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Work in progress. These tutorials are part of an ongoing investigation into multicriticality. The skeleton framework and system assignments are provisional hypotheses, not established results.

The Four Mathematical Skeletons

Independent mathematical structures that recur across physical systems

Across eleven scientific domains, the same four mathematical frameworks appear independently. Each was discovered multiple times by communities that don't read each other's journals. Each produces two emergent propertiesProperties that arise from the interaction of components but don't exist in any individual component. The whole is different from the sum of its parts. at different depths: a structural property ("has") and a functional property ("does"). Together they decompose the multi-critical profileThe combination of independent phase transitions present in a single system. Different systems have different profiles — some have two transitions, others three or four. of any physical system.

How They Work Together

Individual skeletons produce useful properties. When two or more are present in the same physical system, qualitatively new properties emerge that none has alone. The pairwise interactions produce self-organized criticality (S2×S3), topological quantum computation (S1×S4), and Turing universalityA system that can simulate any computation given enough time and space. Conway's Game of Life, the brain at criticality, and random Boolean networks all have this property. at the edge of chaos (S2×S4). The S1×S2 edge is made of something older than any of them: broken symmetry. A symmetry breaks — that is the dynamics — and the order parameter it leaves behind has a shape whose defects are the topology. On that reading it is not a fifth structure but the substance of one edge.

Self-organized criticality Topological quantum computation Edge of chaos S3 Closure S2 Dynamics S1 Topology S4 Computation All four present simultaneously → emergence exceeds the sum

Skeleton 1: Topology

Protection → Fault-tolerant computation
Braid groupsMathematical structures that describe how strands cross over and under each other. The crossing pattern is the information — not the material of the strands., knot invariantsNumbers or polynomials that stay the same no matter how you deform a knot without cutting it. They classify knots by their topology., topological protection. Three strands, two crossings, infinite consequences. Alternating braid crossings create Fibonacci growth, giving the golden ratioφ = 1.618... The growth rate of Fibonacci sequences. Appears wherever alternating three-strand braiding occurs. A binary tracer: either φ shows up or the topology isn't present. φ = 1.618... as a binary tracer: it either shows up in the system or it doesn't.
7 independent communities — from pure mathematics (1860s) to active matter (2010s)

Skeleton 2: Dynamics (Criticality)

Spontaneity → Coherent transport
Phase transitions, power lawsMathematical patterns where small events are common and large events are rare, with no characteristic scale. The signature of systems at a critical point., scale invariance, renormalization groupA mathematical technique that reveals how a system's behavior changes across scales. The tool that made criticality quantitative and earned a Nobel Prize.. The anchor skeleton — the best-understood of the four, documented in the FACETSA cross-domain survey cataloguing how the mathematics of criticality was independently developed across roughly a dozen scientific fields. Preprint, arXiv:2601.22389, with R. Macomber — under review, not yet peer-reviewed. paper (arXiv:2601.22389). Criticality runs through every system in the multi-critical tutorials.
12 independent fields — see the six system tutorials

Skeleton 3: Closure

Self-sustenance → Self-creation
Catalytic closureA network of processes where every catalyst is produced by the network itself. No external catalyst needed. The mathematical definition of self-sustenance., RAF theoryReflexively Autocatalytic and Food-generated set theory. A rigorous graph-theoretic framework for testing whether a network is self-sustaining. Developed by Hordijk and Steel., autopoiesisSelf-creation. A system that produces its own boundary and components. Coined by Maturana and Varela (1971) to define what makes a system 'living.'. A campfire sustains itself. A cell creates itself. No tracer molecule — closure is a structural property of the network, not a numerical signature. Detection requires the full wiring diagram.
8 formalizations across 6 independent discoveries, 4 countries, 50 years

Skeleton 4: Computation

Universality → Coordination
Distributed computation, cellular automataGrid-based systems where each cell follows simple rules based on its neighbors' states. Despite trivial rules, they can produce arbitrarily complex behavior., edge of chaosThe boundary between ordered and chaotic behavior. Where computation concentrates — ordered enough to remember, chaotic enough to explore.. Simple local rules producing global information processing without central control. The edge of chaos IS the critical point — this is where Skeleton 4 meets Skeleton 2.
8 fields — from Turing (1936) to reservoir computing (2000s)

The Emergent Stack

Nine properties, layered. Each requires the ones below it. The order matters — later layers only emerge because earlier layers exist.

Layers 1–4 emerge from closure and dynamics bootstrapping each other. Layers 5–8 emerge from topology and computation bootstrapping each other. Ground anchors everything in physical matter.

All nine present simultaneously = conditions for the Flat G In matter. Right now. S3 Self-sustenance S2 Spontaneity S3 Self-creation S2 Coherent transport S1 Protection S4 Universality S4 Coordination S1 Computation S3 ⊗ S2 S1 ⊗ S4

Skeleton Matrix →

Interactive radar chart. 16 systems × 4 skeletons × 2 depths. Select systems to overlay their profiles and explore which combinations come closest to all nine emergent properties.

See also

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