Superposed Topology - a formulation of Quantum Discrete Morse Theory

Discrete Morse theory compresses a finite cell complex by cancelling matched cells and produces a smaller complex that is homotopy equivalent to the original. Classical reductions are selection-based. A matching is chosen and cancellation follows, so any attempt to place matchings into superposition collapses coherence at readout. This paper introduces a coherent analogue by defining an admissible superposed gradient as a graded operator acting on a Hilbert chain complex, together with a contraction identity and a finiteness condition that makes all reductions algebraic. A critical projector defines quantum Morse numbers as critical subspace dimensions in each degree and yields a reduced differential whose homology agrees with the original. The resulting quantum Morse inequalities bound Betti numbers by quantum Morse numbers and reduce to the classical Morse inequalities when the gradient encodes a deterministic matching. A key distinction from probabilistic mixing is that coherent superpositions produce interference terms inside the critical projector, so phase can change critical statistics while leaving homology invariant.