analytical-viz: capacity planning — symbol legend + plain-English formulas

Section 1 rewritten so a layman can read it without a glossary:

1. **Symbol legend table** listing every letter that appears in the
   three-axis formulas (N, b, N·b, HBM_per_PE, S_kv, kv_bpt,
   users_per_replica, TPOT SLO, step_latency, B_at_SLO, n_users,
   N_replicas, ⌈ ⌉). Each row: Symbol / Plain English / Value now
   for the current sidebar model + chip. For example:
     - N = 'Total number of model parameters (attention weights +
       FFN weights, across ALL layers)' → 6.98 billion params for
       Llama 3 8B
     - kv_bpt = 'KV cache bytes per token per user, summed across all
       layers (= 2·H_kv·d_head·b·layers)' → 128 KB/token

2. **Three-axis table** widened from (Formula / Meaning / Grows with)
   to (In symbols / In plain English / What it says / Value now).
   Axis A now shows the actual substituted arithmetic:
   ⌈ 13.96 GB / 6.0 GB ⌉ = ⌈ 2.33 ⌉ = 3 PEs.
   B and C flag 'see calculator' because they depend on user inputs.

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

tests/analytical_visualization/app.py

@@ -2751,23 +2751,107 @@ with tab_planning:
     # ── Section 1: three-axis sizing formula + calculator ─────────
     st.markdown("### 1. GPU count = max(A, B, C) × N_replicas")
 
