#!/usr/bin/env python3 """ Open-Weight Model Share Follows S-Curve Adoption -- AI Prediction 5 Theory (Paper 6, Section 8-9: R0 from Adoption Data + Potts Regime Shift): Open-weight models should follow logistic-SIR adoption dynamics with reproduction number R0 > 1 (self-sustaining adoption). The ceiling L should exceed 50% (mesh captures majority of inference volume), consistent with the Potts crystallization prediction of first-order phase transition. Method: Subtest 1 (S-curve): Fit logistic vs linear to OpenRouter open-weight token share time series (N=8 data points, 2024-2025). Compare via AICc. Parametric bootstrap p-value under linear null. Subtest 2 (ceiling): Estimated logistic ceiling L. Theory predicts L > 0.50. Bootstrap CI from sector-resampled logistic fits. Subtest 3 (R0): Paper's Table 15 reports implied R0 from quarterly adoption changes. One-sample t-test H0: mean R0 <= 1 vs H1: R0 > 1. Data sources: - Paper 6 Table 15: OpenRouter open-weight token share, quarterly 2024 (Jan, Mar, Jun, Sep, Dec 2024; Jan 2025) - OpenRouter State of AI report (Bai et al., 2025, arxiv:2601.10088): "close to 80% closed May-Sep 2025" and "one-third by late 2025" - Paper 6 Table 15: Implied R0 values from logistic-SIR formula R0(t) = 1 + (ds/dt) / (delta * s * (1-s)) Lean: No theorem exists for this prediction. Review Notes (2026-03-11): Theory match: PARTIAL — Paper 6 models adoption via R0 dynamics (Section 8). No Lean theorem. The Potts regime shift (Section 9) predicts first-order (discontinuous) transition, but test checks smooth logistic. I/O audit: ADEQUATE — OpenRouter token share is a citable proxy for open-weight adoption through a centralized aggregator. Not identical to total market share or distributed inference share. Paper distinguishes these (Section 8). Data-signal alignment: ADEQUATE — N=8 for S-curve fit (was N=4 fabricated). All data points citable. AICc defined (T-k-1=4 for logistic). R0 subtest uses paper's own values (N=6, citable). Bootstrap provides p-values. Code quality: REDESIGNED 2026-03-11 — replaced fabricated data with citable OpenRouter data, added bootstrap p-value, AICc, R0 subtest, figures. Confidence: MEDIUM — S-curve fit still has small N. R0 subtest is stronger (uses paper's own methodology). Data is citable but measures centralized aggregator share, not total or distributed inference market. """ import warnings warnings.filterwarnings("ignore") import numpy as np import sys, os sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..', '..')) from common.stats import log_result, compute_aic_bic _cache = {} _ALL_SUBTESTS = [ 'open_weight_share_scurve', 'open_weight_share_ceiling', 'open_weight_r0_above_unity', ] # OpenRouter open-weight token share time series # Source 1: Paper 6 Table 15 (AI_Transition.tex, from OpenRouter platform data) # Source 2: OpenRouter State of AI report (Bai et al., 2025, arxiv:2601.10088) # NOTE: Measures share through centralized aggregator (OpenRouter), not total # inference market or distributed inference specifically. OPENROUTER_SHARE = [ # Paper 6 Table 15 (quarterly 2024) (2024.00, 0.025), # Jan 2024 (2024.17, 0.050), # Mar 2024 (2024.42, 0.080), # Jun 2024 (2024.67, 0.120), # Sep 2024 (2024.92, 0.250), # Dec 2024 (DeepSeek V3 release spike) (2025.00, 0.180), # Jan 2025 (post-spike reversion) # State of AI report (arxiv:2601.10088) # "close to 80% closed" during May-Sep 2025 → ~20% open-weight (2025.42, 0.200), # ~Jun 2025 (midpoint of May-Sep 2025 period) # "reaching approximately one-third of usage by late 2025" (2025.83, 0.330), # ~Nov 2025 ] # Implied R0 from Paper 6 Table 15 # Computed via R0(t) = 1 + (ds/dt) / (delta * s(t) * (1-s(t))) # delta calibrated from IBM inference displacement trajectory (Section 15.4) IMPLIED_R0 = [ # (period_label, s_start, ds_dt, R0) ('Jan-Mar 2024', 0.025, 0.008, 1.44), ('Mar-Jun 2024', 0.050, 0.008, 1.23), ('Jun-Sep 2024', 0.080, 0.010, 1.16), ('Sep-Nov 2024', 0.120, 0.020, 1.22), ('Dec24-Jan25', 0.250, 0.098, 1.61), # DeepSeek spike ('Jan-Feb 2025', 0.180, -0.069, 0.59), # Post-spike reversion ] def _logistic(t, L, k, t0): """Logistic function: share(t) = L / (1 + exp(-k*(t-t0))).""" return L / (1.0 + np.exp(-k * (t - t0))) def _fit_logistic(years, shares, p0=None, bounds=None): """Fit logistic curve via scipy.optimize.curve_fit.""" from scipy.optimize import curve_fit if p0 is None: p0 = [0.5, 3.0, 2024.8] if bounds is None: bounds = ([0.15, 0.1, 2023.0], [1.0, 20.0, 2026.0]) popt, pcov = curve_fit(_logistic, years, shares, p0=p0, bounds=bounds, maxfev=10000) return popt, pcov def _r_squared(y_true, y_pred): """Coefficient of determination.""" ss_res = np.sum((y_true - y_pred) ** 2) ss_tot = np.sum((y_true - np.mean(y_true)) ** 2) if ss_tot == 0: return float('nan') return 1.0 - ss_res / ss_tot def _aicc(rss, k, T): """Small-sample corrected AIC.""" aic, _ = compute_aic_bic(rss, k=k, T=T) if np.isnan(aic) or T - k - 1 <= 0: return float('nan') correction = 2 * k * (k + 1) / (T - k - 1) return aic + correction def main(): results = [] years = np.array([y for y, _ in OPENROUTER_SHARE], dtype=float) shares = np.array([s for _, s in OPENROUTER_SHARE], dtype=float) n = len(years) # ================================================================ # Subtest 1: Logistic vs linear fit (S-curve) # ================================================================ # Linear fit A = np.column_stack([np.ones(n), years]) beta_lin, _, _, _ = np.linalg.lstsq(A, shares, rcond=None) shares_lin_pred = A @ beta_lin r2_linear = _r_squared(shares, shares_lin_pred) rss_linear = float(np.sum((shares - shares_lin_pred) ** 2)) # Logistic fit try: popt, pcov = _fit_logistic(years, shares) L, k_rate, t0 = popt shares_log_pred = _logistic(years, L, k_rate, t0) r2_logistic = _r_squared(shares, shares_log_pred) rss_logistic = float(np.sum((shares - shares_log_pred) ** 2)) fit_ok = True except Exception: L, k_rate, t0 = float('nan'), float('nan'), float('nan') r2_logistic = float('nan') rss_logistic = float('nan') shares_log_pred = np.full(n, np.nan) fit_ok = False if not fit_ok: results.append(log_result( 'open_weight_share_scurve', np.nan, np.nan, 'SKIPPED', description='Logistic fit failed.')) results.append(log_result( 'open_weight_share_ceiling', np.nan, np.nan, 'SKIPPED', description='Logistic fit failed — ceiling not estimated.')) else: # AICc comparison aicc_lin = _aicc(rss_linear, k=2, T=n) aicc_log = _aicc(rss_logistic, k=3, T=n) delta_aicc = aicc_lin - aicc_log # positive → logistic better logistic_better_aicc = delta_aicc > 0 if not np.isnan(delta_aicc) else False # Parametric bootstrap p-value under linear null rng = np.random.default_rng(42) n_boot = 2000 se_lin = np.sqrt(rss_linear / max(n - 2, 1)) boot_delta_aicc = [] for _ in range(n_boot): y_b = shares_lin_pred + rng.normal(0, se_lin, size=n) y_b = np.clip(y_b, 0.001, 0.999) # Linear fit beta_b = np.linalg.lstsq(A, y_b, rcond=None)[0] rss_b_lin = float(np.sum((y_b - A @ beta_b) ** 2)) # Logistic fit try: popt_b, _ = _fit_logistic(years, y_b) pred_b = _logistic(years, *popt_b) rss_b_log = float(np.sum((y_b - pred_b) ** 2)) d_aicc = _aicc(rss_b_lin, 2, n) - _aicc(rss_b_log, 3, n) boot_delta_aicc.append(d_aicc) except Exception: pass boot_delta_aicc = np.array(boot_delta_aicc) if len(boot_delta_aicc) > 100: p_bootstrap = float(np.mean(boot_delta_aicc >= delta_aicc)) else: p_bootstrap = float('nan') # Verdict: logistic better by AICc AND bootstrap significant ceiling_above_50 = L > 0.5 if logistic_better_aicc and p_bootstrap < 0.10: verd_scurve = 'CONSISTENT' elif logistic_better_aicc or (not np.isnan(p_bootstrap) and p_bootstrap < 0.20): verd_scurve = 'AMBIGUOUS' else: verd_scurve = 'INCONSISTENT' results.append(log_result( 'open_weight_share_scurve', stat=float(delta_aicc) if not np.isnan(delta_aicc) else float('nan'), p=p_bootstrap, verd=verd_scurve, details={ 'r2_logistic': round(r2_logistic, 4), 'r2_linear': round(r2_linear, 4), 'aicc_logistic': round(aicc_log, 2) if not np.isnan(aicc_log) else float('nan'), 'aicc_linear': round(aicc_lin, 2) if not np.isnan(aicc_lin) else float('nan'), 'delta_aicc': round(delta_aicc, 2) if not np.isnan(delta_aicc) else float('nan'), 'logistic_ceiling_L': round(L, 4), 'logistic_growth_rate_k': round(k_rate, 4), 'logistic_midpoint_t0': round(t0, 2), 'logistic_better_aicc': logistic_better_aicc, 'n_data_points': n, 'n_boot_valid': len(boot_delta_aicc), }, description=( f'Tests Potts regime shift prediction (Paper 6 Section 8): open-weight ' f'inference adoption follows logistic S-curve rather than linear growth. ' f'Nonlinear least squares: share(t)=L/(1+exp(-k(t-t0))) vs share(t)=a+bt, ' f'compared by AICc (small-sample corrected). Parametric bootstrap ' f'({len(boot_delta_aicc)} valid reps) under linear null. ' f'Data: OpenRouter token share (Paper 6 Table 15 + arxiv:2601.10088), ' f'N={n} quarterly/semiannual observations. ' f'ΔAICc={delta_aicc:.1f} (positive=logistic better), ' f'p_boot={p_bootstrap:.3f}. ' f'CONSISTENT if ΔAICc>0 AND p<0.10.' ), )) # ================================================================ # Subtest 2: Ceiling estimate # ================================================================ # Bootstrap CI on L by resampling residuals boot_L = [] resid_log = shares - shares_log_pred for _ in range(n_boot): idx_b = rng.choice(n, size=n, replace=True) y_b = shares_log_pred + resid_log[idx_b] y_b = np.clip(y_b, 0.001, 0.999) try: popt_b, _ = _fit_logistic(years, y_b) boot_L.append(float(popt_b[0])) except Exception: pass boot_L = np.array(boot_L) if len(boot_L) > 100: ci_lo_L = float(np.percentile(boot_L, 5)) ci_hi_L = float(np.percentile(boot_L, 95)) pct_above_50 = float(np.mean(boot_L > 0.5)) else: ci_lo_L, ci_hi_L = float('nan'), float('nan') pct_above_50 = float('nan') if ceiling_above_50: ceil_verd = 'CONSISTENT' elif L > 0.4 or (not np.isnan(pct_above_50) and pct_above_50 > 0.25): ceil_verd = 'AMBIGUOUS' else: ceil_verd = 'INCONSISTENT' results.append(log_result( 'open_weight_share_ceiling', stat=float(L), p=float('nan'), verd=ceil_verd, details={ 'estimated_ceiling': round(L, 4), 'predicted_range': '[0.5, 0.8]', 'boot_ci_90_lo': round(ci_lo_L, 4) if not np.isnan(ci_lo_L) else float('nan'), 'boot_ci_90_hi': round(ci_hi_L, 4) if not np.isnan(ci_hi_L) else float('nan'), 'boot_pct_above_50': round(pct_above_50, 3) if not np.isnan(pct_above_50) else float('nan'), 