Smith Normal Forms and Matrix Theory over Ordered Pair of Normalized Real Numbers

Source abstract

Ordered pairs of normalized real numbers form a closed arithmetic system once a normalization function is fixed. Their algebraic and matrix-theoretic extensions are governed by a spectral two-channel structure. The normalization function is not specialized: it is only assumed to be continuous, strictly increasing from $\mathbb{R}$ onto $(0,1)$, and symmetric in the sense that $\varphi(x)+\varphi(-x)=1.$ For an OPNs element $\alpha=(\mu_{\alpha},\nu_{\alpha})$, we define the spectral-coordinate map $S_{\varphi}(\alpha)=(p_{\alpha},q_{\alpha})$, where $p_{\alpha} = -\varphi^{-1}(\mu_{\alpha}) - \varphi^{-1}(\nu_{\alpha}), \qquad q_{\alpha} = \varphi^{-1}(\mu_{\alpha}) - \varphi^{-1}(\nu_{\alpha})$. The scalar theory is reorganized as a two-channel real algebra with two primitive idempotents $e_p$ and $e_q$. Intrinsically, this algebra is a reduced semisimple Artinian real algebra with exactly two primitive central idempotents. The main result of this paper is an idempotent Smith theory for polynomial matrices over OPNs. Although the polynomial coefficient ring $\mathcal{O}_{\varphi}[t]$ has zero divisors, every OPNs polynomial matrix admits a Smith normal form whose invariant factors have the two-channel form $d_i(t)=f_i(t)e_p+g_i(t)e_q, \qquad f_i(t),g_i(t)\in\mathbb{R}[t].$ The divisibility chain, determinantal ideals, finite-presentation classification over $\mathcal{O}_{\varphi}[t]$, and uniqueness of these invariant factors are controlled channel by channel, while the resulting invariant factors remain intrinsic OPNs objects. As the constant-matrix shadow of this Smith theory, every rectangular OPNs matrix is equivalent, under invertible OPNs row and column transformations, to a rectangular diagonal matrix whose only possible diagonal entries are $1_{\mathcal{O}}, \qquad e_p, \qquad e_q, \qquad 0_{\mathcal{O}}.$ The resulting matrix theory gives canonical descriptions of kernels, images, cokernels, linear systems, determinant theory, the Cayley--Hamilton theorem, minimal polynomial theory, rational canonical classification, regular spectral rectangles, similarity classification, and symmetric matrix inertia.It is also shown that the four-pivot geometry is irreducible: the rank pair controls the stable block-sum monoid, the factorization preorder, orbit dimensions, orbit closures, and the associated determinantal ideal chain. In the polynomial theory, projective rank pairs yield explicit nonfree projective modules over $\mathcal{O}_{\varphi}[t]$.