While prior work established a verifier-based polynomial time framework for NP, explicit deterministic machines for concrete NP-complete problems have remained elusive. In this paper, we construct fully specified deterministic Turing Machines (DTMs) for SAT and Subset-Sum within a polynomial-time NP verifier simulation framework. We show that both machines operate in polynomial time and, for satisfiable instances, deterministically generate valid witnesses, thereby extending the framework to deterministic FNP computation without increasing the degree of polynomial complexity. Furthermore, we provide a complete implementation of the framework, including the dynamic computation graph, feasible-graph construction, verification walks, and Turing-machine simulation via edge extensions. The implementation behaves in accordance with the predicted polynomial-time bounds. To ensure transparency and reproducibility, the complete Python implementation and source code are made available in a public online repository.

翻译：尽管先前的研究已为NP问题建立了基于验证器的多项式时间框架，但针对具体NP完全问题的显式确定性机器仍然难以构建。本文在多项式时间NP验证器模拟框架内，为SAT问题和子集和问题构建了完全确定的图灵机。我们证明这两类机器均在多项式时间内运行，并且对于可满足的实例能够确定性地生成有效见证，从而在不增加多项式复杂度阶数的前提下，将框架扩展至确定性FNP计算。此外，我们提供了该框架的完整实现，包括动态计算图、可行图构造、验证游走以及通过边扩展实现的图灵机模拟。该实现的行为符合预测的多项式时间边界。为确保透明度和可复现性，完整的Python实现及源代码已公开于在线代码仓库。