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Achieving nanosecond-scale inference latency for deep neural networks (DNNs) has become a primary architectural concern for latency-critical applications. While Field-Programmable Gate Arrays (FPGAs) offer a promising substrate for low-latency inference, conventional FPGA accelerators remain arithmetic-centric, using LUTs primarily as building blocks for numerical operators and peripheral logic. In contrast, recent LUT-native neural networks treat LUTs as learnable neurons, revealing promising theoretical potential to exploit their intrinsic logic expressivity. However, existing methods are largely confined to algorithmic optimizations, failing to translate this theoretical potential into high-performance FPGA accelerators. Specifically, their differentiable formulations do not faithfully match FPGA LUT primitives, their physically-unaware topologies compromise routability and timing closure, and their lack of automated optimization flow hinders systematic design space exploration (DSE) and efficient hardware implementation. In this paper, we propose FPGN, an end-to-end physically-aware framework that closes the gap between LUT-native learning and latency-optimized FPGA implementation. FPGN addresses these challenges through (i) a hardware-aligned differentiable formulation for training FPGA-native LUT neurons, (ii) a structured LUT-native topology with a streaming hardware architecture to improve routing locality and timing closure, and (iii) a latency-driven compiler that leverages high-fidelity analytical Quality of Results models to automate DSE and hardware generation. Experiments show that FPGN achieves up to 205$\times$ latency reduction compared to representative FPGA-based BNN accelerators and up to 30$\times$ higher LUT efficiency than prior differentiable LUT-native networks, while maintaining competitive inference accuracy.