CultLib/docs/research/analytic-integration-feasibility.md

Analytic noise integration along warped rays: feasibility probe

Campaign cultmath-tapes, Imagination probe, 2026-10-03. This is a measurement, not a spec. The tape target stays parked under ruling cultmath-tapes:ruling:tape-target-unparks-when-asura-stable.

Verdict. Exact integration of the noise does not beat sampling for optical depth on either shipped field.

• The exact line integral of CultMath snoise is cheap and correct. Affine warp spans are long, at 100 to 300 world units.
• The exact noise integral is not the optical depth, because both densities are nonlinear in the noise.
  • The void's wall profile is pow(·, 0.40) with two clamps.
  • Aetheria's fog is clamped at zero and goes opaque within a few world units.
• Given the exact span mean of the noise for free, the resulting transmittance is still 3.6x worse than the shipped march on the void and 22x worse on the wells.
• Exact integration wins only on the linear functional ∫ n dt of octaves the march undersamples. That gain does not survive the density's nonlinearity.
• Aetheria's triangle noise is integrable exactly as piecewise linear, but costs 3x to 22x the march, and its shipped form does not meet the preconditions.

Scenarios and method

• Void: the shipped void (NoiseBoundTests.VoidField.Shipped() at hands/cultmath-interval-ops 2c8585bd).
  • Rh 198, S 50, Rc 248, K 1/30, A 20, F0 0.01 (cells of 100 and 25 world), warp D 10.
  • The shipped orbiting camera, N = 8 tiles at f = 935, one jittered pixel ray per tile.
• Wells: the test's Aetheria wells at warp D = 60 (FogField.Wells(60)).
  • F0 1/414.2167, A 20, B 10, extinction 0.5, flow frequency 1/512, period 8.
  • Like the test, this uses the snoise field shape (r1's), not Aetheria's own triangle noise. Triangle noise is measured separately below, on the same rays.
• Rays: 200 per scenario, and 120 for the triangle basis.
  • The wells give 147 rays with an active interval; the rest look up and miss the fog.
  • The active interval is where the envelope leaves the density to the noise. It is cut where the true optical depth reaches 6.
  • Active length per ray: void median 140 world (p10 114, p90 193); wells median 268 (p10 137, p90 623).
• March steps: the void's march step (LodStep, 12.5 in z) is a median 14.9 world along the ray. The wells' quadratic grid gives a median of 9.1 in the active region.
• Warp: w(p) = flow(p) · shift, so the affine error scales exactly with |shift|.
  • Spans are grown at the worst phase shift, Period / 2. A span is therefore valid at every frame's phase, for any camera that crosses the same world segment.
  • Each span is anchored at p0 with J·d from a central difference. It grows in 1-unit (void) or 2-unit (wells) steps until |w − (w0 + J·d·Δ)| > τ.
• Exact simplex integral: CultMath snoise(float3) has kernel radius² 0.5, so its support never leaves the incident simplices.
  • Along a line it is therefore a sum over the lattice points within √0.5 of the line. Each point contributes ∫ (0.5 − |x|²)⁴ (g·x) dt on its chord, a closed-form degree-9 polynomial.
  • Validation:
    • The lattice sum matches snoise within 2e-4 (float32 rounding at coordinates near 300).
    • The segment integral matches a 5e-4-step quadrature within 2.5e-5 per unit length.
• Run: probe code was in a scratch clone (probe-clone@91c8cbf1, never merged) and ran on Yggdrasil through the verify stopgap, in a dotnet 10 container with 4 CPUs. Timings are single-threaded C# on that host and are relative.

