{
  "accumulator_ceilings": {
    "argued_i128_layernorm_bits": 126,
    "i64_bits": 62
  },
  "batch": 16,
  "exponents_sha256": "87c73b8f34990c8db9a9a48dbc66bf7b1f3502cea83913bc3679e8912ddd8ecb",
  "format": "bob-width-sites-1",
  "geometry": {
    "ctx": 512,
    "d_mlp": 1024,
    "d_model": 256,
    "head_dim": 32,
    "n_head": 8,
    "n_layer": 5,
    "params": 5129216,
    "positions": 8192,
    "vocab": 4097
  },
  "model_config_digest": "4f7f5e807828cea50e906da035011b0d7203203fd6c18f4f654988d06c13598e",
  "optimizer_schedule": {
    "lr_den": 16,
    "lr_num": 1,
    "max_lr_den": 3200,
    "max_lr_num": 200,
    "max_tensor_lr_den": 26214400,
    "max_tensor_lr_num": 104857600,
    "optimizer": "sgd-momentum-7-8",
    "schedule_total": 3128,
    "warmup": 200
  },
  "sites": [
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "token.common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "position.common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "embedding.checked_add",
          "registered_bits": 24,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "residual_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        }
      ],
      "carrier": "checked i64 add, then i32",
      "expression": "token_embedding + position_embedding after common rescale",
      "id": "F01",
      "name": "embed.add",
      "proof": "token source bits=23 plus max(from_exp=-22-to_exp=-19,0)=0 -> 23 <= 62; position source bits=23 plus max(from_exp=-23-to_exp=-19,0)=0 -> 23 <= 62; checked add -> 24 <= 62; output act.resid.bits=23",
      "rounding": "ties-away common rescale; no rounding at add",
      "source": "src/net.rs::forward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "mean_sum",
          "registered_bits": 31,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "sum(x_j), j<d_model",
      "id": "F02",
      "name": "ln.mean.sum",
      "proof": "23 + ceil_log2(256)=8 -> 31 <= 62",
      "rounding": "round-to-nearest, ties away on division by d_model",
      "source": "src/net.rs::ln_forward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "centered_value",
          "registered_bits": 24,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        }
      ],
      "carrier": "checked i64 subtraction, then i32",
      "expression": "x_j - mean",
      "id": "F03",
      "name": "ln.center",
      "proof": "max(x=23, mean<2^23) + carry -> 24 <= 62",
      "rounding": "none",
      "source": "src/net.rs::ln_forward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "square_sum",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -38
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "sum(center_j * center_j), j<d_model",
      "id": "F04",
      "name": "ln.square.sum",
      "proof": "24 + 24 + ceil_log2(256)=8 -> 56 <= 62",
      "rounding": "round-to-nearest, ties away on variance division",
      "source": "src/net.rs::ln_forward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "normalize_product",
          "registered_bits": 54,
          "registered_exponent": {
            "formula": "m_minus_59",
            "kind": "runtime_ln_inv",
            "m_source": "forward_saved_m",
            "output_exponent": -24
          }
        },
        {
          "interval": "strict_signed",
          "name": "normalized_q24",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -24
          }
        }
      ],
      "carrier": "i64 product, then i32",
      "expression": "round(center * inverse_std) into saved Q24 normalized value",
      "id": "F05",
      "name": "ln.normalize",
      "proof": "product: 24 + 30 -> 54 <= 62; normalized_bits(d_model)=30",
      "rounding": "round-to-nearest, ties away",
      "source": "src/net.rs::ln_forward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "final_norm.natural_gamma_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "final_norm.product_common",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "final_norm.beta_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "final_norm.checked_add",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "final_norm.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm1.natural_gamma_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm1.product_common",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm1.beta_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm1.checked_add",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm1.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm1.natural_gamma_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm1.product_common",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm1.beta_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm1.checked_add",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm1.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm1.natural_gamma_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm1.product_common",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm1.beta_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm1.checked_add",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm1.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm1.natural_gamma_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm1.product_common",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm1.beta_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm1.checked_add",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm1.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm1.natural_gamma_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm1.product_common",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm1.beta_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm1.checked_add",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm1.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm2.natural_gamma_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm2.product_common",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm2.beta_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm2.checked_add",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm2.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm2.natural_gamma_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm2.product_common",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm2.beta_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm2.checked_add",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm2.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm2.natural_gamma_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm2.product_common",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm2.beta_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm2.checked_add",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm2.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm2.natural_gamma_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm2.product_common",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm2.beta_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm2.checked_add",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm2.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm2.natural_gamma_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm2.product_common",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm2.beta_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm2.checked_add",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm2.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        }
      ],
      "carrier": "i64 product/add, then i32",
      "expression": "round(normalized * gamma) + beta after registered rescale",
      "id": "F06",
      "name": "ln.affine",
      "proof": "final_norm product bits=53 exp=-43 plus max(from_exp=-43-to_exp=-17,0)=0 -> 53 <= 62; beta bits=23 exp=-24 plus max(from_exp=-24-to_exp=-17,0)=0 -> 23 <= 62; checked add -> 54 <= 62; output bits=23 exp=-17; layer_0.norm1 product bits=53 exp=-43 plus max(from_exp=-43-to_exp=-17,0)=0 -> 53 <= 62; beta bits=23 exp=-28 plus max(from_exp=-28-to_exp=-17,0)=0 -> 23 <= 62; checked add -> 54 <= 62; output bits=23 exp=-17; layer_1.norm1 product bits=53 exp=-43 plus max(from_exp=-43-to_exp=-17,0)=0 -> 53 <= 62; beta bits=23 exp=-31 plus max(from_exp=-31-to_exp=-17,0)=0 -> 23 <= 62; checked add -> 54 <= 62; output bits=23 exp=-17; layer_2.norm1 product bits=53 exp=-43 plus max(from_exp=-43-to_exp=-17,0)=0 -> 53 <= 62; beta bits=23 exp=-31 plus max(from_exp=-31-to_exp=-17,0)=0 -> 23 <= 62; checked add -> 54 <= 62; output bits=23 exp=-17; layer_3.norm1 product bits=53 exp=-43 plus max(from_exp=-43-to_exp=-17,0)=0 -> 53 <= 62; beta bits=23 exp=-31 plus max(from_exp=-31-to_exp=-17,0)=0 -> 23 <= 62; checked add -> 54 <= 62; output bits=23 exp=-17; layer_4.norm1 product bits=53 exp=-43 plus max(from_exp=-43-to_exp=-17,0)=0 -> 53 <= 62; beta bits=23 exp=-31 plus max(from_exp=-31-to_exp=-17,0)=0 -> 23 <= 62; checked add -> 54 <= 62; output bits=23 exp=-17; layer_0.norm2 product bits=53 exp=-43 plus max(from_exp=-43-to_exp=-17,0)=0 -> 53 <= 62; beta bits=23 exp=-31 plus max(from_exp=-31-to_exp=-17,0)=0 -> 23 <= 62; checked add -> 54 <= 62; output bits=23 exp=-17; layer_1.norm2 product bits=53 exp=-43 plus max(from_exp=-43-to_exp=-17,0)=0 -> 53 <= 62; beta bits=23 exp=-31 plus max(from_exp=-31-to_exp=-17,0)=0 -> 23 <= 62; checked add -> 54 <= 62; output bits=23 exp=-17; layer_2.norm2 product bits=53 exp=-43 plus max(from_exp=-43-to_exp=-17,0)=0 -> 53 <= 62; beta bits=23 exp=-31 plus max(from_exp=-31-to_exp=-17,0)=0 -> 23 <= 62; checked add -> 54 <= 62; output bits=23 exp=-17; layer_3.norm2 product bits=53 exp=-43 plus max(from_exp=-43-to_exp=-17,0)=0 -> 53 <= 62; beta bits=23 exp=-31 plus max(from_exp=-31-to_exp=-17,0)=0 -> 23 <= 62; checked add -> 54 <= 62; output bits=23 exp=-17; layer_4.norm2 product bits=53 exp=-43 plus max(from_exp=-43-to_exp=-17,0)=0 -> 53 <= 62; beta bits=23 exp=-30 plus max(from_exp=-30-to_exp=-17,0)=0 -> 23 <= 62; checked add -> 54 <= 62; output bits=23 exp=-17",
