TRANSLATED CONTENT:
# 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

---

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

该论文的核心思想可以被通俗地理解为一个能够**自我完善**的 AI 系统。其递归本质可分解为以下步骤：

#### 1. 定义核心角色：

*   **α-提示词 (生成器)**: 一个“母体”提示词，其唯一职责是**生成**其他提示词或技能。
*   **Ω-提示词 (优化器)**: 另一个“母体”提示词，其唯一职责是**优化**其他提示词或技能。

#### 2. 描述递归的生命周期：

1.  **创生 (Bootstrap)**:
    *   用 AI 生成 `α-提示词` 和 `Ω-提示词` 的初始版本 (v1)。

2.  **自省与进化 (Self-Correction & Evolution)**:
    *   用 `Ω-提示词 (v1)` 去**优化** `α-提示词 (v1)`，得到一个更强大的 `α-提示词 (v2)`。

3.  **创造 (Generation)**:
    *   用**进化后的** `α-提示词 (v2)` 去生成我们需要的**所有**目标提示词和技能。

4.  **循环与飞跃 (Recursive Loop)**:
    *   最关键的一步：将新生成的、更强大的产物（甚至包括新版本的 `Ω-提示词`）反馈给系统，再次用于优化 `α-提示词`，从而启动下一轮进化。

#### 3. 终极目标：

通过这个永不停止的**递归优化循环**，系统在每一次迭代中都进行**自我超越**，无限逼近我们设定的**理想状态**。