Four Explicit Formulas for Solver Loops in LLMs: Technical Proposal & Open Questions

Note: English is not my native language, and while all core ideas and formulas are my own, I used AI tools to help draft and clarify some sections for readability. I welcome corrections or feedback on both content and presentation.

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Background & Motivation

LessWrong has an ongoing concern with the limits of current LLMs: Why do even the best models stumble at robust, multi-step reasoning? Why do "solver loops"—where a system doesn't just react, but recursively self-corrects and updates its understanding—remain so elusive for text-based AIs?

After reviewing past LW threads (see Chain-of-Thought, Metacognitive RL), I believe the problem partly comes from the lack of explicit mathematical machinery for semantic residue, multi-path progression, controlled resets, and focus stabilization within our models.

Here I propose four new mathematical modules—each a concise formula—designed to give LLMs an explicit, repeatable "solver loop". I have tested these ideas in practice (in open-source form), but my main goal is to open the technical details to rationalist critique and further development.

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The Four Formulas

1. BBMC (Void Gem): B = I − G + mc²

• I: Model input embedding
• G: Ground-truth/target embedding
• mc²: "Semantic inertia" (model’s own history/context mass)
• Role:
  This formula quantifies the semantic residue—how far the model's current state diverges from ideal reasoning. Think of B as the “error force” that keeps semantic drift in check.
  In practice: At each reasoning step, compute B, then apply corrective forces to keep the reasoning path close to ground-truth, reducing long-chain hallucinations.

2. BBPF (Progression Gem): BigBig(x) = x + ∑Vi(εi, C) + ∑Wj(Δt, ΔO)Pj

• x: Current model state
• Vi(εi, C): Corrective terms for observed errors under context
• Wj(Δt, ΔO)Pj: Temporal/output-weighted stepwise updates
• Role:
  This formula is a multi-path iterative updater, explicitly aggregating error corrections and temporal influences at each step.
  Instead of predicting one token and moving on, the model now loops over all observed discrepancies, recursively adjusting its trajectory—just like a human solver re-evaluates hypotheses after each calculation.

3. BBCR (Reversal Gem): Collapse → Reset(St, δB) → Rebirth(St+1, δB)

• St: Current semantic state
• δB: Semantic perturbation/variance at time t
• Role:
  Inspired by dynamical systems and Lyapunov stability, this module formalizes a collapse–reset–rebirth process.
  When the model's reasoning collapses (contradiction, stuck chain), it resets to the last stable state and resumes with controlled semantic noise.
  This “hard restart” is not ad-hoc but mathematically triggered, ensuring both robustness and exploration.

4. BBAM (Focus Gem): ãᵢ = aᵢ · exp(−γσ(a))

• aᵢ: Attention weights
• σ(a): Variance of attention weights
• γ: Damping factor
• Role:
  This is a variance-based reweighting of attention, suppressing distractor signals and sharpening focus on consistent reasoning paths.
  As reasoning proceeds, the model's attention gets dynamically “sharpened” to prevent drift—a key to sustaining coherent multi-step logic.

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How Do These Enable Solver Loops?

Combined, these modules give the model explicit tools to:

• Quantify and correct for semantic drift at every step (BBMC)
• Aggregate and apply corrections across multiple reasoning paths (BBPF)
• Recover gracefully from logic failures or hallucinations (BBCR)
• Maintain focused attention over long chains (BBAM)

Empirically:
This approach let LLMs run true solver loops on reasoning benchmarks:
Instead of always moving forward, the system can pause, self-check, backtrack, and correct, before proceeding—very much like human reasoning.

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Why Bring This to LessWrong?

• I believe the lack of explicit mathematical mechanisms for “self-correcting loops” is a root cause of LLMs’ reasoning limits—a topic repeatedly discussed here.
• My hope is that these formulas offer a new concrete proposal to close that gap—either as tools for modelers, or as inspiration for theoretical critique and refinement.

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Open Questions & Call for Critique

• Are there flaws, oversights, or prior works I’ve missed that address solver loops this way?
• Do these modules generalize, or are they “too handcrafted”?
• What counterexamples, pathological cases, or theoretical objections might undermine this approach?

If people are interested in the full math, empirical benchmarks, or open-source code, I’m happy to share links or provide more details.
I welcome all technical, philosophical, or meta-level feedback!

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Again, apologies for any language awkwardness—this is both my first post and my first time writing at this length in English, with AI assistance in editing.