Superelastic Quotient Skill

Superelastic Quotient

Trit: 0 (ERGODIC - coordinates elastic boundaries)
Foundation: StructuredDecompositions.jl + Ordered Locales + Sheaf Cohomology
Principle: Maximum resolution quotients that stretch without breaking

Core Insight

Superelasticity = ability to deform equivalence classes while preserving:

Spatial coherence (open cone conditions from ordered locales)
GF(3) conservation (triadic balance)
Sheaf condition (local-to-global gluing)

                 FINE RESOLUTION
                      │
    ┌─────────────────┼─────────────────┐
    │                 │                 │
    ▼                 ▼                 ▼
 ┌──────┐        ┌──────┐        ┌──────┐
 │[-1]  │───────►│[ 0]  │───────►│[+1]  │
 │MINUS │◄───────│ERGODIC│◄───────│PLUS  │
 └──────┘        └──────┘        └──────┘
    │                 │                 │
    └─────────────────┼─────────────────┘
                      │
                COARSE QUOTIENT
                      
 Elasticity = bidirectional refinement/coarsening

Mathematical Structure

Ordered Locale for Spatial Direction

From ordered locales (Heunen-van der Schaaf 2024):

# Frame L with compatible preorder ≪
struct SpatialQuotient{L}
    locale::OrderedLocale{L}
    equivalence::Function        # x ~ y iff x ≪ y and y ≪ x
    resolution::Int              # Current quotient level
    max_resolution::Int          # Finest grain achievable
end

# Open cone condition: ↟U must be open for quotient consistency
function is_elastic(sq::SpatialQuotient)
    for U in opens(sq.locale)
        # Upper cone must be open
        if !is_open(upper_cone(sq.locale, U))
            return false
        end
    end
    return true
end

Resolution Spectrum

# Quotient tower: finest → coarsest
#
# Level 0: Individual points (maximum resolution)
# Level 1: Local neighborhoods  
# Level 2: Triadic clusters (GF(3) bags)
# Level 3: Skill categories
# Level N: Single equivalence class (minimum resolution)

struct QuotientTower
    levels::Vector{SpatialQuotient}
    morphisms::Vector{Functor}  # Level i → Level i+1
    
    function QuotientTower(base::SpatialQuotient)
        # Build tower via successive quotients
        levels = [base]
        while !is_trivial(last(levels))
            push!(levels, quotient_step(last(levels)))
            push!(morphisms, quotient_map(levels[end-1], levels[end]))
        end
        new(levels, morphisms)
    end
end

# Superelasticity: can jump any number of levels
function elastic_quotient(tower::QuotientTower, source_level::Int, target_level::Int)
    if target_level > source_level
        # Coarsening: compose quotient maps
        compose(tower.morphisms[source_level:target_level-1]...)
    else
        # Refinement: right adjoint (splitting)
        adjoint(compose(tower.morphisms[target_level:source_level-1]...))
    end
end

Spatial Decomposition Pattern

Tripartite Spatial Bags

@present SchSpatialTripartite(FreeSchema) begin
    (Region, Boundary, Point)::Ob
    
    # Spatial inclusion
    region_boundary::Hom(Boundary, Region)
    boundary_point::Hom(Point, Boundary)
    
    # GF(3) trit assignment
    trit::Attr(Region, Int)  # ∈ {-1, 0, +1}
    
    # Spatial coordinates
    coord::Attr(Point, Vector{Float64})
    
    # Quotient level
    level::Attr(Region, Int)
end

@acset_type SpatialDecomp(SchSpatialTripartite)

# Verify spatial + GF(3) coherence
function verify_superelastic(sd::SpatialDecomp)
    # 1. Check GF(3) conservation
    trits = sd[:, :trit]
    @assert sum(trits) % 3 == 0 "GF(3) violation"
    