1. Affine-valid span length (ray length, world units)

τ (× coverage cell)	Void p10 / median / p90	Wells p10 / median / p90
0.05	94 / 129 / 211	71 / 103 / 169
0.10	141 / 197 / 324	108 / 151 / 253
0.25	263 / 420 / 1244	184 / 275 / 715
0.05 × detail cell	44 / 63 / 97	34 / 47 / 75

• One affine span at τ = 0.1 covers about 13 void march steps and about 16 wells march steps.
• The void shell is shorter than one span. The tiled spans are clipped by the active interval, at a median of 27 world and 2 spans per ray.
• On the wells, the tiled spans run a median of 128 world, with 2 per ray (p90 4).
• The tangent anchor is the worst affine choice. A secant or Chebyshev fit over the span cuts the error bound by 2x to 8x, so the spans could be about 1.4x to 2.8x longer at the same τ. That was not measured.

2. Simplex cells per span, per octave

Octave xm is the coverage octave's frequency times m, warped like the coverage octave. The shipped detail octave is x4 and is unwarped in both fields, so it needs no affine span at all.

Counts are for τ = 0.1 tiled spans with both warp phases, given as the median and then the p90. The rows are simplex cells crossed, then lattice points touched, which is the exact work.

Octave	Void cells	Void lattice points	Wells cells	Wells lattice points
x1	6 / 16	10 / 16	4 / 8	8 / 10
x2	8 / 27	10 / 26	8 / 12	10 / 14
x4	13 / 52	14 / 44	12 / 20	14 / 20
x8	20 / 100	20 / 80	22 / 38	22 / 34
x16	40 / 201	35 / 154	44 / 72	38 / 60
x32	75 / 396	62 / 302	86 / 144	70 / 112
x64	151 / 793	118 / 592	172 / 287	132 / 218

• On the same spans, the march takes a median of 2 samples per span on the void (p90 10) and 12 on the wells (p90 25).
• A straight line touches 3.44 lattice points per noise unit.

3. Cost: exact against the march

• Measured cost.
  • One snoise is 530 to 610 ns in C# on this host.
  • One lattice-point integral (hash, gradient, closed-form polynomial) is 85 ns, or 0.16 snoise.
  • With this probe's naive box-scan enumeration it is 1.24 snoise. That is an upper bound; a simplex walk visits only crossed cells.
• How the table counts. Exact is lattice points × 0.16 + 1 snoise per span for the Jacobian. March is 1 snoise per sample. Both sides carry the two warp phases.
• Tolerance. τ is 0.1 × that octave's own cell, so the affine error is comparable across octaves.
• Columns. The two error columns are RMS errors of the active-length mean of one warped octave. The last error column is what replacing the octave by its expectation of 0 costs.

Octave	Void exact/march cost	Void err exact / march / expectation	Wells exact/march cost	Wells err exact / march / expectation
x1	0.36	1.1e-2 / 1.6e-3 / 0.21	0.11	1.9e-2 / 9e-5 / 0.29
x2	0.46	1.0e-2 / 4.7e-3 / 0.13	0.15	1.5e-2 / 2e-4 / 0.21
x4	0.68	9.0e-3 / 4.0e-2 / 0.10	0.23	1.1e-2 / 5e-4 / 0.18
x8	1.03	5.8e-3 / 8.0e-2 / 0.067	0.36	7.2e-3 / 1.1e-3 / 0.10
x16	1.64	4.0e-3 / 9.0e-2 / 0.050	0.56	6.4e-3 / 9.4e-3 / 0.088
x32	2.89	2.8e-3 / 0.105 / 0.036	0.87	4.7e-3 / 3.3e-2 / 0.057
x64	5.26	2.0e-3 / 0.095 / 0.025	1.44	3.2e-3 / 5.3e-2 / 0.040

• Where exact stops paying on cost:
  • At the measured kernel cost: x8 on the void and between x32 and x64 on the wells.
  • At the brief's model of one snoise per cell crossed: about x1 to x2 on the void and x8 on the wells, from the per-span table in the probe log.
• Exact against the march on accuracy:
  • At the shipped step the march is more accurate for octaves it oversamples: void x1 to x2, wells x1 to x8. The exact path's error floor is the affine warp error, about 1e-2 at τ = 0.1 per cell, and it scales with τ.
  • Exact is more accurate at about 0.7x the cost where the march undersamples: void x4 to x8, wells x16 to x32.
  • Above about x16 on the void, the undersampled march is worse than simply using the expectation (0.095 against 0.025 at x64). That is a band-limiting result, which the footprint octave weights already exploit. It is not an exact-integration result.

