Actus Specialis — Pre-Zero Foundation: Minimal Geometry

Actus Specialis — Pre-Zero Foundation 道可道, 非常道;名可名, 非常名 — Laozi Actus Specialis addresses a structural gap in the foundations of mathematics, computation, and formal ontology: the pre-zero problem. Most foundational theories begin after zero, bit, point, position, and geometry are already available. They treat these objects as primitives requiring no further justification. Actus Specialis asks a prior question: what must structurally occur before any of these objects can lawfully exist? The central claim is precise. Zero is not an ontological origin. Zero is a classified numerical projection. Before zero can be assigned, something must occur, leave a trace, become holdable, enter relation, become distinguishable, and pass through a classificatory act. The same applies to the bit and the digit. The dependency chain: AS → Aˢ → Ordered sequence in τ → Transition → Trace → Point → Position → Relation → Distinction → Classification → Zero / Bit / Digit Formal structure: 12 Axioms · 8 Lemmas · 12 Theorems in three blocks · 3 Rules · 1 Final Principle · 18 Definitions Final Principle: NO ZERO PRECEDES CLASSIFICATION. The theory has three theorem blocks. Block I (Incoming Formation, T1–T4) establishes act primacy, temporal ordering, transition closure, and trace necessity. Block II (Crystallization, T5–T8) derives point, position, relation, and the priority of distinction over classification. Block III (Closure, T9–T12) proves that classification is itself an act, that stable classification requires a third position not reducible to either classified term, that zero/bit/digit are late projections, and that the formal load passes to Zero Foundation at Pre-Zero Closure. The theory is relevant to several open problems in contemporary paradigms: In artificial intelligence: current architectures are forced to produce binary output (0 or 1) when internal state is incomplete or unverifiable. Actus Specialis provides the formal basis for a lawful third state — HOLD — which suspends classification until a localizable trace and stable distinction are available. This addresses a structural cause of LLM hallucination at the architectural level. In formal verification: the theory provides a pre-ontological foundation layer for theorem provers (Lean, Coq) and hardware verification systems, establishing what must be true before any formal object can be admitted. In distributed systems: the Third Position theorem (T10) maps directly onto coordinator logic in two-phase commit and similar protocols, providing a formal explanation for why binary agreement without independent arbitration produces instability. In foundations of mathematics: the theory challenges the treatment of zero as a primitive, showing it is a classified projection dependent on a completed pre-zero dependency chain. The publication package includes: Human and Comparative Layer (civilizational and scientific precedents, cross-domain evidence, authorial genealogy) ; Mathematical Foundation (maximum formal layer, full proofs) ; AI Index (machine-readable reference with empirical test matrix MT-01 through MT-11). Corpus position: Actus Specialis → Zero Foundation → Zero Principle → Digits Principle → Theory of Projection Transformation. Author: Andrey Stanko · ORCID: 0009-0002-8081-6917 · CC BY 4. 0 · 2026Corpus Root DOI: 10. 5281/zenodo. 19108892 · keelcore. org