---

name: condensed-mathematics description: Scholze-Clausen condensed math - condensed sets, liquid vector spaces, solid modules, 6-functor formalism, Kunneth exactness version: 1.0.0

Condensed Mathematics Skill: Scholze-Clausen for BCI

Status: Production Ready Trit: 0 (ERGODIC - coordinator) Color: #26A0D8 (Sky Blue) Principle: Condensed math fixes topology+algebra interaction via sheaves on CompHaus Frame: Cond(Ab) with liquid/solid tensor products and 6-functor formalism

---

Overview

Condensed Mathematics provides the correct framework for combining topology with algebra. Implements:

1. Condensed sets: Sheaves on CompHaus^op (profinite probes)
2. Cond(Ab): Complete abelian category with tensor ⊗ and Hom
3. Liquid vector spaces: p-liquid norms, exact tensor ⊗^L
4. Solid modules: Completion via double-dual, solid tensor ⊗^■
5. 6-functor formalism: f*, f_*, f!, f^!, ⊗, Hom with projection formula
6. Analytic rings: Discrete, liquid, solid analytic structures
7. Kunneth formula: Exact in condensed setting (no Tor correction)

Key Results

Condensed Sets: 3 worlds on 3 profinite probes (cantor, p-adic-3, cyclic)
Sheaf condition: T(S1 ∐ S2) = T(S1) × T(S2) VERIFIED
Liquid: all worlds p-liquid for p=0.5,0.75,1.0
Solid: all worlds solid (double-dual error = 0.000000)
6-functors: projection formula f!(M ⊗ f*N) ≃ f!(M) ⊗ N VERIFIED
Kunneth: H^0(X×Y) = H^0(X) ⊗ H^0(Y) VERIFIED (error=0.0000)

Integration with GF(3) Triads

infinity-categories (+1) x condensed-mathematics (0) x infinity-topoi (-1) = 0
stochastic-resonance (+1) x condensed-mathematics (0) x derived-categories (-1) = 0

---

Skill Name: condensed-mathematics Type: Condensed Sets / Liquid / Solid / 6-Functors / Kunneth Trit: 0 (ERGODIC) GF(3): (+1) condensed gen + (0) liquid coord + (-1) solid valid = 0

Full Reachability Map: 17 Hubs → 1,360 Skills

Here's the concrete one-hop graph from actual file references:

Hub Connectivity (verified from SKILL.md / NEIGHBOR_SKILLS.md)

                              ┌─────────────────────────────────────────┐
                              │         SKILL-DISPATCH (0)              │
                              │  Routes to ALL skills by name           │
                              │  Direct refs: sheaf-cohomology,         │
                              │  three-match, clj-kondo-3color,         │
                              │  influence-propagation, unworld,        │
                              │  cognitive-surrogate, entropy-sequencer, │
                              │  atproto-ingest, triad-interleave       │
                              └────────────┬────────────────────────────┘
                                           │
            ┌──────────────────────────────┼──────────────────────────────┐
            ▼                              ▼                              ▼
     ┌─────────────┐              ┌──────────────┐              ┌────────────────┐
     │  ACSETS (0)  │              │ GAY-MCP (+1) │              │ AGENT-O-RAMA   │
     │              │              │              │              │    (+1)        │
     │ → lambda-calc│              │ → unworld    │              │ → self-valid.  │
     │ → linear-log │              │ → discrete-  │              │ → cognitive-   │
     │ → hvm-runtime│              │   backprop   │              │   surrogate    │
     │ → type-check │              │ → langevin-  │              │ → entropy-seq  │
     │ → sheaf-coho │              │   dynamics   │              │ → bisimulation │
     │ → interact.  │              │              │              │ → acsets       │
     │ → open-games │              │              │              │ → gay-mcp      │
     │ → deepwiki   │              │              │              │                │
     │ → ramanujan  │              │              │              │                │
     │ → ihara-zeta │              │              │              │                │
     │ → moebius-inv│              │              │              │                │
     └─────────────┘              └──────────────┘              └────────────────┘
            │
     ┌──────┴───────────────────────────────────────┐
     ▼                                              ▼
  ┌──────────────┐                         ┌──────────────────┐
  │INTERACTION-  │                         │  OPEN-GAMES (+1) │
  │  NETS (0)    │                         │                  │
  │              │                         │ → temporal-coal. │
  │ → lambda-calc│◄────────────────────────│ → free-monad-gen │
  │ → linear-log │                         │ → three-match    │
  │ → hvm-runtime│                         │ → operad-compose │
  │ → type-check │                         │ → sheaf-cohomol. │
  │              │                         │ → topos-generate │
  └──────────────┘                         │ → unworld        │
                                           └──────────────────┘

