Add split-point circuit theory and worked examples to README

[Copilot]

The README did not explain the theoretical structure behind the repo’s split-point environment or show concrete split-point-built examples. This updates the top-level documentation to make the addition-split interpretation, automatic factorization step, and worked examples explicit.

• Theory of split-point-built circuits

  • Clarified that the agent’s primary decision is to choose an addition split point f = g + h.
  • Documented that, after the split is chosen, the environment automatically factors g and h into multiplicative subexpressions to be built.
  • Connected that view to the repo’s path-multiplicity analysis (shortest_path_count, multiple_shortest_paths, shortest_path_samples).

• Worked examples

  • Added an addition-split example for x0 + x1*x2, showing how the environment factors the two sides after the split.
  • Added the requested example for x^2 + 2xy + y^2 + 1, illustrating that the optimal split is (x+y)^2 + 1.
  • Included step tables for each example so the recursive split/factor flow is easy to scan.

• Circuit diagrams

  • Added Mermaid diagrams for both examples.
  • The second diagram explicitly shows that the x+y subgoal is solved once and then reused for both factors of (x+y)^2.

flowchart TD
    f2["x^2 + 2xy + y^2 + 1"] --> s2{{agent picks split}}
    s2 --> sq["(x+y)^2"]
    s2 --> one["1"]
    sq --> ef2{{environment factors}}
    ef2 --> slot1["factor slot 1"]
    ef2 --> slot2["factor slot 2"]
    s3{{agent splits x+y once}} --> x["x"]
    s3 --> y["y"]
    x --> xyres["computed x+y"]
    y --> xyres
    xyres -. reused for .-> slot1
    xyres -. reused for .-> slot2

Loading

Original prompt

Carefully scan the repo to understand the scripts and add the theoretic details and worked out examples of split-point-built circuits to the README.md. Draw circuit diagrams for those examples.

The user has attached the following file paths as relevant context:

• AlphaZero RL Circuit Discovery Plan.md
• README.md

Created from VS Code.

[zengrf]

@copilot Wait, did you really understand the environment? My understanding is we take in a poly, and the agent's sole job is to find addition split points. Once that's done, then the environment automatically factors the two parts and returns a list of factors as subexpressions to be built. Illustrate this with example split points of x^2 + 2xy + y^2 + 1 (the optimal one is to split as (x+y)^2 + 1)

[Copilot]

@copilot Wait, did you really understand the environment? My understanding is we take in a poly, and the agent's sole job is to find addition split points. Once that's done, then the environment autom...

Updated the README to reflect the addition-split environment, not direct gate selection, and added the requested worked example for x^2 + 2xy + y^2 + 1 with optimal split (x+y)^2 + 1, including the factorization/reuse diagram. Addressed in 16f5080.