03-ilang-space / 14-dimension
14dimensionverified
Summary What fixes the leading-order dimension of the witness geometry when the medium has several recording channels, and when that dimension is three.
# Dimension of the witness geometry for several recording channels
- **Subproject:** 03-ilang-space
- **Package:** 14-dimension
- **Version:** v1
- **Mode:** new
- **Date:** 2026-10-08
## Setup and assumptions
**Model (question.md).** A recorded object $a$ with places $h\in H$, $\lvert H\rvert=n$. Medium cells $c=1,\dots,m$, with $d_c=\dim\mathcal H_c$. All types are pairwise different, with one instance each. The initial state is $\lvert\phi\rangle\otimes\lvert\chi\rangle$ with $\lvert\chi\rangle=\bigotimes_c\lvert\chi_c\rangle$ and $\phi_h\neq0$ for all $h$. The contracts are $C=\sum_c\sum_h\lvert h\rangle\langle h\rvert\otimes K^{(c)}_h$ with $K^{(c)}_h$ self-adjoint on cell $c$. The record vector of channel $c$ is $u^{(c)}_h=(K^{(c)}_h-\langle K^{(c)}_h\rangle_c)\lvert\chi_c\rangle$, where $\langle X\rangle_c:=\langle\chi_c\vert X\vert\chi_c\rangle$, and $\dim^{(c)}_0:=\dim_{\mathbb R}\operatorname{aff}_{\mathbb R}\{u^{(c)}_h\}$.
**Quoted inputs.**
- From 05 (setting: one recorded object, one medium object with space $\mathcal H_c$, contract $\sum_h\lvert h\rangle\langle h\rvert\otimes K_h$, start $\lvert\phi\rangle\otimes\lvert\chi\rangle$):
- $g(v,w)=\operatorname{Re}\langle v\vert w\rangle$;
- $h\sim_0h'$ iff $u_h=u_{h'}$;
- $\dim_0=\dim_{\mathbb R}\operatorname{aff}_{\mathbb R}\{u_h\}$;
- (5.10) $\lim_{\lambda\to0^+}\alpha(h,h';\lambda)/\lambda=d_0(h,h')=\lVert u_h-u_{h'}\rVert$, with $\lvert u_h\rangle=(K_h-\langle K_h\rangle)\lvert\chi\rangle$;
- (5.11) $(H_V/{\sim_0},d_0)\cong(\{u_h\},\lVert\cdot\rVert)\subset(\mathcal H_c,g)$;
- (5.13) $\langle\chi\vert u_h\rangle=0$;
- (5.14) $\dim_0\le\min(d_{T(a)}-1,\,2d_c-2)$.
- From 07 ($K_h$ i.i.d. in a basis $e_1=\lvert\chi\rangle,e_2,\dots,e_{d_c}$): (7.7) GUE gives $\dim_0=\min(n-1,2(d_c-1))$ a.s.; (7.11) GOE gives $\dim_0=\min(n-1,d_c-1)$ a.s.
- From 10:
- $\omega(v,w)=\operatorname{Im}\langle v\vert w\rangle$, $Jv=\mathrm iv$, $\Phi=\arg W$;
- (10.8) $\Phi(h_1,\dots,h_k;\lambda)=-\lambda^2\sum_{j=1}^k\omega(u_{h_j},u_{h_{j+1}})+O(\lambda^3)$ for $k\ge3$, with $h_{k+1}:=h_1$;
- (10.12) $\omega(v,w)=g(Jv,w)$, $g(Jv,Jw)=g(v,w)$, $J^2=-\mathbb 1$.
**Assumptions.**
- **S1.** Every statement is of leading order at $\lambda\to0^+$. The contracts do not depend on $\lambda$ (A5).
- **S2 (genericity, item 3).** The records of different channels are drawn independently. Each channel's records are drawn from an absolutely continuous law on the parameter space of its class:
- complex generic: Hermitian $K^{(c)}_h$, as in 07;
- real: real symmetric $K^{(c)}_h$ in a real orthonormal basis $e_1=\lvert\chi_c\rangle,e_2,\dots,e_{d_c}$, as in 07;
- commuting: $K^{(c)}_h=\sum_{j=1}^{d_c}\kappa_h(j)\lvert e_j\rangle\langle e_j\rvert$ in a fixed orthonormal basis $\{e_j\}$ of $\mathcal H_c$, with real $\kappa_h(j)$ drawn and $\langle e_j\vert\chi_c\rangle\neq0$ for all $j$. This is my reading of "generically within the commuting class".
