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Jaehong Oh

ℓPart 1· Chapter 5

Stem

475 words2 min read

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). Ft∪:=⋃F∈FtF.\mathcal{F}_t^{\cup} := \bigcup_{F\in\mathfrak{F}_t} F.

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

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

Definition 5.4 (Stem). Stemt:=(St∪∂tbridge,  Etbridge)\mathrm{Stem}_t := \bigl(\mathcal{S}_t\cup\partial_t^{\mathrm{bridge}},\;E_t^{\mathrm{bridge}}\bigr) where ∂tbridge:={i∈V:∃ (i,j)∈Etbridge}\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∈Sti\in\mathcal{S}_t, then no subset of VV containing ii 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∈FtF\in\mathfrak{F}_t: ∑i∈F∑j∈StWt(i,j)  ≤  cutt(F,Fˉ)  ≤  θ⋅volt(F).\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 cutt(F,Fˉ)\mathrm{cut}_t(F,\bar F) (restricting the outer sum to j∈Stj\in\mathcal{S}_t). The right-hand inequality follows from ϕt(F)≤θ\phi_t(F)\le\theta and condition (F1). □\square

Interpretation. The total relation leaking from a fruit to the stem is at most a fraction θ\theta of the fruit's volume. For θ=0.05\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 F1F_1 and F2F_2.
  • Type B (fruit--stem filaments): edges between a fruit FF and the stem region St\mathcal{S}_t.
  • Type C (pure stem): edges internal to St\mathcal{S}_t.

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