[Roadmap] The Evolution of IU-SR: E-graphs and Pure Geometric Logic

[Concode0]

The Current State of Versor SR ( IU-SR )

The Iterative Geometric Unbending (IU-SR) pipeline introduced in v1.0.0 proved that deterministic symbolic regression is possible using Clifford Algebra. However, as we push it to handle deeper formulas, we've hit a wall.

I recently started applying a Coherence-based grouper and structural feedback loops, but the codebase is becoming a bit tangled. Rather than applying more heuristic patches, I want to step back and rethink the architecture with the community.

Here is the roadmap for the next evolution of IU-SR, heavily focusing on a new radical idea: E-graphs.

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The Main Shift: E-graphs for Tree Optimization

The Problem (Numerical Stability in Deep Formula Chains): For complex equations, the accumulation of rotor operations causes numerical instability (gradient explosion) and combinatorial explosion during symbolic tree search.
The Idea: Could we apply E-graph (Equivalence Graph) techniques (used in compiler optimizations like Rust's egg)? Geometric Algebra is governed by strict deterministic rewrite rules (e.g., $e_1 e_2 = -e_2 e_1$, $R \tilde{R} = 1$). An E-graph could compactly represent the entire space of valid geometric rewrites, allowing the neural network to navigate equivalent expressions without expanding them exponentially.

Ongoing Geometric Refinements

Alongside the E-graph integration, we need to untangle and refine the following core components:

1. Untangling the Coherence Grouper
We are transitioning from Distance Correlation to a pure Coherence-based approach using GA-native phasor relationships. The current implementation is partially applied but structurally tangled. We need a cleaner way to align this grouping logic with the underlying geometric algebra.

2. Adaptive Entry for Implicit Mode
The current criteria for entering Implicit Mode are static (e.g., explicit mode is forced for all first-principles datasets). We need an "intelligent" trigger that analyzes the geometric structure or residual energy to dynamically decide when an implicit representation is required.

3. Elimination of Residual Heuristics
Parts of the refinement phase still rely on non-geometric numerical "tricks" (like standard lstsq). We aim to replace these with purely geometric alternatives—such as isometries, geometric projections, and blade-based rejection.

4. Organic Coupling Between Phases
The transition between Data Prep, Extraction, and Refinement needs to be fluid. We need a better architectural flow to pass geometric constraints back and forth (feedback loops) without tangling the codebase.

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Call for Brainstorming & Collaboration

I am throwing these ideas into the Discussions because solving this requires bridging continuous Deep Learning with discrete Symbolic Computing.

If you have experience with E-graphs or symbolic rewriting, how can we bridge them with PyTorch tensors?
Ideas for stabilizing the new phase-coherence metrics natively in Cl(p,q,r).
Any ideas to replace the remaining heuristics with pure math.
Anything else, if related with this discussions.

You don't need a working PR. If you have a mathematical intuition or a paper we should read, please drop it below.

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Prior Arts & Inspiration:
While searching for ways to stabilize symbolic trees, I found an exploration of E-graphs in Geometric Algebra (e.g., egga by robinka).