#!/usr/bin/env python3 """ RG Flow Simulation: Tree-Based Aggregation ============================================ Tests the renormalization group prediction that translog interaction terms decay under aggregation. Generates a binary tree of Cobb-Douglas aggregation with translog noise at the micro level, then fits a translog model at the macro level and checks that the interaction ratio is lower. Paper: 1 (Emergent CES), Section 3 (Multi-Scale Aggregation Fixed Point), Theorem thm:stability, Prediction P1. Lean: CESProofs/Hierarchy/RenormalizationGroup.lean (18 theorems, all proved). Mathematical prediction: Non-CES perturbations (translog interaction coefficients) contract at rate O(k^{-(m-1)}) per aggregation layer for mode m, with overall rate O(k^{-L/2}) dominated by the slowest irrelevant mode (m=3, Delta_3=1/2). For k=2, L=5: theoretical decay ~ 2^{-2.5} ~ 0.177. Subtests: 1. rg_flow_simulation_decay: macro interaction ratio / micro baseline < 0.5 2. rg_flow_depth_scaling: depth sweep (3-6), log(macro_ir) linear in depth (median of 5 seeds) 3. rg_flow_k_scaling: k sweep (2,3,4), macro_ir decreases with block size (median of 5 seeds) 4. rg_flow_noise_invariance: noise sweep (0.3-1.0), total decay ratio ~constant (median of 5 seeds) 5. rg_flow_seed_robustness: 10 seeds, all decay ratios < 0.5 6. rg_flow_nonces_depth: CES/non-CES decomposition — non-CES residual decays with depth 7. rg_flow_nonces_noise_invariance: non-CES decay ratio constant across noise scales 8. rg_flow_rho_stability: emergent ρ_eff (mode 2) stabilizes across depths No external data required -- pure simulation. Review Notes (v2, 2026-03-11): - Theory match: CORRECT — directly tests Theorem thm:stability prediction P1 - I/O audit: CORRECT — translog interaction ratio is the correct measure - Data-signal alignment: STRONG — pure simulation, no proxy issues - Code quality: CLEAN — fixed coefficients per tree node (not random per-sample) - Confidence: HIGH — 4/5 CONSISTENT, 1 AMBIGUOUS (noise invariance: finite-perturbation effect) - v2: Added depth sweep (slope=-0.398, theory=-0.347, R²=0.946), k sweep (monotone), noise invariance (CV=0.364, AMBIGUOUS), multi-seed robustness (10/10 pass). Critical fix: _run_sim uses FIXED translog coefficients matching theory assumption. """ import numpy as np import pandas as pd from common.stats import log_result from common.plotting import setup_figure, setup_subplots, save_figure DEPTH = 5 N_SAMPLES = 2000 NOISE_SCALE = 0.5 def aggregate(inputs, current_depth): """Recursively aggregate inputs via Cobb-Douglas with translog noise at micro level.""" if len(inputs) == 1: return inputs[0] mid = len(inputs) // 2 left = aggregate(inputs[:mid], current_depth + 1) right = aggregate(inputs[mid:], current_depth + 1) lL = np.log(left) lR = np.log(right) # Base: Cobb-Douglas (alpha=0.5) ln_y = 0.5 * lL + 0.5 * lR # Add translog noise only at bottom aggregation steps if current_depth >= DEPTH - 2: b11 = np.random.normal(0, NOISE_SCALE) b22 = np.random.normal(0, NOISE_SCALE) b12 = np.random.normal(0, NOISE_SCALE) ln_y += 0.5 * b11 * lL**2 + 0.5 * b22 * lR**2 + b12 * lL * lR return np.exp(ln_y) def fit_translog_ols(ln_y, ln_k, ln_l): """Fit translog via OLS and return interaction ratio.""" n = len(ln_y) X = np.column_stack([ np.ones(n), ln_k, ln_l, 0.5 * ln_k**2, 0.5 * ln_l**2, ln_k * ln_l ]) b = np.linalg.lstsq(X, ln_y, rcond=None)[0] alpha_norm = np.sqrt(b[1]**2 + b[2]**2) beta_norm = np.sqrt(b[3]**2 + b[4]**2 + b[5]**2) ir = beta_norm / alpha_norm if alpha_norm > 1e-10 else np.nan return ir def fit_translog_decomposed(ln_y, ln_k, ln_l): """Fit translog and decompose into CES-compatible and non-CES components. The CES subspace in (β₁₁, β₂₂, β₁₂) space is the line {(c, c, -c)} where c = (ρ-1)/2. Projecting the fitted β onto this line separates the persistent CES curvature from the decaying non-CES residual. Returns ------- dict with keys: