R.C.A.P. — Mathematical Formalization for Intelligent Control and Supervision

The R.C.A.P. method (Configurational Pattern Recognition for Pattern Actualization) represents the state of an AI-guided autonomous underwater vehicle through a geometry of four deformable figures. This document is the mathematical and operational appendix to the defensive publication (Zenodo, DOI 10.5281/zenodo.22057555). The essence of the method is the representation, not the mathematics. Each parameter is a vertex at a fixed position, whose side is colored by its danger level. The four figures are read in sequence: Figure 1 is the reference at submersion; Figure 2 shows which parameters have changed; Figure 3 shows by how much, with Figure 1 within it; Figure 4 shows how far one can go, that is, the point of no return. Placed alongside the mission log, they make the post-mission report readable within seconds. The report is recorded at an adaptive frequency, and every color change immediately generates a frame. During the mission the computation is organized into three levels: a geometric engine, a safety kernel and a diagnostic engine. The kernel decides a single VETO, at a fixed cost at every cycle, over the vital parameters alone. The VETO depends neither on Figure 4 nor on the mission AI, and it commands emergency surfacing when the residual time to the point of no return is no longer enough for the maneuver. Instruments are classified as vital (RCAP-I) or mission (RCAP-M) when the mission is planned. The mathematics is a reference implementation that can be improved or replaced. It includes parameter normalization, the time to the point of no return, the lexicographic minimization of Figure 4, the blindness window, validity states and RFC 8785 canonical serialization. The record contains, in Italian and English, the document, a numerical case study, the rcap v1.3 reference software (AGPLv3, 113 automated tests) and the symbol glossary in CSV format. The method is verified at the mathematical (A) and numerical (B) levels. No physical-experimental validation (C) is claimed.