{
  "S1a_real_crossover": {
    "status": "OPERATIONAL CONFIRMATION (crossover unreachable on real LIDC)",
    "strata": {
      "1": {"n": 990, "concentration": 0.806, "spanning": 0.458, "gap": 0.348},
      "2-3": {"n": 816, "concentration": 0.815, "spanning": 0.461, "gap": 0.354},
      "4-8": {"n": 230, "concentration": 0.824, "spanning": 0.468, "gap": 0.357},
      ">8": {"n": 1, "note": "no high-rank lesions in LIDC"}
    },
    "finding": "The concentration-minus-spanning gap is ~CONSTANT (0.35) across lesion patch-count, NOT shrinking. Reason: patch count != feature rank. A compact multi-patch lesion is a COHERENT blob whose tokens share a direction (rank ~1), so it stays in the concentration-wins regime. Real LIDC pathology is UNIFORMLY low-rank; there is no high-rank (multi-focal/scattered) lesion regime to reach the synthetic crossover. Operationally: rank-coverage is NEVER the right objective for real lesions because real lesions never leave the low-rank regime where concentration dominates.",
    "refinement": "To observe the crossover on real data, stratify by the ACTUAL feature-space rank of the lesion tokens (effective rank of P_L Z_lesion), not patch count; or use multi-focal datasets (e.g., metastases). Future work."
  },
  "S3_precondition_predictor": {
    "status": "FAIL (negative, informative)",
    "datasets": {
      "lung_LIDC": {"auroc": 0.871, "tail_gap": 1.139, "bimodality": 0.160},
      "pancreas_MSD": {"auroc": 0.876, "tail_gap": 3.383, "bimodality": 0.265},
      "kidney_KiTS": {"auroc": 0.823, "tail_gap": 2.019, "bimodality": 0.234},
      "liver_MSD": {"auroc": 0.670, "tail_gap": 2.210, "bimodality": 0.235}
    },
    "tail_gap_spearman_with_auroc": 0.2, "bimodality_spearman_with_auroc": 0.2,
    "flags_liver_lowest": false, "predicts_precondition": false,
    "finding": "The membership-distribution proxies that select the LAYER (S2) do NOT predict the DATASET precondition. Liver (worst AUROC 0.67) has a MIDDLING tail_gap, and lung (best) has the LOWEST. The proxies measure RARENESS, not LESION-ness. Liver's failure is that rare tokens exist (vessels, boundaries) but are NOT the (low-contrast) lesion -- a SEMANTIC mismatch, invisible label-free. Predicting localization failure with NO labels is genuinely hard precisely because the failure is 'rare != lesion'.",
    "implication": "The deployment safety trigger needs more than membership geometry: a few labels, anatomical priors, or cross-referencing rare tokens against a normal-tissue model. Pure label-free precondition prediction is an open negative."
  },
  "human_signoff": null
}