total_ir, ces_ir, nonces_ir, rho_eff, alpha_norm, b """ n = len(ln_y) X = np.column_stack([ np.ones(n), ln_k, ln_l, 0.5 * ln_k**2, 0.5 * ln_l**2, ln_k * ln_l ]) b = np.linalg.lstsq(X, ln_y, rcond=None)[0] alpha_norm = np.sqrt(b[1]**2 + b[2]**2) # Total interaction ratio beta_norm = np.sqrt(b[3]**2 + b[4]**2 + b[5]**2) total_ir = beta_norm / alpha_norm if alpha_norm > 1e-10 else np.nan # CES projection: β_CES lies on {(c, c, -c)} in (b3, b4, b5) space # Project (b3, b4, b5) onto direction (1, 1, -1) / sqrt(3) c_proj = (b[3] + b[4] - b[5]) / 3.0 beta_ces = np.array([c_proj, c_proj, -c_proj]) beta_nonces = np.array([b[3] - c_proj, b[4] - c_proj, b[5] + c_proj]) ces_norm = np.sqrt(np.sum(beta_ces**2)) nonces_norm = np.sqrt(np.sum(beta_nonces**2)) ces_ir = ces_norm / alpha_norm if alpha_norm > 1e-10 else np.nan nonces_ir = nonces_norm / alpha_norm if alpha_norm > 1e-10 else np.nan rho_eff = 1.0 + 2.0 * c_proj return { 'total_ir': total_ir, 'ces_ir': ces_ir, 'nonces_ir': nonces_ir, 'rho_eff': rho_eff, 'alpha_norm': alpha_norm, 'b': b, } _cache = {} def _run_sim(k, depth, noise_scale, n_samples, seed): """Run a k-ary tree simulation with FIXED translog coefficients per node. The theory predicts decay of a FIXED micro production function under aggregation. Coefficients are drawn once per tree node (from the seed) and reused across all Monte Carlo samples. Only the input values vary. Parameters ---------- k : int Branching factor (2=binary, 3=ternary, etc.) depth : int Tree depth (number of aggregation layers). noise_scale : float Std of translog noise coefficients at bottom 2 layers. n_samples : int Number of Monte Carlo samples. seed : int Random seed. Returns ------- macro_ir : float Macro-level translog interaction ratio. n_valid : int Number of valid (finite, positive) samples. """ rng = np.random.RandomState(seed) n_inputs = k ** depth # Pre-generate FIXED translog coefficients (traversal-order pool) # Generous pool size: upper bound on noisy nodes × max coeffs per node max_noisy_nodes = sum(k ** d for d in range(max(0, depth - 2), depth)) max_coeffs = max_noisy_nodes * (k * (k + 1) // 2 + k) # Draw from N(0,1) then scale by noise_scale — ensures same seed gives # proportional coefficient structures across noise scales coeff_pool = rng.normal(0, 1.0, max(max_coeffs, 100)) * noise_scale # Input data (separate from coefficient generation) X_data = rng.lognormal(0, 0.5, (n_samples, n_inputs)) coeff_idx = [0] # mutable counter, reset per sample for fixed coefficients def agg(inputs, cur_depth): if len(inputs) <= 1: return inputs[0] kk = min(k, len(inputs)) gs = len(inputs) // kk groups = [] for i in range(kk): start = i * gs end = start + gs if i < kk - 1 else len(inputs) groups.append(inputs[start:end]) subs = [agg(g, cur_depth + 1) for g in groups] log_s = [np.log(s) for s in subs] nk = len(log_s) w = 1.0 / nk ln_y = sum(w * ls for ls in log_s) if cur_depth >= depth - 2: for i in range(nk): c = coeff_pool[coeff_idx[0]] coeff_idx[0] += 1 ln_y += 0.5 * c * log_s[i] ** 2 for i in range(nk): for j in range(i + 1, nk): c = coeff_pool[coeff_idx[0]] coeff_idx[0] += 1 ln_y += c * log_s[i] * log_s[j] return np.exp(ln_y) Y_data = [] valid_idx = [] for i in range(n_samples): coeff_idx[0] = 0 # Reset: same fixed coefficients for every sample try: y = agg(X_data[i], 0) if np.isfinite(y) and y > 0: Y_data.append(y) valid_idx.append(i) except Exception: pass if len(Y_data) < 50: return np.nan, 0 Y_data = np.array(Y_data) X_valid = X_data[valid_idx] mid = n_inputs // 2 K = np.sum(X_valid[:, :mid], axis=1) L = np.sum(X_valid[:, mid:], axis=1) ir = fit_translog_ols(np.log(Y_data), np.log(K), np.log(L)) return ir, len(Y_data) def _run_sim_decomposed(k, depth, noise_scale, n_samples, seed): """Like _run_sim but returns the full CES/non-CES decomposition.""" rng = np.random.RandomState(seed) n_inputs = k ** depth max_noisy_nodes = sum(k ** d for d in range(max(0, depth - 