SilverSight/archive/2026-07-02/docs/FISHER_METRIC_BRIDGE.md

Fisher Metric Bridge — Mathematical Argument

The Problem

fisherMetric50 p i j = δ_ij / p_i (diagonal matrix) fisherMetric p X Y = ∑ k, X_k * Y_k / p_k (bilinear form)

Are these the same? On the full space, yes. On the tangent space, no.

Full Space

For standard basis vectors e_i (where e_i k = δ_ik):

fisherMetric p e_i e_j = ∑ k, (δ_ik * δ_jk) / p_k = δ_ij / p_i

So fisherMetric50 p i j = fisherMetric p e_i e_j. ✓

Tangent Space

For tangent vectors e_i - e_0 and e_j - e_0 (which sum to 0):

fisherMetric p (e_i - e_0) (e_j - e_0) = 
  ∑ k, (e_i k - e_0 k)(e_j k - e_0 k) / p_k

For i ≠ 0, j ≠ 0, i ≠ j:

• k = 0: (-1)(-1)/p_0 = 1/p_0
• k = i: (1)(0)/p_i = 0
• k = j: (0)(1)/p_j = 0
• other: 0

Result: 1/p_0

For i = j ≠ 0:

• k = 0: (-1)(-1)/p_0 = 1/p_0
• k = i: (1)(1)/p_i = 1/p_i
• other: 0

Result: 1/p_i + 1/p_0

So on the tangent space with basis {e_1 - e_0, ..., e_49 - e_0}:

fisherMetric p (e_i - e_0) (e_j - e_0) = δ_ij/p_i + 1/p_0

This is NOT δ_ij/p_i. The 1/p_0 cross-term appears.

The Correct Bridge

fisherMetric50 is the full-space metric tensor (evaluated on standard basis). fisherMetric is the bilinear form on the tangent space.

They are the same Riemannian metric, just expressed in different bases:

Full space:    g_ij = fisherMetric50 p i j = δ_ij / p_i
Tangent space: g_ij = fisherMetric p (e_i - e_0) (e_j - e_0) = δ_ij/p_i + 1/p_0

The chentsov_theorem proves uniqueness on the tangent space. The chentsov_50 theorem uses the full-space representation. The bridge requires showing that the full-space metric is determined by its tangent-space restriction.

Lean Implementation Path

1. Define toOpenSimplex : AminoAcidDistribution → openSimplex 50 (via h_pos)
2. Define fullSpaceMetric p X Y = ∑ k, X k * Y k / p.val k (same as fisherMetric)
3. Show fisherMetric50 p i j = fullSpaceMetric p (basisVec i) (basisVec j) (trivial)
4. Show fullSpaceMetric restricted to tangent space = fisherMetric on tangent vectors
5. Apply chentsov_theorem 50 to get uniqueness on tangent space
6. Lift uniqueness to full space (standard linear algebra)
7. Convert back to fisherMetric50 matrix form

Steps 1-3 are trivial. Step 4 is algebra. Step 5 requires chentsov_theorem (which has 3 internal sorries). Steps 6-7 are standard.