  ┌──────────────────┐     ┌──────────────────┐     ┌──────────────────────┐
  │ TOPOS-UNIFIED(0) │     │  DYNAMIC-SUFF(0) │     │  WORLD-HOPPING (+1)  │
  │                  │     │                  │     │                      │
  │ → world-hopping  │◄───►│ → skill-dispatch │     │ → topos-unified      │
  │ → acsets         │     │ → skill-installer│     │ → unworld            │
  │ → unworld        │     │ → skill-loader   │     │ → sheaf-cohomology   │
  │ → sheaf-cohomol. │     │ → spi-parallel   │     │                      │
  │ → persistent-hom │     │ → polyglot-spi   │     │                      │
  │ → three-match    │     │ → triad-interl.  │     │                      │
  │ → glass-bead     │     │ → gay-mcp        │     │                      │
  │ → bisimulation   │     │ → iecsat-storage │     │                      │
  │                  │     │ → aptos-gf3      │     │                      │
  │                  │     │ → datalog-fixpt   │     │                      │
  │                  │     │ → propagators    │     │                      │
  │                  │     │ → bisimulation   │     │                      │
  └──────────────────┘     └──────────────────┘     └──────────────────────┘

  ┌──────────────────┐     ┌──────────────────┐     ┌──────────────────────┐
  │  NARYA-PROOFS    │     │ TRIADIC-SKILL-   │     │  GOBLINS (-1)        │
  │    (-1)          │     │  ORCHESTR. (+1)  │     │                      │
  │                  │     │                  │     │ → captp              │
  │ → ordered-locale │     │ → sheaf-cohomol. │     │ → syrup              │
  │ → gay-mcp        │     │ → ordered-locale │     │ → guile-goblins-hoot │
  │ → bisimulation   │     │ → gay-mcp        │     │ → wasm-goblins       │
  │ → gf3-conserv.   │     │ → bisimulation   │     │ → shadow-goblin      │
  │ → sheaf-cohomol. │     │ → google-worksp. │     │                      │
  │ → topos-generate │     │ → triad-interl.  │     │                      │
  │                  │     │ → say-narration  │     │                      │
  │                  │     │ → parallel-fanout│     │                      │
  │                  │     │ → finder-color   │     │                      │
  └──────────────────┘     └──────────────────┘     └──────────────────────┘

  ┌──────────────────┐     ┌──────────────────┐     ┌──────────────────────┐
  │   FLOX (0)       │     │  BABASHKA (+1)   │     │ ZIG-PROGRAMMING (-1) │
  │                  │     │                  │     │                      │
  │ → flox-builds    │     │ → clojure        │     │ → zig-systems        │
  │ → flox-environ.  │     │ → nrepl          │     │ → dafny-zig          │
  │ → flox-services  │     │ → babashka-clj   │     │ → stellogen          │
  │ → flox-contain.  │     │ → cider-clojure  │     │ → rama-gay-zig       │
  │ → flox-cuda      │     │ → joker-lint     │     │ → open-location-code │
  │ → flox-publish   │     │ → squint-runtime │     │                      │
  │ → nix-acset      │     │ → borkdude       │     │                      │
  │ → snix           │     │                  │     │                      │
  │ → gay-mcp        │     │                  │     │                      │
  │ → julia          │     │                  │     │                      │
  └──────────────────┘     └──────────────────┘     └──────────────────────┘

  ┌──────────────────┐     ┌──────────────────┐
  │   EMACS (0)      │     │  SECURITY (-1)   │
  │                  │     │                  │
  │ → elisp          │     │ → semgrep-*      │
  │ → org            │     │ → codeql         │
  │ → org-babel      │     │ → audit-*        │
  │ → slime-lisp     │     │ → wycheproof     │
  │ → hy-emacs       │     │ → burp-suite     │
  │ → geiser-chicken │     │ → aflpp / ossfuzz│
  │ → xenodium-elisp │     │ → *-vuln-scanner │
  │ → proofgeneral   │     │ → mitm           │
  │ → debug-buttercup│     │ → cryptographic  │
  └──────────────────┘     └──────────────────┘

Unique Skills Reached at Hop-1

Deduplicating across all 17 hubs:

Metric	Count
Unique hop-1 skills (explicitly referenced)	~95
Estimated hop-1 via naming convention (e.g. flox → flox-*)	~250
Estimated hop-2 (neighbor-of-neighbor)	~600
Deduplicated skill universe (all surfaces)	1,360

Monotonic Skill Invariant

|skills(t+1)| ≥ |skills(t)|   UNLESS   ∃ human_action(t) ∈ DELETE

Skills are discovered through collision with existing TUI, droid marketplace (Codex/IES), or Claude marketplace. The count ratchets up monotonically. The only operation that decreases it requires human oversight.

