The Process
Page 1 of Constitution as Colimit — early-stage research program.
Every physical theory in history has bottomed out at some kind of stuff. Atoms. Fields. Strings. Spacetime itself. There is always something at the base — a substance that everything else is made of.
This framework does not bottom out. It claims there is nothing at the base. What we call reality is not made of anything. It is constituted — parts relating under shared rules to form composites, which themselves become parts of further composites. And what those parts are made of is more constitution. All the way down.
Three axioms define this process. They are not descriptions of something that exists independently. They are what constituting means.
Axiom 1: Distinction
The process differentiates. Something can be different from something else. There is a 0 and a 1.
This sounds trivial. It is not. Before this axiom, not even zero is meaningful — a 0 without a 1 to contrast it with is not a 0. Distinction is not a property that things have. It is the process by which there are things at all.
The distinction is binary — yes or no, this or that — and the reason is physical. Two oscillating systems either synchronize or they do not. A signal either propagates between two points or it does not. The channel is open or closed. Coupled oscillators can lock into phase or drift past each other (Strogatz, 2000; Pikovsky et al., 2001). The signal itself is continuous — it has amplitude, phase, frequency — and real coupled oscillators exhibit a spectrum of behaviors: full locking, partial synchronization, intermittent synchronization, frequency locking without phase locking. The framework’s claim is that for constitutional purposes — whether a stable structural distinction forms — the outcome is effectively binary: either a persistent, stable coupling forms or it does not. This is a modeling choice, not a physical fact about oscillator dynamics. The framework treats it as sufficient for the purpose of generating stable distinctions.
Why binary and not ternary? The framework’s argument: pairwise signal sharing is inherently yes/no. Even apparent three-valued properties — spin comes in three values, color charge comes in three varieties — decompose into trees of binary forks. Three color charges are three non-null states from two independent binary forks (see The Forces). Higher-arity structures at the quantum grain decompose into binary trees. (Three coupled oscillators CAN produce three-phase states — this is a known result in nonlinear dynamics. The framework’s claim is specifically about the pairwise synchronization mechanism that produces constitutional distinctions, not about oscillator dynamics in general.)
Axiom 2: Consistency
The process constrains. Not all distinctions are compatible. Some combinations persist; others do not.
Together, the first two axioms produce a grow-and-prune mechanism. Axiom 1 grows binary trees — every interaction is a fork, every fork produces two branches, and recursive application produces a tree that grows without bound. Every distinction that can be made is made. Axiom 2 prunes: incompatible branches dissolve. What survives — the branches that passed the consistency filter — is what we measure.
Spin values, color charges, particle types: all are surviving branches of this process. The specific numbers at each scale (three spin values, three color charges, four forces) are not derived from axioms in the sense of a mathematical proof from first principles. They are the result of binary branching minus pruning. The axioms determine the mechanism. The mechanism determines the numbers.
Axiom 3: Self-reference
The process applies to itself. Distinctions can be made about distinctions. Relations can relate to relations. Compositions can be composed. The structure folds back on itself, generating depth and complexity from its own operations.
This is the axiom that generates the fractal. Without it, the process produces flat, non-hierarchical configurations — a single layer of distinctions with no capacity to build on themselves. With it, each layer of constitution becomes the material for the next. Taken to its logical conclusion, self-reference implies the process can become aware of itself — but that extrapolation is philosophical, not developed here.
The postulate: perspectivalism
The three axioms define what the process is. A fourth principle — a postulate, not an axiom — defines what observation looks like from inside it.
No constitutional product of the process occupies a privileged perspective. Every observer is constituted by the same three axioms operating at the same scale. Differences between observers are constitutional — different branches, different histories — not fundamental.
This postulate resonates with principles physics already treats as fundamental — though the mappings are conceptual, not formal:
- No preferred perspective resonates with the principle of relativity (no preferred frame). The equivalence principle is a more specific claim about gravity, not a direct consequence of perspectivalism.
- No preferred basis in state space resonates with gauge invariance (no preferred description of forces). Gauge invariance is specifically about local phase redundancy in field theories — a more specific concept than basis-independence.
- Frame-independent probability resonates with Lorentz covariance (laws are the same from every vantage point).
These are structural parallels, not identities. The framework’s perspectivalism is broader and less specific than any one of these physics principles.
Perspectivalism is not derived from the axioms. Axiom 3 makes it natural — if the process is everything and observers are products of the process, no observer sits outside with a special view — but one can construct self-referential systems with preferred frames. The honest status: perspectivalism is strongly motivated by the ontology but functions as an independent physical input.
It is also load-bearing. Perspectivalism constrains the signal space to the complex numbers at the quantum scale (given the additional inputs of finite-dimensional division algebras and commutativity — see the formal section below), constrains the symmetry structure that produces gauge invariance, and provides the frame-independence required for the Born rule’s uniqueness. The three axioms are the engine. Perspectivalism is the constraint that shapes the output into quantum mechanics.
The “process of what?” objection
If there is no substrate, what is the process a process of?
The answer is self-reference. The process does not need a substrate beneath it. It constitutes, and what it constitutes is more constitution. Self-reference closes the loop. The process is not a process of anything external to itself.