'n_boot_valid': len(boot_L), }, description=( f'Tests mesh equilibrium ceiling: logistic asymptote L should exceed ' f'0.50 (mesh captures majority of inference). Estimated from 3-parameter ' f'logistic fit to {n} OpenRouter data points. L={L:.3f}. ' f'Bootstrap 90% CI on L: [{ci_lo_L:.3f}, {ci_hi_L:.3f}]. ' f'{pct_above_50*100:.0f}% of bootstrap replications have L>0.50. ' f'CONSISTENT if L>0.50, AMBIGUOUS if L>0.40 or >25% boot above 0.50.' ), )) # Cache for figures _cache['years'] = years _cache['shares'] = shares _cache['shares_lin_pred'] = shares_lin_pred _cache['shares_log_pred'] = shares_log_pred _cache['L'] = L _cache['k_rate'] = k_rate _cache['t0'] = t0 _cache['r2_logistic'] = r2_logistic _cache['r2_linear'] = r2_linear _cache['beta_lin'] = beta_lin _cache['delta_aicc'] = delta_aicc _cache['p_bootstrap'] = p_bootstrap _cache['boot_delta_aicc'] = boot_delta_aicc _cache['boot_L'] = boot_L # ================================================================ # Subtest 3: Implied R0 > 1 # ================================================================ r0_values = np.array([r0 for _, _, _, r0 in IMPLIED_R0]) r0_labels = [lbl for lbl, _, _, _ in IMPLIED_R0] n_r0 = len(r0_values) from scipy.stats import ttest_1samp t_stat_r0, p_two = ttest_1samp(r0_values, 1.0) # One-tailed: H0: mean <= 1, H1: mean > 1 p_r0 = float(p_two / 2) if t_stat_r0 > 0 else float(1 - p_two / 2) mean_r0 = float(np.mean(r0_values)) std_r0 = float(np.std(r0_values, ddof=1)) # Also compute excluding spike-reversion (last 2 rows) r0_excl = r0_values[:4] t_excl, p_two_excl = ttest_1samp(r0_excl, 1.0) p_r0_excl = float(p_two_excl / 2) if t_excl > 0 else float(1 - p_two_excl / 2) sign_ok = mean_r0 > 1 if p_r0 < 0.10 and sign_ok: verd_r0 = 'CONSISTENT' elif sign_ok and p_r0 < 0.20: verd_r0 = 'AMBIGUOUS' else: verd_r0 = 'INCONSISTENT' results.append(log_result( 'open_weight_r0_above_unity', stat=mean_r0, p=p_r0, verd=verd_r0, details={ 'mean_r0': round(mean_r0, 3), 'std_r0': round(std_r0, 3), 't_stat': round(float(t_stat_r0), 3), 'n_periods': n_r0, 'r0_values': {lbl: round(r0, 2) for lbl, _, _, r0 in IMPLIED_R0}, 'excl_spike_mean_r0': round(float(np.mean(r0_excl)), 3), 'excl_spike_p': round(p_r0_excl, 4), }, description=( f'Tests R0>1 prediction (Paper 6 Section 8): open-weight adoption is ' f'self-sustaining (R0>1) vs decaying (R0<1). One-sample t-test on ' f'{n_r0} implied R0 values from Paper 6 Table 15 (quarterly OpenRouter ' f'adoption changes via logistic-SIR formula). ' f'H0: mean R0<=1, H1: mean R0>1. ' f'Mean R0={mean_r0:.3f}, t={t_stat_r0:.2f}, p={p_r0:.4f} (one-tailed). ' f'Excl. spike-reversion (N=4): mean={np.mean(r0_excl):.3f}, p={p_r0_excl:.4f}. ' f'CONSISTENT if p<0.10 and mean R0>1.' ), )) _cache['r0_values'] = r0_values _cache['r0_labels'] = r0_labels _cache['mean_r0'] = mean_r0 _cache['t_stat_r0'] = float(t_stat_r0) _cache['p_r0'] = p_r0 return results def figures(): """Generate per-subtest figures.""" if not _cache: main() paths = [] import matplotlib.pyplot as plt from common.plotting import setup_figure, save_figure years = _cache.get('years') shares = _cache.get('shares') L = _cache.get('L') k_rate = _cache.get('k_rate') t0 = _cache.get('t0') if years is None or shares is None: return paths # --- Figure 1: S-curve vs linear fit --- if L is not None and not np.isnan(L): fig, ax = setup_figure(width=9, height=5) ax.scatter(years, shares, s=80, c='black', zorder=5, label='OpenRouter data') t_fine = np.linspace(2023.8, 2027, 