Determinantal Diophantine Linear Algebra: Foundations and Lorentz--Fermat Factorization

Abstract

We propose \emph{determinantal Diophantine linear algebra} as a systematic framework for integer and rational solutions of prescribed-determinant equations inside affine linear families of matrices. A datum consists of an integral affine matrix pencil\[ \mathcal A(\mathbf x)=A_0+x_1A_1+\cdots+x_mA_m\]and a level $N$, and the central object is the integral fibre\[ X_{\mathcal A,N}(\Z)=\{\mathbf x\in\Z^m:\det \mathcal A(\mathbf x)=N\}.\]The framework organises several classical subjects---determinantal representations, Smith normal form, norm-form equations, integral points on affine varieties, local lifting, and spectral arithmetic---around one common linear-algebraic object. We establish basic equivalences and closure operations, a differential and $p$-adic lifting theory, Smith-stratum and lattice-index classifications, spectral divisor theorems, and a regular-representation theorem identifying norm equations as a distinguished commutative subclass. We also introduce integral frame landing, identify the global landing transformations with the projective rational matrix group $\GL_n^+(\Q)/\Q_{>0}I_n$, prove the existence of a unique primitive landing level and its exact $n$th-power landing spectrum, and classify complete systems of partial integral measurements through Smith normal form. The determinant can replace one linear measurement under a rational unimodular residual hypothesis. We further derive higher-rank diagonal factorisation, integral spectral gluing for involution pencils, a complete classification of scalar linear clocks on the Lorentz--Fermat orbit, and exact local sieve densities. We then develop an exact defect theory: landing defects are first defined as classes on matrix tori, their metric representatives become closest-vector problems, and minimum-norm corrections are obtained from explicit normal and KKT systems. The determinant residual admits a sensitivity-normalised defect formula, while tangent and orbit projections separate removable geometric error from transverse arithmetic error. Finally, unrestricted zero-level solvability is algorithmically undecidable, since every integral polynomial admits an affine determinantal representation. Thus useful algorithms must exploit structured subclasses rather than the general formalism alone. As a principal example, we develop \emph{Lorentz--Fermat factorization}. The pencil\[ \mathcal F(a,b)=\begin{pmatrix}a&b\\ b&a\end{pmatrix}\]has determinant $a^2-b^2$ and fixed eigenvectors. For an odd semiprime $N=pq$, the nontrivial integral point\[ a=\frac{p+q}{2},\qquad b=\frac{q-p}{2}\]recovers the factors as the eigenvalues $a-b$ and $a+b$. After division by $\sqrt N$, the same matrix is a Lorentz boost, conjugate to the area-preserving stretch $\operatorname{diag}(e^u,e^{-u})$. We prove the exact factor-pair correspondence, count the integral points, derive the Fermat iteration count, and show polynomial-time equivalence between solving this structured determinantal problem and integer factorization. Within the landing formalism, Fermat factorisation is the nearest-point search for the integral landing point associated with the closest nontrivial factor pair in the logarithmic metric; every fixed primitive scalar clock is either a trial-divisor clock or a generalized difference-of-squares clock for a fixed multiple of $N$.