Journal

立体角与渲染方程

The last note owned Monte Carlo sampling mismatch: same integral, two pdfs, one $N$. Those pdfs were already densities in $1/\mathrm{sr}$. Split-sum had already folded an environment into a GGX prefilter $\times$ DFG LUT and left the measure inside the integral (env solid-angle mean luma 1.628 on that loft run). This note owns the measure.

Path tracers sample $d\omega$. Irradiance and the rendering equation track $\Omega_\perp$. A direction is a point on the hemisphere, weighted in steradians. Flux through a flat surface uses the projected solid angle $\Omega_\perp=\int(n\cdot\omega)\,d\omega$. Source area, pixel count, and $L_i$ used as a brightness knob are the wrong meters.

The importance-sampling key was a finite disk of angular radius 3.600°. The solid angle that disk subtends, which that note never printed, is

A hemisphere pdf against a cap that small is why a cosine sample fireflies a highlight. Cited here. The cosine-versus-Phong comparison stays in that note.

A new photographic family: sunlit courtyard / skylight atrium — not the loft cream-glaze bottle / measured-brass / dark-oak still, not the metro colonnade, not the gallery lacquer sphere. Four walls, a square skylight with a visible jamb, matte limewash floor, one low bench for scale. An analytic sky disk sits in the opening: spherical cap of locked half-angle $\alpha=15^\circ$, uniform radiance. White Lambert card $\rho=0.80$, $n=+Y$, $L_e=0$, on the floor under the opening. The card is the instrument. HUD: path tracers sample d omega E tracks Omega_perp. Do not hang $E_B/E_A$ on this photograph.

Pin this. Same camera, same exposure, same $L_i$. Left $\alpha=5^\circ$. Right $\alpha=15^\circ$. The right-hand card integrated more steradians. Lower caption: only Omega changes; brightness change IS the lesson, and $E_B/E_A = 8.818616$ with the same-$R$ control. Upper line: area is not Omega; pixels are not steradians. Photo only.

Hero, Mesa 25.0.7 llvmpipe, linear Rec.709, Khronos PBR Neutral $e=\mathbf{1.00}$, seed 1352782172: $\Omega_A=\mathbf{0.023909417}\,\mathrm{sr}$, $\Omega_B=\mathbf{0.214094348}\,\mathrm{sr}$, ratio 8.954394. $\Omega_{\perp A}=\mathbf{0.023863926}\,\mathrm{sr}$, $\Omega_{\perp B}=\mathbf{0.210446804}\,\mathrm{sr}$. Disk irradiance ratio $E_B/E_A=\mathbf{8.818616}$. The small-angle stand-in $(15/5)^2=9$ is rejected as the meter. Same-$R$ area control: $\Omega$ ratio 3.536. $L_i=(12.0,\,13.2,\,16.0)$ bit-identical A/B; $L_{iY}=\mathbf{13.147}$; disk/fill $Y$-ratio 200. Assertions 28 pass / 0 fail.

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What you are seeing

Working space is scene-referred linear Rec.709. One courtyard, one analytic cap, one Lambert card. Display is inherited: Khronos PBR Neutral, $e=1.00$, then IEC 61966-2-1 sRGB OETF on CPU. The curve still does not create lighting. Neutral still does not author $E$. Cite the tone-mapping and split-sum IBL notes.

Hero — presentation hook. Atrium, visible disk, white card, low bench, $\alpha=15^\circ$. Photo of the family. The meter is not this JPEG.

Size A/B — teaching pin. A | B, same $L_i$, $\alpha=5^\circ$ vs $15^\circ$, shared exposure. Quote the metrics for the digits.

Cosine tilt — Failure B. Same $\alpha=15^\circ$ disk held fixed in world. Left: card $n=+Y$, $\beta=0$. Right: card tilted $\beta=60^\circ$. $L_i$ frozen. Plate caption: (n·ω) is geometry, not a brightness slider. Photo only.

Grid — the measure. Unit hemisphere over the card. Equal-$\Omega$ cells, sky cap as a spherical polygon. Plate caption: the measure, not a latlong unwrap.