      "rounding": "round-to-nearest, ties away at rescale",
      "source": "src/net.rs::ln_forward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.qkv.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.qkv.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.qkv.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.qkv.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.qkv.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.qkv.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.qkv.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.qkv.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.qkv.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.qkv.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.qkv.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.qkv.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.qkv.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.qkv.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.qkv.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.qkv.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.qkv.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.qkv.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.qkv.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.qkv.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.qkv.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.qkv.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.qkv.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.qkv.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.qkv.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "sum(ln1_j * qkv_weight_j), j<d_model",
      "id": "F07",
      "name": "qkv.matmul",
      "proof": "23 + 23 + ceil_log2(256)=8 -> 54 <= 62; output act.qkv.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/net.rs::matmul_fwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "score_accumulator",
          "registered_bits": 51,
          "registered_exponent": {
            "kind": "fixed",
            "value": -36
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "sum(query_j * key_j), j<head_dim",
      "id": "F08",
      "name": "attention.score",
      "proof": "23 + 23 + ceil_log2(32)=5 -> 51 <= 62",
      "rounding": "none before score scaling",
      "source": "src/net.rs::attention_forward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "score_scale_product",
          "registered_bits": 80,
          "registered_exponent": {
            "kind": "fixed",
            "value": -66
          }
        },
        {
          "interval": "strict_signed",
          "name": "rounded_score_q26",
          "registered_bits": 41,
          "registered_exponent": {
            "kind": "fixed",
            "value": -26
          }
        }
      ],
      "carrier": "argued i128 product, checked return to i64",
      "expression": "round(attention_score * registered Q30 scale)",
      "id": "F09",
      "name": "score.scale",
      "proof": "score_scale_q30=273841598 has mag_bits=29; non-accumulating wide product: 51 + 29 -> 80 <= 126",
      "rounding": "round-to-nearest, ties away",
      "source": "src/net.rs::attention_forward"
    },
    {
      "boundaries": [
        {
          "interval": "inclusive_unsigned",
          "name": "exp2_q30",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -30
          }
        }
      ],
      "carrier": "i64 inclusive unsigned Q30",
      "expression": "registered exp2 table/interpolation output",
      "id": "F10",
      "name": "softmax.exp",
      "proof": "registered table range is inclusive 0 <= e <= 2^30",
      "rounding": "table interpolation uses registered integer rounding",
      "source": "src/exp2.rs::exp2_q26"
    },
    {
      "boundaries": [
        {
          "interval": "inclusive_unsigned",
          "name": "exp_sum",
          "registered_bits": 39,
          "registered_exponent": {
            "kind": "fixed",
            "value": -30
          }
        }
      ],
      "carrier": "i64 accumulator with inclusive upper endpoint",
      "expression": "sum(exp_j) over a causal row",
      "id": "F11",
      "name": "softmax.sum",
      "proof": "30 + ceil_log2(512)=9 -> 39 <= 62; exceptional inclusive law: 0 <= sum <= ctx*2^30",
      "rounding": "none",
      "source": "src/net.rs::attention_forward"
    },
    {
      "boundaries": [
        {
          "interval": "inclusive_unsigned",
          "name": "probability_q30",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -30
          }
        }
      ],
      "carrier": "i32 inclusive unsigned Q30",
      "expression": "round((exp << 30) / row_sum)",
      "id": "F12",
      "name": "softmax.probability",
      "proof": "exceptional inclusive law: 0 <= p <= 2^30, including the one-element row endpoint",
      "rounding": "round-to-nearest, ties away",
      "source": "src/net.rs::attention_forward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "attention_value_accumulator",
          "registered_bits": 62,
          "registered_exponent": {
            "kind": "fixed",
            "value": -48
          }
        },
        {
          "interval": "strict_signed",
          "name": "attention_value_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        }
      ],
      "carrier": "i64 accumulator with inclusive Q30 broadcast operand",
      "expression": "sum(probability_j * value_j) over causal row",
      "id": "F13",
      "name": "attention.value",
      "proof": "30 + 23 + ceil_log2(512)=9 -> 62 <= 62; output act.attn_av.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/net.rs::attention_forward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.attention_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.attention_projection.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.attention_projection.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.attention_projection.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.attention_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.attention_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.attention_projection.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.attention_projection.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.attention_projection.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.attention_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.attention_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -43
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.attention_projection.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.attention_projection.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.attention_projection.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.attention_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.attention_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.attention_projection.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.attention_projection.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.attention_projection.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.attention_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.attention_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.attention_projection.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.attention_projection.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.attention_projection.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.attention_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "attention value projection row-dot over d_model",
      "id": "F14",
      "name": "attention.output",
