In [Part 1], I argued that agentic systems—deterministic, stateful, memory-driven—can outperform monolithic AGI for most practical tasks. But I glossed over the how.
How do multiple agents actually coordinate? How do they maintain consistency without becoming a single, fused model?
The answer comes from an unexpected place: multivariable calculus.
The Problem with One Coordinate System
When you solve an integral, you choose a coordinate system:
- Cartesian ((x, y)) for grids and linear structures
- Polar ((r, \theta)) for circles and rotations
- Spherical ((r, \theta, \phi)) for 3D surfaces
Here's the thing: the same problem yields vastly different complexity depending on your choice.
Compute the area of a circle with radius (R):
Cartesian:
[
\int_{-R}^{R} \int_{-\sqrt{R^2-x^2}}^{\sqrt{R^2-x^2}} dy \, dx = \pi R^2
]
Messy. Square roots. Trigonometric substitution. Four pages of algebra.
Polar:
[
\int_{0}^{2\pi} \int_{0}^{R} r \, dr \, d\theta = \pi R^2
]
Clean. Two lines. Same answer.
Intelligence as Coordinate Selection
Current AI forces everything through one coordinate system:
- Vision models: Pixel-space (Cartesian-like grid)
- Language models: Token-space (sequential, linear)
- World models: Geometric-space (3D reconstruction)
But reality doesn't care about your coordinates. A self-driving car doesn't "see" pixels. It doesn't "think" in tokens. It navigates a unified field that requires multiple simultaneous expansions:
| Reality Aspect | Best Coordinate System | Brain Region |
|---|---|---|
| Spatial geometry | Polar/Spherical | Visual cortex |
| Temporal prediction | Sequential/Time | Temporal cortex |
| Physical intuition | Force/Embodied | Motor cortex |
| Social protocol | State-machine/Boolean | Prefrontal cortex |
The brain doesn't fuse these. It runs them in parallel and orchestrates the results.
The Jacobian of Agent Coordination
In calculus, when you switch coordinates, you multiply by the Jacobian determinant to maintain consistency:
[
\iint f(x,y) \, dx \, dy = \iint f(r,\theta) \cdot |J| \, dr \, d\theta
]
Where:
[
J = \begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{vmatrix} = r
]
In agentic systems, the Jacobian is the communication protocol. It ensures that when Agent A (vision) says "obstacle at 3 meters" and Agent B (memory) says "this intersection has blind spots," both map to the same objective reality—even though they computed it through different internal representations.
Real-Time Coordination: The Hierarchical Jacobian
Here's where it gets practical. You can't wait for full consensus in real-time systems.
The brain solves this with hierarchical coordination:
Millisecond scale (Reflex):
└── Agent: Motor controller
└── Input: Sensor state
└── Action: Immediate (no consensus needed)
└── Latency: < 1ms
Centisecond scale (Tactical):
└── Agents: Vision + Memory + Prediction
└── Protocol: Async message passing (ZeroMQ/gRPC)
└── Action: Coordinated response
└── Latency: 10-50ms
Second scale (Strategic):
└── Agents: Full swarm consensus
└── Protocol: Synchronous (Paxos/Raft)
└── Action: Goal reassignment
└── Latency: 100ms+
The "Jacobian"—the protocol ensuring consistency—becomes time-varying:
[
J(t) = \begin{cases}
I & \text{if } t < \tau_{\text{reflex}} \text{ (identity, no transform)} \
J_{\text{async}}(t) & \text{if } \tau_{\text{reflex}} < t < \tau_{\text{strategic}} \
J_{\text{consensus}} & \text{if } t > \tau_{\text{strategic}}
\end{cases}
]
Implementation: A Test Automation Example
Consider a flaky test suite. Multiple agents can coordinate in real-time:
# Vision Agent: Coordinate system = DOM structure (hierarchical)
class VisionAgent:
def observe(self):
return {"locator": "#submit-btn", "state": "visible", "confidence": 0.95}
# Memory Agent: Coordinate system = temporal/experience (sequential)
class MemoryAgent:
def recall(self, locator):
return {"locator": locator,
"historical_flakiness": 0.3,
"avg_retry_time": 2.5}
# Protocol Agent: Coordinate system = deterministic state (Boolean)
class ProtocolAgent:
def should_retry(self, vision, memory):
if vision["confidence"] < 0.9 and memory["historical_flakiness"] > 0.2:
return True
return False
# The "Jacobian"—consensus protocol
def coordinate(vision_agent, memory_agent, protocol_agent, locator):
vision_data = vision_agent.observe()
memory_data = memory_agent.recall(locator)
# This is the coordinate transformation ensuring consistency
decision = protocol_agent.should_retry(vision_data, memory_data)
return decision
Each agent operates in its own "coordinate system"—pixels, history, rules. The protocol ensures they map to the same decision.
Why This Matters for Engineers
1. Modularity without fusion
You don't need to train one giant model that "understands" both vision and history. You need:
- A vision model (existing, pre-trained)
- A memory database (deterministic, queryable)
- A protocol layer (your engineering contribution)
2. Deterministic safety
The Jacobian/protocol is explicit, auditable, and controllable. Unlike the latent space of a fused neural network, you can inspect and modify the consensus logic.
3. Latency optimization
By keeping coordinate systems separate, you can cache, parallelize, and optimize each independently. The vision agent can run on GPU. The memory agent on a fast key-value store. The protocol on edge.
The Conclusion
Intelligence is not about having the biggest, most unified model. It's about having the right coordinate system for the problem—and the protocol to coordinate between them.
Monolithic AI is like solving every integral in Cartesian coordinates because that's what your library supports. It works. It's slow. It's error-prone.
Agentic AI is like having a library that can switch to Polar, Spherical, or any coordinate system the problem demands—then orchestrating the results through a Jacobian that ensures consistency.
The future isn't bigger models. It's better coordinate systems.
Part 3—The Real-Time Jacobian: Hierarchical Control for Millisecond-Scale Agent Coordination
Top comments (0)