A Classical Probabilistic Bit from the Coupling Law α(d, β): Bernoulli Components and an Exact Bit-Flip Involution

The single-constant coupling law α(d,β)=1+d/(d+β), β=6/23, has a one-unit window α ∈ [1,2) that maps exactly onto the unit interval. We record its structure as a classical probabilistic bit (p-bit). The components F=α−1=d/(d+β) and S=2−α=β/(d+β) satisfy F+S=1 and are the probabilities that a two-state variable reads 1 and 0: a Bernoulli variable with bias p(d)=d/(d+β) sweeping [0,1) as d sweeps [0,∞). The scale-reciprocal involution R(d)=β²/d realizes the bit-flip p↦1−p exactly (F∘R=S), with unique fixed point d=β the fair-coin state p=½. At β=6/23 the bias takes exact rational values at integer coordinates. These are algebraic identities in β; no physical p-bit realization is claimed. α(d,β)=1+d/(d+β), β=6/23.--Author: James E. Dunn — Independent Researcher, Hydrogen Lifecycle Research ProgrammeORCID: https://orcid.org/0009-0005-2679-6574Corpus: https://zenodo.org/search?q=creators.orcid:0009-0005-2679-6574License: CC BY 4.0 International