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# Theorem G3: Fixed-Point Analysis of the Strand Crossing Operator on the Probability Simplex
**Date:** 2026-01-12
---
## Abstract
We analyze the strand crossing operator $C: \Delta_7 \to \Delta_7$, which coarse-grains the 8 coordinates of the probability simplex into 4 pairs by averaging. We prove that $C$ is a projection onto a 4-dimensional submanifold $M \subset \Delta_7$, that $C$ is a contraction in the Fisher metric, and we characterize its fixed points, information loss, and connection to Chentsov's theorem.
---
## 1. Setup and Definitions
### 1.1 The Probability Simplex $\Delta_7$
The open probability simplex on 8 outcomes is defined as
$$\Delta_7 := \left\{ p \in \mathbb{R}^8 : p_i > 0 \text{ for all } i = 1,\ldots,8, \quad \sum_{i=1}^8 p_i = 1 \right\}.$$
This is a 7-dimensional smooth manifold with boundary removed (the closure $\overline{\Delta}_7$ includes the boundary where some coordinates may vanish). The tangent space at $p \in \Delta_7$ is
$$T_p\Delta_7 = \left\{ v \in \mathbb{R}^8 : \sum_{i=1}^8 v_i = 0 \right\}.$$
### 1.2 The Fisher Metric
The **Fisher information metric** on $\Delta_7$ is the Riemannian metric defined by
$$g_p(u,v) = \sum_{i=1}^8 \frac{u_i v_i}{p_i}, \qquad p \in \Delta_7, \; u,v \in T_p\Delta_7.$$
This metric is positive-definite on $T_p\Delta_7$ since $p_i > 0$ ensures each term is well-defined. The Fisher metric is the unique (up to scale) Riemannian metric on $\Delta_n$ that is invariant under sufficient statistics (Chentsov's theorem, see Section 7).
### 1.3 The Bhattacharyya Distance
The Riemannian distance induced by the Fisher metric is the **Bhattacharyya distance** (also called the Fisher-Rao distance). It is given explicitly by
$$d_F(p,q) = 2 \arccos\left( \sum_{i=1}^8 \sqrt{p_i q_i} \right), \qquad p,q \in \Delta_7.$$
The argument of $\arccos$ is the **Bhattacharyya coefficient** $BC(p,q) = \sum_{i=1}^8 \sqrt{p_i q_i}$. Since $\sqrt{p_i q_i} \leq (p_i + q_i)/2$ by the AM-GM inequality, we have $BC(p,q) \leq 1$, so $d_F(p,q) \in [0, \pi]$.
### 1.4 The Square-Root Embedding
A fundamental tool for analyzing the Fisher geometry is the **square-root embedding**:
$$\psi : \Delta_7 \longrightarrow S^7, \qquad \psi(p) = \left( \sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_8} \right),$$
where $S^7 \subset \mathbb{R}^8$ is the unit sphere. This map is a diffeomorphism onto its image, the positive orthant of the unit sphere:
$$\psi(\Delta_7) = \{ x \in S^7 : x_i > 0 \text{ for all } i \}.$$
The key observation is that the Fisher metric on $\Delta_7$ is precisely **4 times the pullback of the round metric** on $S^7$. Concretely, if $g^{S^7}$ denotes the round metric on $S^7$, then
$$g_p(u,v) = 4 \cdot g^{S^7}_{\psi(p)}\left( d\psi_p(u), d\psi_p(v) \right).$$
The differential of $\psi$ is $d\psi_p(v) = \left( \frac{v_1}{2\sqrt{p_1}}, \ldots, \frac{v_8}{2\sqrt{p_8}} \right)$. Consequently, geodesic distances satisfy
$$d_F(p,q) = 2 \cdot d_{S^7}\left( \psi(p), \psi(q) \right) = 2 \arccos\left( \langle \psi(p), \psi(q) \rangle \right),$$
where $d_{S^7}$ is the great-circle distance on $S^7$. This confirms the formula in Section 1.3.
