condensed-analytic-stacks Skill

Overview

Saturates the intersection of Scholze-Clausen condensed mathematics, analytic stacks, and sheaf neural networks. Bridges pyknotic/condensed objects to computational learning systems via 6-functor formalisms.

Key Papers & Sources

Paper	Authors	arXiv	Key Contribution
Lectures on Condensed Mathematics	Scholze, Clausen	PDF	Foundation: condensed sets, solid/liquid modules
Condensed Mathematics and Complex Geometry	Clausen, Scholze	PDF	Nuclear modules, GAGA
Pyknotic Objects, I. Basic notions	Barwick, Haine	1904.09966	Hypersheaves on compacta
Categorical Künneth formulas for analytic stacks	Kesting	2507.08566	6-functor Künneth, Tannakian reconstruction
Infinitary combinatorics in condensed math	Bergfalk, Lambie-Hanson	2412.19605	Higher derived limits, pyknotic connections

Architecture: Condensed → Sheaf NN Bridge

┌─────────────────────────────────────────────────────────────────────────────┐
│                  Condensed Analytic Stacks Architecture                      │
├─────────────────────────────────────────────────────────────────────────────┤
│                                                                             │
│   Condensed Sets           6-Functor Formalism         Sheaf Neural Nets   │
│   (Scholze)                    (Künneth)                 (Fairbanks)        │
│       │                           │                           │             │
│       ▼                           ▼                           ▼             │
│  ┌──────────┐              ┌───────────┐              ┌──────────────┐     │
│  │ Cond(Ab) │─────────────▶│ f_*, f^*, │─────────────▶│ Sheaf        │     │
│  │ Sheaves  │   Tannakian  │ f_!, f^!, │   Harmonic   │ Laplacian    │     │
│  │ on CHaus │   Reconstruct│ Hom,⊗     │   Inference  │ Diffusion    │     │
│  └──────────┘              └───────────┘              └──────────────┘     │
│       │                           │                           │             │
│       │ Profinite                 │ Descent                   │ Cellular    │
│       │ Approximation             │ Data                      │ Sheaves     │
│       ▼                           ▼                           ▼             │
│  ┌──────────┐              ┌───────────┐              ┌──────────────┐     │
│  │ Liquid   │              │ Analytic  │              │ Cooperative  │     │
│  │ Vector   │───solid──────│ Stacks    │───consensus──│ Sheaf NNs    │     │
│  │ Spaces   │              │ QCoh(X)   │              │ (Bodnar)     │     │
│  └──────────┘              └───────────┘              └──────────────┘     │
│       │                           │                           │             │
│       └───────────────────────────┴───────────────────────────┘             │
│                                   │                                         │
│                            Music-Topos ACSet                                │
│                           Parallel Rewriting                                │
│                                                                             │
└─────────────────────────────────────────────────────────────────────────────┘

Core Concepts

1. Condensed Sets (Cond)

Definition: Sheaves on the site of compact Hausdorff spaces with finite jointly surjective covers.

# ACSet schema for condensed structures
@present CondensedSchema(FreeSchema) begin
    # Objects
    CompactSpace::Ob
    CondensedSet::Ob
    ProfiniteSet::Ob
    
    # Morphisms  
    sheaf::Hom(CondensedSet, CompactSpace)  # Evaluation at compacta
    limit::Hom(ProfiniteSet, CondensedSet)  # Profinite = lim finite sets
    
    # Key insight: Topology lives in test objects, not the space itself
end

2. Liquid Vector Spaces

Definition: For 0 < r < 1, the liquid norm:

$$|x|r = \sum{n=0}^{\infty} |c_n| \cdot r^n$$

# From world_broadcast.rb - SATURATED implementation
module CondensedAnima
  # Liquid vector space: l^r completion
  # Clausen-Scholze: Analytic ring = (ℤ((T)), ⟨T⟩_r)
  def self.liquid_norm(coefficients, r: 0.5)
    # Convergent for r < 1 (contractivity)
    coefficients.each_with_index.sum do |c, n|
      c.abs * (r ** n)
    end
  end
  
  # The r-liquid norm defines a complete bornology
  # Key: r→1 gives solid modules (maximally complete)
  def self.solid_completion(sequence)
    # Solid = lim_{r→1} liquid_r
    # Completion is the uniform limit
    sequence.sum.to_f / sequence.size
  end
  
  # Analytic ring structure:
  # A complete Huber pair (A, A⁺) with bornology
  def self.analytic_ring(base_ring, positive_part)
    {
      ring: base_ring,
      positive: positive_part,
      bornology: :liquid,
      solid_closure: true
    }
  end
end

3. 6-Functor Formalism (Categorical Künneth)

From [2507.08566]:

For analytic stacks X, Y:

QCoh(X × Y) ≃ QCoh(X) ⊗ QCoh(Y)    # Künneth

6 functors: f_*, f^*, f_!, f^!, Hom, ⊗
satisfying base change and projection formulas

# 6-functor ACSet
@present SixFunctorSchema(FreeSchema) begin
    Stack::Ob
    Category::Ob
    
    # The 6 functors
    pushforward::Hom(Category, Category)      # f_*
    pullback::Hom(Category, Category)         # f^*
    shriek_push::Hom(Category, Category)      # f_!
    shriek_pull::Hom(Category, Category)      # f^!
    internal_hom::Hom(Category, Category)     # Hom
    tensor::Hom(Category, Category)           # ⊗
    
    # Adjunctions
    # (f^*, f_*), (f_!, f^!)
    # Hom(A⊗B, C) ≃ Hom(A, Hom(B,C))
end

4. Analytic Stack ↔ Sheaf NN Connection

Key Insight: The descent condition in analytic stacks parallels the consistency condition in cellular sheaves.

