Robust CHSH Self-Testing with Finite-Energy GKP States

We present a full-oscillator analysis of a finite-energy GKP CHSH test whose observed score yields robust Bell-pair self-testing. Periodically binned position and momentum give the Pauli settings, while a fixed binary coarse-graining of photon number modulo four and its displaced conjugate realize the tilted settings. For a number-filtered GKP source, we retain the finite codeword overlap, define the measurements on all photon-number sectors, and compute the physical correlations without logical post-corrections. With the canonical ideal-logical displacement $d=\sqrt{\pi}$, the CHSH value exceeds the local bound above $4.56$ dB of per-peak squeezing, and Kaniewski's extractability bound becomes nontrivial above $5.02$ dB. Calibrating only $d$ using an independently characterized finite-energy parameter lowers these model thresholds to $4.21$ dB and $4.58$ dB, respectively; at $12$ dB, it raises the score from $2.69486$ to $2.78858$ and the corresponding target-state overlap bound from $0.90758$ to $0.97243$. This calibration is fixed before Bell-test data are collected. The displacement activates the odd modulo-four sectors, so their fixed a priori assignments are a genuine finite-energy component. The large gain is specific to the deterministic phase-bit coarse-graining; independently calibrating the one-bit POVM with randomized odd-sector outcomes gives only a much smaller improvement. These are honest-model predictions, not loss, detection-efficiency, or finite-sample thresholds. In an experiment, a device-independent guarantee for an extracted Bell pair follows by inserting a confidence lower bound on the observed CHSH score into the self-testing theorem.