The classical formulation of the Riemann Hypothesis quietly assumes something very strong: infinite distinguishability. It asks whether a precise analytic inequality holds in the limit where structure can be resolved at arbitrarily fine scales, with no lower bound on resolution. That assumption is not required to state the Prime Number Theorem, not required to test any finite analytic observable, and not required for any physically or computationally realizable process.
What our work shows is simple but important. At every finite resolution—every bounded frequency window, every finite-energy probe, every computable test—the Riemann–Hypothesis positivity inequality holds. There is no breakdown at any finite scale. This result is unconditional and uses only Prime Number Theorem–level arithmetic together with standard functional analysis. The difficulty appears only when one demands that this behavior persist in the singular limit where resolution becomes infinite.
In the framework developed here, the parameter Δ plays the role of a minimal distinguishability unit. You can think of Δ as functioning like a bit—not representing time or dynamics, but resolution itself. Below this scale, differences are no longer meaningfully resolvable. This aligns naturally with the VERSF view of reality as a system with a finite distinguishable state space, and with the Taylor Limit, which places an upper bound on meaningful resolution just as the Planck scale places a lower one. Once Δ is fixed to be nonzero—even arbitrarily small—the arithmetic sampling problem becomes well-posed, bounded, and provably positive.
Seen this way, the Riemann Hypothesis is not really a question about primes “failing” to behave at large heights. It is a question about what happens when one insists on extending distinguishability beyond all physical, computational, and informational limits. The Δ → 0 limit is not just difficult; it corresponds to an ill-posed observable. The classical RH challenge is therefore a question about an idealized infinite-resolution analytic continuation, not about prime distribution at any finite level of reality.
From a computational-universe perspective, this matters. In any system with a finite state space—any universe that can be computed, simulated, or observed—positivity holds. The remaining gap is not arithmetic, but conceptual: whether mathematics should demand truth in a limit that no finite process can ever access.
A useful way to understand the role of Δ in this work is to view the universe as effectively computable at finite resolution. In such a universe, states are not defined by infinitely precise real numbers, but by finite, distinguishable configurations. Computation proceeds by manipulating these configurations step by step, and any meaningful operation must act on a finite amount of information. Infinite precision is not a neutral idealization in this setting; it represents a breakdown of computability.
Within this perspective, Δ plays the role of a resolution unit: the minimum distinguishable scale in logarithmic frequency space. When Δ is positive, the system operates within a finite state space. When Δ is taken to zero, one is no longer refining a computation, but attempting to access an unbounded, non-computable limit.
The main mathematical result of the paper fits naturally into this picture. For every fixed Δ > 0—that is, for every finite resolution or finite number of bits—the explicit-formula quadratic form arising from the Weil trace formula is rigorously nonnegative after removal of a finite-dimensional baseline subspace. At every finite level of distinguishability, the required positivity holds.
The classical Riemann Hypothesis corresponds precisely to the limit Δ → 0, where resolution becomes infinite and the state space ceases to be finite. The analysis shows that this limit is not benign. As Δ shrinks, the prime-sampling functional transitions from a bounded observable to an unbounded distribution. In computational terms, this is the difference between an operation that can be evaluated within a finite machine and one that requires infinite memory or infinite precision. The positivity problem does not simply extend smoothly to infinity; it changes character.
This observation aligns closely with the Taylor Limit, which asserts that there is an upper bound on meaningful distinguishability. Beyond that bound, distinctions no longer correspond to stable or computable states. The failure to extend finite-resolution positivity to infinite resolution is therefore not an accident or a missing inequality. It reflects a deeper constraint: the infinite-resolution object is not part of the same computational state space as its finite-resolution counterparts.
Seen this way, the result is a conditional proof, but the condition is not an arbitrary mathematical assumption. The condition is the assumption that the universe—and mathematics as applied within it—admits infinite distinguishability. If one accepts that assumption, the remaining step is exactly the classical Riemann Hypothesis. If one does not, then the hypothesis describes a property of an idealized object that lies beyond the computational universe.
This reframes the long-standing difficulty of the Riemann Hypothesis in a new light. Positivity holds everywhere it is computable. The obstruction appears only when one attempts to leave the space of finite, distinguishable states. In that sense, RH is not merely a statement about primes; it is a statement about whether infinite-resolution mathematics is meaningfully continuous with the finite world in which all computation—and all verification—takes place.