# Question: 15-bodies

- **Subproject:** 03-ilang-space
- **Package:** 15-bodies
- **Equation tags:** (15.k)
- **Created:** 2026-10-08

## Goal

Determine what a body, its position and the distance between two bodies of one type are in the leading-order witness geometry of a recording medium. Determine whether these notions are well defined, and whether the state of the medium determines them. The notions defined below (background, localized body, position, configuration, two-body distance $D$, state of the medium) are fixed for later packages. They mean only what their definitions say (M1, M4).

**Setting.**
- **Objects.** The bodies are objects of one type $T$, with places $h\in H_T$ and $d_T\ge2$. The medium $c$ is one object of another type, the only instance of its type. The notation $\langle X\rangle:=\langle\chi\vert X\vert\chi\rangle$ and $\lvert u_h\rangle:=(K_h-\langle K_h\rangle)\lvert\chi\rangle$ is that of (5.10).
- **(S1) One body.** One object $b$ of type $T$, the only instance of $T$. The contract is $C=\sum_h\lvert h\rangle\langle h\rvert_b\otimes K_h$, with $K_h$ self-adjoint on $\mathcal H_c$. The start is $\lvert\beta\rangle\otimes\lvert\chi\rangle$ for a place $\beta$.
- **(S2) Two bodies.** Two objects $b_1,b_2$ of type $T$, the only instances of $T$, with family $c_T=\pm1$. The contract is (12.7) with $p_1,p_2$ renamed $b_1,b_2$ and an $h$-independent $\Delta$; by (12.3) and (12.7), this is the general contract of the setting of 12 for which the evolution preserves admissibility. The start is $\lvert\beta_1\beta_2\rangle_{c_T}\otimes\lvert\chi\rangle$, where $\lvert\beta_1\beta_2\rangle_{c_T}$ is the unit vector proportional to $\lvert\beta_1\beta_2\rangle+c_T\lvert\beta_2\beta_1\rangle$, with $\beta_1\neq\beta_2$ if $c_T=-1$.

**Definitions.**
- **Background.** The background of $T$ is the leading-order record geometry $(H_T/{\sim_0},d_0)$ of (5.10) and (5.11), built from the $u_h$. Its points are written $[h]_0$. By (5.10) and (12.12), it is the leading-order geometry of the single-instance view for every product start with $\phi_h\neq0$ for all $h$: $\lvert\phi\rangle\otimes\lvert\chi\rangle$ in (S1), and $\lvert\phi\rangle\otimes\lvert\phi\rangle\otimes\lvert\chi\rangle$ in (S2) with $c_T=+1$.
- **Localized body, position.** In (S1), the body is localized at the point $x$ of the background if every place in $H_V$ of its view lies in $x$. Its position is then $x$.
- **Configuration, two-body distance.** In (S2), the pair is localized at the unordered pair $\{x_1,x_2\}$ of points ($x_1=x_2$ allowed) if every joint place $(h_1,h_2)\in H_V$ of the view of $\{b_1,b_2\}$ satisfies $\{[h_1]_0,[h_2]_0\}=\{x_1,x_2\}$ as multisets. This unordered pair is the configuration. The positions of the two bodies are $x_1$ and $x_2$, and their two-body distance is $D:=d_0(x_1,x_2)$.
- **State of the medium.** The state of the medium is its view $V_{\{c\}}$ (1.5). When it is pure, $V_{\{c\}}=\lvert E\rangle\langle E\rvert$, two such states are the same at leading order if $1-\lvert\langle E\vert E'\rangle\rvert^2=o(\lambda^2)$ as $\lambda\to0^+$.

1. **One body (S1).**
   - (a) Determine the exact state, whether the body is localized at $[\beta]_0$ for every $\lambda$, and the state of the medium.
   - (b) Determine a necessary and sufficient condition under which two places $\beta,\beta'$ give the same state of the medium for every $\lambda$, and one under which they give the same state at leading order. Relate both conditions to the background.

