{
  "status": "PASS",
  "groups": 20,
  "assertions": 5608,
  "results": [
    {
      "article": 1,
      "slug": "stochastic-optimization-noise",
      "check": "Sampling and discrete covariance identities",
      "assertions": 10,
      "witness": {
        "finite_population_variances": [
          5.0,
          1.6666666666666667,
          0.5555555555555556,
          0.0
        ],
        "scalar_stationary_variance": 0.08333333333333333,
        "matrix_covariance": [
          [
            0.015622805157421303,
            0.0009863914506035294
          ],
          [
            0.0009863914506035294,
            0.019337634775857728
          ]
        ]
      },
      "scope": "Enumerates all nonempty subset sizes of a four-gradient population and verifies one stable 2D covariance. No posterior-calibration or generalization evidence.",
      "status": "PASS"
    },
    {
      "article": 2,
      "slug": "implicit-bias-separable-learning",
      "check": "Margins, coordinate changes, and contradictory labels",
      "assertions": 116,
      "witness": {
        "maximum_margin": 0.8944271909999159,
        "trajectory_step_loss_margin": [
          [
            1,
            0.5533380097486379,
            0.4472135954999579
          ],
          [
            10,
            0.2114391196724853,
            0.6472679469278612
          ],
          [
            100,
            0.02618021290683939,
            0.8076515825626189
          ],
          [
            1000,
            0.002528703690596741,
            0.8472075680971195
          ],
          [
            10000,
            0.00025045209849889,
            0.8617419583557351
          ]
        ],
        "direct_gradient": [
          -0.16590611391591695,
          -0.31002551887238755
        ]
      },
      "scope": "Finite two-constraint geometry, explicit gradient steps and 101 contradictory-pair scores. Finite descent is not proof of the cited asymptotic theorems.",
      "status": "PASS"
    },
    {
      "article": 3,
      "slug": "normalization-geometry",
      "check": "Normalization invariance and Jacobian checks",
      "assertions": 10,
      "witness": {
        "jacobian": [
          [
            0.23993835872757036,
            -0.3286112304312372,
            0.08867287170366717
          ],
          [
            -0.3286112304312372,
            0.5007409225618855,
            -0.17212969213064816
          ],
          [
            0.08867287170366717,
            -0.17212969213064816,
            0.08345682042698094
          ]
        ],
        "max_finite_difference_error": 1.384497516632166e-10,
        "companion_outputs": [
          -0.9950371902099893,
          0.9950371902099893
        ]
      },
      "scope": "Three-dimensional normalization and two-sample batch normalization with stated population variance and epsilon; excludes network training and mixed-precision implementation.",
      "status": "PASS"
    },
    {
      "article": 4,
      "slug": "residual-depth-dynamics",
      "check": "Discrete stability, sensitivity, and transient growth",
      "assertions": 51,
      "witness": {
        "general_initial_state": 1.7,
        "non_normal_norm_t1_k4": 1.5168057934430552
      },
      "scope": "24 scalar Euler cases with recurrence and finite-difference sensitivities plus explicit instability, noninvertibility and non-normal examples. Not a residual-network benchmark.",
      "status": "PASS"
    },
    {
      "article": 5,
      "slug": "compute-optimal-scaling",
      "check": "Allocation derivatives, data cap, and inference budget",
      "assertions": 342,
      "witness": {
        "symmetric_optimum": [
          100.0,
          100.0
        ],
        "data_cap_optimum": [
          400.0,
          25
        ],
        "inference_q100_optimum": [
          43.01597090019469,
          132.47179572447453
        ]
      },
      "scope": "Nine power-law coefficient/budget cases and 101 inference charges with exact stationary conditions. Does not validate the loss or cost model empirically.",
      "status": "PASS"
    },
    {
      "article": 6,
      "slug": "contrastive-representation-geometry",
      "check": "Binary view information, nuisance counterexample and contrastive invariants",
      "assertions": 202,
      "witness": {
        "q": 0.25,
        "pair_agreement": 0.625,
        "bayes_accuracy": 0.75,
        "q_half_label_accuracy": 0.5,
        "nuisance_alignment": 1,
        "negative_counts": [
          3,
          15,
          63
        ]
      },
      "scope": "Finite binary channel enumeration and fixed candidate-loss checks; no contrastive encoder is trained.",
      "status": "PASS"
    },
    {
      "article": 7,
      "slug": "masked-reconstruction-semantics",
      "check": "Gaussian conditional risk and exhaustive label-mask enumeration",
      "assertions": 1191,
      "witness": {
        "gaussian_rho": 0.6,
        "optimal_mse": 0.64,
        "copies": 4,
        "mask_probability": 0.5,
        "label_error": 0.03125,
        "enumerated_copy_counts": [
          1,
          2,
          3,
          4,
          5,
          6,
          7,
          8
