An analogy: Consider the task of factoring every number between 1 and finite n using resources limited by some parameter m. We know in the abstract that factoring is computable, and we have strong reasons to believe that a polynomial-time algorithm exists. Yet none of this tells us whether such an algorithm can be carried out within the bound m. The existence of a procedure in principle does not reveal the resource profile of any procedure in practice. Even if the number system over [1,n] has clean analytic regularities—predictable prime density, well-behaved smoothness properties, low descriptive complexity—these features offer no constructive bridge from “a factoring algorithm must exist” to “a factoring algorithm exists under this specific resource limit.” The arithmetic regularities describe the domain; they do not generate the algorithm or its bounds.
An analogy: Consider the task of factoring every number between 1 and finite n using resources limited by some parameter m. We know in the abstract that factoring is computable, and we have strong reasons to believe that a polynomial-time algorithm exists. Yet none of this tells us whether such an algorithm can be carried out within the bound m. The existence of a procedure in principle does not reveal the resource profile of any procedure in practice. Even if the number system over [1,n] has clean analytic regularities—predictable prime density, well-behaved smoothness properties, low descriptive complexity—these features offer no constructive bridge from “a factoring algorithm must exist” to “a factoring algorithm exists under this specific resource limit.” The arithmetic regularities describe the domain; they do not generate the algorithm or its bounds.