... I don't mean quantum effects limiting hardware fabrication sizing. Such small scales have for some time exhibited issues.

Rather, along the lines of imagining the smallest possible divisions of the universe's most tiny physical ruler: its smallest markings would have to be larger than the Planck length. (Yes?) (For the sake of simplicity, assume the division lines have no width.)

Two interrelated questions:

Question 1: Could the divisions of this physical ruler be exactly ℓP or would they have to be only just slightly larger, or some multiple of ℓP ?

E.g. similar conceptually to the Nyquist Frequency in audio, i.e. the sample size would have to exceed ℓP by some factor before the divisions become accurate.

Question 2a: Does being limited to that smallest physical approximation interval ultimately imply that models of Planck-level activities must inherently have some (subtle?) systemic, Heisenberg limitation affecting their combinatoric results ?

(Question 2b: Is question 2a really just a computer science version of aspects of the measurement problem?)

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    $\begingroup$ What? Single precision can do math on 1e-37 level (which is sub-planckian); double precision can do math down to 1e-307 levels (see this excellent article). Outside of this, I am confused what you are actually asking here (specifically towards Q1). $\endgroup$
    – Kyle Kanos
    Sep 29 '14 at 2:23
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    $\begingroup$ The link explains the reasons why precision is the way it is. It is, as far as I can tell, unrelated to the HUP or Planckian limits (assuming they are limits, which may not actually be true). The reason models cannot be verified is because there often are no analytic solutions for the problem. The few analytic solutions that do exist are test cases for codes, and the differences that arise are generally due to algorithms and the aforementioned precision. $\endgroup$
    – Kyle Kanos
    Sep 29 '14 at 2:32
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    $\begingroup$ What do you mean by "analytic solutions for Planck-level models"? I believe what @KyleKanos meant by analytic solution was actually more like an exact closed form solution, without which we'd have to resort to numerically approximating solutions to differential equations, etc. $\endgroup$
    – CuriousKev
    Sep 29 '14 at 2:52
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    $\begingroup$ What @CuriousKev said is correct, by having some closed-form solutions to problems, we can test our numerical algorithms for accuracy (still going to be limited by precision, though). I'm currently unaware of any Planck-level models, but that's mostly because I'm an astrophysicist so I deal with parsec scales, rather than Planck. $\endgroup$
    – Kyle Kanos
    Sep 29 '14 at 2:54
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    $\begingroup$ Here's one I found from a simple Google search: arxiv.org/abs/1311.5712, so yes there are some exact solutions that exist. $\endgroup$
    – Kyle Kanos
    Sep 29 '14 at 3:19

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