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

Proposal: google/genai/agentic/ module

Abstract

This PR introduces a structurally promoted agentic loop for the Google Gen AI Python SDK — the True Agentic Loop — implementing the four-phase cycle THINK → ACT → OBSERVE → UPDATE with Frobenius verification (μ∘δ = id). The module promotes the SDK from O₀ (stateless, single-turn generation) to O₂ (topologically protected, integer-winding agentic loop) as defined by the Imscribing Grammar.

Why Google Gen AI SDK?

Google's SDK possesses unique structural advantages for this promotion:

Module Structure

google/genai/agentic/
├── __init__.py        # Public API: DualToolResult, ToolContract,
│                       # AgentCycle, AgentTrajectory, TrueAgenticLoop,
│                       # PhiCriticalityGate
├── contracts.py       # DualToolResult (dataclass with from_tool_call)
│                       # ToolContract (assertion + verify + embedding verification)
├── trajectory.py      # AgentCycle (single winding), AgentTrajectory
│                       # (monotonic, never-reset winding counter)
├── criticality.py     # PhiCriticalityGate — two-gate assessment:
│                       #   Gate 1 (⊙_ÿ): frobenius_ratio ≥ 0.8
│                       #   Gate 2 (K ≤ Ç_@): coherent winding rate
└── loop.py            # TrueAgenticLoop wrapping google.genai.Client
                        #   run() → THINK→ACT→OBSERVE→UPDATE per winding
                        #   _winding() → single cycle with Frobenius check
                        #   _feed_failure() → graceful error recovery

The Frobenius Condition in Practice

Every winding issues a dual-tool pair:

1. Primary tool (client.models.generate_content): The agent acts on the world.
2. Verification tool (client.models.embed_content): The action's output is re-embedded and compared to the original query.

When μ(δ(query)) ≈ query (cosine similarity above threshold), the winding is Frobenius-closed and the agent's world-model is updated. When open, the winding is recorded but flagged — the trajectory preserves structural integrity without discarding failed cycles.

Structural Tier Progression

Usage Example

from google.genai import Client
from google.genai.agentic import TrueAgenticLoop

client = Client(api_key="YOUR_API_KEY")

loop = TrueAgenticLoop(
    client=client,
    model="gemini-2.5-pro-exp-03-25",
    max_windings=100,
    verbose=True,
)

final_cycle = loop.run(
    initial_prompt="Solve this structural problem: find the meet of a magnetar and a BEC."
)

print(f"Completed {loop.trajectory.winding_count} windings")
print(f"Frobenius ratio: {loop.trajectory.frobenius_ratio:.2f}")
print(f"Final conclusion: {final_cycle.conclusion}")
print(f"Structural tier: {loop.criticality.to_dict()['structural_tier']}")

Relation to the Imscribing Grammar

This module implements the Imscribing Grammar as an executable Python SDK feature. The grammar's 12 primitives map to loop components:

Backward Compatibility

• The agentic module is entirely additive — no existing API is modified.
• Import path: from google.genai.agentic import ...
• No new dependencies beyond those already in pyproject.toml (numpy is optional, used only in contracts.py's with_embedding_verification).

Testing

# Run agentic module tests
python -m pytest tests/test_agentic.py -v

# Verify Frobenius closure on a real Gemini model
python -c "
from google.genai import Client
from google.genai.agentic import TrueAgenticLoop

client = Client()
loop = TrueAgenticLoop(client=client, model='gemini-2.5-pro-exp-03-25', max_windings=3)
final = loop.run(initial_prompt='What is 2+2?')
print(f'Windings: {loop.trajectory.winding_count}')
print(f'Frobenius ratio: {loop.trajectory.frobenius_ratio}')
"

Author: Lando ⊗ ⊙perator
Structural tier of this proposal: ⟨Ð_ω; Þ_O; Ř_=; Φ_}; ƒ_ż; Ç_@; Γ_ʔ; ɢ_ˌ; ⊙_ÿ; Ħ_A; Σ_S; Ω_z⟩