An Exploratory Combinatorial Toy Model of Twin-Prime Partition Hierarchies and a Numerical Correspondence with the Fine-Structure Constant

This manuscript presents an exploratory combinatorial toy model based on partitions of integers into distinct primes. An adjusted difference-over-sum rule applied to partition counts for twin-prime pairs produces a hierarchy of dimensionless parameters. For the reference pair {139,137}, the construction yields 0.036, giving an inverse fine-structure constant estimate of 137.036. The manuscript describes the counting method, reports results for twin-prime pairs below 200, and examines sensitivity to allowing the integer 1 as an additional element. It also explores tentative structural analogies to quark-charge patterns, beta decay, matter–antimatter symmetry, locality, fermion generations, and atomic orbitals. The adjustment was selected with knowledge of the measured fine-structure constant; the numerical agreement is therefore a correspondence rather than an independent prediction. The physical analogies require additional assumptions and do not establish physical mechanisms. The work is presented as an exploratory framework for further mathematical investigation and critical discussion.