Era inflation coefficients I_era: {'1960s': 1.02, '1970s': 0.97, '1980s': 1.0, '1990s': 1.0, '2000s': 0.98, '2010s': 1.03, '2020s': 1.07} ======================================================================== S5.2 Baseline composite score table (spec 1, 16-player calibration pool) ======================================================================== alpha_o alpha_d beta_z atv_z drawdown score player Michael Jordan 1.64 0.59 0.41 1.93 0.10 6.46 Bill Russell -1.50 1.62 -0.34 1.56 0.10 3.28 Tim Duncan -0.52 1.05 -0.70 0.60 0.10 2.77 Kareem Abdul-Jabbar 0.09 0.31 0.58 0.70 0.20 2.06 Hakeem Olajuwon 0.02 0.85 -1.41 -0.53 0.15 0.67 Kobe Bryant -0.46 -0.15 0.71 0.63 0.45 0.18 Shaquille O'Neal 0.14 -0.49 0.71 0.58 0.30 0.14 Magic Johnson 0.20 -0.91 0.80 0.55 0.45 -0.36 Wilt Chamberlain 0.07 1.01 -0.82 -0.93 0.55 -0.42 Nikola Jokic 1.11 -0.63 -1.86 -1.00 0.15 -0.56 LeBron James 0.93 -0.17 2.99 0.12 0.80 -0.59 Larry Bird -0.21 -0.40 0.49 -0.25 0.25 -1.59 David Robinson -0.51 0.84 -0.04 -1.02 0.80 -2.77 Stephen Curry 0.57 -1.35 1.86 0.11 0.40 -2.78 Patrick Ewing -1.46 -0.33 -1.41 -1.53 0.50 -5.06 James Harden -0.29 -1.74 0.48 -1.53 0.85 -6.68 ======================================================================== S5.3 Gatekeeper seat (No.10): the four-candidate race ======================================================================== Hakeem Olajuwon score = +0.67 Nikola Jokic score = -0.56 LeBron James score = -0.59 Stephen Curry score = -2.78 ======================================================================== S5.4 Structural adjustment and Shapley specifications ======================================================================== Structural indices (anchor=1990s): {'1960s': 'RP=1.16/AE=0.82', '1970s': 'RP=1.16/AE=0.88', '1980s': 'RP=1.12/AE=0.96', '1990s': 'RP=1.00/AE=1.00', '2000s': 'RP=0.88/AE=1.02', '2010s': 'RP=0.73/AE=1.10', '2020s': 'RP=0.61/AE=1.25'} v4 (baseline beta + structural adj): {'Hakeem Olajuwon': np.float64(0.73), 'Nikola Jokic': np.float64(-0.83), 'LeBron James': np.float64(-0.78), 'Stephen Curry': np.float64(-2.8)} v3 (Shapley, lambda=0): {'Hakeem Olajuwon': np.float64(0.49), 'Nikola Jokic': np.float64(-1.29), 'LeBron James': np.float64(-0.18), 'Stephen Curry': np.float64(-1.92)} v4-Shapley (both stacked): {'Hakeem Olajuwon': np.float64(0.55), 'Nikola Jokic': np.float64(-1.56), 'LeBron James': np.float64(-0.37), 'Stephen Curry': np.float64(-1.94)} ======================================================================== S6.1 Case backtests ======================================================================== Case 1 Duncan vs James: +2.77 vs -0.59 -> decisive edge PASS Case 2 1990s centers: Olajuwon +0.67 > Robinson -2.77 > Ewing -5.06 Case 3 Curry vs Harden: gap +3.91 (ATV as tiebreaker) ======================================================================== S6.2 Three-cutoff time-split forward test (surprise = rank shift > 3) ======================================================================== Cutoff @2016: surprises = 0 PASS Cutoff @2020: surprises = 0 PASS Cutoff @2022: surprises = 0 PASS ======================================================================== S6.3 Factor ablation tests ======================================================================== [Ablate Superteam-beta] James rank 11 -> 6, gatekeeper gap +1.26 -> +0.07 (discrimination collapses) [Ablate inflation deflation] {'Nikola Jokic': 'rank 10 -> 9', 'Stephen Curry': 'rank 14 -> 13', 'Hakeem Olajuwon': 'rank 5 -> 5'} Jokic-Olajuwon gap 1.23->0.78 ======================================================================== S6.4 Full parameter grid (6075 sets) + S6.11 jackknife & train/test split + S6.6 BMA ======================================================================== Feasible region 88.9% | within: gatekeeper=83.2% tier=98.2% |Jokic-James| median=0.96 Gatekeeper