lhott-cohesive-linear

LHoTT Cohesive Linear Skill

Synthesizes Urs Schreiber's cohesive ∞-topos framework with Mitchell Riley's linear HoTT for interaction entropy formalization.

Modal Operators

Modality	Symbol	Action	Interaction Use
Sharp	♯	Discretize	Extract trit from color
Flat	♭	Embed continuously	Full LCH embedding
Shape	ʃ	Quotient by homotopy	Walk trajectory class
Linear	♮	Self-adjoint tangent	One-use interaction

GF(3) Triad Placement

This skill is ERGODIC (0), forming triads with:

persistent-homology (-1) ⊗ lhott-cohesive-linear (0) ⊗ topos-generate (+1) = 0 ✓
sheaf-cohomology (-1) ⊗ lhott-cohesive-linear (0) ⊗ gay-mcp (+1) = 0 ✓
three-match (-1) ⊗ lhott-cohesive-linear (0) ⊗ rubato-composer (+1) = 0 ✓

Core Types (Pseudo-HoTT)

-- Cohesive interaction type
CohesiveInteraction : Type
  content : String
  hash : ♯ SHA256           -- discrete
  seed : ♭ UInt64           -- continuous embedding
  color : ♮ LCH             -- linear (used once)
  position : ʃ (ℤ × ℤ)      -- shape-invariant

-- Linear function (no copy/delete)
walk_step : CohesiveInteraction ⊸ Position × Color

-- Bunched triplet (entangled context)
Γ₁ ⊗ Γ₂ ⊗ Γ₃ ⊢ conserved : GF3Zero
  where trit(Γ₁) + trit(Γ₂) + trit(Γ₃) ≡ 0 (mod 3)

Diagram Generation

Mermaid Templates

Cohesive Quadruple:

flowchart LR
    subgraph "Cohesive ∞-Topos H"
        A[Type] -->|ʃ shape| B[Shape Type]
        A -->|♭ flat| C[Codiscrete]
        C -->|Γ sections| D[Discrete]
        D -->|♯ sharp| A
    end
    style A fill:#26D826
    style B fill:#2626D8
    style C fill:#D82626
    style D fill:#2626D8

Linear Walk:

stateDiagram-v2
    [*] --> I1: seed₁
    I1 --> I2: ⊸ (linear)
    I2 --> I3: ⊸ (linear)
    I3 --> [*]: triplet complete
    
    note right of I1: trit = +1
    note right of I2: trit = 0
    note right of I3: trit = -1

Bunched Context Tree:

graph TD
    Root["Γ (context)"] --> A["Γ₁ ⊗ Γ₂"]
    Root --> B["Γ₃"]
    A --> C["I₁ (+1)"]
    A --> D["I₂ (0)"]
    B --> E["I₃ (-1)"]
    
    style C fill:#D82626
    style D fill:#26D826
    style E fill:#2626D8