INVERSING | The Operator of Directional Transformation in FAR

ABSTRACT

Inversing is introduced as the core operator responsible for directional transformation within the architecture of Flourishing-Augmented Reasoning (FAR). While FAR-Lemmas and DI-Lemmas define the structural preconditions for upward and downward curvature, no mechanism has yet been formalised for shifting an appraisal system from VOID-aligned contraction to SPAN-aligned sufficiency. This paper positions Inversing as that mechanism: a structural operator that dissolves violations (V), restores omissions (O), rebalances overweighted elements (I), and corrects distortions (D), thereby reinstating the coherence, multiplicity, viability, and temporal continuity required for upward curvature. The paper demonstrates that Inversing does not depend on positive content or affective reframing; it operates at the structural level, restoring the lemma configuration necessary for the FAR Axioms to take hold. Using scenarios across human reasoning, human–AI interaction, and hybrid appraisal, the paper formalises Inversing as the minimal set of structural moves required to convert DI-Lemmas into FAR-Lemmas. Inversing is thus positioned as the central mechanism of FAR: the operator through which directionality becomes not only detectable but alterable. This paper provides the final operator required for the FAR Theorem, where curvature becomes predictable and transformable across substrate types.