Pre-Agent Ontology Skill Skill

Pre-Agent Ontology Skill

Trit: 0 (ERGODIC - coordinates the ontology)

Foundational 5-layer ontology for agent-o-rama. Agents are not primitives—they emerge from derivations, stalks, and sections when gluing succeeds.

Related Skills

unworld: Derivational succession (Layer 1)
sheaf-cohomology: Gluing verification (Layer 2)
bisimulation-game: Observational equivalence (Layer 2)
acsets: Categorical database structure (Layer 2)

5-Layer Hierarchy

Layer 4: EMERGENT        agent, skill, experiment
            ↑
Layer 3: OPERATIONAL     node, emit, aggregation, result
            ↑
Layer 2: SHEAF           stalk, section, cohomology
            ↑
Layer 1: DERIVATIONAL    derivation, chain
            ↑
Layer 0: PRE-ONTOLOGICAL seed, trit, γ (gamma)

Layer 0: Pre-Ontological (Absolute Primitives)

| Term | Type | Definition | |------|------|------------| | seed | uint64 | Deterministic state replacing time | | trit | {-1, 0, +1} | GF(3) charge element | | γ | constant | 0x9E3779B97F4A7C15 (golden ratio bits) |

Layer 1: Derivational

| Term | Type | Definition | |------|------|------------| | derivation | (Seed × Trit) → (Seed × Section) | Fundamental computation unit | | chain | [Seed] | Sequence of derived seeds |

Rule: seed_{n+1} = splitmix64(seed_n ⊕ (trit_n × γ))

Layer 2: Sheaf-Theoretic

| Term | Type | Definition | |------|------|------------| | stalk | Set(Section) | Collection of sections over one trit | | section | local data | Output of derivation, can glue | | cohomology | (H⁰, H¹) | Global sections and obstructions |

Stalk Distribution (2-3-2):

MINUS:   2 elements, trit=-1, role=validator
ERGODIC: 3 elements, trit=0,  role=coordinator
PLUS:    2 elements, trit=+1, role=generator

Verification: 2(-1) + 3(0) + 2(+1) = 0 ✓

Layer 3: Operational

| Term | Sheaf Correspondence | |------|---------------------| | node | section-producer | | emit | stalk transition | | aggregation | gluing (cocycle check) | | result | global section (H⁰ element) |

Layer 4: Emergent

| Term | Definition | |------|------------| | agent | Fiber bundle over trit poset {-1, 0, +1} | | skill | Executable section (self-contained knowledge) | | experiment | Derivation chain evaluation |

Key: Agent = Observation(Bundle(Stalk₋₁, Stalk₀, Stalk₊₁))

5 Stability Invariants

| ID | Invariant | Formula | Verified By | |----|-----------|---------|-------------| | I1 | GF(3) Conservation | Σ trits ≡ 0 (mod 3) | aggregation, trifurcate | | I2 | Determinism | derive(s,t) = derive(s,t) | seed chaining | | I3 | Order Independence (SPI) | parallel(f) = sequential(f) | spi-parallel-verify | | I4 | Gluing (Cocycle) | g_ij ∘ g_jk = g_ik | cohomology check | | I5 | Bisimulation | A ~ B ⟺ ∀obs. obs(A) = obs(B) | bisimulation-game |

Key Primitives

Seed

64-bit unsigned integer replacing temporal state. Same seed → identical derivation chains.

Trit

GF(3) element in {-1, 0, +1}. Forms a field under modular arithmetic.

Derivation

Fundamental computation: (Seed × Trit) → (Seed × Section)

Replaces temporal succession with seed-based chaining.

Stalk

Collection of sections over a single trit value. Organized as fiber bundle.

Section

Local data produced by derivation. Sections glue to form global sections when cocycle condition holds.

Trifurcation Pattern (MANDATORY)

Every operation MUST split into three sub-derivations:

             intent
                │
       ┌────────┼────────┐
       ▼        ▼        ▼
    MINUS    ERGODIC   PLUS
    (-1)      (0)      (+1)
   validate  coordinate generate
       │        │        │
       └────────┼────────┘
                ▼
           aggregate
         (verify Σ=0)

| Sub-Agent | Trit | Role | Skills | |-----------|------|------|--------| | MINUS | -1 | Validator | spi-parallel-verify, bisimulation-game | | ERGODIC | 0 | Coordinator | glass-bead-game, triad-interleave | | PLUS | +1 | Generator | gflownet, self-evolving-agent |

Clojure Implementation

Constants

(def GENESIS-SEED 0x42D)
(def GAMMA 0x9E3779B97F4A7C15)
(def MIX1 0xBF58476D1CE4E5B9)
(def MIX2 0x94D049BB133111EB)

Derivation

(defn derive-seed [seed trit]
  (let [adjusted (bit-xor seed (* trit GAMMA))]
    (splitmix64-next adjusted)))

(defn derivation-chain [genesis-seed trits]
  (reductions derive-seed genesis-seed trits))

GF(3) Arithmetic

(defn gf3-add [a b]
  (let [sum (+ a b)]
    (cond (> sum 1)  (- sum 3)
          (< sum -1) (+ sum 3)
          :else      sum)))

(defn gf3-conserved? [trits]
  (zero? (reduce gf3-add 0 trits)))

Trifurcate Pattern

;; Scatter: emit three roles
(aor/agg-start-node
 "scatter"
 "execute-role"
 (fn [agent-node {:keys [intent]}]
   (aor/emit! agent-node "execute-role" {:intent intent :role :minus})
   (aor/emit! agent-node "execute-role" {:intent intent :role :ergodic})
   (aor/emit! agent-node "execute-role" {:intent intent :role :plus})))

;; Gather: verify conservation
(aor/agg-node
 "execute-role"
 nil
 aggs/+vec-agg
 (fn [agent-node requests _]
   (let [results (mapv execute-role requests)
         trits (mapv :trit results)
         conserved? (gf3-conserved? trits)]
     (aor/result! agent-node {:conserved conserved? :trits trits}))))

Verification Checklist

Before any operation completes:

[ ] GF(3) sum ≡ 0 (mod 3)
[ ] All three trits represented (trifurcation)
[ ] Cocycle condition satisfied (sections glue)
[ ] Deterministic (same seed → same result)
[ ] Order-independent (SPI holds)

Sources

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

general: 734 citations in bib.duckdb

Cat# Integration

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

Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.

Agent Skills: Pre-Agent Ontology Skill

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