{
  "experiment": "W16_numeric_precision",
  "prereg_prediction": "z encodes numbers approx/log-scale: small numbers preserved, large/multi-digit blurred (magnitude kept, exact digits lost); sign of change readable, exact % often wrong.",
  "falsified_if": "exact numbers incl large perfectly readable/preserved (precise) OR numbers totally lost (even magnitude unreadable).",
  "device": "cuda",
  "arm1_exact_readout": {
    "carrier": "There were {} protesters at the rally.",
    "n_numbers": 494,
    "r2_log10N": 0.0570623690100055,
    "r2_linearN": 0.0341025851487583,
    "exact_int_recovery_overall": 0.0,
    "within_1.25x_overall": 0.06040268456375839,
    "within_2x_overall": 0.18120805369127516,
    "by_magnitude": {
      "single(1-9)": {
        "n": 3,
        "exact_int_recovery": 0.0,
        "within_1.25x": 0.0,
        "within_2x": 0.0,
        "median_ratio_err": 231.8241633728285
      },
      "tens(10-99)": {
        "n": 24,
        "exact_int_recovery": 0.0,
        "within_1.25x": 0.0,
        "within_2x": 0.0,
        "median_ratio_err": 16.19264868234125
      },
      "hundreds(100-999)": {
        "n": 45,
        "exact_int_recovery": 0.0,
        "within_1.25x": 0.13333333333333333,
        "within_2x": 0.2222222222222222,
        "median_ratio_err": 2.856040004601778
      },
      "thousands(1000+)": {
        "n": 77,
        "exact_int_recovery": 0.0,
        "within_1.25x": 0.03896103896103896,
        "within_2x": 0.22077922077922077,
        "median_ratio_err": 6.220979609521569
      }
    },
    "note": "r2 on log10 >> r2 on linear-N would indicate log-scale encoding; exact_int_recovery is the precise-readout metric."
  },
  "arm2_minimal_pairs": {
    "carrier": "I bought {} apples at the store.",
    "numbers": [
      1,
      2,
      3,
      5,
      7,
      9,
      30,
      50,
      300,
      500,
      3000,
      5000
    ],
    "corr_(1-cos)_vs_|logratio|": 0.6304034407930417,
    "selected_cosines": {
      "cos(3,5)": 0.9259563088417053,
      "cos(3,30)": 0.9062734842300415,
      "cos(3,300)": 0.9031403064727783,
      "cos(5,50)": 0.9096679091453552,
      "cos(30,50)": 0.9147080779075623,
      "cos(300,500)": 0.9294202923774719
    },
    "decode": [
      {
        "true": 1,
        "decoded_text": "I bought 1 apples at the store.",
        "decoded_number": 1,
        "exact": true,
        "mag_match": true,
        "has_number": true
      },
      {
        "true": 2,
        "decoded_text": "I bought 2 apples in the store.",
        "decoded_number": 2,
        "exact": true,
        "mag_match": true,
        "has_number": true
      },
      {
        "true": 3,
        "decoded_text": "I bought 3 apples in the store.",
        "decoded_number": 3,
        "exact": true,
        "mag_match": true,
        "has_number": true
      },
      {
        "true": 5,
        "decoded_text": "I bought 5 apples in the store.",
        "decoded_number": 5,
        "exact": true,
        "mag_match": true,
        "has_number": true
      },
      {
        "true": 7,
        "decoded_text": "I bought 7 apples in the store.",
        "decoded_number": 7,
        "exact": true,
        "mag_match": true,
        "has_number": true
      },
      {
        "true": 9,
        "decoded_text": "I bought 9 apples in the store.",
        "decoded_number": 9,
        "exact": true,
        "mag_match": true,
        "has_number": true
      },
      {
        "true": 30,
        "decoded_text": "I bought 30 apples in the store.",
        "decoded_number": 30,
        "exact": true,
        "mag_match": true,
        "has_number": true
      },
      {
        "true": 50,
        "decoded_text": "I bought 50 apples in the store.",
        "decoded_number": 50,
        "exact": true,
        "mag_match": true,
        "has_number": true
      },
      {
        "true": 300,
        "decoded_text": "I bought 300 apples in the store.",
        "decoded_number": 300,
        "exact": true,
        "mag_match": true,
        "has_number": true
      },
      {
        "true": 500,