- **Notation.**
- $\mathrm i$ is the imaginary unit;
- $\mathcal H_M:=\bigotimes_c\mathcal H_c$ and $D:=\prod_cd_c$;
- $V:=\{t\in\mathbb R^H:\sum_ht_h=0\}$, with $\dim V=n-1$ and the product $t\cdot s=\sum_ht_hs_h$;
- $d^{(c)}_0(h,h'):=\lVert u^{(c)}_h-u^{(c)}_{h'}\rVert$.
## Derivation
### Step 1. Reduction to the setting of 05
1. The $\lvert h\rangle\langle h\rvert$ are orthogonal projectors. For $c\neq c'$, $K^{(c)}_h$ and $K^{(c')}_h$ act on different tensor factors and therefore commute. Hence $e^{-\mathrm iC\lambda}=\sum_h\lvert h\rangle\langle h\rvert\otimes\bigotimes_ce^{-\mathrm iK^{(c)}_h\lambda}$, and
$$
C=\sum_h\lvert h\rangle\langle h\rvert\otimes K_h,\quad K_h:=\sum_{c=1}^mK^{(c)}_h,\qquad
\lvert\Psi(\lambda)\rangle=\sum_h\phi_h\lvert h\rangle\otimes\bigotimes_{c}e^{-\mathrm iK^{(c)}_h\lambda}\lvert\chi_c\rangle .
\tag{14.1}
$$
2. **Carry-over.** (14.1) is the setting of 05, with the medium object replaced by the part $M=\{c_1,\dots,c_m\}=\bar a$. The contract is a recording contract with self-adjoint $K_h$ on $\mathcal H_M$, and the start is $\lvert\phi\rangle\otimes\lvert\chi\rangle$ with $\lvert\chi\rangle$ a unit vector of $\mathcal H_M$. The rules from which the quantities of 05 and 10 are built do not distinguish a one-object medium from a several-object one:
- the view (1.5)–(1.7) is defined for every part (A7) and depends on the companion only through $\mathcal H_{\bar a}$, its inner product and the branch states;
- the evolution is (14.1);
- (1.11) imposes nothing, because each type has one instance.
The quoted right-hand sides likewise involve the medium only through $(\mathcal H_c,\langle\cdot\vert\cdot\rangle)$, $\lvert\chi\rangle$ and $K_h$. Hence (5.10), (5.11), (5.13), (5.14), (10.8) and (10.12) hold for the whole medium after the replacements $\mathcal H_c\to\mathcal H_M$, $d_c\to D$, $K_h\to\sum_cK^{(c)}_h$ and $\lvert\chi\rangle\to\bigotimes_c\lvert\chi_c\rangle$. Applied to channel $c$ alone (the setting of 05 with medium $c$), they give $u^{(c)}_h$, $d^{(c)}_0$ and $\dim^{(c)}_0$. In particular $\langle\chi_c\vert u^{(c)}_h\rangle=0$ by (5.13).
### Step 2. Record vectors and distance (item 1a)
Let $\iota_c:\mathcal H_c\to\mathcal H_M$ be the map $\iota_c(v):=\lvert\chi_1\rangle\otimes\cdots\otimes\lvert\chi_{c-1}\rangle\otimes v\otimes\lvert\chi_{c+1}\rangle\otimes\cdots\otimes\lvert\chi_m\rangle$. It is a complex-linear isometry. Since the $\lvert\chi_{c'}\rangle$ are unit vectors, $K^{(c)}_h\lvert\chi\rangle=\iota_c(K^{(c)}_h\lvert\chi_c\rangle)$, $\langle\chi\vert K^{(c)}_h\vert\chi\rangle=\langle K^{(c)}_h\rangle_c$ and $\lvert\chi\rangle=\iota_c\lvert\chi_c\rangle$. Inserting $K_h$ of (14.1) into (5.10) gives
$$
u_h=\sum_{c=1}^m\iota_c\bigl(u^{(c)}_h\bigr).
