Description / Abstract: Theoretical Framework: HST is a background-independent, experimentally verifiable unified theory which solves many of the outstanding problems in current physics models of our world. To do this, the theory redefines gravity, time, the speed of light, and mass, while providing resolutions for the Dark Sector (Matter/Energy), the Hubble Tension, the Fine Structure Constant, the internal structure of the nucleon, and the cosmological constant discrepancy. It provides a fractally consistent framework where the 'laws' of physics emerge as the geometric phenomena of a structured vacuum lattice which is relationally consistent. Chronological Registry: This document serves as a formal chronological registry of discoveries and mathematical milestones, established to secure the "HST Genesis" (2012) and the formalization of the Structured Scalar Medium (SSM) (2025–2026). It indexes the foundational identities for the forthcoming series: "Helical Scalar Theory (HST): Series A: The Unified Series." Key Mathematical Pillars Registered: The HST Helical Differentials (dS HST): The first formal coupling of radial displacement and angular rotation into a single metric, identifying mass and inertia as the changing differentials of a spiral's geometry. The Machian Identity (G HST): Derivation of the Gravitational Constant as the residual longitudinal tension of the global manifold, determined by the ratio of the cosmic container's radius to its total mass. The Universal Slip Angle (α HST): Geometric derivation of the Fine Structure Constant as the universal gear ratio required for stable helical propagation. The Borromean Nucleon Triad: Redefinition of the proton as a three-stranded helical knot, identifying "quarks" as individual topological strands of a unified nucleon knot.