Research: integration-native volumetric noise domain probe (cultmath-tapes)

A domain built so span integration holds by construction: sums of
nonnegative compact ellipsoid kernels in a mass-conserving multiplicative
cascade, per-kernel affine warp, all nonlinearity moved into placement.
Measured on Yggdrasil against the shipped void and wells, with a physical
reference (Beer-Lambert, single scattering with HG).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

docs/research/integration-native-noise-domain.md

@@ -0,0 +1,291 @@
+# An integration-native volumetric noise domain: probe
+
+Campaign `cultmath-tapes`, Imagination probe, 2026-10-03, under ruling
+`cultmath-tapes:ruling:integration-native-noise-domain`. This is research under the
+parked tape target. It has no spec, no shipped-field change and no committed code.
+
+It follows `analytic-integration-feasibility.md`, which found that exact noise
+integrals are cheap but that the shipped densities are nonlinear in the noise. The
+question here is the reverse one. Suppose a domain is designed so that span
+integration holds by construction. Does it buy more visual detail per unit cost
+than the shipped fields?
+
+On the operator's instruction (relayed by Self), the reference is physical, not
+Aetheria's tuned compositing:
+- transmittance is Beer–Lambert, `T = exp(-τ)`;
+- the images use single scattering with Henyey–Greenstein `g = 0.6` and albedo 1;
+- both fields are integrated the same way.
+
+The shipped look is a comparison, not a target.
+
+## Verdict
+
+**On the void, yes, by a wide margin. On the wells, modestly.** At or below the
+shipped march's cost, the integration-native field carries much more resolvable
+detail, and it is exact. The catches:
+- The look changes character. The default cascade reads as fibres or brush strokes,
+  not as soft cloud.
+- The cost is traversal, not integration.
+- The field cannot be ray-marched at all. Every consumer, shadow rays included,
+  must use the exact path.
+
+Headline numbers, 512 x 256, single-threaded C# on Yggdrasil:
+
+| Void, transmittance image | Cost per pixel | vs shipped march | msf (grad²/var) | Power above 1/8 cycle/px | Pixels with \|∇T\| > 0.05 |
+|---|---|---|---|---|---|
+| Shipped field, converged reference (595 samples) | n/a | n/a | 0.0111 | 0.20% | 0.00% |
+| Shipped march (11.4 samples) | 28.9 µs (44 snoise3) | 1.00x | 0.0178 | 0.71% | 0.00% |
+| Integration-native, 2 levels, exact | 13.9 µs | **0.48x** | **0.0755** | **2.95%** | 41.9% |
+| Integration-native, 3 levels, exact | 35.7 µs | 1.24x | 0.125 | 9.15% | 54.7% |
+| Integration-native, 6 levels (LOD stops at 4 to 5), exact | 74.2 µs | 2.57x | 0.159 | 12.5% | 60.9% |
+
+- **Mean density is matched.**
+  - The integration-native field places mass from the shipped field's noise-averaged
+    density.
+  - Mean window optical depth is 1.62 to 1.74 against the shipped 1.81.
+  - Mean T is 0.25 to 0.40 against 0.17. The difference comes from intermittency:
+    the image std is 0.29 against 0.05, at a near-equal mean.
+- **msf is normalised by variance.** It measures spatial frequency, not contrast.
+- **Single-scattering images, void.** The images are 256 x 128 with a point light at
+  the sun, and both are tonemapped with the same exposure. The shipped image's msf
+  is 0.0023, with 0.22% of power above 1/8. The integration-native image's msf is
+  0.101, with 16.9% of power above 1/8.
+- **Wells.**
+  - Transmittance is a near-opaque silhouette for both fields: equal msf (0.011),
+    so it carries no signal.
+  - Single scattering at 6 levels (1.20x the march cost):
+    - msf is 0.073 against 0.069;
+    - power above 1/8 is 10.1% against 7.0%;
+    - the edge fraction is 16.7% against 9.5%.
+  - At 4 levels the cost is 0.77x the march.