\tag{14.2}
$$
For $c\neq c'$, $\langle\iota_cv\vert\iota_{c'}w\rangle=\langle v\vert\chi_c\rangle\langle\chi_{c'}\vert w\rangle$, which vanishes for $v=u^{(c)}_h$ by (5.13). The channel images are therefore mutually orthogonal, and
$$
\langle u_h\vert u_{h'}\rangle=\sum_c\langle u^{(c)}_h\vert u^{(c)}_{h'}\rangle,
\qquad
d_0(h,h')=\Bigl(\sum_{c=1}^m d^{(c)}_0(h,h')^2\Bigr)^{1/2}.
\tag{14.3}
$$
Thus $h\sim_0h'$ iff $u^{(c)}_h=u^{(c)}_{h'}$ for every $c$. By (5.11), $(H_V/{\sim_0},d_0)$ is isometric to the set of tuples $(u^{(1)}_h,\dots,u^{(m)}_h)$ in the orthogonal direct sum $\bigoplus_c\chi_c^{\perp}$, because $\sum_c\iota_c$ maps this direct sum isometrically into $\mathcal H_M$.
### Step 3. Dimension, sum condition and general relation (items 1a–c)
1. For a family $\{v_h\}$ in a real vector space, the direction space of $\operatorname{aff}_{\mathbb R}\{v_h\}$ is $\{\sum_ht_hv_h:t\in V\}$. Define the real-linear maps $A_c:V\to\mathcal H_c$, $A_ct:=\sum_ht_hu^{(c)}_h$, and $A:V\to\mathcal H_M$, $At:=\sum_ht_hu_h=\sum_c\iota_c(A_ct)$. Then $\dim^{(c)}_0=\operatorname{rank}A_c$ and $\dim_0=\operatorname{rank}A$.
2. By (14.3), $\lVert At\rVert^2=\sum_c\lVert A_ct\rVert^2$, so $\ker A=\bigcap_c\ker A_c$.
3. **Read-out spaces.** For $w\in\mathcal H_c$ let $f^{(c)}_w(h):=g(w,u^{(c)}_h)$, a real function on $H$, and let $\mathring f:=f-\frac1n\sum_{h}f(h)\in V$ denote the centred version of a function $f$. For $t\in V$ we have $g(w,A_ct)=t\cdot f^{(c)}_w=t\cdot\mathring f^{(c)}_w$. Since $g$ is nondegenerate, $t\in\ker A_c$ iff $t\perp\mathring f^{(c)}_w$ for every $w$. With $\perp$ taken in $V$, this gives
$$
F_c:=(\ker A_c)^{\perp}=\bigl\{\mathring f^{(c)}_w:\ w\in\mathcal H_c\bigr\}\subset V,\qquad \dim F_c=\operatorname{rank}A_c=\dim^{(c)}_0 .
\tag{14.4}
$$
4. Step 2 and the standard identity $(\bigcap_cN_c)^{\perp}=\sum_cN_c^{\perp}$ give
$$
\dim_0=\dim V-\dim\ker A=\dim_{\mathbb R}\bigl(F_1+F_2+\cdots+F_m\bigr).
\tag{14.5}
$$
5. **(b)** The dimension of a sum of subspaces is at most the sum of their dimensions, with equality iff the sum is direct. Hence
$$
\dim_0=\sum_c\dim^{(c)}_0\iff F_1+\cdots+F_m=F_1\oplus\cdots\oplus F_m\iff F_c\cap\sum_{c'\neq c}F_{c'}=\{0\}\ \ \forall c .
\tag{14.6}
$$
In terms of the records: if $h\mapsto\sum_cg(w_c,u^{(c)}_h)$ is constant on $H$ for some $w_c\in\mathcal H_c$, then every $h\mapsto g(w_c,u^{(c)}_h)$ is constant.
6. **(c)** The inclusions $F_c\subseteq\sum_{c'}F_{c'}\subseteq V$ and subadditivity give
$$
\max_c\dim^{(c)}_0\;\le\;\dim_0\;\le\;\min\Bigl(n-1,\ \sum_c\dim^{(c)}_0\Bigr),\qquad
\sum_c\dim^{(c)}_0\le2\sum_c(d_c-1)\le2D-2 .
