# Question: 04-neighbours

- **Subproject:** 03-ilang-space
- **Package:** 04-neighbours
- **Equation tags:** (4.k)
- **Created:** 2026-10-07

## Goal

From the witness angle alone, define which points of a part's view are neighbours, and define the distance along chains of neighbours. Then test both definitions on an explicit chain configuration. The notation introduced below ($\asymp$, $\Gamma_V$, $\sim$, $N_V$, $\ell$) is fixed for later packages. "Neighbour" and "chain distance" are defined notions in the sense of M1.

1. **Related points.** Call two distinct points $x,x'\in X_V$ related, $x\asymp x'$, if $W(x,x')>0$, equivalently $\alpha(x,x')<\pi/2$. Show that $x\asymp x'$ iff the view has a nonzero cross term between them: $(V_A)_{hh'}\neq0$ for (any, equivalently all) $h\in x$, $h'\in x'$. Let $\Gamma_V$ be the graph on $X_V$ whose edges are the related pairs.

2. **Neighbours.** Call two distinct points $x,x'$ neighbours, $x\sim x'$, if $x\asymp x'$ and no third point $y\in X_V$ satisfies
   $$\max\bigl(\alpha(x,y),\alpha(y,x')\bigr)<\alpha(x,x').$$
   Let $N_V$ be the graph on $X_V$ whose edges are the neighbour pairs. Show:
   - (a) $\sim$ is symmetric and well defined in the sense of M6;
   - (b) $\sim$ depends only on the order of the values of $\alpha$: replacing $\alpha$ by $g\circ\alpha$ with any strictly increasing $g$ (and the condition $\alpha<\pi/2$ by $g\circ\alpha<g(\pi/2)$) gives the same relation;
   - (c) $N_V$ and $\Gamma_V$ have the same connected components.

3. **Chain distance.** For $x,x'$ in the same connected component of $\Gamma_V$, define
   $$\ell(x,x'):=\min\Bigl\{\sum_{j=0}^{m-1}\alpha(y_j,y_{j+1})\;:\;y_0=x,\ y_m=x',\ y_j\sim y_{j+1}\ \text{for all } j\Bigr\},$$
   and set $\ell(x,x'):=+\infty$ for points in different components. Show that $\ell$ is a metric on each component, that $\ell\ge\alpha$, and that $\ell(x,x')=\alpha(x,x')$ whenever $x\sim x'$. Show by an example that $\ell$ can exceed $\pi/2$; item 4 may serve.

4. **Chain example.** Take a description with two objects $a$ and $c$ of different types, each the only instance of its type, so that (1.11) imposes no constraint. $T(a)$ has $n\ge2$ places $h_1,\dots,h_n$, and $T(c)$ has places $k_1,\dots,k_{n+m-1}$ with $m\ge1$. Let $A=\{a\}$, and consider the state
   $$\lvert\Psi\rangle=\frac{1}{\sqrt n}\sum_{i=1}^{n}\lvert h_i\rangle\otimes\lvert E_i\rangle,
   \qquad
   \lvert E_i\rangle=\frac{1}{\sqrt m}\sum_{s=i}^{i+m-1}\lvert k_s\rangle ,$$
   in which the companion records the place $h_i$ with a blur of width $m$. Derive the points and their witness angles, the neighbour graph $N_V$, and the chain distance $\ell$, for every $m\ge1$.

## Inputs

From 02-view-content@v1. Notation fixed there: $p_h:=\langle h\vert V_A\vert h\rangle$, $H_V:=\{h:p_h>0\}$, $\lvert\psi_h\rangle:=\bigl(\langle h\rvert\otimes\mathbb 1_{\bar A}\bigr)\lvert\Psi\rangle$, and $W(h,h')=\lvert G_{hh'}\rvert^2$ with $G_{hh'}=\langle E_{h'}\vert E_h\rangle$.

Eq. (2.1):

$$
\lvert\Psi\rangle=\sum_h\lvert h\rangle\otimes\lvert\psi_h\rangle,
\qquad
\lvert\psi_h\rangle=\bigl(\langle h\rvert\otimes\mathbb 1_{\bar A}\bigr)\lvert\Psi\rangle=\sum_r\Psi_{hr}\lvert r\rangle,
\qquad
\sum_h\lVert\psi_h\rVert^2=\lVert\Psi\rVert^2=1 .
$$

Eq. (2.2):

$$
V_A=\sum_{h,h'}\langle\psi_{h'}\vert\psi_h\rangle\,\lvert h\rangle\langle h'\rvert,
\qquad\text{i.e.}\qquad
(V_A)_{hh'}=\langle\psi_{h'}\vert\psi_h\rangle,
\qquad
p_h=\lVert\psi_h\rVert^2 .
$$

From 03-witness-distance@v1. Notation fixed there: $h\approx h'$ iff $W(h,h')=1$; $[h]$ is the class of $h$; $X_V:=H_V/{\approx}$ is the set of points; $\alpha(x,x'):=\arccos\sqrt{W(x,x')}\in[0,\pi/2]$ is the witness angle.

