Exact Angular Counting for Prime Gaps under Gap-Dependent Encoding, with Ramanujan-Based Diagnostics

This paper introduces a geometric–algebraic framework for prime-gap analysis that maps every integer inside a prime gap to a canonical angular coordinate. From this encoding we derive exact counting laws and a per-direction uniformity principle that hold for all data ranges under a standard “full-gap” convention. These are structural equalities—not heuristics—and they do not rely on unproven conjectures. On top of that spine, we build a short-window sieve bridge that transfers information from counts of prime pairs to counts of consecutive gaps at small scales. The bridge makes explicit why the familiar multiplicative corrections from classical analytic number theory naturally reappear when one looks at divisibility properties of gaps. To stabilize numerics, we propose a smooth “helical roughness” weight that closely mimics sharp inclusion–exclusion while reducing variance in experiments. We also supply an LPF/rough-number baseline that explains the multiplicative factors as a local exclusion of small primes, giving a crisp conceptual picture for why those corrections arise. A compact Ramanujan-transform diagnostic then visualizes regularities in the data (for example, the parity peak) as empirical signatures. Crucially, we keep the roles separated: exact identities provide structure, the sieve bridge provides transfer and bounds, and the spectrum provides diagnostics—avoiding over-interpretation and making the pipeline reproducible. What’s new Exact, non-heuristic structure: rigorous counting and uniformity laws linking the angular encoding to gap divisibility. Short-window transfer: a calibrated sieve bridge that carries pair information into consecutive-gap counts at small scales and clarifies the origin of multiplicative corrections. Variance-reduced smoothing: a helical roughness weight that approximates sharp roughness with controlled error while improving numerical stability. Explanatory baseline: a least-prime-factor / rough-number perspective that conceptually explains the observed multiplicative behavior. Clear role separation: identities for structure, sieve for transfer, spectrum for diagnostics—preventing claims that go beyond what the data justify. Why it matters Turns widely used heuristics into testable and numerically stable statements in short windows. Provides a modular toolchain that other researchers can reuse to probe fine-scale questions about prime gaps. How it’s validated Large-scale computations confirm the exact counting laws with perfect agreement across tested ranges, and the short-window predictions align with measured data. Independent, integer-arithmetic checks (without invoking the identities themselves) are included to decouple measurement from theory. For whom Researchers in analytic number theory interested in structure vs. distribution of prime gaps. Computational mathematicians seeking robust, variance-aware instrumentation for large-scale gap analysis. Reproducibility The workflow is organized into three plug-and-play layers—encoding, sieve-bridge, and diagnostics—so results can be replicated, extended to higher ranges, or integrated into other prime-gap studies.