We establish that valid $\Sigma_1$ propositional inference admits reduction to Fibonacci-indexed witness equations. Specifically, modus ponens verification reduces to solving a linear Diophantine equation in $O(M(\log n))$ time, where $M$ denotes integer multiplication complexity. This reduction is transitive: tautology verification proceeds via Fibonacci index arithmetic, bypassing semantic evaluation entirely. The core discovery is a transitive closure principle in $\Phi$-scaled space (Hausdorff dimension $\log_\Phi 2$), where logical consequence corresponds to a search problem over Fibonacci arcs -- a geometric invariant encoded in Zeckendorf representations. This yields a computational model wherein proof verification is achieved through \emph{arithmetic alignment} rather than truth-functional analysis, preserving soundness while respecting incompleteness. The construction synthesizes Lovelace's anticipation of symbolic computation (Note G) with the Turing-Church formalism, revealing a geometric interpretability of logic relative to a $\Sigma_1$ or $\omega$-consistent theory.

翻译：我们证明了有效的Σ₁命题推理可归约为斐波那契索引见证方程。具体而言，肯定前件式的验证可归约为在O(M(log n))时间内求解线性丢番图方程，其中M表示整数乘法复杂度。该归约具有传递性：重言式验证通过斐波那契索引算术进行，完全绕过了语义求值。核心发现是Φ标度空间（豪斯多夫维数为log_Φ 2）中的传递闭包原理，其中逻辑后承对应于斐波那契弧上的搜索问题——这是一个编码在泽肯多夫表示中的几何不变量。这产生了一个计算模型，其中证明验证通过算术对齐而非真值函数分析实现，在保持可靠性的同时尊重不完全性。该构造将洛夫莱斯对符号计算的预见（注释G）与图灵-丘奇形式体系相结合，揭示了相对于Σ₁或ω相容理论的逻辑几何可解释性。