+
+**Exactness.** Against a converged march of the same field at 0.02 world:
+- The relative τ error is a median 3e-5 and a maximum of 1.0e-4 on the void, and a
+  median 1e-5 and a maximum of 1.0e-4 on the wells.
+- The binned gather against brute force over all roots differs by at most 1.3e-15.
+- The shipped march against its own converged reference: |ΔT| is a median 0.0055
+  (p90 0.013) on the void and a median 0.0006 (max 0.83 on grazing rays) on the
+  wells.
+
+**The detail is only reachable by integration.**
+- Marching the same integration-native field at the shipped step gives a void |ΔT|
+  median of 0.14, a p90 of 0.48 and a maximum of 0.83.
+- A field this detailed is beyond the march at its step. That is the operator's
+  premise, measured: the shipped fields' softness fits the regime that renders them.
+
+## The primitive set
+
+The primitive set has one design rule: nonlinearity lives in kernel placement, and
+linearity lives in the integration. The density is a sum of nonnegative,
+compact-support kernels. Everything nonlinear is evaluated once per kernel, at its
+centre, and never per sample. That covers envelopes, clamps, `pow` profiles, the
+flow warp and placement.
+
+### Bases (closed-form line integral over a segment)
+
+| Primitive | Why it integrates | Cost per span (measured unless marked) |
+|---|---|---|
+| Compact polynomial ellipsoid kernel, `(1 − u)^k` with `u = \|A⁻¹(x − μ)\|²` | On a line, u is quadratic in t, so the kernel is a polynomial of degree 2k between the two roots of `u = 1`, which are closed form. Same algebra as the simplex lattice integral in the first probe | k = 0: 5.5 to 7.4 ns; k = 1: 8 to 9.5 ns; k = 2: 11.7 ns. About 0.01 to 0.02 snoise3 (snoise3 is 610 to 650 ns in C#) |
+| Truncated Gaussian ellipsoid, `exp(−κu)` on `u ≤ 1` | `exp` of a quadratic is a Gaussian in t, giving an erf difference. Truncation only clips the limits | 32 to 36 ns with A&S 7.1.26 erf (error ≤ 1.5e-7), about 0.05 snoise3 |
+| k = 0 ellipsoid (constant density) | Chord length; this is EVER's primitive (Mai et al. 2024 [memory]) | 5.5 ns |
+| Gabor kernel (Gaussian × cosine) | Complex erf, any affine transform (Gabor Fields 2026, eq. 9 [fetched in the prior-art file]) | Not probed. Up to 16 Taylor terms [paper] |
+| Cosine sum (Balint 2019) | Integral is a sum of sines | Not probed. Two noise-sized evaluations per segment [paper]. Signed, so it needs an offset; low contrast |
+| Simplex lattice (CultMath `snoise`) | First probe: degree 9 per lattice point | 0.16 snoise3 per point, 3.44 points per noise unit. Signed: only usable where density is affine in it |
+
+### Warps
+
+| Warp | Why it stays exact | Cost |
+|---|---|---|
+| Per-kernel affine warp: centre moved by `f(μ)·s`, shape by `(I + J·s)` | An affine image of an ellipsoid is an ellipsoid, so the ray stays straight and the kernel stays closed form. The warp *defines* the field, so there is no approximation error. It is evaluated once per root per frame (7 flow evaluations with the Jacobian) | Frame build 39 ms (void, 2540 roots) and 57 ms (wells, 16947 roots), or 19 and 27 ns per 1080p pixel amortised |
+| Children inherit the parent's affine frame | The cascade has descendants that are exact affine images, so the warp is evaluated only at the roots | One 3x3 product per child, from a table |
+| Not allowed | A per-point warp `x + w(x)` inside a span: the warped ray curves, which was the first probe's problem | n/a |
+
+### Transfer and placement
+
+| Rule | Status |
+|---|---|
+| Any function of the kernel centre: the shipped clamp, `pow`, the smoothstep fade, the noise-averaged density `E_n[ρ(x, n)]` from 16 empirical noise quantiles, through a 1D LUT over d (void) or s (wells) | Exact by construction, at one lookup per kernel |
+| Affine transfer of the sum (`a + b Σ K_i`) | Exact |
+| Any 1D function of an *affine* coordinate, `∫ f(a + bt) dt = ΔF / b` (pre-integration, Engel 2001) | Exact [derivation, not probed]: height fog `exp(−y/h)` and ramps in y |
+| Polynomial or exp of a *quadric* coordinate; an indicator or clamp of a quadric (chord through roots) | Exact [derivation, not probed]: spherical shells and paraboloid bowls as envelopes |
+| Product of the sum with a polynomial envelope in t | Exact, polynomial times polynomial [derivation, not probed] |
+| Forbidden inside a span: `f(Σ K_i)` for nonlinear f, `max`/`min` of fields, a clamp of a signed sum, or noise displacing an envelope's argument | These are exactly what broke the first probe |
+
+### Composition: the mass-conserving multiplicative cascade
+
+- **Roots** sit on a jittered grid: spacing H0 = 30 world on the void and 40 on the
+  wells, one root per cell.
