Derived Gamma Geometry II: Stable $\infty$-Categories of Gamma-Modules, Derived Monoidal Structures, and Obstructions to Binary Shadows

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Contribution

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Abstract

Let \(\T\) be a commutative ternary \(\Gm\)-semiring in the sense of the triadic, \(\Gm\)-parametrized multiplication \(\{a,b,c\}_{\gamma}\). Building on the affine \(\Gm\)-spectrum \(\SpecG(\T)\), the structure sheaf, and the equivalence between \(\Gm\)-modules and quasi-coherent \(\Gm\)-sheaves on affine \(\Gm\)-schemes, we construct and organize the derived formalism at the level of stable \(\infty\)-categories. Our first contribution is a technically explicit construction of a stable \(\infty\)-category \(\Dinfty(\T,\Gm)\) enhancing the unbounded derived category of \(\Gm\)-modules, obtained by dg-nerve and \(\infty\)-localization of chain complexes. We further explain the derived monoidal structure induced by the ternary \(\Gm\)-tensor product and the corresponding internal \(\RHom\), under standard exactness/projectivity hypotheses. Our second contribution is an obstruction theory to \emph{binary reduction}: we formalize the nonexistence of any conservative ``binary module shadow'' compatible with the cubic localization calculus intrinsic to ternary \(\Gm\)-semirings. In particular, any attempt to represent the triadic \(\Gm\)-action by binary scalars forces \(\Gm\)-mode data to be absorbed into the scalars, hence ceases to be a genuine reduction. Finally, we give a detailed affine derived equivalence between derived quasi-coherent \(\Gm\)-sheaves on \(X=\SpecG(\T)\) and \(\Dinfty(\T,\Gm)\), and we include worked examples illustrating the cubic localization relation and its derived consequences.