise
+// pass so colour changes skip iteration.
+@fragment
+fn fs_color(in: VsOut) -> @location(0) vec4<f32> {
+    let base = in.centered * u.span + u.dc_offset;
+    let dx = dpdx(base);
+    let dy = dpdy(base);
+    let px = length(abs(dx) + abs(dy));
+
+    if u.shadow != 0u {
+        // No data texture to sample neighbours from (this pass never runs
+        // one), so build the same DE height field colorize.wgsl reads from
+        // the texture by aggregating live, at the pixel and its two
+        // neighbours a `dx`/`dy` step away.
+        let here = aggregate_sample(base, dx, dy, px);
+        if here.z != 0.0 {
+            return vec4<f32>(0.1, 0.1, 0.1, 1.0);
+        }
+        let right = aggregate_sample(base + dx, dx, dy, px);
+        let down = aggregate_sample(base + dy, dx, dy, px);
+        let normal = normal_from_heights(here.y, right.y, down.y);
+        return vec4<f32>(shadow_color(normal), 1.0);
+    }
+
+    let aa = max(u.aa_level, 1u);
+    let inv = 1.0 / f32(aa);
+    var acc = vec3<f32>(0.0, 0.0, 0.0);
+    for (var sy: u32 = 0u; sy < aa; sy = sy + 1u) {
+        for (var sx: u32 = 0u; sx < aa; sx = sx + 1u) {
+            let jx = (f32(sx) + 0.5) * inv - 0.5;
+            let jy = (f32(sy) + 0.5) * inv - 0.5;
+            acc = acc + color_sample(iterate_sample(base + jx * dx + jy * dy, px));
         }
     }
     return vec4<f32>(acc / f32(aa * aa), 1.0);

src/view.rs

@@ -82,7 +82,7 @@ impl ViewState {
         let bits = self.precision_bits();
         // Grab-and-drag: moving the mouse right shows content to the left.
         self.center_re = &self.center_re - &big_from_f64(dx * cpp, bits);
-        self.center_im = &self.center_im - &big_from_f64(dy * cpp, bits); // y-down -> imag-up
+        self.center_im = &self.center_im - &big_from_f64(dy * cpp, bits);
     }
 
     /// Zoom by `factor` (<1 zooms in) keeping the complex point currently under
@@ -97,7 +97,7 @@ impl ViewState {
         // off * cpp * (1 - factor). (Derivation: new_c = fixed + (c-fixed)*f.)
         let k = cpp * (1.0 - factor);
         self.center_re = &self.center_re + &big_from_f64(off_x * k, bits);
-        self.center_im = &self.center_im + &big_from_f64(off_y * k, bits); // y flip
+        self.center_im = &self.center_im + &big_from_f64(off_y * k, bits);
         self.half_height *= factor;
     }
 

src/worker.rs

@@ -22,6 +22,12 @@ pub struct RefRequest {
     pub precision: usize,
     pub kind: FractalKind,
     pub power: u32,
+    /// Distortion constant for the Phoenix map (ignored by other kinds).
+    pub phoenix_p: (f64, f64),
+    /// Distortion constant for the Lambda map (ignored by other kinds).
+    pub lambda_l: (f64, f64),
+    /// Complex exponent for the Complex Multibrot kind (ignored by other kinds).
+    pub complex_power: (f64, f64),
 }
 
 pub struct RefResult {
@@ -101,6 +107,9 @@ fn compute(req: &RefRequest) -> Vec<[f32; 2]> {
             req.precision,
             req.kind,
             req.power,
+            req.phoenix_p,
+            req.lambda_l,
+            req.complex_power,
         )
     } else {
         compute_set_reference(
@@ -110,6 +119,9 @@ fn compute(req: &RefRequest) -> Vec<[f32; 2]> {
             req.precision,
             req.kind,
             req.power,
+            req.phoenix_p,
+            req.lambda_l,
+            req.complex_power,
         )
     }
 }

tests/shader_valid.rs

@@ -21,11 +21,44 @@ fn validate(name: &str, src: &str) {
 fn mandelbrot_shader_is_valid() {
     validate(
         "mandelbrot.wgsl",
+        concat!(
+            include_str!("../src/shaders/common.wgsl"),
+            include_str!("../src/shaders/iterate_uniforms.wgsl"),
             include_str!("../src/shaders/mandelbrot.wgsl"),
+        ),
+    );
+}
+
+#[test]
+fn colorize_shader_is_valid() {
+    validate(
+        "colorize.wgsl",
+        concat!(
+            include_str!("../src/shaders/common.wgsl"),
+            include_str!("../src/shaders/iterate_uniforms.wgsl"),
+            include_str!("../src/shaders/colorize.wgsl"),
+        ),
     );
 }
 
 #[test]
 fn blit_shader_is_valid() {
-    validate("blit.wgsl", include_str!("../src/shaders/blit.wgsl"));
+    validate(
+        "blit.wgsl",
+        concat!(
+            include_str!("../src/shaders/common.wgsl"),
+            include_str!("../src/shaders/blit.wgsl"),
+        ),
+    );
+}
+
+#[test]
+fn buddhabrot_shader_is_valid() {
+    validate(
+        "buddhabrot.wgsl",
+        concat!(
+            include_str!("../src/shaders/common.wgsl"),
+            include_str!("../src/shaders/buddhabrot.wgsl"),
+        ),
+    );
 }