#### Four Mathematical Formulas to Unlock Multi-Step Reasoning in AI: WFGY Explained
What separates true reasoning from simple prediction?
While today’s language models can generate fluent text, most still fail at robust, multi-step logical chains—the kind needed for problem-solving, scientific discovery, or even challenging foundational physics.
As part of my “Challenge Einstein” research, I developed an open-source system called WFGY, centered on four explicit mathematical formulas. These modules are designed not only to improve semantic accuracy, but to give LLMs the machinery for genuine iterative reasoning—the so-called “solver loop” long sought in both symbolic and connectionist AI.
The Four Formulas: Building a Solver Loop for AI
1. BBMC (Void Gem): Semantic Residue Correction
Formula: B = I − G + mc²
- I: Input embedding
- G: Ground-truth/target embedding
- mc²: “Semantic inertia” (the model’s context mass)
Purpose:
BBMC measures the semantic “residue” between the model’s output and the ideal ground-truth. This acts like a force that constantly nudges the AI back toward the reasoning path, correcting semantic drift and reducing long-chain hallucinations. It is inspired by physical balancing (energy minimization) to anchor logical inference at each step.
2. BBPF (Progression Gem): Multi-Path Iterative Updates
Formula: BigBig(x) = x + ∑Vi(εi, C) + ∑Wj(Δt, ΔO)Pj
- x: Current state
- Vi(εi, C): Error-based corrections under given context
- Wj(Δt, ΔO)Pj: Temporal and output-weighted adjustments
Purpose:
BBPF allows the model to aggregate feedback and corrections across multiple reasoning paths, not just the current prediction. This mirrors how expert solvers iterate, loop, and refine answers, enabling more robust multi-step inference.
3. BBCR (Reversal Gem): Closed-Loop Reset and Recovery
Formula: Collapse → Reset(St, δB) → Rebirth(St+1, δB)
- St: Current semantic state
- δB: Semantic perturbation at time t
Purpose:
Inspired by dynamical systems and Lyapunov stability, BBCR formalizes a “collapse–reset–rebirth” loop. If the reasoning process breaks down (contradiction, confusion), the system resets to the last stable state and resumes with a controlled update. This provides a mathematical “restart” that ensures stability in long chains and reduces hallucination.
4. BBAM (Focus Gem): Variance-Based Attention Sharpening
Formula: ãᵢ = aᵢ · exp(−γσ(a))
- aᵢ: Attention weight
- σ(a): Variance of attention
- γ: Damping factor
Purpose:
BBAM dynamically sharpens the model’s attention by reducing variance across the attention map, suppressing noisy or distracting paths. This ensures that as reasoning unfolds over multiple steps, the model’s focus remains sharp, improving accuracy and logical consistency.
Why These Formulas Matter
Traditional LLMs are good at “next token prediction,” but true intelligence requires explicit error correction, self-checking, and iterative improvement—just like how scientists or mathematicians solve complex problems.
The four modules of WFGY, when combined, enable LLMs to perform explicit, self-correcting “solver loops” that go far beyond surface-level fluency.
Empirical results:
- Semantic accuracy ↑ 22.4%
- Reasoning success ↑ 42.1%
- Stability ↑ 3.6x (All fully open benchmarks. Details and full math proofs are open source.)
Try It Yourself & Join the Challenge
All formulas, proofs, code, and benchmarks are fully open and free to use.
If you want to see how “mathematical reasoning modules” can transform an LLM’s abilities, or you’re interested in joining the global challenge to rethink AI and even test the limits of Einstein’s theories:
- Full open-access paper: [Zenodo PDF link]
- Project & code: https://github.com/onestardao/WFGY
If you have feedback, critiques, or want to collaborate, let’s push the limits of semantic reasoning—together.
“In an era of AI hype, only truly open, mathematical innovation will advance the frontier.
WFGY is my invitation: Let’s see how far reasoning can go.”
— PSBigBig
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