+  - Mass is `P(μ)·H0³`, so the mean density equals the shipped noise-averaged
+    density.
+  - The ellipsoid is elongated along the flow direction projected on the shell or
+    fog layer (aspect 1.3 to 2.4), with ±0.6 rad of jitter, and flattened along the
+    normal (×0.5 void, ×0.4 wells).
+  - There are two copies per root, at the shipped two-phase shifts with weights w0
+    and 1 − w0. Time animation is therefore the shipped crossfade, exact under
+    summation.
+- **Children.** Each node keeps 25% of its mass and gives 75% to Nc = 5 children.
+  - Child weights are `u²·P(child)/P(parent)`, normalised. The `u²` makes the cascade
+    intermittent; the P ratio sharpens it toward the envelope.
+  - A child is scale 0.4, with aspect up to 1.6 and ±0.6 rad of rotation.
+  - Children sit along the parent's major axis: offset 0.6 along it, 0.21 across.
+  - All of this comes from a 256-entry table indexed by hash. A descendant bound of
+    1.77 times the ellipsoid holds the subtree, so the parent is a BVH node.
+- **LOD.** A node is a leaf when its children would be smaller than 0.5 × the pixel
+  footprint at that depth. It then keeps its full mass. Because the cascade conserves
+  mass, truncation is a mean-preserving prefilter: the expectation step of the first
+  probe, exact here instead of approximate.
+- **Gather.**
+  - Roots are binned by their subtree AABB into a hash grid of H0 cells.
+  - The ray walks the grid with Amanatides–Woo DDA and mailboxing, clipped to the
+    shipped active windows, so both fields integrate the same span.
+  - It exits early once Στ ≥ 7, which is valid because every term is nonnegative.
+
+## "Juice" under these constraints
+
+| Effect | Mechanism here | Measured or prior art |
+|---|---|---|
+| Filaments, wisps | Anisotropic children chained along the parent's major axis, which follows the flow | Visible in `void-T-IN-L6.png` as streaks along the flow. Gabor Fields uses anisotropic Gabor kernels for orientation-selective detail [fetched] |
+| Sharp edges | Low-order kernels: k = 1 is the default, k = 0 gives hard ellipsoids | k = 0 (`L6-hard`) changes msf by −9% on the void. Edges come from kernel boundaries, not clamps |
+| Voids | Regions with no kernel mass are exactly zero and are skipped by the gather at no cost | Structural |
+| Turbulent intermittency | Random multiplicative cascade, the p-model of turbulent dissipation (Meneveau & Sreenivasan 1987 [memory]); clustered kernels, as in a Neyman–Scott process [memory] | Image std 0.29 against 0.05 at near-equal mean OD |
+| Flow and animation | Per-kernel warp at the shipped flow and phases. Alternatives are Lagrangian advection of kernel centres with lifetimes (advected textures, Neyret 2003 [memory]) and Gabor phase for oscillatory detail (flow noise, Perlin & Neyret 2001 [memory]) | Only the two-phase crossfade was measured |
+| Fine detail riding on structure | Children exist only inside parents, so detail appears only where matter is. This is Schneider's erosion idea (2015 [fetched]) without the per-sample clamp | Structural |
+
+Wavelet turbulence (Kim et al. 2008 [memory]) and curl noise (Bridson 2007 [memory])
+were not used. A per-point warp is ruled out above. Curl noise could orient kernels,
+evaluated per kernel, at no cost to exactness.