      "proof": "23 + 23 + ceil_log2(256)=8 -> 54 <= 62; output act.attn_proj_out.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/net.rs::matmul_fwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_input.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_input.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_input.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_input.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_input.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_input.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_input.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_input.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_input.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_input.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_input.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_input.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_input.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_input.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_input.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_input.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_input.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_input.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_input.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_input.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_input.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_input.product_common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_input.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_input.checked_add",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_input.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "MLP input projection row-dot over d_model",
      "id": "F15",
      "name": "mlp.input",
      "proof": "23 + 23 + ceil_log2(256)=8 -> 54 <= 62; output act.mlp_h.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/net.rs::matmul_fwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "slope_product",
          "registered_bits": 34,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "rounded_activation",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        }
      ],
      "carrier": "i64 interpolation product, then i32",
      "expression": "registered GELU table interpolation and rounded output",
      "id": "F16",
      "name": "activation.table",
      "proof": "slope product: 14 + 20 -> 34 <= 62; TABLE_FRAC_BITS=14; registered max slope bits=20; output act.mlp_h.bits=23",
      "rounding": "registered table ties-away rounding",
      "source": "src/gelu.rs::gelu_site_at"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_output.accumulator",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_output.product_common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_output.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_output.checked_add",
          "registered_bits": 57,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_output.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_output.accumulator",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_output.product_common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_output.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_output.checked_add",
          "registered_bits": 57,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_output.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_output.accumulator",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_output.product_common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_output.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_output.checked_add",
          "registered_bits": 57,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_output.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_output.accumulator",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_output.product_common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_output.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_output.checked_add",
          "registered_bits": 57,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_output.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_output.accumulator",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_output.product_common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_output.bias_common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_output.checked_add",
          "registered_bits": 57,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_output.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -21
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "MLP output projection row-dot over d_mlp",
      "id": "F17",
      "name": "mlp.output",
      "proof": "23 + 23 + ceil_log2(1024)=10 -> 56 <= 62; output act.mlp_out.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/net.rs::matmul_fwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "residual.common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "attention.common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "mlp.common",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "residual.checked_add",
          "registered_bits": 24,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "residual_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        }
      ],
      "carrier": "checked i64 add, then i32",
      "expression": "residual + branch after common rescale",
      "id": "F18",
      "name": "residual.add",
      "proof": "residual bits=23; attention source bits=23 plus max(from_exp=-22-to_exp=-19,0)=0 -> 23 <= 62; MLP source bits=23 plus max(from_exp=-21-to_exp=-19,0)=0 -> 23 <= 62; checked add -> 24 <= 62; output act.resid.bits=23",
      "rounding": "ties-away common rescale; no rounding at add",
      "source": "src/net.rs::forward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "logits_accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "logits_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -16
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "row-dot over d_model against tied token embedding",
      "id": "F19",
      "name": "logits",
      "proof": "23 + 23 + ceil_log2(256)=8 -> 54 <= 62; output act.logits.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/net.rs::logits_and_softmax"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "logit_difference",
          "registered_bits": 24,
          "registered_exponent": {
            "kind": "fixed",
            "value": -16
          }
        },
        {
          "interval": "strict_signed",
          "name": "change_base_product",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -46
          }
        },
        {
          "interval": "strict_signed",
          "name": "rounded_lookup_q26",
          "registered_bits": 36,
          "registered_exponent": {
            "kind": "fixed",
            "value": -26
          }
        }
      ],
      "carrier": "i64 product",
      "expression": "(logit - row_max) * LOG2E_Q30",
      "id": "F20",
      "name": "loss.change_base",
      "proof": "difference bits=24; product: 24 + 31 -> 55 <= 62",
      "rounding": "round-to-nearest, ties away into Q26",
      "source": "src/net.rs::logits_and_softmax"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "loss_exp_sum",
          "registered_bits": 43,
          "registered_exponent": {
            "kind": "fixed",
            "value": -30
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "sum(exp(logit - max)), j<vocab",
      "id": "F21",
      "name": "loss.exp_sum",
      "proof": "30 + ceil_log2(4097)=13 -> 43 <= 62",
      "rounding": "none",
      "source": "src/net.rs::logits_and_softmax"
    },
    {
      "boundaries": [
        {
          "interval": "inclusive_unsigned",
          "name": "probability_at_d_logits",
          "registered_bits": 20,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "d_logits_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        }
      ],
      "carrier": "checked i64 subtraction, then i32",
      "expression": "computed probability - one_hot_target",
      "id": "B01",
      "name": "cross_entropy.gradient",
      "proof": "inclusive Q30 probability plus subtraction bit; output bwd.d_logits.bits=23",
      "rounding": "round-to-nearest, ties away in probability division",
      "source": "src/backward.rs::ce_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "d_lnf_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -42
          }
        },
        {
          "interval": "strict_signed",
          "name": "d_lnf_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "sum(dlogit_v * tied_weight_vj), v<vocab",
      "id": "B02",
      "name": "logits.input_gradient",
      "proof": "23 + 23 + ceil_log2(4097)=13 -> 59 <= 62; output bwd.d_lnf_out.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::matmul_bwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "head.natural_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -37
          }
        },
        {
          "interval": "strict_signed",
          "name": "embedding.natural_accumulator",
          "registered_bits": 36,
          "registered_exponent": {
            "kind": "fixed",
            "value": -14
          }
        },
        {
          "interval": "strict_signed",
          "name": "head.common",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -6
          }
        },
        {
          "interval": "strict_signed",
          "name": "embedding.common",
          "registered_bits": 36,
          "registered_exponent": {
            "kind": "fixed",
            "value": -6
          }
        },
        {
          "interval": "strict_signed",
          "name": "branches.checked_add",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -6
          }
        },
        {
          "interval": "strict_signed",
          "name": "wte.gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -6
          }
        }
      ],
