Finiteness and Hierarchical Structure of Open Systems: A Computable Emergence Criterion via the Wuliang Method
- 1W3 Lab
Abstract
Open systems, systems exchanging matter, energy, and information with their environment, span biology, engineering, society, and physics, yet remain among the hardest objects to formalize: far from equilibrium, irreducible, and studied only case by case. Emergence, the central phenomenon of open systems, has lacked a verifiable mathematical criterion since Anderson. This paper advances a counter-intuitive chain of propositions: openness enforces hierarchical structure; hierarchical structure enforces bounded information; and boundedness makes formalization possible. From this we give three structural axioms, openness (O1), stratification (L1), and finiteness (F1), and define emergence as the irreducible excess over assembly prediction during level transitions, computed layer by layer via the homotopy cofiber homology of comparison maps. The framework is named the Wuliang Method: its core construction is the three-layer structure of the constructive operator (evidence layer, normalized aggregation layer, complex modulation layer), its core invariant is the emergence quantity E, and its core property is self-duality (forward generation constructs; reverse generation monitors). The criterion is computable (polynomial time), constructible (explicit three-layer operator), and falsifiable: the same signature appears in two mechanism-free domains, the Internet (TCP/IP stack) and biology (gene regulatory networks). Closed systems are proven to be a degenerate zero-flux special case; static (flux-rank) and dynamic (behavioral-flow) criteria are orthogonal, yielding a four-quadrant classification covering explosive emergence and zero-emergence perpetual motion. To our knowledge this is the first computable emergence criterion that takes the homotopy cofiber as the axis of comparison and assembly prediction as the baseline. This is Paper I (foundations) of a three-part series; Paper II applies the framework to AI architecture deconstruction, Paper III to AI safety. Revision V4.5 (2026-10-07): the corresponding-author block now includes the W3 LAB paper-collection website (https://7fba2d25df5d4627a2297a066267371a.app.workbuddy.host); no technical content changes. The typeset DOCX/PDF files remain the V4.4/V4.3 production versions; the two Markdown files are updated to V4.5.
Related
- Self-Gravitation of Mode Quanta in a Causal Resonator: One-Loop Finiteness in Linearized Quantum Gravity and the Emergence of the Dark-Energy Scale
- Hierarchical Prototype Emergence in Modern Hopfield Models
- Hierarchical emergence of network bursting in a four-cell central pattern generator model
- Order, Structure and Reality: Hierarchical Shell Structure of Integer Partitions
- Probabilistic, Structure-Intrinsic Clustering with an Hierarchical Embedding (PSICHE) in Time and Space Applied In Particle Physics Jet Reconstruction
BibTeX
@misc{wu2026finiteness,
title = {{Finiteness and Hierarchical Structure of Open Systems: A Computable Emergence Criterion via the Wuliang Method}},
author = {Wuliang Wu},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = sep,
doi = {10.5281/zenodo.22682819},
url = {https://doi.org/10.5281/zenodo.22682819},
note = {CC BY-NC},
}
RIS
TY - UNPB
TI - Finiteness and Hierarchical Structure of Open Systems: A Computable Emergence Criterion via the Wuliang Method
AU - Wuliang Wu
PY - 2026
DA - 2026/09/10
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.22682819
UR - https://doi.org/10.5281/zenodo.22682819
AB - Open systems, systems exchanging matter, energy, and information with their environment, span biology, engineering, society, and physics, yet remain among the hardest objects to formalize: far from equilibrium, irreducible, and studied only case by case. Emergence, the central phenomenon of open systems, has lacked a verifiable mathematical criterion since Anderson. This paper advances a counter-intuitive chain of propositions: openness enforces hierarchical structure; hierarchical structure enforces bounded information; and boundedness makes formalization possible. From this we give three structural axioms, openness (O1), stratification (L1), and finiteness (F1), and define emergence as the irreducible excess over assembly prediction during level transitions, computed layer by layer via the homotopy cofiber homology of comparison maps. The framework is named the Wuliang Method: its core construction is the three-layer structure of the constructive operator (evidence layer, normalized aggregation layer, complex modulation layer), its core invariant is the emergence quantity E, and its core property is self-duality (forward generation constructs; reverse generation monitors). The criterion is computable (polynomial time), constructible (explicit three-layer operator), and falsifiable: the same signature appears in two mechanism-free domains, the Internet (TCP/IP stack) and biology (gene regulatory networks). Closed systems are proven to be a degenerate zero-flux special case; static (flux-rank) and dynamic (behavioral-flow) criteria are orthogonal, yielding a four-quadrant classification covering explosive emergence and zero-emergence perpetual motion. To our knowledge this is the first computable emergence criterion that takes the homotopy cofiber as the axis of comparison and assembly prediction as the baseline. This is Paper I (foundations) of a three-part series; Paper II applies the framework to AI architecture deconstruction, Paper III to AI safety. Revision V4.5 (2026-10-07): the corresponding-author block now includes the W3 LAB paper-collection website (https://7fba2d25df5d4627a2297a066267371a.app.workbuddy.host); no technical content changes. The typeset DOCX/PDF files remain the V4.4/V4.3 production versions; the two Markdown files are updated to V4.5.
ER -