Journal

The last note owned correspondence. Color clamp asks whether history looks legal. Depth reject asks whether it is the same surface. Time is out of scope here. Split-sum already hid the samples in a prefilter cube $\times$ DFG LUT; production then multiplies two tables. Tone mapping is inherited — Neutral after resolve, not re-fit. This note unhides the estimator. It owns Monte Carlo sampling mismatch — not VNDF, not a MIS bake-off, not a split-sum re-teach.

Same integral, two pdfs, one $N$. Cosine-weighted hemisphere is the Lambertian ticket. A Phong cosine-power lobe about $R=\mathrm{reflect}(-\omega_o,n)$ is the highlight ticket. On a sharp dielectric they are not interchangeable.

A new photographic family: gallery spotlight. Lacquer dielectric sphere on a short dark-stone plinth, dark slate wall, recessed canvas, painted picture rail. Phong-IS, $N=256$, $s=32$. HUD: Khronos PBR Neutral, PHOTO-ONLY, Phong-IS N=256 s=32. Do not hang $\mathrm{RMSE}_H$ on this photograph. The felt wall sparkles: Lambert sampled by a Phong lobe. That is Failure B on the cover, not a denoiser miss. The clean gallery still is the reference plate below.

Pin this. Same view, same $N=64$, same seed stream. Uniform | cosine | Phong-IS. Only $p$ differs. Wedge crop of the highlight crescent under each column. Caption on the plate: same view, same N=64, same seed, only p differs. Photo only.

Hero, Mesa 25.0.7 llvmpipe, linear Rec.709, Khronos PBR Neutral $e=\mathbf{1.00}$, seed 329363537, key angular radius 3.600°, $N=64$, $s=32$: $\mathrm{RMSE}_H$ cosine 7.655620 vs Phong-IS 1.999657 (pass: phong $<$ cosine). $\mathrm{RMSE}_S$ cosine 0.140667 vs Phong-IS 0.824834 — surround starve, shown honestly. Fireflies ($k=4$) 21536 vs 1653. $\mathrm{ROI}_H=(502,391,555,432)$, $\mathrm{ROI}_S=(349,133,613,319)$. Assertions 24 pass / 0 fail.

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Two questions, two pdfs

The integral does not change. Lighting, geometry, camera, BRDF, exposure, and the per-pixel $\xi$ stream do not change. Only $p(\omega)$ changes. Variance is the photograph.

Cosine. Does this direction match Lambert $\times$ the projected hemisphere?

Phong-IS. Does this direction match the cosine-power highlight about the reflection vector $R$? Not the half-vector. Not Blinn. Not GGX-VNDF.

On a lacquer dielectric under a 3.600° gallery disk they are not the same ticket. Cosine spends almost every sample off the lobe. When a sample does hit the small key, $L_i$ is huge and $p_c$ is only $(n\cdot\omega)/\pi$ — lottery ticket, firefly. Phong-IS puts the inverse-CDF on $R$. The crescent cleans up. The flanks do not automatically follow.

Uniform hemisphere ($p=1/(2\pi)$) is the loud third arm: any-hemisphere sample is not cosine importance. At $N=64$, $s=32$, $\mathrm{RMSE}_H=\mathbf{9.900829}$ vs cosine $\mathbf{7.655620}$; $\mathrm{var}_H=\mathbf{90.04}$ vs $\mathbf{56.93}$. It is not a replacement for cosine. Firefly count at the same row is cosine 21536, uniform 14400, Phong-IS 1653 — quote $\mathrm{RMSE}_H$, not that count, as the loudness ranking.

The scientific claim is narrow: pdf–integrand mismatch is a measurable variance, not a taste. Cosine matches Lambert $\times$ projected hemisphere. Phong-IS matches a cosine-power highlight. Neither matches both at once.

Two facts, never mixed:

1. Beauty plates (hero, 3-up, $N$ ladders, exponent, surround) are CPU MC, Neutral then sRGB OETF. HUD photo-only means do not invent RMSE, firefly count, or $\mathrm{Y_{mean}}$ from the JPEG. 8-bit Neutral+sRGB clips fireflies.
2. Instruments (variance heat, pdf rose, metrics strip, reference) are the float buffer: linear Rec.709 $Y$ residual, pdf rose, RMSE, fireflies, horizon fraction. Quote the metrics.

