Journal

The last note owned a tangent-space height march so a quad reads as carved brick. The silhouette stayed the quad. This one does not pick a new sampler. An HDR environment is folded into two tables: a GGX-prefiltered specular cube and a DFG LUT. The fragment multiplies them. Split-sum makes environment lighting display-ready. Irradiance on/off is the GI-shaped control: distant environment, not local bounce.

Cream stoneware and measured brass on a catcher, loft window behind, full split-sum. The metal carries sharp mullions. The glaze picks up a dim rim and colored fill from the room, not a flat ambient. HUD: Khronos PBR Neutral exposure=1.05, distant environment irradiance, not local multi-bounce GI, photo-only - no metric. Do not hang an energy number on this photograph.

Same camera, same exposure, three shaders. Left SPEC IBL ONLY: ceramic flanks go dark; brass still shows the window. Middle IRRADIANCE ONLY: ceramic fills; brass is black because $k_D=0$. Right FULL SPLIT-SUM. Caption: distant environment ambient, not local multi-bounce GI.

Hero, Mesa 25.0.7 llvmpipe, linear working space, Khronos PBR Neutral, exposure 1.05: white-env Karis prefilter energy 1.000 at $r=0.05,\,0.20,\,0.50,\,1.00$. DFG at $\mathbf{n}\cdot\mathbf{v}=1$, $r\approx 0$: $S=\mathbf{1.000}$, $B=\mathbf{0}$. Flank crop (linear, full / spec-only luma ratio) 5.11. Env solid-angle mean luma 1.628. Assertions 25 pass / 0 fail.

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What you are seeing

The environment is an authored loft HDR (procedural latlong, not a captured EXR): plaster, baseboard, timber-ish beams, a key $+Z$ factory window with jambs and a 4×5 mullion grid, offset backdrop sashes, a door/panel, warm floor. No extra analytic key light on the hero plates. The env does the lighting.

Presentation hook. Product still, $1920\times 1080$, GL photograph, CPU tone map. Glazed ceramic (dielectric $F_0=0.04$, glaze roughness in $0.14$–$0.30$) and measured brass $F_0=(0.910,\,0.778,\,0.423)$. Soft planar contact on the catcher. Photo only.

GI-shaped control. Same objects, same camera eye (1.30, 0.54, 1.98) → (0.05, 0.25, 0.02), fov $30^\circ$. Three-up $1920\times 820$. Metals stay black in the middle panel. That is the model, not a bug. Linear flank crop (ceramic, spec-only vs full) is the meter for “irradiance did something”; the JPEG is still a photograph.

Walk the three panels:

• Spec IBL only. The conductor can live on a prefiltered cube: window, frame, a strip of interior. The dielectric cannot. $F_0=0.04$ leaves the body dark. Studio void.
• Irradiance only. Distant-environment $E(\mathbf{n})$ fills the ceramic. Brass is black ($k_D=0$). No local object bounce, no bottle-in-sphere, no floor color bleeding onto the metal.
• Full split-sum. Specular cube $\times$ DFG, plus irradiance. The plate that is supposed to look like a product shot.

Conductor ladder. Six rungs $r\in\{0.05,\,0.18,\,0.32,\,0.48,\,0.68,\,1.00\}$, measured brass, albedo-only so rung roughness stays the control. Smooth metal still shows mullions. The lobe widens; the highlight tints with $F_0$; the body is not an irradiance-albedo mix-up. lod = roughness * (mips-1). Photo only.

Dielectric ladder. Same rungs, $F_0=0.04$, cream albedo. Grazing Fresnel remains as roughness climbs (that is the DFG). Same env / exposure / TM as the metal ladder. Photo only.

Two facts, never mixed:

1. Beauty plates (heroes, ladders, failures, calibration strip) are GL-rendered split-sum, Khronos PBR Neutral, exposure 1.05, then sRGB OETF. HUD photo-only means do not invent a GL energy theorem from the JPEG.
2. Instruments (mip strip, spp residual, DFG LUT, roughness→mip map, and the CSV) are CPU tables: Karis prefilter, DFG integrate, white-env energy, mip variance, 64-spp vs 512-spp RMSE.

