We introduce a factory–processor interface architecture for modular optical fault-tolerant quantum computation that separates code-dependent encoded-resource preparation from reusable online logical processing on a common Gaussian optical processor. Beyond encoded resources, the interface specifies logical outputs, code-to-mode mapping, measurement roles, frame updates, verification records, and resource-injection boundaries. As a concrete realization, we combine an offline verified Bell-resource factory based on a recursively concatenated C₄ code with physical Gottesman–Kitaev–Preskill (GKP) qubits with an online time-domain measurement-based quantum computation (MBQC) processor. Under the specified additive-displacement-noise model and fixed postselection policy, simulations support the finite-level logical Bell-error ordering P₅<P₂ at 12.25 dB of effective GKP displacement-noise quality, both with and without the prescribed additional post-injection displacement noise. At 15.6 dB and L=2, a separate fixed-sample evaluation gives one-sided 95% Clopper–Pearson upper bounds below 10⁻⁵ for both conditions under the binomial model. The interface supports address-tracked injection, transversal CNOT and logical Bell-measurement constructions, and logical SWAP and Hadamard constructions with tracked code permutations. Given advance resource preparation and sufficient routing parallelism, increasing the concatenation level expands parallel encoded-resource width without increasing the protected online sequence length of these constructions. These results identify a modular route to lower logical error rates while retaining a common Gaussian MBQC operation set with shallow online processing.