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Why: the estimator, then the two pdfs

Working space is scene-referred linear Rec.709. Domain $\Omega^+=\{\omega:n\cdot\omega>0\}$. Below-horizon samples contribute 0 and still count in $N$. Do not renormalize $p$ after a horizon reject.

Same integral

Unbiased solid-angle estimator

Same $f_r$, same $L_i$, same camera, same geometry for every arm. Cosine stays in the numerator. It is not folded into $p_p$.

BRDF (Lambert + Lafortune-normalized Phong)

Dielectric body, not a metal. This run: $\rho_d=(0.155,0.175,0.188)$, $\rho_s=0.12$, default $s=32$. Metal ($\rho_d=0$) would hide Failure B.

$(s+2)$ is the BRDF energy constant. $(s+1)$ is the solid-angle pdf constant. Mixing them biases $\hat L$. White-furnace assert ($L_i\equiv 1$, lobe well above the horizon): $\hat L_o\approx\rho_d+\rho_s$ within a stated band. Science assert, not a hero plate.

No Fresnel, no GGX $D/G$, no Smith. This is not a microfacet product note.

Pdfs (1/sr)

Phong generation is in the $R$-frame. If $n\cdot\omega\le 0$: weight $=0$, increment horizon_frac. RNG is a PCG integer hash, seed 329363537.

Light (analytic; no HDRI)

1-bounce direct. Finite disk key, upper-front-right, angular radius 3.600° from the highlight point, $\mathrm{disk}_r=0.1372$. $L_{\mathrm{key}}=(980,880,720)$ warm. Dim cool fill $L_{\mathrm{fill}}=(1.20,1.38,1.62)$ over $\Omega^+$ so flanks and wall remain measurable.

$L_i(\omega)=L_{\mathrm{key}}$ if the ray hits the disk and is not occluded by sphere or plinth, else $L_{\mathrm{fill}}$. This note samples the reflection pdf only. Do not sample the light pdf. Do not MIS. A Dirac directional is illegal — hemisphere MC cannot hit a delta, and the note would collapse into NEE. Tiny key solid angle is what makes cosine firefly, and what a too-sharp $s$ can miss.

Jacobian / solid angle

The inverse-CDF maps $(\xi_1,\xi_2)\in[0,1)^2$ to $\omega\in\mathbb{S}^2$. $p$ above is already in solid angle. Do not multiply by an extra $d\omega/d\xi$. A half-vector sampler would need the $h\to\omega$ Jacobian $\partial\Omega_h/\partial\Omega_\omega=4(\omega\cdot h)$. That Jacobian is why Blinn / GGX-VNDF is a different note. Cite it. Do not derive it. VNDF is how production samples this family of lobes. It is not the A/B.

Variance used on plates

All of these are linear Rec.709 $Y$, scene-referred, never JPEG, never after Neutral.

Two named ROIs, inclusive pixel rects at beauty $960\times 540$ (bottom-up):

• H — highlight crescent the key paints: $(502,391,555,432)$.
• S — diffuse flank / plinth / foot: $(349,133,613,319)$.

Per-pixel variance of the mean (no ref required):

The heat plate is $\lvert Y_{64}-Y_{\mathrm{ref}}\rvert$ in turbo, cosine vs Phong, same highlight-rim crop, shared p98 scale. Firefly count: $\#\{x:|Y_N-Y_{\mathrm{ref}}|>k\,Y_{\mathrm{ref}}\}$ with $k=4$ on hit pixels. Whole-frame RMSE is a footnote. It averages H and S and hides the lesson.

Display (inherited, not re-derived)

TM is Khronos PBR Neutral, $e=1.00$, frozen across A/B/C. Constants not re-fit. Cite the tone-mapping and split-sum IBL notes. GL_FRAMEBUFFER_SRGB is off; encode is CPU. Measurement is the float buffer, not the JPEG.

