PRH | Essay | 7.3 • On Redundancy of Properties in Mathematical Objects

On Redundancy of Properties in Mathematical Objects Aleksandar Periˇsi´c September 2025 Abstract We propose a general principle about the role of apparently “redundant” properties in the definition of mathematical objects. If a property is unnecessary for the construction of the object but nonetheless impacts its behavior when modified, then the property is either fundamentally simple, or else our axiomatic understanding of the object is incomplete. We apply this principle to the role of the line ℜs = 1 2 in the Riemann Hypothesis, and connect it to entropy considerations that measure how much non-random information primes actually contribute. 1 Introduction In mathematical investigations, one often encounters a property that is present but not required in the axiomatic description of an object. Such a property may at first appear extraneous, yet its modification produces nontrivial consequences for the behavior of the object. This raises a natural question: what is the epistemic status of such a property? Is it structurally deep, or is it deceptively simple? The following lemma is intended as a guiding principle in such contexts. 2 A Guiding Lemma Lemma 1 (Redundancy Principle).Let O be a mathematical object described by a complete or assumed-complete system of axioms A. Suppose there exists a property Psuch that: 1. Pis not required in Afor the definition of O. 2. Changing Pyields a different behavior of O. Then exactly one of the following holds: 1. Pis fundamentally simple, requiring no further structural explanation. 2. Our axiomatic description A is incomplete: there exists additional hidden structure in which Pplays an essential role. 3. We lack the ability to recognize the simplicity of P , treating it as complex despite its basic nature. Remark 1. This principle distinguishes between true redundancy, where a property reflects elementary structure, and apparent redundancy, where the missing information belongs to a deeper axiomatic layer not yet formalized. 1 3 Application: The Line ℜs=1 2 In the investigation of the Riemann Hypothesis, the symmetry axis ℜs = 1 2 emerges naturally from the functional equation of ζ ( s ). From the perspective of the integers and their multiplicative structure, this line is not explicitly required: the integers can be defined without reference to it. Yet the alignment of nontrivial zeros along this line is the central conjecture of analytic number theory. Proposition 1 (Simplicity of the Critical Line under RH).If the Riemann Hypothesis is true, then the condition ℜ ( s ) = 1 2 is simple: it amounts to an arbitrary vertical normalization of ζ ( s ), since replacing ζ ( s )by ζ ( s + k )for any fixed real k would merely shift the critical line without altering the underlying structure of the integers or the Euler product. Thus in this interpretation the line ℜ(s) = 1 2holds no additional information. 4 Relation to the Hopf–Euler–Dirichlet Framework Recent work on Euler–Dirichlet systems via Hopf algebras has shown how multiplicative and additive data can be coupled into a variational principle, with the explicit formula emerging as a stationarity condition. Within this framework, “redundant” parameters such as the location of the critical line can be interpreted as part of the geometry of the underlying factorization Hopf algebra. The lemma above therefore plays the role of a meta-principle: it explains why the critical line can appear simultaneously nonessential to the construction of the integers, yet decisive in the spectral balance between primes and zeros. In Hopf terms, 1 2 is either a structural anchor (simple), an echo of hidden axioms (incomplete), or a trivial alignment masked by analytic difficulty (obscured simplicity). 5 A Categorical Ambition We formalize the Redundancy Principle as a lemma, because proving it at the categorical level would amount to an extraordinarily powerful shortcut: it would convert many seemingly intractable problems into transparent consequences of structural completeness. In particular, for the Riemann Hypothesis, such a categorical proof would compress two centuries of analysis into the framework of a single, long paper. In the absence of such a lemma, we remain confined to brute-force analytic attacks. 6 Entropy vs. Random Baseline: How Much Non-Random Structure Exists? The redundancy theme also appears if we ask how much new information primes actually deliver compared to a random baseline. Consider divisibility tests Xp =1 {p|n} for n≤N . Their entropies sum to about X p≤N H(Xp) = ln N+ ln ln N+O(1), whereas the entropy of n is just H ( n ) = ln N . The ln ln N surplus is redundant correlation, not extra knowledge of n . Thus only ln ln N nats ( log2ln N bits) of genuinely new structure accumulate. Data-hunger law. To gain k more bits of such non-random information, one must extend the range to Nof size N7→ exp 2k·ln N, 2 a double-exponential growth. This explains why deeper prime regularities are so data-hungry. Interpretation. Small primes provide noticeable fractions of a bit, but by p∼ 10 6 the contribution is only 10 −5 bits. Even the discovery of record Mersenne primes translates to just a few dozen bits of surplus information. The redundancy principle explains why: most of the structure is already implicit, and the flow of new non-random data is throttled by the double-log law. 7 Synthesis If we combine both perspectives: 1. Imaginary parts of zeta zeros encode all fine structure of primes. 2. Real parts sit at 1 2by all measures we can test. By the Redundancy Principle, either it is simple and carries no further informational load, or it signals missing axioms yet to be discovered. But if so, we must ask: what scale of axiom could account for such a subtle yet universal constraint? If it is written anywhere, it must be encoded in the statistics of the primes themselves-yet all we seem to uncover are vanishingly rare, one-in-billions statistical fluctuations. What kind of guiding principle would accept such an outlook? Or, put the other way around: if such a principle carries so little information, then what does ”guiding” really mean? And yet, in stark contrast, we know that even a single zero off the critical line would shatter the causal principle: in Euler’s sieve, some composites would begin to appear flagged before any of their factors. How can it be that such a grand violation-expressible in just a few words-carries such an incredibly small amount of measurable information? 8 Conclusion The Redundancy Principle provides a lens for interpreting the epistemic status of properties like ℜs = 1 2 . Viewed alongside entropy analysis, it suggests that primes leave very little room for additional readable information. Whether simplicity, incompleteness, or obscured triviality, the principle clarifies what kinds of explanations are possible and what further work is required. 3