A Nine-Symbol Ledger Alphabet Arising from the Golden-Ratio Cost Ladder in Recognition Science

Recognition Science employs a self-dual cost functional J(x) = 1/2 x + 1/x and a golden-ratio scale hop x →φ±1. We show that the infinite ladder {J(φ±n)}n∈Z compresses to a nine-symbol signed integer alphabet {±4,±3,±2,±1,0} via an affine Lucas-number rounding Ξ(n) = (−1)n round(J(φ|n|)−1)/0.5 . Sign derives from the even/odd parity of n; magnitude follows Lucas gaps (φn + φ−n = Ln). The mapping (i) reduces ledger words in Light-Native Assembly Language to 4-bit integers, (ii) attaches an intrinsic male/female parity bit to every ledger state, and (iii) provides an integer target for future Ecoh-normalised biochemical and spectroscopic measurements.