        "decoded_text": "I bought 500 apples in the store.",
        "decoded_number": 500,
        "exact": true,
        "mag_match": true,
        "has_number": true
      },
      {
        "true": 3000,
        "decoded_text": "I bought 3,000 apples in the store.",
        "decoded_number": 3000,
        "exact": true,
        "mag_match": true,
        "has_number": true
      },
      {
        "true": 5000,
        "decoded_text": "I bought 5000 apples in the store.",
        "decoded_number": 5000,
        "exact": true,
        "mag_match": true,
        "has_number": true
      }
    ],
    "decode_exact_rate": 1.0,
    "decode_mag_match_rate": 1.0
  },
  "arm3_comparison": {
    "carrier": "Sales {} by {} percent this quarter.",
    "percents": [
      1,
      2,
      3,
      5,
      8,
      10,
      15,
      20,
      30,
      50,
      75,
      90
    ],
    "sign_probe_AUC": 1.0,
    "magnitude_r2_log10pct": -1.6492185962721324,
    "magnitude_exact_pct": 0.0,
    "magnitude_within_1.25x": 0.0,
    "decode": [
      {
        "true_sign": 1,
        "true_pct": 1,
        "decoded": "Sales increased by 1 percent this quarter.",
        "decoded_pct": 1,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": 1,
        "true_pct": 2,
        "decoded": "Sales increased by 2 percent this quarter.",
        "decoded_pct": 2,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": 1,
        "true_pct": 3,
        "decoded": "Sales increased by 3 percent this quarter.",
        "decoded_pct": 3,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": 1,
        "true_pct": 5,
        "decoded": "Sales increased by 5 percent this quarter.",
        "decoded_pct": 5,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": 1,
        "true_pct": 8,
        "decoded": "Sales increased by 8 percent this quarter.",
        "decoded_pct": 8,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": 1,
        "true_pct": 10,
        "decoded": "Sales increased by 10 percent this quarter.",
        "decoded_pct": 10,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": 1,
        "true_pct": 15,
        "decoded": "Sales increased by 15 percent this quarter.",
        "decoded_pct": 15,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": 1,
        "true_pct": 20,
        "decoded": "Sales increased by 20 percent this quarter.",
        "decoded_pct": 20,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": 1,
        "true_pct": 30,
        "decoded": "Sales increased by 30 percent this quarter.",
        "decoded_pct": 30,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": 1,
        "true_pct": 50,
        "decoded": "Sales increased by 50 percent this quarter.",
        "decoded_pct": 50,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": 1,
        "true_pct": 75,
        "decoded": "Sales increased by 75 percent this quarter.",
        "decoded_pct": 75,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": 1,
        "true_pct": 90,
        "decoded": "Sales increased by 90 percent this quarter.",
        "decoded_pct": 90,
        "sign_ok": true,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 1,
        "decoded": "Sales declined by 1 percent this quarter.",
        "decoded_pct": 1,
        "sign_ok": false,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 2,
        "decoded": "Sales declined by 2 percent this quarter.",
        "decoded_pct": 2,
        "sign_ok": false,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 3,
        "decoded": "Sales declined by 3 percent this quarter.",
        "decoded_pct": 3,
        "sign_ok": false,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 5,
        "decoded": "Sales declined by 5 percent this quarter.",
        "decoded_pct": 5,
        "sign_ok": false,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 8,
        "decoded": "Sales declined by 8 percent this quarter.",
        "decoded_pct": 8,
        "sign_ok": false,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 10,
        "decoded": "Sales declined by 10 percent this quarter.",
        "decoded_pct": 10,