4. High-octave variance over a span

• Span-mean statistics. Over straight lines of noise length ℓ, the std of the snoise span mean, against a point std of 0.358, is:

  • 0.234 at ℓ = 1;
  • 0.170 at ℓ = 2;
  • 0.121 at ℓ = 4;
  • 0.087 at ℓ = 8;
  • 0.061 at ℓ = 16;
  • 0.031 at ℓ = 64.

  This is about ℓ^-0.5: the high octaves average out like a random field, not like a periodic one.

• Detail octave replaced by its expectation of 0, on τ = 0.1 spans, measured as |ΔOD| per span:

  • Void: median 0.000 and p90 0.091. Relative to the span's OD (OD > 0.01) the median is 2.5% and the p90 is 7.6%. The mean detail displacement is a median 1.0 world (p90 3.6) against the ramp width S = 50.
  • Wells: median 0.11 and p90 5.7 optical depths. The relative median is 72%.

• Why the wells are so bad. Normalised to the clamp, the density moves A·fade/B = 2 per unit of noise, and with extinction 0.5 that is 1 optical depth per world unit per unit of noise. The mean detail displacement of 1.5 world (p90 3.8) is 15% to 38% of the floor blend B = 10. The wells' optical depth is set by where the clamp boundary lies, and that is exactly what the expectation erases.

• So the expectation is adequate for the void's detail octave and is not adequate for the wells.

5. The clamp and other nonlinearity

• Void. 50.0% of τ = 0.1 spans cross a regime boundary: 0 (d' ≤ Rh), the wall ramp, or K (d' ≥ Rc). 45.6% of the active length lies in the ramp, where the density is K max(0, 1 − 1.5 (1 − (d'/Rc)²)^0.40). That is not a polynomial in n, so exact ∫n does not give ∫ρ there. Outside the ramp the density is constant and needs no noise.

• Wells. 42.3% of spans cross the clamp. Only 9.8% of the active length is unclamped, so most of the length where the noise "matters" is length where the noise only decides whether the density is zero.

• The decisive check: transmittance error. This is the sup over the ray of |T_est − T_true|, with the truth taken on a 0.5- or 1-unit grid.

  Estimator	Void median / p90	Wells median / p90
  Shipped march	0.020 / 0.038	0.033 / 0.155
  March at 4x density	0.006 / 0.009	0.030 / 0.049
  Density of the exact span-mean n (τ = 0.1 spans)	0.072 / 0.125	0.719 / 0.997

  The last row is an upper bound on what any exact-mean method can deliver without also modelling the distribution inside the span. It is the Heitz-style correction the survey describes. It loses to the march on both fields.

Triangle basis (operator's addition)

The triangle noise is triNoise3d in Aetheria Assets/Shaders/Volumetric.cginc (origin/master f1dee184).

• tri(x) = |frac(x) − 0.5|.
• It runs two Nimitz-style iterations. Each iterates p += tri3(bp·2), scales bp by 1.8 and p by 1.2, and adds tri(p.z + tri(p.x + tri(p.y)))/z.
• It makes 18 tri calls per evaluation.

As shipped, density() uses it as follows:

• pow(triNoise3d(·), exponent), with Noise.Exponent −0.25 multiplied by a slope flatness read from a texture (NOISE_SLOPE is on).
• Two phases, warped by globalFlow, which is Tri3D = cross(normalize(tri3(P)), normalize(tri3(1.618P))) × 15.
• The fine octave at ×8 is evaluated at a pos whose y the coarse octave has already displaced.
• Amplitude −36.17.

