Paper 05b - Primes as Structural Masonry: Executable Closure from Primitive Zero to Certified FRC Hydrogen-Seed.

Paper 05b - Primes as Structural Masonry: Executable Closure from Primitive Zero to Certified FRC Hydrogen-Seed Abstract This work presents Paper 05b of the Finite Reversible Closure (FRC) Programme, extending the executable prime-closure layer established in Paper 05a into the first certified finite matter-seed ladder. Paper 05a demonstrated that prime support within finite tested domains can be treated through executable finite closure rather than unconstrained arithmetic randomness. The present paper asks the next structural question: what does such prime-survivor closure permit to become structure? Beginning from Primitive 0, where no realised structure may require operational infinity, the paper proceeds through the π/2 orthogonal gate into exact 3:4:5 hard closure. Within this framework, primes are interpreted as survivor residues: finite boundary conditions where reduction fails to destroy structure. A hard 3:4:5 closure is then represented as an FRC quark-seed. Three compatible hard closure-seeds sharing a structural integer anchor form an FRC proton-seed. That proton-seed is tested through finite π/2 lifecycle recurrence, after which a length-5 outer memory-shell is tested against the same shared anchor under strict recurrence across the finite transform set. Where the shell remains hard-lattice realised, boundary-exit free and non-dissolving, the resulting structure is certified as an FRC Hydrogen-seed. The reported result is deliberately bounded. The paper does not claim to replace conventional particle physics or to reproduce Standard Model quarks, protons or hydrogen atoms in their full physical meaning. The terms FRC quark-seed, FRC proton-seed and FRC Hydrogen-seed are operational labels inside the FRC admissibility framework. The proof burden is carried by integer closure, recurrence checks, boundary-exit counts, diagnostic pressure records, drift audits and supporting certificate files, not by visual resemblance or terminology. A completed 50M prime-survivor audit is included as supporting certification. The original prefix run was salvaged from completed chunk summaries over the range 1 to 41,426,000. A summary-only remainder run completed the range 41,426,001 to 50,000,000. The final merge certificate records a continuous gap-free audit from 1 to 50,000,000, with zero corrupt prior chunks, zero corrupt remainder chunks and 100.0% merged completion. The contribution of Paper 05b is therefore not the isolated use of primes, Pythagorean triples, integer lattices or closure language. It is their finite executable integration inside the FRC admissibility programme: Primitive 0 constrains the domain, π/2 supplies the orthogonal gate, 3:4:5 supplies the first hard memory closure, prime survivor residues supply the boundary arithmetic, shared-anchor compatibility admits a proton-seed, lifecycle recurrence tests persistence and an outer memory-shell completes the first certified FRC Hydrogen-seed.