Holonomy-Driven Geometric Quantum Simulator with Representation Lifting

This preprint presents a unified holonomy-first, invariant-only simulator architecture for loop-based quantum operations, integrating two complementary components:(i) an operational framework for discovering, diagnosing, and compiling native holonomic operations from closed-loop executions, and(ii) a minimal semantic language that characterizes what such loop-based operations mean when only conjugacy-invariant data are accessible. The simulator treats holonomy words—compositions of calibrated loop primitives—as the fundamental operational objects, rather than Hamiltonians, generators, or gauge-fixed operators. All semantics are defined at the level of conjugacy classes, represented operationally by invariant fingerprints such as trace moments or overlap-derived scalars. This invariant-first stance enables robust workflows that avoid full operator reconstruction, tomography, or fixed-basis assumptions. On this substrate, the simulator introduces a geometric compilation pipeline that: Extracts interaction structure from short loop compositions using commutator and nested-commutator diagnostics; Organizes noncommutativity and curvature information into a gauge-invariant interaction spectrum; Promotes reproducible, scale-consistent loop words into reusable native operations. A central contribution of the unified framework is a theorem-level semantic obstruction: the default single-system semantic tier is generically incomplete in the presence of entanglement-sensitive loop effects. This is demonstrated by introducing entanglement holonomy, a word-level invariant accessible via joint return amplitudes of entangled probes yet invisible to any reconstruction based solely on single-system class functions or mode-preserving data. As a consequence, the simulator treats the effective representation on which holonomies act as provisional rather than fixed. When invariant closure and scaling tests fail at a given tier, the simulator is forced to lift representation (e.g., from single-particle to multi-particle or encoded subspaces) as an operational necessity rather than a modeling choice. An explicit minimal example of an irreducible two-particle holonomic gate is provided within a quadratic-mode backend. The simulator does not attempt to unify classical and quantum dynamics or to provide a theory of quantum gravity. Instead, it supplies a shared holonomy-based semantic layer in which classical geometric transport and genuinely quantum effects (including entanglement) can be jointly represented and operationally distinguished using the same invariant diagnostics. This unified manuscript synthesizes and extends two previously developed components—an operational framework (GQF, DOI: 10.5281/zenodo.18303288) and a semantic foundation with an obstruction result (MHQL, DOI: 10.5281/zenodo.18332083)—to expose a structural point not apparent when they are treated separately: invariant-first compilation and representation lifting are linked by a principled semantic incompleteness theorem. The result is a rare simulator architecture that is holonomy-driven, gauge-invariant by construction, and representation-adaptive when compelled by invariant evidence.