IndisputableMonolith.NumberTheory.Port
The NumberTheory.Port module aggregates analytic number theory results for the Recognition Science framework. It imports Brun-Titchmarsh, Prime Number Theorem, and Riemann Hypothesis ports to supply explicit bounds on prime sums. Researchers handling attachment-with-margin error budgets cite this module for convergence and tail estimates. The module structures dependencies without adding new proofs.
claimPort of Brun-Titchmarsh inequality, Prime Number Theorem asymptotics, and Riemann Hypothesis statements for prime tail bounds: convergence of $∑_{p} p^{-α}$ for $α>1$ and explicit upper bounds on $∑_{p>N} p^{-α}$ via partial summation.
background
This module sits in the NumberTheory domain and serves as the aggregator for ported results needed by Recognition Science derivations. It imports BrunTitchmarsh, PNT, and RiemannHypothesis submodules to expose tools for handling prime distributions. The local setting supports error budget calculations, as seen in the downstream use for attachment margins.
proof idea
This is a definition module with no proofs. It consists of imports to the BrunTitchmarsh, PNT, and RiemannHypothesis submodules that contain the actual ported statements and arguments.
why it matters in Recognition Science
This module feeds the PrimeTailBounds file, which discharges the PrimeTailBound predicate for the Christmas-route error budget decomposition. It supplies the number theory infrastructure required for convergence and tail bounds on prime sums, connecting directly to the Recognition Science error budget machinery.
scope and limits
- Does not prove the Riemann Hypothesis or any of its equivalents.
- Does not perform numerical evaluations of prime sums or constants.
- Does not extend the results beyond the imported port statements.
- Does not address applications outside the error budget context.