Outlier Cellular Automaton Causal Graphs as a Testbed for a Strand-Based
Ontology
J.Konstapel,11-7-2027.
The Architecture of Persistence: Strands, Knots, and the Algebra of Becoming
What endures? Not the shape, which shifts, but the continuity of relation—a thread of causality weaving through time, knotting here, loosening there, persisting not despite deformation but precisely through it. This is the quiet revelation of Konstapel’s reading of the Outlier cellular automaton: that what we call an object—a replicator, a self—may be a ghost cast by a more fundamental architecture of process.
The cellular automaton’s causal ancestry graph is a vast reticulum, tens of millions of nodes, each a momentary Place where influence converges and diverges. Most of its information is not in the branching—the moments of decision—but in the long, patient chains of transmission: degree-two vertices that merely pass the causal current onward. Contract these transmissive links, and the graph reveals its skeleton: strands connecting knots, an economy of persistence where most of the structure is conduit rather than event.
This is an inversion. Hintze and Bohm had mined the graph for clusters—spatially coherent assemblages that met a rigorous criterion for replication. They found distributed selfhood: replicators realised not by one bounded cluster but by several disjoint clusters acting in coordination. Konstapel proposes that the causal graph itself, read directly after degree-two contraction, renders this result not anomalous but expected. The replicator is the strand, the continuous causal lineage; clusters are mere knots, temporary concentrations of strand-activity that appear and dissolve while the strand continues. Distributed selfhood is self-knotting: one structure knotting itself in several places at once, its unity topological rather than spatial.
The deeper resonance lies in the formal traditions that haunt this argument. Spencer-Brown’s Laws of Form begins with a single act: drawing a distinction. No objects, only the mark that separates marked from unmarked, this from not-this. From this minimal gesture, a logic unfolds—an arithmetic of indication where the mark crossing itself returns to the void. Konstapel draws on this only for its starting point: that relation precedes object, that objects are downstream of distinctions. The Outlier graph, read as strands, is a concrete instance of this formalism: no cell-event has independent identity; each is defined entirely by its causal relations. The mark is the Place; the strand is the traced distinction.
Kauffman’s knot logic pushes further. In Kauffman’s work, the operation of negation—the mark—seen as both value and operator, generates recursively the algebra of Majorana fermions: particles that are their own antiparticles, that interact with themselves to produce or annihilate themselves. The mark, iterated, yields quaternions, braid group representations, the Fibonacci model of topological quantum computing. Topology—how a structure connects to itself and others—carries logical and physical content prior to any metric embedding. Konstapel’s “knots” are not Kauffman’s mathematical knots (closed loops) but borrow the gesture: persistence is topological, defined by connection-pattern, not shape.
Rowlands’ nilpotent Universal Rewrite System offers the most ambitious scaffolding. Here, physical structure emerges from iterated rewriting: a small set of algebraic operations—creation and conservation—acting on nothing generate, through self-similar bifurcation, the Dirac equation and the standard model. The rewrite system is scale-independent, fractal, its total sum always zero. Konstapel proposes that between Rowlands’ abstract rewrite step and his algebraic structures lies a missing intermediate level—Places and strands—and that the Outlier CA’s causal graph makes this level visible: a rewrite system whose causal skeleton can be read directly.
The sequence Konstapel sketches—distinction, rewrite, Place, strand, knot, oscillation, quaternion, Clifford algebra—is offered not as derivation but as structural analogy, a map of questions. The narrower claim is falsifiable: apply degree-two contraction to the published graph; if the reduced graph remains densely branching, the strand reading fails. The quaternionic extension is a further question: if strands carry independent phase relations, their interactions might compose by quaternion algebra. But this is a direction, not a result.
What, then, endures? The strand persists because causality persists. The replicator is not the shape but the relation that generates shape after shape after shape—a continuity of becoming. The Outlier automaton, in Yang’s original discovery and Hintze and Bohm’s rigorous reconstruction, demonstrates that self-replication need not be a property of bounded objects. It can be a property of distributed causal lineages. Konstapel’s contribution is to name these lineages strands and to show that a standard graph-theoretic operation exposes them directly. The poetry is in the inversion: the object is the derivative, the relation is the primitive. The mark persists. The knot tightens and loosens. The strand weaves on.
Annotated Reference List for the Inquiring Reader
Primary Source
Hintze, A., & Bohm, C. (2026). Rethinking self-replication: Detecting distributed selfhood in the Outlier cellular automaton. npj Complexity, 3, 11. https://doi.org/10.1038/s44260-026-00074-2
The empirical bedrock. Hintze and Bohm reconstruct the full causal ancestry graph of Yang’s Outlier cellular automaton—tens of millions of nodes—and apply a rigorous causal criterion for replication. Their central findings: replicators are often distributed across spatially disjoint clusters, and identity persists through substantial shape-change over hundreds of generations. The paper is exemplary in its methodological transparency and its willingness to let the data challenge intuitive notions of what a “replicator” is. For the reader wishing to understand the empirical context, this is the essential starting point.
