Ultraviolet carcinogenesis splits cleanly on the substrate. Because the multistage malignant rate is convex in dose-intensity, squamous-cell carcinoma rises near-linearly with cumulative dose (r=0.999), while melanoma is intermittent-sensitive: the same dose in bursts raises relative risk up to ~5.7×. Chronic exposure builds a screening tan, protecting against melanoma by ~2.6× — the chronic-exposure paradox.
A single convex, multistage malignant rate splits the two ultraviolet cancers. SCC rises near-linearly with cumulative dose (r=0.999); melanoma is intermittent-sensitive (burst relative risk up to ~5.7×); and a chronic-exposure tan protects against melanoma (~2.6×).
One convex rate, two cancers
In this framework a carcinogen is a sustained aberrant drive that lowers the barrier between the normal and malignant basins of the same cell switch; the malignant-crossing rate is Kramers-like, rate ∼ e|h|/spinodal. Because malignancy needs several independent hits, the realised rate is multistage, rateK with K ≈ 5 — a cited biological structure, not a fitted knob. That single convex rate splits the two ultraviolet cancers.
Squamous-cell carcinoma tracks cumulative dose
At a fixed chronic intensity the cumulative malignant-initiation probability accumulates linearly in total dose, so squamous-cell carcinoma rises near-linearly with cumulative ultraviolet: across the low-dose range the dose–response is linear to a correlation of r = 0.999. This matches the recognised near-linear dependence of cutaneous SCC on cumulative, occupational ultraviolet.
Melanoma tracks intermittent bursts
For a fixed cumulative dose, concentrating it into bursts raises the integrated hazard, because a convex rate rewards peaks (Jensen's inequality). Delivering the same dose intermittently rather than spread raises the melanoma relative risk by up to about 5.7× here — the substrate's account of melanoma's sunburn/intermittent-exposure sensitivity.
The chronic-exposure paradox
Chronic exposure is then protective against melanoma. A spread, chronic dose builds a melanin screen (the tan of the previous page) that a fast sunburn cannot, lowering the effective intensity; in this run the tan reduces the chronic-arm hazard by a factor of about 2.6×. That reproduces the otherwise puzzling inverse association between chronic occupational ultraviolet and melanoma.
Grades
The SCC cumulative linearity, the melanoma intermittent-sensitivity, and the chronic-exposure tan paradox are simulation-verified directions [V] Simulation-verified resting on cited epidemiological anchors [L] Cited anchor. The absolute incidences and the absolute relative-risk magnitudes are open [O] Open (obstacle stated): they need a population baseline rate and an absolute dose calibration.