Yiannopoulos, Alexander (2025) Primitive Spectra: Order-Completion and the Arithmetic Structure of Scale. [Preprint]
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Abstract
We propose a structuralist reconstruction of the real continuum, arguing that it is not a primitive ontological stage for physics but an emergent completion of the prime-exponent lattice M. By identifying the reals R as the Dedekind completion of M, we demonstrate that the natural logarithm and exponential functions are not merely analytic tools but inevitable algebraic consequences of identifying the multiplicative structure of number theory with the additive structure of the continuum. Furthermore, we show that the natural base e is the unique structural invariant required to normalize the gauge freedom inherent in this identification. Finally, we refine the Erdős–Kac theorem to argue that the "Triple-Log" scale (ln ln ln n) emerges as a deterministic background limit of arithmetic, providing an intrinsic argument for a natural ultraviolet cutoff in physical theories without external imposition. In the language of Ontic Structural Realism (OSR), this framework suggests that the 'physical continuum' is not an object-like substance, but a structural instantiation of the relation between the additive and multiplicative sectors of arithmetic.
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