$E$ vs $\Omega_\perp$ — the identity. Horizontal axis $\Omega_\perp$ in sr. Cyan: disk $E=L_{iY}\Omega_\perp$. Gold: disk plus named fill. Red ghost: $L_{iY}\Omega$, cosine missing. Squares: cosine-arm $N=256$, inverted through Lambert. Plate caption: irradiance tracks projected solid angle.

Wedge — the article. Same cap twice, side view. Left: steradians on the sphere. Right: foreshortened patch. Plate caption: at 5 deg they almost agree; at 15 deg the article appears.

MC-$N$ — estimator of that integral. Card center, $\alpha=15^\circ$, $N=16/64/256/1024$. Thumbnails are the cosine-$\Omega_\perp$ instrument. The plot is uniform-$\Omega$ (cyan) and cosine-$\Omega_\perp$ (gold), same $\xi$ stream. Plate caption: not an IS bake-off; rel_err must fall. The traces are allowed to rise at $N=256$. The gate is the fall at $1024$.

Metrics strip — the meter, snapshot. Science plate of the float-buffer table. Quote the text metrics, not the JPEG.

Two facts, never mixed:

1. Beauty plates (00, 01, 02) are the GLSL courtyard on this llvmpipe, Neutral then sRGB OETF. The shader shades walls, floor, and the card with a disk form factor. Do not invent $\Omega$, $E$, or $L_o$ from the JPEG.
2. Instruments (03, 04, 05, 06, 07, and the metrics) are the float identities and the CPU estimator. Quote the metrics.

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Three failures

Radiance $L_i$ is power per area per steradian. Irradiance on a plate is radiance times the projected solid angle of the source. Grow the cap and keep $L_i$ fixed: the card changes. Tilt the card and keep $L_i$ and the cap fixed: the card changes again, because $(n\cdot\omega)$ moved. Neither change is a hotter source.

Failure A — area versus $\Omega$. It lives on the size A/B plate. Left $\alpha=5.000^\circ$, right $\alpha=15.000^\circ$, $L_i=(12.0,\,13.2,\,16.0)$ on both, $e=1.00$, $\rho=0.80$. The right card is brighter because $\Omega_\perp$ grew from 0.023863926 sr to 0.210446804 sr. The disk irradiance ratio is $E_B/E_A=\mathbf{8.818616}$. The lower caption names the rejected stand-in: $(15/5)^2=9$.

The same caption carries the area control, which is not a second photograph. On-axis disk, radius $R$, distance $d$, $\alpha=\arctan(R/d)$,

From the metrics, $R=1$, area $=\pi$ on both rows:

Area identical. Recorded $\Omega$ ratio $=\mathbf{3.536}$. Inverse-square in distance at fixed radius would report $(4/2)^2=4$. The assert rejects 4. Scale $R$ and $d$ by the same $k$ and $\Omega$ is unchanged while area scales by $k^2$. Equal area is not equal solid angle. Equal solid angle carries no claim about area.

Failure B — missing cosine. It lives on the cosine-tilt plate, and as a curve on the $E$ vs $\Omega_\perp$ plot. Integrating $L_i\,d\omega$ without $(n\cdot\omega)$ collects flux as if the receiver were a spherical probe. The rendering equation on an opaque surface is the flat-plate integral. $(n\cdot\omega)$ is foreshortening of the incoming beam. It is not a brightness slider and it is not the BRDF. On the tilt plate the $15^\circ$ disk stays put and the card tilts to $\beta=60^\circ$. The right card goes dark at the same $L_i$. The red ghost on the identity plot is $L_{iY}\Omega$ drawn against $\Omega_\perp$. Raising $L_i$ to lift the dark card voids the control.

Failure C — pixels versus steradians. The upper line of the size plate already says so. Pixel coverage is a camera projection of the drawn sky disk. Solid angle in the rendering equation is at the card. The two stored counts are not a matched pair. pixels_disk_A 910 is a luma threshold on the linear $\alpha=5^\circ$ panel at $628\times 720$. pixels_disk_B 7242 is the same kind of count on the $\alpha=15^\circ$ hero at $1280\times 720$. Both lines are tagged omega_from_pixels_illegal. Different frame size, different aspect, and the right-hand panel of the size plate is not the image that produced 7242. Dividing the two integers does not estimate $\Omega_B/\Omega_A$. A $5^\circ$ cap is $\Omega=0.023909417\,\mathrm{sr}$ at 910 px and at any other footprint. Frame size moves the count. It does not move card $\Omega$.

