"""Figures for the numerical audits of the spectral statements (Claims 1 and 2).""" from __future__ import annotations import json import math import os import numpy as np import pandas as pd import plotly.graph_objects as go from analyze import PALETTE, _c, write_fig, linfit RES = "results" def main(): out = {} a = pd.read_csv(f"{RES}/audit_spectrum.csv") dims = sorted(a["d"].unique()) # --- lambda1(A*): diverges for quadratic, converges to 6 for truncated ----- fig = go.Figure() for i, d in enumerate(dims): q = a[(a["act"] == "quad") & (a["d"] == d)].groupby("delta")["lam1"].mean().reset_index() t = a[(a["act"] == "trunc") & (a["M"] == 8.0) & (a["d"] == d)] \ .groupby("delta")["lam1"].mean().reset_index() fig.add_trace(go.Scatter(x=q["delta"], y=q["lam1"], mode="lines+markers", name=f"quad d={d}", line=dict(color=_c(i, len(dims)), dash="dash"))) fig.add_trace(go.Scatter(x=t["delta"], y=t["lam1"], mode="lines+markers", name=f"trunc d={d}", line=dict(color=_c(i, len(dims))))) fig.add_hline(y=6, line_dash="dot", line_color="#111", annotation_text="population λ₁ = 6") fig.update_layout(title="λ₁(A*) vs δ = n/d — quadratic (dashed) vs truncated M=8 (solid)", xaxis_title="δ = n/d", yaxis_title="λ₁(A*)", xaxis_type="log", yaxis_type="log", template="plotly_white", height=470) write_fig(fig, "audit_lam1") # --- the eq. (3.13) error: |lam1-6| + |lam2-2| vs the claimed rate --------- t = a[(a["act"] == "trunc")].copy() t["err"] = (t["lam1"] - 6).abs() + (t["lam2"] - 2).abs() t["rate"] = np.exp(-t["M"] / 3) + t["M"] * np.sqrt(t["d"] / t["n"]) g = t.groupby(["M", "delta", "d"])[["err", "rate"]].mean().reset_index() g["C"] = g["err"] / g["rate"] g.to_csv(f"{RES}/audit_eq313.csv", index=False) fig = go.Figure() Ms = sorted(g["M"].unique()) for i, M in enumerate(Ms): s = g[g["M"] == M] fig.add_trace(go.Scatter(x=s["rate"], y=s["err"], mode="markers", marker=dict(size=9, color=_c(i, len(Ms))), name=f"M={M:g}")) lim = [float(g["rate"].min()) * 0.8, float(g["rate"].max()) * 1.2] for C, dash in ((1.0, "dot"), (0.5, "dash")): fig.add_trace(go.Scatter(x=lim, y=[C * lim[0], C * lim[1]], mode="lines", line=dict(color="#444", dash=dash), name=f"C = {C}")) fig.update_layout( title="Eq. (3.13) audit: |λ₁−6| + |λ₂−2| vs C(e^(−M/3) + M√(d/n))", xaxis_title="e^(−M/3) + M√(d/n)", yaxis_title="|λ₁−6| + |λ₂−2|", xaxis_type="log", yaxis_type="log", template="plotly_white", height=470) write_fig(fig, "audit_eq313") out["eq313_max_C"] = float(g["C"].max()) out["eq313_max_C_largedelta"] = float(g[g["delta"] >= 16]["C"].max()) # --- uniform-in-theta BBP -------------------------------------------------- b = pd.read_csv(f"{RES}/audit_uniform_bbp.csv") bt = b[(b["act"] == "trunc") & (b["M"] == 8.0)].copy() bt["kind"] = np.where(bt["theta"].str.startswith("random"), "random θ", bt["theta"]) fig = go.Figure() kinds = ["random θ", "theta_star", "adversarial"] cols = {"random θ": PALETTE[1], "theta_star": PALETTE[4], "adversarial": PALETTE[6]} for k in kinds: s = bt[bt["kind"] == k] for j, col in enumerate(("lam1", "lam2")): fig.add_trace(go.Scatter( x=s["delta"], y=s[col], mode="markers", marker=dict(size=11, color=cols[k], symbol="circle" if j == 0 else "x"), name=f"{k} — λ{j+1}", legendgroup=k, showlegend=True)) fig.add_hline(y=6, line_dash="dot", line_color="#111") fig.add_hline(y=2, line_dash="dot", line_color="#111") fig.update_layout( title="Uniform-in-θ BBP transition of A(θ): λ₁ (circles) and λ₂ (crosses), truncated M=8", xaxis_title="δ = n/d", yaxis_title="eigenvalue of A(θ)", xaxis_type="log", template="plotly_white", height=470) write_fig(fig, "audit_uniform_bbp") s64 = bt[bt["delta"] == 