A Formal Mathematical Analysis Demonstrating Superior Intake Classification Accuracy in a Child-Level Geometric System Compared to Anglesey County Council v2

This deposit contains a technical paper presenting a strict mathematical comparison between two intake classification systems: (i) a canonical child-level geometric sorter, and (ii) the operational intake procedures used by Anglesey County Council. Using formal mapping functions, structural equivalence criteria, domain–codomain alignment analysis, and error quantification, the paper demonstrates that the child system satisfies determinism, consistency, transparency, and minimal structural error. In contrast, the council system exhibits non-deterministic behaviour, inconsistent mappings, opaque rule structures, and recurrent misclassification patterns. The core mathematical result is expressed through the Misclassification Operator: \[M(x) = “incorrect”\] for all inputs \( x \), capturing the observed behaviour that structurally valid cases are routinely assigned to incompatible categories. Included is an author’s note documenting that the initial intake category has remained uncorrected for over four months despite repeated clarification and no structural change in the case. This record is provided for archival, analytical, and evidentiary purposes. This paper is a direct sequel to *The Little System That Couldn’t*; readers seeking additional structural analysis and classification failure case studies are encouraged to consult that document. v2 (PDF + Interactive Visualizer): Includes the complete formal research paper alongside a self-contained WebGL/HTML Three.js visualizer demonstrating the topological field mapping and real-time state invariants. Disclaimer:This document does not require a disclaimer because it contains no conjecture, no opinion, and no interpretive claims. All statements are strictly structural, mathematical, and derived directly from observable classification behaviour. No part of the analysis relies on inference beyond the defined mapping functions and documented intake outcomes. Keywords and Subjects:intake classification systems, structural mapping analysis, deterministic classifiers, consistency evaluation, domain–codomain alignment, geometric analogy modelling, public-sector decision logic, misclassification operator, administrative error quantification, case-structure representation, operational transparency assessment, non-deterministic mapping behaviour, local authority intake procedures, formal systems analysis, mathematical modelling of classification error, evidentiary documentation of unresolved categorisation, four‑month intake misclassification persistence. Contact: For enquiries or research questions related to this work, email matthewcarlo.research@gmail.com