Perfect Photon-Number Pairing Bounds Homodyne CHSH Violation

A bound on homodyne Bell violation set by the state class, not by the binning. For over two decades, attempts have been made to violate a CHSH inequality using homodyne detection with sign binning alone, since this measurement is nearly lossless and would permit a loophole-free test. The values achieved remain small: 2.048 and 2.076. Two causes are possible — the coarseness of sign binning, or the class of states used. This work separates the two exactly. Because the sign-binned quadrature measurement is a Hermitian operator with spectrum in [-1,1], the maximum over all states is the largest eigenvalue of the Bell operator, and the restriction to perfectly photon-number-correlated states |ψ⟩ = Σ cₙ |n,n⟩ is a projection onto the diagonal subspace. Both maxima are compared at identical angles, so no optimisation over states is involved. Result: the perfectly paired class saturates at S = 2.10198 ± 0.00001, established by convergence to Fock truncation N = 1150, while relaxed pairing reaches 2.22 under the same measurement prescription and continues to rise. The bottleneck is the pairing, not the binning. The gain originates from a targeted admixture of asymmetric photon number at |0,4⟩ and |4,0⟩. Three supporting computations explain the bound: the exact bound a₁ ≤ 0.811876 on the fundamental harmonic of the correlator, which the ideal paired state exhausts to 99.84 %; the completeness of the pairing criterion as an entanglement witness under local symplectic optimisation; and the ineffectiveness of complex amplitudes and Kerr nonlinearity. An attempted proof via subadditivity is documented as having failed, together with the reason no term-by-term estimate can succeed. The bound is presented explicitly as a numerically supported conjecture, not a theorem, and is falsifiable: a state of the form Σ cₙ |n,n⟩ with S > 2.1021 refutes it. Practical consequence for experiment: within the perfectly paired class, further optimisation cannot exceed 2.10198. Higher values require breaking the pairing deliberately. This record contains the manuscript and supplementary material in English and German (PDF and LaTeX sources), and the script reproduce_all.py, which regenerates every tabulated value in eight individually callable sections. Deutsche Fassung des Manuskripts und des Ergänzenden Materials sind in diesem Record enthalten.