analytical-viz: roofline formula table shows substituted numeric form

Each row in the 'Formulas & interpretation' table now shows both the
symbolic form AND the actual numbers plugged in, on a second line in
the Formula column. Row height bumped to fit two lines. Rendering:

  AI:      C / W
           = 8.00e+12 / 2.56e+11
           = 31.25 FLOPs/byte

  L*:      2 · N_active · W / (C · kv_bpt)
           = 2 · 6.98e+09 · 2.56e+11 / (8.00e+12 · 131,072)
           = 3,407 tokens

  B_knee:  B* / (1 - S_kv/L*)
           = 31 / (1 - 8,192/3,407)
           = 31 / (1 - 2.404)   [<= 0 -> no knee]

Makes the derivation traceable at a glance without needing to plug the
numbers back into the abstract formula. Purely a display change; no
runtime behavior shifts, all 57 tests still pass.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

tests/analytical_visualization/app.py

@@ -2320,33 +2320,68 @@ with tab_roofline:
 
     # ── Interpretation table ───────────────────────────────────────
     st.markdown("**Formulas & interpretation**")
+    # Pre-compute substituted-value strings so each row shows both the
+    # symbolic form and the actual numbers plugged in.
+    _C = _default_machine.peak_flops             # FLOPs / s
+    _W = _default_machine.bw_hbm                 # bytes / s
+    _b_elem = model.bytes_per_elem               # bytes / weight
+    _kv_bpt_full = 2 * model.h_kv * model.d_head * model.bytes_per_elem * model.layers
+
+    _sub_ai = (
+        f"= {_C:.2e} / {_W:.2e}\n= {_ai:.2f} FLOPs/byte"
+    )
+    _sub_bstar = (
+        f"= {_C:.2e} · {_b_elem} / (2 · {_W:.2e})\n= {_b_star:.2f}"
+    )
+    if _is_moe:
+        _sub_bmoe = f"= {_b_star:.0f} · 8 (k=8 example)\n= {_b_star * 8:.0f}"
+    else:
+        _sub_bmoe = "N/A (dense preset — sparsity = 1)"
+    _sub_lstar = (
+        f"= 2 · {_n_active:.2e} · {_W:.2e} / ({_C:.2e} · {_kv_bpt_full:,})\n"
+        f"= {_l_star:,.0f} tokens"
+    )
+    _knee_now = knee_batch(_default_machine, model, s_kv)
+    if _knee_now is None:
+        _sub_bknee = (
+            f"= {_b_star:.0f} / (1 − {s_kv:,}/{_l_star:,.0f})\n"
+            f"= {_b_star:.0f} / (1 − {s_kv/_l_star:.3f})   [<= 0 → no knee]"
+        )
+        _bknee_val = "no knee (S_kv ≥ L*)"
+    else:
+        _sub_bknee = (
+            f"= {_b_star:.0f} / (1 − {s_kv:,}/{_l_star:,.0f})\n"
+            f"= {_b_star:.0f} / {1 - s_kv/_l_star:.3f}\n"
+            f"= {_knee_now:.0f}"
+        )
+        _bknee_val = f"{_knee_now:.0f}"
+
     _formula_rows = [
         {"Symbol": "AI",
-         "Formula": "C / W",
+         "Formula": "C / W\n" + _sub_ai,
          "Meaning": "FLOPs per byte of HBM bandwidth. Chip-only.",
          "Your chip": f"{_ai:.1f}"},
         {"Symbol": "B*",
-         "Formula": "C · b / (2 · W)",
+         "Formula": "C · b / (2 · W)\n" + _sub_bstar,
          "Meaning": ("Batch size where weight-fetch time = compute time. "
                      "Below: memory-bound. Above: amortized."),
          "Your chip": f"{_b_star:.0f}"},
         {"Symbol": "B*_moe",
-         "Formula": "B* · (N_total / N_active)",
+         "Formula": "B* · (N_total / N_active)\n" + _sub_bmoe,
          "Meaning": "MoE fetches all experts but computes on active only.",
          "Your chip": ("N/A (dense preset)" if not _is_moe
                        else f"{_b_star * 8:.0f} (k=8 example)")},
         {"Symbol": "L*",
-         "Formula": "2 · N_active · W / (C · kv_bpt)",
+         "Formula": "2 · N_active · W / (C · kv_bpt)\n" + _sub_lstar,
          "Meaning": ("Context where KV-read time = compute time. "
                      "Above: KV-bound regardless of B."),
          "Your chip": f"{_l_star:,.0f} tok"},
         {"Symbol": "B_knee",
-         "Formula": "B* / (1 − S_kv/L*)",
+         "Formula": "B* / (1 − S_kv/L*)\n" + _sub_bknee,
          "Meaning": ("Batch size where cost curve bends onto its floor. "
                      "Diverges at S_kv = L*."),
-         "Your chip": (f"{knee_batch(_default_machine, model, s_kv):.0f}"
-                       if knee_batch(_default_machine, model, s_kv) is not None
-                       else "no knee (S_kv >= L*)")},
+         "Your chip": _bknee_val},
     ]
     st.dataframe(pd.DataFrame(_formula_rows),
-                  width='stretch', hide_index=True)
+                  width='stretch', hide_index=True,
+                  row_height=90)