{
  "awgn_limit_consistency": {
    "code_rate": 0.5,
    "consistent": true,
    "k_probe": 1000000.0,
    "max_abs_diff_db": 0.00068,
    "n": 60,
    "rows": [
      {
        "abs_diff_ceiling_db": 0.00029,
        "abs_diff_converse_db": 1e-05,
        "awgn_converse_wall_db": 2.9115,
        "eps": 0.001,
        "fading_largeK_converse_wall_db": 2.9115
      },
      {
        "abs_diff_ceiling_db": 0.00041,
        "abs_diff_converse_db": 0.00013,
        "awgn_converse_wall_db": 3.9723,
        "eps": 1e-05,
        "fading_largeK_converse_wall_db": 3.9724
      },
      {
        "abs_diff_ceiling_db": 0.0,
        "abs_diff_converse_db": 0.00068,
        "awgn_converse_wall_db": 4.3778,
        "eps": 1e-06,
        "fading_largeK_converse_wall_db": 4.3785
      }
    ]
  },
  "design_around_matrix": {
    "assume_ergodic": {
      "apparent_ergodic_wall_db": 0.1034,
      "bends_slope": false,
      "escape": "assume ergodic capacity instead of outage (the FORFEIT)",
      "forfeit_db": {
        "1e-05": 18.6824,
        "1e-06": 26.3387
      },
      "mechanism": "ergodic capacity ignores BOTH outage and finite blocklength; the apparent wall it budgets is tiny, but the real wall is the baseline \u2014 the difference is forfeited"
    },
    "base_cell": {
      "code_rate": 0.5,
      "diversity_order": 1,
      "k_db": 10.0,
      "n": 60
    },
    "base_slope_db_per_decade": 5.673,
    "only_diversity_bends_the_slope": true,
    "raise_power": {
      "bends_slope": false,
      "escape": "raise transmit power",
      "mechanism": "the wall IS the dB above the ergodic-Shannon cliff a finite-blocklength/outage design must spend \u2014 raising power PAYS the wall, it does not reduce it",
      "reduces_wall": false,
      "residual_wall_db": {
        "1e-05": 18.7858,
        "1e-06": 26.4421
      },
      "slope_db_per_decade": 5.673
    },
    "rayleigh_unreachable_anchor": {
      "bends_slope": false,
      "escape": "pure Rayleigh, NO diversity (L=1) \u2014 the unreachable anchor",
      "mechanism": "single-antenna NLOS: only power, no diversity; the wall is physically unreachable",
      "residual_wall_db": {
        "1e-05": 49.3678,
        "1e-06": 58.9971
      },
      "slope_db_per_decade": 9.873
    },
    "rows": [
      {
        "bends_slope": false,
        "escape": "baseline (no design-around)",
        "mechanism": "n=60, R=0.5, Rician K=10 dB, L=1",
        "residual_wall_db": {
          "1e-05": 18.7858,
          "1e-06": 26.4421
        },
        "slope_db_per_decade": 5.673
      },
      {
        "bends_slope": false,
        "escape": "lower code rate R: 0.5 -> 0.25",
        "mechanism": "halves the spectral-efficiency demand; ADDITIVE shift, slope unchanged",
        "residual_wall_db": {
          "1e-05": 18.6043,
          "1e-06": 26.1366
        },
        "slope_db_per_decade": 5.551
      },
      {
        "bends_slope": false,
        "escape": "longer blocklength n: 60 -> 240",
        "mechanism": "shrinks the FBL dispersion penalty; ADDITIVE, slope unchanged",
        "residual_wall_db": {
          "1e-05": 19.0651,
          "1e-06": 26.8129
        },
        "slope_db_per_decade": 5.771
      },
      {
        "bends_slope": true,
        "escape": "add HARQ retransmissions (L=2 time diversity)",
        "mechanism": "2 independent fade blocks combined; BENDS slope to ~10/2",
        "residual_wall_db": {
          "1e-05": 9.0637,
          "1e-06": 10.7351
        },
        "slope_db_per_decade": 1.646
      },
      {
        "bends_slope": true,
        "escape": "add MIMO / receive diversity (L=4)",
        "mechanism": "4 independent branches (MRC); BENDS slope to ~10/4",
        "residual_wall_db": {
          "1e-05": 6.2522,
          "1e-06": 7.111
        },
        "slope_db_per_decade": 0.934
      }
    ],
    "statement": "power / rate / n move the wall only ADDITIVELY (slope unchanged); ONLY raising the diversity order L bends the wall-vs-reliability SLOPE (10/L); 'assume ergodic' is the forfeit that creates the gap; pure Rayleigh @ eps=1e-6 needs ~59 dB \u2014 unreachable without diversity."
