S-AI-ARP: Artificial Reasoning Processes

Background. Artificial intelligence systems that process language, perform inference, or operate in complex environments share a structural deficiency that has received insufficient formal attention: they lack any intrinsic representation of their own internal cognitive condition. Classical architectures transform inputs into outputs through deterministic or probabilistic pipelines in which neither the confidence of the current state, nor its degree of internal conflict, nor the stability of its trajectory, nor the energy it consumes, plays any governing role in the reasoning process itself. The system does not know whether it is confused, whether it is converging, whether two of its internal components are in contradiction, or whether the cognitive resources it is mobilizing are proportionate to the difficulty of the problem at hand. This absence — not of knowledge about the world, but of knowledge about the system’s own cognitive condition — constitutes the foundational limitation that the present article addresses. The term cognitive, which appears throughout the S-AI corpus with systematic and deliberate precision, designates exactly this domain: the internal states, magnitudes, and dynamics that govern how a system reasons, not merely what it computes. Methods. This article introduces S-AI-ARP (Artificial Reasoning Processes) a formal theory of cognitive reasoning within the Sparse Artificial Intelligence paradigm. S-AI-ARP, the seventh architectural instantiation of S-AI, formalizes the reasoning process as a dynamical convergence from an initial high-entropy cognitive state toward a stable canonical attractor, governed by a hormonal orchestration system and certified by a formally verifiable termination criterion. The framework is organized around three interlocking theoretical pillars. The first is the theory of cognitive magnitudes and cognitive states: six formally defined scalar functions — confidence, entropy, stability, cognitive energy, convergence, and conflict — computed over the internal activity of the system and assembled into a state vector H(t) = [h₁(t), …, hₙ(t)] ∈ [0,1]ⁿ that captures the complete cognitive condition of the system at each instant. The second pillar is the hormonal dynamics of reasoning: a system of seven artificial hormones — Confidexine, Inhibitine, Curiosine, Energexine, Alertine, Clarifine, and Confusionin — whose coupled stochastic differential equations govern the evolution of the cognitive state, and whose stability is analyzed by Lyapunov’s direct method under a formally verifiable deployability condition. The third pillar is formal certification: the Quintuple Equivalence, which establishes that dynamical stability of the hormonal field, monotonic reduction of reasoning entropy, symbolic coherence of committed outputs, Turing-decidability of the reasoning process, and structural convergence of the global cognitive state are not five independent engineering objectives but five equivalent expressions of the single thermodynamic invariant of parsimonious artificial cognition. Results. Six formal theorems are established. ARP.1 proves the global asymptotic stability of the cognitive hormonal subsystem under the deployability condition, guaranteeing that the hormonal field governing reasoning converges to its equilibrium from any initial cognitive state. ARP.2, the Cognitive Entropic Contraction Theorem, establishes the formal equivalence V̇_Lyap(H) ≤ 0 ⟺ Ḣ_P(t) ≤ 0, instantiating the doctrinal invariant in the full generality of the artificial reasoning process. ARP.3 provides a finite-time termination guarantee E[t*] ≤ t_H + t_S < ∞, with all components analytically computable prior to deployment. ARP.4 establishes the Cognitive Decidability-Convergence Equivalence: H(t) → H* ⟺ the ACR metacognitive regime is Turing-decidable. ARP.5 proves the global asymptotic stability of the multi-scale cognitive hierarchy, establishing that the aggregation of local agent cognitive states into a global cognitive state preserves stability under the deployability condition. ARP.6 proves the global asymptotic stability of the complete hybrid system integrating the linguistic, cognitive, hormonal, and certification layers. Conclusions. S-AI-ARP defines a new class of artificial reasoning systems — Cognitive Canonical Reasoning Systems (CCRS) — simultaneously satisfying five formally specified properties: cognitive canonicality, logical validity, exponential convergence, cognitive parsimony, and guaranteed finite-time termination. These five properties are five expressions of the single thermodynamic invariant that governs all parsimonious artificial cognition — the Quintuple Equivalence of Artificial Reasoning: V̇(H) ≤ 0 ⟺ Ṡ(P) ≤ 0 ⟺ s_cons(y(t)) ≥ 0 ⟺ Artificial reasoning is decidable ⟺ ‖ΔH_global‖ → 0 S-AI-ARP constitutes the transversal conceptual framework of the S-AI corpus: the article that explains, formalizes, and unifies the cognitive dimension present in every instantiation of the paradigm, from S-AI-Recursive to S-AI-PTR.