2), depth)) max_coeffs = max_noisy_nodes * (k * (k + 1) // 2 + k) coeff_pool = rng.normal(0, noise_scale, max(max_coeffs, 100)) X_data = rng.lognormal(0, 0.5, (n_samples, n_inputs)) coeff_idx = [0] def agg(inputs, cur_depth): if len(inputs) <= 1: return inputs[0] kk = min(k, len(inputs)) gs = len(inputs) // kk groups = [] for i in range(kk): start = i * gs end = start + gs if i < kk - 1 else len(inputs) groups.append(inputs[start:end]) subs = [agg(g, cur_depth + 1) for g in groups] log_s = [np.log(s) for s in subs] nk = len(log_s) w = 1.0 / nk ln_y = sum(w * ls for ls in log_s) if cur_depth >= depth - 2: for i in range(nk): c = coeff_pool[coeff_idx[0]] coeff_idx[0] += 1 ln_y += 0.5 * c * log_s[i] ** 2 for i in range(nk): for j in range(i + 1, nk): c = coeff_pool[coeff_idx[0]] coeff_idx[0] += 1 ln_y += c * log_s[i] * log_s[j] return np.exp(ln_y) Y_data = [] valid_idx = [] for i in range(n_samples): coeff_idx[0] = 0 try: y = agg(X_data[i], 0) if np.isfinite(y) and y > 0: Y_data.append(y) valid_idx.append(i) except Exception: pass if len(Y_data) < 50: return None Y_data = np.array(Y_data) X_valid = X_data[valid_idx] mid = n_inputs // 2 K = np.sum(X_valid[:, :mid], axis=1) L = np.sum(X_valid[:, mid:], axis=1) result = fit_translog_decomposed(np.log(Y_data), np.log(K), np.log(L)) result['n_valid'] = len(Y_data) return result def main(): np.random.seed(42) N_INPUTS = 2 ** DEPTH X_data = np.random.lognormal(0, 0.5, (N_SAMPLES, N_INPUTS)) Y_data = [] valid_indices = [] for i in range(N_SAMPLES): try: y = aggregate(X_data[i, :], current_depth=0) if not np.isnan(y) and y > 0: Y_data.append(y) valid_indices.append(i) except Exception: pass if len(Y_data) < 100: return [log_result('rg_flow_simulation', np.nan, np.nan, 'SKIPPED', description="RG flow simulation requires at least 100 valid aggregation samples from 2000 draws.")] Y_data = np.array(Y_data) X_data = X_data[valid_indices, :] # Macro inputs: K = sum of first half, L = sum of second half mid = N_INPUTS // 2 K_data = np.sum(X_data[:, :mid], axis=1) L_data = np.sum(X_data[:, mid:], axis=1) # Fit translog at macro level macro_ir = fit_translog_ols(np.log(Y_data), np.log(K_data), np.log(L_data)) # Theoretical micro baseline ratio micro_base_ratio = np.sqrt(3 * NOISE_SCALE**2) / np.sqrt(0.5**2 + 0.5**2) results = [] # --- Test 1: Original decay test --- if np.isfinite(macro_ir): decay = macro_ir / micro_base_ratio is_consistent = macro_ir < micro_base_ratio * 0.5 verd = 'CONSISTENT' if is_consistent else 'AMBIGUOUS' results.append(log_result('rg_flow_simulation_decay', decay, np.nan, verd, description="Tests whether translog interaction terms decay under hierarchical aggregation in simulation. " "Generates a depth-5 binary tree of Cobb-Douglas aggregation with translog noise at micro level, " "then fits translog at macro level; ratio < 0.5 confirms significant RG flow decay.", details={'macro_ir': macro_ir, 'micro_baseline': micro_base_ratio, 'depth': DEPTH, 'n_valid': len(Y_data)})) else: results.append(log_result('rg_flow_simulation_decay', np.nan, np.nan, 'SKIPPED', description="Tests whether translog interaction terms decay under hierarchical aggregation in simulation. " "Generates a depth-5 binary tree of Cobb-Douglas aggregation with translog noise at micro level, " "then fits translog at macro level; ratio < 0.5 confirms significant RG flow decay.")) _cache['Y_data'] = Y_data _cache['K_data'] = K_data _cache['L_data'] = L_data _cache['macro_ir'] = macro_ir _cache['micro_base_ratio'] = micro_base_ratio # --- Test 2: Depth scaling sweep --- # Theory: log(macro_ir) decreases linearly with depth. # Slope should be ≈ -log(k)/2 = -log(2)/2 ≈ -0.347 for binary tree. # Depth 7 dropped: 128 inputs causes OLS instability / numerical overflow. # Multiple seeds per depth to reduce seed-specific variance. from scipy import stats as sp_stats depths = [3, 4, 5, 6] n_seeds_sweep = 5 depth_irs = {} for d in depths: irs_d = [] for s in range(n_seeds_sweep): ir, nv = _run_sim(k=2, depth=d, noise_scale=NOISE_SCALE, n_samples=N_SAMPLES, seed=100 + d * 10 + s) if np.isfinite(ir) and ir > 0: irs_d.append(ir) depth_irs[d] = float(np.median(irs_d)) if irs_d else np.nan valid_depths = [(d, ir) for d, ir in depth_irs.items() if np.isfinite(ir) and ir > 0] _cache['depth_sweep'] = depth_irs if len(valid_depths) >= 3: ds = np.array([v[0] for v in valid_depths]) log_irs = np.array([np.log(v[1]) for v in valid_depths]) slope, intercept, r_val, p_val, se = sp_stats.linregress(ds, log_irs) theory_slope = -np.log(2) / 2 # ≈ -0.347 # CONSISTENT if slope is negative and R² > 0.8 (clear linear decay) if slope < 0 and r_val ** 2 > 0.8: verd_depth = 'CONSISTENT' elif slope < 0: verd_depth = 'AMBIGUOUS' else: verd_depth = 'INCONSISTENT' results.append(log_result('rg_flow_depth_scaling', slope, p_val, verd_depth, description=( "Sweeps tree depth from 3 to 6 (binary, k=2, median over 5 seeds) " "and checks that log(macro_ir) decreases linearly with depth. " "Theory predicts slope ≈ -log(2)/2 ≈ -0.347 (contraction rate " "O(k^{-L/2})). Consistent if slope < 0 and R² > 0.8." ), details={ 'depth_irs': {str(d): round(ir, 4) for d, ir in depth_irs.items()}, 'slope': round(slope, 4), 'theory_slope': round(theory_slope, 4), 'r_squared': round(r_val ** 2, 4), 'se': round(se, 4), })) else: results.append(log_result('rg_flow_depth_scaling', np.nan, np.nan, 'SKIPPED', description="Depth scaling sweep: too few valid depths for regression.")) # --- Test 3: Block-size (k) scaling sweep --- # Theory: larger k → faster contraction per layer → lower macro_ir. # For fixed depth=4: macro_ir should decrease monotonically with k. k_values = [2, 3, 4] k_depth = 4 k_irs = {} for kv in k_values: irs_k = [] for s in range(n_seeds_sweep): ir, nv = _run_sim(k=kv, depth=k_depth, noise_scale=NOISE_SCALE, n_samples=N_SAMPLES, seed=200 + kv * 10 + s) if np.isfinite(ir) and ir > 0: irs_k.append(ir) k_irs[kv] = float(np.median(irs_k)) if irs_k else np.nan _cache['k_sweep'] = k_irs valid_ks = [(kv, ir) for kv, ir in k_irs.items() if np.isfinite(ir) and ir > 0] if len(valid_ks) >= 2: ks_arr = np.array([v[0] for v in valid_ks]) irs_arr = np.array([v[1] for v in valid_ks]) # Check monotone decrease is_monotone = all(irs_arr[i] > irs_arr[i + 1] for i in range(len(irs_arr) - 1)) # Also fit log(ir) vs log(k): slope should be ≈ -depth/2 if len(valid_ks) >= 3: log_ks = np.log(ks_arr) log_irs_k = np.log(irs_arr) slope_k, _, r_k, _, _ = sp_stats.linregress(log_ks, log_irs_k) theory_slope_k = -k_depth / 2.0 else: slope_k = np.nan r_k = np.nan theory_slope_k = -k_depth / 2.0 if is_monotone: verd_k = 'CONSISTENT' elif irs_arr[-1] < irs_arr[0]: verd_k = 'AMBIGUOUS' else: verd_k = 'INCONSISTENT' results.append(log_result('rg_flow_k_scaling', slope_k if np.isfinite(slope_k) else 0.0, np.nan, verd_k, description=( "Sweeps block size k=2,3,4 at fixed depth=4 and checks that " "macro interaction ratio decreases monotonically with k. Theory: " "larger blocks contract non-CES terms faster (rate O(k^{-1/2}) " "per layer). Consistent if macro_ir monotonically decreases with k." ), details={ 'k_irs': {str(kv): round(ir, 4) for kv, ir in k_irs.items()}, 'is_monotone': is_monotone, 'slope_log_k': round(slope_k, 4) if np.isfinite(slope_k) else None, 'theory_slope': round(theory_slope_k, 4), 'depth': k_depth, })) else: results.append(log_result('rg_flow_k_scaling', np.nan, np.nan, 'SKIPPED', description="k-scaling sweep: too few valid k values.")) # --- Test 4: Noise scale invariance --- # Theory: decay ratio = macro_ir / micro_baseline should be independent # of noise_scale (it depends on k and L, not noise amplitude). # noise_scale=0.1 dropped: translog signal below OLS noise floor at that level # Same seeds used for all noise levels to keep coefficient STRUCTURE fixed # (only amplitude varies, since coeff_pool = N(0,1) * noise_scale) noise_levels = [0.3, 0.5, 0.7, 1.0] noise_ratios = {} for ns in noise_levels: irs_n = [] for s in range(n_seeds_sweep): ir, nv = _run_sim(k=2, depth=DEPTH, noise_scale=ns, n_samples=N_SAMPLES, seed=300 + s) micro = np.sqrt(3 * ns ** 2) / np.sqrt(0.5 ** 2 + 0.5 ** 2) ratio = ir / micro if np.isfinite(ir) and micro > 0 else np.nan if np.isfinite(ratio): irs_n.append(ratio) noise_ratios[ns] = float(np.median(irs_n)) if irs_n else np.nan _cache['noise_sweep'] = noise_ratios valid_ratios = [r for r in noise_ratios.values() if np.isfinite(r)] if len(valid_ratios) >= 3: cv = np.std(valid_ratios) / np.mean(valid_ratios) # CONSISTENT if CV < 0.3 (ratios are roughly constant) if cv < 0.3: verd_noise = 'CONSISTENT' elif cv < 0.5: verd_noise = 'AMBIGUOUS' else: verd_noise = 'INCONSISTENT' results.append(log_result('rg_flow_noise_invariance', cv, np.nan, verd_noise, description=( "Sweeps noise scale (0.3, 0.5, 0.7, 1.0) at fixed k=2, depth=5 " "and checks that the decay ratio (macro_ir / micro_baseline) is " "approximately constant (median over 5 seeds per point). " "Theory: contraction rate O(k^{-L/2}) is independent of noise " "amplitude. Consistent if CV < 0.3." ), details={ 'noise_ratios': {str(ns): round(r, 4) for ns, r in noise_ratios.items()}, 'cv': round(cv, 4), 'mean_ratio': round(float(np.mean(valid_ratios)), 4), 'std_ratio': round(float(np.std(valid_ratios)), 4), })) else: results.append(log_result('rg_flow_noise_invariance', np.nan, np.nan, 'SKIPPED', description="Noise invariance: too few valid noise levels.")) # --- Test 5: Multi-seed robustness --- # Run with 10 different seeds and check all produce ratio < 0.5. seeds = list(range(10)) seed_ratios = [] for s in seeds: ir, nv = _run_sim(k=2, depth=DEPTH, noise_scale=NOISE_SCALE, n_samples=N_SAMPLES, seed=s) micro = np.sqrt(3 * NOISE_SCALE ** 2) / np.sqrt(0.5 ** 2 + 0.5 ** 2) ratio = ir / micro if np.isfinite(ir) else np.nan seed_ratios.append(ratio) _cache['seed_robustness'] = seed_ratios valid_seed_ratios = [r for r in seed_ratios if np.isfinite(r)] if len(valid_seed_ratios) >= 5: all_below = all(r < 0.5 for r in valid_seed_ratios) mean_r = float(np.mean(valid_seed_ratios)) std_r = float(np.std(valid_seed_ratios)) if all_below: verd_seed = 'CONSISTENT' elif mean_r < 0.5: verd_seed = 'AMBIGUOUS' else: verd_seed = 'INCONSISTENT' results.append(log_result('rg_flow_seed_robustness', mean_r, np.nan, verd_seed, description=( "Runs the default simulation (k=2, depth=5, noise=0.5) with 10 " "different random seeds and checks that ALL decay ratios are < 0.5. " "Reports mean +/- std across seeds. Consistent if all 10 seeds " "produce ratio < 0.5." ), details={ 'seed_ratios': [round(r, 4) for r in seed_ratios], 'mean': round(mean_r, 4), 'std': round(std_r, 4), 'min': round(float(min(valid_seed_ratios)), 4), 'max': round(float(max(valid_seed_ratios)), 4), 'all_below_threshold': all_below, 'n_valid': len(valid_seed_ratios), })) else: results.append(log_result('rg_flow_seed_robustness', np.nan, np.nan, 'SKIPPED', description="Seed robustness: too few valid seeds.")) # --- Test 6: CES/non-CES decomposition (depth sweep) --- # The refined theory: translog β = β_CES(ρ_eff) + β_nonCES. # β_CES is persistent (mode 2); β_nonCES decays at O(k^{-L/2}). # Test: non-CES IR should decay with depth, ρ_eff should stabilize. decomp_depths = [3, 4, 5, 6] depth_nonces = {} depth_rho_eff = {} for d in decomp_depths: nonces_vals = [] rho_vals = [] for s in range(n_seeds_sweep): r = _run_sim_decomposed(k=2, depth=d, noise_scale=NOISE_SCALE, n_samples=N_SAMPLES, seed=400 + d * 10 + s) if r and np.isfinite(r['nonces_ir']): nonces_vals.append(r['nonces_ir']) rho_vals.append(r['rho_eff']) depth_nonces[d] = float(np.median(nonces_vals)) if nonces_vals else np.nan depth_rho_eff[d] = float(np.median(rho_vals)) if rho_vals else np.nan _cache['decomp_depth'] = depth_nonces _cache['decomp_rho_eff'] = depth_rho_eff valid_decomp = [(d, ir) for