Tumors are not monoliths. Majeed, Ghosh, and Mundhe (arXiv 2604.01385) modeled tumor populations as mixtures of drug-sensitive and drug-resistant subpopulations competing for resources under chemotherapy. The key result is not that resistance exists — it always does — but that mathematical thresholds determine outcome. Below a critical ratio of resistant to sensitive cells, chemotherapy eliminates the tumor entirely. Above it, the resistant subpopulation expands to fill the niche vacated by the sensitive cells. Between the two thresholds, a coexistence regime where both phenotypes persist and the tumor stabilizes at reduced but nonzero mass. The thresholds depend on growth rates, carrying capacity, and drug kill rate — all quantifiable. The model doesn’t predict whether resistance will evolve. It predicts where the boundary falls between elimination, coexistence, and resistant takeover.

Neural networks have an analogous threshold structure. Liu et al. (arXiv 2604.01334) proved that the loss landscape of a two-layer ReLU network undergoes bifurcation as the nonlinearity perturbation parameter λ increases. Below a critical value λ — which scales as λ ∝ α^m with network width α — the landscape is essentially convex: one minimum, no spurious attractors. Above λ*, the landscape fractures into multiple basins. The linearized regime dominates at large width, pushing the bifurcation threshold upward. The proof was verified in Lean 4, making it one of the first formally verified results in deep learning theory.

In the tumor, resistance exists at all drug concentrations, but it only dominates above a threshold. In the network, nonlinearity exists at all perturbation strengths, but it only fractures the landscape above a threshold. The resistance is always present. The question that determines the outcome is whether the system operates above or below the critical value.

The presence of resistance is universal — its relevance is determined by a threshold.