      "carrier": "i64 accumulators and checked branch add",
      "expression": "sum_n(dlogit_nv * final_norm_nj) plus embedding-path gradient",
      "id": "B03",
      "name": "tied_embedding.weight_gradient",
      "proof": "head 23 + 23 + ceil_log2(8192)=13 -> 59 <= 62; embedding path 23 + ceil_log2(8192)=13 -> 36 <= 62; head common exponent source bits=59 plus max(from_exp=-37-to_exp=-6,0)=0 -> 59 <= 62; embedding common exponent source bits=36 plus max(from_exp=-14-to_exp=-6,0)=0 -> 36 <= 62; checked add -> 60 <= 62; wte grad.bits=23",
      "rounding": "single ties-away gradient rescale",
      "source": "src/backward.rs::backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "final_norm.local_g_product",
          "registered_bits": 46,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "final_norm.local_g",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "final_norm.dgamma_accumulator",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "final_norm.gamma_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -10
          }
        }
      ],
      "carrier": "i64 product, checked i32 local value",
      "expression": "rescale(dy_j * gamma_j) to registered local g",
      "id": "B04",
      "name": "final_norm.local_g",
      "proof": "local product: 23 + 23 -> 46 <= 62; narrowed dgamma accumulation: 23 + 24 + ceil_log2(8192)=13 -> 60 <= 62; output bwd.d_ln_g.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::ln_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "final_norm_gsum",
          "registered_bits": 31,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "sum(local_g_j), j<d_model",
      "id": "B05",
      "name": "final_norm.row_gsum",
      "proof": "23 + ceil_log2(256)=8 -> 31 <= 62",
      "rounding": "round-to-nearest, ties away on mean division",
      "source": "src/backward.rs::ln_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "final_norm_bsum",
          "registered_bits": 61,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        }
      ],
      "carrier": "argued i128 LayerNorm row accumulator",
      "expression": "sum(local_g_j * normalized_j), j<d_model",
      "id": "B06",
      "name": "final_norm.row_bsum",
      "proof": "23 + 30 + ceil_log2(256)=8 -> 61 <= 126",
      "rounding": "round-to-nearest, ties away on mean division",
      "source": "src/backward.rs::ln_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "final_norm.normalized_bmean_product",
          "registered_bits": 83,
          "registered_exponent": {
            "kind": "fixed",
            "value": -68
          }
        },
        {
          "interval": "strict_signed",
          "name": "final_norm.post_q48_projection",
          "registered_bits": 28,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "final_norm.three_term",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "final_norm.inverse_product",
          "registered_bits": 60,
          "registered_exponent": {
            "formula": "m_plus_g_minus_x_minus_59",
            "kind": "runtime_ln_inv",
            "m_source": "final_norm_saved_m",
            "output_exponent": -14
          }
        },
        {
          "interval": "strict_signed",
          "name": "final_norm.dx",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -14
          }
        }
      ],
      "carrier": "argued i128 intermediate, checked i64 terms, then i32",
      "expression": "normalized-gradient formula for final norm input",
      "id": "B07",
      "name": "final_norm.dx",
      "proof": "normalized*bmean needs 83 <= 126; runtime normalized-energy premise sum(n^2)<=d*(2^24+1)^2; post-Q48 projection needs d_ln_g.bits=23 + conservative sqrt-width=5 -> 28 <= 62; three-term subtraction needs 30 <= 62; final term*inverse needs 60 <= 62; output bwd.d_resid.bits=23",
      "rounding": "round-to-nearest, ties away",
      "source": "src/backward.rs::ln_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_projection.common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_projection.common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_projection.common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_projection.common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_projection.common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "dY @ W^T, inner width d_model",
      "id": "B08",
      "name": "mlp_output.input_gradient",
      "proof": "23 + 23 + ceil_log2(256)=8 -> 54 <= 62; output bwd.d_mlp_h.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::matmul_bwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_projection.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -35
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_projection.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_projection.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -35
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_projection.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_projection.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -35
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_projection.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_projection.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -35
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_projection.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_projection.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -35
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_projection.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "X^T @ dY, inner width batch*ctx",
      "id": "B09",
      "name": "mlp_output.weight_gradient",
      "proof": "23 + 23 + ceil_log2(8192)=13 -> 59 <= 62; max grad.bits=23",
      "rounding": "single ties-away gradient rescale",
      "source": "src/backward.rs::matmul_bwd"
    },
    {
      "boundaries": [
        {
          "interval": "gelu_derivative_q29",
          "name": "gelu_derivative_q29",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -29
          }
        },
        {
          "interval": "strict_signed",
          "name": "derivative_product",
          "registered_bits": 53,
          "registered_exponent": {
            "kind": "fixed",
            "value": -48
          }
        },
        {
          "interval": "strict_signed",
          "name": "rounded_d_mlp_h",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -19
          }
        }
      ],
      "carrier": "i64 product, then i32",
      "expression": "dY * registered GELU derivative",
      "id": "B10",
      "name": "activation.derivative",
      "proof": "23 + 30 -> 53 <= 62; output bwd.d_mlp_h.bits=23",
      "rounding": "registered table and ties-away product rounding",
      "source": "src/gelu.rs::gelu_backward_at_site"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_input.accumulator",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -42
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_input.common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_input.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_input.accumulator",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -42
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_input.common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_input.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_input.accumulator",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -42
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_input.common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_input.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_input.accumulator",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -42
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_input.common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_input.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_input.accumulator",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -42
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_input.common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_input.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "dH @ W^T, inner width d_mlp",
      "id": "B11",
      "name": "mlp_input.input_gradient",
      "proof": "23 + 23 + ceil_log2(1024)=10 -> 56 <= 62; output bwd.d_ln2_out.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::matmul_bwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_input.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -36
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.mlp_input.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -11
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_input.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -36
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.mlp_input.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -12
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_input.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -36
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.mlp_input.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -11
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_input.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -36
