Domain Convergence Research

Investigating cross-domain bridges between eleven scientific fields

The central claim: a small set of mathematical structures — criticality, topology, autocatalysis, and computation — keep reappearing, independently, across very different sciences. This collection maps those recurrences and tests which are genuine structural bridges rather than coincidences.

This collection documents a systematic investigation into structural relationships between criticalityThe study of phase transitions — sudden qualitative changes in a system's behavior at threshold values. Power laws, scale invariance, and universality are its signatures., topologyMathematical properties preserved under continuous deformation. Braiding, knot invariants, and topological protection. Information stored in topology survives local noise., autocatalysisA network of reactions where every catalyst is produced by the network itself. Self-sustaining systems, from campfires to cells. Formalized as RAF (Reflexively Autocatalytic Food-generated) theory., computation, and seven other scientific domains. The work combines interactive tutorials with computational experiments, tracking both positive findings and null results.

Interactive Tutorials

Domain Convergence

Eleven domains, five clusters, seven bridges, and the emergent property stack. The central map of how criticality, topology, computation, autocatalysis, and related fields connect.

Criticality Bridges

Six physical systems where critical phenomena couple domains. Detailed analysis of how power-law correlations, universality, and scale invariance create cross-domain bridges.

Physical Systems Chart

Interactive matrix mapping physical systems against domain clusters. Filter by system, domain, or bridge type to explore the evidence landscape.

Anyon Fusion Algebra

Fibonacci anyon fusion rules, braiding, and topological quantum computation. Interactive fusion tree visualization with step-by-step algebraic operations.

Autocatalytic Closure for the Working Mathematician

RAF sets via greatest fixed point algorithm. From group theory intuitions through the formal definition to proved emergence — a mathematician's path to autocatalytic closure.

Where the Topology Lives

Two layers of topological structure: the model layer (chain-level operations) and the architecture layer (training-derived topology). Analysis of what survives tokenization.

The Four Mathematical Skeletons

Topology, dynamics, closure, computation — four independent mathematical frameworks that recur across eleven scientific domains. Each was independently discovered by multiple communities. Includes interactive tutorials for all four skeletons (S1–S4) plus the radar-chart matrix for cross-system comparison.

Tutorial Series

Multi-Critical Systems

Six physical systems where multiple independent phase transitions converge. Glass, active matter, cardiac, protein, brain, tokamak — each taught from plain language to its multi-critical structure.

Critical Lines and Multicritical Points

Three interactive tutorials on how a system can be critical at a point, along a line, or only nearly: Kosterlitz–Thouless, eight models from Ashkin–Teller to walking Potts, and the techniques that tell them apart. Live Monte Carlo toys, equation traces, section quizzes, retakeable final tests, a combined test and an exportable scoreboard.

Jumps, Slowing, and the Line They Sit On

A growing autocatalytic chemistry absorbs most additions quietly, then one addition makes it jump. Is that jump a point on a critical line? Two linked interactive pages: an early-warning signal that watches recovery time and variance before a jump, and correctly stays silent for a structural jump; and a procedure that brackets the jump in two controls and walks the line of jumps to the point where it vanishes.

Poised-State Candidate Systems

Twelve candidate systems evaluated against the five-feature poised-state framework. Animated radar charts, interactive deep dives, KT transition hypothesis, and cross-system pattern analysis.

The Poised State and Its Critics

Has anyone seriously studied the poised state? A guided reading path through thirty years of claim and rebuttal — the edge of chaos and its deflation, the criticality hypothesis and the latent-variable objection, RAF theory, and the quantum claim nobody engaged. Includes an interactive latent-variable machine.

Computational Research

Anyon Family Sweep

Which non-abelian anyon families support complex systems? Systematic evaluation of SU(2)_k, SU(3)_k, SO(3)_k, D(S₃), and Haagerup categories for self-catalysis, RAF closure, and computational universality. Finding: sharp phase transition at k=2 to k=3.

Braiding Phase Space Search

NKS enumeration of 1,364 braiding rules on Fibonacci anyons, classified by Wolfram's four complexity classes. Finding: unitary evolution on finite Hilbert space cannot produce genuine chaos—all "Class IV" candidates are quasi-periodic.

Lattice Braiding Search

Spatially extended braiding dynamics on 60-anyon chains. 1,570 simulations across 12 spatial patterns. Finding: product-state approximation cannot capture genuine complexity—it lives in quantum entanglement.

Key Findings

Self-catalysisA reaction where a product catalyzes its own production. In anyon physics: when a particle type appears in its own fusion products (N^j_jj ≥ 1). and RAF closureA network satisfying two conditions: every reaction is catalyzed internally, and every reactant can be built from simple 'food' molecules. The mathematical test for self-sustenance. are generic for k≥3 in SU(2)_k Chern-Simons theoryA topological quantum field theory that describes anyon behavior in 2D systems. The level k determines which anyon types exist and how they fuse. Higher k = more complex anyons., not special to Fibonacci anyonsThe simplest non-abelian anyons capable of universal quantum computation. They appear at Chern-Simons level k=3 and have quantum dimension φ (the golden ratio).. Verified by explicit construction of fusion rules for k=2 through k=6.
Sharp phase transition at k=2 to k=3: self-catalysis and computational universality share the same algebraic root (non-trivial self-coupling N^j_jj ≥ 1). Below threshold, neither property holds; at or above, both hold.
Fibonacci anyons sit at the edge of chaosThe boundary between ordered and chaotic behavior where computation is maximized. For anyon fusion algebras, Fibonacci sits at this boundary — minimal self-catalysis, minimal universality. for fusion algebras: minimal self-catalysis, minimal universality, golden ratio dimension growth. Analogous to K=2 in Boolean networks.
No genuine Class IV behavior in braiding phase space: unitary evolution on finite Hilbert space is linear and norm-preserving, giving Lyapunov exponentA measure of how fast nearby trajectories diverge. Positive = chaos (trajectories separate exponentially). Zero = no chaos (trajectories stay together). Exactly zero for unitary quantum evolution. exactly zero. The 360 "Class IV" candidates (n=3) are quasi-periodic with irrational eigenvalue phases.
The three senses of “poised” have been studied separately and have opposite outcomes — the edge-of-chaos wing deflated, biological criticality contested, the quantum claim unengaged — and a 77-paper literature survey found no source studying the conjunction as one object, under any name. They share no order parameter, no observable and no test, so they cannot be cited as mutually supporting. See The Poised State and Its Critics.
Bridge mechanism taxonomy: approximately 4 of 7 identified bridges operate on rule-space structural identity rather than parameter coincidence or dynamical coupling.
The AC bridgeBridge letters: A = Topology, B = Criticality, C = Autocatalysis / Closure, D = Computation. A–C links Topology and Autocatalysis / Closure. (topology ↔ autocatalysis) has physical content beyond the algebraic structural identity. Anyon emergence from the substrate (Nayak et al. 2008) IS physical generation in Kauffman's sense — the food set is energy input, the generation is collective excitation, the closure is fusion algebra. FQHE systems are already AC demonstrators; the gap is formal recognition and cross-community publication, not finding a new system.

Proven (explicit construction)

Structural (well-supported)

Hypothesis (under investigation)

Null result (informative)