+    # Symbol legend — spell out every letter that appears in the formulas
+    # below with a plain-English meaning + the value for the current
+    # sidebar model + chip. Makes the terse formulas readable at a glance.
+    _n_active_here = total_active_params(model)
+    _hbm_pe_bytes_here = int(_default_machine.pe_hbm_gb * 1e9)
+    _kv_bpt_here = 2 * model.h_kv * model.d_head * model.bytes_per_elem * model.layers
+    _weight_bytes_here = _n_active_here * model.bytes_per_elem
+
+    st.markdown("**Symbol legend** — what each letter in the formulas means")
+    _legend_rows = [
+        {"Symbol": "N",
+         "Plain English": "Total number of model parameters (attention "
+                          "weights + FFN weights, across ALL layers)",
+         "Value now": f"{_n_active_here/1e9:.2f} billion params"},
+        {"Symbol": "b",
+         "Plain English": "Bytes per parameter (2 for BF16, 1 for FP8, "
+                          "0.5 for INT4)",
+         "Value now": f"{model.bytes_per_elem} bytes (BF16)"},
+        {"Symbol": "N · b",
+         "Plain English": "Total bytes needed to store one full copy of "
+                          "the model weights",
+         "Value now": f"{_weight_bytes_here/1e9:.2f} GB"},
+        {"Symbol": "HBM_per_PE",
+         "Plain English": "Amount of high-bandwidth memory on ONE chip "
+                          "(GPU / PE)",
+         "Value now": f"{_default_machine.pe_hbm_gb:.1f} GB"},
+        {"Symbol": "S_kv",
+         "Plain English": "Context length in tokens (how much history "
+                          "one user's KV cache holds)",
+         "Value now": f"{s_kv:,} tokens (sidebar)"},
+        {"Symbol": "kv_bpt",
+         "Plain English": "KV cache bytes per token per user, summed "
+                          "across all layers "
+                          "(= 2 · H_kv · d_head · b · layers)",
+         "Value now": f"{_kv_bpt_here:,} B/token "
+                       f"({_kv_bpt_here/1024:.0f} KB/token)"},
+        {"Symbol": "users_per_replica",
+         "Plain English": "How many concurrent users one replica serves "
+                          "(= per-replica batch size)",
+         "Value now": "(from calculator below)"},
+        {"Symbol": "TPOT SLO",
+         "Plain English": "Latency budget per output token (Time Per "
+                          "Output Token — the SLA the operator commits to)",
+         "Value now": "(from calculator below)"},
+        {"Symbol": "step_latency",
+         "Plain English": "Time for one decode forward pass at a given B "
+                          "(weight fetch + compute + KV read)",
+         "Value now": "See Chip Roofline tab"},
+        {"Symbol": "B_at_SLO",
+         "Plain English": "Largest batch size that still finishes one "
+                          "decode step within the SLO budget",
+         "Value now": "(from calculator below)"},
+        {"Symbol": "n_users",
+         "Plain English": "Total concurrent users you need to serve "
+                          "across the whole deployment",
+         "Value now": "(from calculator below)"},
+        {"Symbol": "N_replicas",
+         "Plain English": "Number of complete model copies deployed "
+                          "(each is a self-contained instance)",
+         "Value now": "(from calculator below)"},
+        {"Symbol": "⌈ x ⌉",
+         "Plain English": "'Ceiling' — round UP to the next whole "
+                          "number (you can't have 3.2 GPUs)",
+         "Value now": "e.g. ⌈ 3.2 ⌉ = 4"},
+    ]
+    st.dataframe(pd.DataFrame(_legend_rows), width='stretch',
+                  hide_index=True)
+
+    st.markdown("**The three axes** — plain-English formulas")
     _axes_rows = [
         {"Axis": "A. Capacity floor",
-         "Formula": "⌈ N·b / HBM_per_PE ⌉",
-         "Meaning": ("Min PEs to hold ONE replica's weights alone. "
-                     "The bare floor — below this weights don't fit."),
-         "Grows with": "Model size (N)"},
+         "In symbols": "⌈ N · b / HBM_per_PE ⌉",
+         "In plain English": "⌈ (weight bytes) ÷ (HBM per chip) ⌉",
+         "What it says": "You need at least this many chips just to "
+                          "hold ONE copy of the weights. Below this, "
+                          "the model literally doesn't fit.",
+         "Value now":
+             f"⌈ {_weight_bytes_here/1e9:.2f} GB / "
+             f"{_default_machine.pe_hbm_gb:.1f} GB ⌉ = "
+             f"⌈ {_weight_bytes_here/_hbm_pe_bytes_here:.2f} ⌉ = "
+             f"**{-(-_weight_bytes_here // _hbm_pe_bytes_here)} PEs**"},
         {"Axis": "B. KV headroom",
-         "Formula": "⌈ (N·b + users_per_replica · S_kv · kv_bpt) / HBM_per_PE ⌉",
-         "Meaning": ("Min PEs to hold weights + all KV of the users "
-                     "assigned to this replica."),
-         "Grows with": "Users × context"},
+         "In symbols":
+             "⌈ (N·b + users_per_replica · S_kv · kv_bpt) / HBM_per_PE ⌉",
+         "In plain English":
+             "⌈ (weight bytes + all users' KV cache) ÷ (HBM per chip) ⌉",
+         "What it says": "Weights ALSO need room for every user's KV "
+                          "history. As users × context grows, KV eats "
+                          "the remaining HBM until you need more chips.",
+         "Value now": "(depends on users + context — see calculator)"},
         {"Axis": "C. Throughput SLO",
-         "Formula": ("N_replicas = ⌈ n_users / B_at_SLO ⌉, where "
-                     "B_at_SLO = largest B s.t. step_latency ≤ TPOT SLO"),
-         "Meaning": ("Enough replicas so each carries ≤ B_at_SLO users "
-                     "and meets per-token latency SLO."),
-         "Grows with": "Users, or tighter SLO"},
+         "In symbols":
+             "N_replicas = ⌈ n_users / B_at_SLO ⌉",
+         "In plain English":
+             "Replicas = ⌈ total users ÷ users each replica handles "
+             "within the latency budget ⌉",
+         "What it says": "Even if capacity fits, a single replica can "
+                          "only serve so many users before per-token "
+                          "latency blows the SLO. More users → more "
+                          "replicas.",
+         "Value now": "(depends on SLO — see calculator)"},
     ]
     st.dataframe(pd.DataFrame(_axes_rows), width='stretch',
                   hide_index=True)