Stem

Prerequisites: Chapter 4 (Fruit).

5.1 Definitions#

If a fruit is a "strongly cohesive cluster", then the stem is the weak connective tissue between fruits.

Definition 5.1 (Fruit union). ${\mathcal{F}}_t^{\cup }:={\bigcup }_{F\in {\mathfrak{F}}_t}F.$

Definition 5.2 (Stem region). ${\mathcal{S}}_t:=V\setminus {\mathcal{F}}_t^{\cup }.$ Nodes that belong to no fruit.

Definition 5.3 (Bridge edge). $E_t^{\mathrm{bridge}}:=\{(i,j)\in E_t:\nexists F\in {\mathfrak{F}}_t\text{ with }i,j\in F\}.$ Edges whose endpoints do not co-occur in any single fruit.

Definition 5.4 (Stem). ${\mathrm{Stem}}_t:=({\mathcal{S}}_t\cup {\partial }_t^{\mathrm{bridge}},E_t^{\mathrm{bridge}})$ where ${\partial }_t^{\mathrm{bridge}}:=\{i\in V:\exists (i,j)\in E_t^{\mathrm{bridge}}\}$.

5.2 Properties#

Proposition 5.5 (Characterisation of stem nodes). If $i\in {\mathcal{S}}_t$, then no subset of $V$ containing $i$ simultaneously satisfies conditions (F1) and (F2) of Definition 4.5.

A stem node is one that is "not strongly bound to any cluster". This can happen because:

1. Its relations are spread evenly across multiple fruits, or
2. There is no cohesive structure around it at all.

Proposition 5.6 (Weakness of stem connections). For any fruit $F\in {\mathfrak{F}}_t$: ${\sum }_{i\in F}{\sum }_{j\in {\mathcal{S}}_t}{W}_t(i,j)\le {\mathrm{cut}}_t(F,\bar{F})\le \theta \cdot {\mathrm{vol}}_t(F).$

Proof. The left-hand side is a partial sum of ${\mathrm{cut}}_t(F,\bar{F})$ (restricting the outer sum to $j\in {\mathcal{S}}_t$). The right-hand inequality follows from ${\varphi }_t(F)\le \theta$ and condition (F1). $\square$

Interpretation. The total relation leaking from a fruit to the stem is at most a fraction $\theta$ of the fruit's volume. For $\theta =0.05$, less than 5% leaks out.

5.3 The No-Boundary Principle#

This is a central philosophical choice of the theory.

Principle 5.7 (No-boundary principle). In the mathematical description of a fruit, the stem (exterior) is never modelled explicitly. The exterior's existence and properties are detected only through interior traces (doors).

Why not boundary conditions?#

Modelling the stem explicitly would require:

1. Arbitrary assumptions about the stem's relational structure.
2. Forced boundary conditions at the fruit--stem interface.
3. Inevitable arbitrariness from "inventing the exterior".

The alternative (Uhlenbeck approach):

1. Do not define the exterior.
2. Detect it only through singularities (doors) inside the fruit.
3. Singularities emerge naturally as energy-concentration points, not as imposed boundary data.

This is the subject of Chapter 6.

5.4 Classification of Stem Components (Remark)#

The stem can be subdivided:

• Type A (inter-fruit bridges): edges connecting distinct fruits $F_1$ and $F_2$.
• Type B (fruit--stem filaments): edges between a fruit $F$ and the stem region ${\mathcal{S}}_t$.
• Type C (pure stem): edges internal to ${\mathcal{S}}_t$.

By the no-boundary principle, all three types are treated as "exterior" from the perspective of any individual fruit.