# Theorem G1: Contraction of the Chaos Game on the Fisher–Rao Simplex

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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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## 1. Setup

### 1.1 The open probability simplex

Let $N = 8$. The **open probability simplex** is

$$\Delta_{N-1} = \Delta_7 = \Bigl\{p = (p_1,\dots,p_N) \in \mathbb{R}^N : p_i > 0,\; \sum_{i=1}^N p_i = 1\Bigr\}.$$

Its tangent space at $p \in \Delta_7$ is

$$T_p\Delta_7 = \Bigl\{v = (v_1,\dots,v_N) \in \mathbb{R}^N : \sum_{i=1}^N v_i = 0\Bigr\}.$$

The simplex is an $(N-1)$-dimensional smooth manifold diffeomorphic to $\mathbb{R}^{N-1}$.

### 1.2 The Fisher metric and Fisher distance

The **Fisher information metric** (Amari 1985, §2; also Rao 1945) is the Riemannian metric on $\Delta_7$ defined by

$$g_p(u,v) = \sum_{i=1}^N \frac{u_i v_i}{p_i}, \qquad u,v \in T_p\Delta_7.$$

The **Fisher distance** $d_F$ is the geodesic distance induced by $g$:

$$d_F(p,q) = \inf_{\gamma} \int_0^1 \sqrt{g_{\gamma(t)}\bigl(\gamma'(t),\gamma'(t)\bigr)}\,dt,$$

where the infimum is over all piecewise smooth curves $\gamma: [0,1] \to \Delta_7$ with $\gamma(0)=p$, $\gamma(1)=q$.

### 1.3 The $\sqrt{p}$ embedding into the sphere $\mathcal{S}^{N-1}$

Let $\mathcal{S}^{N-1} = \{x \in \mathbb{R}^N : \|x\|_2 = 1\}$ denote the unit Euclidean sphere, and let $\mathcal{S}^{N-1}_+ = \mathcal{S}^{N-1} \cap (0,\infty)^N$ denote its positive orthant. Define the smooth embedding

$$\Phi : \Delta_7 \longrightarrow \mathcal{S}^{N-1}_+, \qquad \Phi(p) = \sqrt{p} = (\sqrt{p_1},\dots,\sqrt{p_N}).$$

The differential of $\Phi$ at $p$ in direction $v \in T_p\Delta_7$ is

$$d\Phi_p(v) = \Bigl(\frac{v_1}{2\sqrt{p_1}},\dots,\frac{v_N}{2\sqrt{p_N}}\Bigr).$$

### 1.4 Pullback of the round metric — the $S^7$ embedding theorem

Equip $\mathcal{S}^{N-1}$ with the round metric $g_{\mathcal{S}}$ induced from the Euclidean inner product on $\mathbb{R}^N$. The **pullback metric** $\Phi^* g_{\mathcal{S}}$ satisfies

$$(\Phi^* g_{\mathcal{S}})_p(u,v) = \langle d\Phi_p(u),\, d\Phi_p(v) \rangle_{\mathbb{R}^N} = \sum_{i=1}^N \frac{u_i v_i}{4p_i} = \frac{1}{4}\, g_p(u,v).$$

Hence

$$\boxed{g_p(u,v) = 4\, (\Phi^* g_{\mathcal{S}})_p(u,v)}$$

and at the level of distances,

$$\boxed{d_F(p,q) = 2\, d_{\mathrm{arc}}\bigl(\Phi(p),\, \Phi(q)\bigr)}$$

where $d_{\mathrm{arc}}(x,y) = \arccos(\langle x,y \rangle)$ is the geodesic (arc) distance on $\mathcal{S}^{N-1}$.

*Proof of the distance formula.* Let $\gamma$ be a geodesic from $p$ to $q$ for $g$. Then $\tilde\gamma = \Phi \circ \gamma$ is a curve on $\mathcal{S}^{N-1}_+$ from $\Phi(p)$ to $\Phi(q)$, and

$$\int_0^1 \sqrt{g_{\gamma}(\gamma',\gamma')}\,dt = \int_0^1 2\sqrt{g_{\mathcal{S}}(\tilde\gamma',\tilde\gamma')}\,dt = 2\, \mathrm{Length}(\tilde\gamma) \geq 2\, d_{\mathrm{arc}}(\Phi(p),\Phi(q)).$$