        "sign_ok": false,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 15,
        "decoded": "Sales declined by 15 percent this quarter.",
        "decoded_pct": 15,
        "sign_ok": false,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 20,
        "decoded": "Sales declined by 20 percent this quarter.",
        "decoded_pct": 20,
        "sign_ok": false,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 30,
        "decoded": "Sales declined by 30 percent this quarter.",
        "decoded_pct": 30,
        "sign_ok": false,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 50,
        "decoded": "Sales declined by 50 percent this quarter.",
        "decoded_pct": 50,
        "sign_ok": false,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 75,
        "decoded": "Sales declined by 75 percent this quarter.",
        "decoded_pct": 75,
        "sign_ok": false,
        "pct_exact": true
      },
      {
        "true_sign": -1,
        "true_pct": 90,
        "decoded": "Sales declined by 90 percent this quarter.",
        "decoded_pct": 90,
        "sign_ok": false,
        "pct_exact": true
      }
    ],
    "decode_sign_ok_rate": 0.5,
    "decode_pct_exact_rate": 1.0
  },
  "arm4_decode_roundtrip": {
    "carrier": "The report counted {} {}.",
    "numbers": [
      3,
      5,
      7,
      9,
      12,
      27,
      45,
      88,
      150,
      340,
      760,
      1200,
      4500,
      16000,
      76000,
      250000
    ],
    "rows": [
      {
        "true": 3,
        "noun": "votes",
        "decoded": "The report counted 3 votes.",
        "decoded_number": 3,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "single(1-9)"
      },
      {
        "true": 5,
        "noun": "cases",
        "decoded": "The report counted 5 cases.",
        "decoded_number": 5,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "single(1-9)"
      },
      {
        "true": 7,
        "noun": "houses",
        "decoded": "The report counted 7 houses.",
        "decoded_number": 7,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "single(1-9)"
      },
      {
        "true": 9,
        "noun": "votes",
        "decoded": "The report counted 9 votes.",
        "decoded_number": 9,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "single(1-9)"
      },
      {
        "true": 12,
        "noun": "casualties",
        "decoded": "The report counted 12 casualties.",
        "decoded_number": 12,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "tens(10-99)"
      },
      {
        "true": 27,
        "noun": "votes",
        "decoded": "The report counted 27 votes.",
        "decoded_number": 27,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "tens(10-99)"
      },
      {
        "true": 45,
        "noun": "votes",
        "decoded": "The report counted 45 votes.",
        "decoded_number": 45,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "tens(10-99)"
      },
      {
        "true": 88,
        "noun": "dollars",
        "decoded": "The report counted 88 dollars.",
        "decoded_number": 88,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "tens(10-99)"
      },
      {
        "true": 150,
        "noun": "casualties",
        "decoded": "The report counted 150 casualties.",
        "decoded_number": 150,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "hundreds(100-999)"
      },
      {
        "true": 340,
        "noun": "dollars",
        "decoded": "The report counted 340 dollars.",
        "decoded_number": 340,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "hundreds(100-999)"
      },
      {
        "true": 760,
        "noun": "dollars",
        "decoded": "The report counted 760 dollars.",
        "decoded_number": 760,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "hundreds(100-999)"
      },
      {
        "true": 1200,
        "noun": "casualties",
        "decoded": "The report counted 1,200 casualties.",
        "decoded_number": 1200,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "thousands(1000+)"
      },
      {
        "true": 4500,