This is not mysticism. The logical structure resembles a cellular automaton: rules generate patterns, and the patterns are nothing other than the rules applied recursively. But the analogy is imperfect — a cellular automaton’s rules still require something to apply to (a lattice of cells). The framework’s stronger claim is that no such substrate is needed. Whether this is coherent or merely describes an abstract structure awaiting instantiation is an open philosophical question.
Scope
These are not physics axioms. They are axioms of constitution itself. Every system at every scale — physical, chemical, biological, social — is constituted from parts relating under shared rules. The process that produces quarks from sub-quark structure is the same process that produces cells from molecules, ecosystems from organisms, and societies from individuals. What changes between scales is the signal space and the measurement apparatus, not the process.
The quantum scale is where the framework makes its most precise contact with existing science — the formal apparatus produces genuine mathematical results that connect to quantum mechanics. But the framework’s scope is not physics. It is constitution at every scale. A food web and a proton are both colimits — composites formed from parts relating under shared rules. The axioms produce both.
There are no constitutions — only the labels we make for them.
Convergence
In 1929, Alfred North Whitehead published Process and Reality — a philosophical system built on process rather than substance. His “Category of the Ultimate” resonates with the three axioms, though the mapping requires interpretive charity on at least the self-reference term. His “Many” (the irreducible plurality of entities) is distinction. His “Creativity” (the many become one and are increased by one) is consistency — and the colimit construction is this formula exactly. His recursive application is self-reference.
The convergence from completely different starting points — 1929 process philosophy without mathematical formalism, and 2026 category-theoretic physics without Whitehead as input — is evidence that the shape is natural to the problem.
But this framework goes further. Whitehead’s “actual occasions” were still things — minimal entities that experience their world. This framework removes even that residual substance. There are no occasions. There is only the process of occasioning. Whitehead kept one noun at the bottom of his ontology. This framework keeps none.
The formal version
The three axioms map directly to the three primitives of category theory:
| Axiom | Category theory | What it provides |
|---|---|---|
| Distinction | Objects | Things that can be different from each other |
| Consistency | Morphisms | Constrained relationships between objects |
| Self-reference | Functors | Structure-preserving maps between categories — the process applied to its own outputs |
Each entity at every scale is modeled as a coalgebra for a parameterized Moore machine functor. The signal space varies by scale — electromagnetic at the quantum grain, chemical at the cellular grain, behavioral at the social grain. The formation of higher-scale entities from lower-scale interactions is modeled as the categorical colimit: a universal construction that captures how parts constitute a whole.
For connected diagrams, the colimit is lossy — internal structure is quotient-ed away. (Disconnected diagrams produce coproducts, which preserve information.) Self-similar — colimits compose associatively across scales. And universal — the category has all colimits (Theorem 1: cocompleteness, via Adamek & Rosicky 1994).
Perspectivalism proposed constrains the signal space to ℂ at the quantum grain. The argument: “no preferred direction” requires a continuous symmetry group, which requires dimension ≥ 2 over ℝ. The only non-trivial finite-dimensional field extension of ℝ is ℂ (fundamental theorem of algebra: odd-degree polynomials always have real roots, so no odd extensions exist; ℂ is algebraically closed, so no extensions beyond it exist). Frobenius’s theorem provides independent confirmation: the only finite-dimensional associative division algebras over ℝ are ℝ, ℂ, and the quaternions ℍ. Commutativity eliminates ℍ; perspectivalism eliminates ℝ. Only ℂ remains. Unstated assumptions: this argument requires two inputs not derived from the axioms: (1) the signal space must be a division algebra (every nonzero element has a multiplicative inverse — why should physical signals have multiplicative inverses?), and (2) the signal space must be commutative (why not quaternions?). Both are physically motivated but neither is formally entailed by perspectivalism.
Newman’s objection (1928): purely structural knowledge collapses to a trivial claim about cardinality. The framework’s answer: the constitutional relations are not arbitrary logical mappings but actual causal, dynamical relations with physical consequence. The objectivity test is convergence — the same criterion science uses for any empirical claim. Multiple independent perspectives triangulating to the same underlying pattern is what makes structure real.
Full proofs, theorem statements, and the enrichment proof supplement are available in the GitHub repository.
Sources
Coupled oscillator synchronization -- Strogatz, S.H. (2000), Nonlinear Dynamics and Chaos, Westview Press; Pikovsky, A., Rosenblum, M. & Kurths, J. (2001), Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press. Division algebras -- Frobenius, G. (1878), “Über lineare Substitutionen und bilineare Formen,” J. Reine Angew. Math. 84, 1–63. Field extensions of ℝ -- follows from the fundamental theorem of algebra; see Artin, M. (1991), Algebra, Prentice Hall. Newman’s objection -- Newman, M.H.A. (1928), “Mr. Russell’s ‘Causal Theory of Perception,’” Mind 37(146), 137–148. Process philosophy -- Whitehead, A.N. (1929), Process and Reality, Macmillan. Coalgebra theory -- Rutten, J.J.M.M. (2000), “Universal coalgebra: a theory of systems,” Theoretical Computer Science 249(1), 3–80. Cocompleteness -- Adamék, J. & Rosický, J. (1994), Locally Presentable and Accessible Categories, Cambridge University Press, Theorem 2.75.