We formalize the operational, mathematical, and structural architecture of the Poly-Invariant System (PIS), an advanced algebraic framework that maps finite tuples of non-zero numbers to an order-independent multi-dimensional signature vector space. Traditional coordinate spaces suffer from inherent positional and sequence vulnerabilities; the PIS framework resolves these challenges by executing simultaneous symmetric operations across an n-dimensional space to construct an indestructible macro-signature. While the underlying mappings intersect with classical elementary symmetric polynomials and Vieta's root-coefficient identities, the PIS framework uniquely reformulates these invariants into an operational vector configuration coupled with a normalized structural ratio and a subtractive tension manifold. We systematically develop the mechanical principles of the Dimensional Padding Lemma to safeguard low-variable datasets against division-by-zero singularities. Crucially, we introduce and formalize two novel structural axioms authored by the researcher: The Fundamental Law of Association and The Inverse Law of Association. Using graph-theoretic vocabulary, we prove that under these analytical laws, the PIS framework transitions from a pure information encoder into a non-directional relational network topology, where the signature vector possesses latent relational energy capable of instantaneously materializing complete, transitive element-to-element association fields among decoded entities. To ensure engineering viability, we perform an asymptotic complexity profile and address the numerical precision challenges of Wilkinson's Polynomial Phenomenon by introducing a scalable 3D block-clustering architecture. Finally, we derive the exact algebraic hypersurface defining the system's Discriminant Loci, establishing the precise bifurcation boundaries and root-collision manifolds that govern coordinate reconstruction, offering profound applications for lossless data engineering, non-linear geometric visualization, cryptographic systems, and network modeling.
Siyan Stephen Christian (Mon,) studied this question.