{
  "domain": {
    "N": 1000,
    "K": 40,
    "bases": [
      2,
      3,
      5,
      8,
      10,
      12,
      16
    ],
    "templates": [
      "AM",
      "MA",
      "MS",
      "MM",
      "MD"
    ]
  },
  "baseline": {
    "histories": 161582,
    "family_labeled_incidence": 40111,
    "numeric_incidence": 38506,
    "strong_points": 1547,
    "family_pair_witnesses": 1663,
    "triple_points": 58,
    "z891_histories": 118,
    "max_mid": 1600,
    "outside_histories": 1551
  },
  "base10_level": {
    "projection": "base10_prefix0",
    "occupied_cells": 14,
    "family_pair_candidates": 38,
    "family_pair_true_cells": 15,
    "family_pair_false_cells": 23,
    "false_only_multifamily_cells": 4,
    "raw_cross_family_position_pairings": 225950731,
    "projection_collision_load": 225949068,
    "exact_witnesses_constant": 1663,
    "pair_purity": 7.360011594740094e-06
  },
  "base10_result_centered": {
    "projection": "base10_anchor_exact_z",
    "occupied_cells": 2111,
    "family_pair_candidates": 2888,
    "family_pair_true_cells": 595,
    "family_pair_false_cells": 2293,
    "false_only_multifamily_cells": 982,
    "raw_cross_family_position_pairings": 323213,
    "projection_collision_load": 321550,
    "exact_witnesses_constant": 1663,
    "pair_purity": 0.005145213837314712
  },
  "base10_prefix": [
    {
      "projection": "base10_prefix0",
      "occupied_cells": 14,
      "family_pair_candidates": 38,
      "family_pair_true_cells": 15,
      "family_pair_false_cells": 23,
      "false_only_multifamily_cells": 4,
      "raw_cross_family_position_pairings": 225950731,
      "projection_collision_load": 225949068,
      "exact_witnesses_constant": 1663,
      "pair_purity": 7.360011594740094e-06
    },
    {
      "projection": "base10_prefix1",
      "occupied_cells": 504,
      "family_pair_candidates": 279,
      "family_pair_true_cells": 248,
      "family_pair_false_cells": 31,
      "false_only_multifamily_cells": 25,
      "raw_cross_family_position_pairings": 14632396,
      "projection_collision_load": 14630733,
      "exact_witnesses_constant": 1663,
      "pair_purity": 0.00011365192686146548
    },
    {
      "projection": "base10_prefix2",
      "occupied_cells": 10032,
      "family_pair_candidates": 1725,
      "family_pair_true_cells": 1638,
      "family_pair_false_cells": 87,
      "false_only_multifamily_cells": 87,
      "raw_cross_family_position_pairings": 21798,
      "projection_collision_load": 20135,
      "exact_witnesses_constant": 1663,
      "pair_purity": 0.07629140288099825
    },
    {
      "projection": "base10_prefix3",
      "occupied_cells": 38399,
      "family_pair_candidates": 1663,
      "family_pair_true_cells": 1663,
      "family_pair_false_cells": 0,
      "false_only_multifamily_cells": 0,
      "raw_cross_family_position_pairings": 1663,
      "projection_collision_load": 0,
      "exact_witnesses_constant": 1663,
      "pair_purity": 1.0
    },
    {
      "projection": "base10_prefix4",
      "occupied_cells": 38506,
      "family_pair_candidates": 1663,
      "family_pair_true_cells": 1663,
      "family_pair_false_cells": 0,
      "false_only_multifamily_cells": 0,
      "raw_cross_family_position_pairings": 1663,
      "projection_collision_load": 0,
      "exact_witnesses_constant": 1663,
      "pair_purity": 1.0
    }
  ],
  "seven_base_joint": {
    "projection": "joint_seven_base_levels",
    "occupied_cells": 438,
    "family_pair_candidates": 252,
    "family_pair_true_cells": 214,
    "family_pair_false_cells": 38,
    "false_only_multifamily_cells": 30,
    "raw_cross_family_position_pairings": 30351416,
    "projection_collision_load": 30349753,
    "exact_witnesses_constant": 1663,
    "pair_purity": 5.4791512857258456e-05
  },
  "exact": {
    "projection": "exact_values",
    "occupied_cells": 38506,
    "family_pair_candidates": 1663,
    "family_pair_true_cells": 1663,
    "family_pair_false_cells": 0,
    "false_only_multifamily_cells": 0,
    "raw_cross_family_position_pairings": 1663,
    "projection_collision_load": 0,
    "exact_witnesses_constant": 1663,
    "pair_purity": 1.0
  },
  "false_coalescence_891": {
    "result": 891,
    "MA_mid": 870,
    "MS_mid": 900,
    "q10_level_both": [
      2,
      2
    ],
    "MA_example": {
      "template": "MA",
      "a": 29,
      "b": 30,
      "c": 21,
      "m": 870,
      "z": 891
    },
    "MS_example": {
      "template": "MS",
      "a": 25,
      "b": 36,
      "c": 9,
      "m": 900,
      "z": 891
    },
    "genuine_891_bridges": [
      [
        27,
        891
      ],
      [
        33,
        891
      ]
    ]
  },
  "interpretation_notes": [
    "projection_collision_load counts family-ordered cross-family pairs of distinct exact positions within one projected cell",
    "candidate cell-pair false counts are not monotone under refinement",
    "all counts depend on N=1000 K=40 and five fixed templates; no inference to arbitrary real arithmetic",
    "exact incidence is radix invariant under bijective numeral encoding, but tower-level projection is not injective"
  ]
}
PASS: direct/factorized enumeration; false confluence signatures; refinement monotonicity; cross-radix conditions