2. **Two bodies (S2), either family.**
   - (a) Determine the exact state. Determine whether the pair stays localized at $\{[\beta_1]_0,[\beta_2]_0\}$, whether it becomes correlated with the medium, and the state of the medium.
   - (b) Determine whether the positions and $D$ are well defined (M6), in particular whether they depend on the description labels, on the split (12.6) or on $\Delta$.
   - (c) Determine the single-instance view and the view of $\{b_1,b_2\}$ in this state, and whether their witness data depend on $D$.
   - (d) Determine a necessary and sufficient condition, in terms of the record vectors, under which two starts give the same state of the medium at leading order, and a sufficient condition under which they give the same state for every $\lambda$. Relate the first condition to the pair geometry (12.13).
   - (e) Determine whether $D$ is in general a function of the state of the medium at leading order. Determine a necessary and sufficient condition on the record vectors under which the state of the medium at leading order distinguishes all configurations.

3. **Examples.** (S2) with $\Delta=0$, for either family; $i$ is an index.
   - (a) **Line.** The places are $h_1,\dots,h_n$ with $n\ge2$, and $K_{h_i}=i\,X$, where $X$ is self-adjoint on $\mathcal H_c$ and $\sigma_X:=(\langle X^2\rangle-\langle X\rangle^2)^{1/2}>0$.
   - (b) **Window.** The places are $h_1,\dots,h_n$ with $n\ge2$, and the window width is $m\ge1$. The medium has the places $\{0,1\}^L$ with $L:=n+m-1$, and $\lvert\chi\rangle=\lvert0\cdots0\rangle$. The operator $\sigma_x^{[l]}$ maps each place of $c$ to the place with bit $l$ flipped, and $K_{h_i}:=\sum_{l=i}^{i+m-1}\sigma_x^{[l]}$.

   For each example, determine $D$ and $d_0$ of the background, the pairs of configurations that give the same state of the medium (for every $\lambda$, and at leading order), and whether $D$ is a function of the state of the medium.

## Inputs

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

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):

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

Eq. (2.11):

$$
W(h)=1,\qquad W(h,h')=\lvert G_{hh'}\rvert^2=\frac{\lvert(V_A)_{hh'}\rvert^2}{p_hp_{h'}} .
$$

Eq. (2.16). Here $\pi$ is a permutation of the objects with $T(\pi(a))=T(a)$, and $(h^\pi)_{\pi(a)}:=h_a$. For every admissible state:

$$
p^{\pi(A)}_{h^\pi}=p^A_h,\qquad H_V^{\pi(A)}=\{h^\pi:h\in H_V^A\},\qquad W^{\pi(A)}(h_1^\pi,\dots,h_k^\pi)=W^A(h_1,\dots,h_k).
$$

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

Eq. (3.4). Here $a\neq b$ are objects of $A$ of the same type $T$, and $h^{(ab)}$ is $h$ with the entries $a,b$ exchanged.

$$
\lvert\psi_h\rangle=\bigl(\langle h^{(ab)}\rvert\otimes\mathbb 1_{\bar A}\bigr)P_{ab}\lvert\Psi\rangle=c_T\,\lvert\psi_{h^{(ab)}}\rangle .
$$

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

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

From 05-recording-contract@v1. These hold in the setting of 05: one recorded object, one medium object, and the recording contract $\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$. (5.6), (5.10) and (5.11) are for the start $\lvert\phi\rangle\otimes\lvert\chi\rangle$ with $\phi_h\neq0$ for all $h$. Notation fixed there: $g(v,w):=\operatorname{Re}\langle v\vert w\rangle$, and $h\sim_0h'$ iff $u_h=u_{h'}$.

Eq. (5.1):

$$
e^{-iC\lambda}=\sum_h\lvert h\rangle\langle h\rvert\otimes e^{-iK_h\lambda}.
$$

Eq. (5.6):

$$
W(h,h';\lambda)=\bigl\lvert\langle\chi\vert e^{iK_{h'}\lambda}e^{-iK_h\lambda}\vert\chi\rangle\bigr\rvert^2 .
$$

Eq. (5.10):

$$
\lim_{\lambda\to0^+}\frac{\alpha(h,h';\lambda)}{\lambda}=d_0(h,h')=\bigl\lVert u_h-u_{h'}\bigr\rVert,
\qquad \lvert u_h\rangle=\bigl(K_h-\langle K_h\rangle\bigr)\lvert\chi\rangle .
$$

Eq. (5.11):

$$
\bigl(H_V/{\sim_0},\,d_0\bigr)\;\cong\;\bigl(\{u_h:h\in H_V\},\,\lVert\cdot\rVert\bigr)\subset(\mathcal H_c,g).
$$