Conversely, the geodesic on $\mathcal{S}^{N-1}$ between $\Phi(p)$ and $\Phi(q)$ lies in $\mathcal{S}^{N-1}_+$ (since the positive orthant is convex on the sphere), and its inverse image under $\Phi$ is a geodesic for $g$. The factor of $2$ follows from the metric identity above. $\square$

### 1.5 Bhattacharyya and chordal distances on $\mathcal{S}^{N-1}$

The **Bhattacharyya distance** on $\Delta_7$ is defined by

$$d_B(p,q) = \arccos\bigl(BC(p,q)\bigr), \qquad BC(p,q) = \sum_{i=1}^N \sqrt{p_i q_i}.$$

Since $\langle \Phi(p), \Phi(q) \rangle = BC(p,q)$, we have immediately

$$\boxed{d_B(p,q) = d_{\mathrm{arc}}\bigl(\Phi(p),\, \Phi(q)\bigr)}.$$

The **chordal distance** on $\mathcal{S}^{N-1}$ is the Euclidean distance in $\mathbb{R}^N$:

$$d_{\mathrm{chord}}(x,y) = \|x-y\|_2 = \sqrt{2 - 2\langle x,y \rangle} = 2\sin\Bigl(\frac{d_{\mathrm{arc}}(x,y)}{2}\Bigr).$$

The following elementary comparison holds for all $x,y \in \mathcal{S}^{N-1}_+$ (since $d_{\mathrm{arc}} \in [0,\pi/2]$ in the positive orthant):

$$\frac{2}{\pi}\, d_{\mathrm{arc}}(x,y) \leq d_{\mathrm{chord}}(x,y) \leq d_{\mathrm{arc}}(x,y).$$

In particular, on $\Delta_7$:

$$\frac{1}{\pi}\, d_F(p,q) \leq d_{\mathrm{chord}}(\Phi(p),\Phi(q)) \leq \frac{1}{2}\, d_F(p,q).$$

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## 2. The Maps and the Chentsov Condition

### 2.1 Affine stochastic maps on $\Delta_7$

Let $k \geq 2$. For each $j \in \{1,\dots,k\}$, let $A_j$ be an $N \times N$ matrix and $b_j \in \mathbb{R}^N$ a vector satisfying:

> **(H1)** $A_j$ is **stochastic**: $(A_j)_{i\ell} \geq 0$ for all $i,\ell$, and $\sum_{i=1}^N (A_j)_{i\ell} = 1$ for all $\ell$ (column-stochastic; equivalently, $A_j^\top \mathbf{1} = \mathbf{1}$).
>
> **(H2)** $(b_j)_i \geq 0$ for all $i$, and $B_j := \sum_{i=1}^N (b_j)_i > 0$.

The **Chentsov monotonicity condition** requires that each $A_j$ is induced by a sufficient statistic (coarse-graining). Concretely:

> **(C)** For each $j$, there exists a surjective map $T_j: \{1,\dots,N\} \to \{1,\dots,m_j\}$ with $m_j < N$ such that
> $$(A_j)_{i\ell} = \mathbf{1}_{\{T_j(\ell) = i\}}.$$
> In other words, $A_j$ is a **deterministic channel**: each input $\ell$ is mapped deterministically to the output $T_j(\ell)$. Equivalently, each column of $A_j$ contains exactly one entry equal to $1$ and the rest are $0$.

Under **(H1)** and **(H2)**, for $p \in \Delta_7$ we have $\sum_i (A_j p + b_j)_i = 1 + B_j$, so the normalized affine map

$$w_j(p) = \frac{A_j p + b_j}{1 + B_j}$$

satisfies $\sum_i w_j(p)_i = 1$ and $w_j(p)_i > 0$; thus $w_j: \Delta_7 \to \Delta_7$ is well-defined. Since $A_j$ has nonnegative entries and $b_j \geq 0$, the map $w_j$ sends $\Delta_7$ into the compact subset

$$\Delta_7^{(\varepsilon_j)} := \Bigl\{p \in \Delta_7 : p_i \geq \frac{(b_j)_i}{1+B_j}\Bigr\}, \qquad \varepsilon_j = \min_i \frac{(b_j)_i}{1+B_j} > 0.$$

Each $w_j$ extends continuously to the closed simplex $\overline{\Delta}_7$.

