Hormone-driven cancer is Kramers escape over a lowered R19 barrier
Hormone-driven cancer is Kramers escape over an R19 barrier that a sustained hormone drive lowers. The dimensionless dose-response shape ΔV/ΔV0 versus h/hsp is identical for every γ (max spread 0.0) — universal: the tissue master gene sets only the barrier scale, never the shape. Site shapes are sim-verified; epidemiological anchors are measured; absolute incidence is open.
A carcinogen is a sustained aberrant drive that lowers the escape barrier out of the healthy well; transformation is Kramers escape over it, with relative risk RR = rate(dose)/rate(0). The barrier ratio ΔV(γ,h)/ΔV0 as a function of h/hsp is identical across γ (max spread 0.0) — the dose-response shape is universal [F], so γ sets only the absolute barrier scale.
The cancer kernel is the same R19 switch
A cell fate is the R19 double well; a carcinogen is a sustained drive that lowers the barrier out of the healthy (metastable) well, and malignant transformation is the thermally-activated escape over it. The exact escape barrier is the saddle-minus-metastable energy of the tilted well.
![\mathrm{RR}(\mathrm{dose}) = \frac{\exp[-\Delta V(h)/D]}{\exp[-\Delta V(0)/D]}](../eq/rep-onc-002.svg)
The dose-response shape is universal in γ
The dimensionless barrier ratio as a function of drive-over-spinodal is identical for every tissue γ — the master gene sets the absolute barrier height but never the shape of the hormone dose-response. This is a forced result with no free parameters.
| h / hsp | ΔV / ΔV0 | max spread across γ |
|---|---|---|
| 0.00 | 1.00000 | 0.0 |
| 0.20 | 0.71014 | 0.0 |
| 0.40 | 0.45804 | 0.0 |
| 0.60 | 0.24771 | 0.0 |
| 0.80 | 0.08705 | 0.0 |
| 0.95 | 0.01083 | 0.0 |
Three sites, three falsifiable shapes
Breast — cumulative oestrogen, monotone
Cumulative oestrogen exposure is sustained drive, so relative risk rises monotonically with exposure years until the barrier floor is reached. The WHI combined-MHT figure (RR ≈ 1.26) is the measured anchor for the exposure-to-drive scale; risk rises from 1.0 to about 5.2 across the modelled range.
| oestrogen-exposure years | h / hsp | relative risk |
|---|---|---|
| 0 | 0.000 | 1.000 |
| 5 | 0.312 | 2.067 |
| 10 | 0.624 | 3.633 |
| 15 | 0.935 | 5.156 |
| 20 | 0.999 | 5.294 |
| 25 | 0.999 | 5.294 |
| 30 | 0.999 | 5.294 |
Prostate — androgen, concave and saturating
Androgen drive lowers the barrier with diminishing returns, and once the drive reaches the spinodal the barrier floor is zero, so risk plateaus. Past the fold (drive ≥ 1.0× hsp) the relative risk holds flat at 5.3 — the saturation model (more testosterone is not more risk above threshold).
| androgen drive (× hsp) | barrier reduction | relative risk |
|---|---|---|
| 0.0 | 0.000 | 1.000 |
| 0.2 | 0.290 | 1.621 |
| 0.4 | 0.542 | 2.468 |
| 0.6 | 0.752 | 3.504 |
| 0.8 | 0.913 | 4.580 |
| 1.0 | 1.000 | 5.295 |
| 1.3 | 1.000 | 5.295 |
| 1.6 | 1.000 | 5.295 |
Cervical — HPV×smoking, saturating synergy
HPV lowers the barrier height (softens the well) while smoking adds mutagenic drive. Their synergy index is near 1.0 (multiplicative) at low joint exposure and falls to 0.602 (sub-multiplicative) at high exposure, because the barrier floor is zero — a falsifiable prediction that synergies saturate.
| joint exposure | RR HPV | RR smoking | RR both | synergy index |
|---|---|---|---|---|
| 0.05 | 1.224 | 1.135 | 1.376 | 0.990 |
| 0.10 | 1.473 | 1.283 | 1.819 | 0.963 |
| 0.20 | 2.030 | 1.621 | 2.853 | 0.867 |
| 0.30 | 2.636 | 2.017 | 3.929 | 0.739 |
| 0.40 | 3.248 | 2.468 | 4.825 | 0.602 |
Grades (C3): the shape and universality are forced/sim-verified [F]/[V]; the epidemiological anchors are measured [L]; absolute population incidence is open [O]. The single lattice noise D=0.15 sets absolute rates only — every RR ratio, slope sign, plateau and synergy statement is invariant to it. This is not clinical advice.