We tried to formalize what emergence actually is — not as a vague "more than the sum of its parts" but as a specific mathematical construction. The answer we landed on: every entity is a coalgebra for a Moore machine functor (internal state + signal exchange), and constitution of higher-level entities from lower-level interactions is the categorical colimit.
The construction does three things we didn't expect. First, it produces a sustainability criterion with three failure modes (fragmentation, condensation, rate-exceeds-modulation) that appear across ecology, neuroscience, and cancer biology — and a null model comparison shows the food web result is structural, not a degree-sequence artifact. Second, when you instantiate the framework at quantum grain with the empirical observation that signals carry phase, perspectivalism (no privileged observer) forces U(1) phase invariance, the category becomes C-enriched, and you get a distributive amplitude sum with the algebraic form of a path integral. Third, the framework's ontology turns out to be structurally identical to relational quantum mechanics — which we didn't set out to recover.
The quantum alignment is incomplete — three steps are proved, one critical step (discrete sum to Feynman path integral) is open, and the paper says so explicitly. 13 pressure points are named. The framework is open for challenge.
The construction does three things we didn't expect. First, it produces a sustainability criterion with three failure modes (fragmentation, condensation, rate-exceeds-modulation) that appear across ecology, neuroscience, and cancer biology — and a null model comparison shows the food web result is structural, not a degree-sequence artifact. Second, when you instantiate the framework at quantum grain with the empirical observation that signals carry phase, perspectivalism (no privileged observer) forces U(1) phase invariance, the category becomes C-enriched, and you get a distributive amplitude sum with the algebraic form of a path integral. Third, the framework's ontology turns out to be structurally identical to relational quantum mechanics — which we didn't set out to recover.
The quantum alignment is incomplete — three steps are proved, one critical step (discrete sum to Feynman path integral) is open, and the paper says so explicitly. 13 pressure points are named. The framework is open for challenge.
Paper (PDF + LaTeX), analysis code, and data: https://github.com/helyn-research/constitution-as-colimit