# Polynomial Social Cognition > Social cognition via polynomial functors - "Agent" IS a polynomial p(y) = Σᵢy^{Aᵢ} with positions (observations) and directions (actions). Integrates Spivak/Niu Poly, Feigenbaum bifurcation, Mode 1/Mode 2, and GF(3) grading. - Skill: `plurigrid/polynomial-social-cognition` (Agent Skill, multi-file: 2 files) - Install (CLI): `npx skillmds@latest add plurigrid/polynomial-social-cognition` - Raw SKILL.md: https://api.skillmd.com/api/skills/plurigrid/polynomial-social-cognition/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: AI & ML - Author: plurigrid (https://skillmd.com/u/plurigrid) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/plurigrid/polynomial-social-cognition --- # 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? | Old concept | Polynomial equivalent | |-------------|----------------------| | Agent | `p ∈ Poly` | | Agent state | Position `i ∈ p(1)` | | Agent action | Direction `a ∈ p[i]` | | Relationship | Lens morphism `p → q` | | Coalition | Coproduct `p + q` | | Hierarchy | Composition `p ◁ q` | | Market | Tensor `p ⊗ q` | | Institution | Quotient `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: | Scale | n | Polynomial | New Structure Required | |-------|---|------------|----------------------| | Dyad | 2 | `p ⊗ p` | None (base case) | | Triad | 3 | `p ⊗ p ⊗ p + Coalition` | Coproduct `+` | | Band | 15 | `Σ_{rep} p^⊗n` | Σ-indexing | | Tribe | 50 | `Π_{myth} (Σ_{rep} p^⊗n)` | Π-structure | | Institution | 150+ | `p/~` | Quotient by role | The Feigenbaum constant δ ≈ 4.669 measures the **rate** at which polynomial complexity must increase. ## Polynomial Operations ```python 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 ```clojure ;; 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: ```python # 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 | Old Term | New Term | Polynomial | |----------|----------|------------| | Agent | Poly / Interface | `p(y)` | | State | Position | `p(1)` | | Action | Direction | `p[i]` | | Relationship | Lens | `p → q` | | Social network | Poly diagram | Composition of lenses | ## 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.*