Morphogenesis Across n: Overlays, Emergence Thresholds, and Weak Self-Similarity in the Partition Graph

TL;DR

This paper introduces a formal framework for analyzing the growth and structural persistence of partition graphs across different levels, focusing on overlays, thresholds, and self-similarity.

Contribution

It develops Ferrers-translation maps as graph embeddings to study structural overlays and thresholds, providing a rigorous language for growth and motif persistence in partition graphs.

Findings

Finite motifs persist across levels under translation overlays.

Stable emergence thresholds exist for overlay-monotone properties.

Monotonicity established for extremal local invariants and threshold results for specific motifs.

Abstract

We study the partition graphs $G_n$ as a growing family of discrete geometric objects and introduce a formal framework for comparing their structures across different levels. The main tool is a family of Ferrers-translation maps \[ T_\tau:G_n\to G_{n+k},\qquad (T_\tau(\lambda))'=\lambda'+\tau', \] defined for fixed partitions $\tau\vdash k$. We prove that these maps are induced graph embeddings, giving a rigorous notion of translation overlay: an induced copy of $G_n$ inside $G_{n+k}$. As a consequence, every finite rooted induced motif persists to all higher levels under translation overlays, and every overlay-monotone finitely witnessed property has a stable emergence threshold. We apply this framework to obtain monotonicity for the extremal local invariants $\Delta_n$, $\Omega_n$, and $S_n$, and to establish strict threshold statements for a canonical family of theorem-safe motifs drawn from boundary, axial, and rear morphology. This yields a conservative structural language for discussing growth across $n$ while keeping exact transport separate from stronger typed or visual interpretations. We also record a compact atlas framework for first appearances, repeated patterns, and comparative growth profiles. In this way the paper isolates a theorem-level core for persistence and thresholds, and complements it with a weaker notion of self-similarity based on recurring finite motifs and repeated local fragments.