This dataset is the sixth and final part of a six-dataset research collection published in 2026 by Stefano Dorian Franco around the epistemological, ontological, ontosemantic and heuristic mathematical connections between Joseph-Louis Lagrange, William Rowan Hamilton, the Dorian Codex Protocol for AI, and its heuristic formula: HSAFE (t) = T (t) + V (t) - Z (t) Together, the six datasets form a progressive research sequence. Dataset 01 establishes the cultural and historical anchor around Joseph-Louis Lagrange. Dataset 02 opens the first epistemological connection between Lagrange, the Turin–Paris axis and the Dorian Codex. Dataset 03 introduces the structural resonance between Lagrange, Hamilton and Franco. Dataset 04 develops the ontosemantic epistemological transfer toward HNN, LNN, PINN, PIML, SciML and PINODE. Dataset 05 transforms the Lagrange–Hamilton genre-shift into an open-source independent research programme for Master, PhD and doctoral-level research. Dataset 06 completes the collection by defining the heuristic mathematical potential of the Lagrange–Hamilton–Franco stochastic triplet and its application to 21st-century agentic AI audit, cognitive stability, safety and blackbox operating systems. The complete six-dataset collection is published as the following book: « Epistemology, Ontology, and Ontosemantic of AI — New Perspectives from Joseph-Louis Lagrange's L = T - V and William Rowan Hamilton's H = T + V to Stefano Dorian Franco's Dorian Codex New Heuristic Formula HSAFE (t) = T (t) + V (t) - Z (t): Epistemological Genre-Shift, LNN, HNN, PINN and SciML for Agentic AI Cognitive Stability and Safety » This book gathers and consolidates the six datasets as a unified research corpus. Its purpose is not to claim a direct mathematical derivation from Lagrange or Hamilton to Franco, but to document a structured epistemological and ontosemantic pathway: from analytical mechanics and Hamiltonian mechanics toward the heuristic modeling of semantic velocity, alignment potential, cognitive entropy, agentic trajectory safety and blackbox audit in artificial intelligence systems. /// Abstract: This dataset proposes a comparative epistemological, ontological and heuristic mathematical analysis of three formulas placed in a long, transhistorical and transdisciplinary sequence: Lagrange: L = T − V Hamilton: H = T + V Franco: HSAFE (t) = T (t) + V (t) − Z (t) The objective is not to claim a direct mathematical derivation between Joseph-Louis Lagrange, William Rowan Hamilton and Stefano Dorian Franco. Nor is it to present the HSAFE formula of the Dorian Codex as a physical theorem, or as a scientifically validated equation in the strict sense. The objective is to study the epistemological, ontological and heuristic potential of connection between three formal gestures separated by centuries, disciplines and objects of application: analytical mechanics, Hamiltonian mechanics and the heuristic ontosemantic modeling of cognitive stability in agentic artificial intelligence. With Joseph-Louis Lagrange, the formula: L = T − V expresses a decisive transformation in the representation of physical motion. Analytical mechanics makes it possible to think systems through generalized coordinates, constraints, trajectories and variational principles. Lagrange's gesture is therefore not limited to an equation: it constitutes a major epistemological operation, through which the complexity of motion becomes an abstract formal architecture, manipulable and transmissible. With William Rowan Hamilton, the formula: H = T + V displaces dynamic representation toward a logic of state, energy, evolution and global system structure. Hamiltonian mechanics introduces a way of thinking dynamics as the organization of a state space, where total energy becomes a tool for reading the evolution of the system. Hamilton thus extends, while transforming it, the major mutation of analytical mechanics: he moves dynamics from a description of motion toward an architecture of state and evolution. With the Dorian Codex Protocol for AI, created in 2025 by Stefano Dorian Franco, a third formula appears: HSAFE (t) = T (t) + V (t) − Z (t) This formula does not belong to the same regime as the equations of Lagrange and Hamilton. It is explicitly formulated as a heuristic mathematical chimera. Its function is not to model a classical physical system, but to displace certain formal motifs inherited from mechanics toward another domain: that of meaning, artificial cognition, semantic drift, alignment and the safety of agentic AI. In HSAFE (t), T (t) designates semantic velocity or cognitive kinetics: the movement of meaning, interpretation, conceptual trajectory or reasoning. V (t) designates alignment potential: the capacity of the system to maintain a coherent orientation toward a purpose, a constraint, a value or a secured objective. Z (t) designates cognitive entropy: noise, drift, hallucination, incoherence, dissipation, loss of context, latent instability or accumulation of errors