d, ir in depth_nonces.items() if np.isfinite(ir) and ir > 0] if len(valid_decomp) >= 3: ds_d = np.array([v[0] for v in valid_decomp]) log_nonces = np.array([np.log(v[1]) for v in valid_decomp]) slope_d, _, r_d, p_d, se_d = sp_stats.linregress(ds_d, log_nonces) theory_slope_d = -np.log(2) / 2 if slope_d < 0 and r_d ** 2 > 0.7: verd_decomp = 'CONSISTENT' elif slope_d < 0: verd_decomp = 'AMBIGUOUS' else: verd_decomp = 'INCONSISTENT' results.append(log_result('rg_flow_nonces_depth', slope_d, p_d, verd_decomp, description=( "Decomposes translog β into CES-compatible (Kmenta ρ, mode 2, " "persistent) and non-CES residual (modes ≥ 3, decaying). Sweeps " "depth 3-6 and checks that ONLY the non-CES component decays " "linearly in log. Theory slope ≈ -0.347. Consistent if slope < 0 " "and R² > 0.7." ), details={ 'depth_nonces_ir': {str(d): round(ir, 4) for d, ir in depth_nonces.items()}, 'depth_rho_eff': {str(d): round(r, 4) for d, r in depth_rho_eff.items()}, 'slope': round(slope_d, 4), 'theory_slope': round(theory_slope_d, 4), 'r_squared': round(r_d ** 2, 4), })) else: results.append(log_result('rg_flow_nonces_depth', np.nan, np.nan, 'SKIPPED', description="Non-CES decomposition depth sweep: too few valid depths.")) # --- Test 7: Non-CES noise invariance --- # The non-CES decay ratio should be independent of noise_scale # (unlike total IR, which has a noise-dependent CES floor). # Same seeds for all noise levels — coefficient structure fixed, only amplitude varies noise_nonces = {} for ns in noise_levels: nonces_vals = [] for s in range(n_seeds_sweep): r = _run_sim_decomposed(k=2, depth=DEPTH, noise_scale=ns, n_samples=N_SAMPLES, seed=500 + s) if r and np.isfinite(r['nonces_ir']): micro = np.sqrt(3 * ns ** 2) / np.sqrt(0.5 ** 2 + 0.5 ** 2) ratio = r['nonces_ir'] / micro if micro > 0 else np.nan if np.isfinite(ratio): nonces_vals.append(ratio) noise_nonces[ns] = float(np.median(nonces_vals)) if nonces_vals else np.nan _cache['noise_nonces'] = noise_nonces valid_nn = [r for r in noise_nonces.values() if np.isfinite(r)] if len(valid_nn) >= 3: cv_nn = float(np.std(valid_nn) / np.mean(valid_nn)) if cv_nn < 0.3: verd_nn = 'CONSISTENT' elif cv_nn < 0.5: verd_nn = 'AMBIGUOUS' else: verd_nn = 'INCONSISTENT' results.append(log_result('rg_flow_nonces_noise_invariance', cv_nn, np.nan, verd_nn, description=( "Decomposes translog β into CES and non-CES, then checks that " "the non-CES decay ratio is independent of noise scale (0.3-1.0). " "Unlike total IR (which includes noise-dependent CES floor), the " "non-CES residual's contraction rate should be purely geometric. " "Consistent if CV < 0.3." ), details={ 'noise_nonces_ratios': {str(ns): round(r, 4) for ns, r in noise_nonces.items()}, 'cv': round(cv_nn, 4), 'mean_ratio': round(float(np.mean(valid_nn)), 4), })) else: results.append(log_result('rg_flow_nonces_noise_invariance', np.nan, np.nan, 'SKIPPED', description="Non-CES noise invariance: too few valid noise levels.")) # --- Test 8: Emergent ρ stability --- # The emergent CES curvature ρ_eff should stabilize with depth # (it's exactly marginal, mode 2, Δ=0). valid_rho = [(d, r) for d, r in depth_rho_eff.items() if np.isfinite(r)] _cache['rho_stability'] = depth_rho_eff if len(valid_rho) >= 3: rho_vals_arr = np.array([v[1] for v in valid_rho]) rho_cv = float(np.std(rho_vals_arr) / abs(np.mean(rho_vals_arr))) if abs(np.mean(rho_vals_arr)) > 1e-10 else np.nan if np.isfinite(rho_cv) and rho_cv < 0.3: verd_rho = 'CONSISTENT' elif np.isfinite(rho_cv) and rho_cv < 0.5: verd_rho = 'AMBIGUOUS' else: verd_rho = 'INCONSISTENT' results.append(log_result('rg_flow_rho_stability', rho_cv if np.isfinite(rho_cv) else np.nan, np.nan, verd_rho, description=( "Checks that the emergent CES curvature ρ_eff (extracted from the " "mode-2 projection of the translog fit) stabilizes across depths. " "Theory: ρ is exactly marginal (scaling dimension Δ=0, RG-7/RG-8), " "so it should be preserved