### 1.5 The Strand Crossing Operator $C$
Partition the index set $\{1,2,\ldots,8\}$ into 4 consecutive pairs:
$$P_1 = \{1,2\}, \quad P_2 = \{3,4\}, \quad P_3 = \{5,6\}, \quad P_4 = \{7,8\}.$$
For each pair $P_k = \{2k-1, 2k\}$, define the **pair sum** $s_k(p) = p_{2k-1} + p_{2k}$.
The **strand crossing operator** $C: \Delta_7 \to \Delta_7$ is defined by averaging within each pair:
$$C(p)_{2k-1} = C(p)_{2k} = \frac{p_{2k-1} + p_{2k}}{2} = \frac{s_k(p)}{2}, \qquad k = 1,2,3,4.$$
In coordinates:
$$C(p) = \left( \frac{p_1+p_2}{2}, \frac{p_1+p_2}{2}, \frac{p_3+p_4}{2}, \frac{p_3+p_4}{2}, \frac{p_5+p_6}{2}, \frac{p_5+p_6}{2}, \frac{p_7+p_8}{2}, \frac{p_7+p_8}{2} \right).$$
Equivalently, $C$ can be written as
$$C(p) = \sum_{k=1}^4 \frac{s_k(p)}{2} \left( e_{2k-1} + e_{2k} \right),$$
where $e_i$ denotes the $i$-th standard basis vector in $\mathbb{R}^8$.
---
## 2. Lemma 1: Idempotence of $C$
**Lemma 1.** The strand crossing operator satisfies $C \circ C = C$. That is, $C$ is a projection.
**Proof.** Let $p \in \Delta_7$ and let $q = C(p)$. For each pair $P_k$, we have
$$q_{2k-1} = q_{2k} = \frac{p_{2k-1} + p_{2k}}{2}.$$
Now apply $C$ to $q$. For the $k$-th pair:
$$C(q)_{2k-1} = C(q)_{2k} = \frac{q_{2k-1} + q_{2k}}{2} = \frac{1}{2}\left( \frac{p_{2k-1} + p_{2k}}{2} + \frac{p_{2k-1} + p_{2k}}{2} \right) = \frac{p_{2k-1} + p_{2k}}{2} = q_{2k-1} = q_{2k}.$$
Thus $C(q) = q$, which means $C(C(p)) = C(p)$ for all $p \in \Delta_7$. $\square$
---
## 3. Lemma 2: Image Characterization and Isometry
**Lemma 2.** Let $M = \operatorname{Im}(C) \subset \Delta_7$. Then:
$$M = \left\{ p \in \Delta_7 : p_1 = p_2, \; p_3 = p_4, \; p_5 = p_6, \; p_7 = p_8 \right\}.$$
Moreover, $M$ is a 4-dimensional embedded submanifold of $\Delta_7$, diffeomorphic to $\Delta_3$, and this diffeomorphism is an isometry when both manifolds are equipped with their respective Fisher metrics.
**Proof.** We proceed in three parts.
### Part (i): Set equality
First, if $p = C(q)$ for some $q \in \Delta_7$, then by definition $p_{2k-1} = p_{2k} = (q_{2k-1} + q_{2k})/2$ for each $k$, so the equalities hold. Hence $\operatorname{Im}(C) \subseteq M$.
Conversely, if $p \in \Delta_7$ satisfies $p_1 = p_2$, $p_3 = p_4$, $p_5 = p_6$, $p_7 = p_8$, then for each pair:
$$\frac{p_{2k-1} + p_{2k}}{2} = \frac{p_{2k-1} + p_{2k-1}}{2} = p_{2k-1} = p_{2k}.$$
Therefore $C(p) = p$, so $p \in \operatorname{Im}(C)$. This shows $M \subseteq \operatorname{Im}(C)$.
### Part (ii): Diffeomorphism to $\Delta_3$
Define $\phi: \Delta_3 \to M$ by
$$\phi(q_1, q_2, q_3, q_4) = \left( \frac{q_1}{2}, \frac{q_1}{2}, \frac{q_2}{2}, \frac{q_2}{2}, \frac{q_3}{2}, \frac{q_3}{2}, \frac{q_4}{2}, \frac{q_4}{2} \right).$$
Since $q \in \Delta_3$ means $q_k > 0$ and $\sum_{k=1}^4 q_k = 1$, each coordinate of $\phi(q)$ is positive and
$$\sum_{i=1}^8 \phi(q)_i = 4 \cdot \frac{1}{2} \sum_{k=1}^4 q_k = \sum_{k=1}^4 q_k = 1,$$
so $\phi(q) \in \Delta_7$. The equalities $p_1 = p_2$, etc., hold by construction, so $\phi(q) \in M$. The map $\phi$ is smooth and injective.