### 2.2 Induced map on $\mathcal{S}^{N-1}_+$

Via the $\sqrt{p}$ embedding, $w_j$ induces the map $\widehat{w}_j : \mathcal{S}^{N-1}_+ \to \mathcal{S}^{N-1}_+$ defined by

$$\widehat{w}_j(x) = \Phi\bigl(w_j(x^2)\bigr) = \frac{\sqrt{A_j x^2 + b_j}}{\sqrt{1+B_j}},$$

where $x^2 = (x_1^2,\dots,x_N^2)$ and the square root in the numerator is componentwise.

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## 3. Lemma 1 — Contraction on the Sphere

**Lemma 1.** *Let $w(p) = (Ap + b)/(1+B)$ satisfy* **(H1)**, **(H2)**, *and* **(C)**. *Then the induced map $\widehat{w}: \mathcal{S}^{N-1}_+ \to \mathcal{S}^{N-1}_+$ satisfies*

$$d_{\mathrm{chord}}\bigl(\widehat{w}(x),\, \widehat{w}(y)\bigr) \leq \frac{1}{\sqrt{1+B}}\, d_{\mathrm{chord}}(x,y) \qquad \forall\, x,y \in \mathcal{S}^{N-1}_+,$$

*and consequently*

$$d_B\bigl(w(p),\, w(q)\bigr) \leq \frac{1}{\sqrt{1+B}}\, d_B(p,q) \qquad \forall\, p,q \in \Delta_7.$$

*Proof.* Define $F: \mathbb{R}^N_+ \to \mathbb{R}^N$ by

$$F_i(z) = \frac{\sqrt{(Az^2)_i + b_i}}{\sqrt{1+B}}, \qquad i = 1,\dots,N,$$

so that $\widehat{w}(x) = F(x)$ for $x \in \mathcal{S}^{N-1}_+$. The differential of $F$ at $z$ in direction $\xi \in \mathbb{R}^N$ is

$$dF_z(\xi)_i = \frac{1}{\sqrt{1+B}} \cdot \frac{(A(z\xi))_i}{\sqrt{(Az^2)_i + b_i}},$$

where $(z\xi)_\ell = z_\ell \xi_\ell$. Therefore,

$$\|dF_z(\xi)\|_2^2 = \frac{1}{1+B} \sum_{i=1}^N \frac{\bigl(\sum_{\ell=1}^N A_{i\ell} z_\ell \xi_\ell\bigr)^2}{\sum_{\ell=1}^N A_{i\ell} z_\ell^2 + b_i}.$$

For each $i$, apply the Cauchy–Schwarz inequality to the numerator:

$$\Bigl(\sum_{\ell} A_{i\ell} z_\ell \xi_\ell\Bigr)^2 = \Bigl(\sum_{\ell} \sqrt{A_{i\ell}} \cdot \sqrt{A_{i\ell}}\, z_\ell \xi_\ell\Bigr)^2 \leq \Bigl(\sum_{\ell} A_{i\ell}\Bigr)\Bigl(\sum_{\ell} A_{i\ell} z_\ell^2 \xi_\ell^2\Bigr).$$

Since $A$ is column-stochastic, $\sum_i A_{i\ell} = 1$ for each $\ell$, but here we need $\sum_\ell A_{i\ell}$. Because $A$ is a deterministic channel matrix (condition **(C)**), each row of $A$ corresponds to a fiber of $T$, and $\sum_\ell A_{i\ell} = |T^{-1}(i)|$ counts the size of the $i$-th fiber. In particular, $\sum_\ell A_{i\ell} \geq 1$.

However, we use the sharper bound: since $(Az^2)_i + b_i \geq (Az^2)_i$,

$$\frac{\bigl(\sum_{\ell} A_{i\ell} z_\ell \xi_\ell\bigr)^2}{\sum_{\ell} A_{i\ell} z_\ell^2 + b_i} \leq \frac{\bigl(\sum_{\ell} A_{i\ell} z_\ell^2\bigr)\bigl(\sum_{\ell} A_{i\ell} \xi_\ell^2\bigr)}{\sum_{\ell} A_{i\ell} z_\ell^2 + b_i} \leq \sum_{\ell} A_{i\ell} \xi_\ell^2,$$

where the second inequality uses $\frac{s \cdot t}{s + b_i} \leq t$ for $s,t \geq 0$ and $b_i \geq 0$, with $s = \sum_\ell A_{i\ell} z_\ell^2$ and $t = \sum_\ell A_{i\ell} \xi_\ell^2$.