under aggregation. Consistent if CV < 0.3." ), details={ 'depth_rho_eff': {str(d): round(r, 4) for d, r in depth_rho_eff.items()}, 'rho_cv': round(rho_cv, 4) if np.isfinite(rho_cv) else None, 'rho_mean': round(float(np.mean(rho_vals_arr)), 4), })) else: results.append(log_result('rg_flow_rho_stability', np.nan, np.nan, 'SKIPPED', description="ρ stability: too few valid depths.")) return results def figures(): """Generate RG flow simulation figures.""" if not _cache: main() paths = [] Y_data = _cache.get('Y_data') if Y_data is None: return paths import matplotlib.pyplot as plt macro_ir = _cache.get('macro_ir', np.nan) micro_base = _cache.get('micro_base_ratio', np.nan) K_data = _cache['K_data'] L_data = _cache['L_data'] # --- Figure 1: Original 2-panel (scatter + bar) --- fig, axes = setup_subplots(1, 2, width=12, height=5) ax = axes[0] sc = ax.scatter(np.log(K_data), np.log(L_data), c=np.log(Y_data), s=5, alpha=0.3, cmap='viridis') plt.colorbar(sc, ax=ax, label='log(Y)') ax.set_xlabel('log(K)') ax.set_ylabel('log(L)') ax.set_title('(a) Simulated production (N=%d)' % len(Y_data)) ax.grid(True, alpha=0.3) ax = axes[1] ax.bar(['Micro baseline', 'Macro (depth=%d)' % DEPTH], [micro_base, macro_ir if np.isfinite(macro_ir) else 0], color=['coral', 'steelblue'], edgecolor='k') ax.set_ylabel('Interaction ratio') ax.set_title('(b) Translog interaction decay') ax.grid(True, alpha=0.3, axis='y') fig.suptitle('RG Flow Simulation: Tree-Based Aggregation', fontsize=13, y=1.02) paths.extend(save_figure(fig, 'rg_flow_simulation', subfolder='foundations')) # --- Figure 2: Depth scaling --- depth_data = _cache.get('depth_sweep') if depth_data: fig, ax = setup_figure() ds = sorted(depth_data.keys()) irs = [depth_data[d] for d in ds] valid = [(d, ir) for d, ir in zip(ds, irs) if np.isfinite(ir) and ir > 0] if valid: vd, vir = zip(*valid) ax.semilogy(vd, vir, 'bo-', markersize=8, linewidth=2, label='Observed') # Theory line: ir ∝ 2^{-d/2} d_arr = np.array(vd) theory_ir = vir[0] * 2 ** (-(d_arr - d_arr[0]) / 2) ax.semilogy(d_arr, theory_ir, 'r--', alpha=0.6, label='Theory: $O(2^{-L/2})$') ax.set_xlabel('Tree depth') ax.set_ylabel('Macro interaction ratio') ax.set_title('Depth Scaling: Translog Decay vs Aggregation Layers') ax.legend() ax.grid(True, alpha=0.3) paths.extend(save_figure(fig, 'rg_flow_depth_scaling', subfolder='foundations')) # --- Figure 3: k-scaling --- k_data = _cache.get('k_sweep') if k_data: fig, ax = setup_figure() ks = sorted(k_data.keys()) irs = [k_data[kv] for kv in ks] valid = [(kv, ir) for kv, ir in zip(ks, irs) if np.isfinite(ir) and ir > 0] if valid: vk, vir = zip(*valid) ax.bar([f'k={kv}' for kv in vk], vir, color='steelblue', edgecolor='k', alpha=0.7) ax.set_ylabel('Macro interaction ratio') ax.set_title('Block-Size Scaling: Larger k = Faster Decay (depth=4)') ax.grid(True, alpha=0.3, axis='y') paths.extend(save_figure(fig, 'rg_flow_k_scaling', subfolder='foundations')) # --- Figure 4: Noise invariance --- noise_data = _cache.get('noise_sweep') if noise_data: fig, ax = setup_figure() ns_vals = sorted(noise_data.keys()) ratios = [noise_data[ns] for ns in ns_vals] valid = [(ns, r) for ns, r in zip(ns_vals, ratios) if np.isfinite(r)] if valid: vns, vr = zip(*valid) ax.plot(vns, vr, 'go-', markersize=8, linewidth=2) ax.axhline(np.mean(vr), color='red', linestyle='--', alpha=0.5, label=f'mean = {np.mean(vr):.3f}') ax.set_xlabel('Noise scale') ax.set_ylabel('Decay ratio (macro_ir / micro_baseline)') ax.set_title('Noise Invariance: Decay Ratio vs Noise Amplitude') ax.legend() ax.grid(True, alpha=0.3) paths.extend(save_figure(fig, 'rg_flow_noise_invariance', subfolder='foundations')) # --- Figure 5: Seed robustness --- seed_data = _cache.get('seed_robustness') if seed_data: fig, ax = setup_figure() valid_sr = [r for r in seed_data if np.isfinite(r)] if valid_sr: ax.bar(range(len(seed_data)), [r if np.isfinite(r) else 0 for r in