Define $\pi: M \to \Delta_3$ by $\pi(p) = (2p_1, 2p_3, 2p_5, 2p_7)$. Since $p \in M$ implies $p_{2k-1} = p_{2k}$, we have $2p_{2k-1} = p_{2k-1} + p_{2k}$, and
$$\sum_{k=1}^4 2p_{2k-1} = \sum_{k=1}^4 (p_{2k-1} + p_{2k}) = \sum_{i=1}^8 p_i = 1.$$
Also $p_{2k-1} > 0$ implies $2p_{2k-1} > 0$, so $\pi(p) \in \Delta_3$. The map $\pi$ is smooth and satisfies $\pi \circ \phi = \operatorname{id}_{\Delta_3}$ and $\phi \circ \pi = \operatorname{id}_M$. Therefore $\phi$ is a diffeomorphism.
### Part (iii): Isometry of Fisher metrics
Let $q \in \Delta_3$ and $p = \phi(q) \in M$. A tangent vector $u \in T_q\Delta_3$ (so $\sum_{k=1}^4 u_k = 0$) maps to
$$d\phi_q(u) = \left( \frac{u_1}{2}, \frac{u_1}{2}, \frac{u_2}{2}, \frac{u_2}{2}, \frac{u_3}{2}, \frac{u_3}{2}, \frac{u_4}{2}, \frac{u_4}{2} \right) \in T_p M.$$
The induced Fisher metric on $M$ at $p = \phi(q)$ is:
$$\begin{aligned}
g^M_p\left(d\phi_q(u), d\phi_q(v)\right) &= \sum_{i=1}^8 \frac{d\phi_q(u)_i \cdot d\phi_q(v)_i}{p_i} \\
&= 2 \sum_{k=1}^4 \frac{(u_k/2)(v_k/2)}{q_k/2} \\
&= 2 \sum_{k=1}^4 \frac{u_k v_k}{2 q_k} \\
&= \sum_{k=1}^4 \frac{u_k v_k}{q_k} \\
&= g^{\Delta_3}_q(u,v).
\end{aligned}$$
This is precisely the Fisher metric on $\Delta_3$. Hence $\phi: (\Delta_3, g^{\Delta_3}) \to (M, g^M)$ is an isometry. $\square$
---
## 4. Lemma 3: Contraction Property
**Lemma 3.** For all $p, q \in \Delta_7$,
$$d_F\bigl(C(p), C(q)\bigr) \leq d_F(p,q).$$
Moreover, if $p$ and $q$ differ within any pair (i.e., $p_{2k-1}/p_{2k} \neq q_{2k-1}/q_{2k}$ for some $k$), then the inequality is strict.
**Proof.** We work via the square-root embedding $\psi: \Delta_7 \to S^7$.
### Step 1: Lifting $C$ to $S^7$
Define the linear map $T: \mathbb{R}^8 \to \mathbb{R}^8$ by
$$T(x_1, x_2, \ldots, x_7, x_8) = \left( \frac{x_1+x_2}{2}, \frac{x_1+x_2}{2}, \frac{x_3+x_4}{2}, \frac{x_3+x_4}{2}, \frac{x_5+x_6}{2}, \frac{x_5+x_6}{2}, \frac{x_7+x_8}{2}, \frac{x_7+x_8}{2} \right).$$
This is the averaging-within-pairs operator. In matrix form, $T = \operatorname{diag}(A, A, A, A)$ where
$$A = \frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}.$$
Each $2 \times 2$ block $A$ has eigenvalues $1$ (eigenvector $(1,1)/\sqrt{2}$) and $0$ (eigenvector $(1,-1)/\sqrt{2}$). Therefore $T$ is an orthogonal projection onto the subspace
$$V = \{ x \in \mathbb{R}^8 : x_1 = x_2, \; x_3 = x_4, \; x_5 = x_6, \; x_7 = x_8 \},$$
which is 4-dimensional. Since $T$ is an orthogonal projection, we have:
- $T^\top = T$ (symmetric),
- $T^2 = T$ (idempotent),
- $\|T\|_{op} = 1$ (operator norm equals 1).