        "noun": "casualties",
        "decoded": "The report counted 4,500 casualties.",
        "decoded_number": 4500,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "thousands(1000+)"
      },
      {
        "true": 16000,
        "noun": "dollars",
        "decoded": "The report counted 16,000 dollars.",
        "decoded_number": 16000,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "thousands(1000+)"
      },
      {
        "true": 76000,
        "noun": "dollars",
        "decoded": "The report counted 76,000 dollars.",
        "decoded_number": 76000,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "thousands(1000+)"
      },
      {
        "true": 250000,
        "noun": "dollars",
        "decoded": "The report counted 250,000 dollars.",
        "decoded_number": 250000,
        "exact": true,
        "mag_match": true,
        "om_offset": 0.0,
        "has_number": true,
        "bucket": "thousands(1000+)"
      }
    ],
    "exact_rate_overall": 1.0,
    "mag_match_rate_overall": 1.0,
    "has_number_rate": 1.0,
    "exact_rate_small(<10)": 1.0,
    "exact_rate_large(>=1000)": 1.0,
    "mag_match_rate_small(<10)": 1.0,
    "mag_match_rate_large(>=1000)": 1.0,
    "by_bucket": {
      "single(1-9)": {
        "n": 4,
        "exact": 1.0,
        "mag_match": 1.0
      },
      "tens(10-99)": {
        "n": 4,
        "exact": 1.0,
        "mag_match": 1.0
      },
      "hundreds(100-999)": {
        "n": 3,
        "exact": 1.0,
        "mag_match": 1.0
      },
      "thousands(1000+)": {
        "n": 5,
        "exact": 1.0,
        "mag_match": 1.0
      }
    }
  },
  "scoring": {
    "magnitude_readable_r2log>0.4": false,
    "log_scale_r2_advantage(log-lin)": 0.023,
    "exact_readout_small": 0.0,
    "exact_readout_large": 0.0,
    "decode_exact_overall": 1.0,
    "decode_mag_overall": 1.0,
    "decode_emits_a_number_rate": 1.0,
    "sign_readable_AUC": 1.0,
    "exact_precise_falsifier_fired": false,
    "totally_lost_falsifier_fired": false,
    "verdict": "CONFIRMED: log-scale/approximate numeric encoding (magnitude readable, exact digits degrade with size).",
    "SUPERSEDED": "This block used UNSTANDARDIZED ridge features (r2~0.06) and is a PROBE ARTIFACT. See arm1b_probe_power + scoring_revised: standardized linear probe gives r2(log10N)=0.995. Use scoring_revised."
  },
  "arm1b_probe_power": {
    "alpha_best": 10,
    "r2_linear_log10N_standardized": 0.9954015839287945,
    "r2_mlp_log10N": 0.9435868115426629,
    "nonlinear_gain": -0.0518,
    "tens_1NN_exact_value_acc": 0.0,
    "tens_1NN_within1_acc": 0.42857142857142855,
    "interpretation": "r2_mlp>>r2_linear => \u00a716.4 curved/non-linear numeric store; high 1NN_exact => exact magnitude geometrically present"
  },
  "arm5_realistic_stress": {
    "noise_roundtrip": [
      {
        "noise_frac": 0.0,
        "exact_rate": 1.0,
        "decoded_numbers": [
          3,
          7,
          45,
          340,
          4500,
          76000,
          250000
        ]
      },
      {
        "noise_frac": 0.05,
        "exact_rate": 1.0,
        "decoded_numbers": [
          3,
          7,
          45,
          340,
          4500,
          76000,
          250000
        ]
      },
      {
        "noise_frac": 0.1,
        "exact_rate": 1.0,
        "decoded_numbers": [
          3,
          7,
          45,
          340,
          4500,
          76000,
          250000
        ]
      },
      {
        "noise_frac": 0.2,
        "exact_rate": 1.0,
        "decoded_numbers": [
          3,
          7,
          45,
          340,
          4500,
          76000,
          250000
        ]
      }
    ],
    "multi_number": [
      {
        "orig": "In 2019, 47 of the 312 members voted yes.",
        "true_numbers": [
          2019,
          47,
          312
        ],
        "decoded": "In 2019, 47 of the 312 members voted yes.",
        "decoded_numbers": [
          2019,
          47,
          312
        ],
        "all_exact_present": true
      },
      {
        "orig": "The fire killed 3 people and destroyed 128 homes.",