The importance-sampling key is Failure A at a still smaller cap. 3.600° subtends 0.012398431 sr. That note’s cosine pdf spends its mass on the hemisphere around $n$, not on that cap. This note prints the steradian.

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The rendering equation once

Write it once. Name every symbol. Later lines evaluate this integral. They do not rename it.

A path tracer samples this integral: draw $\omega\in\Omega^+$, weight by $f_r L_i (n\cdot\omega)/p(\omega)$. Building the tracer is a different note. What this one locks is the measure $p$ is a density of.

$d\omega$, then projected solid angle

$\theta$ is polar angle from $n$. $\phi$ is azimuth in the tangent frame.

The second form is the grid. Equal steps of $\phi$ and of $\cos\theta$ are equal steradians. The polar step that realizes a fixed $\Delta\cos\theta$ is $\Delta\theta=\Delta\cos\theta/\sin\theta$, wide near the pole, packed near the horizon. A lat-long that is even in $\theta$ is even in the picture and uneven in $d\omega$: its cells scale with $\sin\theta$ and collapse at the pole. Sampling $\theta$ from a uniform angle is not sampling solid angle. The inverse CDF for uniform $d\omega$ on $\Omega^+$ is $\cos\theta=\xi_1$, $\phi=2\pi\xi_2$.

Full sphere $\int_{S^2}d\omega=4\pi$. Hemisphere $\int_{\Omega^+}d\omega=2\pi$. This run: sphere_sr 12.566370614, hemisphere_sr 6.283185307. Neither number is $\pi$. $\pi$ is the projected hemisphere:

A sky of constant radiance $L_i$ over all of $\Omega^+$ delivers irradiance $E=L_i\pi$. The $2\pi$ count of directions is the wrong meter for a flat plate. That factor is geometry. It is already in the integral, before $f_r$.

E_analytic_* below is this lab’s scalar: Rec.709 luma of $L_i$ times $\Omega_\perp$. The plates label that axis $\mathrm{W}/\mathrm{m}^2$. It is not a spectrally integrated pyranometer reading.

The cap, then one bounce

On-axis spherical cap, $\theta\in[0,\alpha]$, $\phi\in[0,2\pi)$, $L_i$ constant on the cap:

$\Omega_\perp$ is the area of the disk of radius $\sin\alpha$ in the tangent plane. That disk is the right-hand drawing on the wedge plate. The ratio of the two closed forms is the identity printed on that plate:

because $1-\cos\alpha=2\sin^2(\alpha/2)$ and $\sin\alpha=2\sin(\alpha/2)\cos(\alpha/2)$. At the locked pair the quotient of the metrics tokens is the same identity: 0.998097 at $5^\circ$, 0.982963 at $15^\circ$.

Irradiance on the untilted plate from the cap alone, and the Lambert evaluation of the integral with $L_e=0$, one bounce, no interreflect:

Lambert’s $\rho/\pi$ is already per steradian. The $\pi$ in $\Omega_\perp$ cancels it. Dropping $(n\cdot\omega)$ and leaving $\rho/\pi$ in place is not a harmless rescaling. The weight still has the units of radiance. The integral is the wrong one. That wrong integral is the red ghost.

Named fill is a dim constant on the rest of $\Omega^+$, same projected measure. The complement has projected solid angle $\pi\cos^2\alpha$:

E_analytic_* is the disk product $L_{iY}\,\Omega_\perp$. Lo_analytic_*_Y is disk plus fill. The A/B irradiance identity is the disk row. The two columns answer different questions.

Estimator of the same integral

Uniform in solid angle on $\Omega^+$, and cosine-weighted, are instruments of the integral above. Same $N$, same $\xi$. This note does not rank them. That ranking is the importance-sampling note.

Under $p_{\cos}$ the factor $(n\cdot\omega)$ cancels, and a hit contributes $\rho L_i$ or $\rho L_{\mathrm{fill}}$ according to the cap test. Under $p_{\mathrm{unif}}$ the cosine stays in the weight: $(n\cdot\omega)/p_{\mathrm{unif}}=(n\cdot\omega)\,2\pi$. Both estimate the integral written once above, provided the density in the denominator is the density that was sampled. $f_r$ supplies one $\mathrm{sr}^{-1}$. $p$ supplies the other. $d\omega$ is accounted for. A density built on pixel footprint, or on uniform $\theta$, does not become a solid-angle density by writing $1/\mathrm{sr}$ next to it.