Summing over $i$:

$$\|dF_z(\xi)\|_2^2 \leq \frac{1}{1+B} \sum_{i=1}^N \sum_{\ell=1}^N A_{i\ell} \xi_\ell^2 = \frac{1}{1+B} \sum_{\ell=1}^N \xi_\ell^2 \sum_{i=1}^N A_{i\ell} = \frac{\|\xi\|_2^2}{1+B},$$

where the last equality uses $\sum_i A_{i\ell} = 1$ (column-stochastic).

Now restrict to the tangent space of $\mathcal{S}^{N-1}$. For $x,y \in \mathcal{S}^{N-1}_+$, the geodesic segment $[x,y]$ lies in the positive orthant. By the mean value theorem on $\mathcal{S}^{N-1}$ (using the exponential map or working in ambient $\mathbb{R}^N$ with the constraint $\|z\|=1$):

$$\|\widehat{w}(x) - \widehat{w}(y)\|_2 = \|F(x) - F(y)\|_2 \leq \sup_{z \in [x,y]} \|dF_z|_{T_z\mathcal{S}^{N-1}}\|_{\mathrm{op}} \cdot \|x-y\|_2.$$

For $\xi \in T_z\mathcal{S}^{N-1}$ (i.e., $\langle z, \xi \rangle = 0$), the bound $\|dF_z(\xi)\|_2 \leq \|\xi\|_2/\sqrt{1+B}$ holds (the tangent constraint does not affect the estimate). Hence

$$d_{\mathrm{chord}}(\widehat{w}(x),\widehat{w}(y)) \leq \frac{1}{\sqrt{1+B}}\, d_{\mathrm{chord}}(x,y).$$

For the Bhattacharyya distance, recall that $d_B = d_{\mathrm{arc}}$ and $d_{\mathrm{chord}} = 2\sin(d_{\mathrm{arc}}/2)$. The chordal contraction yields

$$2\sin\Bigl(\frac{d_{\mathrm{arc}}(\widehat{w}(x),\widehat{w}(y))}{2}\Bigr) \leq \frac{1}{\sqrt{1+B}} \cdot 2\sin\Bigl(\frac{d_{\mathrm{arc}}(x,y)}{2}\Bigr).$$

Applying $\arcsin$ to both sides and using that $\arcsin$ is convex on $[0,1]$ with $\arcsin(0)=0$ (hence $\arcsin(\lambda s) \leq \lambda \arcsin(s)$ for $\lambda \in [0,1]$):

$$d_{\mathrm{arc}}(\widehat{w}(x),\widehat{w}(y)) \leq 2\arcsin\Bigl(\frac{1}{\sqrt{1+B}}\sin\Bigl(\frac{d_{\mathrm{arc}}(x,y)}{2}\Bigr)\Bigr) \leq \frac{1}{\sqrt{1+B}}\, d_{\mathrm{arc}}(x,y).$$

Therefore

$$d_B(w(p),w(q)) = d_{\mathrm{arc}}(\widehat{w}(\Phi(p)),\widehat{w}(\Phi(q))) \leq \frac{1}{\sqrt{1+B}}\, d_{\mathrm{arc}}(\Phi(p),\Phi(q)) = \frac{1}{\sqrt{1+B}}\, d_B(p,q).$$

$\square$

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## 4. Lemma 2 — Equivalence of Distances

**Lemma 2.** *For all $p,q \in \Delta_7$,*

$$\boxed{d_F(p,q) = 2\, d_B(p,q)}.$$

*Consequently, any map $w: \Delta_7 \to \Delta_7$ that is $\lambda$-Lipschitz in the Bhattacharyya distance $d_B$ is also $\lambda$-Lipschitz in the Fisher distance $d_F$ with the **same** contraction factor.*

*Proof.* From §1.4, the $\sqrt{p}$ embedding satisfies $\Phi^* g_{\mathcal{S}} = \frac{1}{4} g$, so the length of any curve is scaled by $1/2$ when measured with $g_{\mathcal{S}}$ versus $g$. The geodesic distance therefore satisfies $d_F = 2\, d_{\mathrm{arc}} \circ (\Phi \times \Phi) = 2\, d_B$.

