---
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