\tag{14.7}
$$
The second inequality is (5.14) for each channel. The last follows from $\prod_c(1+(d_c-1))\ge1+\sum_c(d_c-1)$. For $m=2$, (14.5) reads exactly $\dim_0=\dim^{(1)}_0+\dim^{(2)}_0-\dim(F_1\cap F_2)$. The lower bound is attained for identical channels (all $F_c$ equal), and the upper bound whenever (14.6) holds.
### Step 4. Loop data of several channels
Taking imaginary parts in (14.3) gives $\omega(u_h,u_{h'})=\sum_c\omega(u^{(c)}_h,u^{(c)}_{h'})$. With (10.8) and $h_{k+1}:=h_1$,
$$
\Phi(h_1,\dots,h_k;\lambda)=-\lambda^2\sum_{c=1}^m\Omega^{(c)}(h_1,\dots,h_k)+O(\lambda^3),\qquad
\Omega^{(c)}:=\sum_{j=1}^k\omega\bigl(u^{(c)}_{h_j},u^{(c)}_{h_{j+1}}\bigr),
\tag{14.8}
$$
so channel $c$ contributes $-\lambda^2\Omega^{(c)}$. Now suppress the channel index on $K$. The operator $K-\langle K\rangle_c$ is self-adjoint, the product $\langle K_h\rangle_c\langle K_{h'}\rangle_c$ is real, and $\langle K_hK_{h'}\rangle_c^*=\langle K_{h'}K_h\rangle_c$. Hence
$$
\langle u^{(c)}_h\vert u^{(c)}_{h'}\rangle=\langle K_hK_{h'}\rangle_c-\langle K_h\rangle_c\langle K_{h'}\rangle_c,
\qquad
\omega\bigl(u^{(c)}_h,u^{(c)}_{h'}\bigr)=\frac{1}{2\mathrm i}\bigl\langle[K_h,K_{h'}]\bigr\rangle_c .
\tag{14.9}
$$
### Step 5. Commuting records in one channel (item 2)
1. **Spectral form.** Pairwise commuting self-adjoint operators on the finite-dimensional space $\mathcal H_c$ have a joint spectral decomposition $K^{(c)}_h=\sum_{j=1}^J\kappa_h(j)P_j$. Here $P_1,\dots,P_J$ are the orthogonal projectors onto the joint eigenspaces, with $\sum_jP_j=\mathbb 1$ and $J\le d_c$. The $\kappa_h(j)$ are real, and the tuples $(\kappa_h(j))_{h\in H}$ are distinct for distinct $j$. Let $p_j:=\langle\chi_c\vert P_j\vert\chi_c\rangle=\lVert P_j\chi_c\rVert^2\ge0$, so $\sum_jp_j=1$. Let $S:=\{j:p_j>0\}$, $r_\chi:=\lvert S\rvert$ and $\bar\kappa_h:=\sum_jp_j\kappa_h(j)=\langle K^{(c)}_h\rangle_c$.
2. **Gram matrix.** Since $\langle\chi_c\vert P_jP_{j'}\vert\chi_c\rangle=\delta_{jj'}p_j$, (14.9) becomes
$$
\langle u^{(c)}_h\vert u^{(c)}_{h'}\rangle=\sum_{j}p_j\bigl(\kappa_h(j)-\bar\kappa_h\bigr)\bigl(\kappa_{h'}(j)-\bar\kappa_{h'}\bigr),
\qquad d^{(c)}_0(h,h')^2=\operatorname{Var}_p\bigl(\kappa_h-\kappa_{h'}\bigr).
\tag{14.10}
$$
The Gram matrix is the covariance matrix of the functions $j\mapsto\kappa_h(j)$ under the weights $p$. It is real, symmetric and positive semidefinite.
3. **Loop term.** The Gram matrix is real; equivalently, $[K_h,K_{h'}]=0$ in (14.9). Hence $\omega(u^{(c)}_h,u^{(c)}_{h'})=0$ for all $h,h'$, and
$$
\Omega^{(c)}(h_1,\dots,h_k)=0\qquad\text{for every closed polygon}.
\tag{14.11}
$$
A commuting channel therefore contributes nothing to the $\lambda^2$ term of the loop phases (14.8).