Eq. (3.1):

$$
\varrho_h:=\lvert E_h\rangle\langle E_h\rvert=\frac{\lvert\psi_h\rangle\langle\psi_h\rvert}{p_h},
\qquad
h\approx h'\;\Longleftrightarrow\;W(h,h')=1\;\Longleftrightarrow\;\varrho_h=\varrho_{h'} .
$$

Eq. (3.2):

$$
X_V:=H_V/{\approx},\qquad
\varrho_x:=\varrho_h,\qquad
W(x,x'):=W(h,h')=\operatorname{Tr}(\varrho_x\varrho_{x'})\qquad(h\in x,\ h'\in x'),
$$

Eq. (3.3). Here $\pi$ is a type-preserving permutation of the objects, and the map below is the canonical bijection induced by $h\mapsto h^\pi$. It is the identity on $X_V$ when $\pi$ permutes only companion objects or only same-type instances inside $A$. The points and $W$ are well defined in the sense of M6.

$$
X_V^A\to X_V^{\pi(A)},\quad [h]\mapsto[h^\pi],\qquad
W^{\pi(A)}\bigl([h_1^\pi],[h_2^\pi]\bigr)=W^A\bigl([h_1],[h_2]\bigr).
$$

Eq. (3.5). Here $\alpha$ is a metric on $X_V$.

$$
\cos\alpha(x,x')=\sqrt{W(x,x')}=\frac{\lvert(V_A)_{hh'}\rvert}{\sqrt{p_h\,p_{h'}}}
=\frac{\lvert\langle h\vert V_A\vert h'\rangle\rvert}{\sqrt{\langle h\vert V_A\vert h\rangle\langle h'\vert V_A\vert h'\rangle}},\qquad h\in x,\ h'\in x' .
$$

## Assumptions

None beyond the base problem. The part $A$ and its companion $\bar A$ are both nonempty, and $\lambda$ is fixed.

## Scope

- In scope: the relation $\asymp$, the neighbour relation $\sim$ and the chain distance $\ell$ for one part at fixed $\lambda$, and the chain example of item 4.
- Out of scope:
  - other neighbour rules and comparisons between them;
  - whether the contract system couples only neighbours (locality);
  - dimension and limits of growing families;
  - evolution, several parts, and the phases of the cyclic products with $k\ge3$;
  - any physical or spatial meaning beyond the definitions (M1, M4).

## Depth

- Item 1: short argument.
- Item 2 (a) and (b): short argument.
- Item 2 (c): short argument. Standard facts about minimum spanning trees, such as the cut property, may be used without proof.
- Item 3: short argument. Standard facts about shortest-path metrics on graphs with positive edge weights may be used without proof.
- Item 4: derive.

## Expected result

- Item 1: the equivalence $x\asymp x'\iff(V_A)_{hh'}\neq0$.
- Item 2: statements (a)–(c).
- Item 3: $\ell$ is a metric on each component (an extended metric on $X_V$), $\ell\ge\alpha$, and $\ell=\alpha$ on neighbour pairs; an example with $\ell>\pi/2$.
- Item 4: closed forms for the witness angles, the neighbour graph and $\ell$, separately for $m=1$ and for $m\ge2$. If the computation does not give the following, report the discrepancy:
  - for $m\ge2$: the $n$ points $x_i=[h_i]$ are distinct, with $\cos\alpha(x_i,x_j)=\max(0,m-\lvert i-j\rvert)/m$;
  - $N_V$ is the chain $x_1\sim x_2\sim\dots\sim x_n$;
  - $\ell(x_i,x_j)=\lvert i-j\rvert\arccos(1-1/m)$;
  - for $m=1$ no two points are related.

Give every main result a tag $(4.k)$.

Consistency checks, at most three, chosen from:
- for a product state, $X_V$ is a single point, and there are no neighbours;
- in item 4 with $m=1$, all branch states are orthogonal and every component of $\Gamma_V$ is a single point;
- in item 4 with $m\ge n$, every two points are related, and $N_V$ is still the chain.

## Code

None.