feat: improve CLI

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

CLAUDE.md

@@ -31,10 +31,16 @@ cargo install wasm-bindgen-cli --version 0.2.128   # must match the wasm-bindgen
 python3 -m http.server -d dist 8080
 ```
 
-Native debug env vars (see `src/app.rs`, near the top of `FractalApp::new`):
-`MANDEL_KIND`, `MANDEL_POWER`, `MANDEL_JULIA="re,im"`, `MANDEL_SHARE="<fragment>"`,
-`MANDEL_VIEW="re,im,half_height[,iterations]"`, `MANDEL_DE=1`,
-`MANDEL_BUDDHABROT=1`, `MANDEL_EXPORT=1` (+ `MANDEL_EXPORT_PATH=out.png`).
+Native CLI flags (`src/cli.rs`, applied in `FractalApp::apply_cli`): `--kind`,
+`--power`, `--julia re,im`, `--phoenix-p re,im`, `--lambda-l re,im`,
+`--palette`, `--share <fragment>`,
+`--view re,im,half_height[,iterations]`, `--de`, `--buddhabrot`,
+`--buddha-palette`. `--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, `--export-path out.png`) 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
@@ -56,11 +62,27 @@ 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/fractal/kind.rs` — the `FractalKind` enum (Mandelbrot, Burning Ship,
+  Tricorn, Multibrot, Celtic, Perpendicular, Buffalo, Phoenix, Lambda,
+  Complex Multibrot) plus everything that only needs to switch on it:
+  `label`/`description`/`formula` (UI text), `share_tag`/`from_share_tag`
+  (share-link encoding), `default_set_view` (per-kind starting view), and the
+  `ALL` array used to enumerate every kind.
+- `src/fractal/reference.rs` — `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
@@ -70,8 +92,8 @@ pixel is a handful of `f32` complex multiplies.
   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 here must match the matching
-  `FractalKind` variant's discriminant exactly.
+  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
@@ -89,28 +111,31 @@ pixel is a handful of `f32` complex multiplies.
   `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).
+  `default_view_for` (wraps `FractalKind::default_set_view`, adding the
+  kind-independent Julia case). `JULIA_PRESETS` and `SET_PRESETS` are sized as
+  `[T; FractalKind::<last variant> as usize + 1]` — adding a new `FractalKind`
+  means bumping both (and adding an empty `&[]` slot to each if the kind has
+  none), plus adding it to `FractalKind::ALL` in `kind.rs`.
 - `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), `mandelbrot.wgsl` (matching
-`KIND_*` const + `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.
+Touches, in order: `kind.rs` (enum variant + `ALL` slot + `label`/
+`description`/`formula`/`share_tag`/`from_share_tag`/`default_set_view`
+arms), `reference.rs` (CPU iteration formula arm, 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`), `app.rs` (`JULIA_PRESETS`/`SET_PRESETS` slot,
+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
 

Cargo.toml

@@ -14,6 +14,7 @@ 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"] }

src/cli.rs

@@ -17,10 +17,26 @@ pub struct Cli {
     #[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>,
 
+    /// Coloring palette index.
+    #[arg(long, value_name = "INDEX")]
+    pub palette: Option<u32>,
+
+    /// Phoenix constant p for the Phoenix kind (z -> z^2 + c + p*z_prev).
+    #[arg(long, value_name = "RE,IM")]
+    pub phoenix_p: Option<String>,
+
+    /// Lambda constant λ for the Lambda kind (z -> λ*z*(1 - z)).
+    #[arg(long, value_name = "RE,IM")]
+    pub lambda_l: Option<String>,
+
     /// Restore a view from a share-link fragment (the part after '#').
     #[arg(long, value_name = "FRAGMENT")]
     pub share: Option<String>,
@@ -41,13 +57,23 @@ pub struct Cli {
     #[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 (default: fractal-<timestamp>.png).
