# Theorem G3: Fixed-Point Analysis of the Strand Crossing Operator on the Probability Simplex

**Date:** 2026-01-12

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> **Domain note:** All Fisher metric computations require \(p_i > 0\) for all \(i\). Raw byte frequencies produce \(p_i = 0\) for unseen classes. Apply **Laplace smoothing**: \(F(E)_i = (f_i + 1)/(|E| + 8)\) to ensure strict positivity. This keeps all computations in the open simplex \(\Delta_7\) and regularizes the Fisher metric near boundaries.

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## 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.

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## 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 $\mathbb{R}^8$.

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## 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 $p \in \Delta_7$. $\square$

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## 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. $\square$

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## 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 there exists a pair $k$ such that $(p_{2k-1}, p_{2k})$ is NOT proportional to $(q_{2k-1}, q_{2k})$ — i.e., $p_{2k-1}/p_{2k} \neq q_{2k-1}/q_{2k}$ — then the inequality is strict. If every pair has proportional sub-vectors, Cauchy-Schwarz attains equality for all $k$ and $d_F(C(p), C(q)) = d_F(p,q)$.

**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 equality if and only if $x^{(k)}$ 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 strictness: if there exists a pair $k$ such 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 $k$ and the total inequality is strict. If every pair has proportional sub-vectors, Cauchy-Schwarz attains equality for all $k$ and $d_F(C(p), C(q)) = d_F(p,q)$. $\square$

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## 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 unless every pair has proportional sub-vectors.

### (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. $\square$

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## 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. If the pairings are **nested** (each $P_{j+1}$ coarsens the connected components formed by $P_1,\ldots,P_j$), then the composition $C_{\mathcal{P}} = C_m \circ C_{m-1} \circ \cdots \circ C_1$ is a projection: $C_{\mathcal{P}}^2 = C_{\mathcal{P}}$. For non-nested pairings, $C_{\mathcal{P}}$ is a contraction (claim 3) but may not be idempotent.

2. For nested pairings, $\operatorname{Im}(C_{\mathcal{P}})$ 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 $P_j$. For non-nested pairings the image depends on the order of composition and lacks this simple uniformity characterization.

3. $C_{\mathcal{P}}$ is a contraction in the Fisher metric.

4. If the pairings are **nested** (each $P_{j+1}$ coarsens the connected components formed by $P_1,\ldots,P_j$), 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 contraction (Lemma 3), so any composition is a contraction. For nested pairings, each $P_{j+1}$ coarsens the connected components formed by $P_1,\ldots,P_j$; consequently $C_{j+1}$ only averages across coordinates already made equal by the preceding composition, so $M_{j+1} \subset M_j$, yielding the filtration. Moreover, if $p \in \operatorname{Im}(C_{\mathcal{P}})$ for a nested sequence, then $p$ is constant on each connected component of $G_{\mathcal{P}}$, so applying any $C_j$ does not change $p$; hence $C_{\mathcal{P}}(p) = p$, giving idempotence. For non-nested pairings, the composition of projections need not be idempotent. $\square$

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## 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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## 8. Connection to the Chaos Game (Theorem G1)

The strand crossing operator $C$ is structurally related to the iterated function system maps $\{w_j\}$ of Theorem G1 (see [`G1_CHAOS_GAME_CONTRACTION.md`](G1_CHAOS_GAME_CONTRACTION.md)). The two documents analyze different facets of the same mathematical family: column-stochastic coarse-graining on the probability simplex.

### 8.1 Structural correspondence

Each map $w_j(p) = (A_j p + b_j)/(1+B_j)$ of G1 becomes a pure column-stochastic action $w_j(p) = A_j p$ in the limit $\varepsilon \to 0^+$ (where the offset $b_j \to 0$, so $B_j \to 0$). If $A_j$ is chosen as a **pair-averaging matrix** that coarse-grains a single pair $P_j = \{2j-1, 2j\}$ — i.e. the $8 \times 8$ matrix with

$$(A_j)_{2j-1,\ell} = (A_j)_{2j,\ell} = \begin{cases} \frac12 & \ell \in P_j, \\ \delta_{\ell,i} & \text{for } i \notin P_j, \end{cases}$$

then $w_j(p) \to A_j p$ as $\varepsilon \to 0^+$, and the action of $A_j$ on a single pair is precisely the restriction of $C$ to that pair. G3's full operator $C$ is the simultaneous application of four such pair-averaging maps, or equivalently the composition $C = C_4 \circ C_3 \circ C_2 \circ C_1$ where $C_j$ averages pair $P_j$. The correspondence is independent of the specific partition of $\{1,\dots,8\}$ into four pairs; any partition yields the same algebraic structure.

### 8.2 Contraction type: projection vs. strict contraction

G3's $C$ is a **linear projection** ($C^2 = C$) and a **non-strict contraction** ($\operatorname{Lip}(C) = 1$). Its operator norm $\|A_C\|_{\mathrm{op}} = 1$ on the sphere reflects idempotence: once mass is equalized within a pair, reapplying $C$ leaves it unchanged.

G1's $w_j$ are **strict contractions** ($\operatorname{Lip}(w_j) = 1/\sqrt{1+B_j} < 1$). The strictness is a direct consequence of the offset $b_j > 0$: the regularization creates a positive lower bound on every coordinate, eliminating the score-function invariances on fibers that would otherwise become non-contracting directions.

### 8.3 The $\varepsilon \to 0^+$ limit

As $\varepsilon \to 0^+$, $B_j \to 0$ and the G1 contraction factor $\lambda_j = 1/\sqrt{1+B_j} \to 1$. The maps $w_j$ converge to pure column-stochastic action $w_j(p) = A_j p$, and G1's strict contraction degenerates to the same non-strict contraction that G3 exhibits ($\operatorname{Lip} = 1$). In this sense, G3's $C$ is the **$\varepsilon = 0$ limiting case** of the G1 construction, and the regularization $\varepsilon > 0$ is what upgrades the projection to a proper contraction for the chaos game.

Readers of G1 should note that the pair-averaging matrix $A_j$ described above does not satisfy G1's condition **(C)** (Chentsov deterministic channel) — it is a probabilistic channel, not a deterministic one. However, the column-stochastic structure alone suffices for the $1$-Lipschitz bound, and the $\varepsilon$-regularization in G1 independently provides the strict contraction guarantee. The structural analogy between $C$ and the $\varepsilon \to 0$ limit of $w_j$ holds for the entire column-stochastic family, irrespective of the deterministic-channel restriction in G1's formal hypotheses.

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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.*
