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)

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.