From HLV Geometry to Electroweak-Like Gauge Structure: Boundary Spectral Operators, Rank-2 Emergence Gates, and the Conditional U(2) → SU(2) × U(1) Bundle

This work develops a conservative geometry-to-operator programme for investigating whether electroweak-like gauge structure can emerge from the native geometric and spectral architecture of the Helix–Light–Vortex (HLV) framework. Earlier HLV electroweak work established a conditional downstream result: if an isolated rank-2 complex boundary-mode bundle is supplied, its natural frame redundancy is U(2), with the familiar decomposition into SU(2) and U(1) sectors. That result, however, does not establish that HLV geometry itself generates the required rank-2 sector. The present work makes this missing step explicit. It connects native HLV carrier geometry, a typed geometric/symbolic language, semantic operator maps, and boundary spectral operators into a controlled derivation chain: HLV geometry → typed semantics → boundary operator H∂ → spectral projector → rank-2 boundary bundle → U(2) → SU(2) ⊕ U(1). A central distinction is maintained between geometric syntax, operator semantics, mathematical bundle structure, numerical evidence, and physical interpretation. Geometry alone is not assumed to determine a unique physical operator. Instead, candidate geometry-to-operator maps must be explicitly declared and subjected to provenance locks, null-model comparisons, robustness tests, spectral-gap criteria, and falsification gates. The manuscript reformulates the unresolved electroweak problem as a prospective rank-2 emergence test: can a fixed native HLV boundary operator produce an isolated and robust two-dimensional spectral subspace that survives comparison with crystalline, randomized, and generic quasiperiodic controls? The previous EW2 bundle result is retained as a conditional mathematical consistency result. No claim is made that the Standard Model electroweak sector has been derived, that the physical electroweak couplings or symmetry-breaking mechanism have been recovered, or that HLV-specific electroweak physics has been experimentally established. The resulting framework identifies a precise falsifiable boundary between what has already been mathematically demonstrated and what must still be derived from native HLV geometry before an electroweak interpretation can be physically justified.