Fixed Cost vs Pixel Cost: A Resolution Ladder With R^2 = 0.99993

Frame time was a clean linear function of shaded pixels, and an earlier 'quartering pixels saved ~20%' figure did not survive the full curve.

Core Finding:We measured frame time at six render extents from 1080p to native 4K and fit T = F + k·P. On the three native rungs, p50 frame time fit with R^2 = 0.999931, and three resampled rungs landed on the same line within 0.04 ms. Dropping from 4K to 1080p cut frame time by 48%, not the roughly 20% an earlier spot measurement had suggested.

Linear fit on p50

R^2 = 0.999931

Fit on p50: F = 2.6416 ms, k = 0.5707 ms/Mpix, R^2 = 0.999931. p99 fit R^2 = 0.999999.

4K to 1080p frame-time reduction

48%

4K to 1440p: 1.548x ratio, about a 35.7% frame-time reduction. 4K to 1080p: a 48% reduction.

p50 frame time, native 4K

7.373 ms

p50 frame time: 1080p 3.812 ms; 1440p 4.763 ms; native 4K fullscreen 7.373 ms.

Pixel term share at native 4K

64%

At native 4K, pixel term 4.74 ms = 64% of the frame; at 1080p it is 31%.

Empirical Findings Table

Frame time at six render extents, 1080p to native 4K

Telemetry Metric	Measured Value	Context & Observability
Linear fit on p50	R^2 = 0.999931	Fit on p50: F = 2.6416 ms, k = 0.5707 ms/Mpix, R^2 = 0.999931. p99 fit R^2 = 0.999999.
4K to 1080p frame-time reduction	48%	4K to 1440p: 1.548x ratio, about a 35.7% frame-time reduction. 4K to 1080p: a 48% reduction.
p50 frame time, native 4K	7.373 ms	p50 frame time: 1080p 3.812 ms; 1440p 4.763 ms; native 4K fullscreen 7.373 ms.
Pixel term share at native 4K	64%	At native 4K, pixel term 4.74 ms = 64% of the frame; at 1080p it is 31%.
Held-out rung residuals	±0.04 ms	Held-out rungs land within ±0.04 ms of the fit (residuals +0.001, -0.013, +0.039 ms).
Pixel cost equals fixed cost	4.63 Mpix	Crossover where pixel cost equals fixed cost: 4.63 Mpix.
Present pass share at 4K	97%	At 4K, the present pass (GPU shading plus present) was 7.129 ms of the 7.373 ms frame (97%).
Native 4K p99	7.807 ms	Every rung met the 8.33 ms (120 FPS) target at p50 and p99; native 4K p99 was 7.807 ms.

Engineering Action Directives

For this scene on this GPU, resolution behaved as a near-linear lever, and the cost tracked shaded pixels rather than the swapchain size. A half-scale render presented into a 4K swapchain cost 3.813 ms, within 0.001 ms of native 1080p at 3.812 ms.
The earlier ~20% figure was most likely confounded by vsync quantization or a windowed-mode clip that hid the true render extent. Logging the true shaded extent per rung was what made the curve clean.

Methodology Boundaries & Limits

The fit has only three points; a high R^2 on three points says the points are collinear, not that the model holds across all resolutions.
One light scene at one camera anchor on one GPU. Later builds with more content measured much slower 4K frames, so these absolute numbers describe this scene only.
The source of the earlier ~20% figure is not in this record, so the confound is an explanation, not a demonstrated cause.