  },
  "explicit_non_claim": "An EXACT finite-blocklength MODELING result, RE-ANCHORED to the standardized URLLC/NTN reliability ladder (eps in {1e-3,1e-5,1e-6}). The two-sided wall band [Shannon-1959 sphere-packing CONVERSE floor, RCU achievability CEILING] and the wall-vs-reliability SLOPE LAW (fading ~10/L dB/decade; AWGN ~sqrt(2 ln(1/eps))) are genuine finite-n bounds under the stated AWGN / maximal-power-sphere and quasi-static Rician/Rayleigh fading (CSIR, no CSIT; L-branch MRC) model. They bound what ANY code can do IN THE MODEL; they are NOT the normal approximation, NOT a claim about a specific competitor product, and NOT a legal or freedom-to-operate opinion.",
  "headline": {
    "by_channel": {
      "awgn": {
        "forfeited_margin_1e3_to_1e6_db_ceiling": 1.5358,
        "forfeited_margin_1e3_to_1e6_db_floor": 1.4664,
        "rungs": {
          "1e-03": {
            "band_gap_db": 0.2672,
            "band_ordered": true,
            "converse_wall_db": 2.9115,
            "rcu_ceiling_wall_db": 3.1787
          },
          "1e-05": {
            "band_gap_db": 0.307,
            "band_ordered": true,
            "converse_wall_db": 3.9723,
            "rcu_ceiling_wall_db": 4.2793
          },
          "1e-06": {
            "band_gap_db": 0.3367,
            "band_ordered": true,
            "converse_wall_db": 4.3778,
            "rcu_ceiling_wall_db": 4.7145
          }
        }
      },
      "rayleigh": {
        "forfeited_margin_1e3_to_1e6_db_ceiling": 29.5889,
        "forfeited_margin_1e3_to_1e6_db_floor": 29.5889,
        "rungs": {
          "1e-03": {
            "band_gap_db": 0.2619,
            "band_ordered": true,
            "converse_wall_db": 29.4081,
            "rcu_ceiling_wall_db": 29.67
          },
          "1e-05": {
            "band_gap_db": 0.2619,
            "band_ordered": true,
            "converse_wall_db": 49.3678,
            "rcu_ceiling_wall_db": 49.6296
          },
          "1e-06": {
            "band_gap_db": 0.2619,
            "band_ordered": true,
            "converse_wall_db": 58.9971,
            "rcu_ceiling_wall_db": 59.2589
          }
        }
      },
      "rician_k10db": {
        "forfeited_margin_1e3_to_1e6_db_ceiling": 17.0892,
        "forfeited_margin_1e3_to_1e6_db_floor": 17.0848,
        "rungs": {
          "1e-03": {
            "band_gap_db": 0.2569,
            "band_ordered": true,
            "converse_wall_db": 9.3573,
            "rcu_ceiling_wall_db": 9.6141
          },
          "1e-05": {
            "band_gap_db": 0.2596,
            "band_ordered": true,
            "converse_wall_db": 18.7858,
            "rcu_ceiling_wall_db": 19.0454
          },
          "1e-06": {
            "band_gap_db": 0.2613,
            "band_ordered": true,
            "converse_wall_db": 26.4421,
            "rcu_ceiling_wall_db": 26.7033
          }
        }
      }
    },
    "code_rate_bits_per_cu": 0.5,
    "n": 60,
    "statement": "6G-URLLC NTN-LOS (Rician K=10 dB) reliability wall: [9.36, 9.61] dB @ eps=1e-3 (soft)  ->  [18.79, 19.05] dB @ eps=1e-5 (TR 38.913)  ->  [26.44, 26.70] dB @ eps=1e-6 (6G). A 1e-3-budgeted NTN-LOS design forfeits +17.08 dB of link margin to reach 1e-6."