4. **Largest $\dim^{(c)}_0$.** Since $P_j\chi_c=0$ for $j\notin S$, we have $u^{(c)}_h=\sum_{j\in S}(\kappa_h(j)-\bar\kappa_h)P_j\lvert\chi_c\rangle=\Theta(\Pi\kappa_h)$, where:
- $\kappa_h\in\mathbb R^S$ denotes the restriction to $S$;
- $\Theta:\mathbb R^S\to\mathcal H_c$, $x\mapsto\sum_{j\in S}x_jP_j\lvert\chi_c\rangle$, is injective, because the vectors $P_j\lvert\chi_c\rangle$, $j\in S$, are nonzero and mutually orthogonal;
- $\Pi y:=y-(\sum_{j\in S}p_jy_j)\mathbf 1$ is a projector on $\mathbb R^S$ with kernel $\mathbb R\mathbf 1$.
Fix $h_0$ and let $L:=\operatorname{span}_{\mathbb R}\{\kappa_h-\kappa_{h_0}\}$. The direction space of $\operatorname{aff}\{u^{(c)}_h\}$ is $\Theta\Pi(L)$, and $\dim\Pi(L)=\dim(L+\mathbb R\mathbf 1)-1$. Let $[\cdots]_S$ denote the real $r_\chi\times n$ matrix with the columns shown. Then
$$
\dim^{(c)}_0=\operatorname{rank}_{\mathbb R}\bigl[\mathbf 1,\ \kappa_h-\kappa_{h_0}\ (h\neq h_0)\bigr]_{S}-1\ \le\ \min(n,r_\chi)-1\ \le\ \min(n,d_c)-1 .
\tag{14.12}
$$
The first bound is attained iff the matrix has full rank $\min(n,r_\chi)$. The value $\min(n-1,d_c-1)$ is attained iff, in addition, $r_\chi=d_c$: all joint eigenspaces are one-dimensional, and $\lvert\chi_c\rangle$ has a nonzero component along every joint eigenvector. The value depends on $p$ only through its support $S$.
### Step 6. Generic dimension of several channels (item 3)
1. **Carriers.** In each class, the records of channel $c$ lie in a real subspace $S_c\subseteq\chi_c^\perp$ of dimension $\delta_c$:
- complex generic: $S_c=\chi_c^\perp$ by (5.13), so $\delta_c=2(d_c-1)$;
- real: $u^{(c)}_h=\sum_{k\ge2}(K^{(c)}_h)_{k1}e_k$ has real coefficients, so $S_c=\operatorname{span}_{\mathbb R}\{e_2,\dots,e_{d_c}\}$ and $\delta_c=d_c-1$;
- commuting (S2): $u^{(c)}_h=\sum_j(\kappa_h(j)-\bar\kappa_h)\langle e_j\vert\chi_c\rangle\lvert e_j\rangle$, so $S_c=\{\sum_jx_j\langle e_j\vert\chi_c\rangle\lvert e_j\rangle:\ x\in\mathbb R^{d_c},\ \sum_jp_jx_j=0\}$ with all $p_j=\lvert\langle e_j\vert\chi_c\rangle\rvert^2>0$, and $\delta_c=d_c-1$.
Conversely, every family $(v_h)\in S_c^H$ is realized within the class:
- complex generic and real: by $K^{(c)}_h=\lvert v_h\rangle\langle\chi_c\rvert+\lvert\chi_c\rangle\langle v_h\rvert$, which is real symmetric for real $v_h$ and gives $\langle K^{(c)}_h\rangle_c=0$ and $u^{(c)}_h=v_h$;
- commuting: by $\kappa_h=x_h$, where $v_h=\sum_jx_h(j)\langle e_j\vert\chi_c\rangle\lvert e_j\rangle$, which gives $\bar\kappa_h=0$.
2. **Rank.** By Step 3, $\dim_0=\operatorname{rank}A$, with $A:V\to\bigoplus_cS_c$, a space of real dimension $\sum_c\delta_c$.
- In fixed real bases, the entries of $A$ are linear in the draw. $\operatorname{rank}A\ge r$ iff some $r\times r$ minor, a polynomial in the draw, is nonzero.
- Let $r_*:=\min(n-1,\sum_c\delta_c)$. Choose the $v_h$ in step 1 so that $r_*$ of the tuples $(v^{(c)}_h-v^{(c)}_{h_0})_c$, $h\neq h_0$, are linearly independent. Then some $r_*\times r_*$ minor is a nonzero polynomial.
