Quantum Riemann Machine: Coherent Channel Compatibility and Environment Cost

Quantum programmes can agree on the operation performed within each branch whilst differing in the interference produced when those branches are recombined. I propose a Quantum Riemann Machine as a finite architecture for checking whether specified implementations admit a common quantum realisation and determining its environmental cost. Its geometric input records the continuation of Kraus frames around loops; its physical consistency condition is the positive Gram condition of channel gluing. For three branches, the eigenphases of a single loop give an exact coherence threshold and pure-environment spectrum. I extend this calculation to one reference branch and an arbitrary number of mutually calibrated branches, allowing their loop transports to be noncommuting. The singular values of the sum of these transports determine the complete spectrum through explicit cubic polynomials, and hence determine feasibility and the minimum environment dimension. A four-branch example has feasible restrictions to every three branches but no joint realisation. Reorienting one transport while preserving its eigenvalues restores feasibility, demonstrating information that separate loop spectra do not retain. Leaving one comparison unspecified instead permits an environment of dimension six, whereas prescribing it can require dimension eight or make the programme impossible. Spectral error bounds and constructive compression retain every local branch channel exactly. These certificates distinguish incompatible interference specifications from insufficient environmental resources and quantify the consequences of reducing an implementation, connecting its transport description to the design and verification of coherently controlled quantum programmes.