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.mlp_input.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -11
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_input.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -36
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.mlp_input.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -12
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "X^T @ dH, inner width batch*ctx",
      "id": "B12",
      "name": "mlp_input.weight_gradient",
      "proof": "23 + 23 + ceil_log2(8192)=13 -> 59 <= 62; max grad.bits=23",
      "rounding": "single ties-away gradient rescale",
      "source": "src/backward.rs::matmul_bwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.norm2.local_g_product",
          "registered_bits": 46,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm2.local_g",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm2.dgamma_accumulator",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -38
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm2.gamma_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -13
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm2.local_g_product",
          "registered_bits": 46,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm2.local_g",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm2.dgamma_accumulator",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -38
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm2.gamma_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -13
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm2.local_g_product",
          "registered_bits": 46,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm2.local_g",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm2.dgamma_accumulator",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -38
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm2.gamma_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -13
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm2.local_g_product",
          "registered_bits": 46,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm2.local_g",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm2.dgamma_accumulator",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -38
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm2.gamma_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -13
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm2.local_g_product",
          "registered_bits": 46,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm2.local_g",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm2.dgamma_accumulator",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -38
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm2.gamma_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -13
          }
        }
      ],
      "carrier": "i64 product, checked i32 local value",
      "expression": "rescale(dy_j * gamma_j) to registered local g",
      "id": "B13",
      "name": "norm2.local_g",
      "proof": "local product: 23 + 23 -> 46 <= 62; narrowed dgamma accumulation: 23 + 24 + ceil_log2(8192)=13 -> 60 <= 62; output bwd.d_ln_g.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::ln_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "norm2.gsum",
          "registered_bits": 31,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "norm2.bsum",
          "registered_bits": 61,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        }
      ],
      "carrier": "i64 gsum and argued i128 LayerNorm bsum",
      "expression": "sum(g) and sum(g * normalized), j<d_model",
      "id": "B14",
      "name": "norm2.row_sums",
      "proof": "gsum 23 + ceil_log2(256)=8 -> 31 <= 62; bsum 23 + 30 + ceil_log2(256)=8 -> 61 <= 126",
      "rounding": "round-to-nearest, ties away on division",
      "source": "src/backward.rs::ln_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "norm2.normalized_bmean_product",
          "registered_bits": 83,
          "registered_exponent": {
            "kind": "fixed",
            "value": -68
          }
        },
        {
          "interval": "strict_signed",
          "name": "norm2.post_q48_projection",
          "registered_bits": 28,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "norm2.three_term",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "norm2.inverse_product",
          "registered_bits": 60,
          "registered_exponent": {
            "formula": "m_plus_g_minus_x_minus_59",
            "kind": "runtime_ln_inv",
            "m_source": "norm2_saved_m",
            "output_exponent": -14
          }
        },
        {
          "interval": "strict_signed",
          "name": "norm2.dx",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -14
          }
        }
      ],
      "carrier": "argued i128 intermediate, checked i64 terms, then i32",
      "expression": "normalized-gradient formula before residual-gradient add",
      "id": "B15",
      "name": "norm2.dx",
      "proof": "normalized*bmean 83 <= 126; runtime normalized-energy premise sum(n^2)<=d*(2^24+1)^2; post-Q48 projection 28 <= 62; three-term subtraction 30 <= 62; final product 60 <= 62; output bwd.d_resid.bits=23",
      "rounding": "round-to-nearest, ties away",
      "source": "src/backward.rs::ln_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.attention_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.attention_projection.common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.attention_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.attention_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.attention_projection.common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.attention_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.attention_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.attention_projection.common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.attention_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.attention_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.attention_projection.common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.attention_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.attention_projection.accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.attention_projection.common",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.attention_projection.output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "dY @ W^T, inner width d_model",
      "id": "B16",
      "name": "attention_output.input_gradient",
      "proof": "23 + 23 + ceil_log2(256)=8 -> 54 <= 62; output bwd.d_attn_av.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::matmul_bwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.attention_projection.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -34
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.attention_projection.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.attention_projection.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -34
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.attention_projection.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.attention_projection.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -34
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.attention_projection.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.attention_projection.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -34
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.attention_projection.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.attention_projection.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -34
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.attention_projection.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "X^T @ dY, inner width batch*ctx",
      "id": "B17",
      "name": "attention_output.weight_gradient",
      "proof": "23 + 23 + ceil_log2(8192)=13 -> 59 <= 62; max grad.bits=23",
      "rounding": "single ties-away gradient rescale",
      "source": "src/backward.rs::matmul_bwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "value_gradient_accumulator",
          "registered_bits": 62,
          "registered_exponent": {
            "kind": "fixed",
            "value": -48
          }
        },
        {
          "interval": "strict_signed",
          "name": "d_qkv_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        }
      ],
      "carrier": "i64 accumulator with inclusive Q30 broadcast operand",
      "expression": "sum_i(probability_ij * dA_iv), causal extent",
      "id": "B18",
      "name": "value.gradient",
      "proof": "30 + 23 + ceil_log2(512)=9 -> 62 <= 62; output bwd.d_qkv.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::attention_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "probability_gradient_accumulator",
          "registered_bits": 51,
          "registered_exponent": {
            "kind": "fixed",
            "value": -36
          }
        },
        {
          "interval": "strict_signed",
          "name": "d_p_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "dA_i dot V_j over head_dim",
      "id": "B19",
      "name": "probability.gradient",