Yang, [initial]. (2024). [Original report of repeating/self-replicating structures in the Outlier cellular automaton rule, as cited in Hintze & Bohm, 2026.]
The initial discovery, cited in Hintze and Bohm but not independently located at the time of Konstapel’s writing. The interested reader should verify this source directly; its bibliographic details remain to be confirmed.
The Calculus of Distinctions
Spencer-Brown, G. (1969). Laws of Form. George Allen and Unwin.
A cult classic that straddles mathematics and philosophy. Spencer-Brown begins with a single primitive operation—drawing a distinction—and derives an algebra of indications whose models include Boolean logic, propositional calculus, and (in the re-entrant forms) finite automata. The book is notoriously enigmatic; its central insight is that distinction, not object, is the foundational act of cognition and logic. The “mark” (a simple cross) denotes both the act of distinguishing and the state of being distinct. A1, the Law of Calling (marking twice is indistinguishable from marking once), and A2, the Law of Crossing (crossing from marked to unmarked and back returns to the original), generate the entire system. For the reader interested in how process precedes object, this is the ur-text .
Bricken, W. (2017). Distinction is Sufficient: Iconic and Symbolic Perspectives on Laws of Form. Cybernetics & Human Knowing, 24(3-4), 29-74.
A thoughtful exegesis that unpacks Spencer-Brown’s iconic algebra, showing how it challenges the dualism underlying Western rationality. Bricken emphasises that Laws of Form offers a postsymbolic territory where form condenses complexity, objects unite with processes, and absence is a primary conceptual tool. The paper is particularly useful for readers who find Spencer-Brown’s original opaque; it traces misunderstandings that arise when translating iconic forms into symbolic notation .
Knot Logic and Topological Foundations
Kauffman, L. H. (2016). Knot logic and topological quantum computing with Majorana fermions. In J. Chubb et al. (Eds.), Logic and Algebraic Structures in Quantum Computing (pp. 223-335). Cambridge University Press.
A magisterial survey connecting knot theory, quantum computing, and the algebra of fermions. Kauffman shows that the mark—negation, distinction—viewed recursively, generates the fusion algebra of Majorana fermions, the quaternions, and braid group representations. The paper explores knot-theoretic recoupling theory, unitary solutions to the Yang-Baxter equation, and the Fibonacci model for topological quantum computing. The core philosophical gesture: topology (how things connect) precedes metric geometry, and logical operations can be seen as particles that interact with themselves. Essential for understanding the conceptual background of Konstapel’s “knots” and “strands” .
Kauffman, L. H. (1995). Knot logic. In L. H. Kauffman (Ed.), Knots and Applications. World Scientific.
The earlier articulation of knot logic, exploring how topological invariants—knotting, linking, framing—carry logical and physical content independent of metric embedding. While Konstapel draws on the general gesture rather than Kauffman’s specific formalism, this work establishes the broader programme of topological foundations .
The Universal Rewrite System
Rowlands, P. (2007). Zero to Infinity: The Foundations of Physics. World Scientific.
A bold and ambitious work proposing that physics emerges from a universal computational rewrite system: a small set of algebraic operations acting on nothing generate, through iterated application, the Dirac equation, the standard model, and the structure of the vacuum. Rowlands employs nilpotent and Clifford-algebraic methods; the core claim is that physical structure is the outcome of rewriting, not a postulate. The book is dense and controversial; it aims to show that the foundations of physics can be derived from an information-theoretic process. Konstapel draws on this as a source of formal vocabulary and as a parallel: the Outlier CA’s causal graph is a concrete instance of a rewrite system made visible .
Rowlands, P. (2017). The universal rewrite system. Journal of Physics: Conference Series, 845(1), 012024. (and related papers)
Subsequent papers developing the universal rewrite system, showing its application to formal language theory, Turing machines, and quantum computing. The system is characterised by duality, self-similarity, scale-independence, and bifurcation at every stage. Rowlands and collaborators argue that the rewrite structure generates Clifford algebras, quaternions, and the algebraic foundations of quantum mechanics. For the reader interested in the computational ontology underlying Konstapel’s proposal, these papers provide the technical detail .
The Outlier CA and the Empirical Context
Langton, C. G. (1984). Self-reproduction in cellular automata. Physica D: Nonlinear Phenomena, 10(1-2), 135-144.
A classic precursor to the Outlier work. Langton demonstrated that self-replication could emerge in simple cellular automata, though his replicators were typically bounded structures with a clear spatial morphology. The Outlier rule, by contrast, produces distributed replicators—a significant departure from Langton’s paradigm. This paper provides historical context for understanding why Hintze and Bohm’s findings are so striking .