+    /// Output path for --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)]
@@ -65,6 +91,8 @@ pub enum KindArg {
     Buffalo,
     Phoenix,
     Lambda,
+    #[value(alias = "cmulti")]
+    ComplexMultibrot,
 }
 
 impl From<KindArg> for FractalKind {
@@ -79,6 +107,7 @@ impl From<KindArg> for FractalKind {
             KindArg::Buffalo => FractalKind::Buffalo,
             KindArg::Phoenix => FractalKind::Phoenix,
             KindArg::Lambda => FractalKind::Lambda,
+            KindArg::ComplexMultibrot => FractalKind::ComplexMultibrot,
         }
     }
 }

src/fractal/buddhabrot.rs

@@ -46,7 +46,11 @@ pub struct BuddhabrotUniforms {
     /// (yellow core, blue halo), 2 = grayscale. Display-only, like `exposure`
     /// — excluded from `ContentKey` so changing it doesn't reset accumulation.
     pub palette: u32,
-    pub _pad: [u32; 3],
+    /// 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
@@ -62,6 +66,7 @@ struct ContentKey {
     bailout_sq: f32,
     kind: u32,
     power: u32,
+    complex_power: [f32; 2],
     r_cap: u32,
     g_cap: u32,
     b_cap: u32,
@@ -78,6 +83,7 @@ impl From<&BuddhabrotUniforms> for ContentKey {
             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,
@@ -114,7 +120,13 @@ 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(include_str!("../shaders/buddhabrot.wgsl").into()),
+            source: wgpu::ShaderSource::Wgsl(
+                concat!(
+                    include_str!("../shaders/common.wgsl"),
+                    include_str!("../shaders/buddhabrot.wgsl"),
+                )
+                .into(),
+            ),
         });
 
         let uniform_buffer = device.create_buffer(&wgpu::BufferDescriptor {

src/fractal/kind.rs

@@ -0,0 +1,165 @@
+//! `FractalKind`: the enum selecting which iteration formula is in use, plus
+//! everything that only needs to switch on it (UI label/description/formula
+//! text, share-link tag, default parameter-plane view). The CPU/GPU orbit
+//! math itself lives in `reference.rs` (CPU reference orbit) and
+//! `shaders/mandelbrot.wgsl` (GPU perturbation delta) since both must also
+//! stay in sync with `common.wgsl`'s `KIND_*` constants.
+
+/// 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 = 0,
+    /// `z -> (|Re z| + i|Im z|)^2 + c`.
+    BurningShip = 1,
+    /// `z -> conj(z)^2 + c` (the Mandelbar).
+    Tricorn = 2,
+    /// `z -> z^power + c` (power >= 2).
+    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 {
+    /// Every kind, in declaration/discriminant order. Sized arrays keyed by
+    /// `kind as usize` (`JULIA_PRESETS`, `SET_PRESETS`) must have one slot per
+    /// entry here.
+    pub const ALL: [FractalKind; 10] = [
+        FractalKind::Mandelbrot,
+        FractalKind::BurningShip,
+        FractalKind::Tricorn,
+        FractalKind::Multibrot,
+        FractalKind::Celtic,
+        FractalKind::Perpendicular,
+        FractalKind::Buffalo,
+        FractalKind::Phoenix,
+        FractalKind::Lambda,
+        FractalKind::ComplexMultibrot,
+    ];
+
+    pub fn description(&self) -> &'static str {
+        match self {
+            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)."
+            }
+        }
+    }
+
+    /// UI label for this kind (combo box / info panel heading).
+    pub fn label(&self) -> &'static str {
+        match self {
+            FractalKind::Mandelbrot => "Mandelbrot",
+            FractalKind::BurningShip => "Burning Ship",
+            FractalKind::Tricorn => "Tricorn",
+            FractalKind::Multibrot => "Multibrot",
+            FractalKind::Celtic => "Celtic",
+            FractalKind::Perpendicular => "Perpendicular",
+            FractalKind::Buffalo => "Buffalo",
+            FractalKind::Phoenix => "Phoenix",
+            FractalKind::Lambda => "Lambda",
+            FractalKind::ComplexMultibrot => "Complex Multibrot",
+        }
+    }
+
+    /// The iteration formula in human-readable notation (mirrors the doc
+    /// comments on the variants above). `power` is only used by Multibrot;
+    /// `complex_power` only by Complex Multibrot.