Reference (not a sampling arm)

$L_{\mathrm{ref}}$ = analytic Lambert $\times$ fill + Phong-IS of specular $\times$ fill (32 spp) + disk NEE (96 spp). Independent seed 1374605335. Arms never sample the light pdf. This is how the box gets a high-$N$-quality ref without 4096 spp hemisphere MC on llvmpipe. The reference plate outlines ROI H/S. Not the cover.

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Teaching pin: three pdfs, one view

The 3-up is the policy compare. Identical camera, geometry, materials, lights, exposure, $N=64$, seed 329363537. Only $p$ differs.

If the full-frame columns look like “Phong is better,” the surround plate is the correction. Trust the wedge for Failure A. Trust the honesty crop for Failure B. Quote $\mathrm{RMSE}_H$ / $\mathrm{RMSE}_S$ from the metrics, not from the 8-bit panel.

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Unique artifact: variance heat on the highlight rim

This is the thing this note exists to draw. False-color $\lvert Y_{64}-Y_{\mathrm{ref}}\rvert$, linear Rec.709 $Y$, turbo, same rim crop, shared p98 scale. HUD: INSTRUMENT FINGERPRINT, linear Rec.709 Y residual not PNG.

Cosine heat rings the crescent: a yellow-green residual blob with red firefly speckles. Phong-IS drops there — mostly blue, sparse specks, the outline of a lobe that actually got samples. Heat leaves the rim and may appear on the flank. That swap is the fingerprint. Not another RGB triangle. Not a DFG LUT remake.

The pdf rose is the secondary science plate. Hemisphere overlay, linear pdf in 1/sr, turbo: cosine about $n$ vs Phong about the same $R$, $s=32$. Polar insets under each. Caption on the plate: not Blinn, not GGX VNDF. Beside the photo, never the cover.

Quote the metrics, not the JPEG.

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Ladders: $N$, then $s$

Failure A. Cosine at $N=4/16/64/256$, same seed-stream prefixes. Fireflies decay slowly on the rim. At $N=256$ the body is quieter; the crescent is still not a Phong crescent. $\mathrm{RMSE}_H$: 27.886635 / 15.989179 / 7.655620 / 3.837384.

Contrast. Phong-IS at the same $N$ rungs, matching pdf. The rim cleans earlier. Crescent readable by $N=16$, tighter at 64/256. Surround stays dark with sparkles — Failure B riding along. $\mathrm{RMSE}_H$: 8.377738 / 4.037456 / 1.999657 / 1.010789.

Exponent control. Phong-IS, fixed $N=64$, $s=8/32/128$, matching pdf. Wide lobe $\approx$ cosine on H ($s=8$, $\mathrm{RMSE}_H=1.783442$, still beats cosine). Default $s=32$ is the pass plate. Sharp $s=128$ vs this 3.600° key: $\mathrm{RMSE}_H=7.367447$ — ROI H is the $s=32$ crescent, so the extra rim is starved. That is the exponent control, not the pass predicate. High $s$ also hurts S.

Do not retune exposure or Neutral to hide cosine fireflies on a beauty plate. Fireflies are the lesson. The heat plate is where they are measured.

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Failure B: Phong-IS starves the surround

This is the honesty plate. Tight crop of flank / plinth / foot. Cosine $N=64$ | Phong-IS $s=32$ | Phong-IS $s=128$. Starve is allowed to win the crop. HUD: PHOTO-ONLY ON BEAUTY RMSE FROM FLOAT Y. Caption: Failure B flank / plinth / foot starve is allowed.

At $N=64$, $s=32$: $\mathrm{RMSE}_S$ cosine 0.140667 vs Phong-IS 0.824834. At $s=128$: 0.860317. Both above cosine. The pdf lives around $R$. Lambert still needs samples on the rest of $\Omega^+$. High $s$ will not feed the plinth.

