Sheaf Cohomology

Sheaf Cohomology Skill: Local-to-Global Verification

Status: ✅ Production Ready Trit: -1 (MINUS - validator/constraint) Color: #2626D8 (Blue) Principle: Local consistency → Global correctness Frame: Čech cohomology with descent conditions

Overview

Sheaf Cohomology validates that locally consistent data/code patches glue correctly into globally consistent structures. Uses:

1. Čech cohomology: H^n(U, F) obstruction classes
2. Nerve of coverage: N(U) simplicial complex from open cover
3. Descent conditions: Cocycle conditions for morphisms
4. tree-sitter integration: AST-level local consistency

Correct by construction: If local patches satisfy cocycle conditions, global structure is guaranteed.

Core Formula

H⁰(U, F) = ker(d⁰)           # Global sections (agree everywhere)
H¹(U, F) = ker(d¹)/im(d⁰)    # Obstruction to gluing
H²(U, F) = ker(d²)/im(d¹)    # Higher obstructions

For code verification:

# Three patches (files/modules) are consistent iff:
# On U_ij ∩ U_jk ∩ U_ik: g_ij ∘ g_jk = g_ik  (cocycle condition)

cocycle_satisfied?(patch_i, patch_j, patch_k)
  == (compose(g_ij, g_jk) == g_ik)

Why Sheaf Cohomology for Code?

1. Module boundaries: Each module is an "open set"
2. Import/export: Transition functions between patches
3. Type consistency: Cocycle = type compatibility
4. Refactoring safety: H¹ = 0 means safe global transform

Gadgets

1. ČechCoverVerifier

Verify local consistency across code patches:

verifier = SheafCohomology::CechCoverVerifier.new(
  coverage: [:module_a, :module_b, :module_c]
)
verifier.add_transition(:module_a, :module_b, transition_ab)
verifier.add_transition(:module_b, :module_c, transition_bc)
verifier.add_transition(:module_a, :module_c, transition_ac)

verifier.cocycle_satisfied?  # => true if g_ab ∘ g_bc = g_ac
verifier.h1_obstruction      # => 0 if globally consistent

2. NerveConstructor

Build simplicial complex from coverage:

nerve = SheafCohomology::NerveConstructor.new(
  opens: file_modules,
  intersections: shared_interfaces
)
nerve.simplices(0)  # => vertices (modules)
nerve.simplices(1)  # => edges (shared interfaces)
nerve.simplices(2)  # => triangles (triple overlaps)
nerve.euler_characteristic  # => χ(N(U))

3. DescentVerifier

Check morphism descent conditions:

descent = SheafCohomology::DescentVerifier.new
descent.add_local_section(:patch_a, section_a)
descent.add_local_section(:patch_b, section_b)
descent.verify_descent!  # => raises if descent fails
descent.global_section   # => glued global section

4. TreeSitterSheaf

Integration with tree-sitter for AST verification:

sheaf = SheafCohomology::TreeSitterSheaf.new(
  language: :clojure,
  coverage: :function_boundaries
)
sheaf.parse_file("src/core.clj")
sheaf.local_consistency_check  # => per-function type consistency
sheaf.global_gluing_check      # => cross-function compatibility
sheaf.h1_obstructions          # => list of gluing failures

Commands

# Verify sheaf consistency
just sheaf-check

# Check specific coverage
just sheaf-coverage src/

# Compute cohomology obstructions
just sheaf-h1

# Integration with tree-sitter
just sheaf-ast src/*.clj

API

require 'sheaf_cohomology'

# Create verifier
verifier = SheafCohomology::Verifier.new(
  trit: -1,
  coverage_strategy: :ast_boundaries
)

# Add patches from tree-sitter
verifier.add_patches_from_ast(parsed_files)

# Check consistency
result = verifier.verify!
result[:h0]  # Global sections (fully consistent)
result[:h1]  # Gluing obstructions
result[:h2]  # Higher obstructions
result[:gf3_conserved]  # GF(3) sum = 0 with triad

Integration with GF(3) Triads

Forms valid triads with ERGODIC (0) and PLUS (+1) skills:

sheaf-cohomology (-1) ⊗ acsets (0) ⊗ gay-mcp (+1) = 0 ✓
sheaf-cohomology (-1) ⊗ unworld (0) ⊗ rama-gay-clojure (+1) = 0 ✓
sheaf-cohomology (-1) ⊗ glass-bead-game (0) ⊗ rubato-composer (+1) = 0 ✓

Mathematical Foundation

Čech Complex

C⁰ = ∏_i F(U_i)
C¹ = ∏_{i<j} F(U_i ∩ U_j)
C² = ∏_{i<j<k} F(U_i ∩ U_j ∩ U_k)

Differential Maps

d⁰: C⁰ → C¹:  (d⁰s)_{ij} = s_j|_{U_ij} - s_i|_{U_ij}
d¹: C¹ → C²:  (d¹g)_{ijk} = g_jk - g_ik + g_ij

Cocycle Condition

g ∈ Z¹(U, F) ⟺ d¹g = 0 ⟺ g_ij + g_jk = g_ik on triple overlaps

Example Output

─── Sheaf Cohomology Verification ───
Coverage: 5 modules, 8 interfaces, 3 triple overlaps

Local Consistency:
  ✓ module_a ∩ module_b: types compatible
  ✓ module_b ∩ module_c: types compatible
  ✓ module_a ∩ module_c: types compatible

Čech Cohomology:
  H⁰ = 1 (connected components)
  H¹ = 0 (no gluing obstructions)
  H² = 0 (no higher obstructions)

Global Structure: ✓ CONSISTENT
GF(3) Trit: -1 (MINUS/Validator)

Skill Name: sheaf-cohomology Type: Local-to-Global Verification Trit: -1 (MINUS) Color: #2626D8 (Blue) GF(3): Forms valid triads with ERGODIC + PLUS skills

Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

Graph Theory

• networkx [○] via bicomodule
  • Universal graph hub

Bibliography References

• category-theory: 139 citations in bib.duckdb

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 8. Degeneracy

Concepts: redundancy, fallback, multiple strategies, robustness

GF(3) Balanced Triad

sheaf-cohomology (−) + SDF.Ch8 (−) + [balancer] (−) = 0

Skill Trit: -1 (MINUS - verification)

Secondary Chapters

• Ch1: Flexibility through Abstraction
• Ch2: Domain-Specific Languages
• Ch10: Adventure Game Example
• Ch7: Propagators

Connection Pattern

Degeneracy provides fallbacks. This skill offers redundant strategies.

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:

Trit: -1 (MINUS)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #FF6B6B

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.