Tumor containment for Norton-Simon models

The paper’s Proposition 4 (containment minimizes resistance) is proved correctly by comparing R−N trajectories: it establishes the pointwise chain ÑidMTD ≤ ÑMTD ≤ Ñalt ≤ ÑCont ≤ ÑnoTreat, and then invokes a robust comparison lemma (Lemma 10) to conclude RCont(t) ≤ Ralt(t) for all t (see the statement of Model 3 and assumptions, the definition of containment/alt, the proposition statement, and the proof with Eqs. (1)–(2) in the paper: ). By contrast, the candidate solution’s “first‑crossing” proof via S(R) reparametrization contains a critical flaw: it incorrectly assumes that containment “never exceeds Ntol” and that SCont(R)=Ntol−R for all R≥Rc, which fails after the stabilization phase ends (post‑failure N may exceed Ntol under containment, and SCont is then no longer Ntol−R). This invalidates the key step used to force Salt(r) ≤ SCont(r) at the alleged first crossing. The conclusion matches the paper, but the argument as written is not valid without substantial repair.