The same starve is why Phong $\mathrm{Y_{mean}}\approx 0.059$ vs cosine $\approx 0.146$ at $N=64$. Combined Phong white furnace ($\rho_d+\rho_s$) undershoots: measured 0.1930 vs expect 0.2917. At $s=32$ the inverse-CDF never places the Lambert tail in float32 ($\xi=\mu^{s+1}$ underflows). Spec-only Phong furnace hits $\rho_s=\mathbf{0.1200}$ exactly — that is the $s+1$ vs $s+2$ check. Cosine combined hits 0.2923 vs 0.2917. Logged, not hidden. Unbiased in theory; float inverse-CDF is not a full-hemisphere Lambert sampler at this $s$.

Phong $\mathrm{RMSE}_S$ is not monotone in $N$ on this one seed: 0.824834 at 64, 1.754730 at 256. Rare disk hits on the plinth are high-leverage. Residual heat, not a second theorem.

Hero is Phong-IS, so the felt wall sparkles. That is this failure on the cover. The clean gallery photograph is the reference plate.

Veach / MIS is how you stop choosing. One closer sentence, not a bake-off hero.

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

Float buffer, Mesa llvmpipe. RMSE / var / fireflies from linear Rec.709 $Y$, never JPEG. Firefly $k=4$. Seed 329363537. Ref seed 1374605335. Hash = pcg.

Pass predicate, $N=64$, $s=32$:

$\mathrm{RMSE}_H(\mathrm{phong})<\mathrm{RMSE}_H(\mathrm{cosine})$: 1.999657 $<$ 7.655620. PASS.

$N$ ladder, $s=32$, $\mathrm{RMSE}_H$:

Exponent, Phong-IS, $N=64$:

White furnace ($L_i\equiv 1$, $R=n$, lobe off-horizon):

Header constants, as written:

Hero rounding used in the lede: $\mathrm{RMSE}_H$ 7.66 / 2.00; $\mathrm{RMSE}_S$ 0.141 / 0.825; fireflies 21536 / 1653; key 3.600°; seed 329363537; 24 pass / 0 fail. Do not invent RMSE from the hero, the 3-up, or the surround crop. Those frames are photo-only. The metrics strip is a science snapshot of the same table; quoted numbers are the float buffer.

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Controls

Every A/B/C plate shares bit-identical camera, geometry, materials, lights, exposure, and per-pixel $\xi$ stream. Inverse-CDF differs; directions differ. Do not reuse a cosine direction under a Phong weight.

Seed

One published integer: 329363537. Same stream across arms. A residual that flips with a re-seed is RNG, not a theorem.

Horizon

Weight 0, count in $N$, log horizon_frac. No pdf renormalize. Renormalizing biases the estimator toward the upper hemisphere. Phong $s=32$: 0.025246. Cosine / uniform: 0 (generation in the $n$-frame). Wider lobe ($s=8$) $\to$ 0.084079; sharper ($s=128$) $\to$ 0.002089.

White furnace

$L_i=1$, lobe off-horizon. Cosine combined and Phong spec-only are the gated asserts. Combined Phong shortfall is the float inverse-CDF tail, not a loosened band.

Firefly $k$

$k=4$ on linear $Y$. Do not score fireflies on the JPEG. Neutral+OETF hides them.

Light angular size

Stated 3.600°. Small enough that cosine misses, large enough that Phong-$s=32$ hits. Delta light $\to$ experiment becomes NEE. Fat light $\to$ cosine looks fine and the note dies.

Material

Dielectric, both $\rho_d$ and $\rho_s$ live. Plinth / wall / floor are Lambert. Metal-only hides surround starve.

ROI

H = crescent the key actually paints. S = flank + plinth, no highlight pixels. Whole-frame RMSE is not a substitute.

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Two paths, do not mix the instruments

The 3-up is a photograph of the control and the source of the teaching. Quote the metrics for RMSE and fireflies. Do not quote the 8-bit panel as 7.655620.

