The Synaptic Continuity Protocol (SCP) is a physical and information-theoretic framework for transferring consciousness between a biological and a synthetic neural substrate without interrupting awareness. The protocol combines (i) parallel fabrication of a synthetic body via molecular manufacturing (10^21 atoms/s, 81 days), (ii) a fiber-optic neural bridge (8.6×10^18 bit/s, 8600 fibers) maintaining real-time causal synchrony, and (iii) parallel synaptic migration (10^6 steps, ~17 minutes) in which the biological brain shadows its replacement. The paper develops the full quantitative argument: thermodynamic and information-theoretic feasibility bounds, the bridge physical layer (capacity, latency, power), a neuromorphic substrate preserving causal topology (integrated-information constraint), sensory-motor and endocrine integration, a formal continuity theory (causal-mirror theorem, natural-discontinuity argument), verification acceptance criteria with statistical design, fault tolerance with checkpoint rollback, ethics and governance, a systematic objection-and-rebuttal table, and a 40-60 year roadmap with cost estimates. SCP is the companion to the Quantum Flash Tomograph (Osiej, 2026), used only as non-destructive quality-assurance scanning of the sacrificial verification twin; SCP itself requires no destructive step.