Fix inverted sign in formal/informal LDA analysis gradient

The model (analysis/bicorder_model.json) maps a positive LDA score to
cluster 2 = Institutional/Bureaucratic = "formal", but ldaScoreToScale
added the score (5 + score*4/3), sending formal/institutional protocols
toward 9 (informal) and vice versa. bicorder.json defines this gradient
as 1 = formal, 9 = informal, so the score must be subtracted.

- Flip the sign: value = 5 - (ldaScore * 4/3); correct the doc comment to
  state the model's actual sign convention
- Rename calculateBureaucratic -> calculateFormalInformal and update the
  stale analysisOrder comment, matching bicorder.json's formal/informal terms

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

bicorder-app/src/App.svelte

@@ -60,8 +60,8 @@
     });
 
     // Analysis screens (shown in both shortform and longform)
-    // Show useful/not-useful gradient first (index 3), then the others
+    // Show the useful gradient first (index 3), then the others
-    const analysisOrder = [3, 0, 1, 2]; // useful/not-useful, hardness, polarization, bureaucratic
+    const analysisOrder = [3, 0, 1, 2]; // useful, hardness, polarization, formal/informal
     analysisOrder.forEach((index) => {
       screens.push({ type: 'analysis', index, gradient: data.analysis[index] });
     });
@@ -303,25 +303,30 @@
 
   function ldaScoreToScale(ldaScore: number | null): number | null {
     /**
-     * Convert LDA score to 1-9 scale.
+     * Convert LDA score to the analysis[2] "formal vs informal" 1-9 scale.
-     * LDA scores typically range from -4 to +4 (8 range)
+     * LDA scores typically range from -4 to +4 (8 range); target is 1-9.
-     * Target scale is 1 to 9 (8 range)
      *
-     * Formula: value = 5 + (ldaScore * 4/3)
+     * The model's sign convention (see analysis/bicorder_model.json):
-     * - LDA -3 or less → 1 (bureaucratic)
+     *   positive LDA → cluster 2 = Institutional/Bureaucratic = "formal"
-     * - LDA 0 → 5 (boundary)
+     *   negative LDA → cluster 1 = Relational/Cultural = "informal"
-     * - LDA +3 or more → 9 (relational)
+     * bicorder.json defines this gradient as 1 = formal, 9 = informal, so a
+     * positive LDA score must map toward 1 (formal). The score is therefore
+     * subtracted, not added.
+     *
+     * Formula: value = 5 - (ldaScore * 4/3)
+     * - LDA +3 or more → 1 (formal / institutional / bureaucratic)
+     * - LDA 0          → 5 (boundary, characteristics of both families)
+     * - LDA -3 or less → 9 (informal / relational / cultural)
      */
     if (ldaScore === null) return null;
 
-    // Scale: value = 5 + (ldaScore * 1.33)
+    const value = 5 - (ldaScore * 4.0 / 3.0);
-    const value = 5 + (ldaScore * 4.0 / 3.0);
 
     // Clamp to 1-9 range and round
     return Math.round(Math.max(1, Math.min(9, value)));
   }
 
-  function calculateBureaucratic(): number | null {
+  function calculateFormalInformal(): number | null {
     // Collect all diagnostic gradients with their set and gradient info
     const ratings: Record<string, number> = {};
 
@@ -329,7 +334,7 @@
       const setName = diagnosticSet.set_name;
       diagnosticSet.gradients.forEach((gradient) => {
         if (gradient.value !== null) {
-          // Create dimension name in format: SetName_left_vs_right
+          // Dimension name must match the model's keys: SetName_left_vs_right
           const dimensionName = `${setName}_${gradient.term_left}_vs_${gradient.term_right}`;
           ratings[dimensionName] = gradient.value;
         }
@@ -343,10 +348,10 @@
       // Get prediction from classifier (need detailed: true to get ldaScore)
       const result = classifier.predict(ratings, { detailed: true });
 
-      // Convert LDA score to 1-9 scale
+      // Convert LDA score to the 1-9 formal/informal scale
       return ldaScoreToScale(result.ldaScore);
     } catch (error) {
-      console.error('Error calculating bureaucratic/relational score:', error);
+      console.error('Error calculating formal/informal score:', error);
       return null;
     }
   }
@@ -362,8 +367,8 @@
           // Polarized/Centrist
           data.analysis[1].value = calculatePolarization();
         } else if (index === 2) {
-          // Bureaucratic/Relational (LDA classifier)
+          // Formal/Informal (LDA classifier)
-          data.analysis[2].value = calculateBureaucratic();
+          data.analysis[2].value = calculateFormalInformal();
         }
       }
     });