Title

Paper 05a — Finite Closure of Primes via an Executable Quadratic Basis
Execution Layer of the FRC Programme (Following Paper 05)

Abstract

This work presents the executable realisation of finite closure within the Finite Reversible Closure (FRC) Programme, extending the structural framework established in Paper 05.

Paper 05 defines composite excitations as arising from holonomy and occupation patterns and introduces a finite admissible quadratic basis governing closure channels within the integer lattice. However, that basis achieves only near-complete coverage of primes within finite domains, leaving a finite residual set.

In this work (Paper 05a), the residual is analysed as a structured boundary condition rather than randomness. Through a finite extension of admissible quadratic forms, the closure basis is expanded from 27 to 49 forms.

The resulting finite basis is evaluated computationally and verified against an independent sieve to achieve complete in-range coverage of all primes up to $100{,}000{,}000$, with zero residual across both geometric and resistant channels.

This demonstrates that, within the tested domain, prime support arises from finite admissible closure and does not require unconstrained or infinite generative rules.

All code and outputs are provided to ensure full reproducibility and transparency.

Introduction

The Finite Reversible Closure (FRC) Programme establishes a reality-constrained framework in which all realised structure must arise through finite, executable processes. Primitive 0 enforces that no operational infinity may exist: structure must complete through admissible closure.

Within this framework, Paper 05 defines composite excitations as arising from holonomy and occupation patterns, and introduces a finite quadratic basis that governs closure channels within the integer lattice. These channels partition structure into admissible modes, notably geometric and resistant classes.

While this basis captures the majority of prime support within finite domains, it does not achieve complete closure. A finite residual set remains. The central question is whether this residual reflects intrinsic randomness requiring unbounded extension, or whether it is itself a structured and finitely resolvable feature of the lattice.

Paper 05a addresses this directly by treating the residual as a boundary condition of incomplete closure. Through a constrained search over admissible quadratic forms, the residual is absorbed into an extended finite basis.

This document therefore represents the execution layer of closure theory: transforming structural description into verified finite completion.

Appendices A–E define the basis, characterise the residual, resolve the extension, execute the computation, and verify full closure within the tested domain.