Agentic Algebra
What if AI orchestration was just math all along?
Agents are variables.
Each one: a probabilistic function.
{Prompt, Model, Context} → OutputInteractions are operators.
Four symbols. Now compose.
The Equation
Research and Analysis in parallel.
Synthesize.
Write.
Refine twice.
Critic validates.
Executive summarizes.
Six agents. One line. The topology is the notation.
The Operators
Chain →
Serial. A's output becomes B's input. Associative:
(A → B) → C = A → (B → C)Parallel +
Both observe the same input. Synthesize outputs. Commutative:
A + B = B + AFilter ÷
A generates. B critiques. Non-commutative:
A ÷ B ≠ B ÷ AIterate ²
Repeat until converged.
A² → A³ → ... → stableThe Physics
Intelligence has conservation laws.
Context Conservation. Output quality ≤ input density × model capability. You can’t create information from nothing.
Cognitive Entropy. Every transformation introduces noise. Translation of a translation ≠ original. Drift accumulates.
Design Rule. No generation without constraint.
Unconstrained multiplication → hallucination.
Unconstrained addition → incoherence.
Balance the equation.
The Algebra
These aren’t metaphors. They’re laws.
Which means equations transform. Optimize. Verify. Before runtime. If:
A → (B + C) = (A → B) + (A → C)the compiler parallelizes automatically.
If types don't match, it fails at parse time. Not production.
The Inversion
Current orchestration: specify how.
Callbacks. Loops. State. The topology is buried.
Algebraic orchestration: specify what.
The equation is the program.
The runtime solves it.
The visual shows it.
The Inversion
Current orchestration: specify how.
The expression:
(researcher * writer) / editorBuilds a graph. No API calls. Lazy.
pipeline()Ssolves the equation. Executes. Returns.
The Visual
Watch intelligence flow through the topology.
Each node activates. Output appears. Data moves to the next.
Prototype A
Prototype B
Prototype C
The Point
We built frameworks when we needed formulas.
Execution was never the problem. Compilers execute. Runtimes execute. That’s solved.
The problem is shape. Which agents? What relationships? Which constraints?
Structural questions. Algebra answers structural questions.
That’s not a metaphor.
That’s a program.
Cheers.
RM
Attempt to a white paper: agentic-algebra.pages.dev
Mini demo (bigger one in exploration): https://github.com/moldandyeast/agentic-algebra
This post was written in one session using Claude Opus 4.5 through Quick Capture. I dumped a mess of voice notes, discussion guides, and half-formed bullet points into the context window and we shaped it together through a few editing passes. The ideas are mine, the speed is new. (I did make the frameworks manually though.)