SilverSight/docs/fundamental_math/G3_EIGENSOLID_FIXED_POINT.md

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). 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.