We propose an effective framework in which geometric structure arises from the interplay between continuous and discrete dynamics. The model combines a continuous flow generated by a Serre-type operator with a discrete mutation acting on a two-component state space, and incorporates an Ising order parameter to capture phase alignment. The central object is a defect functional δ (Ψ) =∥eΔτDSΨ−MΨ∥2, () = \| e^ DS - M \|², δ (Ψ) =eΔτDSΨ−MΨ2, which measures the mismatch between the continuous and discrete evolutions. This defect plays a dual role: it acts as an effective energy density and as a regulator in a functional renormalization group (FRG) flow. States that minimise the defect define dynamically selected stable sectors, while generic configurations remain unstable under the flow. To resolve degeneracy in the symmetric construction, we introduce a two-branch structure with distinct effective flow rates. This asymmetric formulation separates dynamically stable modes and induces nontrivial structure in the defect landscape. The mismatch between branch dynamics generates an anisotropic response, which we interpret as an effective curvature. An effective metric is defined by the Hessian of the defect field, gij=∂i∂jδ, g₈₉ = ᵢ ⱼ, gij=∂i∂jδ, providing a geometric description of the state space. Within this interpretation, alignment between the continuous flow and discrete mutation corresponds to defect-free configurations, while deviations generate curvature-like effects in the induced geometry. Rather than postulating spacetime as a background, the present framework suggests that geometric structure can emerge as an effective description of dynamical consistency conditions between different modes of evolution. While the model is formulated at a minimal level, it captures a mechanism by which stable sectors, energy-like quantities, and geometric responses arise from a unified defect-based dynamics.
Jeong Min Yeon (Thu,) studied this question.