Escher's drawings, Bach's canons, Gödel's theorems, the self-attention of transformers — four
seemingly unrelated domains share a single three-stage formal model:
reduction → self-reference → emergence.
The claim that emergence occurs once self-reference crosses a critical complexity, and that this
structure is formally isomorphic across domains, is the
Threshold Recursive Isomorphism Hypothesis (TRIH).
109kthesis characters · 70 references
One way of thinking
Reduction → self-reference → emergence
A picture that contains itself, recursively
In Drawing Hands two hands sketch each other, and in Print Gallery a figure inside the picture is looking at the very picture that contains him. The reduced units of line and plane are recombined into a structure that draws itself. Beginning from the symmetry rules of wallpaper groups and hyperbolic distance, an irreducible visual infinity emerges.
Palindrome in time, endless rising, a theme meeting itself
The Crab Canon is a palindrome in time, and the Endlessly Rising Canon (BWV 1079) climbs forever. In Contrapunctus XIV the name B-A-C-H is embedded as pitches, and in the six-voice Ricercar the King's theme meets itself across six voices. From the reduced units of counterpoint an auditory infinity emerges — a formal parallel to the causal self-attention mask of the Transformer.
A formal system naming itself through Gödel numbering
The sentence G states of itself, "this statement is unprovable." Through Gödel numbering the reduced system of arithmetic points back at itself, and an undecidable statement emerges. Turing's halting problem is the computational version of the same self-application, and the second incompleteness theorem shows that self-reference is not the same as self-completeness.
Self-attention lets tokens reference themselves within the sequence
Self-attention makes each token attend to itself and its past within the sequence. A triple line of evidence — induction heads (Olsson 2022), monosemantic features (Templeton 2024) and attribution graphs (Anthropic 2025) — directly observes the self-referential circuit, and beyond a threshold scale emergent abilities (Wei 2022) appear. This is the decisive evidence that lifts what was once mere analogy into a claim demonstrated at the circuit level.
§4 Ricercar ↔ Transformer is treated as a formal parallel — the stronger claim of isomorphism is reserved for the visual and logical domains.
Isomorphic figures
Three domains, one self-referential structure
§4 Bach · §5 Gödel · §6 Transformer — self-made conceptual figures (SVG) of each domain's self-referential structure.
§4 · Bach · Crab CanonOne voice forward, the other in retrograde — a crab's walk across the axis of symmetry.
§5 · Gödel · numberingA string of symbols is encoded as prime exponents into a single number G, which loops back to name its own proposition.
§6 · Transformer · induction headA two-layer circuit where attention looks back to the B after a prior A to predict the next token — self-attention's self-reference.
Publications & public records
Verifiable scholarly records
2026
Self-Reference and the Possibility of Mind — Proposing TRIH
Self-Reference and the Possibility of Mind
유승훈 (Seunghun Yu) · Published on Zenodo (Jun 2026) · preparing journal submissions (Leonardo, Cognitive Science)
Across four domains — the visual (Escher), the auditory (Bach), the logical (Gödel) and the computational (Transformer) — self-referential structure is treated under a single three-stage formal model: reduction → self-reference → emergence. The Threshold Recursive Isomorphism Hypothesis (TRIH) holds that emergence occurs once self-reference crosses a critical complexity, and that its structure is formally isomorphic across domains. In the Transformer's self-attention, a triple line of circuit-level evidence — induction heads, monosemantic features and attribution graphs — makes the self-referential circuit directly observable. This lifts a discourse once confined to analogy into a claim demonstrable at the circuit level.