  },
  "label": "EXACT two-sided finite-blocklength wall RE-ANCHORED to the standardized URLLC/NTN reliability ladder (eps in {1e-3,1e-5,1e-6,...}): per-eps band [Shannon-1959 sphere-packing CONVERSE floor, RCU achievability CEILING] over the AWGN and quasi-static Rician/Rayleigh fading (L-branch MRC) channels, with the wall-vs-reliability SLOPE LAW (fading ~ 10/L dB/decade; AWGN ~ sqrt(2 ln(1/eps)), sub-linear). A genuine finite-n bound, NOT the normal approximation; reduces to the AWGN ladder as the Rician K-factor -> infinity.",
  "method": {
    "amortization": "the eps-/K-/L-independent per-realization error-vs-dB tables are built ONCE per (n,R) and re-inverted for every (eps, K, diversity L) \u2014 the whole ladder costs one table + cheap brentq re-inversions.",
    "channel_model": "complex AWGN, n complex uses <-> real AWGN d=2n dims, ||x||^2=d*s; quasi-static Rician/Rayleigh fading (CSIR, no CSIT); L-branch MRC for diversity.",
    "converse_floor": "EXACT Shannon-1959 sphere-packing converse error averaged over the fading law \u2014 common/finite_blocklength.fading_error_from_table on the _sp_error_db_table (the optimistic FLOOR no code beats).",
    "diversity": "L-branch MRC over i.i.d. Rician fades via diversity_power_quantile (ncx2 additivity, renormalized to unit mean -> isolates diversity, no array gain); L=1 = single antenna, L->inf -> AWGN.",
    "rcu_ceiling": "RCU (Random-Coding Union, PPV 2010 Thm 16) achievability error averaged over the SAME fading law \u2014 the tightened CEILING a real code faces.",
    "slope_law": "wall(eps) fit both linear-in-decades (fading: slope -> 10/L) and linear-in-sqrt(2 ln(1/eps)) (AWGN: sub-linear); R^2 reported."
  },
  "monotonicity_lemma": {
    "proven": true,
    "statement": "WALL-vs-RELIABILITY LAW: re-anchoring the two-sided band [converse floor, RCU ceiling] to the URLLC ladder (eps 1e-3->1e-5->1e-6) keeps it monotone increasing on BOTH edges and ordered/non-empty at every eps incl. the deepest target; and one decade step of the fading converse wall = d_fbl (AWGN/FBL growth) + d_out_L (fading/outage growth, >=0), so the fading step >= the AWGN step (fading is the real wall at deep eps) and is non-increasing in the diversity order L (only diversity bends the slope; L->inf recovers the sub-linear AWGN law). Magnitudes (10/L, sqrt(ln(1/eps))) are the v23 numerics.",
    "tier": "z3-lra"
  },
  "montecarlo_crosscheck": {
    "abs_z_score": 1.118,
    "agree": true,
    "cliff_db": 22.6143,
    "code_rate": 0.5,
    "eps": 1e-06,
    "k_linear": 10.0,
    "mc_std_error": 1.395e-07,
    "montecarlo_error": 1.156e-06,
    "n": 60,
    "n_samples": 48000000,
    "quadrature_error": 1e-06
  },
  "robust_sweep": {
    "all_bands_ordered": true,
    "band_tight_gap_below_0p6_db": true,
    "blocklengths": [
      60,
      120,
      240
    ],
    "cells": [
      {
        "band_gap_db": 0.498,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 29.057,
        "eps": 0.001,
        "k_db": null,
        "k_linear": 0.0,
        "n": 60,
        "rcu_ceiling_wall_db": 29.5551
      },
      {
        "band_gap_db": 0.498,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 49.0169,
        "eps": 1e-05,
        "k_db": null,
        "k_linear": 0.0,
        "n": 60,
        "rcu_ceiling_wall_db": 49.5149
      },
      {
        "band_gap_db": 0.4978,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 58.6466,
        "eps": 1e-06,
        "k_db": null,
        "k_linear": 0.0,
        "n": 60,
        "rcu_ceiling_wall_db": 59.1445
      },
      {
        "band_gap_db": 0.497,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 21.6494,
        "eps": 0.001,
