Carcinogen Dose–Response (Kramers)

A carcinogen lowers the R19 barrier, raising the malignant crossing rate. This volume does not assume the Kramers law — it simulates the stochastic R19 field and counts crossings, and both a convex super-linear dose–response and the Kramers exponential emerge from the dynamics. Benzene→AML (IARC Group 1) anchors it. Grade [V] shape+law / [L] anchor / [O] steepness.

The dose–response is measured by direct stochastic barrier-crossing simulation: it is convex/super-linear (convex=True) and the Kramers law emerges (log-rate linear in barrier, R²=0.973). Sites: AML←benzene/radiation/alkylators; lymphoma←radiation/immunosuppression/EBV. Absolute steepness (noise scale D) is uncalibrated and graded [O].

Why the curve bends up

The kernel is one line: a carcinogen erodes the barrier, and the Kramers rate depends exponentially on the remaining barrier. A small extra dose near the top of the range removes proportionally more barrier, so the crossing rate accelerates — the curve is convex, not linear.

At the spinodal the barrier reaches zero and the rate diverges: with no barrier left the transformation is deterministic rather than thermally activated. This is the mathematical face of a saturating carcinogen.

The shape is measured, not assumed

Rather than write down the Kramers formula and trust it, the volume simulates the stochastic R19 field directly: a population of healthy cells under overdamped Langevin dynamics with a carcinogen drive that erodes the barrier, and the malignant crossing rate is counted from how many cells cross the ridge. Two things emerge from the dynamics. First, the measured dose–response is convex and super-linear (convex=True; the measured relative rate runs roughly 1 → 10 → 370 → 1782 across the dose fractions). Second, the logarithm of the measured rate is linear in the independently-computed effective barrier (R²=0.973) with an Arrhenius slope of -19.0, recovering the expected −1/D=-20.0 — so the Kramers exponential law is an emergent property of the simulated substrate, not an assumed kernel.

Anchors, honestly graded

Benzene→acute myeloid leukaemia is the cleanest occupational anchor (IARC Group 1, ppm-years exposure metric); lymphoma is anchored to ionizing radiation, chronic immunosuppression, and EBV drive. The shape and the Kramers law are [V] (simulation-measured) and the anchors are [L], but the absolute incidence and the steepness depend on the cellular-noise scale D, which the volume does not calibrate — stated [O], no fabricated relative-risk numbers.