+    pub fn formula(&self, power: u32, complex_power: (f64, f64)) -> String {
+        match self {
+            FractalKind::Mandelbrot => "z = z² + c".to_string(),
+            FractalKind::BurningShip => "z = (|Re(z)| + i|Im(z)|)² + c".to_string(),
+            FractalKind::Tricorn => "z = conj(z)² + c".to_string(),
+            FractalKind::Multibrot => format!("z = z^{power} + c"),
+            FractalKind::Celtic => "z = |Re(z²)| + i·Im(z²) + c".to_string(),
+            FractalKind::Perpendicular => "z = (x² − y²) − 2x|y|i + c".to_string(),
+            FractalKind::Buffalo => "z = |Re(z²)| − i|Im(z²)| + c".to_string(),
+            FractalKind::Phoenix => "z = z² + c + p·z_prev".to_string(),
+            FractalKind::Lambda => "z = λ·z(1 − z)".to_string(),
+            FractalKind::ComplexMultibrot => {
+                format!("z = z^({:.3}{:+.3}i) + c", complex_power.0, complex_power.1)
+            }
+        }
+    }
+
+    /// Short tag used to identify this kind in a share-link fragment.
+    pub fn share_tag(&self) -> &'static str {
+        match self {
+            FractalKind::Mandelbrot => "mandel",
+            FractalKind::BurningShip => "burning",
+            FractalKind::Tricorn => "tricorn",
+            FractalKind::Multibrot => "multi",
+            FractalKind::Celtic => "celtic",
+            FractalKind::Perpendicular => "perp",
+            FractalKind::Buffalo => "buffalo",
+            FractalKind::Phoenix => "phoenix",
+            FractalKind::Lambda => "lambda",
+            FractalKind::ComplexMultibrot => "cmulti",
+        }
+    }
+
+    /// Inverse of `share_tag`; unknown tags fall back to `None` so the caller
+    /// can decide the default (matches historical share-link behavior).
+    pub fn from_share_tag(tag: &str) -> Option<FractalKind> {
+        Some(match tag {
+            "mandel" => FractalKind::Mandelbrot,
+            "burning" => FractalKind::BurningShip,
+            "tricorn" => FractalKind::Tricorn,
+            "multi" => FractalKind::Multibrot,
+            "celtic" => FractalKind::Celtic,
+            "perp" => FractalKind::Perpendicular,
+            "buffalo" => FractalKind::Buffalo,
+            "phoenix" => FractalKind::Phoenix,
+            "lambda" => FractalKind::Lambda,
+            "cmulti" => FractalKind::ComplexMultibrot,
+            _ => return None,
+        })
+    }
+
+    /// Default parameter-plane (Mandelbrot-mode) view for this kind, as
+    /// `(center_re, center_im, half_height)`. The Julia (dynamical) plane
+    /// doesn't vary by kind, so it isn't covered here.
+    pub fn default_set_view(&self) -> (f64, f64, f64) {
+        match self {
+            FractalKind::Mandelbrot => (-0.5, 0.0, 1.25),
+            FractalKind::BurningShip => (-0.5, -0.5, 1.3),
+            FractalKind::Tricorn => (-0.25, 0.0, 1.7),
+            FractalKind::Multibrot => (0.0, 0.0, 1.5),
+            FractalKind::Celtic => (-0.5, 0.0, 1.6),
+            FractalKind::Perpendicular => (-0.5, 0.0, 1.5),
+            FractalKind::Buffalo => (-0.5, 0.5, 1.5),
+            FractalKind::Phoenix => (-0.5, 0.0, 1.5),
+            FractalKind::Lambda => (-0.5, 0.0, 2.4),
+            FractalKind::ComplexMultibrot => (0.0, 0.0, 1.5),
+        }
+    }
+}

src/fractal/mod.rs

@@ -2,14 +2,17 @@
 //! egui paint callback.