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Why: the reflection integral, then Karis

Working space is linear radiance. Cubemaps are linear. Tone map is a named display step after shading, not baked into the tables.

Reflection equation (specular)

Microfacet specular (Cook–Torrance):

NDF is isotropic GGX / Trowbridge–Reitz. $\alpha=r^2$ in the NDF only:

A hemisphere of HDR samples times that BRDF, every fragment, is the reference. Production does not do that.

Split-sum (Karis / UE4-style)

Factor the integral into prefiltered incident radiance (depends on $L_i$ and roughness, not on $F_0$) and a view-dependent BRDF integral against a white environment (depends on $\mathbf{n}\cdot\mathbf{v}$ and roughness, not on $L_i$):

with $\mathbf{R}=\mathrm{reflect}(-\mathbf{v},\mathbf{n})$. That multiply is the production trick. This note photographs it and names the two stored tables.

Term A — prefiltered radiance

Stored in cubemap mips. Karis weighting, $V=N=R$, no second $G$ in the prefilter:

Importance-sample the GGX NDF in half-vector space, reflect to $\mathbf{l}$, weight by $\mathbf{n}\cdot\mathbf{l}$. Constant $L_i=1$ must return 1. That is the white-env energy row: 1.000 at four roughness keys.

This run: specular cube 128, 8 mips, CPU authored, uploaded per level. glGenerateMipmap is not called. Production spp is 64 (doubled on coarser mips). A separate 512 sky cube, mip0 only, GL_LINEAR, is the backdrop; it is not the GGX chain.

Roughness $\to$ mip (linear, stated)

Eight mips $\Rightarrow$ $\lambda=7r$. This is not a GGX solid-angle match. It is the default so the roughness ladder is readable. Shader: lod = rough * uMaxMip.

Term B — DFG / EnvBRDF LUT

2D: $X=\mathbf{n}\cdot\mathbf{v}$, $Y=$ roughness. Single-scatter. Schlick $F$ split into $F_0$ and the white-Fresnel remainder. Smith–GGX IBL geometry with

LUT RGB is (scale, bias). Integrate against a white environment; the env cube never enters this table.

Dielectric $F_0=0.04$. Conductor $F_0=$ linear RGB albedo (measured brass here; no dielectric coat). At $\mathbf{n}\cdot\mathbf{v}=1$, $r\approx 0$: $S\to 1$, $B\to 0$. CSV: $S=0.999984$, $B=0$. LUT range [0, 1].

Distant irradiance (the locked GI control)

Lambert:

with $k_D=(1-F)(1-\mathrm{metal})$ so metals do not pick up a diffuse body. $E$ is a low-frequency irradiance cubemap from the same distant environment. This run: cube 64, 192 spp cosine, then a 3-pass $3\times 3$ blur on mip0 only so window mullions do not reprint as wood grain on Lambertian flanks. Specular mips are not blurred.

Honesty line: irradiance here is distant environment lighting preintegrated over the hemisphere. It is not local multi-bounce GI, not path tracing, not DDGI, not lightmaps, not SSGI.

On/off is the three-up above. No extra directional fill on the hero.

Display

TM is Khronos PBR Neutral (linear in, linear display-referred out). Exposure 1.05, identical on every comparison row. TM is not baked into the cubemaps. GL_FRAMEBUFFER_SRGB is off; encode is CPU.

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Unique artifacts: the mip strip, the LUT, the map

This is the thing this note exists to draw. CPU GGX prefilter, $+Z$ face, mip $0\ldots 7$, roughness $=$ mip $/$ (mips$-1$). Mip0 is a 4×5 factory window with dark frames. Then the mullions dissolve into the lobe. Caption: not glGenerateMipmap. $+Z$ luma variance is the monotone blur meter: 70.9 → 43.9 → 23.3 → 9.34 → 3.54 → 2.33 → 0.76 → 0.

