The Subset Theorem — Quantum Application v4.0

The Subset Theorem — Quantum Application v4.0 is a mathematical preprint developing a formal quantum-kinematic application of a nine-dimensional integer-lattice framework. It constructs isometric embeddings of finite-dimensional quantum systems into ℓ²(ℤ⁹), identifies the associated corner algebra P₀ℬ(ℓ²(ℤ⁹))P₀ ≅ Mₙ(ℂ), embeds density operators, and extends finite-dimensional unitary operations. For a specified normalized Markov kernel, it derives Fourier-multiplier tensor factorization, spectral formulas for the infinite lattice and finite periodic quotients, and a signed-permutation symmetry action. It also presents a nine-party GHZ state construction and discrete shift-operator uncertainty relations. These are formal mathematical arguments within their stated assumptions; they do not establish that the framework is mathematically novel or physically realized. Independent human mathematical review, a comprehensive literature novelty assessment, and full bibliography authentication have not been completed. No privileged physical ℤ⁹ substrate, quantum-tunneling identification, quantum technology, or experimentally demonstrated physical result is established by this manuscript.