 
 pub mod buddhabrot;
+pub mod kind;
 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,
-};
+pub use kind::FractalKind;
+pub use reference::{compute_reference, compute_set_reference};
+#[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

@@ -10,33 +10,9 @@
 //! * 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 super::kind::FractalKind;
 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 = 0,
-    /// `z -> (|Re z| + i|Im z|)^2 + c`.
-    BurningShip = 1,
-    /// `z -> conj(z)^2 + c` (the Mandelbar).
-    Tricorn = 2,
-    /// `z -> z^power + c` (power >= 2).
-    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,
-}
-
 /// Reference orbit escapes once |Z|^2 exceeds this. Kept larger than the pixel
 /// bailout so pixels escaping alongside the reference can still reach their
 /// bailout before the stored orbit runs out.
@@ -57,6 +33,7 @@ pub fn compute_reference(
     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();
@@ -72,6 +49,9 @@ pub fn compute_reference(
     // 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);
 
@@ -142,6 +122,10 @@ pub fn compute_reference(
                 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).
@@ -178,6 +162,32 @@ 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)]
@@ -190,10 +200,21 @@ pub fn compute_set_reference(
     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, phoenix_p, lambda_l,
+        &zero,
+        &zero,
+        center_re,
+        center_im,
+        max_iter,
+        precision,
+        kind,
+        power,
+        phoenix_p,
+        lambda_l,
+        complex_power,
     )
 }
 
@@ -216,6 +237,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         // Independent naive f64 orbit.
@@ -255,6 +277,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
         assert_eq!(points.len(), 501, "interior orbit should not escape");
     }
@@ -273,6 +296,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-1.75_f64, -0.03_f64);
@@ -302,6 +326,7 @@ mod tests {
             3,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (0.3_f64, 0.2_f64);
@@ -337,6 +362,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (mut zr, mut zi) = (0.15_f64, -0.1_f64);
@@ -366,6 +392,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-0.6_f64, 0.4_f64);
@@ -396,6 +423,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-0.7_f64, -0.2_f64);
@@ -426,6 +454,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-1.2_f64, -0.35_f64);
@@ -448,8 +477,17 @@ mod tests {
         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));
+        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);
@@ -469,4 +507,50 @@ mod tests {
             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

@@ -37,6 +37,7 @@ fn geom_differs(a: &Uniforms, b: &Uniforms) -> bool {
         || 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
@@ -87,6 +88,9 @@ pub struct Uniforms {
     /// 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,
     // 0 = classic colors, 1 = shadows
@@ -160,7 +164,14 @@ 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 {
@@ -207,6 +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,
+                },
             ],
         });
 
@@ -222,6 +246,10 @@ impl FractalRenderer {
                     binding: 1,
                     resource: ref_buffer.as_entire_binding(),
                 },
+                wgpu::BindGroupEntry {
+                    binding: 2,
+                    resource: lights_buffer.as_entire_binding(),
+                },
             ],
         });
 
@@ -288,7 +316,14 @@ impl FractalRenderer {
         // Colourise pass: data texture + colour uniforms → colour texture.
         let colorize_shader = device.create_shader_module(wgpu::ShaderModuleDescriptor {
             label: Some("colorize"),
-            source: wgpu::ShaderSource::Wgsl(include_str!("../shaders/colorize.wgsl").into()),
+            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 {
@@ -362,7 +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 =
@@ -588,6 +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"),
@@ -608,6 +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,
@@ -620,6 +675,10 @@ impl ExportRender {
                     binding: 1,
                     resource: ref_buffer.as_entire_binding(),
                 },
+                wgpu::BindGroupEntry {
+                    binding: 2,
+                    resource: lights_buffer.as_entire_binding(),
+                },
             ],
         });
 
@@ -757,6 +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).

src/fractal/share.rs

@@ -25,6 +25,8 @@ pub struct ShareState {
     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).