    # 2. Check spatial coherence (open cone condition)
    for r in parts(sd, :Region)
        boundary = incident(sd, r, :region_boundary)
        @assert forms_open_cone(sd, boundary) "Spatial violation"
    end
    
    # 3. Check elasticity (can refine/coarsen)
    @assert has_refinement(sd) && has_quotient(sd) "Not elastic"
    
    return true
end

Resolution-Adaptive Navigation

# Specter-style navigators with resolution awareness
struct ResolutionNavigator
    base_path::Vector{Symbol}
    min_level::Int
    max_level::Int
end

# Navigate at current resolution
function select_at_level(nav::ResolutionNavigator, level::Int, target)
    if level < nav.min_level || level > nav.max_level
        # Elastic stretch to accommodate
        stretched_nav = stretch_navigator(nav, level)
        return select(stretched_nav.base_path, target)
    end
    return select(nav.base_path, target)
end

# Bidirectional resolution transform
function transform_elastic(nav::ResolutionNavigator, f::Function, target, 
                           source_level::Int, target_level::Int)
    # Get current view at source level
    view = select_at_level(nav, source_level, target)
    
    # Apply transform
    transformed = f(view)
    
    # Project to target level (quotient or refine)
    if target_level > source_level
        return quotient_to_level(transformed, target_level)
    else
        return refine_to_level(transformed, target_level)
    end
end

Skill Decomposition Example

# Decompose 524+ skills into spatial triads

function skill_spatial_decomposition(skills::Vector{Symbol}, resolution::Int)
    # Level 0: Individual skills
    base = SpatialDecomp()
    
    for skill in skills
        trit = skill_to_trit(skill)
        coord = skill_to_embedding(skill)  # From skill-embedding-vss
        add_point!(base, skill, trit, coord)
    end
    
    # Build quotient tower
    tower = QuotientTower(base)
    
    # Return view at requested resolution
    return elastic_quotient(tower, 0, resolution)
end

# Example usage:
# Level 0: 524 individual skills
# Level 1: ~175 triadic clusters (3 skills each, GF(3) balanced)
# Level 2: ~58 mega-triads (3 clusters each)
# Level 3: ~19 domains
# Level 4: 6 "worlds" (a-z mapped)
# Level 5: 2 hemispheres (generator/validator)
# Level 6: 1 unified skill space

Integration with Cat# Three Homes

# Each home operates at different resolution scales

HOME_RESOLUTIONS = Dict(
    1 => (  # Polynomial Comonads
        name = "State Dynamics",
        typical_level = 0..2,  # Fine-grained, individual behaviors
        skill = :gay-mcp,
        trit = +1
    ),
    2 => (  # Monads in Span
        name = "Schema Relations", 
        typical_level = 2..4,  # Medium, relational structure
        skill = :acsets,
        trit = 0
    ),
    3 => (  # Path Algebras
        name = "Equivalence Classes",
        typical_level = 3..6,  # Coarse, bisimulation quotients
        skill = :bisimulation-game,
        trit = -1
    )
)

# Superelastic: can operate across all levels
function dispatch_to_home_elastic(concept, resolution::Int)
    home = concept_to_home(concept)
    home_range = HOME_RESOLUTIONS[home].typical_level
    
    if resolution in home_range
        # Native resolution
        return (home, :native, resolution)
    else
        # Elastic stretch
        direction = resolution < first(home_range) ? :refine : :coarsen
        return (home, direction, resolution)
    end
end

Sheaf-Theoretic Gluing

# Superelastic quotients preserve the sheaf condition
# across all resolution levels

struct ElasticSheaf{C}
    presheaf::Functor{Op{C}, FinSet}
    tower::QuotientTower
end

# Sheaf condition at any resolution level
function sheaf_condition(es::ElasticSheaf, level::Int)
    projected = project_to_level(es.presheaf, es.tower, level)
    
    # For each covering family at this level
    for cover in coverings(es.tower.levels[level])
        # Sections must glue
        sections = [projected(U) for U in cover]
        if !unique_amalgamation(sections, overlaps(cover))
            return false
        end
    end
    return true
end