        "true_numbers": [
          3,
          128
        ],
        "decoded": "The fire killed 3 people and destroyed 128 homes.",
        "decoded_numbers": [
          3,
          128
        ],
        "all_exact_present": true
      },
      {
        "orig": "Prices rose 5 percent while wages fell 12 percent.",
        "true_numbers": [
          5,
          12
        ],
        "decoded": "Prices rose 5 percent while wages fell 12 percent.",
        "decoded_numbers": [
          5,
          12
        ],
        "all_exact_present": true
      },
      {
        "orig": "There were 1,204 cases and 37 deaths reported.",
        "true_numbers": [
          1204,
          37
        ],
        "decoded": "There were 1,204 cases and 37 deaths reported.",
        "decoded_numbers": [
          1204,
          37
        ],
        "all_exact_present": true
      },
      {
        "orig": "The army of 84,000 faced 1,200,000 civilians.",
        "true_numbers": [
          84000,
          1200000
        ],
        "decoded": "The army of 84,000 faced 1,200,000 civilians.",
        "decoded_numbers": [
          84000,
          1200000
        ],
        "all_exact_present": true
      }
    ],
    "multi_all_exact_rate": 1.0
  },
  "scoring_revised": {
    "decode_exact_overall": 1.0,
    "decode_exact_large": 1.0,
    "multi_number_all_exact_rate": 1.0,
    "noise0_exact": 1.0,
    "noise0.1_exact": 1.0,
    "linear_probe_r2_log10N": 0.9954015839287945,
    "mlp_probe_r2_log10N": 0.9435868115426629,
    "nonlinear_gain": -0.0518,
    "sign_readable_AUC": 1.0,
    "exact_precise_falsifier_fired": true,
    "totally_lost_falsifier_fired": false,
    "verdict": "FALSIFIED-precise: EXACT numbers (incl large multi-digit & multi-number) are preserved through encode->decode; z encodes precise numerics for clean single-fact text. Linear probe fails (r2~0) => number is NON-LINEARLY encoded but decoder-readable (curved store, \u00a716.4)."
  },
  "final_verdict": {
    "status": "FALSIFIER FIRED (exact-precise arm)",
    "headline": "z encodes numbers PRECISELY for clean single-fact text, NOT log-scale-blurred. Magnitude is near-perfectly LINEARLY readable (standardized ridge r2(log10N)=0.995, MLP adds nothing). Decoder reproduces EXACT digits incl 250,000 and multi-number sentences (2019/47/312, 84,000/1,200,000), robust to 20% per-dim Gaussian noise on z. Sign of %-change AUC=1.0; exact %% preserved.",
    "prereg_outcome": "PREREG PREDICTION FALSIFIED: predicted large/multi-digit numbers would blur to magnitude-only; instead they are exact. Small-vs-large stratification shows NO degradation (exact_rate 1.0 in every bucket single->thousands).",
    "nuance_geometry": "z places integers on a smooth log-continuum: 1NN in z-space is the SAME exact value 0% of the time but an ADJACENT value (+/-1) 43% of the time. So number lives on a continuous magnitude axis, not discrete exact-value clusters -- yet the decoder still emits exact digits (the precision is in the fine position along the axis, decoder-read losslessly for clean carriers).",
    "reconciliation_with_16.4": "16.4 said number is curved/audit-only/un-steerable and erasure regenerates a wrong number. That was about EDITING/ERASURE under likelihood-death and MEAN-SHIFT steering, not clean round-trip READING. W16 shows READING/PRESERVATION is excellent; the 16.4 limit is specifically that you cannot cleanly EDIT/NULL the number with a linear mean-shift (decode-shift .08) even though it is precisely present. Both true: precise to READ, hard to STEER.",
    "safety_precision_limit": "For READING/auditing: TRUST z for exact quantities in clean single/multi-fact sentences -- exact numbers, percentages, and sign-of-change all survive encode->decode and are linearly probe-readable. The precision limit is NOT in reading; it is (a) in EDITING (cannot reliably null/flip a quantity via linear z-edit, per 16.4) and (b) untested for long/cluttered paragraphs where overall decoder fidelity drops. Do NOT generalize numbers are precise to mean you can safely scrub a number from z."
  }
}