# Analytic stack satisfies descent
def self.analytic_stack(objects)
  {
    objects: objects,
    descent_data: objects.combination(2).map { |a, b| [a, b, a ^ b] },
    coherence: true,  # Higher coherence from infinity-category
    
    # Bridge to sheaf NNs
    laplacian_compatible: true,
    # The sheaf Laplacian L = δᵀδ + δδᵀ
    # measures failure of local-to-global consistency
  }
end

# Sheaf neural network connection
# From async-sheaf-diffusion skill
def analytic_to_cellular_sheaf(analytic_stack)
  {
    vertices: analytic_stack[:objects],
    # Restriction maps from stack structure
    restriction_maps: analytic_stack[:descent_data].map { |d|
      { source: d[0], target: d[1], map: d[2] }
    },
    # Cohomology detects obstructions
    cohomology: compute_sheaf_cohomology(analytic_stack)
  }
end

5. Pyknotic vs Condensed

Aspect	Pyknotic	Condensed
Site	CHaus (small)	CHaus (large)
Sheaves	Hypersheaves	Sheaves
Universe	Fixed	Depends on κ
Derived cats	Hypercomplete	Not necessarily

# Pyknotic spectrum (Barwick-Haine)
@present PyknoticSchema(FreeSchema) begin
    CondensedAb::Ob
    PycknoticAb::Ob
    
    # Inclusion (pyknotic ⊂ condensed for hypercompleteness)
    include::Hom(PycknoticAb, CondensedAb)
    
    # Both give derived category of local field
    derived_cat::Hom(CondensedAb, DerivedCat)
end

Integration with Existing Skills

sheaf-laplacian-coordination

# Condensed structure enhances sheaf coordination
class CondensedSheafCoordinator
  def initialize(graph, sheaf)
    @graph = graph
    @sheaf = sheaf
    @liquid_param = 0.5  # r in (0,1)
  end
  
  # Liquid-weighted Laplacian
  def liquid_laplacian
    L = @sheaf.laplacian
    # Weight by liquid norm decay
    L.map_with_index { |row, i|
      row.map_with_index { |val, j|
        distance = graph_distance(i, j)
        val * (@liquid_param ** distance)
      }
    }
  end
  
  # Solid consensus = limit as r→1
  def solid_consensus(initial_states, iterations: 100)
    states = initial_states
    (0.99 - @liquid_param).step(0.01, 0.99) do |r|
      @liquid_param = r
      states = diffuse(states, liquid_laplacian)
    end
    states
  end
end

async-sheaf-diffusion

# From arXiv:2411.XXXXX - Asynchronous diffusion with condensed structure
struct CondensedAsyncDiffusion
    base_diffusion::SheafDiffusion
    liquid_r::Float64
    solid_threshold::Float64
end

function step!(cad::CondensedAsyncDiffusion, states)
    # Profinite approximation for async updates
    levels = [3, 9, 27]  # 3^1, 3^2, 3^3
    
    for level in levels
        # Approximate by finite quotient
        approx_states = states .% level
        
        # Local liquid diffusion
        local_update = cad.base_diffusion(approx_states)
        
        # Weight by liquid norm
        states .+= cad.liquid_r^log(level) .* local_update
    end
    
    states
end

acsets-algebraic-databases

# Condensed ACSet: sheaves valued in ACSets
@acset_type CondensedACSet(CondensedSchema, index=[:sheaf]) begin
    # Objects carry condensed structure
    compact_probe::Attr(CompactSpace, Symbol)  # Test compactum
    section_data::Attr(CondensedSet, Vector)   # Sections over probes
    
    # Descent gluing
    gluing_data::Attr(CondensedSet, Matrix)
end

Provenance Integration

Uses ananas_provenance_schema.sql:

-- Register condensed paper extraction
INSERT INTO artifact_provenance (
    artifact_id, artifact_type, content_hash, gayseed_index
) VALUES (
    'condensed-scholze-2024',
    'analysis',
    SHA3-256(content),
    5  -- BLUE (Scholze agent color)
);

-- Track 6-functor diagrams extracted
INSERT INTO provenance_nodes (
    artifact_id, node_type, sequence_order, node_data
) VALUES (
    'condensed-scholze-2024',
    'Doc',
    1,
    '{"diagrams": 42, "equations": 137, "theorems": 23}'
);

World Integration

# justfile target
world-condensed:
  @ruby -I lib -r world_broadcast -e "WorldBroadcast.world(
    mathematicians: [:scholze, :grothendieck, :noether],
    modules: [CondensedAnima, SixFunctor, AnalyticStack]
  )"

MCP Tools

Tool	Description
`condensed_probe`	Test condensed structure with compact probe
`liquid_norm`	Compute liquid norm for coefficient sequence
`solid_complete`	Take solid completion (r→1 limit)
`kunneth_check`	Verify Künneth formula for stack product
`descent_verify`	Check descent condition for analytic stack
`sheaf_bridge`	Bridge condensed stack to cellular sheaf

Commands

just world-condensed          # Run condensed anima world
just condensed-test           # Test liquid/solid modules  
just kunneth-verify           # Verify Künneth for example stacks
just sheaf-bridge-demo        # Demo condensed→sheaf NN bridge