The fifth editorial, After Tools, Problems Remain, argued that human judgment must not be surrendered to powerful tools. The sixth, Problem as Foundation, brought research back to real objects, contradictions, and boundaries. This editorial takes the next step: once a real problem has been established, what does it mean for research to genuinely move forward? A problem governs the starting point; innovation governs the advance. No problem means empty novelty; no advance means that the problem remains where it began.Placed in the historical arc from Science: the endless frontier (1945) to science: a new golden age (2026), this editorial defines innovation as a substantive increment on a real problem. Such an increment may enlarge knowledge, capability, or the accessible boundary of inquiry: we may know what was previously unknown, do what was previously impossible, or bring a problem previously beyond observation, computation, design, or verification into reach. Zero-to-one opens an entry point, whereas one-to-N develops that entry point into reliable, usable, and transferable knowledge or capability. Pseudo-innovation is not a third mode of advance, but an appearance of novelty without a substantive increment.Innovation may occur in problem formulation, method, mechanism, system capability, or paradigm. Real problems may arise from national missions, engineering practice, and clinical needs, but also from fundamental questions whose use cannot yet be foreseen. Artificial intelligence is expanding the accessible domain of research by entering hypothesis generation, model discovery, experimental design, and validation; at the same time, it lowers the threshold for work to look innovative. A defensible claim must therefore identify the real problem, state the baseline and increment, present evidence commensurate with the claim, and define its boundary. For Applied Mathematics and Mechanics, the final question remains whether a study has produced a real advance on a real problem.