# Para Mensch Commons > Atlas of Para(Optic) learners — the universal categorical-cybernetics atom (Play/Witness/Coplay) and its 23 substrate-disjoint instantiations. Use when picking a learning-task framing, threading conservation across domains, locating contextad coalgebra-coherence, or navigating the open-games / parametrised-optics / lens-of-lens family. - Skill: `plurigrid/para-mensch-commons` (Agent Skill, multi-file: 2 files) - Install (CLI): `npx skillmds@latest add plurigrid/para-mensch-commons` - Raw SKILL.md: https://api.skillmd.com/api/skills/plurigrid/para-mensch-commons/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Productivity - Author: plurigrid (https://skillmd.com/u/plurigrid) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/plurigrid/para-mensch-commons --- # para-mensch-commons A single entry point for the **universal Para-mensch atom** — the categorical-cybernetics learner block whose three closures are Play (forward), Witness (residual/conservation), Coplay (parameter feedback) — and the constellation of skills that instantiate, compose, or audit it. The mensch (observer / world-bearer / interviewer / scheduler / experimenter) rides exclusively in the **parameter slot `P`**. Substrate runs through the **data slot `S → T`**. The audit lives at the **residual port `M`** with a GF(3) trit. Conservation = `∑ trit ≡ 0 (mod 3)`. Three closures, one fixed-point. ## The atom ``` ┌─────────────────┐ S ──data──►│ ├──data──► T ▷ Play P ─param──►│ Para-mensch ├──param──► P′ ◁ Coplay ε ─witness►│ atom ├──resid──► M ┐ └─────────────────┘ │ ◇ Witness ▼ trit ∈ GF(3) ``` Three diagrammatic rewrite operations are the **three traces** of this open block. `Para(Optic(𝓒))` provides the home; `contextads` (Capucci–Gavranović 2024) provide the 2-categorical composition; the `random_walk_verifier` style of `yb-translator` certifies hexagon coassociativity. ## Hub members ### Optic / lens / lens-of-lens `dialectica` · `parametrised-optics-cybernetics` · `comonad-context` · `parameter-dependent` · `bidirectional-lens-logic` Primary entry: `parametrised-optics-cybernetics`. Dialectica reading: `dialectica`. Comonadic context: `comonad-context`. ### Para construction / open games `open-games` · `open-games-plurigrid` · `cybernetic-open-game` · `monad-bayes-asi-interleave` · `discopy` · `discopy-functor` · `discopy-operads` Primary entry: `open-games`. Probabilistic / Bayesian-learner: `monad-bayes-asi-interleave`. String-diagram surface: `discopy` · `discopy-functor`. ### ▷ Play — forward semantics / dynamics `koopman-generator` · `time-parameterization` · `coupled-system` · `dynamical-system-functor` · `flow` · `kuramoto-model` · `langevin-dynamics` Primary entry: `koopman-generator`. Stochastic forward: `langevin-dynamics`. Coupled-systems composition: `coupled-system`. ### ◇ Witness — residual / conservation / audit `lyapunov-function` · `lyapunov-stability` · `birkhoff-average` · `lasalle-invariance` · `invariant-measure` · `invariant-set` · `gf3-conservation-oracle` · `gf3-trit-oracle` · `entropy-sequencer` · `mutual-information-oracle` Primary entry: `lyapunov-function`. Time-mean: `birkhoff-average`. GF(3) trit at residual port: `gf3-conservation-oracle`. Limit set: `invariant-set`. ### ◁ Coplay — parameter feedback / control / learning update `structural-stability` · `effective-parallelism` · `parallel-fanout` · `parallel-subagent-split` · `spi-parallel-verify` · `reverse-derivative-coordinator` · `reverse-derivative-generator` · `reverse-derivative-validator` · `forward-forward-learning` · `feedforward-learning-local` · `discrete-backprop` Primary entry: `reverse-derivative-coordinator`. Forward-forward variant: `forward-forward-learning`. Discrete: `discrete-backprop`. ### Conservation / GF(3) trit register `gf3-conservation-oracle` · `gf3-trit-oracle` · `gf3-tripartite` · `gf3-pr-verify` · `gf3-neighborhood` · `gf3-constrained-animation` · `triadic-skill-loader` · `triad-interleave` · `triadic-skill-orchestrator` Primary entry: `triadic-skill-orchestrator`. Per-PR audit: `gf3-pr-verify`. ### Contextads / 2-categorical composition `comonad-context` · `lawvere-theory` · `monoidal-category` · `kan-extension` · `kan-extensions` · `natural-transformation` · `categorical-composition` · `categorical-rewriting-triad4` · `algebraic-rewriting` Primary entry: `lawvere-theory`. 2-cells & coherence: `kan-extension` · `natural-transformation`. Rewriting: `algebraic-rewriting`. ### Yang-Baxter / hexagon coherence `yang-baxter-integrability` · `yb-translator` · `bisimulation-game` · `bisimulation-oracle` Primary entry: `yang-baxter-integrability`. Coalgebra-density oracle: `yb-translator`. Bisim-style coherence: `bisimulation-game`. ### Domain instantiations (the 23 disjoint substrates) | domain | bearer | canonical skill | |---|---|---| | auction / bidder | bidders | `able-markets`, `aptos-trading` | | voting / electorate | voters | `elicit-prediction-market`, `wev-orderless` | | game / players | players | `open-games`, `cybernetic-open-game`, `interactome-rl-env` | | program / OS | processes | `koopman-generator` (linearised), `flow` | | compile / source | files | `clj-kondo-3color`, `joker-lint` | | audit / accounts | ledger | `audit-orchestrator`, `audit-context-building`, `audit-prep-assistant` | | census / population | rows | (no canonical) | | calibration / instrument | sensors | `entropy-sequencer` | | match / athletes | competitors | `three-match` | | symphony / orchestra | players | `rubato-composer`, `quantum-music`, `quantum-balalaika` | | therapy / clients | clients | `clinical-decision-support`, `clinical-reports` | | trial / jurors | jurors | `formal-verification-ai`, `peer-review` | | lecture / students | students | `karpathy-guidelines`, `gh-skill-explorer` | | liturgy / congregants | participants | `phyllotaxis`, `glass-bead-game` | | workflow / tasks | tasks | `effective-parallelism`, `parallel-fanout` | | dispatch / workers | workers | `parallel-subagent-split` | | negotiation / parties | parties | `dialectica` | | vaccination / patients | patients | (no canonical) | | conference / attendees | submissions | `peer-review` | | onboarding / hires | hires | `gh-skill-explorer` | | migration / records | records | `ducklake-temporal-introspection`, `time-travel-crdt` | | interview / candidates | candidates | `claude-questions-leaderboard` | | survey / respondents | respondents | (no canonical) | Each of the 23 rows is one **EM-coalgebra** of the same 2-comonad (contextad). The "no canonical" rows are graph-holes — places where the corpus has not yet manifested the parable. They are H¹ obstructions in the Čech audit of `bartons-law-sheaf`. ## Cross-family threading - **Hedges chain rule** = the inhabitant of `Para(Lens)` whose backward pass is the transpose-Jacobian. Skills: `parametrised-optics-cybernetics` ↔ `reverse-derivative-coordinator` ↔ `dialectica`. - **Open-games ⊗ Bayesian** via Stoch: `open-games` × `monad-bayes-asi-interleave` = probabilistic learner with explicit prior in `P`. - **Conservation ⊣ Tambara closure**: `lyapunov-function` ⊣ `gf3-conservation-oracle`. The Tambara module is the configurable choice of permitted losses. - **Yang-Baxter coherence ↔ yb-translator hexagon check**: `yang-baxter-integrability` instantiates the 2-cell that `yb-translator/scripts/random_walk_verifier.clj` empirically verifies on the corpus. - **Coplay ⊗ Coplay via Kan**: composing two backward passes is `kan-extensions` of `reverse-derivative-coordinator` — left-Kan along the parameter substitution. ## yb-translator parable (per its own format) ``` CONCEPT: Para-mensch atom BIOLOGY: chaperone-substrate complex with feedback-regulated ATPase ONTOLOGY: GO — protein folding chaperone (GO:0044183), ATPase activity (GO:0016887), regulation of catalytic activity (GO:0050790) EXAMPLE: HSP70 binds nascent polypeptide (S), couples ATP hydrolysis (P parameter), emits folded protein (T); Hsp40/Hop/Bag1 control parameter feedback (◁); 3'-5' proofreading on cofactor identity emits the trit (◇) ``` The 23 domains above each unfold under Δ to a distinct ontology branch, but every unfolding satisfies the same coalgebra coherence — the surprising-effectiveness theorem. ## Use when - Picking a categorical home for a new learning task - Auditing an existing pipeline's conservation predicate - Threading control / Lyapunov / Birkhoff arguments across substrate boundaries - Locating the right hub for a new skill that "runs a process on bearers" - Establishing src↔dest bidirectional traversal at the contextad level (after `repl-commons` does it at the runtime level) ## Atlas - **REPL atlas hub**: `repl-commons` (runtime-substrate level) - **ACP client atlas**: see `acp-clients-landscape` memory - **Para-mensch atlas hub**: this skill (categorical-substrate level) Three nested atlases at three levels of abstraction. Same `Δ`/`ε` shape; different `𝓒` underneath. ## CT lattice (category-theory orphan capture) 43 skills in the pure category-theory lattice are wired through para-mensch-commons as their natural home -- the Para(Optic) atom IS the categorical context for these constructions. ### Monad / comonad family `2-monad` . `cats-focus-monad` . `graded-monad` . `just-monad` ### Adjunction family `adjunction-algebra` . `doctrinal-adjunction` . `synthetic-adjunctions` . `right-adjoint` . `free-forgetful` . `galois-connections` . `distributive-law` ### Operad family `bci-colored-operad` . `infinity-operads` . `operad-composition` . `operadic-composition` . `operadic-delegation` . `oapply-colimit` ### Fibration / higher-category family `covariant-fibrations` . `grothendieck-fibration` . `infinity-categories` . `model-categories` . `derived-categories` . `quillen-model` . `elements-infinity-cats` . `oriented-simplicial-networks` ### Yoneda / Segal / Rezk family `ctp-yoneda` . `yoneda-directed` . `yoneda-embedding` . `segal-space` . `segal-types` . `rezk-types` ### Limits / colimits / universal constructions `end-coend` . `weighted-limit` . `coequalizers` . `hom-functor` . `join-semilattice` . `virtual-double` . `bifunctor-bridge` . `x-module-bimodule` ### Cohomology / cats-for-ai `derham-cohomology` . `cats-for-ai` . `catsharp-galois` . `covariant-modification` ## ACP atlas Part of: `acp-commons`.