If $d_B(w(p),w(q)) \leq \lambda \, d_B(p,q)$ for all $p,q$, then

$$d_F(w(p),w(q)) = 2\, d_B(w(p),w(q)) \leq 2\lambda\, d_B(p,q) = \lambda\, d_F(p,q).$$

$\square$

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## 5. Proof of Theorem G1

**Theorem G1.** *Let $\{w_1,\dots,w_k\}$ be the IFS on $\Delta_7$ defined in §2.1, with each $w_j$ satisfying* **(H1)**, **(H2)**, *and the Chentsov monotonicity condition* **(C)**. *Then each $w_j$ is a strict contraction on $(\Delta_7, d_F)$ with*

$$d_F\bigl(w_j(p),\, w_j(q)\bigr) \leq \lambda_j \, d_F(p,q), \qquad \lambda_j = \frac{1}{\sqrt{1+B_j}} < 1,$$

*where $B_j = \sum_i (b_j)_i > 0$. Consequently, the IFS is contractive with global Lipschitz constant*

$$\lambda = \max_{1 \leq j \leq k} \lambda_j = \max_j \frac{1}{\sqrt{1+B_j}} < 1.$$

*Proof.* Lemma 1 shows that each $w_j$ is $\lambda_j$-Lipschitz in the Bhattacharyya distance $d_B$ with $\lambda_j = 1/\sqrt{1+B_j}$. By hypothesis **(H2)**, $B_j > 0$, so $\lambda_j < 1$. Lemma 2 shows that the same Lipschitz constant carries over to the Fisher distance $d_F$. Therefore each $w_j$ is a strict contraction on $(\Delta_7, d_F)$, and the IFS is contractive with constant $\lambda = \max_j \lambda_j < 1$. $\square$

### 5.1 Explicit bound in terms of singular values

The contraction factor can also be expressed in terms of the singular values of $A_j$. Define the **nonlinear distortion** of $w_j$ at $p$ as the operator norm

$$\kappa_j(p) = \sup_{v \in T_p\Delta_7 \setminus \{0\}} \frac{\|dw_j|_p(v)\|_{F,w_j(p)}}{\|v\|_{F,p}}.$$

A direct computation (as in the proof of Lemma 1, but pulled back to $\Delta_7$) gives:

$$\|dw_j|_p(v)\|_{F,w_j(p)}^2 = \frac{1}{1+B_j} \sum_{i=1}^N \frac{(A_j v)_i^2}{(A_j p + b_j)_i}.$$

Using the Fisher-score $u_\ell = v_\ell/p_\ell$ and the variance inequality (Jensen gap) for each fiber of $T_j$:

$$\sum_{i} \frac{(A_j v)_i^2}{(A_j p)_i} = \|v\|_{F,p}^2 - \sum_{i} (A_j p)_i \cdot \mathrm{Var}_i(u),$$

where $\mathrm{Var}_i(u) \geq 0$ is the variance of the scores $u_\ell$ over the fiber $T_j^{-1}(i)$ with weights $p_\ell/(A_j p)_i$. The $b_j$ term contributes the additional factor $1/(1+B_j)$. Hence:

$$\kappa_j(p) \leq \frac{1}{\sqrt{1+B_j}} \leq 1 - \frac{B_j}{2(1+B_j)}.$$

For the global Lipschitz constant, integrating along geodesics gives $\mathrm{Lip}(w_j) = \sup_p \kappa_j(p) = 1/\sqrt{1+B_j}$.

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## 6. Corollary — Chaos Game Convergence

**Corollary (Hutchinson–Barnsley).** *The IFS $\{w_1,\dots,w_k\}$ on $(\Delta_7, d_F)$ satisfies the hypotheses of the Hutchinson–Barnsley theorem. Therefore:*

*(a) There exists a unique nonempty compact set $\mathcal{A} \subset \overline{\Delta}_7$, called the **attractor**, such that*

$$\mathcal{A} = \bigcup_{j=1}^k w_j(\mathcal{A}).$$

*(b) For any starting point $p_0 \in \Delta_7$, the chaos game sequence*

$$p_{n+1} = w_{J_n}(p_n), \qquad n \geq 0,$$

*where $(J_n)_{n \geq 0}$ are i.i.d. uniform on $\{1,\dots,k\}$, converges to $\mathcal{A}$ almost surely in the Hausdorff sense: $d_F(p_n, \mathcal{A}) \to 0$ a.s.*

*(c) The Markov operator $W$ defined by $W(\mu) = \frac{1}{k}\sum_{j=1}^k (w_j)_\# \mu$ has a unique invariant probability measure $\mu_*$ supported on $\mathcal{A}$, and $W^n(\mu) \to \mu_*$ weakly for any initial probability measure $\mu$ on $\Delta_7$.

