Quantum Mathematical Language: Programming Quantum Algorithms via Subspace Reflections and Relativistic Causal Geometry

"Geometry is space; algebra is time." Atiyah's dictum exposes a core limitation of conventional quantum programming: encoding computation as sequential gates ($U_k \cdots U_2 U_1$) obscures the holistic spatial geometry underlying algorithmic intent. We present Quantum Mathematical Language (QMath), where computation is symmetry breaking: an intrinsic geometric twist $c$ generated by hyperplane reflections and unfolded into causal partial orders via Bessis's Garside categories. With this categorical compilation target, QMath introduces projector-based subspace reflections (reflect/phase, generalizing Householder operators) alongside explicit perspective shifts (view/transform, surfacing a landmark of human algorithmic reasoning). Because generalized reflections provide unitary completeness while reflection pairs capture planar rotations, QMath concisely rewrites canonical algorithms, directly exposing speedup mechanisms as subspace angles or Fourier cancellations. Integrating linear typing, non-unitary spectral filtering, and Garside normalization, QMath lowers coordinate-free geometry into canonical causal DAGs.