From Undefined to Universe: The Logical Necessity of Self-Referential Paradox as the Ontological Origin of Existence — Jia, Baolong's PR-ER-LE Framework: A Single-Premise Answer to the Leibniz Question

Description / Abstract We present a constructive proof that a single premise — "There exists at least one undefined syntactic position" — logically entails the emergence of a self-sustaining dynamical universe. Within the JiaBaolong Universal Axiom System, we formalize the derivation chain: Undefined → Self-Reference → Paradox (PR) → Resolution (LE) → Structure (ER) → New Undefined → Loop Through four theorems, we prove each step is a logical necessity: 1. Self-reference is ineliminable (Theorem 3.1): In any system of syntactic positions with referential capacity, self-reference cannot be forbidden without self-contradiction. 2. Self-referential Undefined produces paradox (Theorem 3.3): An undefined position referencing itself necessarily generates a Liar-type paradox, isomorphic to Gödel's self-referential constructions. 3. Paradox necessitates resolution (Theorem 3.5): In a non-degenerate system, paradox must be resolved (LE), as collapse and stasis are excluded. 4. Resolution generates structure (Theorem 3.7): Each resolution creates new relations (ER), which are again subject to self-reference — forming a self-sustaining PR ⇔ ER ⇔ LE closed loop. This reduces the JiaBaolong two-axiom system to a single premise: "There exists at least one undefined syntactic position." We argue this constitutes the most parsimonious answer to the Leibniz Question ("Why is there something rather than nothing?"): Nothing is not a stable state — it is a self-detonating seed from which existence necessarily emerges. We further discuss implications across three domains: · Physics: The PR-ER-LE framework maps to quantum mechanics (LE ↔ Born rule), general relativity (ER ↔ spacetime geometry), and the four fundamental forces (topological effects of graph dynamics). · Computation: The Mediumless Computation theorems prove that PR-ER-LE dynamics are fundamentally incompatible with deterministic hardware (Anti-GPU, Anti-Multi-Core, Anti-Cache), resolving the Simulation Hypothesis negatively. · Consciousness: Qualia are identified as the macroscopic PR loop's first-person encounter with its own computational friction, providing a precise necessary condition for subjective experience. Computational verification via the Trinity Graph Process (TGP) demonstrates — through 500+ simulations at scales up to N = 80,000 — that the complete evolutionary chain VOID → TOPOLOGY → CHAOS → MATTER → CHEMISTRY → REPLICATION → SELECTION emerges spontaneously from PR ⇔ ER ⇔ LE dynamics, with universal constants (α_SOC = 1.4700 ± 0.0001, R = 3.1226 ± 0.0048) invariant across 8 independent universe ensembles. Keywords · Undefined · Self-referential paradox · JiaBaolong Axiom · PR-ER-LE framework · Computational ontology · Leibniz Question · Lazy Evaluation · Emergence · Trinity Graph Process · Mediumless Computation · Consciousness · Computational cosmology · Philosophy of mathematics · Simulation Hypothesis