Fermion Masses from a Prime Sieve: the Two-Scale Picture, the Twin-Prime Slab Closure, and a Pre-Registered Audit of the Generation Problem

This paper audits the fermion-mass programme of the prime-sieve framework — the programme that inherited the electromagnetic sector’s core-energy formula when the ff account was sealed. Its claim profile is deliberately asymmetric to the main paper’s, and we state it plainly: two positive closures, one quantified open channel, one archived problem. (i) Two-scale picture established: the bare core (~ℓ0 , setting the on-site interaction) and the dressed cloud (𝜉 ≫ ℓ0 , setting the physical mass) are distinct scales; the mass equation 𝑚 = 𝜇𝑞2/(2𝜋𝜉3 ) is numerically verified pointwise, the Compton self-consistency band is endogenous, and the cubic amplifier 𝑚𝑖/𝑚𝑗 = (𝑞𝑖/𝑞𝑗 ) 2 (𝜉𝑗/𝜉𝑖 ) 3 gives the ratio structure. (ii) Twin-prime slab closure established: the framework’s stiffness gap 2𝜋/𝜇𝑞2 = 6.100 is matched by the twin-prime (𝑔 = 2) slab’s core-energy amplification 6.164–6.25 (2.5%); and the self consistency condition 𝑟𝑐 ∗ (𝑔) = √ratio(𝑔) 𝜇𝑞2/2𝜋 is lattice-legal only for 𝑔 = 2 (𝑟𝑐 ∗ = 1.012ℓ0 ; all other gaps give 0.43–0.47ℓ0 , sub-lattice) — a derived selection rule localizing every fermion core in a twin-prime slab, with the self-consistent mass 𝑚∗ = 0.988𝑡 and zero free parameters. (iii) The 10−22 hierarchy open, quantified: the overall scale is reduced to a “0.62 problem” — the bare core energy must be 0.62× its registered value, with the Mott essential singularity (the framework’s only exponential) converting 22 orders of magnitude into a factor 1.6; the shell-geometry review (simple-cubic → simple-hexagonal, the true shell model) delivers 𝑅 = 0.706–0.720, a pre-registered near miss — right direction, 13% short, overshooting to 0.41 at the coordination upper bound (FCC). (iv) The generation problem archived: a fully pre-registered campaign — seven endogenous routes (C1, C2/R1–R3, R3-flavour, G-B2/F1) and six revival candidates (R4-1…4, R5, R5ff) — all closed against frozen judgment bands; the campaign produced the ff* window (the required interaction sequence 𝑈 = (0.630, 0.758, 0.848)𝑡 with descending quanta (0.064, 0.045)𝑡), an inversion theorem excluding internal excitation ladders, and a sealing theorem (E15 × Mott-line incompatibility): within the framework’s current basic layer, no endogenous three generation mechanism exists. Resurrection requires one of two registered basic-layer events. Every step was pre-registered before computation; every miss is recorded with its numbers; twelve scripts reproduce every figure.