- A nonzero polynomial vanishes only on a Lebesgue-null set, so this minor is nonzero almost surely under S2. Since $\operatorname{rank}A\le r_*$ always,
$$
\dim_0=\min\Bigl(n-1,\ \sum_{c=1}^m\delta_c\Bigr)\quad\text{generically},\qquad
\delta_c=\begin{cases}2(d_c-1)&\text{complex generic},\\ d_c-1&\text{real or commuting}.\end{cases}
\tag{14.13}
$$
For $m=1$ this reproduces (7.7), (7.11) and the attained bound (14.12).
3. **Loop data.**
- Real channel: the $u^{(c)}_h$ are real combinations of the orthonormal $e_k$. Their Gram matrix is real, so $\Omega^{(c)}=0$.
- Commuting channel: $\Omega^{(c)}=0$ by (14.11).
- Complex generic channel with $d_c\ge2$, and a triangle of distinct places $h_1,h_2,h_3$: $\Omega^{(c)}$ is a polynomial in the draw. It equals $\omega(e_2,\mathrm ie_2)=1$ at $v_{h_1}=e_2$, $v_{h_2}=\mathrm ie_2$, $v_{h_3}=0$. If, in addition, the records of all other channels vanish, the total $\sum_c\Omega^{(c)}$ takes the same value.
By the polynomial argument, applied to all (finitely many) triangles at once,
$$
\text{the }\lambda^2\text{ term of }\Phi(h_1,h_2,h_3;\lambda)\text{ is generically }\neq0\iff\text{some channel is complex generic with }d_c\ge2;\ \text{otherwise it is }0 .
\tag{14.14}
$$
### Step 7. Combinations with $\dim_0=3$ (item 3)
Cells with $d_c=1$ have $\delta_c=0$ and $u^{(c)}_h=0$; they are omitted. Since $n-1\ge3$, (14.13) gives $\dim_0=3$ generically iff
$$
\sum_c\delta_c=3\quad(n\ge5),\qquad\sum_c\delta_c\ge3\quad(n=4).
\tag{14.15}
$$
For $n\ge5$: a complex generic channel has even $\delta_c\ge2$. So at most one such channel appears, with $d_c=2$, and the remaining $\delta=1$ comes from one real or commuting qubit. The table lists all solutions; R denotes a real or commuting channel, chosen per channel.
| channels (class, $d_c$) | $\delta_c$ | $D=\prod_cd_c$ | $\lambda^2$ loop term (14.14) |
|---|---|---|---|
| GUE(2) + R(2) | 2+1 | 4 | generically nonzero |
| R(4) | 3 | 4 | zero |
| R(3) + R(2) | 2+1 | 6 | zero |
| R(2) + R(2) + R(2) | 1+1+1 | 8 | zero |
For $n=4$: every combination with $\sum_c\delta_c\ge3$ gives $\dim_0=3$.
**Minimal total dimension.** We use $D=\prod_c(1+(d_c-1))\ge1+\sum_c(d_c-1)$, with equality only when there is a single nontrivial cell.
- $n\ge5$: the table gives $D_{\min}=4$, both with loop data (GUE(2)+R(2)) and without (R(4)). $D\le3$ allows only a single nontrivial cell with $d_c\le3$. Its $\delta_c$ lies in $\{1,2,4\}$, which excludes $\delta_c=3$.
- $n=4$, with loop data: $D_{\min}=3$, attained by a single complex generic qutrit ($\delta=4$). $D=2$ allows only $\delta\le2$.
- $n=4$, without loop data: all channels are R, so $\sum_c(d_c-1)\ge3$ and hence $D\ge4$. This is attained by R(4).
In summary,
$$
D_{\min}=4\ \text{with and without loop data}\ (n\ge5);\qquad D_{\min}=3\ \text{with},\ D_{\min}=4\ \text{without}\ (n=4).
\tag{14.16}
$$
### Step 8. Cubic lattice (item 4a)
Since $\langle0\vert\sigma_x\vert0\rangle=0$ and $\sigma_x\lvert0\rangle=\lvert1\rangle$, we get $u^{(1)}_h=x\lvert1\rangle$, $u^{(2)}_h=y\lvert1\rangle$ and $u^{(3)}_h=z\lvert1\rangle$. With (14.2), (14.3), and the cells ordered $1,2,3$:
$$
u_h=x\lvert100\rangle+y\lvert010\rangle+z\lvert001\rangle,\qquad
d_0(h,h')=\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}.
