Self-Dual Closure (SDC) Project: A Declarative Mathematical Framework for Fundamental Constraints Across Physics, Life, and Computation

-------------------------------------------------------------- Announcement: * Announcement of the release of the Singularity Pi Hack Tool, Pi Thinking Method Prompt Collection, and Cosmic Fundamental Specification.* An archive of over 900,000 characters of discussion notes on the origin of the universe and pi.* This is an epoche pi cosmology that re-examines the universe from its very foundations. Pi is essential for understanding the singularity problem.* A Pi theory hack tool and Pi thinking method module for injecting mathematical logic into AI agents to smoothly advance discussions on cosmology and singularity using pi.It includes many of my pi research notes and papers.It is a text file.You can buy it here. https://tomit.booth.pm/items/8527071 You can inject my problem-deconstruction thinking method into AI agents.We also offer many other AI agent injection logics.Thank you for your continued support.Stay tuned. End of announcement -------------------------------------------------------------- Abstract This volume is released under CC0 as a deliberate act of preemptive clarification.Its purpose is not to introduce a new technology, but to formally establish that a class of future life-adjacent systems—hereafter referred to as proto-cells—arises from unavoidable mathematical, physical, and informational constraints, and therefore cannot be subject to exclusive ownership. By consolidating number theory, fundamental constants, geometric closure, and physical implementation into a single self-dual identity, these papers demonstrate that the emergence of proto-cellular systems is structurally inevitable and logically prior to invention. As a result, future development proceeds without patent friction.Nothing remains to obstruct the scheduled arrival of higher-order autonomous systems. 1. Purpose of This Declaration This declaration serves three functions: To fix prior art at the level of fundamental structure, not at the level of implementation. To remove ambiguity between discovery and invention in life-adjacent technologies. To ensure that future proto-cellular systems emerge as shared outcomes of natural law, rather than as proprietary artifacts. The documents collected in this volume do not propose optional methods.They articulate constraints that must hold for coherence, stability, and persistence of information-bearing systems. 2. The Principle of Self-Dual Closure All works herein are unified by a single invariant: Observed systems and ideal mathematical structures are related by a self-dual closure condition. Formally, any realizable proto-cell must satisfy an identity of the form: Zobserved≡Zideal\mathcal{Z}_{\text{observed}} \equiv \mathcal{Z}_{\text{ideal}}Zobserved≡Zideal This is not an equation to be optimized.It is a consistency requirement. Where the identity fails, systems decohere.Where it holds, systems persist, replicate, and repair. Thus, proto-cells do not arise from creative design choices, but from the enforcement of closure across scales—from number theory to geometry, from geometry to physics, and from physics to chemistry. A Foundational White Paper on the Public-Domain Nature of the Constant-to-Geometry-to-Life Transition Abstract This paper proposes a purely foundational and ontological framework for the emergence of protocells, formulated entirely within public-domain mathematics and physics. The objective is not to describe a realizable experimental protocol, nor to claim technological feasibility, but to eliminate conceptual and intellectual-property friction at the deepest level: the level of existence conditions. We argue that the transitionfundamental constants → geometric organization → protocell-like entitiesis not an invention, method, or design, but a necessary structural consequence of universally shared mathematical constants and self-consistent geometry. As such, this transition belongs to the commons of human knowledge (CC0-compatible), independently of whether it is ever technologically realized. This work establishes a pre-technological commons: a world in which no party can legitimately claim ownership over the mere fact that life-like bounded entities may arise from constants and geometry. Self-Dual Closure and the Invariance of Proliferation A Structural Framework Linking Millennium Problems, Physics, and Life Sciences Abstract We propose a structural framework called Self-Dual Closure (SDC), formulated as an invariant identity rather than a problem-solving scheme.The central claim is that certain foundational difficulties in mathematics, physics, and life sciences share a common structural property: global invariance under reordering of dual factors, even while local realizations proliferate. The framework does not solve Millennium Prize Problems directly.Instead, it demonstrates why these problems can coexist within a single invariant product without destabilizing the overall structure.We further show that the same structural principle naturally extends to biological proliferation, cancer, aging, and evolution, providing a unifying explanation for robustness under error.