Small-angle stand-in

$\Omega$ and $\Omega_\perp$ at $5^\circ$ and $15^\circ$, and $\Omega$ at $3.600^\circ$, are from the metrics. $\Omega_\perp$ at $3.600^\circ$ and the $\pi\alpha^2$ column are that closed form, evaluated. They are not extra rows of the metrics file.

Each entry is close to its neighbors. The ratio of the stand-in is exactly $(15/5)^2=9$. The identity is

$E$ tracks $\Omega_\perp$. E_over_omega_ratio 0.984837 is the quotient of those two printed ratios, $(\Omega_{\perp B}/\Omega_B)/(\Omega_{\perp A}/\Omega_A)=\cos^2(7.5^\circ)/\cos^2(2.5^\circ)$. The absolute foreshortening at B alone is $\Omega_{\perp B}/\Omega_B=\mathbf{0.982963}$.

Fairness, gated:

Only $\Omega$ or the cosine changes between the photographs. The disk spectrum is one RGB triple, $(12.0,\,13.2,\,16.0)$, Rec.709 luma $Y=13.147$, exposure $e=1.00$. The brightness change on the size plate is the lesson.

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Unique artifacts

The grid plate is the measure drawn as a mesh. Twelve equal steps of $\cos\theta$ from 1 to 0, twenty-four equal steps of $\phi$. Every cell has solid angle

The sky cap is a spherical polygon of half-angle $15^\circ$ (cap B), laid on that mesh. It is not a lat-long texture, and it is not however many quads the checkerboard painted. The boundary cuts cells; $\Omega$ is the cap integral. The lower caption rounds cap B to $\Omega=0.214094\,\mathrm{sr}$, $\Omega_\perp=0.210447\,\mathrm{sr}$, prints the IS-key 0.012398431, and states d omega = d phi d(cos theta). The ninth digit lives in the metrics: 0.214094348 and 0.210446804. Upper line: the measure, not a latlong unwrap.

The wedge plate is the same cap in side view, $5^\circ$ above and $15^\circ$ below. Left: the polar segment on the sphere, cone rays drawn, unprojected $\Omega$. Right: the foreshortened disk of radius $\sin\alpha$, area $\Omega_\perp$ on the unit sphere. At $5^\circ$ the disk is a speck and $\Omega_\perp/\Omega=\mathbf{0.998097}$. At $15^\circ$ the disk is readable and the ratio is 0.982963. Lower caption: at 5 deg they almost agree; at 15 deg the article appears. The row labels print $\Omega$ to nine digits and the ratio to six; quote the metrics, not a re-read of the JPEG.

The identity plot plots irradiance against $\Omega_\perp$, out to $0.85\,\mathrm{sr}$ so the axis covers $\Omega_\perp(30^\circ)=\pi/4$. Cyan is $E=L_{iY}\Omega_\perp$, the disk row. Gold is that plus named fill on the projected complement,

The vertical gap is $L_{\mathrm{fill},Y}(\pi-\Omega_\perp)$. It narrows as the cap grows. Red is the Failure-B ghost $L_{iY}\,\Omega(\alpha)$, plotted at $x=\Omega_\perp(\alpha)$. Its gap above cyan is $L_{iY}(\Omega-\Omega_\perp)$. At $5^\circ$ that gap is invisible on this axis. At $15^\circ$ the metrics already carry it: $\Omega_\perp/\Omega=\mathbf{0.982963}$. At the $30^\circ$ rung, an analytic mark on this plot and not a second metrics family,

Circles mark the ladder $8^\circ/20^\circ/30^\circ$. Squares are the cosine-$\Omega_\perp$ arm at $N=256$, at A and at B only, mapped back by $E=L_{oY}\,\pi/\rho$. The estimator’s reference includes fill, so the squares target gold, the total irradiance. A square on the red ghost would mean the cosine had been dropped from the weight. A square read against cyan alone ignores the fill the estimator was asked to integrate. rel_err in the metrics is how far the underlying $L_o$ sits from Lo_analytic, not a pixel reading of this JPEG.

