A Deterministic Linear-Time Framework for 3SAT via Local Sieve and Additive Backtracking

Overview This repository contains the formal proof and algorithmic framework for a deterministic, linear-time resolution strategy for 3SAT instances. By exploiting high-density clause intersections, the framework bypasses the traditional exponential branching cascades ($O(2^n)$) typical of standard satisfiability solvers. The methodology decomposes complex 3SAT instances into structured, localized components where variable states are restricted and resolved additively. Core Architecture The proposed pipeline operates over three distinct, deterministic phases: 1. High-Density Connected Components: Input clauses are partitioned into clause groups ($\mathcal{G}$) based on a strict minimum $2$-variable overlap constraint, mathematically mapping onto the connected components of a clause-intersection graph. 2. Local Structural Sieve & Unit Propagation: For each component, a static structural metric deterministically selects a $2$-variable pivot backbone ($V_c$). Evaluating the $4$ invariant configurations ($\text{TT, TF, FT, FF}$) over $V_c$ either satisfies clauses directly or cleanly reduces them to single-literal unit clauses. Remaining isolated variables are resolved deterministically via **unit propagation**, populating a pre-filtered local solution list ($\mathcal{S}_G$) capped at a maximum of $4$ valid profiles. 3. Sequential Additive Global Synthesis: The framework sequentially reconciles the filtered local lists. Crucially, because distinct clause groups are structurally proven to share at most a single variable, localized variable mismatches cannot trigger a geometric backtracking cascade into the past history. Key Breakthroughs & Complexity Bounds Phase 1 Complexity: Constructing the local sieved lists executes in strict linear time relative to the number of input clauses: $T_{\text{Phase 1}} = \sum O(|G_i|) = O(m)$. Phase 2 Complexity: The global synthesis backtracking mechanism is structurally proven to behave additively rather than multiplicatively. The maximum number of backtracking steps ($B$) is strictly bounded by the summation of the alternative pool choices: $B \le \sum (S_i - 1) \le 3k = O(k)$. Complete Linear Resolution: Combining both phases establishes a deterministic, global runtime bound of $O(n + m)$, structurally flattening the traditional exponential search tree.