Deterministic Assignment-Free Spectral Operator for SAT Decision: A Fully-Specified Polynomial-Time Framework with Complete Algorithm, Proof Architecture, and Implementation Guide

This work presents a deterministic, assignment-free spectral decision framework for CNF-SAT. The method compiles a Boolean formula into a finite-dimensional operator whose spectral behavior distinguishes SAT from UNSAT instances without searching over assignments. Satisfiable formulas produce a simple zero-frequency pole with quantifiable residue at the output port; unsatisfiable formulas exhibit a positive spectral gap. A masked resolvent-slope observable separates these cases with an explicit inverse-polynomial margin. The entire framework is non-relativizing, non-algebrizing, and based only on finite linear algebra and complex analysis. What This Paper Provides: A complete operator construction using a fixed catalogue of gadgets Explicit, fully reproducible block-Laplacian and skew matrices A deterministic “canonical pinning’’ scheme that requires no assignment search Proof that the SAT/UNSAT distinction appears as a spectral dichotomy A full error budget with polynomial bounds A rigorous, step-by-step algorithm with all constants specified A practical engineering variant for experimentation A code-ready appendix (CompileOperator, ScaledOperator, full pseudocode) A reviewer’s checklist for independent verification What You Can Do With This Document: Directly implement the algorithm in Python, C++, or MATLAB Test SAT/UNSAT separation on small instances Reproduce every construction exactly (no ambiguity in any matrix or index) Compare theoretical predictions with numeric behavior Use the operator as a spectral augmentation module for SAT solvers Why This Work Is Significant: This is one of the first fully explicit and assignment-free operator frameworks designed to attack SAT decision spectrally. The paper provides a complete and self-contained technical toolkit: Finite catalog of gadgets Deterministic wiring rules Conditioning/normalization policy Exact spectral masks Fully specified resolvent sampling scheme All content is expressed without LaTeX-dependent notation so the results survive translation across repositories, models, and codebases. Included in the PDF: Overview of the spectral compilation method Local semantic proofs for all gadgets Clause-defect and port-mass lower bounds Jackson–Fejér masked resonator construction Full spectral algorithm (proof-grade) Practical heuristic algorithm (engineer-ready) CDCL-integration module Assignment extractor Worked example Reviewer's checklist Intended Audience: Researchers in complexity theory SAT solver developers Spectral/linear algebra researchers Physicists working on operator methods Anyone exploring non-traditional SAT algorithms Important Note:This document is presented as a complete technical contribution. The claims regarding SAT decision and polynomial-time solvability should be evaluated through peer review and independent implementation. See also: The Tier–Omega Monad: Trans-Recursive Completion of the Everything Equation (DOI: 10.5281/zenodo.17859631), which establishes the unique trans-recursive invariant that terminates all possible meta-law recursion and completes the Everything Equation. This work provides the structural boundary underlying the entire framework developed in the associated papers.