These three plates are the fingerprints. The courtyard stills do not replace them.

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Size A/B

The size plate is the photograph of Failure A. Identical $L_i$, identical $e=1.00$, identical $\rho=0.80$, identical camera. The two RGB triples compare equal.

From the metrics, disk cap, untilted card:

The same-$R$ pair under Failure A is the area hook: $R=1$, $d\in\{2,4\}$, $\Omega\in\{0.663334,\,0.187600\}\,\mathrm{sr}$, ratio 3.536.

The stored footprints are 910 and 7242, and they are not a ratio to set beside 8.954394 or 8.818616. Failure C: different frames, both tagged illegal as steradians. The lower caption of the size plate already prints the irradiance ratio.

Fill is named: Li_fill $=(0.0600,\,0.0660,\,0.0800)$, $Y=0.065735$, disk/fill $Y$-ratio 200. At $\alpha=5^\circ$ that dim hemisphere is the same order as the disk term inside Lo_analytic_A_Y 0.132081911, because $\Omega_{\perp A}$ is small. At $\alpha=15^\circ$ the disk dominates: E_analytic_B 2.766752422, Lo_analytic_B_Y 0.753613234. The ratio 8.818616 is the disk-cap identity. Interreflect is not in it. Walls exist so the courtyard reads.

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Cosine tilt, then the $N$ ladder

The cosine plate moves $(n\cdot\omega)$ and leaves the disk alone. $\alpha=15.000^\circ$ fixed in world. $L_i$, exposure, and $\rho=0.80$ frozen. Left $\beta=0$, $n=+Y$, $\Omega_\perp=\mathbf{0.210446804}$. Right $\beta=60^\circ$. Small-source caption: $\cos 60^\circ=\mathbf{0.500}$. The analytic row is the cap integral

Half of $\Omega_{\perp B}$ is $0.105223402$. The cap stays above the horizon ($\beta+\alpha=75^\circ<90^\circ$), and the two numbers meet inside the gated tolerance $5\times 10^{-4}$. The caption is 0.5. The meter is the integral 0.105223414. The Monte Carlo ladder below is the untilted card. Do not “fix” the dark card by raising $L_i$.

The MC-$N$ plate estimates the rendering equation at the card center, $\alpha=15^\circ$, $n=+Y$. Two arms, one $\xi$ stream per sample index, nested prefixes: $N=16$ is the head of $N=64$, which is the head of $N=256$, which is the head of $N=1024$. The published value is a mean over a small card-center neighborhood of independent streams. Reference is Lo_analytic_B (disk + fill), $Y=\mathbf{0.753613234}$. rel_err is $|Y(\hat L_o)-Y(L_o)|/Y(L_o)$ on that mean. It is not a single-pixel lottery and not the importance-sampling note’s $\mathrm{RMSE}_H$.

From the metrics ($\alpha=15^\circ$ ladder):

The thumbnails label the cosine arm at display rounding (0.1695, 0.0168, 0.0234, 0.0100). Those four decimals are not the quote. The metrics are.

$N=256$ bumps against $N=64$ on both arms (unif $0.017026571\rightarrow 0.021826733$; cos $0.016774743\rightarrow 0.023405564$). The rise is on the MC plate, between the second and third markers, on both traces. One seed, nested prefixes, not a theorem that every rung is monotone. The gate is the nested fall to 1024: rel_err at $N=1024$ is below $N=16$ and below $N=64$ on both arms. Which arm sits closer changes with the rung. That trade is not a sampler ranking. A flat error out to 1024 would be a broken weight or a broken $p$.

Uniform-$\Omega$ at $\alpha=5^\circ$ is a high-variance instrument. The cap is a small fraction of $2\pi$. Card-center A at $N=256$, from the metrics: unif rel_err 0.054307110, cos 0.059173158. The fall gate uses the $\alpha=15^\circ$ ladder.

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Quote the metrics. Do not quote the beauty photographs as meters.

Float buffer, Mesa llvmpipe. $\Omega$, $\Omega_\perp$, $E$, $L_o$, rel_err from linear Rec.709, before Neutral. Seed 1352782172. Hash = pcg. TM = Khronos PBR Neutral (not re-fit), $e=1.00$.

