fix(barycenter): native directed path via shortest_path_length [br-r37-c1-ecqmz]

Skill: /porting-to-rust (was 30+ min ago at the boundary; same
pattern as br-r37-c1-89n9d / br-r37-c1-wojl3 — drop a directed-graph
delegate to a buggy Rust path, reuse a correct fnx primitive).

Found via differential probe extending the directed-collapse-defect
audit: fnx.barycenter on a strongly-connected DiGraph cycle5+chord
returns [0, 2], nx returns [0]. Manual computation:

  bc[0] = 0+1+1+2+3 = 7   ← minimum, the only barycenter
  bc[1] = 4+0+1+2+3 = 10
  bc[2] = 3+4+0+1+2 = 10
  bc[3] = 2+3+4+0+1 = 10
  bc[4] = 1+2+2+3+0 = 8

The Rust _raw_barycenter has the same gr.undirected() collapse as
_raw_diameter / _raw_radius — computes barycentricity on the
undirected projection where node 2 sits symmetrically with node 0.

Fix: in fnx.barycenter, after the strongly-connected guard, the
directed branch now computes barycentricity natively via
``fnx.shortest_path_length(G)`` (already directed-aware) and finds
the argmin. Same pattern as the existing ``attr=`` Python branch in
the same wrapper. Rust _raw_barycenter stays for undirected.

Verified bit-exact against nx 3.6.1 on:
  - directed cycle5: [0,1,2,3,4] (vertex-transitive)
  - directed cycle5+chord: [0]    (was [0,2] — wrong)
  - directed K4: [0,1,2,3]
  - directed cycle3: [0,1,2]
  - undirected K4 / P5: unchanged

Wider test surface: 112 barycenter-family tests pass. Golden
snapshot regenerated to capture the corrected directed barycenter
output for the dicycle5 / dicycle5+chord fixtures added in
br-r37-c1-zr9e0.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

Author: Dicklesworthstone

python/franken_networkx/__init__.py

@@ -9612,11 +9612,27 @@ def barycenter(G, weight=None, attr=None, sp=None, *, backend=None, **backend_kw
         # NetworkXNoPath in that case. Route directed inputs through
         # the Python path below when the graph is not strongly
         # connected so the parity error surfaces correctly.
-        if G.is_directed() and not is_strongly_connected(G):
-            raise NetworkXNoPath(
-                f"Input graph {G} is disconnected, so every induced subgraph "
-                "has infinite barycentricity."
-            )
+        if G.is_directed():
+            if not is_strongly_connected(G):
+                raise NetworkXNoPath(
+                    f"Input graph {G} is disconnected, so every induced subgraph "
+                    "has infinite barycentricity."
+                )
+            # br-r37-c1-ecqmz: _raw_barycenter has the directed-collapse
+            # defect (gr.undirected() before computing) — returns wrong
+            # barycenter on directed cycle5+chord (gives [0, 2] instead
+            # of nx's [0]). Compute natively via shortest_path_length
+            # which IS directed-aware.
+            smallest = float("inf")
+            result: list = []
+            for v, dists in shortest_path_length(G):
+                bc = sum(dists.values())
+                if bc < smallest:
+                    smallest = bc
+                    result = [v]
+                elif bc == smallest:
+                    result.append(v)
+            return result
         return _raw_barycenter(G)
     if attr is None:
         return _call_networkx_for_parity(

tests/python/goldens/review_mode_algorithms.json

@@ -5549,8 +5549,7 @@
           ]
         ],
         "barycenter": [
-          0,
-          2
+          0
         ]
       }
     }