Abstract
<jats:p>Clonogenic survival under fractionated irradiation flattens after one to two weeks at a level that depends on fraction size. The linear–quadratic (LQ) model cannot reproduce this, because its exponent is a linear functional of delivered dose and therefore carries no memory of how the dose was given. We construct the smallest kill law that can, from six axioms: two clonogenic states, first-order kill in each, a constant ratio between their kill rates, a causal damage-driven transfer between them, conservation of total transfer, and one-way transfer over the interval considered. We prove the axioms independent, and prove that they determine the structural results without specifying either the hazard or the transfer kernel. These give a five-variable initial-value problem — a lethal hazard carrying a Lea–Catcheside repair memory, a two-stage cascade converting accumulated damage into transfer, and the two clonogenic compartments — which we solve in closed form for arbitrary dose rate. Two exact properties carry the results. The transfer term cancels identically when the two compartments are equally radiosensitive, so survival then depends only on the repair-weighted cumulative lethality; delivery order therefore changes cell kill if and only if a tolerance gap exists, provided repair is complete between deliveries. And once transfer is complete the per-fraction log-kill falls from L to rL, so late dose is devalued by r, tumour volume is amplified by 1/r, and cytoreduction is worth 1/r times its conventional value. For instantaneous fractions separated by intervals long compared with the repair and signal time constants, the system reduces exactly to a two-by-two linear map, verified to machine precision. Fitted to clonogenic survival at five fraction sizes the model returns a gate time constant of 1.75 fractions and a tolerance ratio of 0.063, with a held-out quantity implying 1.84 and excluding zero delay; we also report that it generalises slightly worse than a hand-added floor when whole fraction sizes are withheld. Away from that limit the formulation yields results a fraction-based law cannot express. Continuous low-dose-rate irradiation is penalised twice, by repair and by allowing the transfer time to act during delivery, and a seven-day implant delivering 60 Gy is predicted to be equivalent to thirty 2 Gy fractions. For radionuclide therapy, cell kill is predicted to rise by more than four log-kill as the effective half-life falls from thirty days to six hours at matched absorbed dose, so absorbed dose alone is an insufficient basis for comparing agents. The heterogeneity of uptake in radionuclide therapy also supplies the one configuration in which the tolerance ratio becomes measurable in vivo, because the dose-rate contrast between regions is known from the dosimetric imaging itself. The construction is calibrated on one cell line and the tolerance ratio has never been measured on a clonogenic endpoint; the claims that survive this are structural.</jats:p>