# A Formalization of Recursive Self-Optimizing Generative Systems

**tukuai**
Independent Researcher
GitHub: [https://github.com/tukuai](https://github.com/tukuai)

## Abstract

We study a class of recursive self-optimizing generative systems whose objective is not the direct production of optimal outputs, but the construction of a stable generative capability through iterative self-modification. The system generates artifacts, optimizes them with respect to an idealized objective, and uses the optimized artifacts to update its own generative mechanism. We provide a formal characterization of this process as a self-mapping on a space of generators, identify its fixed-point structure, and express the resulting self-referential dynamics using algebraic and λ-calculus formulations. The analysis reveals that such systems naturally instantiate a bootstrapping meta-generative process governed by fixed-point semantics.

---

## 1. Introduction

Recent advances in automated prompt engineering, meta-learning, and self-improving AI systems suggest a shift from optimizing individual outputs toward optimizing the mechanisms that generate them. In such systems, the object of computation is no longer a solution, but a *generator of solutions*.

This work formalizes a recursive self-optimizing framework in which a generator produces artifacts, an optimization operator improves them relative to an idealized objective, and a meta-generator updates the generator itself using the optimization outcome. Repeated application of this loop yields a sequence of generators that may converge to a stable, self-consistent generative capability.

Our contribution is a compact formal model capturing this behavior and a demonstration that the system admits a natural interpretation in terms of fixed points and self-referential computation.

---

## 2. Formal Model

Let (\mathcal{I}) denote an intention space and (\mathcal{P}) a space of prompts, programs, or skills. Define a generator space
$$
\mathcal{G} \subseteq \mathcal{P}^{\mathcal{I}},
$$
where each generator (G \in \mathcal{G}) is a function
$$
G : \mathcal{I} \to \mathcal{P}.
$$

Let (\Omega) denote an abstract representation of an ideal target or evaluation criterion. We define:
$$
O : \mathcal{P} \times \Omega \to \mathcal{P},
$$
an optimization operator, and
$$
M : \mathcal{G} \times \mathcal{P} \to \mathcal{G},
$$ a meta-generative operator that updates generators using optimized artifacts.

Given an initial intention (I \in \mathcal{I}), the system evolves as follows:
$$
P = G(I),
$$
$$
P^{*} = O(P, \Omega),
$$
$$
G' = M(G, P^{*}).
$$

---

## 3. Recursive Update Operator

The above process induces a self-map on the generator space:
$$
\Phi : \mathcal{G} \to \mathcal{G},
$$
defined by
$$
\Phi(G) = M\big(G,; O(G(I), \Omega)\big).
$$

Iteration of (\Phi) yields a sequence ({G_n}*{n \ge 0}) such that
$$G*{n+1} = \Phi(G_n).
$$

The system’s objective is not a particular (P^{*}), but the convergence behavior of the sequence ({G_n}).

---

## 4. Fixed-Point Semantics

A *stable generative capability* is defined as a fixed point of (\Phi):
$$G^{*} \in \mathcal{G}, \quad \Phi(G^{*}) = G^{*}.
$$

Such a generator is invariant under its own generate–optimize–update cycle. When (\Phi) satisfies appropriate continuity or contractiveness conditions, (G^{*}) can be obtained as the limit of iterative application:
$$G^{*} = \lim_{n \to \infty} \Phi^{n}(G_0).
$$

This fixed point represents a self-consistent generator whose outputs already encode the criteria required for its own improvement.

---

## 5. Algebraic and λ-Calculus Representation

The recursive structure can be expressed using untyped λ-calculus. Let (I) and (\Omega) be constant terms, and let (G), (O), and (M) be λ-terms. Define the single-step update functional:
$$
\text{STEP} ;\equiv; \lambda G.; (M;G)\big((O;(G;I));\Omega\big).
$$

Introduce a fixed-point combinator:
$$
Y ;\equiv; \lambda f.(\lambda x.f(x,x))(\lambda x.f(x,x)).
$$

The stable generator is then expressed as:
$$G^{*} ;\equiv; Y;\text{STEP},
$$
satisfying
$$G^{*} = \text{STEP};G^{*}.
$$

This formulation makes explicit the self-referential nature of the system: the generator is defined as the fixed point of a functional that transforms generators using their own outputs.

---

## 6. Discussion

The formalization shows that recursive self-optimization naturally leads to fixed-point structures rather than terminal outputs. The generator becomes both the subject and object of computation, and improvement is achieved through convergence in generator space rather than optimization in output space.

Such systems align with classical results on self-reference, recursion, and bootstrapping computation, and suggest a principled foundation for self-improving AI architectures and automated meta-prompting systems.

---

## 7. Conclusion

We presented a formal model of recursive self-optimizing generative systems and characterized their behavior via self-maps, fixed points, and λ-calculus recursion. The analysis demonstrates that stable generative capabilities correspond to fixed points of a meta-generative operator, providing a concise theoretical basis for self-improving generation mechanisms.

---

### Notes for arXiv submission

* **Category suggestions**: `cs.LO`, `cs.AI`, or `math.CT`
* **Length**: appropriate for extended abstract (≈3–4 pages LaTeX)
* **Next extension**: fixed-point existence conditions, convergence theorems, or proof sketches

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## 附录：高层次概念释义 (Appendix: High-Level Conceptual Explanation)

The core idea of this paper can be popularly understood as an AI system capable of **self-improvement**. Its recursive nature can be broken down into the following steps:

#### 1. Define Core Roles:

*   **α-Prompt (Generator)**: A "parent" prompt whose sole responsibility is to **generate** other prompts or skills.
*   **Ω-Prompt (Optimizer)**: Another "parent" prompt whose sole responsibility is to **optimize** other prompts or skills.

#### 2. Describe the Recursive Lifecycle:

1.  **Bootstrap**:
    *   Use AI to generate initial versions (v1) of `α-Prompt` and `Ω-Prompt`.

2.  **Self-Correction & Evolution**:
    *   Use `Ω-Prompt (v1)` to **optimize** `α-Prompt (v1)`, obtaining a more powerful `α-Prompt (v2)`.

3.  **Generation**:
    *   Use the **evolved** `α-Prompt (v2)` to generate **all** target prompts and skills we need.

4.  **Recursive Loop**:
    *   The most crucial step: feed the newly generated, more powerful products (including new versions of `Ω-Prompt`) back into the system, again for optimizing `α-Prompt`, thereby initiating the next round of evolution.

#### 3. Ultimate Goal:

Through this never-ending **recursive optimization loop**, the system **self-transcends** in each iteration, infinitely approaching the **ideal state** we set.