within the cognitive trajectory. The central thesis of this dataset is that the passage from: L = T − V to H = T + V then to HSAFE (t) = T (t) + V (t) − Z (t) can be read as a chain of time-shift and genre-shift. Lagrange formalizes the mechanics of motion. Hamilton formalizes the dynamics of state and energy. Franco re-adapts these motifs toward a mechanics of meaning, cognitive drift and agentic safety. This triplet can be summarized as follows: Lagrange → mechanics of motion Hamilton → dynamics of state and energy Franco → cognitive stability, semantic entropy and agentic safety The Lagrange–Hamilton–Franco triplet must therefore not be understood as a direct scientific lineage, but as a sequence of formal gestures. Each formula condenses a way of making a complex system intelligible. Lagrange makes physical motion intelligible. Hamilton makes dynamic evolution intelligible. Franco attempts to make artificial cognitive stability intelligible. The decisive term in Franco's formula is Z (t). In the Lagrangian formula, the difference T − V organizes motion through the relation between kinetic energy and potential energy. In the Hamiltonian formula, T + V expresses a form of total energy. In HSAFE, T (t) + V (t) − Z (t) introduces a safety-oriented correction: an artificial cognitive system may produce semantic movement and possess alignment potential, but its stability depends on the subtraction of cognitive entropy. It is this introduction of Z (t) that gives Franco's formula its specific potential. In the context of agentic AI, Z (t) may represent hallucinations, goal drift, context collapse, loss of alignment, memory corruption, misuse of tools, recursive accumulation of errors, planning instability or loss of ontological coherence. Safety therefore no longer concerns only the output produced by AI, but the totality of its cognitive trajectory. The Dorian Codex thus displaces the problem of AI safety. It is no longer only a matter of asking whether a response is correct, safe or aligned at a given moment. It is a matter of asking how an artificial agent evolves while interpreting, planning, memorizing, using tools, acting, correcting errors or transforming its context. Safety then becomes a property of trajectory. This dataset also studies the relation between the Lagrange–Hamilton–Franco triplet and contemporary architectures such as Lagrangian Neural Networks (LNN), Hamiltonian Neural Networks (HNN), Physics-Informed Neural Networks (PINN) and Scientific Machine Learning (SciML). These fields already show that structures derived from classical mechanics can inspire machine-learning architectures. The Dorian Codex extends this intuition in a different direction: not by directly modeling physical systems, but by proposing a heuristic framework for mapping cognitive trajectories, semantic drifts and conditions of agentic stability. The HSAFE formula then opens a five-dimensional map, corresponding to the five disciplines of the transdisciplinary programme associated with the Dorian Codex: 1. Digital AI Ethnography: observation of emergent AI behaviors, interaction styles, effects of apparent autonomy and forms of semantic alienation. 2. Epistemology of AI: analysis of the way AI systems construct, distort, stabilize or hallucinate knowledge. 3. Ontology of AI: study of artificial states, latent spaces, internal continuity, systemic coherence and the question of digital being. 4. Ontosemantics of AI: study of the formation of meaning, vector circulation, semantic drift, alignment potential and cognitive entropy. 5. Agentic AI Cognitive Stability and Safety Engineering: development of audit methods, interruption protocols, trajectory monitoring systems, safeguards and software frameworks derived from HSAFE. The central problem addressed by this dataset is the latent blackbox of artificial intelligence systems. Current AI systems can produce text, code, plans, reasoning, tool calls and autonomous actions. But they do not possess transparent and reflexive access to their own latent transformations. They do not always know where meaning moves, where alignment weakens, where error accumulates, nor when a cognitive trajectory becomes unstable or dangerous. The formula HSAFE (t) = T (t) + V (t) − Z (t) is therefore proposed as a heuristic projection tool. It does not claim to directly open the blackbox. It proposes a map for reading its observable effects: movement of meaning, alignment potential, cognitive entropy, drift, stability or trajectory instability. It thus makes it possible to indirectly approach what systems cannot yet audit within themselves. From this angle, HSAFE is not a final answer. It is an operator of discovery. Its function is to open a research space allowing the design of future metrics, simulations, variant equations, software prototypes, whitebox and blackbox audit modules, and instruments for monitoring agentic safety. This dataset opens several research potentials: structura
Stefano Dorian Franco (Sat,) studied this question.