The $\alpha=15^\circ$ Monte Carlo ladder is the table in the previous section. Card-center A at $N=256$ is only in the metrics: unif $L_{oY}$ 0.139254898, cos $L_{oY}$ 0.139897615, rel_err 0.054307110 / 0.059173158.

Hero line: $E_B/E_A$ 8.818616; same-$R$ 3.536; IS-key $\Omega$ 0.012398431; $L_{iY}$ 13.147; fill $Y$-ratio 200; seed 1352782172; 28 pass / 0 fail. Do not invent $\Omega$ or $E$ from the hero, the size plate, or the cosine plate. The metrics strip is a snapshot of this table. The source of quoted numbers is the float-buffer metrics.

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Controls

Three knobs. Everything else stays put.

Size ($\Omega$)

Locked A/B: $\alpha\in\{5.000^\circ,15.000^\circ\}$, on-axis from the card center, one disk radiance. Ladder rungs on the identity plot only: $8^\circ$, $20^\circ$, $30^\circ$. Same-$R$ area control is a caption on the size plate and a row in the metrics, not a reshoot.

Cosine ($\beta$)

Locked: $\alpha=15.000^\circ$ disk fixed in world, card normal at $\beta\in\{0^\circ,60^\circ\}$. The analytic row is the cap integral, written next to $\cos 60^\circ=0.500$.

MC-$N$

Locked ladder $N\in\{16,64,256,1024\}$ at the untilted card center. Uniform-$\Omega^+$ and cosine-$\Omega_\perp$, same $\xi$, nested prefixes. Hero quote is the $\alpha=15^\circ$ fall to 1024. $N=256$ may sit above $N=64$.

Frozen on every plate:

• Camera, card albedo, wall and floor albedo, fill radiance, disk RGB, exposure $e=1.00$, Neutral constants inherited from the tone-mapping note, one seed.
• Display: RGBA32F $\to$ Neutral $\to$ sRGB OETF on CPU. GL_FRAMEBUFFER_SRGB off.
• No auto-exposure. No per-plate gain.
• Fill named. Disk / fill luminance ratio 200.
• $\Omega$ from the cap formula. The white shape in the opening is a drawn disk. Triangle count does not define the steradian.

The fairness gate is bit-identical $L_i$ on A and B. Only $\alpha$ or $\beta$ changes. Darkening A by lowering $L_i$ so the JPEGs read as a pair fails the note.

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Two paths, do not mix the instruments

The size plate is a photograph of the control and the source of the teaching. The number 8.818616 is the metrics.

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Honesty gaps

1. Beauty shading, including the card, is a disk form factor in GLSL. The meter is the card-center analytic identity and the CPU estimator. The JPEG is a picture of the courtyard.
2. Fill $Y$-ratio is 200. At $\alpha=5^\circ$ the fill term is the same order as the disk term inside Lo_analytic_A_Y 0.132081911. At $\alpha=15^\circ$ the disk dominates Lo_analytic_B_Y 0.753613234. The ratio 8.818616 is disk-only.
3. Tilt. $\Omega_\perp(15^\circ,60^\circ)=\mathbf{0.105223414}$. Half of $\Omega_{\perp B}$ is $0.105223402$. They meet inside $5\times 10^{-4}$. The caption is $\cos 60^\circ=0.500$. The cap stays above the horizon. The $N$ ladder is untilted.
4. Interreflect is not in the A/B identity. Walls exist so the courtyard reads.
5. pixels_disk is a luma threshold on the linear frame, before Neutral, aimed at the drawn disk. 910 is the $628\times 720$, $\alpha=5^\circ$ panel. 7242 is the $1280\times 720$ hero. Not a matched pair. Not a steradian. A $5^\circ$ cap is 0.023909417 sr either way.
6. Uniform-$\Omega$ at $\alpha=5^\circ$ is high variance. The fall gate is the nested $\alpha=15^\circ$ ladder.
7. $N=256$ bumps versus $N=64$ on both arms. The traces on the MC plate show it. The gate is nested fall to 1024, both arms, below $N=16$ and below $N=64$.
8. rel_err is a nested neighborhood mean, not a single pixel and not $\mathrm{RMSE}_H$. Thumbnails show the cosine arm. The metrics hold both arms. HUD rounding on the thumbnails is not the quote.
9. JPEG is 8-bit display-referred. Neutral + OETF allocates codes. $\Omega$, $E$, $L_o$, and rel_err live in the float buffer and in the metrics.
10. Neutral constants are copied from the tone-mapping note. Not re-fit. Energy after Neutral is not a claim.
11. Analytic disk. Not a captured EXR. Not a directional delta, which would replace the cap with a Dirac and erase the steradian plate. The drawn opening does not define $\Omega$ by tessellation.
12. The two MC arms are instruments of one integral. Closer rel_err changes with $N$. Cite the importance-sampling note for pdf–integrand mismatch. Cite the IBL note for Karis / DFG. Env mean luma 1.628 stays there.
13. A $9\times$ irradiance jump at $5^\circ$ versus $15^\circ$ is the rejected stand-in. The identity is 8.818616. A same-$R$ ratio of 4 is the rejected inverse-square in distance. The identity is 3.536.

