The idea about fundamental unit of time - usually assumed to be given by Planck time about 10^{-44} seconds - is rather naive if taken to mean discretization of time. The article proposes a variant of this idea and identifies the fundamental chronon as a fundamental periodicity of dynamics and deduces for it a value about 10^{-33} seconds from observed bounds to the variation of dynamical periods. This value is by 11 orders of magnitude longer than Planck time. Personally I am a little bit skeptic about reliability of these bounds since very short times are involved.

Planck time is deduced from a mere dimensional analysis argument by feeding in speed of light c, Planck constant h, and Newton's constant G so that it is rather ad hoc noton. Therefore it has been surprising to me at least how seriously people have taken it. Moreover, Planck length appears in theories - in superstring theory in particular - typically as an ad hoc formal parameter with no direct geometric interpretation. In general relativity this leads to non-renormalizability and it is not possible to quantize gravitation in this framework. In superstring theories it led to landscape catastrophe: even a smallest change in the physics at Planck length scales changes completely the physics at long length scales: butterfly effect in the theory space.

For these reasons the question whether there exists some fundamental length-/time unit or several of them is a key problem of recent day physics. Could there be some fundamental length scale or possibly several of them with a clear geometric interpretation? In TGD Universe this is indeed the case.

- Planck length is derived quantity and CP
_{2}length scale defines the fundamental length, which from p-adic mass calculation for electron mass roughly 10^{4}times longer than Planck length. Space-time is continuuous but CP_{2}length serves as a fundamental unit of length, kind of length stick. - p-Adic length scale hypothesis (PLH) predicts actually infinite hierarchy of length/time units as p-adic length scale hypothesis stating that these units are proportional to sqrt(p), p preferred p-adic prime. p-Adic length scale hypothesis in this general form emerges both from M
^{8}-H duality and p-adic mass calculations. - A stronger form of PLH states that certain primes near powers of 2 are physically favored so that in the most general case one obtains a hierarchy of length and time units coming as half octaves. This form of hypothesis not well-understood although it conforms with period doubling in chaotic systems. Also powers of other small primes are possible and there is some evidence for the powers of 3. This would relate the preferred length scales of physics in long scales to CP
_{2}scale. - TGD predicts second length scale hierarchy corresponding to the hierarchy of effective values h
_{e}ff=nh_{0}of Planck constant (h=6h_{0}) labelling phases of ordinary matter behaving like dark matter. n corresponds number theoretically to the dimension for extension of rationals. This makes possible a hierarchy of quantum coherence length coming as n/6-multiples of the ordinary Compton length. Quantum coherence in long length scales is the most important implication and the coherence of living matter would be due to quantum coherence at magnetic body - distinguishing between TGD and Maxwellian and QFT view about classical fields. There is considerable evidence for the existence of h_{eff}hierarchy from various anomalies, in particular from those in living matter. - Also now the challenge is testing of this hypothesis in macroscopic length scales: we cannot directly access short scales. The idea is simple: measure ratios of p-adic mass scales. They do not depend on CP
_{2}scale nor on the value of n. The ratios of dark quantum scales - say dark Compton lengths - are typically given by the ratio n_{1}/n_{2}of integers involved.This allows precise tests by measuring mass and length scale ratios rather than masses and length scales. For instance, the possibility of scaled variants of hadron physics and electroweak physics allow to test the hypothesis. There are indeed indications for scaled up variants of mesons with mass scale differing by a factor 512 from that for ordinary hadrons. The Compton lengths would be same as for ordinary hadrons for n/6=512: the dark Compton scale for p-adically scaled up meson be same as ordinary Compton length making possible resonant coupling. If the valence electron of atom is dark, its Bohr radius is scaled up by (n/6)

^{2}: these states might be misinterpreted as Rydberg states.

- Physics as number theory vision predicts that hierarchy of extensions of rationals defines evolutionary hierarchy. A generalization of real numbers to adeles labelled by extensions of rationals is assumed. For given extension of rationals adeles form a book like structure having as pages real numbers and extensions of p-adic number fields induced by extension of rationals. The pages are glued together along the back of the book consisting of points in given extension of rationals common to reals and extensions of all p-adic number fields. This hierarchy corresponds to evolutionary hierarchy and the dimension n of extension has identification as effective Planck constant h
_{eff}. - Space-time itself becomes a book-like structure. Real space-time surfaces are replaced with adelic surfaces, which containing real sheet and p-adic sheets glued together along the back of a book consisting of points with imbedding space coordinates in given extension of rationals. The points of space-time surface with coordinates in given extension of rationals form a discrete cognitive representaton, which is unique and improves with the dimension of extension so that at the limit of algebraic numbers it is dense set of space-time surface.
- I call this discretization identifiable as intersection of sensory world (reals) and cognitive worlds as cognitive representation. The discretizaton reflects the limitations of cognition which must always discretize. In M
^{8}picture space-time surface are "roots" of octonionic polynomials and the polynomial defines the extension of rationals via its roots. At M^{8}level there are also essentially unique imbedding space coordinates making discretization unique.

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https://math.stackexchange.com/questions/3286548/can-we-remove-any-prime-number-with-this-strange-process

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