Binary Measure, Arithmetic Half-Frequency, and Spectral Statistics: Finite Dyadic Pushforwards, Deterministic Sign Records, and Zeta-Zero Comparisons

The Bare Binary Biconditional connected exchange-invariant binary probability, the fair Bernoulli product law, almost sure half-frequency, and Lebesgue measure under dyadic coding. This paper completes the three-study half-probability convergence program related to that construction while keeping the studies’ logical roles separate. The first is a direct finite-depth image of the exact measure identity: 400,000 depth-40 binary records sampled with parameter 𝑝 = 1/2 give a histogram consistent with uniform measure on [0,1], while 𝑝 = 0.35 gives the nonuniform finite-resolution pattern of a measure known analytically to be singular with respect to Lebesgue measure. The second examines three deterministic arithmetic binary records through an arithmetic cutoff of 3,000,000: the Liouville sign, the nonzero Möbius sign, and the residue class modulo 4 of odd primes. Their observed head frequencies are respectively 0.499843, 0.500029, and 0.499426, but their eligible-trial counts, exact ties, and sign-change textures differ substantially. These deterministic records are not modeled as independent Bernoulli trials. The third reproduces the established Montgomery–Odlyzko spectral-statistics comparison using 1,200 cached nontrivial zeta zeros, with 1,190 zeros retained after a low-height cutoff and 1,189 nearest-neighbor spacings. It is included as a calibration and as wider deterministic-random context, not as evidence for the binary measure theorem. Together, the three studies provide a reproducible illustrated analysis of the earlier synthesis without claiming to prove it computationally, establish asymptotic rates from finite data, or derive a number-theoretic conclusion. The separate question of observation-record convergence is addressed independently.