GR-raytracing/README.md

# GR 4D ray tracing — Phase 0 prototype

`minkowski_sky` is a deliberately small, CPU-only, single-frame Phase 0
benchmark. It renders point sources from a sky catalog through an analytic
backend. It is not a sky texture: each source remains a direction, temperature,
and amplitude until its sub-pixel Gaussian PSF is splatted.

Build and render the default 1280 x 720 image:

```sh
make run
```

The program first creates `assets/sky_grid_5deg.csv` when it is missing. The
synthetic catalog places stars every 2 degrees on the union of longitude and
latitude lines spaced 10 degrees apart; the two poles are stored only once.
The eight octants (four 90-degree longitude sectors in each hemisphere)
alternate red `temperature_K = 3000` and blue `temperature_K = 12000`.
Red stars use `amplitude = 1`; blue stars use `amplitude = 0.00141095580387`,
which equalizes their CIE/linear-sRGB luminance under the renderer's blackbody
integration. Longitude boundaries belong to the sector to their east and the
equator to the northern hemisphere, so boundary stars have a deterministic
color.
The default output is PNG at `output/imgs/minkowski_sky.png`.

## Movie PNG sequence (Phase A)

Movie mode consumes a canonical observer-track CSV rather than a fixed camera.
Each row stores coordinate time, proper time, Cartesian position, and the full
four-by-four tetrad (21 columns total).  Generate the first reproducible
Minkowski benchmark—two coordinate seconds at 30 fps, accelerating from rest
to about `0.95c`—then render its numbered PNG frames:

```sh
make clean && make
mkdir -p output/imgs
./build/minkowski_sky --write-minkowski-accel-track output/minkowski_accel_2s.csv \
  --duration 2 --fps 30 --proper-acceleration 1.52
./build/minkowski_sky --observer-track output/minkowski_accel_2s.csv \
  --frames-dir output/imgs --frames-prefix minkowski_accel \
  --start-time 0 --duration 2 --fps 30 --exposure 1e-5
```

This writes `minkowski_accel_000000.png` through
`minkowski_accel_000060.png`.  The renderer treats the CSV as its observer
input; the acceleration generator is only a reproducible flat-spacetime test
fixture.  Movie mode collects all current frame-mesh vertices into a single
SoA ray pool, activates rays as a newest-to-oldest coordinate-time scan reaches
their observer event, and advances active rays to each slab boundary.  The
analytic backends use logical slabs with no metric I/O; nmesh slab loading is
the next backend step.  `--slab-duration` sets the coordinate-time width
(default `64`) for this current fixed-mesh pass.  The current
synthetic test catalog uses global default exposure `1e-3`; the accelerated
benchmark explicitly uses `1e-5` because its physical Doppler blue shift
otherwise clips the later frames.

PNG is the default output and the default build links `libpng`:

```sh
make clean && make
mkdir -p output/imgs
./build/minkowski_sky --output output/imgs/minkowski_sky.png
```

If `libpng` is unavailable, rebuild with `make clean && make ENABLE_PNG=0`.
That intentionally selects the binary-PPM fallback, whose default path is
`output/imgs/minkowski_sky.ppm`; pass a `.ppm` path for explicit output.

Run the flat-spacetime geodesic regression with:

```sh
make test
```

This also checks that the Kerr--Schild metric remains finite at `r=2M` and
that the central ray from the default Schwarzschild camera is classified as
captured.

Build an independent analytic Schwarzschild executable in Cartesian ingoing
Kerr--Schild coordinates (regular at the horizon), then render the test catalog
to PNG:

```sh
make clean && make SPACETIME=schwarzschild
mkdir -p output/imgs
./build/schwarzschild_sky --catalog assets/sky_grid_5deg.csv \
  --width 640 --height 360 --coarse-cell-pixels 8 --fov-deg 60 \
  --output output/imgs/schwarzschild_test_catalog.png
```

`SPACETIME=minkowski` (the default) and `SPACETIME=schwarzschild` select source
files at compile time, so each executable contains exactly one metric provider.
The Schwarzschild demonstration uses mass `M=1` and places a static camera at
coordinate radius `30` by default.  `--look-ra-deg` and `--look-dec-deg`
define the direction from the camera to the hole; the camera is placed at the
opposite direction from the origin and its local forward axis points radially
inward.  Use `--observer-radius R` to select any `R > 2`; it is a coordinate
radius in Cartesian Kerr--Schild coordinates.  The backend escapes at `r=256`
and declares capture at `r=1.5`, safely inside the horizon at `r=2`.  Those
rendering thresholds are Phase-1 demonstration values, not settled production
refinement or integration settings.