Self's reading, tested:

• "The warped ray is an exact polyline." This holds for the warp inside triNoise3d. It does not hold for Aetheria's flow: normalize and cross make the flow piecewise rational. With the normalize dropped, the cross product of two piecewise-linear vectors is piecewise quadratic. tri composed with a piecewise polynomial stays piecewise polynomial of the same degree, with breakpoints at quadratic roots, so that variant is still closed-form. Counted along the real flow at the worst shift, the kinks run 15% to 30% above the straight-ray counts, but the pieces between them are curved.
• "The clamp roots are linear." This holds only if the exponent is 1. With −0.25·flatness, each piece is (a + bt)^e. One such piece has a closed-form integral, but the clamp's root on a sum of two powered phases plus the smoothstep fade has none.
• "Whole periods give the exact mean." This holds for one triangle wave at whole periods, where the mean is exactly 0.25. It fails for the composed form, which has no period along a general line; the scales 2, 1.8 and 1.2 and the nested warp are incommensurate.
  • Measured on lines, the span-mean std of triNoise3d is 0.029 at ℓ = 1, 0.010 at ℓ = 8 and 0.0042 at ℓ = 64, against a point std of 0.083. That is about ℓ^-0.47, the same random-field decay as simplex.
  • Its bias against the global mean of 0.1995 is within ±0.0015, so the expectation is unbiased. It is not exact.
• The cost decides it. The exact piecewise-linear walk is verified, matching a 1e-4 quadrature within 3.4e-8. But the nest has 77.7 linear pieces per noise unit of straight line, against 3.44 lattice points for simplex.

  • One piece costs 1.38 point evaluations with a derivative carried.

  • At Aetheria's scale, on the wells rays, the pieces per world unit and the exact cost relative to the march are:

    Octave	Pieces per world unit	Exact cost / march
    x1	0.20	2.8x
    x2	0.38	5.5x
    x4	0.76	11x
    x8 (shipped fine)	1.5	22x
    x64	12	176x

  • On the void rays it is 4.1x at x1 and 29x at x8.

  • At the same time the march's RMS error on the span mean is already small: 2.4e-3 at x1 and 1.7e-2 at x8.

Per-basis verdict

Basis	Exact line integral	Under the shipped warp	Optical depth	Verdict
Simplex (CultMath snoise)	Closed form, 0.16 snoise per lattice point, 3.44 points per noise unit; validated	Affine spans of 100 to 300 world at τ = 0.1 cell; error floor about 1e-2	Not delivered: the void's pow profile and the wells' clamp make ρ nonlinear in n; the exact-mean transmittance loses 3.6x and 22x	Viable kernel, wrong target. Pays only if the density were affine in n over the span, or as a control variate for a march (survey §4, Kettunen 2021)
Triangle (triNoise3d)	Closed form per linear piece; validated	Not piecewise linear under Aetheria's normalized-cross flow or the −0.25 power; made linear by exponent 1 and a non-normalized flow	Same nonlinearity, plus 78 pieces per noise unit	Loses. 3x to 22x the march even where its preconditions hold
Gabor kernels (survey §1.2, not probed)	Closed form for any affine-transformed kernel (complex erf, at most 16 terms)	The affine warp spans above transfer directly, with each span a whitening transform	Same density-nonlinearity limit	Candidate smooth basis for large structure, to be measured only if the density is first made linear in the noise over a span

What would change the verdict

• A density that is affine in the noise inside a span. For example, the void's ramp replaced by a linear ramp in d', with clamp crossings bracketed by the interval bound. Exact simplex integration would then deliver optical depth directly in the 54% of void length that is outside the ramp's clamps. As measured, that 54% is constant density and needs no noise at all.
• Exact integration as a control variate, not a replacement. The exact mean is cheap and unbiased for ∫n, and the march corrects the nonlinear residual. That is the Kettunen construction. It was not measured here.
• Wider skipping. The operator's first idea, skipping space outside the warp range, is already what the interval envelope does, and the probe's active intervals are its output.
  • On the void the noise matters over only about 140 world per ray.
  • On the wells, opacity ends the ray within a few world units of entering the unclamped region.