A forced mixing invariant for the Rogers-Ramanujan modular matrices

Definitions and mathematical statement

The core of source problem 1.4 asks whether diagonalizing transformation matrices of vector-valued modular forms is mathematically informative. Its Rogers--Ramanujan example has diagonal $T$ for $\tau\mapsto\tau+1$ and nondiagonal $S$ for $\tau\mapsto-1/\tau$. The extracted word "digi-talization" is interpreted as diagonalization from this context. No general criterion for "interesting" is specified.

The attached aid gives a precise weight-zero, two-dimensional formulation. Let $\rho:\operatorname{PSL}_2({\mathbb{Z}})\to\operatorname{GL}_2({\mathbb{C}})$ be irreducible, with matrices $S,T$ satisfying $S^2=(ST)^3=I$. In a $T$-eigenbasis write $T=\operatorname{diag}(a,b)$ with $a\ne b$. It states \[ S_{12}S_{21}=\frac{a^2-ab+b^2}{(a-b)^2},\qquad \det(ST-TS)=a^2-ab+b^2. \] The determinant is conjugacy invariant; a nonzero value certifies failure of simultaneous diagonalization. For the Rogers--Ramanujan exponents $-1/60$ and $11/60$, so $a=e^{-2\pi i/60}$ and $b=e^{22\pi i/60}$, it reports \[ S_{12}S_{21}=\frac{5-\sqrt5}{10}>0. \] In a symmetric normalization, put $\varphi=(1+\sqrt5)/2$. One compatible golden-ratio matrix and its Lee--Yang fusion diagonalization are \[ S=\frac1{\sqrt{\varphi+2}}\begin{pmatrix}-\varphi&1\\1&\varphi\end{pmatrix},\qquad S^{-1}\begin{pmatrix}0&1\\1&1\end{pmatrix}S =\operatorname{diag}(-\varphi^{-1},\varphi). \] Basis order and normalization affect individual entries, whereas the determinant certificate does not. These specific results do not classify transformation matrices in arbitrary dimensions or weights.

The source extraction also includes the introduction of the following section, "Bloch group method problems," rather than only problem 1.4. Retaining it as contextual material, for positive definite symmetric $A\in M_r({\mathbb{Q}})$, $B\in{\mathbb{Q}}^r$, $C\in{\mathbb{Q}}$, it introduces the Nahm sum \[ f_{A,B,C}(\tau)=\sum_{\mathbf n\in{\mathbb{Z}}_{\ge0}^r}\frac{q^{\frac12\mathbf n^t A\mathbf n+B^t\mathbf n+C}}{(q)_{n_1}\cdots(q)_{n_r}},\quad (q)_m=\prod_{j=1}^m(1-q^j),\quad(q)_0=1. \] This is a conventional reconstruction of an OCR-corrupted displayed formula. The historical text attributes to Zagier a rank-one classification with seven choices, including $(2,0,-1/60)$ and $(2,1,11/60)$, using asymptotics as $q\to1$. It describes Nahm's proposed modularity--Bloch-torsion relation and notes that several choices of $B,C$ can occur for one $A$, motivated by dilogarithms. The phrase "image of $A$ in the Bloch group" is not defined here; this appended text is not a precise, independently verified general conjecture statement.

Short English statement

The Rogers--Ramanujan example has an intrinsic mixing certificate determined by its two $T$-eigenvalues. A symmetric $S$ normalization also diagonalizes a fusion matrix. The extraction's appended Nahm-sum introduction is retained separately as historical context rather than folded into the matrix question.

Sources

• Local UnsolvedMath registry, permanent ID $20000731$, source code catalogue entry; canonical aim-analytic-number-theory-notes.json, notes[94].
• AIM, "Mock modular forms in combinatorics and arithmetic geometry," March 8--12, 2010, problem 1.4, source-reported PDF page 1; list edited by Sharon Anne Garthwaite: https://aimath.org/WWN/mockmodular/mockmodular.pdf. The extraction includes a page header and the next section's introduction. No PDF inspection is asserted for this draft.
• The appended source text names Werner Nahm, Conformal Field Theory and Torsion Elements of the Bloch Group, described as available on the arXiv, without an identifier; and Don Zagier, "The Dilogarithm Function," in Frontiers in number theory, physics, and geometry II. These references concern that contextual section.
• Attached research attempt 1: partial result, recorded run status unknown, rigor proved, novelty confidence low; registry status partially solved, classification PARTIAL-PROGRESS. This machine-generated aid is not peer reviewed and requires independent verification before citation.