Prime Distribution in Base-2 Exponential Rows: Symmetry, Predictive Hunt Zones, and Effective Bounds

Abstract

We develop a unified framework for studying the distribution of prime numbers in base-2 exponential (dyadic) rows. Using explicit estimates from prime number theory, we prove unconditionally that the normalized left–right prime-count imbalance in any dyadic interval [x, 2x] satisfies O(1/log x), and that any pre-fixed, prime-independent predictive window of relative width α captures a fraction α + O(1/log x) of the primes in the interval. Explicit effective bounds are obtained, including a symmetry bound of 1.20/log x and a predictive-window capture error of 3.10/log x for x ≥ 599. The framework is extended to finite window families, weighted windows, arithmetic progressions, and suitable short intervals, together with computable finite-ε thresholds and an explicit reproducibility protocol. The historical Fradkin Symmetry Conjecture is proved in its relative count-imbalance form, with σ_k = O(1/((k − 1) log 2)). We also investigate an auxiliary energetic formulation based on E(n) = log log n, rigorously distinguishing proved functional-analytic results from conditional statements and open conjectures. In particular, the stronger energetic characterization of individual primes remains open.

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