@@ -37,20 +39,7 @@ 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 {
-                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",
-            }
-        ));
+        s.push_str(&format!("&f={}", self.kind.share_tag()));
         s.push_str(&format!("&pw={}", self.power));
         s.push_str(&format!(
             "&re={}&im={}&hh={}&it={}",
@@ -60,8 +49,12 @@ impl ShareState {
         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
     }
@@ -79,18 +72,7 @@ impl ShareState {
             julia: map.get("m").map(|m| *m == "j").unwrap_or(false),
             kind: map
                 .get("f")
-                .map(|f| match *f {
-                    "mandel" => FractalKind::Mandelbrot,
-                    "multi" => FractalKind::Multibrot,
-                    "burning" => FractalKind::BurningShip,
-                    "tricorn" => FractalKind::Tricorn,
-                    "celtic" => FractalKind::Celtic,
-                    "perp" => FractalKind::Perpendicular,
-                    "buffalo" => FractalKind::Buffalo,
-                    "phoenix" => FractalKind::Phoenix,
-                    "lambda" => FractalKind::Lambda,
-                    _ => FractalKind::Mandelbrot,
-                })
+                .and_then(|f| FractalKind::from_share_tag(f))
                 .unwrap_or(FractalKind::Mandelbrot),
             power: map.get("pw").and_then(|s| s.parse().ok()).unwrap_or(2),
             center_re: (*map.get("re")?).to_string(),
@@ -109,6 +91,10 @@ impl ShareState {
                 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),
@@ -134,6 +120,7 @@ mod tests {
             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,
@@ -149,6 +136,7 @@ mod tests {
         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);
     }

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::Bgra8Unorm;
+    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/main.rs

@@ -16,6 +16,8 @@ mod view;
 #[cfg(not(target_arch = "wasm32"))]
 mod cli;
 #[cfg(not(target_arch = "wasm32"))]
+mod headless;
+#[cfg(not(target_arch = "wasm32"))]
 mod worker;
 
 use app::FractalApp;
@@ -49,11 +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

@@ -47,30 +47,21 @@ struct Uniforms {
     // Tonemap colour style: 0 = classic (R/G/B = raw caps), 1 = nebula
     // (yellow core, blue halo), 2 = grayscale.
     palette: u32,
-    // Padding to a 16-byte multiple. 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").
+    // 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,
-    _pad1: u32,
-    _pad2: 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;
 
-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;
-
 @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>>;
@@ -91,10 +82,6 @@ fn rand01(seed: u32) -> f32 {
     return f32(hash_u32(seed)) * (1.0 / 4294967295.0);
 }
 
-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);
-}
-
 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) {
@@ -126,6 +113,8 @@ fn advance(z: vec2<f32>, zp: vec2<f32>, c: vec2<f32>) -> vec2<f32> {
     } 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
 }
@@ -232,12 +221,7 @@ fn cs_main(@builtin(global_invocation_id) gid: vec3<u32>) {
 
 @vertex
 fn vs_main(@builtin(vertex_index) idx: u32) -> @builtin(position) vec4<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 vec4<f32>(verts[idx], 0.0, 1.0);
+    return vec4<f32>(fullscreen_triangle_pos(idx), 0.0, 1.0);
 }
 
 @fragment

src/shaders/colorize.wgsl

@@ -10,147 +10,28 @@
 // at the fragment's integer pixel coordinate (nearest — iteration data must not
 // be linearly filtered across escape boundaries).
 
-// Must match `Uniforms` in mandelbrot.wgsl / the Rust `Uniforms` struct.
-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,
-    kind: u32,
-    power: u32,
-    dc_offset: vec2<f32>,
-    phoenix_p: vec2<f32>,
-    lambda_l: vec2<f32>,
-    de_coloring: u32,
-    shadow: u32,
-};
-
-struct Light {
-    azimuth: f32,
-    altitude: f32,
-    color: u32,
-    _pad: u32
-};
-
 @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>;
 
-// Smooth cyclic palettes (Inigo Quilez cosine palettes). Must match the palette
-// in mandelbrot.wgsl.
