This work proposes a structural bridge between the analytic data contained in the Riemann zeta function and the geometric logic underlying Yau’s solution of the Calabi conjecture. The starting point is the classical decomposition of the zeta function into a pole term and a Mellin transform of the integer-lattice discrepancy. In this form, the zeta function already records a tension between continuous volume and discrete lattice counting. The Euler product further extracts the commutative prime-exponent semigroup, which may be interpreted as a toric exponent shadow. The central idea is not to claim a proof of the Riemann Hypothesis, nor to derive Calabi–Yau geometry directly from the zeta function. Rather, the paper interprets the zeta-derived lattice discrepancy as an arithmetic volume mismatch and asks how such mismatch could be lifted into a stable categorical and geometric register. To do this, the commutative toric exponent shadow is refined into a noncommutative prime-address monoid, where ordered prime words remember insertion histories forgotten by the Euler product. Each prime word is then assigned to an object in a Calabi–Yau register category and read through its Yoneda access profile. Yau’s theorem enters as a regularisation paradigm. In the Calabi conjecture, Ricci-flatness is obtained not by directly forcing curvature to vanish, but by solving a complex Monge–Ampère equation that matches volume forms inside a fixed Kähler class. The associated a priori estimates provide control over the potential, the metric, and higher derivatives. In the present framework, these estimates are reinterpreted as resolution bounds for the Yoneda lens: the C0C⁰C0 estimate stabilises the field of view, the C2C²C2 estimate prevents geometric collapse or blow-up, and higher estimates stabilise higher categorical operations. Thus the proposed framework should be read as a categorical-geometric lift of zeta-derived arithmetic data. The zeta function supplies the analytic skeleton: pole structure, lattice discrepancy, and prime exponent data. The noncommutative prime-address system, Yoneda lens, and Yau-regularised Calabi–Yau register provide a possible language for reading this skeleton through stable geometric resolution. The critical line and Ricci-flatness are not identified, but they are treated as parallel balance principles: the former as an analytic self-dual axis, the latter as a geometric zero-bias condition. This creates a precise research programme for studying whether zeta symmetry, toric exponent data, and categorical resolution can be organised within a common geometric framework. We first extract from the Euler product a commutative toric exponent semigroup. Its noncommutative word refinement is not forced by the zeta function, but is introduced as an enrichment that remembers prime insertion order. A Calabi–Yau register category is then proposed only after imposing additional geometric constraints, such as trivial canonical class or a suitable toric Calabi–Yau condition.
Jeong Min Yeon (Wed,) studied this question.