        ]
      },
      "scope": "Second-moment algebra and exhaustive masks for up to eight repeated bits; no image reconstruction model is trained.",
      "status": "PASS"
    },
    {
      "article": 8,
      "slug": "vision-transformer-inductive-bias",
      "check": "Patch collision, all three-token permutations and analytic attention counts",
      "assertions": 71,
      "witness": {
        "patch_input": [
          1.0,
          2.0,
          3.0,
          4.0
        ],
        "collision_input": [
          1.0,
          2.0,
          4.0,
          3.0
        ],
        "shared_embedding": [
          1.0,
          2.0
        ],
        "fixed_position_defect": 2.177542430560174,
        "global_pairs_at_P16": 38416
      },
      "scope": "Finite token-attention algebra and specified operation counts; no image benchmark or accelerator profiling.",
      "status": "PASS"
    },
    {
      "article": 9,
      "slug": "equivariance-under-symmetry",
      "check": "C4 group projection, symmetry mismatch and reflection-sensitive target",
      "assertions": 398,
      "witness": {
        "group_projected_matrix": [
          [
            1.5,
            0.0
          ],
          [
            0.0,
            1.5
          ]
        ],
        "lambda_half_target_error": 0.0625,
        "lambda_half_equivariance_error": 0.25,
        "oriented_triple_product": 1,
        "reflected_triple_product": -1
      },
      "scope": "Finite-group identities, analytic planar maps and a labeled tetrahedron; no continuous-image or trained EGNN certification.",
      "status": "PASS"
    },
    {
      "article": 10,
      "slug": "graph-depth-oversmoothing",
      "check": "Two-node spectral decay, feature reinjection and exhaustive mean-tree collisions",
      "assertions": 394,
      "witness": {
        "reinjection_beta": 0.2,
        "contrast_eigenvalue": 0.8,
        "stationary_gain": 0.5555555555555556,
        "leaf_assignments": 256,
        "distinct_root_values": 9,
        "collision_a": [
          1,
          0,
          0,
          0,
          0,
          0,
          0,
          0
        ],
        "collision_b": [
          0,
          1,
          0,
          0,
          0,
          0,
          0,
          0
        ],
        "shared_root_value": 0.125
      },
      "scope": "Linear two-node recurrence and all 256 binary assignments to eight leaves; no universal nonlinear-GNN theorem or training benchmark.",
      "status": "PASS"
    },
    {
      "article": 11,
      "slug": "diffusion-score-error",
      "check": "Noise-to-score identity and exact biased OU mean/variance recurrences",
      "assertions": 117,
      "witness": {
        "T": 2,
        "step_counts": [
          4,
          8,
          16,
          32,
          64,
          128
        ],
        "biases": [
          0,
          0.1,
          0.25,
          -0.2
        ],
        "limiting_mean_b025": 0.31606027941427883,
        "refinement": [
          {
            "b": 0.0,
            "mean_error_at_N128": 0.0
          },
          {
            "b": 0.1,
            "mean_error_at_N128": 0.00028834504988073384
          },
          {
            "b": 0.25,
            "mean_error_at_N128": 0.0007208626247024452
          },
          {
            "b": -0.2,
            "mean_error_at_N128": 0.0005766900997614677
          }
        ]
      },
      "scope": "Scalar Gaussian fields and listed Euler discretizations only; no learned predictor or image sampler is tested.",
      "status": "PASS"
    },
    {
      "article": 12,
      "slug": "flow-matching-transport",
      "check": "Conditional covariance and independently known Gaussian flow endpoints",
      "assertions": 106,
      "witness": {
        "independent_midpoint_variance": 0.5,
        "anticorrelated_midpoint_variance": 0,
        "Euler_N1_scale": 0.0,
        "Heun_N1_scale": 0.5,
        "Euler_N64_error": 0.019885668352616848,
        "Heun_N64_error": 2.5536292165284635e-06
      },
      "scope": "Gaussian coupling identities and two explicitly coded scalar solvers; the singular anticorrelated path is only a counterexample, not a valid regular flow.",
      "status": "PASS"
    },
    {
      "article": 13,
      "slug": "variational-posterior-collapse",
      "check": "Finite binary KL decomposition and constrained Gaussian loading optimum",
      "assertions": 344,
      "witness": {
        "binary_channel_grid": 101,
        "independent_encoder_rate_nats": 0.3680642071684971,
        "independent_encoder_information_nats": 0.0,
        "lambda3_noise3_loading_energy": 0,
        "Gaussian_parameter_pairs": 20,
        "candidate_energies_per_pair": 161
      },
      "scope": "Exact finite information sums and scalar Gaussian expected likelihood. No deep encoder, decoder, or mutual-information estimator is trained.",
      "status": "PASS"
    },
    {
      "article": 14,
      "slug": "low-rank-adaptation-capacity",
      "check": "Rotated SVD residuals, factor gauge invariance, and weighted linear-output error",
      "assertions": 47,
      "witness": {
        "matrix_dimension": 8,
        "ranks": [
          0,