From 12-identical-probes@v1. These hold in the setting of 12: probes $p_1,p_2$ of one type $T$ with $d_T\ge2$, a medium $c$ that is the only instance of its type, the contract $C=\sum_h\lvert h\rangle\langle h\rvert_{p_1}\otimes K^{(1)}_h+\sum_h\lvert h\rangle\langle h\rvert_{p_2}\otimes K^{(2)}_h$, and $\Delta_h:=K^{(1)}_h-K^{(2)}_h$, written $\Delta$ when it does not depend on $h$, with $K_h:=K^{(1)}_h$. (12.12)–(12.14) hold for $c_T=+1$, an $h$-independent $\Delta$ and the start $\lvert\phi\rangle\otimes\lvert\phi\rangle\otimes\lvert\chi\rangle$ with $\phi_h\neq0$ for all $h$. (12.12) is for the view of $\{p_1\}$; (12.13) and (12.14) are for the view of $\{p_1,p_2\}$, with companion $\{c\}$.

Eq. (12.3):

$$
e^{-iC\lambda}\mathcal H_{\rm adm}\subseteq\mathcal H_{\rm adm}\ \ \forall\lambda
\iff(P_{12}CP_{12}-C)\,\mathcal H_{\rm adm}=0
\iff\Delta_h=\Delta\ \text{ for all }h .
$$

Eq. (12.6), the freedom of splitting the same $C$ into its two contract terms:

$$
K^{(1)}_h\to K^{(1)}_h+X,\qquad K^{(2)}_h\to K^{(2)}_h-X,\qquad \Delta_h\to\Delta_h+2X,\qquad X=X^\dagger\ \text{on }\mathcal H_c .
$$

Eq. (12.7), for $h$-independent $\Delta$:

$$
C=\sum_h\bigl(\lvert h\rangle\langle h\rvert_{p_1}+\lvert h\rangle\langle h\rvert_{p_2}\bigr)\otimes K_h-\mathbb 1_{p_1p_2}\otimes\Delta .
$$

Eq. (12.12):

$$
d_0(h,h')=\lim_{\lambda\to0^+}\frac{\alpha^{\{p_1\}}(h,h';\lambda)}{\lambda}=\bigl\lVert u_h-u_{h'}\bigr\rVert,\qquad\lvert u_h\rangle=\bigl(K_h-\langle K_h\rangle\bigr)\lvert\chi\rangle .
$$

Eq. (12.13):

$$
d_0\bigl((h_1,h_2),(h_1',h_2')\bigr)=\bigl\lVert u_{h_1}+u_{h_2}-u_{h_1'}-u_{h_2'}\bigr\rVert .
$$

Eq. (12.14):

$$
W\bigl((h_1,h_2),(h_2,h_1);\lambda\bigr)=1\qquad\forall\lambda .
$$

## Assumptions

- The setting and the definitions under "Goal". All contracts are independent of $\lambda$ (A5).
- "Leading order" refers to $\lambda\to0^+$. Statements "for every $\lambda$" are exact.

## Scope

- In scope:
  - one body, and two bodies of one type, recorded by one medium, with the localized starts of (S1) and (S2);
  - the leading order, and the exact statements asked for;
  - the two examples.
- Out of scope:
  - contracts that move bodies, and motion;
  - bodies of different types, and more than two bodies;
  - whether the background is the geometry of some view for $c_T=-1$;
  - distances other than $d_0$, such as the finite-$\lambda$ witness angle or chain distances;
  - the cost;
  - any physical or spatial meaning beyond the definitions (M1, M4).

## Depth

- Item 1: short argument.
- Item 2: derive.
- Item 3: derive.

## Expected result

- Item 1: a closed form for the state, and two equivalences.
- Item 2: a closed form for the state; statements on localization, correlation and well-definedness; the two views; an equivalence and a sufficient condition; a yes/no answer and an equivalence.
- Item 3: closed forms for $D$ and $d_0$; the pairs of configurations with the same state of the medium; a yes/no answer for each example.

Give every main result a tag $(15.k)$.

Consistency checks, at most three: for example $\beta_1=\beta_2$ for $c_T=+1$, a change of split (12.6), and $m=1$ in the window example.

## Code

None.