Para Mensch Commons

---
name: para-mensch-commons
description: "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."
---

# 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`.