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: $\mathcal{D}(F,t) := \bigl(\{W_t(i,j)\}_{i,j\in F},\;\{g_t(i,j)\}_{i,j\in F},\;\{d_t(i)\}_{i\in F}\bigr).$

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 $\epsilon_0$ 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). $\varepsilon^{\mathrm{hol}}_{F,t}(i):=\sum_{(i,j,k)\in\mathcal{L}_\partial(F)}W_t(i,j)\,W_t(j,k)\;\omega_t(i,j,k)$ where $\omega_t(i,j,k)=d_G(\Omega_t(i,j,k),e)^2$.

Proposition 6.9 (Gauge invariance of holonomy door energy). $\varepsilon^{\mathrm{hol}}_{F,t}(i)$ 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). $\mathbf{e}(F,t):=\{e_p\}_{p\in\Sigma_\tau(F,t)},\qquad e_p:=b_{F,t}(p).$

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 $\|\delta W\|_\infty:=\max_{i,j}|W_t'(i,j)-W_t(i,j)|$ be small. Then:

(i) $|b'_{F,t}(i)-b_{F,t}(i)|\le|V|\cdot\|\delta W\|_\infty$ for all $i\in F$.

(ii) If $b_{F,t}(i)\ge\tau+\epsilon$, then $i\in\Sigma'$ (door persists).

(iii) If $b_{F,t}(i)\le\tau-\epsilon$, then $i\notin\Sigma'$ (non-door persists).

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

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

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