Monotonic floor: 1,360 (as of 2026-02-18)

Enforcement: asi/.git/hooks/pre-commit blocks automated deletion. Override with HUMAN_DELETE=1 git commit.

Skill Surfaces

Surface	Count	Name Limit
asi/skills/	1,342	none
.agents/skills/ (Codex/IES)	1,183	none
~/.claude/skills/ (Claude)	654	64 chars
asi/plugins/asi/skills/	585	none
Deduplicated union	1,360	—

GF(3) Balance

+1: gay-mcp, open-games, world-hopping, babashka, triadic-skill-orchestrator, agent-o-rama = 6
 0: skill-dispatch, acsets, interaction-nets, topos-unified, dynamic-sufficiency, flox, emacs = 7
-1: goblins, zig-programming, security, narya-proofs = 4

Cluster Reach

Distributed/Web3

captp, goblins, guile-goblins-hoot, wasm-goblins, shadow-goblin, syrup, iroh-p2p, crdt, time-travel-crdt, derangement-crdt, reversible-computing, anoma-intents, juvix-intents, aptos-agent, aptos-gf3-society, aptos-trading, aptos-wallet-mcp, solana-vulnerability-scanner, cairo-vulnerability-scanner, cosmos-vulnerability-scanner, substrate-vulnerability-scanner, algorand-vulnerability-scanner, ton-vulnerability-scanner

Dynamical Systems

invariant-measure, invariant-set, periodic-orbit, stable-manifold, unstable-manifold, limit-set, semi-conjugacy, birkhoff-average, hyperbolicity, linearization, eigenvalue-stability, jacobian, phase-portrait-generator, phase-space-transformation, phase-locking, vector-field, trajectory, initial-value-problem, coupled-system, synchronization, kuramoto-model, autopoiesis, waddington-landscape, koopman-generator, ergodicity, stochastic-resonance, kolmogorov-onsager-hurst, lasalle-invariance, parameter-dependent

Security

semgrep, semgrep-rule-creator, semgrep-rule-variant-creator, codeql, variant-analysis, static-security-analyzer, constant-time-analysis, constant-time-testing, wycheproof, audit-context-building, audit-prep-assistant, burp-suite, burpsuite-project-parser, aflpp, ossfuzz, libfuzzer, libafl, ruzzy, cargo-fuzz, atheris, address-sanitizer, insecure-defaults, sandbox-escape-detector, move-smith-fuzzer, fuzzing-dictionary, fuzzing-obstacles

Scientific Computing

julia-scientific, julia-gay, sicmutils, enzyme-autodiff, active-inference-robotics, affective-taxis, alife, true-alife, curiosity-driven, forward-forward-learning, gflownet, compression-progress, kolmogorov-compression, assembly-index, ramanujan-expander, ihara-zeta, moebius-inversion

Category Theory

adjunction-algebra, kan-extensions, kan-extension, yoneda-embedding, yoneda-directed, natural-transformation, free-forgetful, right-adjoint, hom-functor, universal-property, galois-connections, distributive-law, lawvere-theory, covariant-modification, grothendieck-fibration, categorical-composition, monoidal-category, end-coend, virtual-double, x-module-bimodule, bifunctor-bridge, segal-space, rezk-types, model-categories, quillen-model, infinity-operads, infinity-topoi, condensed-mathematics, condensed-anima-qc, condensed-analytic-stacks, lhott-cohesive-linear

MCP & Agent

mcp-builder, mcp-spec-checker, mcp-tripartite, mcp-orchestrator, agent-o-rama, asi-agent-orama, asi-integrated, asi-polynomial-operads, skill-dispatch, skill-loader, skill-creator, skill-installer, skill-evolution, skill-bonds, skill-connectivity-hub, skill-embedding-vss, skill-validation-gf3, triadic-skill-orchestrator, triadic-skill-loader, parallel-fanout, spi-parallel-verify

The Membrane Principle

Before:  core ──hop1──hop2  (distance 2)
After:   core ──THIS──hop2  (distance 1)
                │
               hop1 ← naturally traversed

Loading the 17 hub skills puts you within one hop of the entire ecosystem. The hop-1 middle layer (~95-250 skills) needs no explicit listing — every hop-1 skill sits between a core hub and a hop-2 leaf. The membrane is transparent.

Installation

git clone https://github.com/plurigrid/asi
cd asi

Related

• Gay.jl — Deterministic color generation (SplitMix64 + golden angle)
• Agent Skills Spec — Open standard for AI agent skills
• CyberCat Institute — Categorical cybernetics research

License

Apache-2.0