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Honesty gaps

1. Offline spp strip on OSMesa / llvmpipe. Named $N$ on a still. Not 60 Hz. Not “interactive 1 spp.” Not a real-time path tracer. Not hardware RT. Not a production IBL baker.
2. JPEG is 8-bit display-referred. Neutral+OETF clips fireflies. RMSE / var / firefly count are linear $Y$ on the float buffer.
3. Hero is Phong-IS. Felt wall sparkles. Failure B on the cover, not a denoiser miss. Clean still is the reference plate.
4. $L_{\mathrm{ref}}$ uses disk NEE. That is not an A/B/C arm. The note still samples reflection pdfs only.
5. Combined Phong white furnace undershoots (0.1930 vs 0.2917). Inverse-CDF at $s=32$ cannot place the Lambert tail in float32. Spec-only hits $\rho_s$ exactly; cosine combined hits $\rho_d+\rho_s$.
6. Phong $\mathrm{RMSE}_S$ is not monotone in $N$ on this seed (64: 0.824834, 256: 1.754730). Rare disk hits on the plinth. Residual heat, not a second theorem.
7. $s=128$ $\mathrm{RMSE}_H=7.367447$ is the exponent control against a 3.600° key and an $s=32$ ROI H, not a failed pass predicate.
8. Beauty $960\times 540$ ($\le 1280\times 720$). Composites $1280\times 720$. Not full-frame 4k spp.
9. Analytic sphere. $n$ from the implicit surface, not a faceted mesh that sparkles under a sharp lobe. One sphere, one plinth, dark gallery.
10. No Fresnel, no GGX, no Smith, no HDRI, no MIS arm. 1-bounce disk + constant fill.
11. Whole-frame RMSE is not the lesson. It would average H and S.
12. Not DLSS / OIDN / SVGF / ReSTIR. Offline spp strip, not a 1-spp product.
13. Neutral constants copied from the tone-mapping note / Khronos PBR Neutral. Not re-fit.
14. Cousin only: VNDF is how production samples this family of lobes; Veach is how you stop starving Lambert.

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Mesa / llvmpipe — what this run can claim

Can claim: on this OSMesa / llvmpipe build, the same integral under cosine vs Phong-IS at published $N$ and $s$ produces these $\mathrm{RMSE}_H$ / $\mathrm{RMSE}_S$ / firefly numbers; the heat plate is that residual; the 3-up is the same view with only $p$ changed.

Cannot claim: hardware RT, a real-time budget, “Phong is the correct production sampler,” energy identity after Neutral, anything measured off the JPEG. Discrete-GPU metrics, occupancy, bandwidth, or “this is how the hardware works.”

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Assertions

This run: 24 pass / 0 fail.

No assert tolerances were loosened to hide cosine fireflies or Phong surround starve.

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Out of scope

Full MIS product bake-off as hero — Veach is one closer sentence, no balance-heuristic plates. GGX VNDF deep dive — cousin sentence only; no Smith $G$, no half-vector Jacobian derivation, no roughness $\to$ mip. Re-deriving Neutral TM, Karis prefilter, DFG LUT, split-sum energy tables — cite IBL + TM. Spectral path tracer. DLSS / SVGF / OIDN / ReSTIR as the denoiser of this still. Multi-bounce GI, next-event estimation as a sampling arm, area-light LTC. Shadow-map bias. Toksvig / anisotropic GGX / sRGB-vs-linear texture decode. A second photographic family as cover or journal hero. Real-time path-tracer claims, hardware RT cores, occupancy, bandwidth.

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Estimator lock

Lo   = (1/N) sum  f_r(w_k, w_o) * L_i(w_k) * (n·w_k) / p(w_k)
p_c  = (n·ω)/π                         // ONB around n
p_p  = (s+1)/(2π) (ω·R)^s              // ONB around R; n·ω≤0 → weight 0, still in N
f_r  = ρd/π + ρs (s+2)/(2π) (ω·R)_+^s  // s+1 is pdf; s+2 is BRDF
PNG  = sRGB_OETF( Neutral(e * Lo) )    // e=1.00, inherited

Locked: same integral, two pdfs, one $N$; cosine in the numerator; horizon weight 0 and still in $N$; Neutral inherited, not re-fit. Pin the gallery still as the presentation. Pin the 3-up as the teaching. Pin the variance heat as the fingerprint. Pin the surround crop as the honesty plate. The tickets are not interchangeable.