Most people hear “infinite precision” and assume it’s just harmless mathematical cleanliness. This paper argues it isn’t. The moment you allow infinite distinguishability—the idea that reality meaningfully contains endlessly finer differences—you quietly change what counts as a scientific question. You move from statements that can be settled by measurement or computation to statements that only live inside an idealized limit. That matters because almost every deep “infinity problem” in science shows the same pattern: everything behaves well at every finite scale, and the trouble appears only when you demand the limit be taken literally.
The paper’s key move is to make that hidden assumption explicit and give it a name: distinguishability. Science isn’t just “what exists,” it’s what can be reliably told apart using finite resources—finite time, finite energy, finite memory. Your parameter Δ captures this: it represents a minimum resolvable “step” in state space. Below Δ, two values may be different on paper, but they’re operationally the same—no experiment, computer, or observer can ever exploit the difference. That doesn’t weaken science; it strengthens it by drawing a bright line between meaningful structure and formal decoration.
Once you see the world this way, a lot of famous “mysteries” become easier to classify. Infinities in quantum field theory, singularities in gravity, instability in inverse problems, and the weirdness of delta-function probing all share a common feature: they are triggered by pushing a theory beyond its distinguishability regime. Renormalization, for example, works so well precisely because it quietly restores finite resolution—physics at any finite scale is well-defined, and the “infinite” answer is a sign you asked an ill-posed question.
The paper is also important because it connects this to everyday experience of time. Time is not something you perceive directly; you perceive difference: one state distinguishable from the next. If nothing changes in a way that can be told apart, time has no operational content. That’s why your opening example lands: “infinitely slow” motion isn’t an extreme kind of motion—over any finite observation it becomes indistinguishable from rest. In the same way, “infinitely fine” resolution isn’t an extreme kind of precision—it’s the point where precision stops being a physical notion at all.
Finally, this frames a new standard for what counts as a scientifically decidable claim. Your Distinguishability Criterion says: if a claim only becomes true-or-false in an infinite-resolution limit, and if its predicted structure is unstable under any finite coarse-graining, then it may still be mathematically definable—but it’s not something finite science can ever settle. That’s a big deal, because it doesn’t shut down inquiry; it redirects it. It encourages science to focus on resolution-stable truths—the ones that survive when you acknowledge the world is finite, computable, and constrained—rather than treating idealized infinities as if they were physical terrain.