      "proof": "23 + 23 + ceil_log2(32)=5 -> 51 <= 62; output bwd.d_p.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::attention_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "softmax_rowdot_accumulator",
          "registered_bits": 62,
          "registered_exponent": {
            "kind": "fixed",
            "value": -47
          }
        }
      ],
      "carrier": "i64 accumulator with inclusive Q30 operand",
      "expression": "sum_j(dP_j * probability_j) over causal row",
      "id": "B20",
      "name": "softmax.rowdot",
      "proof": "23 + 30 + ceil_log2(512)=9 -> 62 <= 62",
      "rounding": "round-to-nearest, ties away to row mean",
      "source": "src/backward.rs::attention_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "softmax_subtraction",
          "registered_bits": 24,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "softmax_product",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -47
          }
        },
        {
          "interval": "strict_signed",
          "name": "rounded_softmax_gradient",
          "registered_bits": 24,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        }
      ],
      "carrier": "i64 product, then checked value",
      "expression": "probability_j * (dP_j - rowdot_mean)",
      "id": "B21",
      "name": "softmax.gradient",
      "proof": "subtraction bits=24; product 30 + 24 -> 54 <= 62",
      "rounding": "round-to-nearest, ties away by Q30",
      "source": "src/backward.rs::attention_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "rsqrt_product",
          "registered_bits": 52,
          "registered_exponent": {
            "kind": "fixed",
            "value": -47
          }
        },
        {
          "interval": "strict_signed",
          "name": "rounded_score_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -17
          }
        },
        {
          "interval": "strict_signed",
          "name": "d_score_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -22
          }
        }
      ],
      "carrier": "i64 product, then i32",
      "expression": "softmax gradient * registered reciprocal-sqrt Q30 scale",
      "id": "B22",
      "name": "score.unscale",
      "proof": "rsqrt_head_dim_q30=189812531 has mag_bits=28; 24 + 28 -> 52 <= 62; output bwd.d_score.bits=23",
      "rounding": "round-to-nearest, ties away by Q30 and site rescale",
      "source": "src/backward.rs::attention_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "query_gradient_accumulator",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "d_qkv_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "dScore @ K over causal context extent",
      "id": "B23",
      "name": "query.gradient",
      "proof": "23 + 23 + ceil_log2(512)=9 -> 55 <= 62; output bwd.d_qkv.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::attention_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "key_gradient_accumulator",
          "registered_bits": 55,
          "registered_exponent": {
            "kind": "fixed",
            "value": -40
          }
        },
        {
          "interval": "strict_signed",
          "name": "d_qkv_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -18
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "dScore^T @ Q over causal context extent",
      "id": "B24",
      "name": "key.gradient",
      "proof": "23 + 23 + ceil_log2(512)=9 -> 55 <= 62; output bwd.d_qkv.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::attention_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.branch_accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.three_branch_add",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.combined_common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.d_ln1_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.branch_accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.three_branch_add",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.combined_common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.d_ln1_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.branch_accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.three_branch_add",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.combined_common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.d_ln1_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.branch_accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.three_branch_add",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.combined_common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.d_ln1_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.branch_accumulator",
          "registered_bits": 54,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.three_branch_add",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -41
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.combined_common",
          "registered_bits": 56,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.d_ln1_output",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "combined Q/K/V gradient times fused projection weights",
      "id": "B25",
      "name": "qkv.input_gradient",
      "proof": "each Q/K/V branch 23 + 23 + ceil_log2(256)=8 -> 54 <= 62; fixed three-branch add needs ceil_log2(3)=2 -> 56 <= 62; output bwd.d_ln1_out.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::matmul_bwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.qkv.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -35
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.qkv.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -10
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.qkv.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -35
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.qkv.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -11
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.qkv.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -35
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.qkv.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -11
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.qkv.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -35
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.qkv.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -11
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.qkv.gradient_accumulator",
          "registered_bits": 59,
          "registered_exponent": {
            "kind": "fixed",
            "value": -35
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.qkv.weight_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -11
          }
        }
      ],
      "carrier": "i64 accumulator",
      "expression": "X^T @ dQKV, inner width batch*ctx",
      "id": "B26",
      "name": "qkv.weight_gradient",
      "proof": "23 + 23 + ceil_log2(8192)=13 -> 59 <= 62; max grad.bits=23",
      "rounding": "single ties-away gradient rescale",
      "source": "src/backward.rs::matmul_bwd"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "layer_0.norm1.local_g_product",
          "registered_bits": 46,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm1.local_g",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm1.dgamma_accumulator",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -38
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_0.norm1.gamma_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -12
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm1.local_g_product",
          "registered_bits": 46,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm1.local_g",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm1.dgamma_accumulator",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -38
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_1.norm1.gamma_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -13
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm1.local_g_product",
          "registered_bits": 46,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm1.local_g",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm1.dgamma_accumulator",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -38
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_2.norm1.gamma_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -14
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm1.local_g_product",
          "registered_bits": 46,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm1.local_g",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm1.dgamma_accumulator",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -38
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_3.norm1.gamma_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -14
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm1.local_g_product",
          "registered_bits": 46,
          "registered_exponent": {
            "kind": "fixed",
            "value": -39
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm1.local_g",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm1.dgamma_accumulator",
          "registered_bits": 60,
          "registered_exponent": {
            "kind": "fixed",
            "value": -38
          }
        },
        {
          "interval": "strict_signed",
          "name": "layer_4.norm1.gamma_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -13
