Quantum Cantor–Bendixson Machine: Exact and Simultaneous Boundary Repair

Quantum programmes described through successive approximations need a consistent way to retain their limiting behaviour when individual approximating settings are discarded. I propose a Quantum Cantor–Bendixson Machine that certifies and repairs quantum operations at boundaries of a classical control space. For continuous matrix-valued observables over a compact zero-dimensional metric space and a finite retained boundary, the smallest completely bounded repair cost is exactly twice the largest boundary leakage, and a continuous unital completely positive map attains it. I extend this result to simultaneous finite boundary constraints, identifying when their costs combine without an additional penalty. For independent operations on any finite product of convergent-sequence spaces, a product repair preserves every Cantor–Bendixson derivative, including the infinite intermediate boundaries, at the exact cost already forced by the final boundary. A coherent qubit example establishes why the corresponding cost for a quantum subspace can instead scale as the square root of leakage. Computable operations can have noncomputable optimal repair costs: their possible leakages are exactly the lower semicomputable reals in the unit interval. An explicit tail certificate permits finite bounds and a computable repair arbitrarily close to the optimum. These results specify when one operation can respect several topological reductions, what this consistency costs, and which additional information makes its certification effective.