Deterministic AST-Gated Objective Functions for Singularity-Free Automatic Differentiation and Safe Reinforcement Learning (RICIS-III v8.3)

Gradient-based optimization and deep reinforcement learning (RL) frequently encounter catastrophic failures at singular points of computation graphs. In classical IEEE-754 hardware floating-point pipelines, indeterminate states (0/0, 0 × ∞) trigger undefined numerical representations (NaN, Inf) or hardware traps (SIGFPE), while reward hacking exploits policy divergence across unregularized poles. Traditional workarounds — such as heuristic gradient clipping, artificial denominator offsets (ε-padding), or penalty discriminator networks — distort the underlying optimization landscape and incur dynamic branching overheads. This paper introduces the formal architecture of RICIS-III (v8.3), a deterministic framework that resolves computational singularities via Abstract Syntax Tree (AST) provenance tracking and an algebraic Laurent ε-monomial calculus. By enforcing Absolute Continuity (L₀) and Identity (L₁), singular states are represented as typed zeros (0_F = F·ε) and indexed infinities (inf_F = F·ε⁻¹), enabling static O(1) reduction without dynamic limit evaluation. We formalize a protected objective functional, Goal_P(θ), protected by three multiplicative structural gates: zero-order balance (Ψ_Zero), divergence balance (Ψ_Inf), and canonical AST isomorphism (δ_AST). If an agent attempts metric exploitation or gradient hacking, the objective collapses to an identity-failure zero (0_IdentityFailure), natively locking parameter updates (∇_θ Goal_P = 0⃗) without CPU branching. Furthermore, we specify an active barrier-repulsion recovery protocol to prevent policy freezing. Core foundational identities are formally verified via an executable Lean 4 kernel.