Mathematics of a Finite Universe. Version 2.0

The description for Version 2.0 should make one thing clear from the beginning: **this is not just a revised text, but a conceptual correction of the original MFU formulation.** A fish in an aquarium does not know the world beyond the glass. Its mathematics may describe structures far beyond what it can directly reach, but the fish itself remains limited by the physical resources available inside the tank. Version 2.0 of *Mathematics of a Finite Universe* (MFU) develops this metaphor further and substantially revises the original framework. The first version introduced fixed ontological limits: a minimal distinguishable scale **ε_ont** and a maximal informational bound **Ω_ont**. Version 2.0 abandons these as universal numerical boundaries. Instead, it introduces a more careful distinction between formal mathematics and the operational accessibility of mathematical structures to a physical observer. The notion of **ontological zero** is reformulated as an operational status. A difference may remain formally nonzero while becoming physically indistinguishable under a given procedure and finite resources. The relevant threshold **ε_X** may depend on the quantity under study, the measurement or computational procedure, available information and energy, and the required resolution. Likewise, **ontological infinity** is no longer identified with numbers exceeding some fixed **Ω_ont**. It now describes a structure whose complete operational distinction would require more information than the observer can physically access: > **I(X) > I_max** The Riemann zeta function remains the central example. Version 2.0 introduces a working **information-cost function I_ζ(T)**, representing the resources required to distinguish and individually encode all nontrivial zeros up to height **T**. MFU does not present the growth of this function as a theorem of zeta-function theory. It is explicitly introduced as an operational postulate: > **lim (T → ∞) I_ζ(T) = ∞** which, together with finite **I_max**, leads to an operational horizon **T_*(I_max)**. Another major revision concerns the **Riemann Hypothesis** itself. Version 2.0 no longer suggests that physical limits imply undecidability or unprovability. It clearly separates mathematical proof from exhaustive enumerative verification. A finite proof may in principle establish a property of an infinite structure without checking every zero individually, while complete element-by-element verification remains inaccessible to an observer with finite resources. The notion of **trans-empiricality** is also refined. It is now treated as a relational status depending on the structure, the verification procedure, and the finite resources of the observer, rather than as an intrinsic limitation of mathematics. Version 2.0 therefore shifts the emphasis of MFU. It no longer attempts to replace mathematical infinity with fixed physical cutoffs. Instead, it asks where the boundary lies between > **formal mathematical structure** and > **its operational accessibility to a physical observer.** The aquarium metaphor remains, but the meaning of the glass has changed. It is no longer a numerical wall placed inside mathematics itself. It is the informational horizon of an observer who may reason beyond that boundary symbolically, while never being able to carry an infinite amount of distinguishable information across it. Follow my work and related discussions on Facebook: [Facebook]