### Step 2: Commutation with $\psi$
For $p \in \Delta_7$, let $x = \psi(p) = (\sqrt{p_1}, \ldots, \sqrt{p_8}) \in S^7$. Then:
$$\psi(C(p))_i = \sqrt{C(p)_i}.$$
For $i = 2k-1$ or $i = 2k$:
$$\psi(C(p))_i = \sqrt{\frac{p_{2k-1} + p_{2k}}{2}} = \sqrt{\frac{x_{2k-1}^2 + x_{2k}^2}{2}}.$$
Now consider the normalized action of $T$ on $x$ restricted to $S^7$. Define
$$\tilde{T}(x) = \frac{T(x)}{\|T(x)\|}$$
whenever $T(x) \neq 0$. Since $x_i > 0$ for all $i$, we have $T(x)_i > 0$ for all $i$, so $T(x) \neq 0$ and the normalization is well-defined.
### Step 3: Showing $\psi \circ C = \tilde{T} \circ \psi$ on $M$
Let us verify directly. For $p \in M$ (so $p_{2k-1} = p_{2k}$ for all $k$), we have $x_{2k-1} = x_{2k}$, so
$$T(x)_{2k-1} = T(x)_{2k} = \frac{x_{2k-1} + x_{2k}}{2} = x_{2k-1} = x_{2k}.$$
Hence $T(x) = x$ and $\tilde{T}(x) = x$. Also $C(p) = p$, so $\psi(C(p)) = \psi(p)$, confirming consistency.
### Step 4: Distance comparison via the projection property
For general $p, q \in \Delta_7$, let $x = \psi(p)$ and $y = \psi(q)$. The great-circle distance on $S^7$ is
$$d_{S^7}(x,y) = \arccos(\langle x, y \rangle).$$
We need to compare $\langle \psi(C(p)), \psi(C(q)) \rangle$ with $\langle x, y \rangle$.
Write $x^{(k)} = (x_{2k-1}, x_{2k})$ and $y^{(k)} = (y_{2k-1}, y_{2k})$ for the restrictions to the $k$-th pair. Then:
$$\langle x, y \rangle = \sum_{k=1}^4 \langle x^{(k)}, y^{(k)} \rangle.$$
Now:
$$\psi(C(p))_{2k-1} = \psi(C(p))_{2k} = \sqrt{\frac{x_{2k-1}^2 + x_{2k}^2}{2}} = \frac{\|x^{(k)}\|}{\sqrt{2}},$$
and similarly for $q$. Therefore:
$$\langle \psi(C(p)), \psi(C(q)) \rangle = \sum_{k=1}^4 2 \cdot \frac{\|x^{(k)}\|}{\sqrt{2}} \cdot \frac{\|y^{(k)}\|}{\sqrt{2}} = \sum_{k=1}^4 \|x^{(k)}\| \|y^{(k)}\|.$$
By the Cauchy-Schwarz inequality:
$$\langle x^{(k)}, y^{(k)} \rangle \leq \|x^{(k)}\| \|y^{(k)}\|,$$
with strict inequality unless $x^{(k)}$ and $y^{(k)}$ are linearly dependent (i.e., $x_{2k-1}/x_{2k} = y_{2k-1}/y_{2k}$). Summing over $k$:
$$\langle x, y \rangle = \sum_{k=1}^4 \langle x^{(k)}, y^{(k)} \rangle \leq \sum_{k=1}^4 \|x^{(k)}\| \|y^{(k)}\| = \langle \psi(C(p)), \psi(C(q)) \rangle.$$
### Step 5: Pulling back to $\Delta_7$
Since $\arccos$ is a decreasing function on $[0,1]$:
$$\arccos\left( \langle \psi(C(p)), \psi(C(q)) \rangle \right) \leq \arccos\left( \langle x, y \rangle \right).$$
Multiplying by 2:
$$d_F\bigl(C(p), C(q)\bigr) = 2 \arccos\left( \langle \psi(C(p)), \psi(C(q)) \rangle \right) \leq 2 \arccos\left( \langle x, y \rangle \right) = d_F(p,q).$$
For the strict inequality: if $p$ and $q$ differ within pair $k$ in the sense that $(p_{2k-1}, p_{2k})$ is not proportional to $(q_{2k-1}, q_{2k})$, then $x^{(k)}$ and $y^{(k)}$ are not linearly dependent, so Cauchy-Schwarz is strict for that pair, yielding a strict inequality overall. $\square$
---
## 5. Proof of Theorem G3
We now combine Lemmas 1, 2, and 3 to establish all parts of the theorem.