          }
        }
      ],
      "carrier": "i64 product, checked i32 local value",
      "expression": "rescale(dy_j * gamma_j) to registered local g",
      "id": "B27",
      "name": "norm1.local_g",
      "proof": "local product: 23 + 23 -> 46 <= 62; narrowed dgamma accumulation: 23 + 24 + ceil_log2(8192)=13 -> 60 <= 62; output bwd.d_ln_g.bits=23",
      "rounding": "single ties-away rescale",
      "source": "src/backward.rs::ln_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "norm1.gsum",
          "registered_bits": 31,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "norm1.bsum",
          "registered_bits": 61,
          "registered_exponent": {
            "kind": "fixed",
            "value": -44
          }
        }
      ],
      "carrier": "i64 gsum and argued i128 LayerNorm bsum",
      "expression": "sum(g) and sum(g * normalized), j<d_model",
      "id": "B28",
      "name": "norm1.row_sums",
      "proof": "gsum 23 + ceil_log2(256)=8 -> 31 <= 62; bsum 23 + 30 + ceil_log2(256)=8 -> 61 <= 126",
      "rounding": "round-to-nearest, ties away on division",
      "source": "src/backward.rs::ln_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "norm1.normalized_bmean_product",
          "registered_bits": 83,
          "registered_exponent": {
            "kind": "fixed",
            "value": -68
          }
        },
        {
          "interval": "strict_signed",
          "name": "norm1.post_q48_projection",
          "registered_bits": 28,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "norm1.three_term",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -20
          }
        },
        {
          "interval": "strict_signed",
          "name": "norm1.inverse_product",
          "registered_bits": 60,
          "registered_exponent": {
            "formula": "m_plus_g_minus_x_minus_59",
            "kind": "runtime_ln_inv",
            "m_source": "norm1_saved_m",
            "output_exponent": -14
          }
        },
        {
          "interval": "strict_signed",
          "name": "norm1.dx",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -14
          }
        }
      ],
      "carrier": "argued i128 intermediate, checked i64 terms, then i32",
      "expression": "normalized-gradient formula before embedding-gradient add",
      "id": "B29",
      "name": "norm1.dx",
      "proof": "normalized*bmean 83 <= 126; runtime normalized-energy premise sum(n^2)<=d*(2^24+1)^2; post-Q48 projection 28 <= 62; three-term subtraction 30 <= 62; final product 60 <= 62; output bwd.d_resid.bits=23",
      "rounding": "round-to-nearest, ties away",
      "source": "src/backward.rs::ln_backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "token.accumulator",
          "registered_bits": 36,
          "registered_exponent": {
            "kind": "fixed",
            "value": -14
          }
        },
        {
          "interval": "strict_signed",
          "name": "position.accumulator",
          "registered_bits": 36,
          "registered_exponent": {
            "kind": "fixed",
            "value": -14
          }
        },
        {
          "interval": "strict_signed",
          "name": "position.wpe_gradient",
          "registered_bits": 23,
          "registered_exponent": {
            "kind": "fixed",
            "value": -11
          }
        }
      ],
      "carrier": "two disjoint i64 accumulators",
      "expression": "token and position path gradient accumulation at disjoint destinations",
      "id": "B30",
      "name": "embedding.gradient_add",
      "proof": "token 23 + ceil_log2(8192)=13 -> 36 <= 62; position 23 + ceil_log2(8192)=13 -> 36 <= 62; disjoint destinations, max -> 36 <= 62; max grad.bits=23",
      "rounding": "single ties-away gradient rescale",
      "source": "src/backward.rs::backward"
    },
    {
      "boundaries": [
        {
          "interval": "strict_signed",
          "name": "h.0.attn.proj.bias.multiplier_product",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -7
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.attn.proj.bias.raw_momentum_add",
          "registered_bits": 31,
          "registered_exponent": {
            "kind": "fixed",
            "value": -7
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.attn.proj.bias.momentum",
          "registered_bits": 27,
          "registered_exponent": {
            "kind": "fixed",
            "value": -7
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.attn.proj.weight.multiplier_product",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.attn.proj.weight.raw_momentum_add",
          "registered_bits": 31,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.attn.proj.weight.momentum",
          "registered_bits": 27,
          "registered_exponent": {
            "kind": "fixed",
            "value": -9
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.attn.qkv.bias.multiplier_product",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -10
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.attn.qkv.bias.raw_momentum_add",
          "registered_bits": 31,
          "registered_exponent": {
            "kind": "fixed",
            "value": -10
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.attn.qkv.bias.momentum",
          "registered_bits": 27,
          "registered_exponent": {
            "kind": "fixed",
            "value": -10
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.attn.qkv.weight.multiplier_product",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -10
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.attn.qkv.weight.raw_momentum_add",
          "registered_bits": 31,
          "registered_exponent": {
            "kind": "fixed",
            "value": -10
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.attn.qkv.weight.momentum",
          "registered_bits": 27,
          "registered_exponent": {
            "kind": "fixed",
            "value": -10
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.ln1.bias.multiplier_product",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -12
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.ln1.bias.raw_momentum_add",
          "registered_bits": 31,
          "registered_exponent": {
            "kind": "fixed",
            "value": -12
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.ln1.bias.momentum",
          "registered_bits": 27,
          "registered_exponent": {
            "kind": "fixed",
            "value": -12
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.ln1.weight.multiplier_product",
          "registered_bits": 30,
          "registered_exponent": {
            "kind": "fixed",
            "value": -12
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.ln1.weight.raw_momentum_add",
          "registered_bits": 31,
          "registered_exponent": {
            "kind": "fixed",
            "value": -12
          }
        },
        {
          "interval": "strict_signed",
          "name": "h.0.ln1.weight.momentum",
          "registered_bits": 27,
          "registered_exponent": {
            "kind": "fixed",
            "value": -12
          }
        },
        {
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        {
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        {
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          "registered_exponent": {
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          "registered_exponent": {
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        },
        {
          "interval": "strict_signed",
          "name": "wte.subtraction",
          "registered_bits": 52,
          "registered_exponent": {
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        {
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          "name": "wte.weight",
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          "registered_exponent": {
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        }
      ],
      "carrier": "checked i64 product/subtraction, then i32",
      "expression": "weight - round(lr_num * momentum / lr_den)",
      "id": "B32",
      "name": "weight.update",