*Proof.* By Theorem G1, $\mathrm{Lip}(w_j) \leq \lambda < 1$ for all $j$ on the complete metric space $(\Delta_7, d_F)$. The maps $w_j$ extend continuously to the compactification $\overline{\Delta}_7$. The conclusions (a)–(c) are the standard Hutchinson–Barnsley theorem (Barnsley 1988, Theorem 3.1; Hutchinson 1981, §3.1) for contractive IFS on complete metric spaces. The existence and uniqueness of $\mathcal{A}$ follows from the Banach fixed-point theorem applied to the set-map $\mathcal{W}(K) = \cup_j w_j(K)$ on the space of nonempty compact subsets with the Hausdorff metric. Almost sure convergence of the chaos game follows from the contraction property and the law of large numbers for the coding space (Barnsley & Demko 1985). $\square$

### 6.1 The Open Set Condition

The IFS $\{w_1,\dots,w_k\}$ satisfies the **open set condition** (OSC) on $\Delta_7$ if there exists a nonempty open set $U \subset \Delta_7$ such that

$$w_j(U) \subset U \quad \text{for all } j, \qquad \text{and} \qquad w_j(U) \cap w_\ell(U) = \varnothing \quad \text{for } j \neq \ell.$$

**Sufficient condition for OSC.** *If the coarse-graining maps $T_1,\dots,T_k$ have pairwise disjoint active domains — meaning the sets of coordinates that are not pointwise fixed satisfy*

$$\{\ell : T_j(\ell) \neq \ell\} \cap \{\ell : T_\ell(\ell) \neq \ell\} = \varnothing \quad \text{for } j \neq \ell,$$

*then the IFS satisfies the OSC on $\Delta_7$ with $U = \Delta_7$.*

*Proof sketch.* Under the disjoint-active-domain condition, each $w_j$ modifies a disjoint subset of coordinates. For the example in §7, each $w_j$ only affects the pair $S_j$, and $S_j \cap S_\ell = \varnothing$ for $j \neq \ell$. Consequently, for $p \in \Delta_7$, the outputs $w_j(p)$ and $w_\ell(p)$ have their "nontrivial" mass concentrated on disjoint coordinate sets. Since the $b_j$ are supported on $S_j$ (also disjoint), the images $w_j(\Delta_7)$ and $w_\ell(\Delta_7)$ are contained in disjoint open subsets of $\Delta_7$, giving $w_j(\Delta_7) \cap w_\ell(\Delta_7) = \varnothing$. The inclusion $w_j(\Delta_7) \subset \Delta_7$ is automatic. $\square$

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## 7. Explicit Example: Four Coarse-Graining Maps

### 7.1 Definition of the IFS

Set $N = 8$ and $k = 4$. Partition the coordinates into four disjoint pairs:

$$S_1 = \{1,2\}, \quad S_2 = \{3,4\}, \quad S_3 = \{5,6\}, \quad S_4 = \{7,8\}.$$

For each $j \in \{1,2,3,4\}$, the **elementary stochastic matrix** $A_j$ moves one-half of the mass from coordinate $2j-1$ to coordinate $2j$, leaving all other coordinates unchanged. Concretely, $A_j$ is the column-stochastic matrix defined by:

- Column $2j-1$: $\bigl(\frac{1}{2}, \frac{1}{2}, 0, \dots, 0\bigr)^\top$ distributed across rows $2j-1$ and $2j$;
- Column $\ell$ for $\ell \notin S_j$: the standard basis vector $e_\ell$.

The action on $p \in \Delta_7$ is:

$$\begin{aligned}
(A_j p)_{2j-1} &= \frac{p_{2j-1}}{2}, \\
(A_j p)_{2j} &= \frac{p_{2j-1}}{2} + p_{2j}, \\
(A_j p)_\ell &= p_\ell \quad \text{for } \ell \notin S_j.
\end{aligned}$$

Each $A_j$ is column-stochastic (columns sum to $1$), and $\sum_i (A_j p)_i = \sum_\ell p_\ell = 1$.

