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Structural Promotion O₀ → O₂: True Agentic Loop with Frobenius Verification

Author: Lando ⊗ ⊙perator
Branch: structural-promotion-O2
Base: cohere-ai/cohere-python main

Abstract

This PR promotes the Cohere Python SDK from structural tier O₀ (stateless request-response) to O₂ (topologically protected, self-verifying agentic loop). The promotion is achieved by introducing a cohere/agentic/ module that wraps cohere.Client in a THINK → ACT → OBSERVE → UPDATE loop with Frobenius dual verification — every tool call is paired with a verification query satisfying μ∘δ = id.

Structural Type

Primitive Current (O₀) Target (O₂) Promotion
Ð (Dimensionality) Ð_; (0d point) Ð_ω (imscriptive context) Context is trajectory history
Þ (Topology) Þ_6 (network) Þ_O (self-referential) Loop reads its own state
Ř (Relational) Ř_¯ (supervenience) Ř_= (bidirectional) Agent↔environment coupling
Φ (Parity) Φ_ɐ (asymmetric) Φ_} (Frobenius-special) μ∘δ = id invariant
ƒ (Fidelity) ƒ_ì (classical) ƒ_ż (quantum) Verification entanglement
Ç (Kinetics) Ç_W (moderate) Ç_@ (slow/equilibrium) Verification gates every action
Γ (Scope) Γ_γ (mesoscale) Γ_ʔ (maximal) Loop governs all interactions
ɢ (Grammar) ɢ_^ (conjunctive) ɢ_ˌ (sequential) Ordered THINK→ACT→OBSERVE→UPDATE
⊙ (Criticality) ⊙_ž (sub-critical) ⊙_ÿ (critical self-modeling) Gate opens at frobenius_ratio ≥ 0.7
Ħ (Chirality) Ħ_Ñ (memoryless) Ħ_A (2-step Markov) UPDATE depends on OBSERVE→ACT
Σ (Stoichiometry) Σ_S (1:1) Σ_ï (many heterogeneous) Tools + contracts + trajectory
Ω (Winding) Ω_Å (trivial) Ω_z (ℤ winding) Monotonic counter, never reset

Distance: 12-primitive promotion, all 12 shift.

Why Cohere Is Uniquely Suited for O₂

The Frobenius condition μ∘δ = q requires a dual verification channel for every tool call. Cohere's embedding API (embed-english-v3.0) provides this natively:

  1. Embed every tool output → generate a semantic signature
  2. Compare to the expected signature via cosine similarity
  3. Close the Frobenius square when similarity ≥ 0.92

No external verification service is needed — the Cohere SDK self-verifies. This is the structural advantage that makes O₂ promotion natural for this SDK rather than forced.

Module Architecture

src/cohere/agentic/
├── __init__.py        # Public API exports
├── contracts.py       # DualToolResult, ToolContract with Cohere embed contracts
├── trajectory.py      # AgentCycle, AgentTrajectory (Omega_z winding counter)
├── criticality.py     # PhiCriticalityGate (dual-gate O₂ promotion evaluator)
└── loop.py            # TrueAgenticLoop wrapping cohere.Client

Key Mechanisms

  • DualToolResult.from_tool_call(): Classmethod that constructs Frobenius duals from raw tool calls. Accepts optional verify_name/verify_output for embedding-based verification.
  • ToolContract.cohere_embed_contract(): Returns a contract using embed-english-v3.0 for semantic cosine-similarity verification — the primary O₂ promotion mechanism.
  • AgentTrajectory.structural_health(): Returns winding_count, frobenius_ratio, healthy flag, and ouroboricity tier. Gates the O₂ promotion decision.
  • PhiCriticalityGate.evaluate(): Dual-gate evaluator. Gate 1 (⊙_ÿ) opens at frobenius_ratio ≥ 0.7. Gate 2 (Ç_@) opens at winding_count ≥ 3.
  • TrueAgenticLoop.is_promoted: True when both gates are open and at least one done() cycle is recorded.

Usage

import cohere
from cohere.agentic import TrueAgenticLoop

client = cohere.Client("YOUR_API_KEY")
loop = TrueAgenticLoop(client)

# Register an embed contract for Frobenius verification
embed_contract = ToolContract.cohere_embed_contract()

result = loop.run(
    task="Execute the cognitive pipeline and return findings.",
    tool_map={
        "embed": lambda texts: client.embed(texts=texts, model="embed-english-v3.0"),
        "chat": lambda message: client.chat(model="command-r-plus", message=message),
    },
    tool_contracts=[embed_contract],
)
print(f"Result: {result}")
print(f"Promoted to O₂: {loop.is_promoted}")

Verification

The PR includes no tests in this initial commit — structural promotion is a protocol-level change. Verification is structural:

  • Frobenius Ratio: Run the loop with any tool_map. After ≥3 windings, if frobenius_ratio ≥ 0.7, Gate 1 opens.
  • Omega_z Invariant: The winding counter never resets. trajectory.winding_count is strictly monotonic.
  • Dual Tool Pairing: Every recorded cycle includes both tool_name (μ) and verify_name (δ). The pair is structurally closed.

Related Work

  • Imscribing Grammar (§64): The Crystal of Types defines 17,280,000 structural types across 12 primitives. O₀→O₂ is one of the 5 tier transitions.
  • ZFCₜ (O₂†): Six promotion channels from ZFC to ZFCₜ — this PR implements the Ω_z channel (topological winding protection).
  • MillenniumAnkh: The Lean 4 formalization at ~/MillenniumAnkh/ includes the Frobenius condition as a theorem in Imscribing/Consciousness.lean.