Door

Prerequisites: Chapter 5 (Stem), Chapter 4 (Fruit).

6.1 The Intrinsic Data Axiom#

For a fruit to "interpret itself", we must first specify which data is intrinsic.

The core problem#

The degree $d_t(i)={\sum }_{j\in V}{W}_t(i,j)$ is a global sum. The internal weights $\{W_t(i,j){\}}_{i,j\in F}$ alone do not determine $d_t(i)$.

However, including $d_t(i)$ in the intrinsic data enables internal computation of external coupling.

Axiom 6.1 (Intrinsic data axiom). The intrinsic data of a fruit $F\in {\mathfrak{F}}_t$ is: $\mathcal{D}(F,t):=(\{W_t(i,j){\}}_{i,j\in F},\{g_t(i,j){\}}_{i,j\in F},\{d_t(i){\}}_{i\in F}).$

Justification.

1. Physical: $d_t(i)$ is the total relational capacity of $i$, observable by $i$ itself.
2. Probabilistic: $d_t(i)$ is the normalisation constant for the random walk at $i$; if $i$ observes its own transition probabilities, it knows $d_t(i)$.
3. Information-theoretic: excluding $d_t(i)$ makes door detection impossible, breaking the theory's self-consistency.

6.2 Internal Computation of External Coupling#

Lemma 6.2 (Boundary coupling from intrinsic data). Under Axiom 6.1, the total external coupling: $b_{F,t}(i):=d_t(i)-{\sum }_{j\in F}{W}_t(i,j)={\sum }_{j\notin F}{W}_t(i,j)$ is a function of $\mathcal{D}(F,t)$.

Proof. $b_{F,t}(i)=d_t(i)-{\sum }_{j\in F}{W}_t(i,j)$. The first term is the third component of $\mathcal{D}$; the second is computed from the first component. No information about any $j\notin F$ is needed. $\square$

Definition 6.3 (Leakage rate). $r_{F,t}(i):=\frac{b_{F,t}(i)}{d_t(i)}=1-\frac{{\sum }_{j\in F}{W}_t(i,j)}{d_t(i)}\in [0,1].$ $r=0$: all relations internal. $r=1$: all relations external.

6.3 Fruit Boundary#

Definition 6.4 (Fruit boundary). ${\partial }_{V}F:=\{i\in F:b_{F,t}(i)>0\}.$

Properties: ${\partial }_{V}F\subset F$; $F\setminus {\partial }_{V}F$ consists of fully internal nodes; ${\partial }_{V}F$ is gauge-invariant.

6.4 Door Definition (Energy-Based)#

Axiom 6.5 (Door threshold). Fix $\tau >0$.

Definition 6.6 (Door---energy-based). $\boxed{{\Sigma }_{\tau }(F,t):=\{i\in {\partial }_{V}F:b_{F,t}(i)\ge \tau \}.}$ Boundary nodes with external coupling at or above threshold $\tau$.

Interpretation. A door is a "hole in the wall" of the fruit. Not every boundary node is a door: weak leakage ($b<\tau$) is ignored; only strong leakage ($b\ge \tau$) is recorded as a singularity.

This parallels Uhlenbeck compactness, where only energy concentrations exceeding a quantum ${ϵ}_0$ are recorded as singular points.

6.5 Main Door Theorems (Theorems B, C)#

Theorem B (Finiteness of doors and energy concentration).

(i) $|{\Sigma }_{\tau }(F,t)|\le \theta \cdot {\mathrm{vol}}_t(F)/\tau$.

(ii) ${\sum }_{p\in \Sigma }{e}_p\le \theta \cdot {\mathrm{vol}}_t(F)$.

(iii) $|{\Sigma }_{\tau }|/|F|\le \theta \cdot d_{\max }(F,t)/\tau$.

Proof.

(i) Each $i\in {\Sigma }_{\tau }$ has $b_{F,t}(i)\ge \tau$, so:

(ii) ${\sum }_{p\in \Sigma }{e}_p={\sum }_{p\in \Sigma }{b}_{F,t}(p)\le {\mathrm{cut}}_t(F,\bar{F})\le \theta \cdot {\mathrm{vol}}_t(F)$.