\tag{14.17}
$$
- **Distance.** The map $h\mapsto u_h$ is injective. So $\sim_0$ is trivial, and $(H,d_0)$ is isometric to $\{1,\dots,N\}^3\subset\mathbb R^3$ with the Euclidean distance.
- **Dimension.** Each $\dim^{(c)}_0=1$, since $N\ge2$. By (14.4), $F_1=\mathbb R\mathring x$, $F_2=\mathbb R\mathring y$ and $F_3=\mathbb R\mathring z$ (since $g(w,x\lvert1\rangle)=x\operatorname{Re}\langle w\vert1\rangle$). If $ax+by+cz$ is constant on $\{1,\dots,N\}^3$ with $N\ge2$, then $a=b=c=0$. So the sum of the $F_c$ is direct, and $\dim_0=3$ by (14.6).
- **Loop.** The Gram matrix $\langle u_h\vert u_{h'}\rangle=xx'+yy'+zz'$ is real. All three channels are commuting ($\sigma_x$ only), consistent with (14.11).
Hence, for every lattice triangle,
$$
\dim_0=3,\qquad\Phi(h_1,h_2,h_3;\lambda)=O(\lambda^3)\quad(\lambda^2\text{ term }=0).
\tag{14.18}
$$
### Step 9. Plane plus line (item 4b)
Since $\sigma_y\lvert0\rangle=\mathrm i\lvert1\rangle$ and $\langle0\vert\sigma_y\vert0\rangle=0$, we get $u^{(1)}_h=(x+\mathrm iy)\lvert1\rangle$ and $u^{(2)}_h=z\lvert1\rangle$. Therefore
$$
u_h=(x+\mathrm iy)\lvert10\rangle+z\lvert01\rangle,\qquad
\langle u_h\vert u_{h'}\rangle=xx'+yy'+zz'+\mathrm i\,(xy'-yx'),
\tag{14.19}
$$
$$
d_0(h,h')=\sqrt{(x-x')^2+(y-y')^2+(z-z')^2},
\tag{14.20}
$$
- **Distance.** As in Step 8, $(H,d_0)$ is isometric to $\{1,\dots,N\}^3\subset\mathbb R^3$.
- **Dimension.** $w=\lvert1\rangle$ gives $f^{(1)}_w=x$, and $w=\mathrm i\lvert1\rangle$ gives $f^{(1)}_w=\operatorname{Re}[-\mathrm i(x+\mathrm iy)]=y$. So $F_1=\operatorname{span}\{\mathring x,\mathring y\}$ and $\dim^{(1)}_0=2$, while $F_2=\mathbb R\mathring z$ and $\dim^{(2)}_0=1$. The sum is direct, so $\dim_0=3$ by (14.6).
- **Loop.** By (14.19), $\omega(u_h,u_{h'})=xy'-yx'$. Only channel 1 contributes, since channel 2 is commuting (14.11). By (14.8), with $h_j=(x_j,y_j,z_j)$,
$$
\Phi(h_1,h_2,h_3;\lambda)=-\lambda^2\bigl[(x_2-x_1)(y_3-y_1)-(x_3-x_1)(y_2-y_1)\bigr]+O(\lambda^3).
\tag{14.21}
$$
The bracket is the integer $2A_{xy}$, where $A_{xy}$ is the signed area of the projection of the triangle onto the $(x,y)$ coordinates. It is nonzero iff the projected points are not collinear. For example, the term is $-\lambda^2$ for $h_1=(x,y,z)$, $h_2=(x+1,y,z)$, $h_3=(x,y+1,z)$, and $0$ for every triangle with constant $x$ or constant $y$.
## Result
- **Carry-over (Step 1).** (5.10)–(5.14), (10.8) and (10.12) hold for the composite medium after the replacements $\mathcal H_c\to\mathcal H_M$, $K_h\to\sum_cK^{(c)}_h$, $\chi\to\bigotimes_c\chi_c$.