        "k_db": 5.0,
        "k_linear": 3.162278,
        "n": 60,
        "rcu_ceiling_wall_db": 22.1464
      },
      {
        "band_gap_db": 0.498,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 41.478,
        "eps": 1e-05,
        "k_db": 5.0,
        "k_linear": 3.162278,
        "n": 60,
        "rcu_ceiling_wall_db": 41.976
      },
      {
        "band_gap_db": 0.4979,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 51.1065,
        "eps": 1e-06,
        "k_db": 5.0,
        "k_linear": 3.162278,
        "n": 60,
        "rcu_ceiling_wall_db": 51.6044
      },
      {
        "band_gap_db": 0.4711,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 9.4102,
        "eps": 0.001,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 60,
        "rcu_ceiling_wall_db": 9.8814
      },
      {
        "band_gap_db": 0.4852,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 18.6043,
        "eps": 1e-05,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 60,
        "rcu_ceiling_wall_db": 19.0894
      },
      {
        "band_gap_db": 0.4943,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 26.1366,
        "eps": 1e-06,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 60,
        "rcu_ceiling_wall_db": 26.6309
      },
      {
        "band_gap_db": 0.4622,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 6.4119,
        "eps": 0.001,
        "k_db": 13.0,
        "k_linear": 19.952623,
        "n": 60,
        "rcu_ceiling_wall_db": 6.8742
      },
      {
        "band_gap_db": 0.4647,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 10.0291,
        "eps": 1e-05,
        "k_db": 13.0,
        "k_linear": 19.952623,
        "n": 60,
        "rcu_ceiling_wall_db": 10.4938
      },
      {
        "band_gap_db": 0.4645,
        "band_ordered": true,
        "code_rate": 0.25,
        "converse_wall_db": 11.9462,
        "eps": 1e-06,
        "k_db": 13.0,
        "k_linear": 19.952623,
        "n": 60,
        "rcu_ceiling_wall_db": 12.4107
      },
      {
        "band_gap_db": 0.2619,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 29.4081,
        "eps": 0.001,
        "k_db": null,
        "k_linear": 0.0,
        "n": 60,
        "rcu_ceiling_wall_db": 29.67
      },
      {
        "band_gap_db": 0.2619,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 49.3678,
        "eps": 1e-05,
        "k_db": null,
        "k_linear": 0.0,
        "n": 60,
        "rcu_ceiling_wall_db": 49.6296
      },
      {
        "band_gap_db": 0.2619,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 58.9971,
        "eps": 1e-06,
        "k_db": null,
        "k_linear": 0.0,
        "n": 60,
        "rcu_ceiling_wall_db": 59.2589
      },
      {
        "band_gap_db": 0.2617,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 21.988,
        "eps": 0.001,
        "k_db": 5.0,
        "k_linear": 3.162278,
        "n": 60,
        "rcu_ceiling_wall_db": 22.2497
      },
      {
        "band_gap_db": 0.2619,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 41.8287,
        "eps": 1e-05,
        "k_db": 5.0,
        "k_linear": 3.162278,
        "n": 60,
        "rcu_ceiling_wall_db": 42.0906
      },
      {
        "band_gap_db": 0.2619,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 51.4569,
        "eps": 1e-06,
        "k_db": 5.0,
        "k_linear": 3.162278,
        "n": 60,
        "rcu_ceiling_wall_db": 51.7188
      },
      {
        "band_gap_db": 0.2569,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 9.3573,
        "eps": 0.001,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 60,
        "rcu_ceiling_wall_db": 9.6141
      },
      {