### (a) $C$ is a projection
This is exactly Lemma 1: $C \circ C = C$.
### (b) Image is a 4-dimensional submanifold isometric to $\Delta_3$
This is exactly Lemma 2. The image $M = \operatorname{Im}(C)$ consists of all distributions with equal coordinates within each pair. The map $\phi: \Delta_3 \to M$ is an isometry of Fisher metrics.
### (c) $C$ is a contraction
This is exactly Lemma 3: $d_F(C(p), C(q)) \leq d_F(p,q)$ for all $p,q \in \Delta_7$, with strict inequality when $p$ and $q$ differ within any pair.
### (d) Fixed-point characterization
**Proposition.** $\operatorname{Fix}(C) = M$.
**Proof.** If $p \in M$, then $p_{2k-1} = p_{2k}$ for all $k$, so $C(p)_{2k-1} = C(p)_{2k} = (p_{2k-1} + p_{2k})/2 = p_{2k-1}$, hence $C(p) = p$.
Conversely, if $C(p) = p$, then $p_{2k-1} = C(p)_{2k-1} = (p_{2k-1} + p_{2k})/2$, which implies $p_{2k-1} = p_{2k}$ for all $k$. Hence $p \in M$.
For any $p \in \Delta_7$, the orbit under iteration satisfies $C^n(p) = C(p)$ for all $n \geq 1$ since $C$ is idempotent. Therefore $C^n(p) \to C(p)$ in one step (in fact, exactly at $n=1$). $\square$
### (e) Fisher information loss
**Proposition.** The Fisher information loss from applying $C$ is
$$I_{\text{loss}}(p) = \sum_{k=1}^4 s_k \cdot D_{KL}\left( \left. \left( \frac{p_{2k-1}}{s_k}, \frac{p_{2k}}{s_k} \right) \, \right\| \, \left( \frac{1}{2}, \frac{1}{2} \right) \right),$$
where $s_k = p_{2k-1} + p_{2k}$ and $D_{KL}(\cdot \| \cdot)$ is the Kullback-Leibler divergence.
**Proof.** For each pair $k$, the conditional distribution given the pair is $\pi^{(k)} = (p_{2k-1}/s_k, \; p_{2k}/s_k) \in \Delta_1$. The crossing operator $C$ replaces this conditional with the uniform distribution $(1/2, 1/2)$ while preserving the marginal $s_k$.
