The condition that a cubic achieves a minimum must be compatible. But $ t^3 $ has no minimum. Therefore, the only possibility is that the model is still $ d(t) = t^3 $, and the phrase minimum depth refers to the **lowest recorded value in a physical window**, but mathematically, over $ \mathbb{R} $, $ t^3 $ has no minimum.

["Understanding Why $ t^3 $ Cannot Achieve a Global Minimum: A Mathematical Perspective", "When analyzing functions and their minima, a common principle emerges: a function must attain a minimum only if it is bounded below and achieves that bound over its entire domain. However, in the case of the simple cubic function $ f(t) = t^3 $, this condition appears paradoxical—because while the function is defined for all real numbers ($ t \in \mathbb{R} $), it does not possess a global minimum. This discrepancy invites deeper exploration into mathematical behavior, domain implications, and the philosophy of minima.", "### $ t^3 $ Has No Global Minimum Over $ \mathbb{R} $", "Mathematically, the function $ f(t) = t^3 $ behaves monotonically increasing across the entire real line. As $ t \ o -\infty $, $ f(t) \ o -\infty $, and as $ t \ o +\infty $, $ f(t) \ o +\infty $. Crucially, for any proposed candidate for a global minimum—say, $ -100 $—there always exists some $ t < 0 $ such that $ (-100)^3 = -1,000,000 $, far below $ -100 $. Thus, no finite value minimizes $ t^3 $ everywhere.", "This reveals a fundamental distinction: a function has a global minimum on $ \mathbb{R} $ only if it is bounded below and achieves its minimum at some point. The cubic function fails this criterion, despite being smooth and continuous.", "### The Condition for a Minimum: Boundedness and Attainment", "In optimization and calculus, a necessary condition for a function to attain a minimum over a domain is boundedness below—but this is not sufficient. To guarantee the existence of a global minimum, the function must also attain that bound within its domain. Since $ \mathbb{R} $ is unbounded, only special cases suffice—like functions that asymptotically settle or repeat. The cubic function lacks such behavior.", "### The Interpretation of “Minimum Depth” in Physical Contexts", "The idea of “minimum depth” sometimes arises in applied modeling—say, in sculpture, engineering, or geophysics. When phrased mathematically, “lowest recorded value in a physical window” suggests that, although $ t^3 $ dips infinitely low, real-world constraints or measurement contexts define a practical minimum within observed bounds. Here, the phrase “minimum depth” is not a mathematical minimum over $ \mathbb{R} $, but rather the lowest observed or meaningful value within a restricted domain or physical system.", "This subtle shift—from abstract entire-space minimization to constrained physical observation—opens a more nuanced interpretation: the cubic’s unboundedness becomes meaningful only when bounded by non-mathematical factors like spatial limits or detection thresholds. In such windows, effective minima emerge from domain restrictions or context, not pure calculus.", "### Conclusion: Compatibility of Conditions and Domain Matters", "The apparent contradiction dissolves once we recognize that a function’s capacity to achieve a minimum depends critically on the mathematical domain and the physical interpretation. $ t^3 $ fails to reach a global minimum over $ \mathbb{R} $ because it has no upper or lower bound and monotonically spans all real outputs. Yet, in a physical window constrained by real-world limits, “minimum depth” borrows $ t^3 $’s behavior metaphorically—denoting a lowest measurable or meaningful value, rather than a scalar minimum.", "Thus, compatibility hinges on domain definition: the cubic model’s absence of a global minimum over $ \mathbb{R} $ doesn’t invalidate the concept of minima in bounded contexts—only their interpretation shifts across mathematical rigor and practical application.", "---", "TL;DR:\nThe cubic $ t^3 $ has no global minimum over all real numbers because it decreases without bound. However, “minimum depth” in physical contexts refers not to a mathematical minimum across $ \mathbb{R} $, but to the lowest observed value within a constrained domain—showing how precise definitions unify rigorous analysis with real-world interpretation."]