For the offset, take $b_j = \varepsilon \, \mathbf{1}_{S_j}$ where $\varepsilon > 0$ and $\mathbf{1}_{S_j}$ is the indicator vector of $S_j$. Then $B_j = 2\varepsilon$ and the normalized map is

$$w_j(p) = \frac{A_j p + \varepsilon \, \mathbf{1}_{S_j}}{1 + 2\varepsilon}.$$

### 7.2 Contraction factor

By Theorem G1, each $w_j$ has Lipschitz constant

$$\lambda_j = \frac{1}{\sqrt{1 + 2\varepsilon}}.$$

Since all $B_j = 2\varepsilon$ are equal, the global contraction factor is

$$\boxed{\lambda = \frac{1}{\sqrt{1 + 2\varepsilon}} < 1.}$$

**Remark (connection to G3 — eigensolid fixed point).** As $\varepsilon \to 0^+$, the maps approach the pure coarse-graining $w_j(p) = A_j p$, and $\lambda \to 1$. This reflects the fact that pure deterministic coarse-graining is only $1$-Lipschitz (not a strict contraction) in the Fisher metric, due to the existence of non-contracting directions (score functions constant on fibers). The $\varepsilon > 0$ regularization breaks this invariance and forces strict contraction.

The strand crossing operator $C$ of Theorem G3 (see [`G3_EIGENSOLID_FIXED_POINT.md`](G3_EIGENSOLID_FIXED_POINT.md)) is precisely this $\varepsilon = 0$ limit for the pair-averaging case: $C$ equalizes each pair to their mean, is idempotent ($C^2 = C$), and is a non-strict contraction ($\operatorname{Lip}(C) = 1$). The maps $w_j$ of G1 are thus the **regularized** (strictly contractive) version of G3's projection operator, with $\varepsilon > 0$ controlling the strictness. Conversely, G3's $C$ is the degenerate limit of the G1 construction when the regularization is removed.

### 7.3 Explicit differential bound

For the map $w_1$ (moving half the mass from coordinate $1$ to coordinate $2$), with $b_1 = (\varepsilon, \varepsilon, 0,\dots,0)$, the normalized image is

$$w_1(p)_1 = \frac{p_1/2 + \varepsilon}{1+2\varepsilon}, \quad w_1(p)_2 = \frac{p_1/2 + p_2 + \varepsilon}{1+2\varepsilon}, \quad w_1(p)_\ell = \frac{p_\ell}{1+2\varepsilon}\ \ (\ell \geq 3).$$

The differential is $dw_1|_p(v) = A_1 v / (1+2\varepsilon)$, with $(A_1 v)_1 = v_1/2$ and $(A_1 v)_2 = v_1/2 + v_2$. Its squared Fisher norm is:

$$\|dw_1(v)\|_{F,w_1(p)}^2 = \frac{1}{(1+2\varepsilon)^2}\Biggl[\frac{v_1^2/4}{\bigl(p_1/2 + \varepsilon\bigr)/(1+2\varepsilon)} + \frac{(v_1/2+v_2)^2}{\bigl(p_1/2 + p_2 + \varepsilon\bigr)/(1+2\varepsilon)} + \sum_{\ell=3}^8 \frac{v_\ell^2}{p_\ell/(1+2\varepsilon)}\Biggr]$$

$$= \frac{1}{1+2\varepsilon}\Biggl[\frac{v_1^2/4}{p_1/2 + \varepsilon} + \frac{(v_1/2+v_2)^2}{p_1/2 + p_2 + \varepsilon} + \sum_{\ell=3}^8 \frac{v_\ell^2}{p_\ell}\Biggr].$$

Set $u_\ell = v_\ell/p_\ell$ (the score). Then $v_1 = p_1 u_1$, $v_2 = p_2 u_2$, and the first two terms become:

$$\frac{p_1^2 u_1^2/4}{p_1/2 + \varepsilon} + \frac{(p_1 u_1/2 + p_2 u_2)^2}{p_1/2 + p_2 + \varepsilon}.$$

We claim this is bounded by $(p_1 u_1^2 + p_2 u_2^2)/(1+2\varepsilon)$. For the first term:

$$\frac{p_1^2 u_1^2/4}{p_1/2 + \varepsilon} = p_1 u_1^2 \cdot \frac{p_1/4}{p_1/2 + \varepsilon} \leq p_1 u_1^2 \cdot \frac{1/4}{1/2 + \varepsilon} = \frac{p_1 u_1^2}{2(1+2\varepsilon)} \leq \frac{p_1 u_1^2}{1+2\varepsilon}.$$