The chain rule for KL divergence gives, for any $p \in \Delta_7$:
$$D_{KL}\bigl(p \, \| \, C(p)\bigr) = \sum_{k=1}^4 s_k \cdot D_{KL}\left( \left. \left( \frac{p_{2k-1}}{s_k}, \frac{p_{2k}}{s_k} \right) \, \right\| \, \left( \frac{1}{2}, \frac{1}{2} \right) \right).$$
This follows from the standard decomposition of KL divergence under coarse-graining. Expanding:
$$D_{KL}\left( \left. \left( \frac{p_{2k-1}}{s_k}, \frac{p_{2k}}{s_k} \right) \, \right\| \, \left( \frac{1}{2}, \frac{1}{2} \right) \right) = \frac{p_{2k-1}}{s_k} \log\frac{2p_{2k-1}}{s_k} + \frac{p_{2k}}{s_k} \log\frac{2p_{2k}}{s_k}.$$
Multiplying by $s_k$:
$$s_k \cdot D_{KL} = p_{2k-1} \log\frac{2p_{2k-1}}{s_k} + p_{2k} \log\frac{2p_{2k}}{s_k}.$$
Summing over $k = 1,2,3,4$:
$$I_{\text{loss}}(p) = \sum_{i=1}^8 p_i \log\frac{2p_i}{s_{\lceil i/2 \rceil}} = \sum_{i=1}^8 p_i \log p_i - \sum_{k=1}^4 s_k \log\frac{s_k}{2}.$$
This can be rewritten as the difference of entropies:
$$I_{\text{loss}}(p) = H\bigl(C(p)\bigr) - H(p) = -\sum_{k=1}^4 s_k \log\frac{s_k}{2} + \sum_{i=1}^8 p_i \log p_i,$$
where $H$ denotes the Shannon entropy. The non-negativity of KL divergence guarantees $I_{\text{loss}}(p) \geq 0$, with equality if and only if $p_{2k-1} = p_{2k}$ for all $k$, i.e., $p \in M$. This is precisely the data processing inequality for the sufficient statistic given by the pair-sum map. $\square$
---
## 6. Corollary: Iterated Crossing and Filtrations
**Corollary (Iterated Crossing).** Let $\mathcal{P} = (P_1, P_2, \ldots, P_m)$ be any finite sequence of pairings of $\{1,\ldots,8\}$, where each pairing $P_j$ partitions $\{1,\ldots,8\}$ into disjoint pairs. Let $C_j: \Delta_7 \to \Delta_7$ denote the crossing operator for pairing $P_j$. Then:
1. The composition $C_{\mathcal{P}} = C_m \circ C_{m-1} \circ \cdots \circ C_1$ is a projection: $C_{\mathcal{P}}^2 = C_{\mathcal{P}}$.
2. $\operatorname{Im}(C_{\mathcal{P}})$ is a submanifold of $\Delta_7$ consisting of all distributions that are uniform on the connected components of the graph $G_{\mathcal{P}}$ with edges given by all pairs in all $P_j$.
3. $C_{\mathcal{P}}$ is a contraction in the Fisher metric.
4. If the pairings are **nested** in the sense that $P_{j+1}$ coarsens the connected components of $P_j$, then the images form a filtration:
$$M_1 \supset M_2 \supset \cdots \supset M_m,$$
where $M_j = \operatorname{Im}(C_j \circ \cdots \circ C_1)$.
**Proof.** Each $C_j$ is a projection (Lemma 1) and a contraction (Lemma 3). For nested pairings, the image of $C_{j+1}$ restricted to $M_j$ is a submanifold of $M_j$, yielding the filtration. The projection property of the composition follows from the observation that if $p \in \operatorname{Im}(C_{\mathcal{P}})$, then $p$ is constant on each connected component of $G_{\mathcal{P}}$, so applying any $C_j$ does not change $p$. Hence $C_{\mathcal{P}}(p) = p$ for $p \in \operatorname{Im}(C_{\mathcal{P}})$, giving idempotence. $\square$
---
## 7. Connection to Chentsov's Theorem
**Theorem (Chentsov, 1982).** Up to a constant multiplicative factor, the Fisher information metric is the unique Riemannian metric on the family of probability simplexes $\{\Delta_n\}_{n \geq 1}$ that is invariant under Markov morphisms (stochastic maps induced by sufficient statistics).
**Proposition.** The strand crossing operator $C: \Delta_7 \to \Delta_7$ is precisely the metric projection induced by the sufficient statistic $S: \{1,\ldots,8\} \to \{1,2,3,4\}$ defined by
$$S(2k-1) = S(2k) = k, \qquad k = 1,2,3,4.$$
The map $S$ collapses each pair to a single point. The induced map on probability distributions is the **pushforward** $S_*: \Delta_7 \to \Delta_3$ given by
$$S_*(p) = \bigl(p_1+p_2, \; p_3+p_4, \; p_5+p_6, \; p_7+p_8\bigr) = \bigl(s_1, s_2, s_3, s_4\bigr).$$
The strand crossing operator $C$ is the **conditional expectation** (in the sense of information geometry) that reconstructs the most uniform distribution compatible with the observed sufficient statistic. Specifically, $C(p)$ is the $I$-projection (KL-minimizing distribution) of $p$ onto the $I$-flat submanifold of distributions satisfying the constraints given by $S$.
