The Inverse Problem of Perception - Disclusterization

First, we provide a rigorous justification of the AFI equation system against classical variational physics, isolating the seven structural commitments that distinguish a navigability theory from least-action mechanics and showing that the AFI extension is minimal — every removed term re-collapses to a known passive-physics result, and every added term carries a non-redundant operational role that physics alone cannot supply. Second, we introduce disclusterization: the inverse operator S−1 that recovers admissible possibilities, hidden options, and the perception map itself from observed cluster assignments and observed choices. Disclusterization is shown to be a well-posed Bayesian recovery problem under three identifiability conditions — cluster coverage, witness diversity, and behavioural injection — and to be ill-posed in their absence. The resulting operator gives any AFI controller three operational capabilities its forward-only formulation lacked: cluster auditability (detecting silent over-clustering and over-splitting), drift detection through parameter recovery, and unblocking — the targeted refinement of S to expose admissible system states that the current cluster lattice silently hides. We close with a high-quality physical-simulation protocol covering twelve canonical regimes — Fermat refraction, brachistochrone descent, Stefan moving-boundary phase change, Fourier diffusion, radiative coupling, bottleneck pedestrian flow, bacterial chemotaxis, diffusion-limited aggregation, Kepler 2-body orbit, site percolation at the critical threshold, the Bak–Tang–Wiesenfeld sandpile at self-organised criticality, and LQR-vs-AFI policy selection on a cart-pole — each instrumented with a pre-declared baseline and a falsifiability test, and with an explicit safety doctrine: enable by default, block only on hard structural, fire, electrical, sanitary, or legal predicates.

The honest classification is: AFI plus disclusterization remains a candidate general law of navigability and a strong universal descriptive framework. The contribution of this paper is not to claim universal predictive law — that status is still unproven — but to add the inverse machinery that makes the forward law auditable, falsifiable, and safe to deploy.

Keywords. navigability, possibility fields, perception clustering, inverse problems, identifiability, rate-distortion, Lambert-W, phase-change navigation, falsifiability.