        "band_gap_db": 0.2596,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 18.7858,
        "eps": 1e-05,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 60,
        "rcu_ceiling_wall_db": 19.0454
      },
      {
        "band_gap_db": 0.2613,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 26.4421,
        "eps": 1e-06,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 60,
        "rcu_ceiling_wall_db": 26.7033
      },
      {
        "band_gap_db": 0.2552,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 6.057,
        "eps": 0.001,
        "k_db": 13.0,
        "k_linear": 19.952623,
        "n": 60,
        "rcu_ceiling_wall_db": 6.3123
      },
      {
        "band_gap_db": 0.2558,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 9.6788,
        "eps": 1e-05,
        "k_db": 13.0,
        "k_linear": 19.952623,
        "n": 60,
        "rcu_ceiling_wall_db": 9.9346
      },
      {
        "band_gap_db": 0.2557,
        "band_ordered": true,
        "code_rate": 0.5,
        "converse_wall_db": 11.6376,
        "eps": 1e-06,
        "k_db": 13.0,
        "k_linear": 19.952623,
        "n": 60,
        "rcu_ceiling_wall_db": 11.8933
      },
      {
        "band_gap_db": 0.1863,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 29.5413,
        "eps": 0.001,
        "k_db": null,
        "k_linear": 0.0,
        "n": 60,
        "rcu_ceiling_wall_db": 29.7276
      },
      {
        "band_gap_db": 0.1863,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 49.5008,
        "eps": 1e-05,
        "k_db": null,
        "k_linear": 0.0,
        "n": 60,
        "rcu_ceiling_wall_db": 49.6872
      },
      {
        "band_gap_db": 0.1863,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 59.1301,
        "eps": 1e-06,
        "k_db": null,
        "k_linear": 0.0,
        "n": 60,
        "rcu_ceiling_wall_db": 59.3165
      },
      {
        "band_gap_db": 0.1863,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 22.1174,
        "eps": 0.001,
        "k_db": 5.0,
        "k_linear": 3.162278,
        "n": 60,
        "rcu_ceiling_wall_db": 22.3036
      },
      {
        "band_gap_db": 0.1863,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 41.9618,
        "eps": 1e-05,
        "k_db": 5.0,
        "k_linear": 3.162278,
        "n": 60,
        "rcu_ceiling_wall_db": 42.1481
      },
      {
        "band_gap_db": 0.1863,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 51.59,
        "eps": 1e-06,
        "k_db": 5.0,
        "k_linear": 3.162278,
        "n": 60,
        "rcu_ceiling_wall_db": 51.7763
      },
      {
        "band_gap_db": 0.1853,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 9.3523,
        "eps": 0.001,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 60,
        "rcu_ceiling_wall_db": 9.5376
      },
      {
        "band_gap_db": 0.1858,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 18.8646,
        "eps": 1e-05,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 60,
        "rcu_ceiling_wall_db": 19.0504
      },
      {
        "band_gap_db": 0.1861,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 26.5613,
        "eps": 1e-06,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 60,
        "rcu_ceiling_wall_db": 26.7474
      },
      {
        "band_gap_db": 0.1857,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 5.9246,
        "eps": 0.001,
        "k_db": 13.0,
        "k_linear": 19.952623,
        "n": 60,
        "rcu_ceiling_wall_db": 6.1103
      },
      {
        "band_gap_db": 0.1858,
        "band_ordered": true,
        "code_rate": 0.75,
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        "eps": 1e-06,