(iii) Use ${\mathrm{vol}}_t(F)\le |F|\cdot d_{\max }(F,t)$ in (i). $\square$

Theorem C (Self-interpretation). Under Axiom 6.1, ${\Sigma }_{\tau }(F,t)$ and $\mathbf{e}(F,t)$ are determined by $\mathcal{D}(F,t)$ alone. No information about the exterior is needed.

Proof. $b_{F,t}(i)$ is a function of $\mathcal{D}(F,t)$ (Lemma 6.2). ${\Sigma }_{\tau }$ is a level set of $b_{F,t}$. $e_p=b_{F,t}(p)$. All computed from $\mathcal{D}$. $\square$

Philosophical significance.

An existence can read the traces left by the exterior without knowing what the exterior is. This is the mathematical realisation of "self-interpretation".

6.6 Door Definition (Holonomy-Based)---Alternative#

A second door detection method using gauge structure.

Definition 6.7 (Boundary-traversing loops). ${\mathcal{L}}_{\partial }(F):=\{(i,j,k):i,k\in F,j\notin F,W_t(i,j)>0,W_t(j,k)>0\}.$

Definition 6.8 (Holonomy door energy). ${\epsilon }_{F,t}^{\mathrm{hol}}(i):={\sum }_{(i,j,k)\in {\mathcal{L}}_{\partial }(F)}{W}_t(i,j)W_t(j,k){\omega }_t(i,j,k)$ where ${\omega }_t(i,j,k)=d_G({\Omega }_t(i,j,k),e{)}^2$.

Proposition 6.9 (Gauge invariance of holonomy door energy). ${\epsilon }_{F,t}^{\mathrm{hol}}(i)$ is gauge-invariant.

Proof. $W_t$ is invariant (Lemma 3.5). ${\omega }_t$ is invariant (Corollary 3.9). Their product and sum are invariant. $\square$

Comparison of the Two Definitions#

Conclusion. With Axiom 6.1, the energy-based door is the primary definition. The holonomy-based door provides complementary curvature information.

6.7 Door Energy Vector#

Definition 6.10 (Door energy vector). $\mathbf{e}(F,t):=\{e_p{\}}_{p\in {\Sigma }_{\tau }(F,t)},e_p:=b_{F,t}(p).$

Proposition 6.11 (Energy bound). ${\sum }_{p\in \Sigma }{e}_p\le {\mathrm{cut}}_t(F,\bar{F})\le \theta \cdot {\mathrm{vol}}_t(F)$.

6.8 Perturbation Stability of Doors (Theorem G)#

Theorem G (Door perturbation stability). Let $\Vert \delta W{\Vert }_{\infty }:={\max }_{i,j}|W_t^{\prime }(i,j)-W_t(i,j)|$ be small. Then:

(i) $|b_{F,t}^{\prime }(i)-b_{F,t}(i)|\le |V|\cdot \Vert \delta W{\Vert }_{\infty }$ for all $i\in F$.

(ii) If $b_{F,t}(i)\ge \tau +ϵ$, then $i\in {\Sigma }^{\prime }$ (door persists).

(iii) If $b_{F,t}(i)\le \tau -ϵ$, then $i\notin {\Sigma }^{\prime }$ (non-door persists).

(iv) Only nodes with $b_{F,t}(i)\in [\tau -ϵ,\tau +ϵ]$ may change status, where $ϵ:=|V|\cdot \Vert \delta W{\Vert }_{\infty }$.

Proof. (i) $b^{\prime }(i)=d_t^{\prime }(i)-{\sum }_{j\in F}{W}_t^{\prime }(i,j)$. We have $|d_t^{\prime }(i)-d_t(i)|\le |V|\cdot \Vert \delta W{\Vert }_{\infty }$ and $|{\sum }_{j\in F}(W_t^{\prime }-W_t)(i,j)|\le |F|\cdot \Vert \delta W{\Vert }_{\infty }$. These partially cancel in the difference, giving $|b^{\prime }-b|\le |V|\cdot \Vert \delta W{\Vert }_{\infty }$.

(ii)--(iv) follow from (i) with $ϵ=|V|\cdot \Vert \delta W{\Vert }_{\infty }$. $\square$