Sample-count instrument, CPU both sides. Production spp vs 512 spp at $r=0.25/0.50/0.80$, $+Z$, $32^2$, plus a relative-error heatmap. RMSE 0.777 / 1.045 / 1.465. That is 64-spp vs 512-spp on an HDR window with a sun disc, not production-mip vs analytic, and not a GL filtering theorem. Residual fireflies at 64 spp are expected.

Scale $S$ and bias $B$, $128^2$, CPU GGX, 64 spp. Axes labeled. $I_{\mathrm{BRDF}} = F_0 \cdot S(\mathbf{n}\cdot\mathbf{v},\,a) + B(\mathbf{n}\cdot\mathbf{v},\,a)$, single-scatter. The smooth, face-on corner is the CSV row: $S\approx 1$, $B=0$.

$\lambda(r)=r\cdot(\mathrm{mips}-1)$ as a plot, plus mip thumbnails of the same $+Z$ window. Caption: not a solid-angle match; the ladder stays readable. GGX $\alpha=r^2$ lives in the NDF, not in this map.

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Quote the CSV. Do not quote the beauty photographs as energy.

Photoreal gallery, Mesa llvmpipe:

Hero rounding used in the lede: energy 1.000; $S=\mathbf{1.000}$, $B=\mathbf{0}$; flank ratio 5.11; env luma 1.628. The ratio is still $\gg 1$. Do not invent a GL energy or RMSE theorem from the hero or the roughness ladders. Those frames are photo-only.

Highlight RMS grows on the first metal rungs, then saturates once the lobe covers the sphere ($r\gtrsim 0.32$: 54.1, 46.9, 39.2, 39.2). Width after that is not a useful meter. Mip variance is.

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Failures / controls

Wrong mip

Left: roughness $0.75$, force mip0. Sparkly sandpaper; the window is still a window. Middle: same roughness, correct $\lambda$. Right: roughness $0.08$, force max mip. Pewter blob; a smooth conductor has been given the roughest table. Photo only. The metal ladder is the honest map; this plate is the knife.

No DFG LUT ($F=1$)

Same prefilter cube, same exposure. Left: skip the LUT, output the prefiltered radiance as if $F=1$. Metals wash grey; the brass $F_0$ is gone; the dielectric loses the grazing rim and the body energy is wrong. Right: $F_0 S+B$. Photo only.

LDR clip vs HDR

Same TM, same exposure. Left: env authored with $L_i\le 1$, then prefiltered. Window highlight and interior both collapse; the metal loses punch. Right: float HDR env. TM is not in the cubemaps; clipping the source is the failure. Photo only.

Instrument strip (not a hero)

Smooth metal / rough metal / Lambertian with spec off. Caption: calibration balls, not the product shot. Photo only. If this were the cover, the note would be a void-sphere demo.

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Two paths, do not mix the instruments

The GI-control three-up is a photograph of the control and the source of the flank-ratio CSV row. Quote the CSV. Do not quote the 8-bit panel as 5.11.

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Honesty gaps

1. Distant-environment irradiance $\neq$ local GI. No bottle-to-brass bounce, no floor color in the metal, no path tracing, no DDGI, no lightmaps, no SSGI. Metals are black in the irradiance-only panel because $k_D=0$.
2. Single-scatter split-sum. No multi-scatter energy compensation. Grazing metals can pick up a white-ish LUT bias term.
3. Roughness→mip is linear. $\lambda=r\cdot(\mathrm{mips}-1)$, not a GGX solid-angle match.
4. Prefilter RMSE is 64-spp vs 512-spp at matched roughness on the HDR window, not production-mip vs analytic. Sun disc → residual fireflies (RMSE 0.777–1.465). Documented; not a GL filtering theorem.
5. Highlight RMS grows on the first metal rungs, then saturates. Mip variance is the monotone blur meter.
6. Irradiance blur is a 3-pass presentation filter on $E(\mathbf{n})$ so mullions do not reprint as wood grain on dielectrics. Specular prefilter mips are not blurred.
7. Contact is a planar cosine term, not a shadow map or a ray trace.
8. Env is procedural loft HDR, not a captured EXR. Solid-angle mean luma 1.628.
9. Cubemap sampling is llvmpipe RGBA16F with CPU-authored mips. Seamless cubemap is requested; face-edge quality is still a Mesa caveat.
10. No hardware IBL unit and no real-time convolution cost claim. Prefilter, irradiance, and DFG are CPU.
11. Residual turned-form spec bands on the bottle are the loft windows wrapping a surface of revolution. Milder than a marble-chalk urn, not a second lobe.
12. JPEG is visualization. Measurement is the CSV. Do not FFT or energy-integrate a beauty frame.