          1,
          2,
          3,
          4,
          5,
          6,
          7,
          8
        ],
        "rho100_unweighted_error": 100,
        "rho100_weighted_error": 16,
        "factor_gauge_matrix": [
          [
            2,
            1
          ],
          [
            0,
            3
          ]
        ]
      },
      "scope": "Constructed matrices and a centered linear-input criterion only. No claim about trained adapters, optimizer behavior, or accelerator savings.",
      "status": "PASS"
    },
    {
      "article": 15,
      "slug": "sparse-expert-routing",
      "check": "Exhaustive top-one routing and matched-marginal correlated counterexample",
      "assertions": 287,
      "witness": {
        "assignments_enumerated": 256,
        "tokens": 8,
        "experts": 2,
        "capacities": [
          1,
          2,
          3,
          4,
          5,
          6,
          7,
          8
        ],
        "iid_drop_at_capacity4": 0.13671875,
        "correlated_drop_at_capacity4": 0.5,
        "expected_loads_both": [
          4,
          4
        ]
      },
      "scope": "Eight-token, two-expert top-one routing with a drop policy; integer counting and expectations only, not a trained MoE or distributed-system benchmark.",
      "status": "PASS"
    },
    {
      "article": 16,
      "slug": "adversarial-robustness-bounds",
      "check": "Linear corner extrema, stationary-gradient counterexample, and smoothing quantile",
      "assertions": 127,
      "witness": {
        "dimensions": 4,
        "corners": 16,
        "epsilon_values": 51,
        "margin_at_epsilon_point25": 0.5,
        "gradient_steps": 20,
        "stationary_loss": 0.0,
        "endpoint_loss": 1.0,
        "p_A_at_normal_quantile_1": 0.8413447460685428
      },
      "scope": "All 16 corners of a 4D linear box at 51 radii; 20 ascent iterations on delta^2; supplied Gaussian probabilities. No trained network or attack benchmark.",
      "status": "PASS"
    },
    {
      "article": 17,
      "slug": "calibration-decision-quality",
      "check": "Calibrated-but-uninformative population, bin merging, and cost-optimal actions",
      "assertions": 516,
      "witness": {
        "cases": 20,
        "group_positive_counts": [
          1,
          9
        ],
        "constant_Brier": 0.25,
        "informed_Brier": 0.09,
        "coarse_probability_ECE": 0.0,
        "fine_probability_ECE": 0.25,
        "cost_at_equal_error_costs": {
          "pooled": 0.5,
          "informed": 0.1
        }
      },
      "scope": "Assigned finite outcomes and population expectations; binary event-probability bins, not multiclass confidence ECE. No validation-set fitting or real-user outcomes.",
      "status": "PASS"
    },
    {
      "article": 18,
      "slug": "distribution-shift-identifiability",
      "check": "Change of measure, clipping bias, prior inversion, and nonidentifiability witnesses",
      "assertions": 503,
      "witness": {
        "target_risk": 0.5,
        "source_risk": 0.1,
        "clipped_weighted_risk": 0.2,
        "weight_ESS_fraction": 0.36,
        "target_prior_positive": 0.3,
        "observed_positive_fraction": 0.38,
        "indistinguishable_world_risks": [
          0.0,
          1.0
        ]
      },
      "scope": "Finite known distributions and fixed conditional operators; no learned density ratios, target labels, or neural adaptation benchmark.",
      "status": "PASS"
    },
    {
      "article": 19,
      "slug": "preference-optimization-objectives",
      "check": "Finite KL optimum, reward-offset invariance, and cyclic population preferences",
      "assertions": 41,
      "witness": {
        "policy_grid_points": 10001,
        "beta_values_checked": [
          0.25,
          0.5,
          1.0,
          2.0,
          4.0
        ],
        "optimal_preferred_probability_beta1": 0.75,
        "cycle_pair_probabilities": [
          0.6666666666666666,
          0.6666666666666666,
          0.6666666666666666
        ],
        "cycle_logodds_sum": 2.0794415416798353,
        "scalar_score_required_sum": 0.0
      },
      "scope": "Assigned finite rewards and three deterministic ranking groups. Grid comparison is a numerical consistency check, not neural convergence or measured human alignment.",
      "status": "PASS"
    },
    {
      "article": 20,
      "slug": "mechanistic-feature-identifiability",
      "check": "Basis invariance, eight-pattern superposition errors, and unused decodable signal",
      "assertions": 735,
      "witness": {
        "rotation_angles": 101,
        "hidden_inputs": 4,
        "binary_patterns": 8,
        "rectified_errors_by_active_count": {
          "0": 0.0,
          "1": 0.0,
          "2": 0.5,
          "3": 3.0
        },
        "expected_rectified_error_p_half": 0.5625,
        "dense_and_empty_encode_same": true
      },
      "scope": "Fixed linear interfaces and assigned binary features; no large-model feature extraction, sparse-autoencoder training, or completeness claim.",
      "status": "PASS"
    }
  ],
  "scope": "Finite declared models and numerical identities; no new neural training or benchmark measurements."
}