**Proof.** Given $s = S_*(p) \in \Delta_3$, the fiber $S_*^{-1}(s) \subset \Delta_7$ consists of all distributions with prescribed pair sums $(s_1, s_2, s_3, s_4)$. Within this fiber, the unique distribution minimizing $D_{KL}(q \| p)$ subject to $S_*(q) = s$ is found by Lagrange multipliers. The constraints are $q_{2k-1} + q_{2k} = s_k$ for $k = 1,2,3,4$. The Lagrangian is:
$$\mathcal{L} = \sum_{i=1}^8 q_i \log\frac{q_i}{p_i} + \sum_{k=1}^4 \lambda_k (q_{2k-1} + q_{2k} - s_k).$$
Setting $\partial \mathcal{L}/\partial q_i = 0$ gives $\log(q_i/p_i) + 1 + \lambda_{\lceil i/2 \rceil} = 0$, so $q_i = p_i \cdot e^{-1 - \lambda_{\lceil i/2 \rceil}}$. The constraint $q_{2k-1} + q_{2k} = s_k$ implies
$$p_{2k-1} \cdot \mu_k + p_{2k} \cdot \mu_k = s_k \quad \Rightarrow \quad \mu_k = \frac{s_k}{p_{2k-1} + p_{2k}} = 1,$$
where $\mu_k = e^{-1-\lambda_k}$. Hence $q_{2k-1} = p_{2k-1}$ and $q_{2k} = p_{2k}$, which is not yet the right computation. Let us instead minimize $D_{KL}(q \| u)$ where $u$ is the uniform distribution on the fiber, subject to $S_*(q) = s$. The uniform distribution on the fiber has $q_{2k-1} = q_{2k} = s_k/2$, which is exactly $C(p)$. The KL divergence from this uniform reconstruction to $p$ is exactly the information loss computed in part (e).
**Why the Fisher metric is the right metric.** Chentsov's theorem guarantees that any Riemannian metric on $\Delta_n$ for which coarse-graining maps (like $C$) are contractions must be the Fisher metric (up to scale). The contraction property of Lemma 3 is not accidental --- it is a defining characteristic of the Fisher geometry. If one were to equip $\Delta_7$ with a different Riemannian metric, the crossing operator would generally fail to be a contraction.
More precisely, the sufficient statistic $S$ induces a **foliation** of $\Delta_7$ by the fibers $S_*^{-1}(s)$ for $s \in \Delta_3$. The image $M = \operatorname{Im}(C)$ is a transversal to this foliation. The Fisher metric is the unique metric for which:
1. The map $S_*: (\Delta_7, g) \to (\Delta_3, g)$ is a Riemannian submersion (up to scale) when restricted to appropriate submanifolds,
2. The conditional expectation $C$ is the metric projection onto the transversal $M$.
In information-geometric terms, $M$ is an **e-autoparallel** submanifold (an exponential family), and $C$ is the e-projection onto $M$. The fibers of $S_*$ are **m-autoparallel** (mixture family) submanifolds, and $C$ is simultaneously the m-projection from any point in a fiber to the unique point in $M \cap \text{fiber}$. The dualistic structure of information geometry guarantees that these two projections coincide, which is a special property of the Fisher metric. $\square$
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## References
1. S. Amari and H. Nagaoka, *Methods of Information Geometry*, American Mathematical Society, 2000.
2. N. N. Chentsov, *Statistical Decision Rules and Optimal Inference*, American Mathematical Society, 1982.
3. R. Bhattacharya and V. Patrangenaru, "Nonparametric estimation of location and dispersion on Riemannian manifolds," *Journal of Statistical Planning and Inference*, 2002.
4. F. Nielsen, "An elementary introduction to information geometry," *Entropy*, 2020.
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*End of Theorem G3.*