For a local radial boost relative to that static camera, pass
`--observer-inward-speed V`, where `0 <= V < 1` is measured in the static
observer's orthonormal frame and positive values point toward the hole.  The
default is `0`, preserving the static camera.

For rays that asymptote to the future horizon in coordinate-time backward
integration, the Schwarzschild demo also terminates at
`log(alpha p^0) = 8`.  This is the normalized-momentum horizon diagnostic
already evolved by the integrator; it is disabled by default and does not
replace the AH-calibrated spatial capture criterion planned for nmesh data.

Useful options:

```sh
./build/minkowski_sky --width 1920 --height 1080 --fov-deg 30 \
  --catalog assets/sky_grid_5deg.csv --output output/imgs/frame.png
./build/minkowski_sky --catalog assets/2mass/processed/2mass_psc_m31_0p5deg_stars.csv \
  --look-ra-deg 10.6847083 --look-dec-deg 41.26875 --fov-deg 1.8 \
  --exposure 1e15 --output output/imgs/2mass_m31.png
./build/minkowski_sky --catalog assets/2mass/processed/2mass_psc_m44_1p0deg_stars.csv \
  --look-ra-deg 129.99165 --look-dec-deg 19.54139 --fov-deg 2.0 \
  --exposure 1e15 --width 1920 --height 1920 --output output/imgs/2mass_m44.png
./build/minkowski_sky --write-catalog assets/sky_grid_5deg.csv
```

The camera is a fixed inertial observer at coordinate position `(0,0,0)`,
with a tetrad whose forward direction is coordinate `-Z` and whose vertical
direction is `+Y`. The frame first triangulates the image plane, then traces
only its vertices backwards. Escaped endpoints form a triangulation on the
source sky. For every locally invertible triangle, catalog stars inside its
spherical source triangle are interpolated back to the image triangle and
splatted as PSFs. Consequently multiple image triangles naturally create
multiple images of the same star.

The ray state evolves `(x^i, Pi_i, log(alpha p^0))` in coordinate time with
RK4 using the 3+1 equations in Bohn et al. II.A, until the spacetime backend
classifies the ray. `spacetime.c` is the only module containing the Minkowski
metric or its infinity criterion; frame, observer, and integrator use only
`SpacetimeSource` and `MetricData`. The initial regular mesh size is exposed
as `--coarse-cell-pixels`; it is a Phase-0 sampling knob, not a settled
production refinement threshold.

### Adaptive image mesh refinement

Adaptive refinement is disabled by default (`--refine-max-level 0`), so the
existing coarse-mesh renders remain unchanged.  When enabled, its defaults are
an absolute direction error of `1e-3` degrees, relative error `0.1`, minimum
long edge `0.5` pixels, minimum area `0.25` pixel-squared, and a provisional
minimum discrete-Jacobian magnitude of `1e-3`.  Each value can be overridden
independently:

```text
--refine-max-level N
--refine-angle-abs-deg D
--refine-angle-rel R
--refine-jacobian-min J
--refine-min-edge-pixels P
--refine-min-area-pixels2 A
```

`N` caps the triangle refinement level.  Let `e` be the angle between the
traced longest-edge midpoint direction and the normalized endpoint
interpolation, and let `s` be the angle between those two endpoint **camera
directions**.  Both are evaluated internally in radians; the absolute CLI
threshold `D` is specified in degrees and converted before comparison.  `s`
is the angular geometric size of the image triangle's test edge, not a
source-sky/lens-map length.  A locally escaped triangle is split only when
**both** `e > D_rad` (the converted `--refine-angle-abs-deg D`) and
`e / max(s, 1e-15) > --refine-angle-rel`.  `P` and `A`
prevent selecting a leaf already at or below the requested image-plane
long-edge and area scales.
Triangles whose three vertices disagree between capture and escape are split
independently of the direction-error thresholds, allowing the mesh to follow a
shadow boundary.

