Energy = Geometry × Speed: A Heuristic Unification Framework Derived from the Hourglass Operator Manifold

We propose a structural reading of energy as the product of geometric coupling configuration (G) and kinematic traversal rate (v): E = G × v. We examine how this lens re-encodes formulas across multiple physical domains and identify conjectural bridges to open problems. The framework explicitly distinguishes three epistemic tiers: EXACT CONSISTENCY MAPPING (the known formula is recovered exactly under the stated identification of G and v), INTERPRETIVE READING (coherent re-encoding, not a derivation), and CONJECTURAL EXTENSION (research directions only). The principal unresolved mathematical target is Φ(system) → (G, v): a non-arbitrary rule that maps a physical system to a uniquely preferred (G, v) decomposition without reference to the expected answer. Until this selection rule exists, the framework is a structural lens and organized research program, not a generative physical theory. The package includes the manuscript, a consistency checker (energy_geometry_speed.py), a 5D dimensional enforcement engine (dimension_system.py, covering M, L, T, Q, Θ), a decomposition-and-ranking engine (gv_engine.py), an adversarial test suite (adversarial_tests.py), illustrative conjectural modules for open problems (open_problem_probes.py), and a full findings report (report.md). Key findings: E=hf and E=mc² are exact consistency mappings under stated identifications. The Lorentz factor γ does not emerge from E=G×v. The kinetic ½ factor is not derivable from the template alone. The naive identification G(n=2)=h fails at Planck scale by approximately 10¹⁹. UNIQUELY_RECOVERED count: 0. Part of the AXONLang Labs LLC mathematical framework series. Companion to the Hourglass Framework (doi.org/10.5281/zenodo.19120026) and Three Consequences (doi.org/10.5281/zenodo.19152117). Soli Deo Gloria