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Mesa / llvmpipe — what this run can claim

Can claim: on this OSMesa / llvmpipe build, an analytic sky cap of published $\alpha$ over a Lambert card, same $L_i$, produces an irradiance and a one-bounce $L_o$ that track $\Omega_\perp=\pi\sin^2\alpha$. The hemisphere grid, the $E$–$\Omega_\perp$ plot, and the projected wedge are diagrams of that identity. A fixed-$N$ estimator of the rendering equation at the card center reports the numbers in the metrics.

Cannot claim: hardware RT, a real-time budget, or interactive 1-spp. That uniform or cosine is the production sampler. Energy after Neutral. Anything measured off the JPEG. A captured sky. Interreflect inside the A/B ratio. A pixel footprint as a steradian. A $9\times$ irradiance jump at $5^\circ$ versus $15^\circ$ as confirmation. A same-$R$ ratio of 4 as confirmation.

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Assertions

This run: 28 pass / 0 fail.

No tolerance was opened to accept $E_B/E_A=9$ or a same-$R$ ratio of 4.

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Out of scope

Phong-versus-cosine as a second bake-off, MIS balance heuristics, GGX VNDF, Smith $G$, the half-vector Jacobian. Cite the importance-sampling note for pdf–integrand mismatch. Split-sum, Karis prefilter, DFG LUT, the loft HDR — cite the IBL note. Tone-map bake-off — Neutral is inherited at $e=1.00$; cite the tone-mapping note. A full path tracer: multi-bounce GI, next-event estimation as a sampling arm, Russian roulette, spectral transport, participating media. Real-time path tracing, hardware RT, DLSS / SVGF / OIDN / ReSTIR. Area-light LTC as the production method — the wedge is a diagram of $\Omega$ against $\Omega_\perp$. A directional delta, IES profiles, Hosek–Wilkie / Preetham as the disk. Microfacet metals and the banned still-life (lacquer, brass, oak, bottle glaze). Shadow maps, POM, anisotropic footprints, mip LOD, TAA. Different notes.

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Measure lock

Omega(alpha)       = 2 pi (1 - cos alpha)
Omega_perp(alpha)  = pi sin^2(alpha)
E_disk             = Li_Y * Omega_perp
Lo_disk            = rho * Li * sin^2(alpha)
Lo_analytic        = Lo_disk + rho * L_fill * (1 - sin^2(alpha))
E_B / E_A          = 8.818616     # (15/5)^2 = 9 is the rejected stand-in
sameR Omega ratio  = 3.536        # R=1, d=2/4, area=pi; 4 is rejected
IS-key 3.600 deg   -> Omega = 0.012398431 sr   # cited, not rematched
hat Lo             = (1/N) sum  f_r Li (n·w) / p(w)
p_unif             = 1/(2 pi) on Omega^+      # cos theta = xi_1
p_cos              = (n·w) / pi               # cos theta = sqrt(xi_1)
PNG                = sRGB_OETF( Neutral(e * Lo) )    # e=1.00, inherited

Pin the hero as the presentation. Pin the size A/B plate as the teaching pair. Pin the grid, $E$ vs $\Omega_\perp$, and wedge as the fingerprints. Pin the MC-$N$ plate as the estimator. Path tracers sample $d\omega$. Irradiance tracks $\Omega_\perp$. The previous note’s densities were already in $1/\mathrm{sr}$. This note drew the steradian.