- **Item 1.**
- (a) $u_h=\sum_c\iota_c(u^{(c)}_h)$ (14.2), with mutually orthogonal channel images. $d_0^2=\sum_c(d^{(c)}_0)^2$ (14.3). $\dim_0=\dim(F_1+\cdots+F_m)$ (14.5), with the read-out spaces $F_c$ of (14.4).
- (b) $\dim_0=\sum_c\dim^{(c)}_0$ iff the $F_c$ are linearly independent (14.6).
- (c) $\max_c\dim^{(c)}_0\le\dim_0\le\min(n-1,\sum_c\dim^{(c)}_0)$ (14.7).
- Loop phases: the $\lambda^2$ term is a sum of channel contributions (14.8), with $\omega=\frac1{2\mathrm i}\langle[K_h,K_{h'}]\rangle_c$ (14.9).
- **Item 2.** The Gram matrix is a real covariance matrix (14.10). $\dim^{(c)}_0\le\min(n,r_\chi)-1\le\min(n,d_c)-1$, with the rank formula and attainment condition of (14.12). The channel's $\lambda^2$ loop term vanishes (14.11).
- **Item 3.**
- Generically, $\dim_0=\min(n-1,\sum_c\delta_c)$ (14.13).
- The $\lambda^2$ loop term is generically nonzero iff there is a complex generic channel with $d_c\ge2$ (14.14).
- $\dim_0=3$ iff $\sum_c\delta_c=3$ for $n\ge5$, or $\sum_c\delta_c\ge3$ for $n=4$ (14.15); the solutions are listed in the table of Step 7.
- $D_{\min}$ is given in (14.16).
- **Item 4.**
- (a) Euclidean $d_0$, $\dim_0=3$, zero $\lambda^2$ loop term (14.17), (14.18).
- (b) Euclidean $d_0$, $\dim_0=3$, and loop phase $-\lambda^2\cdot2A_{xy}$ (14.19)–(14.21).
## Consistency checks
1. **Scaling.** The evolution (14.1) depends only on $\lambda K$. Under $K^{(c)}_h\to sK^{(c)}_h$ with $s>0$:
- $u\to su$ and $d_0\to sd_0$, so $\lambda d_0$ is invariant under $\lambda\to\lambda/s$;
- $\omega\to s^2\omega$, so $\lambda^2\Omega^{(c)}$ is invariant in (14.8) and (14.21);
- $\dim_0$, $\dim^{(c)}_0$ and $\delta_c$ are unchanged.
2. **Single channel ($m=1$).** Here $\iota_1=\mathrm{id}$, and (14.2), (14.3) reduce to (5.10). (14.5) gives $\dim_0=\dim^{(1)}_0$, the two bounds in (14.7) coincide, and (14.6) holds trivially. (14.8) reduces to (10.8), and (14.13) reduces to (7.7) and (7.11).
3. **Loop reversal.** Reversing $(h_1,\dots,h_k)$ flips the sign of every $\Omega^{(c)}$, because $\omega$ is antisymmetric (it is the imaginary part of a Hermitian form). In (14.21), exchanging $h_2\leftrightarrow h_3$ flips the sign of the bracket. As a cross-check of (14.9) in Step 9, $[x\sigma_x+y\sigma_y,\,x'\sigma_x+y'\sigma_y]=2\mathrm i(xy'-yx')\sigma_z$ and $\langle0\vert\sigma_z\vert0\rangle=1$ reproduce $\omega=xy'-yx'$.
## Open issues
- The carry-over (Step 1) assumes that the constructions of 05 and 10 refer to the medium only as the companion part of $a$ (view and branch states). Only their quoted equations were available here.
- Two parts of S2 are readings of "generically within a class": the independence of the channels, and the definition of the generic commuting class.
- Only the $\lambda^2$ term of the loop phases is determined. For real and commuting channels the loop phases are $O(\lambda^3)$; higher orders are not addressed.
- For $n=4$, $\dim_0=3=n-1$ is the maximal possible value. The condition (14.15) is then only a threshold.
## Methods used
- tensor-product isometric embeddings, orthogonal decomposition
- real affine spans, rank–nullity, orthogonal complements of intersections and sums of subspaces
- splitting of a Hermitian form into Euclidean and symplectic parts
- joint spectral decomposition of commuting self-adjoint operators; covariance matrices
- genericity through nonvanishing polynomials (minors), absolute continuity
- shoelace formula for signed areas