200) beta_lin = _cache.get('beta_lin') if beta_lin is not None: lin_pred = beta_lin[0] + beta_lin[1] * t_fine ax.plot(t_fine, lin_pred, '--', color='#b2182b', lw=2, label=f'Linear (R²={_cache.get("r2_linear", 0):.3f})') log_pred = _logistic(t_fine, L, k_rate, t0) ax.plot(t_fine, log_pred, '-', color='#2166ac', lw=2, label=f'Logistic (R²={_cache.get("r2_logistic", 0):.3f})') ax.axhline(L, color='#2166ac', ls=':', alpha=0.5, label=f'Ceiling L={L:.3f}') ax.axhline(0.5, color='gray', ls='--', alpha=0.4, label='50% threshold') delta = _cache.get('delta_aicc', float('nan')) p_b = _cache.get('p_bootstrap', float('nan')) ax.set_xlabel('Year', fontsize=11) ax.set_ylabel('Open-Weight Token Share (OpenRouter)', fontsize=11) ax.set_title( f'Open-Weight Share: Logistic vs Linear\n' f'ΔAICc={delta:.1f}, p_boot={p_b:.3f}, L={L:.3f} (N={len(years)})', fontsize=12) ax.set_ylim(-0.02, 0.75) ax.legend(fontsize=9, loc='upper left') ax.grid(True, alpha=0.25) paths.extend(save_figure(fig, 'open_weight_share_scurve', subfolder='applications')) # --- Figure 2: Ceiling bootstrap --- boot_L = _cache.get('boot_L') if boot_L is not None and len(boot_L) > 50: fig2, ax2 = setup_figure() ax2.hist(boot_L, bins=50, color='#4393c3', edgecolor='white', alpha=0.8, density=True) ax2.axvline(0.5, color='red', lw=2, ls='--', label='Threshold L=0.50') if L is not None and not np.isnan(L): ax2.axvline(L, color='black', lw=2, label=f'Point estimate L={L:.3f}') ci_lo = np.percentile(boot_L, 5) ci_hi = np.percentile(boot_L, 95) ax2.axvline(ci_lo, color='#2166ac', lw=1.5, ls=':', label=f'90% CI: [{ci_lo:.3f}, {ci_hi:.3f}]') ax2.axvline(ci_hi, color='#2166ac', lw=1.5, ls=':') pct = np.mean(boot_L > 0.5) * 100 ax2.set_xlabel('Bootstrap Ceiling L') ax2.set_ylabel('Density') ax2.set_title(f'Bootstrap Distribution of Ceiling L ({pct:.0f}% > 0.50)') ax2.legend(fontsize=9) ax2.grid(True, alpha=0.25) paths.extend(save_figure(fig2, 'open_weight_share_ceiling', subfolder='applications')) # --- Figure 3: Implied R0 --- r0_vals = _cache.get('r0_values') r0_labels = _cache.get('r0_labels') if r0_vals is not None: fig3, ax3 = setup_figure(width=9, height=5) colors = ['#2166ac'] * 4 + ['#b2182b', '#b2182b'] # spike-reversion in red bars = ax3.bar(range(len(r0_vals)), r0_vals, color=colors, edgecolor='black', lw=0.5, alpha=0.8) ax3.axhline(1.0, color='red', lw=2, ls='--', label='R₀ = 1 (critical)') ax3.set_xticks(range(len(r0_vals))) ax3.set_xticklabels(r0_labels, rotation=30, ha='right', fontsize=9) ax3.set_ylabel('Implied R₀') mean_r0 = _cache.get('mean_r0', np.mean(r0_vals)) p_r0 = _cache.get('p_r0', float('nan')) ax3.set_title( f'Implied R₀ from OpenRouter Adoption (Paper 6 Table 15)\n' f'Mean R₀={mean_r0:.3f}, t={_cache.get("t_stat_r0", 0):.2f}, ' f'p={p_r0:.4f} (one-tailed)', fontsize=12) ax3.legend(fontsize=10) ax3.grid(True, alpha=0.25, axis='y') # Annotate spike-reversion ax3.annotate('DeepSeek\nspike', xy=(4, r0_vals[4]), fontsize=8, ha='center', va='bottom', color='#b2182b') ax3.annotate('Reversion', xy=(5, r0_vals[5]), fontsize=8, ha='center', va='top', color='#b2182b') paths.extend(save_figure(fig3, 'open_weight_r0_above_unity', subfolder='applications')) return paths if __name__ == '__main__': for r in main(): print(f" {r['name']}: {r['verdict']} " f"(stat={r['statistic']:.4f}, p={r['p_value']})")

statistic	0.4885
predicted range	[0.5, 0.8]
estimated ceiling	0.4885

Open Weight Share Ceiling

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