-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));
-}
-
 @vertex
 fn vs_main(@builtin(vertex_index) idx: u32) -> @builtin(position) vec4<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 vec4<f32>(verts[idx], 0.0, 1.0);
-}
-
-fn load(x: i32, y: i32) -> vec3<f32> {
-    let dist = textureLoad(data_tex, vec2<i32>(x, y), 0).g;
-    return vec3<f32>(f32(x), f32(y), dist);
-}
-
-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));
+    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 {
-        if textureLoad(data_tex, vec2<i32>(i32(pos.x), i32(pos.y)), 0).b != 0. {
+        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 d = array<vec3<f32>, 3>(load(i32(pos.x), i32(pos.y)), load(i32(pos.x + 1), i32(pos.y)), load(i32(pos.x), i32(pos.y + 1)));
-
-            let normal = normalize(cross(d[1] - d[0], d[2] - d[0]));
-
-            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 vec4<f32>(color, 1.0);
+            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);
@@ -158,8 +39,7 @@ fn fs_main(@builtin(position) pos: vec4<f32>) -> @location(0) vec4<f32> {
         let de = d.g;
         let interior_frac = d.b;
 
-        let t = fract(ci * u.color_scale + u.color_offset);
-        var col = palette(u.palette_id, t) * de;
+        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);

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

@@ -12,46 +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,
-    shadow_palette_id: u32,
-    aa_level: u32,
-    // Iteration formula (see the KIND_* constants below).
-    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>,
-    // 0 = escape-time coloring, 1 = distance-estimation shading.
-    de_coloring: u32,
-    // 0 = classic colors, 1 = shadows
-    shadow: u32,
-};
-
-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;
-
 @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>,
@@ -61,12 +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.
@@ -74,16 +34,17 @@ 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.
@@ -123,6 +84,47 @@ 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.
@@ -163,6 +165,8 @@ fn advance_delta(z: vec2<f32>, e: vec2<f32>) -> vec2<f32> {
         // 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 (and Phoenix square part)
 }
@@ -183,30 +187,13 @@ fn fprime(z: vec2<f32>) -> vec2<f32> {
     } 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
@@ -345,22 +332,15 @@ fn color_sample(s: Sample) -> vec3<f32> {
     if !s.escaped {
         return vec3<f32>(0.0, 0.0, 0.0);
     }
-    let t = fract(s.ci * u.color_scale + u.color_offset);
-    return palette(u.palette_id, t) * s.de;
+    return classic_color(s.ci, s.de);
 }
 
-// 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));
-
+// 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);
     let inv = 1.0 / f32(aa);
     var ci_sum = 0.0;
@@ -382,7 +362,22 @@ fn fs_data(in: VsOut) -> @location(0) vec4<f32> {
     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 vec4<f32>(ci_avg, de_avg, interior_frac, 1.0);
+    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
@@ -395,6 +390,21 @@ fn fs_color(in: VsOut) -> @location(0) vec4<f32> {
     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);

src/worker.rs

@@ -26,6 +26,8 @@ pub struct RefRequest {
     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 {
@@ -107,6 +109,7 @@ fn compute(req: &RefRequest) -> Vec<[f32; 2]> {
             req.power,
             req.phoenix_p,
             req.lambda_l,
+            req.complex_power,
         )
     } else {
         compute_set_reference(
@@ -118,6 +121,7 @@ fn compute(req: &RefRequest) -> Vec<[f32; 2]> {
             req.power,
             req.phoenix_p,
             req.lambda_l,
+            req.complex_power,
         )
     }
 }

tests/shader_valid.rs

@@ -21,24 +21,44 @@ fn validate(name: &str, src: &str) {
 fn mandelbrot_shader_is_valid() {
     validate(
         "mandelbrot.wgsl",
-        include_str!("../src/shaders/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", include_str!("../src/shaders/colorize.wgsl"));
+    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",
-        include_str!("../src/shaders/buddhabrot.wgsl"),
+        concat!(
+            include_str!("../src/shaders/common.wgsl"),
+            include_str!("../src/shaders/buddhabrot.wgsl"),
+        ),
     );
 }