rate by lambda: {0.5: np.float64(0.327), 0.7: np.float64(0.827), 0.85: np.float64(0.979), 1.0: np.float64(1.0), 1.2: np.float64(1.0)} Constraint jackknife: Drop [Case1]: feasible 91.8% gate=81.1% tier=98.2% Drop [Case2]: feasible 88.9% gate=83.2% tier=98.2% Drop [Case3]: feasible 97.2% gate=84.7% tier=98.3% Drop [EwingOrder]: feasible 88.9% gate=83.2% tier=98.2% Drop [JordanFirst]: feasible 88.9% gate=83.2% tier=98.2% Train/test split [IS=Cases1,3 | OOS=Case2,Ewing,Jordan]: OOS blind pass 100.0% Reverse split: OOS blind pass 88.9% BMA posterior: P(gatekeeper)=82.8% P(tier)=98.3% Chosen set distance to feasible-region centroid = 0.062, closer than 100% of feasible sets ======================================================================== S6.4 cont. Structural damping theta sensitivity (both beta philosophies) ======================================================================== theta=0.2: v4 gatekeeper=Olajuwon PASS(margin +1.39) Shapley gatekeeper=Olajuwon PASS theta=0.4: v4 gatekeeper=Olajuwon PASS(margin +1.51) Shapley gatekeeper=Olajuwon PASS theta=0.6: v4 gatekeeper=Olajuwon PASS(margin +1.63) Shapley gatekeeper=Olajuwon PASS ======================================================================== S6.5 Monte Carlo uncertainty quantification (3000 draws, params +/-20% + input noise) ======================================================================== P(gatekeeper = Olajuwon): 97.8% P(Duncan > James): 100.0% P(Jokic ~ James same tier): 60.0% P(both > Curry): 100.0% P(Curry > Harden): 100.0% ======================================================================== S6.7 Multicollinearity diagnostics (VIF) and factor dimensionality ======================================================================== VIF: {'alpha_o': np.float64(3.09), 'alpha_d': np.float64(2.69), 'beta': np.float64(2.71), 'atv': np.float64(2.83), 'media': np.float64(6.29), 'allnba': np.float64(2.7)} -> media highly collinear; quantitative support for Media-beta down-weighting (S4.4) Eigenvalue variance shares: [0.493 0.254 0.138 0.068 0.03 0.016] top-3 dims ~ 0.89 ======================================================================== S6.8 Convergent validity + S6.9 External validity ======================================================================== Convergent validity: model net alpha vs playoff WS/48 (not in model), Pearson r = 0.769 External validity: vs media consensus ranks, Spearman rho = 0.632; excluding James, rho = 0.739 Largest divergence = James (consensus #2 vs model #11): the design intent of beta treatment; the divergence is explainable ======================================================================== S6.10 Leave-one-player-out LOPO (16 runs) ======================================================================== Conclusions fully retained: 16/16 failures: none ======================================================================== S6.12 Six out-of-sample players (frozen-parameter scoring) ======================================================================== Dirk Nowitzki score=-3.17 overall 18/22 Kevin Garnett score=-3.04 overall 17/22 Kevin Durant score=-4.22 overall 20/22 Giannis Antetokounmpo score=-1.09 overall 12/22 Jerry West score=-2.71 overall 14/22 Moses Malone score=-3.78 overall 19/22 Core conclusions undisturbed: gatekeeper still Olajuwon -> True ======================================================================== S6.13 Opponent strength: orthogonality check + ATV-v2 variant ======================================================================== Opponent win pct vs phi: Pearson r = 0.169 (near-orthogonal -> independent information dimension) ATV-v2 gatekeeper race: {'Hakeem Olajuwon': np.float64(0.93), 'Nikola Jokic': np.float64(-0.67), 'LeBron James': np.float64(-0.56), 'Stephen Curry': np.float64(-2.67)} Gatekeeper margin: baseline +1.23 -> v2 +1.49 (widens) ======================================================================== S6.14a Five-cutoff time splits (adding 2012 and 2024) ======================================================================== Cutoff @2012: surprises = 0 PASS Cutoff @2024: surprises = 0 PASS (Five cutoffs total with 2016/2020/2022 from S6.2) ======================================================================== S6.14b Random-subsample cross-validation (200 draws of 12/16 sub-pools) ======================================================================== Conclusion retention over evaluable sub-pools: 151/151 = 100.0% ======================================================================== S6.14c Placebo / negative controls (permutation, 500 each) ======================================================================== Permute [championship terminal value ATV]: gatekeeper rate 45.4% (real data 97.8%; large collapse = factor carries real information) Permute [defensive honors]: gatekeeper rate 65.4% (real data 97.8%; large collapse = factor carries real information) ======================================================================== S6.14d Noise dose-response (measurement noise scaled 0.5x-3x, 400 each) ======================================================================== Noise x0.5: gatekeeper retention 100.0% Noise x1.0: gatekeeper retention 99.5% Noise x1.5: gatekeeper retention 98.5% Noise x2.0: gatekeeper retention 91.0% Noise x3.0: gatekeeper retention 78.0% ======================================================================== S6.14e Alternative beta measurement: NBA75-teammate basis (measurement robustness) ======================================================================== Correlation with baseline Team-beta load: r = 0.884 beta75-basis gatekeeper race: {'Hakeem Olajuwon': np.float64(0.27), 'Nikola Jokic': np.float64(-0.32), 'LeBron James': np.float64(-0.26), 'Stephen Curry': np.float64(-2.3)} Gatekeeper=Olajuwon PASS, Duncan>James: True, Jordan first: True ======================================================================== S6.14f Weak-supporting-cast credit: solo-deep-run (CARRY) variant (the other direction of beta) ======================================================================== Race after +0.3*z(carry): {'Hakeem Olajuwon': np.float64(0.81), 'Nikola Jokic': np.float64(0.02), 'LeBron James': np.float64(-0.45), 'Stephen Curry': np.float64(-3.08)} Gatekeeper=Olajuwon PASS (credits: Olajuwon carry=1 / Jokic=2 / James=1; Duncan>James: True) kappa=0.15: gatekeeper=Olajuwon PASS kappa=0.45: gatekeeper=Olajuwon PASS kappa=0.6: gatekeeper=Olajuwon PASS ======================================================================== S6.14g Cross-specification concordance: Kendall's W ======================================================================== Kendall's W across seven specifications = 0.971 (1 = perfect agreement) ======================================================================== S6.15a Specification-curve/multiverse analysis + continuous parameter-space check (4500 random specs) ======================================================================== Gatekeeper margin: sign consistency P(>0)=90.2%, median +0.80, 5th pct -0.24, IQR[+0.42,+1.09] Duncan-James margin: P(>0)=100.0%, median +3.45, 5th pct +2.13 Continuous sampling agrees with the discrete grid -> no hidden flip pockets between grid points ======================================================================== S6.15b Permutation hypothesis tests (formal p-values, 1000 each) ======================================================================== H0[atv information exchangeable]: p = 0.3207 (fail to reject) H0[defsc information exchangeable]: p = 0.0480 (reject H0) H0[beta_team information exchangeable]: p = 0.0130 (reject H0) Fisher combined test: chi2=17.0 (df=6), joint p = 9.15e-03 ======================================================================== S6.15c Breakdown analysis (minimal input perturbation that flips the gatekeeper conclusion) ======================================================================== Hakeem Olaju playoff BPM down: cannot flip within full range (8) Hakeem Olaju defensive honors down: requires 16.16 (= 2.4 cross-sectional sigma) to flip Hakeem Olaju ATV down: requires 1.53 (= 0.6 cross-sectional sigma) to flip Hakeem Olaju drawdown up: cannot flip within full range (1) LeBron James playoff BPM up: cannot flip within full range (8) LeBron James teammate load down: requires 8.26 (= 2.2 cross-sectional sigma) to flip Nikola Jokic ATV up: requires 1.51 (= 0.6 cross-sectional sigma) to flip ======================================================================== S6.15d AMIP-style sample breakdown (how many players must be removed to flip) ======================================================================== Requires greedy removal of k = 4 non-candidate players to flip (last removed: James Harden) ======================================================================== S6.15e Bootstrap confidence intervals for ranks (input-noise bootstrap, 2000 draws, 95% rank intervals) ======================================================================== 95% rank intervals for key players (composite-score order, controls included): Michael Jordan median rank 1 95%CI[1, 1] Kobe Bryant median rank 7 95%CI[5, 10] Hakeem Olajuwon median rank 5 95%CI[5, 7] Nikola Jokic median rank 10 95%CI[8, 11] LeBron James median rank 10 95%CI[7, 11] Stephen Curry median rank 13 95%CI[13, 14] ======================================================================== S7.1 Data-endogenous identification of lambda: validation loss curve L(lambda) ======================================================================== L(lambda): {0.3: 0.225, 0.5: 0.003, 0.7: 0.0, 0.85: 0.0, 1.0: 0.0, 1.2: 0.0, 1.4: 0.02} -> zero-loss plateau lambda in [0.7,1.2]; chosen 0.85 lies inside ======================================================================== S7.2 Shapley share spec: full recheck at lambda=0 + residual sensitivity + st threshold ======================================================================== lambda=0 gatekeeper race: {'Hakeem Olajuwon': np.float64(0.49), 'Nikola Jokic': np.float64(-1.29), 'LeBron James': np.float64(-0.18), 'Stephen Curry': np.float64(-1.92)} Cases: Duncan>James True, Olajuwon>Robinson True, Curry-Harden gap 5.05, Jordan first True ATV share-spec comparison: {'Hakeem Olajuwon': '2.47->2.06', 'Nikola Jokic': '1.30->1.30', 'LeBron James': '4.08->1.75', 'Stephen Curry': '4.05->1.98'} Residual sensitivity (1215 sets, lambda eliminated): gatekeeper 87.6% tier 99.6% st=0.0: gatekeeper=LeBron James st=0.2: gatekeeper=LeBron James st=0.3: gatekeeper=LeBron James st=0.4: gatekeeper=Olajuwon st=0.5: gatekeeper=Olajuwon st=0.7: gatekeeper=Olajuwon ======================================================================== S7.3 Data-driven weights: independent convergence of entropy method and CRITIC ======================================================================== [Entropy] gatekeeper=Olajuwon: True Spearman rho=0.921 [CRITIC] gatekeeper=Olajuwon: True Spearman rho=0.906 ======================================================================== Final ranking (paper S5.1 two-stage structure): seats 1-9 locked tier (intra-tier by composite score): Michael Jordan(+6.46) > Bill Russell(+3.28) > Tim Duncan(+2.77) > Kareem Abdul-Jabbar(+2.06) > Kobe Bryant(+0.18) > Shaquille O'Neal(+0.14) > Magic Johnson(-0.36) > Wilt Chamberlain(-0.42) > Larry Bird(-1.59) Seat 10 gatekeeper = Hakeem Olajuwon (+0.67) Seats 11-12 same tier = Nikola Jokic (-0.56) / LeBron James (-0.59) Seat 13 = Stephen Curry (-2.78) Result files: results_final_en.csv, grid_margins_en.csv