Independently of the midpoint geometry test, an all-escaped triangle also
computes the discrete lens Jacobian
`J = Omega_source / Omega_image`.  Both signed solid angles use
`2 atan2(dot(a, cross(b,c)), 1 + dot(a,b) + dot(b,c) + dot(c,a))`, with the
ordered camera directions for `Omega_image` and their traced infinity
directions for `Omega_source`.  A J-driven split requires **both** a shared
image edge whose incident triangles have opposite nonzero signs of `J` and
`min(abs(J_left), abs(J_right)) < --refine-jacobian-min`.  It then requests
that shared edge on both leaves.  Thus `|J|` bounds the fold selection instead
of widening it as a standalone critical-curve band.  A negative sign is
physical parity and is retained.  The `1e-3` default is deliberately
provisional and should be tuned with the small Schwarzschild refinement
diagnostic before being treated as a production threshold.

For a short Schwarzschild diagnostic that permits at most one actual split
generation, for example:

```sh
./build/schwarzschild_sky --catalog assets/sky_grid_5deg.csv \
  --width 48 --height 48 --coarse-cell-pixels 24 --fov-deg 40 \
  --refine-max-level 1 --refine-angle-abs-deg 0.001 \
  --refine-angle-rel 0.001 --refine-jacobian-min 0.001 \
  --refine-min-edge-pixels 1 \
  --refine-min-area-pixels2 1 --draw-mesh \
  --output output/imgs/schwarzschild_refinement.png
```

For movies, each refinement generation completes the full newest-to-oldest
time-slab sweep before any probe becomes a mesh vertex.  Newly added vertices
are therefore traced only by the next generation; the renderer never returns
to a slab that has already been released.  At the start of every generation,
newly inserted vertices and geometry-only longest-edge probes for its new
leaves are collected together, so both ray sets use the same parallel
`RayPool` pass.

`--look-ra-deg` and `--look-dec-deg` rotate that fixed tetrad so its forward
axis is the corresponding catalog direction; their defaults reproduce the
original `-Z` view. `--exposure` converts a catalog's physical flux
normalization to the prototype HDR scale. The current synthetic catalog is
calibrated for default exposure `1e-3`; a 2MASS blackbody normalization in
steradians requires a much larger display exposure such as the example above.
The optics path
integrates each fitted Planck spectrum through CIE 1931 color-matching functions
and converts the resulting radiance to linear sRGB; it does not use an empirical
color-temperature RGB approximation.

Point sources use a flux-normalized circular Moffat PSF by default
(`--psf-fwhm-pixels 2.7 --psf-moffat-beta 4.5`). The FWHM matches the former
1.15-pixel Gaussian core while the Moffat wings remain continuous; both values
are display/optics calibration parameters.

The default renderer builds one immutable, process-wide 64-by-64 sub-pixel
Moffat lookup kernel.  Its weights are pixel-area integrals and are bilinearly
interpolated between phase tables.  The renderer reports its build time and
cached/direct-fallback image counts.  Pass `--psf-direct` to use the slower
8-point quadrature reference evaluator for regression comparisons; an image
whose required HDR-tail support exceeds the cache radius selects that reference
path automatically.

The PSF-cache completion line is printed before tracing and catalog splatting
begin.  For long renders, pass `--verbose` to print catalog-prefetch state,
splat-worker local heartbeats (8, 16, 32, ... completed triangles per worker),
and image-write boundaries.  The worker heartbeats use neither global progress
accounting nor cross-worker synchronization.
Verbose ray-trace output reports the initial mesh trace and refinement stages
for single frames; movie mode additionally reports each generation's sample
count and each time slab's activation and terminal-ray summary.
Movie renders always print one summary per time slab; `--verbose` also prints
the ray counts before each slab is loaded.

For PSF validation only, `make psf-hdr-test` builds
`build/minkowski_psf_hdr_test`, a separate binary with a `--hdr-output PATH`
option.  It writes the pre-tone-mapping RGB framebuffer as a three-plane,
32-bit float FITS image.  Values remain linear HDR at the renderer's arbitrary
scale; no tone mapping or per-frame normalization is applied.  The ordinary
binaries do not contain this option or writer.

The FITS header describes a synthetic 8640-by-5760, 36-by-24 mm full-frame
sensor with 4.1667 um pixels.  Each render records its active centered crop and
derives `FOCALLEN` from that render's width and horizontal `--fov-deg`; these
camera fields support plate-solving workflows but do not calibrate flux.

Pass `--draw-mesh` to alpha-composite image-plane triangle edges as
one-pixel-wide 0.5 linear-gray diagnostic lines at 0.5 opacity. The line
rasterizer uses coverage-based antialiasing.