This note provides a methodological clarification regarding the interpretation of the numerical results presented in “Coarse SRM–PPA Transport: Numerical Realization, Emergent Geometry, and Structural Limits. ” 1. Scope of the Einstein-like Correspondence The preprint reports a proportionality between geometry-like and stress-like quantities of the form ⟨Gij⟩ ∼ κ⟨Tij⟩ + b It is clarified that this relation holds at the level of coarse-grained observables, specifically: spatial averages over the lattice, time-sampled snapshots, regression across configurations. This correspondence is therefore kinematic and emergent, not imposed. 2. Absence of Local Constitutive Law A subsequent field-level analysis was performed to test the stronger condition: ⟨Gij⟩∼κ⟨Tij⟩+b The result yields: R² ≈ 0. 05 indicating no statistically significant local correlation. This demonstrates that: the system does not admit a pointwise stress–geometry relation, the observed Einstein-like behavior is not a local field equation, geometry-like quantities depend on nonlocal and higher-order structure. 3. Consistency with Raychaudhuri Results This finding is fully consistent with the previously reported failure of Raychaudhuri-type closure: R² ∈ 0. 025, 0. 079 Both results indicate that: the system contains geometric ingredients (expansion, shear, vorticity), but lacks a self-consistent dynamical closure mechanism. 4. Refined Interpretation The system is therefore best described as operating in a: "Pre-Raychaudhuri regime" characterized by: emergence of structured fields, existence of coarse geometric correspondences, absence of local constitutive relations, absence of closed dynamical equations. 5. Conceptual Implication The results establish a clear separation between: "Coarse emergent phenomenology and local dynamical closure. " This implies that additional microscopic structure (referred to as motif algebra) is required to: generate invariant couplings, produce local field equations, enable Raychaudhuri-type dynamics. 6. Status of the Results No claims of the original preprint are invalidated by this clarification. Instead, the numerical program now more precisely identifies: what can emerge from coarse transport, and what fundamentally cannot. This refinement strengthens the internal consistency and delineates the next stage of the research program.
Dauren Sarsenov (Fri,) studied this question.