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

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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

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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