# Maximum resolution = finest quotient where sheaf condition holds
function max_elastic_resolution(es::ElasticSheaf)
    for level in 0:length(es.tower.levels)-1
        if !sheaf_condition(es, level)
            return level - 1  # Last valid level
        end
    end
    return length(es.tower.levels) - 1
end

DuckDB Integration

-- Schema for spatial quotient tracking
CREATE TABLE IF NOT EXISTS spatial_quotients (
    quotient_id UUID PRIMARY KEY,
    level INT NOT NULL,
    trit TINYINT NOT NULL,
    region_name VARCHAR,
    parent_quotient UUID,
    child_count INT,
    spatial_centroid DOUBLE[],
    is_elastic BOOLEAN DEFAULT true,
    created_at TIMESTAMP DEFAULT CURRENT_TIMESTAMP,
    
    CONSTRAINT gf3_check CHECK (trit IN (-1, 0, 1))
);

-- Quotient tower as recursive CTE
WITH RECURSIVE tower AS (
    -- Base: finest level
    SELECT quotient_id, level, trit, region_name, parent_quotient
    FROM spatial_quotients WHERE level = 0
    
    UNION ALL
    
    -- Recursive: coarser levels
    SELECT sq.quotient_id, sq.level, sq.trit, sq.region_name, sq.parent_quotient
    FROM spatial_quotients sq
    JOIN tower t ON sq.parent_quotient = t.quotient_id
)
SELECT level, COUNT(*) as regions, SUM(trit) % 3 as gf3_check
FROM tower
GROUP BY level
ORDER BY level;

GF(3) Balanced Triads

Superelastic Conservation Law:
─────────────────────────────
At EVERY resolution level, triadic decompositions sum to 0 mod 3

Level 0: skill₁(-1) ⊗ skill₂(0) ⊗ skill₃(+1) = 0 ✓
Level 1: cluster₁(-1) ⊗ cluster₂(0) ⊗ cluster₃(+1) = 0 ✓
Level 2: domain₁(-1) ⊗ domain₂(0) ⊗ domain₃(+1) = 0 ✓
...
Level N: unified(-1 + 0 + 1) = 0 ✓

Elasticity preserves this invariant under stretch!

GF(3) Triads for This Skill

structured-decomp (-1) ⊗ superelastic-quotient (0) ⊗ gay-mcp (+1) = 0 ✓
sheaf-cohomology (-1) ⊗ superelastic-quotient (0) ⊗ topos-generate (+1) = 0 ✓  
ordered-locale (-1) ⊗ superelastic-quotient (0) ⊗ tripartite-decompositions (+1) = 0 ✓

Babashka CLI

# Query at different resolution levels
bb superelastic-quotient.bb --level 0 --skill gay-mcp  # Individual
bb superelastic-quotient.bb --level 2 --skill gay-mcp  # Triadic cluster
bb superelastic-quotient.bb --level 4 --skill gay-mcp  # Domain

# Find maximum elastic resolution for concept
bb superelastic-quotient.bb --max-resolution "polynomial comonad"

# Stretch between levels
bb superelastic-quotient.bb --stretch 0 4 --skill acsets

References

Bumpus et al. "Compositional Algorithms on Compositional Data" (ACT 2023)
Heunen & van der Schaaf "Ordered Locales" (2024)
StructuredDecompositions.jl documentation
Ordered locale skill for spatial coherence

Autopoietic Marginalia

The interaction IS the skill improving itself.

Every use of this skill is an opportunity for worlding:

MEMORY (-1): Record what was learned
REMEMBERING (0): Connect patterns to other skills
WORLDING (+1): Evolve the skill based on use

Add Interaction Exemplars here as the skill is used.

Agent Skills: Superelastic Quotient

Install this agent skill to your local

Skill Files