The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers for which the inflation propensity remains so unpredictable it could be used to generate reliable proof-of-work algorithms for the cryptocurrency industry; it has so far resisted every attempt at linearizing its behavior. Here, we establish an ad hoc equivalent of modular arithmetics for Collatz sequences based on five arithmetic rules that we prove apply to the entire Collatz dynamical system and for which the iterations exactly define the full basin of attractions leading to any odd number. We further simulate these rules to gain insight into their quiver geometry and computational properties and observe that they linearize the proof of convergence of the full rows of the binary tree over odd numbers in their natural order, a result which, along with the full description of the basin of any odd number, has never been achieved before. We then provide two theoretical programs to explain why the five rules linearize Collatz convergence, one specifically dependent upon the Axiom of Choice and one on Peano arithmetic.

Introduced by Gunter Pauli, the Blue Economy, namely bio-inspired industrial ecology or self-profitable circular economy, is a remarkable example of the way the knowledge flow can fundamentally alter micro, meso and macroeconomics, and be converted into cash flow. Its reception is also a case of limited rationality in management and economics, and of resistance to change in general. Here I simplify the Blue Economy to the following equation: waste + knowledge = asset. I then explore the implications of this equation in terms of venture capitalism (microeconomics) accounting (micro-mesoeconomics) and in terms of GDP (macroeconomics). I finally discuss its possible impact on politico-economic decision-making and its clear continuity with the knowledge economy. One fertile question thus arises: what could be the micro-mesomacro-economic stygmergies of the Blue Economy?