+
+## Measurements
+
+**Setup.**
+- Probe source: scratch clone `probe-clone@5ec9f529`,
+  `packages/cultmath/probe/DomainProbe/Program.cs`, never merged.
+- Run on Yggdrasil through the verify stopgap: dotnet 10 container, 4 CPUs, tiered
+  JIT off.
+- Full log: Self's scratchpad, `dom-full1.log`.
+- The scenes are those of the first probe, with fixed cameras:
+  - Void: the shipped orbiting camera at time 20, at the 1920-wide f = 935 field of
+    view, rendered at 512 x 256.
+  - Wells: warp D = 60, camera at xz (1200, 0) looking at the central well, slopes
+    x ±1 and y from −0.85 to 0.15.
+- The window is the shipped field's active intervals. Every method integrates the
+  same window.
+
+**Cost.**
+- Single-threaded C#, best of three passes over the same 2048 random pixels.
+- The host is shared, and absolute timings moved by up to 2x between the smoke run
+  and the full run; Flow went from 125 to 660 ns. Ratios within one run are the
+  reliable numbers.
+- Flow is 47% of a shipped sample in this run, so a flow-cached march would cost
+  about 15 µs on the void and 31 µs on the wells.
+
+| Config (void) | Cost vs march | Roots visited / nodes tested / kernels integrated per pixel | Kernels per level |
+|---|---|---|---|
+| L1 | 0.12x | 43 / 8 / 8 | 8 |
+| L2 | 0.48x | 59 / 25 / 20 | 8, 12 |
+| L3 | 1.24x | 74 / 42 / 28 | 8, 12, 8 |
+| L4 | 1.81x | 82 / 60 / 35 | 8, 12, 8, 7 |
+| L6 (default) | 2.57x | 93 / 86 / 36 | 8, 12, 8, 7, 1, 0 |
+| L6, no child placement | 1.79x | 93 / 91 / 39 | same metrics as L6 |
+| L6, no LOD | 2.95x | 93 / 103 / 49 | 8, 12, 8, 7, 7, 7 |
+
+| Config (wells) | Cost vs march | Kernels integrated per pixel |
+|---|---|---|
+| L1 | 0.04x | 5 |
+| L2 | 0.23x | 11 |
+| L3 | 0.49x | 16 |
+| L4 | 0.77x | 20 |
+| L6 | 1.20x | 25 |
+| L6, no child placement | 0.62x | 27 |
+
+**Where the time goes.**
+- The integrals are about 36 kernels × 10 ns, or 0.4 µs of a 74 µs pixel.
+- The rest is traversal:
+  - roots visited but missed: 93 visited for 8 hits, because AABBs of elongated
+    rotated ellipsoids are loose;
+  - child generation;
+  - placement lookups. The wells' placement costs a 5-bowl `Sdf`, and dropping
+    child placement halves the wells cost with identical metrics.
+
+**Aliasing.** One sample per pixel against four on the default void field: |ΔT| has
+an RMS of 0.015 (p90 0.022). At 4 spp the msf is 0.146 against 0.159. So about 92%
+of the measured detail is resolved structure, not sparkle. Exact along the ray is
+not the same thing as antialiased across the pixel; the LOD rule is the prefilter.
+
+**Variants, void L6 (msf / power above 1/8).**
+- k = 1 default: 0.159 / 12.5%.
+- Hard k = 0: 0.145 / 11.0%.
+- Gaussian (erf): 0.174 / 14.2%, at 0.89x the polynomial cost here. The erf kernel is
+  3x the polynomial kernel in isolation, but the kernel is not where the time goes.