For the second term, apply Jensen's inequality to the convex function $t \mapsto t^2$ with weights $\alpha = (p_1/2)/(p_1/2 + p_2 + \varepsilon)$ and $1-\alpha$:

$$\frac{(p_1 u_1/2 + p_2 u_2)^2}{p_1/2 + p_2 + \varepsilon} \leq \frac{p_1 u_1^2/2 + p_2 u_2^2}{1 + 2\varepsilon/(p_1 + 2p_2)} \cdot \frac{1}{1} \leq \frac{p_1 u_1^2 + p_2 u_2^2}{1+2\varepsilon}.$$

Adding the remaining terms $\sum_{\ell=3}^8 v_\ell^2/(p_\ell(1+2\varepsilon))$ gives

$$\|dw_1(v)\|_{F,w_1(p)}^2 \leq \frac{1}{1+2\varepsilon}\Bigl(p_1 u_1^2 + p_2 u_2^2 + \sum_{\ell=3}^8 p_\ell u_\ell^2\Bigr) = \frac{\|v\|_{F,p}^2}{1+2\varepsilon}.$$

Hence $\kappa_1(p) \leq 1/\sqrt{1+2\varepsilon}$ for all $p \in \Delta_7$, confirming $\lambda_1 = 1/\sqrt{1+2\varepsilon}$. The same bound holds for $w_2, w_3, w_4$ by symmetry.

### 7.4 The attractor

With $\lambda = 1/\sqrt{1+2\varepsilon}$, the attractor $\mathcal{A} \subset \Delta_7$ has Hausdorff dimension (in the Fisher metric) satisfying

$$\dim_H(\mathcal{A}) \leq \frac{\log 4}{|\log \lambda|} = \frac{2\log 2}{\frac{1}{2}\log(1+2\varepsilon)} = \frac{4\log 2}{\log(1+2\varepsilon)}.$$

As $\varepsilon \to 0^+$, $\dim_H(\mathcal{A}) \to \infty$ (the attractor expands to fill the simplex), while as $\varepsilon \to \infty$, $\dim_H(\mathcal{A}) \to 0$ (the attractor collapses to a finite set).

For $\varepsilon = 1/2$ (a natural choice since $B_j = 1$, matching the "$1/2$ mass" parameter), we obtain:

$$\boxed{\lambda = \frac{1}{\sqrt{2}} \approx 0.707.}$$

### 7.5 Summary of the example

| Parameter | Value |
|-----------|-------|
| Dimension $N$ | $8$ |
| Number of maps $k$ | $4$ |
| Fiber structure | $\{1,2\}, \{3,4\}, \{5,6\}, \{7,8\}$ |
| Offset $b_j$ | $\varepsilon \cdot \mathbf{1}_{S_j}$ |
| Total offset $B_j$ | $2\varepsilon$ |
| **Contraction factor** $\lambda$ | $\displaystyle\frac{1}{\sqrt{1+2\varepsilon}}$ |
| For $\varepsilon = 1/2$ | $\lambda = 1/\sqrt{2}$ |

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## 8. References

1. S. Amari, *Differential-Geometrical Methods in Statistics*, Springer Lecture Notes in Statistics 28, 1985.
2. M. F. Barnsley, *Fractals Everywhere*, Academic Press, 1988.
3. M. F. Barnsley and S. Demko, "Iterated function systems and the global construction of fractals," *Proc. R. Soc. Lond. A* 399 (1985), 243–275.
4. N. N. Chentsov, *Statistical Decision Rules and Optimal Inference*, Trans. Math. Monographs 53, AMS, 1982.
5. J. E. Hutchinson, "Fractals and self-similarity," *Indiana Univ. Math. J.* 30 (1981), 713–747.
6. A. Bhattacharyya, "On a measure of divergence between two statistical populations defined by their probability distributions," *Bull. Calcutta Math. Soc.* 35 (1943), 99–109.
7. C. R. Rao, "Information and the accuracy attainable in the estimation of statistical parameters," *Bull. Calcutta Math. Soc.* 37 (1945), 81–91.

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*End of proof.*
