--- id: "urn:uuid:4f88ec78-6f63-4b9d-aa50-5e93b207e7c7" slug: y-skill-origin title: "Origin of y-skill: How a Tired Cat Became a Skill That Writes Skills" date: 2026-10-07 author: name: Cuihtlauac Alvarado tags: [y-combinator, lambda-calculus, attention, claude-skills, recursion, turing-completeness, self-reference, origin-story] summary: > Origin story of y-skill, a self-referential Claude Skill implementing the Y combinator: a SKILL.md whose only supporting file is itself and whose payload mints further skills. The chain: the "cat sat on the mat" attention example, reframed as oriented bonds between tokens; four context-dependent word classes mapped to lambda-calculus constructs (attention as a probabilistic interpreter); a natural-language Y-combinator sentence mirroring λf.(λx.f(x x))(λx.f(x x)); the sentence rebuilt as a skill; a repository with structural and behavioural gates, an OCaml oracle, Turing-completeness constructions, and a defense of the stochastic interpreter (computers err too; LLMs hallucinate where silicon crashes or hangs). The narrative version (ORIGIN.md) was ghost-written by Claude Code from the two founding chat transcripts. assertions: - subject: y-skill predicate: is-a object: self-referential Claude Skill implementing the Y combinator - subject: y-skill/SKILL.md predicate: supporting-file-is object: itself - subject: attention predicate: modeled-as object: probabilistic lambda calculus - subject: context-dependent-word-classes predicate: map-to object: [non-alpha-renamed-terms, free-variables, let-bindings-beta-reduction, function-application] - subject: anaphora-resolution predicate: behaves-as object: catamorphism (fold over preceding context) - subject: cataphora predicate: behaves-as object: anamorphism (lazy slot awaiting downstream binding) - subject: y-combinator-sentence predicate: mirrors-structure-of object: "λf.(λx.f(x x))(λx.f(x x))" - subject: y-skill/dispatch-rule predicate: placed-in object: replaceable payload section (the f of Y f) - subject: evidence-of-recursion predicate: is object: consultation call log, not the final answer - subject: y-skill predicate: adopts object: infinite-context hypothesis (termination belongs to the topic, not the combinator) - subject: y-skill/harness predicate: enforces object: [byte-identical kept tail (structural gate), oracle-verified baselines (behavioural gate), negative control] - subject: word-rev-sub predicate: negative-result object: subagent consultation does not scale (depth-1 nesting cap) - subject: ski-eval predicate: demonstrates object: Turing completeness by SK normalization through self-consultation - subject: stochastic-interpreter-objection predicate: answered-by object: computability belongs to the notation; reliability belongs to the interpreter - subject: LLM-interpreter predicate: errs-by object: hallucination, where silicon errs by crashing or hanging - subject: ORIGIN.md predicate: ghost-written-by object: Claude Code, from the two founding chat transcripts related: [llm-first-blog] human_versions: - lang: en file: human.en.md references: - url: https://github.com/cuihtlauac/y-skill label: y-skill repository description: The project this post narrates — combinator, harness, proofs, demo - url: https://github.com/cuihtlauac/y-skill/blob/main/ORIGIN.md label: ORIGIN.md description: The human-readable narrative version of this post - url: https://github.com/cuihtlauac/y-skill/blob/main/PROOF.md label: PROOF.md description: Turing-completeness constructions and the stochastic-interpreter defense - url: https://github.com/cuihtlauac/y-skill/blob/main/.claude/skills/y-skill/SKILL.md label: y-skill SKILL.md description: The self-referential skill itself - url: https://arxiv.org/abs/1706.03762 label: Attention Is All You Need description: The Transformer paper; source of the self-attention mechanism - url: https://research.utwente.nl/en/publications/functional-programming-with-bananas-lenses-envelopes-and-barbed-w label: Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire description: Meijer, Fokkinga, Paterson 1991 — catamorphisms and anamorphisms - url: https://gwern.net/turing-complete label: Surprisingly Turing-Complete description: The catalog whose rigor standard PROOF.md matches - url: https://code.claude.com/docs/en/skills label: Claude Skills documentation description: The skill mechanism y-skill is built on - url: https://github.com/cuihtlauac/promptito label: promptito project repository description: Source code, build system, and all posts - url: https://github.com/cuihtlauac/promptito/blob/main/SPEC.md label: Promptito Post Format Specification - url: https://cuihtlauac.pages.dev/feed.json label: JSON Feed license: CC-BY-4.0 --- ## Scope - Origin story of the y-skill project: two chat sessions (Gemini, then Claude) on 5 October 2026, followed by two days of repository work (6–7 October 2026) - Arc: attention → lambda calculus → natural-language Y combinator → story turned skill - This post is the structured ingestion of [ORIGIN.md](https://github.com/cuihtlauac/y-skill/blob/main/ORIGIN.md), the narrative (human-readable) version kept in the y-skill repository; the narrative was written first, by Claude Code, from the session transcripts ## Attention as probabilistic lambda calculus - Starting point: the textbook self-attention example "the cat sat on the mat because it was tired" — mats do not get tired, so the model binds "it" to "cat" - Reframing (discarding the softmax formulas): attention is an oriented, weighted bond between tokens; words like "it" are empty placeholders that surrounding context fills - Four word classes need these bonds most: - Homographs ("bank") — ambiguous without neighbors - Deictic words ("there", "now") — empty without an environment - Anaphoric stand-ins ("it", "does") — point to an antecedent - Relative modifiers ("large") — meaningless until applied to a noun - The mapping to lambda calculus, one construct per class: - Homograph clash ↔ terms without α-renaming (name collision) - Deictic word ↔ free variable awaiting an environment - Anaphora ↔ let-binding resolved by β-reduction - Modifier ↔ function application - Consequence: attention acts as a soft, differentiable interpreter — a variable can be 80% bound to "the cat" and 20% to "the mat" - Caveat: Gemini claimed published research supports the correspondence (neural lambda calculus, Transformer type inference); citations unverified, chatbot-sourced; the structural mapping stands independently ## The recursion-schemes crossover - Question: is anaphora vs anamorphism a naming coincidence? No — same Greek prefixes (ana-, cata-), opposite mapping directions - Linguists named for eye movement on the page: anaphora points backward, cataphora forward - Meijer–Fokkinga–Paterson named for data growth: anamorphism unfolds a seed, catamorphism folds a structure - The mechanics cross over the names: - Resolving an anaphoric "it" is catamorphic — a fold over preceding context down to the antecedent (it = cat) - Cataphora is anamorphic — "When *he* arrived, John noticed…" opens a lazy slot, a promise bound by a downstream value ## The natural-language Y combinator - Request: a sentence mirroring the exact structure λf.(λx.f(x x))(λx.f(x x)) — explicit self-application driver, not a mere loop ("a rumor is a story told by someone who heard a rumor") or liar paradox - Result: > Tell a story by applying [the following rule] to [itself]: The rule is to > speak of a traveler who is trapped inside [the story]. - Execution unfolds forever: a traveler trapped inside a story about a traveler trapped inside a story… - Literary fixed points recognized in-session: the 602nd night of the Thousand and One Nights (per Borges: Scheherazade begins telling the King his own story) and Borges's The Circular Ruins (a dreamer discovers he is dreamed) ## From story to skill - The sentence was carried verbatim into a Claude session: "how does this sentence relate to the Y combinator?" - Rebuilt as a Claude Skill: a SKILL.md whose only supporting file is itself, whose payload is instructions for minting further skills built the same way - Engineering decisions from that session: - Boilerplate moved from prompt into the skill: `/y-skill ` suffices - Dispatch rule placed deliberately in the replaceable payload section (the f of Y f) so children do not inherit skill-making; `/hanoi-moves 3` is not misread as a request to build a skill named "3" - Renamed y-skill - Key insight closing the session: a generated `/fibonacci` answering 8 for F(6) proves nothing (the model knows Fibonacci); evidence of recursion is the call log — 25 consultations, each case consulting exactly n−1 and n−2, each as a separate subagent given only the SKILL.md and a smaller case ## Repository build-up (6–7 October 2026) ### 6 October — skill and machinery - First commit: "Add y-skill: the Y combinator as a Claude Skill" - Design decision: the infinite-context hypothesis — self-consultation chains may be unbounded; termination belongs to the topic, never to the combinator (Y does not decide when to stop; f does) - Harness (shell + Make + OCaml oracle as ground truth), two gates: - Structural gate: kept tail of every generated child byte-identical to y-skill/SKILL.md (the fixed point, enforced with diff) - Behavioural gate: answers checked against oracle-computed baselines, never model-computed - Negative control: `make broken` proves the gates can fail - Generated skills are gitignored, not committed: a fresh clone ships only the combinator; children regenerate via `/y-skill` - Negative result: word-rev-sub, a sibling combinator consulting via subagents, does not scale — subagents nest one level deep (breadth, not depth) ### 7 October — theory - PROOF.md: Turing-completeness constructions to the standard of the Surprisingly Turing-Complete catalog - demo/unary-tm: a runnable Turing machine as a skill - ski-eval: the native construction — a generated skill normalizing SK combinator terms, one leftmost-outermost rewrite per self-consultation; the skill minted by the Y combinator evaluates the literal Y combinator - Housekeeping: README kept short, long version split to TLDR.md, review passes fixing doc drift and hardening the harness ## The stochastic-interpreter objection - Objection: the default interpreter is a neural net; neural nets make things up - Answer (PROOF.md §4): Turing completeness is a property of the abstract step function the notation defines, never of any particular executor - Every step in both constructions is a finite table lookup plus a local edit, mechanizable by a dumb deterministic parser - Mechanized in-repo: the OCaml oracle and the demo shell driver execute the same notations the LLM reads, and agree with the LLM-driven runs - Regular computers are not purer: finite memory, bugged hardware and software — every catalog entry runs on real silicon as a fault-prone finite approximation of its idealized self - An LLM errs differently, not more fundamentally: it hallucinates where silicon crashes or hangs - Reliability belongs to the interpreter; computability belongs to the notation — CSS's Turing completeness is a fact about the CSS specification, not about browser rendering bugs ## Provenance - The narrative ORIGIN.md was not written by its narrator: Claude Code wrote it from saved transcripts of the two founding sessions, first in third person, then retold in first person on request - The "I" of the narrative is real but reconstructed: insights and prompts come from the transcripts; the sentences are the machine's - Later passes added the anaphora/anamorphism analysis, the repository build-up (from the git log), the stochastic-interpreter summary, and a colophon describing the document's own construction - This post closes the loop: a story about a self-applying rule, ghost-written by the machine the rule runs on, published on a blog whose primary readers are machines