Applying the Poincaré theorems and Euler integral in the qubit-based signal and sensing framework of the quantum computing

Applying the Poincaré theorems and Euler integral in the qubit based signal and sensing framework can be as Qubit State Space. Defined by rational amplitudes or unit-fraction-based norms. Signal as Path: A time-evolving qubit state (signal) as a path through this discrete manifold.Use Euler integration or Poincaré recurrence to describe or detect repeating patterns, signal stability and minimal sensing requirements.Signal values across this grid can be analyzed via Euler calculus. Signal coverage, gaps, and redundancy are reflected in topological counts and we can apply Euler integrals to compress, summarize, or filter the signal.