Selected quotations and author highlights:

[Page 2, Section 1, Introduction] wrote:"nature makes dynamically the most effective choice for the radix in which its computations take place."

[Page 2, Section 1, Introduction] wrote:"By demanding that this radix works most efficiently physical laws are derived in a unified way."

[Page 5, Section 2, Radix economy and the Lagrangian action] wrote:"matrix representation characterizes each number in each radix in a unique way."

[Page 8] wrote:"least (classical) action paths as being those for which the radix[integer part of dimensionless Lagrangian actionS/h]is most efficient,i.e.those paths for which the radix [integer part of S/h] has the least economy."

[Page 9, Section 3, The principle of least radix economy] wrote:"there is, however, a fundamental physical radixthe one which most efficiently works at each scale. Furthermore,the fundamental radix establishes the form that physical laws do have at each scale. We propose that nature has her own dynamical means to specify the most efficient radix."

[Page 21, Section 5, Breaking of spacetime commutativity and finite systems] wrote:"In atomic models, the physical radix η coincides with the so-calledprincipal quantum number n."

[Page 21, Section 5, Breaking of spacetime commutativity and finite systems] wrote:"(i.e. the most significant digit of the fractional part of the action {S/h}) corresponds to theazimuthal quantum number ℓdescribing the orbitals (electronic subshells)."

[Page 24, Section 6, Statistics of action] wrote:"the optimal radix η corresponds to observing chains whose most probable outcome in η ’experiments’ are η quanta of action."

[Page 26] wrote:"Because of the above interpretation of the radix economy C as an entropy-like quantity, this relationship relates the action η to the “entropy C(1, η) of a single particle” (described by the quantum of action in the unary radix) and coincides with the connection between action and entropy made by de Broglie in his book “Thermodynamics of the isolated particle”"

[Page 27, Section 7, The unary radix: de Broglie relationships, special relativity, relativistic wave equations and spin from the quantum of action.] wrote:"η is the number of quanta of action"

[Page 28] wrote:"motion can be expressed as a chain containing a combination of n zeroes and n ones (from the discussion in Section 6). We must think in those chains as containing the information that codifies the state of the system in spacetime. Each digit is thus related to a “quantum of information”, which can correspond to a quantum of action (’1’ state) or to a vacuum state (’0’ state)."

[Page 29] wrote:"Special relativity and relativistic quantum mechanics are consequences of the principle of least radix economy and the physical implications of the unary radix."

[Page 36, Section 8, Derivation of the Boltzmann’s principle] wrote:"there seem to be more available microstates by the fact that a path with η nonprime can subdivide into subunits"

given by the divisors of η.

[Page 38, Section 10, Conclusion] wrote:"A new derivation of Lorentz time dilation and Einstein’s special relativity has been accomplished from statistical arguments involving chains of zeroes and ones (instead of the traditional geometric-kinematic approach) that arise from realizing that the physical radix coincides with the mode of certain binomial distributions (whose form has been established)."

[Page 39] wrote:"A physical number is not only just a number accompanied by physical units. The number is also given in a certain radix (a fact that has previously been overlooked even when physicists always like to speak about “orders of magnitude”) and this radix is physically important, as we have shown, even when physical laws at a certain scale (i.e. when the dynamical variables do not change in many orders of magnitude) are not affected by how actual numbers are indeed represented."

**References:**- Garcia-Morales, V. Quantum Mechanics and the Principle of Least Radix Economy.
*Found Phys*45, 295–332 (2015). https://doi.org/10.1007/s10701-015-9865-x

https://link.springer.com/article/10.1007/s10701-015-9865-x - https://arxiv.org/pdf/1401.0963v4

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