        "k_db": 5.0,
        "k_linear": 3.162278,
        "n": 240,
        "rcu_ceiling_wall_db": 51.9482
      },
      {
        "band_gap_db": 0.045,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 9.4489,
        "eps": 0.001,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 240,
        "rcu_ceiling_wall_db": 9.4939
      },
      {
        "band_gap_db": 0.045,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 19.0965,
        "eps": 1e-05,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 240,
        "rcu_ceiling_wall_db": 19.1415
      },
      {
        "band_gap_db": 0.0451,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 26.8539,
        "eps": 1e-06,
        "k_db": 10.0,
        "k_linear": 10.0,
        "n": 240,
        "rcu_ceiling_wall_db": 26.899
      },
      {
        "band_gap_db": 0.045,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 5.7826,
        "eps": 0.001,
        "k_db": 13.0,
        "k_linear": 19.952623,
        "n": 240,
        "rcu_ceiling_wall_db": 5.8276
      },
      {
        "band_gap_db": 0.045,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 9.442,
        "eps": 1e-05,
        "k_db": 13.0,
        "k_linear": 19.952623,
        "n": 240,
        "rcu_ceiling_wall_db": 9.487
      },
      {
        "band_gap_db": 0.045,
        "band_ordered": true,
        "code_rate": 0.75,
        "converse_wall_db": 11.4663,
        "eps": 1e-06,
        "k_db": 13.0,
        "k_linear": 19.952623,
        "n": 240,
        "rcu_ceiling_wall_db": 11.5113
      }
    ],
    "code_rates": [
      0.25,
      0.5,
      0.75
    ],
    "eps_ladder": [
      0.001,
      1e-05,
      1e-06
    ],
    "gap_shrinks_with_blocklength": true,
    "k_grid": [
      {
        "k_db": null,
        "label": "Rayleigh (NLOS, no diversity) \u2014 worst case"
      },
      {
        "k_db": 5.0,
        "label": "Rician K=5 dB (moderate LOS)"
      },
      {
        "k_db": 10.0,
        "label": "Rician K=10 dB (LOS-dominated LEO, NTN headline)"
      },
      {
        "k_db": 13.0,
        "label": "Rician K=13 dB (strong LOS)"
      }
    ],
    "max_band_gap_db": 0.498,
    "max_gap_by_blocklength_db": {
      "60": 0.498,
      "120": 0.2411,
      "240": 0.1176
    },
    "min_converse_wall_db": 5.7826,
    "min_wall_cell": {
      "band_gap_db": 0.045,
      "band_ordered": true,
      "code_rate": 0.75,
      "converse_wall_db": 5.7826,
      "eps": 0.001,
      "k_db": 13.0,
      "k_linear": 19.952623,
      "n": 240,
      "rcu_ceiling_wall_db": 5.8276
    },
    "n_cells": 108,
    "no_table_rail_hits": true,
    "rail_hits": [],
    "statement": "across all 108 standardized (n,R,eps,K) cells the wall is >= 5.78 dB and the two-sided sandwich gap stays tight <= 0.498 dB (worst at the short-block low-rate corner n=60/R=0.25), shrinking with blocklength (0.498 -> 0.118 dB as n: 60 -> 240) \u2014 the pin does not loosen at deep eps.",
    "worst_gap_cell": {
      "band_gap_db": 0.498,
      "band_ordered": true,
      "code_rate": 0.25,
      "converse_wall_db": 29.057,
      "eps": 0.001,
      "k_db": null,
      "k_linear": 0.0,
      "n": 60,
      "rcu_ceiling_wall_db": 29.5551
    }
  },
  "slope_law": {
    "awgn_slope_db_per_decade": 0.488,
    "awgn_sqrt_log_r2": 0.99905,
    "code_rate_bits_per_cu": 0.5,
    "eps_ladder": [
      0.001,
      0.0001,
      1e-05,
      1e-06
    ],
    "law": {
      "awgn_sublinear_below_fading": true,
      "diversity_bends_slope_to_10_over_L": true,
      "fading_fits_decade_linear_r2_above_0p99": true,
      "rayleigh_slope_approx_10_per_decade": true
    },
    "n": 60,
    "rayleigh_L1_slope_db_per_decade": 9.873,
