polynomial-social-cognition

The Core Insight

"Agent" was hiding POLYNOMIAL STRUCTURE

What we called "Agent" is actually a polynomial functor:

p(y) = Σ_{i ∈ p(1)} y^{p[i]}

p(1) = positions (states, observations, what I can perceive)
p[i] = directions (actions, responses from position i)

Why Polynomial?

p ∈ Poly

i ∈ p(1)

a ∈ p[i]

p → q

p + q

p ◁ q

p ⊗ q

p/~

Mode 1 vs Mode 2 as Polynomials

Mode 1 (Innate, Fast, Subcortical):
    p₁(y) = 3y³
    
    3 positions: {threat, safe, ambiguous}
    3 directions each: {approach, freeze, avoid}
    
    FIXED polynomial - phylogenetically determined
    Fast because structure is precomputed (50ms)

Mode 2 (Interpretive, Slow, Prefrontal):
    p₂(y) = Σ_{c ∈ Context} y^{Actions(c)}
    
    Positions indexed by CONTEXT (variable)
    Directions depend on interpretation (learned)
    
    VARIABLE polynomial - ontogenetically acquired
    Slow because structure must be computed (200ms+)

GF(3) as Polynomial Grading

The trit polynomial:

    𝟛(y) = y⁻¹ + y⁰ + y¹  =  y⁻¹ + 1 + y
         
    3 positions: {-1, 0, +1}
    Valence determines which sub-polynomial is active
    
Social polynomial as GF(3)-graded:

    p(y) = p₋₁(y) + p₀(y) + p₊₁(y)
    
    p₋₁ = avoid polynomial (threat responses)
    p₀  = neutral polynomial (observation)
    p₊₁ = approach polynomial (affiliation)
    
Conservation: morphisms preserve grading
    f: p → q  implies  Σ trits(p) ≡ Σ trits(q) (mod 3)

Feigenbaum Bifurcation = Polynomial Insufficiency

At each social scale transition, the current polynomial operations become insufficient:

p ⊗ p

p ⊗ p ⊗ p + Coalition

+

Σ_{rep} p^⊗n

Π_{myth} (Σ_{rep} p^⊗n)

p/~

The Feigenbaum constant δ ≈ 4.669 measures the rate at which polynomial complexity must increase.

Polynomial Operations

from discopy.monoidal import Ty, Box

# Types
Poly = Ty('𝑝')           # Generic polynomial
Pos = Ty('Pos')          # Positions
Dir = Ty('Dir')          # Directions

# Operations
tensor = Box('⊗', Poly @ Poly, Ty('𝑝⊗𝑞'))      # Parallel (Market)
compose = Box('◁', Poly @ Poly, Ty('𝑝◁𝑞'))     # Wiring (Authority)
cosum = Box('+', Poly @ Poly, Ty('𝑝+𝑞'))       # Coproduct (Communal)
quotient = Box('/~', Poly, Ty('𝑝/∼'))          # Quotient (Institution)

# Relational models as polynomial morphisms
communal = Box('Communal', Poly @ Poly, Ty('(𝑝+𝑞)/∼'))
authority = Box('Authority', Poly @ Poly, Ty('𝑝◁𝑞'))
equality = Box('Equality', Poly @ Poly, Ty('𝑝⊗𝑞+swap'))
market = Box('Market', Poly @ Poly, Ty('𝑝⊗𝑞'))

Trampoline as Lens Morphism

The Mode 1 ↔ Mode 2 trampoline is a lens:

    (get, put): p₂ ⇄ p₁
    
    get: p₂(1) → p₁(1)
        Compress context-indexed positions to 3 valence states
        (This is Mode 2 → Mode 1: FORGET)
        
    put: Σᵢ p₂[i] × p₁[get(i)] → p₂[i]
        Elaborate 3 actions to context-appropriate responses
        (This is Mode 1 → Mode 2: GENERATE)

Wounded Cue = Partial Lens

Clean signal:   p → q is a LENS (total, invertible-ish)
Wounded signal: p → q is a PARTIAL lens (gaps in fiber)

Error correction strategies:
    Interpolate: Extend by nearby fibers (secure attachment)
    Mute: Send to terminal polynomial 1 (avoidant)
    Raw: Output the partiality as data (anxious)

Connection to Spivak/Niu Poly

From David Spivak and Nelson Niu's work:

• Poly is the category of polynomial functors
• Morphisms are lenses (charts + cocharts)
• ⊗ (tensor) and ◁ (composition) form a duoidal structure
• Polynomial comonads model dynamical systems
• The arena perspective: positions = observations, directions = moves

DiscoHy Integration

;; discohy polynomial social cognition
(import discopy.monoidal [Ty Box])

(setv Poly (Ty "𝑝"))
(setv Mode1 (Ty "3y³"))
(setv Mode2 (Ty "Σ_c y^A(c)"))

;; Relational models
(setv communal (Box "Communal" (@ Poly Poly) (Ty "(𝑝+𝑞)/∼")))
(setv authority (Box "Authority" (@ Poly Poly) (Ty "𝑝◁𝑞")))
(setv market (Box "Market" (@ Poly Poly) (Ty "𝑝⊗𝑞")))

;; Bifurcation = adding polynomial structure
(defn bifurcate [p new-structure]
  (Box (+ "+" new-structure) p (Ty (+ "𝑝+" new-structure))))

Mathpix Integration (for paper extraction)

This skill is designed for extracting polynomial/categorical structures from papers:

# Extract polynomial formulas from paper images
from mathpix import extract_latex

# Look for patterns like:
# p(y) = Σ...
# Poly morphisms
# Lens diagrams
# Duoidal structure

POLYNOMIAL_PATTERNS = [
    r'p\(y\)\s*=',           # Polynomial definition
    r'\\Sigma.*y\^',          # Σ-indexed
    r'\\otimes|\\triangleleft', # Tensor/composition
    r'Poly|Arena|Lens',       # Category names
]

GF(3) Triads

polynomial-social-cognition (0) ⊗ discohy-streams (-1) ⊗ gay-mcp (+1) = 0 ✓
feigenbaum-bifurcation (-1) ⊗ polynomial-social-cognition (0) ⊗ unworld (+1) = 0 ✓

References

• Spivak, D. & Niu, N. - Polynomial Functors: A General Theory of Interaction
• Shapiro, B. & Spivak, D. - Dynamic Categories, Dynamic Operads (arXiv:2205.03906)
• Fiske, A. - Structures of Social Life (1991) - Relational models
• Friston, K. - Active inference as polynomial dynamics

Files

• /Users/bob/iii/src/feigenbaum_social_render.py

  - DisCoPy implementation

• /Users/bob/iii/src/feigenbaum_extended.py

  - Bifurcation dynamics

• /Users/bob/iii/src/discohy_feigenbaum_social.hy

  - Hy implementation

• /Users/bob/iii/src/feigenbaum_diagrams/

  - Rendered diagrams

Key Terminology Update

p(y)

p(1)

p[i]

p → q

See Also

• discohy-streams

  - Operadic color streams

• unworld

  - Derivational pattern generation

• gay-mcp

  - GF(3) conservation

• asi-polynomial-operads

  - Polynomial + operads + open games

• parametrised-optics-cybernetics

  - Para(C) and agency

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.