<p>This paper presents a unified exact discrete-conjugate framework for the generation, modification, and analysis of generalized worm profiles and the corresponding wheel. Existing methods often rely on curve fitting (e.g., B-splines), which degrades normal vector fidelity. In this paper, we propose a fitting-free approach that operates directly on discrete points with analytical normals. The framework follows a clear computational sequence. First, we establish a generalized mathematical model based on a dimensionality reduction strategy to derive exact axial profiles. Second, we superimpose a dual-purpose polynomial modification strategy that supports both interactive design and inverse compensation via the least squares method. Singular value decomposition can also be integrated to improve matrix conditioning; numerical examples demonstrate that by utilizing a fourth-order polynomial, residual profile errors are suppressed to approximately 0.17&#xa0;μm. Third, we incorporate a dual-threshold feasibility criterion to evaluate grinding wheel manufacturability. Based on the identified physical limits and mathematical singularities on the wheel, the starting and ending positions of the worm profile are determined; for a high-lead case, the valid active profile percentage was identified as 73.99%. Finally, we use analytical tangent derivation to construct wheel fillets and verify the corresponding contact points against singularity thresholds. This procedure ensures that <i>G</i>¹ continuity is achieved within valid geometric regions and serves as a safeguard for worm manufacturing.</p>