diff --git a/lang/el-compiler/src/codegen.el b/lang/el-compiler/src/codegen.el
index 8b72adf..a723ab9 100644
--- a/lang/el-compiler/src/codegen.el
+++ b/lang/el-compiler/src/codegen.el
@@ -2633,6 +2633,12 @@ fn builtin_arity(name: String) -> Int {
     if str_eq(name, "__engram_neighbors_filtered") { return 3 }
     if str_eq(name, "__engram_activate") { return 2 }
     if str_eq(name, "__engram_activate_json") { return 2 }
+    if str_eq(name, "__engram_geo_descriptor_json") { return 1 }
+    if str_eq(name, "__engram_geo_overlap_json") { return 2 }
+    if str_eq(name, "__engram_geo_subtract_json") { return 3 }
+    if str_eq(name, "__engram_geo_combine_json") { return 2 }
+    if str_eq(name, "__engram_geo_distance_json") { return 2 }
+    if str_eq(name, "__engram_geo_analogy_json") { return 2 }
     if str_eq(name, "__engram_scan_nodes_json") { return 2 }
     if str_eq(name, "__generate") { return 1 }
     // Filesystem
@@ -2732,6 +2738,12 @@ fn builtin_arity(name: String) -> Int {
     if str_eq(name, "engram_scan_nodes_json") { return 2 }
     if str_eq(name, "engram_neighbors_json") { return 3 }
     if str_eq(name, "engram_activate_json") { return 2 }
+    if str_eq(name, "engram_geo_descriptor_json") { return 1 }
+    if str_eq(name, "engram_geo_overlap_json") { return 2 }
+    if str_eq(name, "engram_geo_subtract_json") { return 3 }
+    if str_eq(name, "engram_geo_combine_json") { return 2 }
+    if str_eq(name, "engram_geo_distance_json") { return 2 }
+    if str_eq(name, "engram_geo_analogy_json") { return 2 }
     if str_eq(name, "engram_stats_json") { return 0 }
     // LLM
     if str_eq(name, "llm_call") { return 2 }
diff --git a/lang/elc.c b/lang/elc.c
index 865c5ec..a5855fe 100644
--- a/lang/elc.c
+++ b/lang/elc.c
@@ -6748,6 +6748,24 @@ el_val_t builtin_arity(el_val_t name) {
   if (str_eq(name, EL_STR("__engram_activate_json"))) {
     return 2;
   }
+  if (str_eq(name, EL_STR("__engram_geo_descriptor_json"))) {
+    return 1;
+  }
+  if (str_eq(name, EL_STR("__engram_geo_overlap_json"))) {
+    return 2;
+  }
+  if (str_eq(name, EL_STR("__engram_geo_subtract_json"))) {
+    return 3;
+  }
+  if (str_eq(name, EL_STR("__engram_geo_combine_json"))) {
+    return 2;
+  }
+  if (str_eq(name, EL_STR("__engram_geo_distance_json"))) {
+    return 2;
+  }
+  if (str_eq(name, EL_STR("__engram_geo_analogy_json"))) {
+    return 2;
+  }
   if (str_eq(name, EL_STR("__engram_scan_nodes_json"))) {
     return 2;
   }
@@ -6991,6 +7009,24 @@ el_val_t builtin_arity(el_val_t name) {
   if (str_eq(name, EL_STR("engram_activate_json"))) {
     return 2;
   }
+  if (str_eq(name, EL_STR("engram_geo_descriptor_json"))) {
+    return 1;
+  }
+  if (str_eq(name, EL_STR("engram_geo_overlap_json"))) {
+    return 2;
+  }
+  if (str_eq(name, EL_STR("engram_geo_subtract_json"))) {
+    return 3;
+  }
+  if (str_eq(name, EL_STR("engram_geo_combine_json"))) {
+    return 2;
+  }
+  if (str_eq(name, EL_STR("engram_geo_distance_json"))) {
+    return 2;
+  }
+  if (str_eq(name, EL_STR("engram_geo_analogy_json"))) {
+    return 2;
+  }
   if (str_eq(name, EL_STR("engram_stats_json"))) {
     return 0;
   }
diff --git a/lang/runtime/engram.el b/lang/runtime/engram.el
index 60015cd..b940d0f 100644
--- a/lang/runtime/engram.el
+++ b/lang/runtime/engram.el
@@ -119,6 +119,37 @@ fn engram_activate_json(query: String, limit: Int) -> String {
     return __engram_activate_json(query, limit)
 }
 
+// --- Geometry operators (§5) ---
+// Relational-neighborhood algebra over centered engram descriptors. Each takes
+// comma-separated seed-id set(s); a centered neighborhood is grown from those
+// seeds (ad-hoc NOCACHE path) and the operator's JSON result is returned.
+// Delegates to the native __engram_geo_*_json seeds (el_seed.c); in the heavy
+// engram runtime the same bare names resolve directly to el_runtime.c symbols.
+
+fn engram_geo_descriptor_json(seeds: String) -> String {
+    return __engram_geo_descriptor_json(seeds)
+}
+
+fn engram_geo_overlap_json(a_seeds: String, b_seeds: String) -> String {
+    return __engram_geo_overlap_json(a_seeds, b_seeds)
+}
+
+fn engram_geo_subtract_json(a_seeds: String, b_seeds: String, mode: String) -> String {
+    return __engram_geo_subtract_json(a_seeds, b_seeds, mode)
+}
+
+fn engram_geo_combine_json(a_seeds: String, b_seeds: String) -> String {
+    return __engram_geo_combine_json(a_seeds, b_seeds)
+}
+
+fn engram_geo_distance_json(a_seeds: String, b_seeds: String) -> String {
+    return __engram_geo_distance_json(a_seeds, b_seeds)
+}
+
+fn engram_geo_analogy_json(a_seeds: String, b_seeds: String) -> String {
+    return __engram_geo_analogy_json(a_seeds, b_seeds)
+}
+
 // --- Generation ---
 
 fn generate(form: String) -> String {
diff --git a/lang/runtime/engram_geometry.c b/lang/runtime/engram_geometry.c
index c559939..0e27911 100644
--- a/lang/runtime/engram_geometry.c
+++ b/lang/runtime/engram_geometry.c
@@ -1415,7 +1415,8 @@ int engram_geo_reify_index_finalize(GeoReifyIndex* ix){
             r->centroid_unit=malloc((size_t)r->dim*sizeof(float));
             if(r->centroid_unit){
                 double nrm=0; for(int d=0;d<r->dim;d++){ double v=(double)r->centroid_raw[d]-ix->mean[d]; r->centroid_unit[d]=(float)v; nrm+=v*v; }
-                nrm=sqrt(nrm); if(nrm>1e-12) for(int d=0;d<r->dim;d++) r->centroid_unit[d]=(float)(r->centroid_unit[d]/nrm);
+                nrm=sqrt(nrm);
+                if(nrm>1e-12){ for(int d=0;d<r->dim;d++) r->centroid_unit[d]=(float)(r->centroid_unit[d]/nrm); }
                 else { free(r->centroid_unit); r->centroid_unit=NULL; }
             }
         }