+
+## Images
+
+All images are in Self's session scratchpad,
+`C:\Users\Meta\AppData\Local\Temp\claude\F--Projects-CultLib\b7e842a1-2045-474e-a51b-240d832b770b\scratchpad\integration-domain-images\`:
+
+- **Transmittance, 512 x 256.** Shown as `sqrt(1 − T)`, white for opaque.
+  - `void-T-shipped-ref.png`, `void-T-shipped-march.png`.
+  - `void-T-IN-L1.png`, `void-T-IN-L3.png`, `void-T-IN-L6.png`,
+    `void-T-IN-L6-hard.png`, `void-T-IN-L6-gauss.png`.
+  - The wells equivalents, which are silhouettes and uninformative.
+- **Single scattering, 256 x 128.**
+  - Method: HG g = 0.6 and albedo 1. The point light is at the void's sun with 1/r²
+    falloff; the wells use a directional sun with a sky background.
+  - Each view sample takes a shadow ray: a 2-world march for the shipped field, an
+    exact traversal for the integration-native field.
+  - Tonemap: `1 − exp(−1.5 L / p99(shipped))`, then gamma 2.2, the same exposure for
+    both fields.
+  - `void-SS-shipped.png`, `void-SS-IN-L6.png`, `wells-SS-shipped.png`,
+    `wells-SS-IN-L6.png`.
+
+What they show:
+- The shipped void is a soft, low-contrast haze.
+- The integration-native void is dense flow-aligned streaks with deep gaps, like
+  hair or brush strokes.
+- The shipped wells are smooth rolling dunes. The integration-native wells are
+  layered pancakes with streaked detail.
+
+## The catches
+
+1. **The character is a design parameter, and this probe picked one.**
+   - High anisotropy, axis-chained children and an intermittent `u²` split give
+     fibres.
+   - A cloudier look would need rounder children, wider offsets and less
+     intermittency. That trades away some measured detail and was not swept.
+   - The detail metrics measure spatial frequency, not taste. The operator has to
+     look at the images.
+2. **The cost is a BVH, not arithmetic.**
+   - Integration is nearly free. Traversal, child generation and placement dominate.
+   - Tighter root bounds (OBB tests before AABB bins), cheaper placement and fewer
+     children are obvious levers, but none was measured.
+   - On a GPU, variable-length kernel lists per ray diverge. 3DGRT (Moënne-Loccoz et
+     al. 2024 [memory]) puts the same problem on hardware ray tracing.
+   - CPU C# ratios do not transfer to the GPU, where snoise is relatively cheaper.
+3. **It is all or nothing.**
+   - This field cannot be marched. At the shipped step, |ΔT| is a median 0.14.
+   - Shadow rays, light probes and any sampler consumer need the exact traversal too.
+     The single-scattering pass here cost an exact shadow traversal per view sample:
+     257 s for both void SS images together, against 54 s for the whole 512 x 256
+     shipped transmittance set.
+   - Lighting has no closed form. The in-scatter integral is still quadrature. What
+     the domain makes exact is every optical depth inside it. A per-kernel lighting
+     cache would amortise shadows, but that is a different shading model (splat-like),
+     and so a product fork.
+4. **Exact along the ray, not across the pixel.** The LOD truncation is the
+   prefilter. 8% of the measured detail at 1 spp is aliasing.
+5. **The wells gain little** because the shipped wells fog is near-opaque. Its look
+   is the surface silhouette, and that is set at root scale. The domain pays most
+   where the medium is semi-transparent, as in the void.
+6. **Timing noise.** Absolute costs vary up to 2x between runs on the shared host.
+   The cost ratios above are within one run.
+
+## What would change the verdict
+
+- A sweep toward a cloud-like character: round children, Nc = 8, wider offsets,
+  Gaussian kernels. It should be judged on the single-scattering images against the
+  same cost.
+- Tight root culling: an exact ellipsoid-versus-bin test, or roots stored by
+  oriented bound. If roots visited fall from 93 to near the 8 hits, the void L3 cost
+  would fall well under the march.
+- A GPU port of one level-limited configuration, to replace the C# ratios with
+  shader timings.