    "rows": [
      {
        "converse_walls_db": [
          2.9115,
          3.4956,
          3.9723,
          4.3778
        ],
        "decade_fit_r2": 0.99328,
        "decade_slope_db_per_decade": 0.488,
        "deep_slope_db_per_decade": 0.406,
        "diversity_order": 1,
        "diversity_order_estimate": 20.509,
        "expected_diversity_order": null,
        "expected_slope_10_over_L": null,
        "k_factor": null,
        "label": "AWGN (no fading)",
        "sqrt_log_coef_db": 0.953,
        "sqrt_log_fit_r2": 0.99905
      },
      {
        "converse_walls_db": [
          9.3573,
          13.3135,
          18.7858,
          26.4421
        ],
        "decade_fit_r2": 0.97904,
        "decade_slope_db_per_decade": 5.673,
        "deep_slope_db_per_decade": 7.656,
        "diversity_order": 1,
        "diversity_order_estimate": 1.763,
        "expected_diversity_order": null,
        "expected_slope_10_over_L": null,
        "k_factor": 10.0,
        "label": "Rician K=10 dB (NTN-LOS)",
        "sqrt_log_coef_db": 10.957,
        "sqrt_log_fit_r2": 0.96208
      },
      {
        "converse_walls_db": [
          29.4081,
          39.4064,
          49.3678,
          58.9971
        ],
        "decade_fit_r2": 0.99992,
        "decade_slope_db_per_decade": 9.873,
        "deep_slope_db_per_decade": 9.629,
        "diversity_order": 1,
        "diversity_order_estimate": 1.013,
        "expected_diversity_order": 1,
        "expected_slope_10_over_L": 10.0,
        "k_factor": 0.0,
        "label": "Rayleigh, diversity L=1",
        "sqrt_log_coef_db": 19.22,
        "sqrt_log_fit_r2": 0.99814
      },
      {
        "converse_walls_db": [
          16.0611,
          21.111,
          26.1077,
          30.9276
        ],
        "decade_fit_r2": 0.99989,
        "decade_slope_db_per_decade": 4.96,
        "deep_slope_db_per_decade": 4.82,
        "diversity_order": 2,
        "diversity_order_estimate": 2.016,
        "expected_diversity_order": 2,
        "expected_slope_10_over_L": 5.0,
        "k_factor": 0.0,
        "label": "Rayleigh, diversity L=2",
        "sqrt_log_coef_db": 9.656,
        "sqrt_log_fit_r2": 0.99831
      },
      {
        "converse_walls_db": [
          9.6371,
          12.3427,
          14.9427,
          17.4142
        ],
        "decade_fit_r2": 0.99959,
        "decade_slope_db_per_decade": 2.593,
        "deep_slope_db_per_decade": 2.471,
        "diversity_order": 4,
        "diversity_order_estimate": 3.856,
        "expected_diversity_order": 4,
        "expected_slope_10_over_L": 2.5,
        "k_factor": 0.0,
        "label": "Rayleigh, diversity L=4",
        "sqrt_log_coef_db": 5.051,
        "sqrt_log_fit_r2": 0.99903
      }
    ],
    "statement": "the wall's growth rate vs reliability is set by the DIVERSITY order L: fading rises ~10/L dB per decade of reliability (Rayleigh L=1 -> 10), while AWGN grows only ~sqrt(2 ln(1/eps)) (sub-linear, ~0.45 dB/decade) \u2014 so at the real URLLC target the fading/outage penalty dominates the FBL penalty by an order of magnitude, and only raising L bends the slope."
  },
  "tier": "numeric-exact",
  "title": "WALL vs RELIABILITY \u2014 the two-sided FBL wall re-anchored to the URLLC ladder + slope law",
  "validation": {
    "all_cells_ordered": true,
    "awgn_limit_consistent": true,
    "awgn_sublinear": true,
    "band_tight_gap_below_0p6_db": true,
    "diversity_bends_slope": true,
    "gap_shrinks_with_blocklength": true,
    "lemma_proven": true,
    "montecarlo_agree": true,
    "no_table_rail_hits": true,
    "only_diversity_bends": true,
    "rayleigh_slope_approx_10": true
  },
  "wall_ladder_ok": true
}