      "proof": "optimizer=sgd-momentum-7-8; h.0.attn.proj.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.0.attn.proj.weight tensor numerator=13107200 needs ceil_log2(constant)=24; product with momentum.bits=27 -> 51 <= 62; subtraction max(weight.bits=23,delta.bits=51)+1 -> 52 <= 62; output weight.bits=23; h.0.attn.qkv.bias tensor numerator=26214400 needs ceil_log2(constant)=25; product with momentum.bits=27 -> 52 <= 62; subtraction max(weight.bits=23,delta.bits=52)+1 -> 53 <= 62; output weight.bits=23; h.0.attn.qkv.weight tensor numerator=1638400 needs ceil_log2(constant)=21; product with momentum.bits=27 -> 48 <= 62; subtraction max(weight.bits=23,delta.bits=48)+1 -> 49 <= 62; output weight.bits=23; h.0.ln1.bias tensor numerator=13107200 needs ceil_log2(constant)=24; product with momentum.bits=27 -> 51 <= 62; subtraction max(weight.bits=23,delta.bits=51)+1 -> 52 <= 62; output weight.bits=23; h.0.ln1.weight tensor numerator=25600 needs ceil_log2(constant)=15; product with momentum.bits=27 -> 42 <= 62; subtraction max(weight.bits=23,delta.bits=42)+1 -> 43 <= 62; output weight.bits=23; h.0.ln2.bias tensor numerator=104857600 needs ceil_log2(constant)=27; product with momentum.bits=27 -> 54 <= 62; subtraction max(weight.bits=23,delta.bits=54)+1 -> 55 <= 62; output weight.bits=23; h.0.ln2.weight tensor numerator=12800 needs ceil_log2(constant)=14; product with momentum.bits=27 -> 41 <= 62; subtraction max(weight.bits=23,delta.bits=41)+1 -> 42 <= 62; output weight.bits=23; h.0.mlp.fc.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.0.mlp.fc.weight tensor numerator=819200 needs ceil_log2(constant)=20; product with momentum.bits=27 -> 47 <= 62; subtraction max(weight.bits=23,delta.bits=47)+1 -> 48 <= 62; output weight.bits=23; h.0.mlp.proj.bias tensor numerator=104857600 needs ceil_log2(constant)=27; product with momentum.bits=27 -> 54 <= 62; subtraction max(weight.bits=23,delta.bits=54)+1 -> 55 <= 62; output weight.bits=23; h.0.mlp.proj.weight tensor numerator=13107200 needs ceil_log2(constant)=24; product with momentum.bits=27 -> 51 <= 62; subtraction max(weight.bits=23,delta.bits=51)+1 -> 52 <= 62; output weight.bits=23; h.1.attn.proj.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.1.attn.proj.weight tensor numerator=13107200 needs ceil_log2(constant)=24; product with momentum.bits=27 -> 51 <= 62; subtraction max(weight.bits=23,delta.bits=51)+1 -> 52 <= 62; output weight.bits=23; h.1.attn.qkv.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.1.attn.qkv.weight tensor numerator=819200 needs ceil_log2(constant)=20; product with momentum.bits=27 -> 47 <= 62; subtraction max(weight.bits=23,delta.bits=47)+1 -> 48 <= 62; output weight.bits=23; h.1.ln1.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.1.ln1.weight tensor numerator=12800 needs ceil_log2(constant)=14; product with momentum.bits=27 -> 41 <= 62; subtraction max(weight.bits=23,delta.bits=41)+1 -> 42 <= 62; output weight.bits=23; h.1.ln2.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.1.ln2.weight tensor numerator=12800 needs ceil_log2(constant)=14; product with momentum.bits=27 -> 41 <= 62; subtraction max(weight.bits=23,delta.bits=41)+1 -> 42 <= 62; output weight.bits=23; h.1.mlp.fc.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.1.mlp.fc.weight tensor numerator=409600 needs ceil_log2(constant)=19; product with momentum.bits=27 -> 46 <= 62; subtraction max(weight.bits=23,delta.bits=46)+1 -> 47 <= 62; output weight.bits=23; h.1.mlp.proj.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.1.mlp.proj.weight tensor numerator=13107200 needs ceil_log2(constant)=24; product with momentum.bits=27 -> 51 <= 62; subtraction max(weight.bits=23,delta.bits=51)+1 -> 52 <= 62; output weight.bits=23; h.2.attn.proj.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.2.attn.proj.weight tensor numerator=6553600 needs ceil_log2(constant)=23; product with momentum.bits=27 -> 50 <= 62; subtraction max(weight.bits=23,delta.bits=50)+1 -> 51 <= 62; output weight.bits=23; h.2.attn.qkv.bias tensor numerator=104857600 needs ceil_log2(constant)=27; product with momentum.bits=27 -> 54 <= 62; subtraction max(weight.bits=23,delta.bits=54)+1 -> 55 <= 62; output weight.bits=23; h.2.attn.qkv.weight tensor numerator=819200 needs ceil_log2(constant)=20; product with momentum.bits=27 -> 47 <= 62; subtraction max(weight.bits=23,delta.bits=47)+1 -> 48 <= 62; output weight.bits=23; h.2.ln1.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.2.ln1.weight tensor numerator=6400 needs ceil_log2(constant)=13; product with momentum.bits=27 -> 40 <= 62; subtraction max(weight.bits=23,delta.bits=40)+1 -> 41 <= 62; output weight.bits=23; h.2.ln2.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.2.ln2.weight tensor numerator=12800 needs ceil_log2(constant)=14; product with momentum.bits=27 -> 41 <= 62; subtraction max(weight.bits=23,delta.bits=41)+1 -> 42 <= 62; output weight.bits=23; h.2.mlp.fc.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.2.mlp.fc.weight tensor numerator=819200 needs ceil_log2(constant)=20; product with momentum.bits=27 -> 47 <= 62; subtraction max(weight.bits=23,delta.bits=47)+1 -> 48 <= 62; output weight.bits=23; h.2.mlp.proj.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.2.mlp.proj.weight tensor numerator=13107200 needs ceil_log2(constant)=24; product with momentum.bits=27 -> 51 <= 62; subtraction max(weight.bits=23,delta.bits=51)+1 -> 52 <= 62; output weight.bits=23; h.3.attn.proj.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.3.attn.proj.weight tensor numerator=13107200 needs ceil_log2(constant)=24; product with momentum.bits=27 -> 51 <= 62; subtraction max(weight.bits=23,delta.bits=51)+1 -> 52 <= 62; output weight.bits=23; h.3.attn.qkv.bias tensor numerator=104857600 needs ceil_log2(constant)=27; product with momentum.bits=27 -> 54 <= 62; subtraction max(weight.bits=23,delta.bits=54)+1 -> 55 <= 62; output weight.bits=23; h.3.attn.qkv.weight tensor numerator=819200 needs ceil_log2(constant)=20; product with momentum.bits=27 -> 47 <= 62; subtraction max(weight.bits=23,delta.bits=47)+1 -> 48 <= 62; output weight.bits=23; h.3.ln1.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.3.ln1.weight tensor numerator=6400 needs ceil_log2(constant)=13; product with momentum.bits=27 -> 40 <= 62; subtraction max(weight.bits=23,delta.bits=40)+1 -> 41 <= 62; output weight.bits=23; h.3.ln2.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.3.ln2.weight tensor numerator=12800 needs ceil_log2(constant)=14; product with momentum.bits=27 -> 41 <= 62; subtraction max(weight.bits=23,delta.bits=41)+1 -> 42 <= 62; output weight.bits=23; h.3.mlp.fc.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.3.mlp.fc.weight tensor numerator=819200 needs ceil_log2(constant)=20; product with momentum.bits=27 -> 47 <= 62; subtraction max(weight.bits=23,delta.bits=47)+1 -> 48 <= 62; output weight.bits=23; h.3.mlp.proj.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.3.mlp.proj.weight tensor numerator=13107200 needs ceil_log2(constant)=24; product with momentum.bits=27 -> 51 <= 62; subtraction max(weight.bits=23,delta.bits=51)+1 -> 52 <= 62; output weight.bits=23; h.4.attn.proj.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.4.attn.proj.weight tensor numerator=13107200 needs ceil_log2(constant)=24; product with momentum.bits=27 -> 51 <= 62; subtraction max(weight.bits=23,delta.bits=51)+1 -> 52 <= 62; output weight.bits=23; h.4.attn.qkv.bias tensor numerator=26214400 needs ceil_log2(constant)=25; product with momentum.bits=27 -> 52 <= 62; subtraction max(weight.bits=23,delta.bits=52)+1 -> 53 <= 62; output weight.bits=23; h.4.attn.qkv.weight tensor numerator=819200 needs ceil_log2(constant)=20; product with momentum.bits=27 -> 47 <= 62; subtraction max(weight.bits=23,delta.bits=47)+1 -> 48 <= 62; output weight.bits=23; h.4.ln1.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.4.ln1.weight tensor numerator=12800 needs ceil_log2(constant)=14; product with momentum.bits=27 -> 41 <= 62; subtraction max(weight.bits=23,delta.bits=41)+1 -> 42 <= 62; output weight.bits=23; h.4.ln2.bias tensor numerator=26214400 needs ceil_log2(constant)=25; product with momentum.bits=27 -> 52 <= 62; subtraction max(weight.bits=23,delta.bits=52)+1 -> 53 <= 62; output weight.bits=23; h.4.ln2.weight tensor numerator=12800 needs ceil_log2(constant)=14; product with momentum.bits=27 -> 41 <= 62; subtraction max(weight.bits=23,delta.bits=41)+1 -> 42 <= 62; output weight.bits=23; h.4.mlp.fc.bias tensor numerator=26214400 needs ceil_log2(constant)=25; product with momentum.bits=27 -> 52 <= 62; subtraction max(weight.bits=23,delta.bits=52)+1 -> 53 <= 62; output weight.bits=23; h.4.mlp.fc.weight tensor numerator=409600 needs ceil_log2(constant)=19; product with momentum.bits=27 -> 46 <= 62; subtraction max(weight.bits=23,delta.bits=46)+1 -> 47 <= 62; output weight.bits=23; h.4.mlp.proj.bias tensor numerator=52428800 needs ceil_log2(constant)=26; product with momentum.bits=27 -> 53 <= 62; subtraction max(weight.bits=23,delta.bits=53)+1 -> 54 <= 62; output weight.bits=23; h.4.mlp.proj.weight tensor numerator=13107200 needs ceil_log2(constant)=24; product with momentum.bits=27 -> 51 <= 62; subtraction max(weight.bits=23,delta.bits=51)+1 -> 52 <= 62; output weight.bits=23; ln_f.bias tensor numerator=819200 needs ceil_log2(constant)=20; product with momentum.bits=27 -> 47 <= 62; subtraction max(weight.bits=23,delta.bits=47)+1 -> 48 <= 62; output weight.bits=23; ln_f.weight tensor numerator=102400 needs ceil_log2(constant)=17; product with momentum.bits=27 -> 44 <= 62; subtraction max(weight.bits=23,delta.bits=44)+1 -> 45 <= 62; output weight.bits=23; wpe tensor numerator=819200 needs ceil_log2(constant)=20; product with momentum.bits=27 -> 47 <= 62; subtraction max(weight.bits=23,delta.bits=47)+1 -> 48 <= 62; output weight.bits=23; wte tensor numerator=13107200 needs ceil_log2(constant)=24; product with momentum.bits=27 -> 51 <= 62; subtraction max(weight.bits=23,delta.bits=51)+1 -> 52 <= 62; output weight.bits=23",
      "rounding": "round-to-nearest, ties away",
      "source": "src/optim.rs::sgd_step_at_sites"
    }
  ]
}