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Mesa / llvmpipe — what this run can claim

Can claim: on this OSMesa / llvmpipe build, a CPU Karis GGX prefilter of an authored HDR loft conserves white-env energy (1.000 at four roughness keys), a CPU DFG LUT hits $S\approx 1$, $B=0$ at face-on smooth, and a GLSL 330 split-sum shader times those tables plus a distant-environment irradiance cube produces these photographs. Turning irradiance off darkens dielectric flanks (linear crop ratio 5.11). Wrong mip, missing LUT, and an LDR-clipped env are visible failures. Photographs of those paths are photographs of this software rasterizer.

Cannot claim: NVIDIA / AMD / Intel hardware IBL units, real-time convolution cost, occupancy, or bandwidth. That seamless cubemap filtering on llvmpipe equals a discrete GPU. That glGenerateMipmap would have been a GGX chain (it is not; we did not call it). That distant $E(\mathbf{n})$ is local multi-bounce GI. That linear roughness→mip is a solid-angle match. That an FFT or energy integral of an 8-bit sRGB JPEG is a spectrum of the signal. Discrete-GPU metrics, or “this is how the hardware works.”

Science path does not depend on GALLIVM_PERF: CPU latlong, CPU prefilter, CPU DFG.

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Assertions

This run: 25 pass / 0 fail.

No assert tolerances were loosened for the photoreal plates.

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Out of scope

Local path-traced GI, photon maps, irradiance caching of scene bounce. DDGI, lightmaps, SSGI/SSR as a substitute for the cube. Multi-bounce local solvers of any kind. Anisotropic GGX, Toksvig AA of the NDF, sheen, clearcoat, layered metals. Area lights / LTC. Deep tone-map bake-off (TM is one named operator). Shadow-map bias. Real-time convolution on a “hardware IBL unit.” Re-deriving the mipmaps chirp, the anisotropic ellipse, or the POM height march: cite continuity; this is a BRDF-integral approximation, not a new sampler and not new geometry.

Album sequence:

1. Perspective-correct — what you interpolate
2. Z-fighting — how depth is encoded
3. Mipmaps — which frequencies survive an isotropic minify
4. Anisotropic — which UV ellipse that pixel covers
5. Parallax occlusion — fake depth from a height field; the silhouette is still the quad
6. Split-sum IBL — environment lighting as a multiply: GGX-prefiltered cube $\times$ DFG, plus distant $E(\mathbf{n})$

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Fragment lock

float lod = rough * uMaxMip;                 // linear: r * (mips-1)
vec3 pre  = textureLod(uPrefilter, R, lod).rgb;
vec2 dfg  = texture(uDFG, vec2(Nv, rough)).rg;
spec = pre * (F0 * dfg.x + vec3(dfg.y));     // Karis split
// kD = (1-F)*(1-metal);  diff = kD * albedo/PI * E(n)

Prefilter writes every mip. Irradiance is one low-res cube. DFG does not see $L_i$. Pin the hero as the presentation. Pin the three-up as the GI-control photograph. Pin the mip strip plus the DFG LUT plus the roughness→mip map as the unique artifacts. The formula is the caption. Distant $E(\mathbf{n})$ is why the ceramic flanks filled, and why the brass stayed black without a diffuse body.