R-EGIC & From Logic to Learning: The One-Bit Existence Guard and the Guard-Stamped Learning Architecture for AI Safety

<b>This work now includes </b><b>R-EGIC and</b><b> a prospective analysis of Guard-Stamped Learning and its future extensions, demonstrating the practical application of the core logical theory.</b>This manuscript details a formal framework that connects a minimalist foundational logic to a practical architecture for formally-grounded AI safety. It posits that system integrity is best achieved by enforcing existence as a verifiable property: an object may only be used in a computation after a formal proof of its validity has been "stamped" into a trusted ledger.The theoretical foundation is the <b>One-Bit Existence Guard (OBEG)</b>, a minimal logic built on three layers: the <b>Existence Guard Calculus (EGC)</b> for operational semantics, <b>Guarded First-Order Logic (GFOL)</b> for a sound logical syntax, and <b>Guard-Fused Predication (GFFOL)</b> to ensure all claims operate on explicitly validated entities. This structure provides a robust logical basis for "Existence-Centric Computing", a paradigm focused on enforcing invariants at the point of object invocation.To validate this abstract logic against a high-stakes problem, we present its premier application: <b>Guard-Stamped Learning (GSL)</b>, a proposed architecture for online AI safety. GSL treats each gradient update (<code>Δθ</code>) as an object requiring formal certification against a safety constitution before use. The core of this document explores a <b>prospective analysis</b> of GSL's feasibility, which is shown to be contingent on hypothesized efficiency gains from very recent or anticipated breakthroughs in computational complexity (e.g., Cook &amp; Mertz, 2024; Williams, 2025). We demonstrate how these results could make per-step formal verification computationally viable.Finally, this manuscript is extended with the chapter <b>"Beyond Δθ: Guard-Stamped Artefacts and a Roadmap to Modular Safety"</b> This new analysis rigorously explores a roadmap for applying the GSL toolkit to other agent components, outlining the specific constraints and open research problems on the path toward a comprehensively guarded modular agent.This integrated work is intended for researchers in foundational logic, theoretical computer science, and critically, those in AI safety, formal methods, and control theory seeking novel, formally-grounded approaches to building trustworthy intelligent systems.