64.0] out["bbp_delta64"] = dict(lam1_min=float(s64["lam1"].min()), lam1_max=float(s64["lam1"].max()), lam2_min=float(s64["lam2"].min()), lam2_max=float(s64["lam2"].max()), ov_min=float(s64["sq_overlap_v1"].min()), ov_max=float(s64["sq_overlap_v1"].max())) # --- indicator mass ------------------------------------------------------- c = pd.read_csv(f"{RES}/audit_indicator.csv") ct = c[c["act"] == "trunc"].copy() ct["kind"] = np.where(ct["theta"].str.startswith("random"), "random θ", ct["theta"]) fig = go.Figure() for i, k in enumerate(["random θ", "theta_star", "adversarial"]): s = ct[ct["kind"] == k] fig.add_trace(go.Box(x=s["M"], y=s["ratio"], name=k, marker_color=[PALETTE[1], PALETTE[4], PALETTE[6]][i])) fig.add_hline(y=1.0, line_dash="dot", line_color="#111", annotation_text="bound with C = 1") fig.update_layout( title="Uniform indicator-mass bound: measured mass ÷ (e^(−M/2) + √(d/n)·log(n/d))", xaxis_title="M", yaxis_title="ratio", template="plotly_white", height=440, boxmode="group") write_fig(fig, "audit_indicator") out["indicator_max_ratio"] = float(ct["ratio"].max()) out["indicator_max_ratio_adv"] = float(ct[ct["kind"] == "adversarial"]["ratio"].max()) # --- Theorem 3.2 deficit bound -------------------------------------------- th = pd.read_csv(f"{RES}/thm32_bound.csv") gg = th.groupby(["M", "delta"])[["deficit", "rate", "C_implied"]].mean().reset_index() fig = go.Figure() Ms = sorted(gg["M"].unique()) for i, M in enumerate(Ms): s = gg[gg["M"] == M] fig.add_trace(go.Scatter(x=s["rate"], y=s["deficit"], mode="markers+lines", marker=dict(size=10, color=_c(i, len(Ms))), line=dict(color=_c(i, len(Ms))), name=f"M={M:g}")) lim = [float(gg["rate"].min()) * 0.9, float(gg["rate"].max()) * 1.1] fig.add_trace(go.Scatter(x=lim, y=lim, mode="lines", line=dict(color="#444", dash="dot"), name="C = 1")) fig.update_layout( title="Theorem 3.2 audit: realised deficit 1 − |⟨θ_∞,θ*⟩| vs e^(−M/2) + (d/n)^(1/5), d=512", xaxis_title="e^(−M/2) + (d/n)^(1/5)", yaxis_title="1 − |⟨θ_∞, θ*⟩|", xaxis_type="log", yaxis_type="log", template="plotly_white", height=470) write_fig(fig, "audit_thm32") out["thm32_max_C_Mge4"] = float(th[th["M"] >= 4]["C_implied"].max()) out["thm32_max_C_all"] = float(th["C_implied"].max()) # --- smooth vs hard truncation robustness --------------------------------- p = f"{RES}/sweep_smooth.csv" if os.path.exists(p): sm = pd.read_csv(p).groupby(["d", "delta"])["sq_overlap"].mean().reset_index() hd = pd.read_csv(f"{RES}/sweep_trunc.csv").groupby(["d", "delta"])["sq_overlap"] \ .mean().reset_index() dims2 = sorted(sm["d"].unique()) fig = go.Figure() for i, d in enumerate(dims2): s = sm[sm["d"] == d].sort_values("delta") h = hd[hd["d"] == d].sort_values("delta") fig.add_trace(go.Scatter(x=s["delta"], y=s["sq_overlap"], mode="lines+markers", name=f"smooth d={d}", line=dict(color=_c(i, len(dims2))))) fig.add_trace(go.Scatter(x=h["delta"], y=h["sq_overlap"], mode="lines", name=f"hard d={d}", line=dict(color=_c(i, len(dims2)), dash="dot"))) fig.update_layout( title="Robustness: C^∞ truncation (eq. 3.10, solid) vs hard truncation (eq. 4.3, dotted)", xaxis_title="δ = n/d", yaxis_title="squared overlap", template="plotly_white", height=470) write_fig(fig, "audit_smooth_vs_hard") mrg = sm.merge(hd, on=["d", "delta"], suffixes=("_smooth", "_hard")) mrg["absdiff"] = (mrg["sq_overlap_smooth"] - mrg["sq_overlap_hard"]).abs() mrg.to_csv(f"{RES}/smooth_vs_hard.csv", index=False) out["smooth_vs_hard_maxdiff_delta_ge_4"] = float(mrg[mrg["delta"] >= 4]["absdiff"].max()) out["smooth_vs_hard_meandiff_delta_ge_4"] = float(mrg[mrg["delta"] >= 4]["absdiff"].mean()) with open(f"{RES}/audit_summary.json", "w") as f: json.dump(out, f, indent=2, default=float) print(json.dumps(out, indent=2, default=float)) if __name__ == "__main__": main()