Autoconvex Structures as Fixed Point Traps in Logical Systems

Marko Chalupa
01 Mai 2025

Abstract

We introduce a formal framework for autoconvex structures—logical configurations in which every attempted semantic divergence from a target proposition increases the likelihood of systemically enforced re-acceptance. We interpret these structures as fixed point traps, highlighting their role in logical convergence, semantic recursion, and reflective consistency. The paper formalizes the construct within a proof-theoretic and type-theoretic context and identifies potential applications in automated reasoning, metamathematics, and AI alignment.

1. Introduction

In the study of logical systems, fixed points often characterize stable interpretations, recursive functions, and self-referential truths. In this paper, we generalize this idea by formalizing a class of logical structures—termed autoconvex—which force convergence not by truth value alone but by the exhaustion of coherent alternative paths. These are distinct from traditional fixed points in that they are emergent from drift, rather than defined by idempotence.

This paper does not merely propose a theory.
It describes the form in which theories organize their own return.

2. Preliminaries and Definitions

We assume a classical logical framework with a decidable underlying logic. Let ℒ denote a propositional language over a set of formulas Φ.

Drift Path

Given a formula φ ∈ Φ, a drift path is a sequence of derivations D₀, D₁, …, Dₙ such that each Dᵢ attempts to refute or bypass φ.

SnapReturn

A drift path exhibits SnapReturn if for all drift derivations Dᵢ, there exists a structural homomorphism h: Dᵢ → φ such that continued drift leads to logical contradiction or repetition.

Autoconvex Structure

A logical structure S ⊆ Φ is autoconvex with respect to φ if every semantically coherent path that diverges from φ projects back onto φ, either through contradiction or drift exhaustion.

3. Fixed Point Interpretation

Autoconvex structures can be seen as fixed point traps in the semantic space: once entered, all interpretive operations inevitably return to the anchor formula φ. Unlike classical fixed points, autoconvex fixed points arise through dynamic structural recursion.

∀M, δ(M) ⇒ ρ(M), with ρ(M) = M

This relation forms a quasi-idempotent closure under recursive refutation.

4. Formal Model

We formalize the drift structure using type theory. Consider a context Γ ⊢ φ where the direct derivation is blocked. We define a structure:

Phase1: Γ ⊢ ¬φ
Phase2: Γ ⊢ ¬¬φ
SnapReturn: Phase1 ∧ Phase2 ⇒ ⊥

This constitutes the drift trap: both negation and negation-of-negation force collapse, structurally reinforcing φ.

5. Implications and Applications

Autoconvex traps have implications in systems where semantic coherence is enforced: theorem provers, fixed point logics, and even machine learning models with probabilistic convergence. These structures simulate a logical version of energetic minimization: all semantic tension flows back into the core proposition.

In AI safety and formal verification, autoconvexity may serve as a scaffold for alignment strategies based on contradiction collapse and consistency pressure. From a computational logic perspective, these structures are gold-level assets for systems tasked with self-consistency under drift and recursion. Their emergent fixed point dynamics provide a novel framework for studying stability in non-axiomatic inference environments.

6. Conclusion

We introduced autoconvex structures as formal traps within logic systems that reflect a generalized fixed point behavior emerging from structural inevitability. Future work will analyze autoconvexity in modal, temporal, and higher-order logics, and explore automated detection of such traps in formal systems.

Autoconvex Framework · Marko Chalupa · 2025 · CC BY 4.0