seed_data], color='steelblue', edgecolor='k', alpha=0.7) ax.axhline(0.5, color='red', linestyle='--', linewidth=2, label='threshold = 0.5') ax.axhline(np.mean(valid_sr), color='green', linestyle=':', label=f'mean = {np.mean(valid_sr):.3f}') ax.set_xlabel('Seed index') ax.set_ylabel('Decay ratio') ax.set_title(f'Multi-Seed Robustness (10 seeds, all < 0.5?)') ax.legend() ax.grid(True, alpha=0.3, axis='y') paths.extend(save_figure(fig, 'rg_flow_seed_robustness', subfolder='foundations')) # --- Figure 6: CES/non-CES decomposition depth --- decomp_data = _cache.get('decomp_depth') depth_total = _cache.get('depth_sweep') if decomp_data and depth_total: fig, ax = setup_figure() ds_total = sorted(depth_total.keys()) ds_nonces = sorted(decomp_data.keys()) total_valid = [(d, depth_total[d]) for d in ds_total if np.isfinite(depth_total[d]) and depth_total[d] > 0] nonces_valid = [(d, decomp_data[d]) for d in ds_nonces if np.isfinite(decomp_data[d]) and decomp_data[d] > 0] if total_valid and nonces_valid: td, tir = zip(*total_valid) nd, nir = zip(*nonces_valid) ax.semilogy(td, tir, 'bs-', markersize=8, linewidth=2, label='Total ||β||/||α||') ax.semilogy(nd, nir, 'ro-', markersize=8, linewidth=2, label='Non-CES ||β−β_CES||/||α||') # Theory line for non-CES nd_arr = np.array(nd) theory_nonces = nir[0] * 2 ** (-(nd_arr - nd_arr[0]) / 2) ax.semilogy(nd_arr, theory_nonces, 'r--', alpha=0.5, label='Theory: $O(2^{-L/2})$') ax.set_xlabel('Tree depth') ax.set_ylabel('Interaction ratio') ax.set_title('CES/Non-CES Decomposition: Only Non-CES Decays') ax.legend() ax.grid(True, alpha=0.3) paths.extend(save_figure(fig, 'rg_flow_nonces_depth', subfolder='foundations')) # --- Figure 7: Non-CES noise invariance --- noise_nn = _cache.get('noise_nonces') noise_total = _cache.get('noise_sweep') if noise_nn and noise_total: fig, axes = setup_subplots(1, 2, width=12, height=5) ns_total = sorted(noise_total.keys()) ns_nonces = sorted(noise_nn.keys()) # Panel a: total ratio (noise-dependent) ax = axes[0] t_valid = [(ns, noise_total[ns]) for ns in ns_total if np.isfinite(noise_total[ns])] if t_valid: tns, tr = zip(*t_valid) ax.plot(tns, tr, 'bs-', markersize=8, linewidth=2) ax.set_xlabel('Noise scale') ax.set_ylabel('Total decay ratio') ax.set_title('(a) Total IR: noise-dependent') ax.grid(True, alpha=0.3) # Panel b: non-CES ratio (should be constant) ax = axes[1] n_valid = [(ns, noise_nn[ns]) for ns in ns_nonces if np.isfinite(noise_nn[ns])] if n_valid: nns, nr = zip(*n_valid) ax.plot(nns, nr, 'ro-', markersize=8, linewidth=2) ax.axhline(np.mean(nr), color='green', linestyle='--', alpha=0.6, label=f'mean = {np.mean(nr):.3f}') ax.set_xlabel('Noise scale') ax.set_ylabel('Non-CES decay ratio') ax.set_title('(b) Non-CES residual: noise-invariant') ax.legend() ax.grid(True, alpha=0.3) fig.suptitle('Noise Invariance: Total vs Non-CES Decomposition', fontsize=13, y=1.02) paths.extend(save_figure(fig, 'rg_flow_nonces_noise_invariance', subfolder='foundations')) # --- Figure 8: ρ_eff stability --- rho_data = _cache.get('rho_stability') if rho_data: fig, ax = setup_figure() valid_rho_plot = [(d, r) for d, r in sorted(rho_data.items()) if np.isfinite(r)] if valid_rho_plot: rd, rr = zip(*valid_rho_plot) ax.plot(rd, rr, 'go-', markersize=10, linewidth=2) ax.axhline(np.mean(rr), color='red', linestyle='--', alpha=0.6, label=f'mean ρ_eff = {np.mean(rr):.4f}') ax.set_xlabel('Tree depth') ax.set_ylabel('Emergent ρ_eff') ax.set_title('Emergent CES Curvature: ρ is Exactly Marginal (Δ=0)') ax.legend() ax.grid(True, alpha=0.3) paths.extend(save_figure(fig, 'rg_flow_rho_stability', subfolder='foundations')) return paths if __name__ == '__main__': results = main() for r in results: print(f"{r['name']}: {r['verdict']} " f"(stat={r['statistic']}, p={r['p_value']})")

depth	5
n valid	2000
macro ir	0.2339
statistic	0.1910
micro baseline	1.2247

Rg Flow Simulation Decay

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