Verify sparse-ball shell probabilities with radial controls

Reviewed-on: #7

docs/FIGURE_GALLERY.md

@@ -37,3 +37,9 @@
 [配对报告](../results/finetune-comparison/REPORT.md) · [CDF](../results/finetune-comparison/figures/paired_effective_heads_cdf.png) · [逐层分布](../results/finetune-comparison/figures/paired_effective_heads_by_layer.png) · [配对变化](../results/finetune-comparison/figures/paired_effective_heads_delta.png)
 
 同目录保留PDF/SVG。图表随代码、原始数据和SHA清单一同提交，不依赖聊天附件。
+
+## 高维壳与稀疏性对照
+
+![稀疏球壳对照](../results/sparse-shell/sparse_shell_controls.png)
+
+[条件定理、反例与数据说明](../results/sparse-shell/REPORT.md) · [PDF](../results/sparse-shell/sparse_shell_controls.pdf) · [SVG](../results/sparse-shell/sparse_shell_controls.svg)

results/sparse-shell/REPORT.md

@@ -0,0 +1,22 @@
+# 稀疏分布与高维壳：条件定理及反例
+
+在同一外半径R下，若X在d维球内均匀分布，P(||X||<=aR)=a^d。若X均匀分布于嵌入该空间的k维子空间球，则概率为a^k。0<a<1、k<d时严格有a^k>a^d。
+
+因此用户猜想在上述模型下成立。k仍很大时也有薄壳，只是由有效维度k决定，而非环境维度d。
+
+| r/R | 128维均匀球（理论） | 16维子空间均匀球（理论） | 内部概率倍数 |
+|---|---:|---:|---:|
+| 0.8 | 3.9402e-13 | 0.0281475 | 7.14367e+10 |
+| 0.9 | 1.39008e-06 | 0.185302 | 133303 |
+| 0.95 | 0.00140806 | 0.440127 | 312.576 |
+| 0.99 | 0.276252 | 0.851458 | 3.08218 |
+
+模拟构造4组实际128维向量，每组20000个，再测量其范数：全维均匀球、16维子空间均匀球、同样16稀疏但采用原128维球半径分布、以及全部固定在半径R的16稀疏点。前两者验证条件定理；后两者说明仅凭稀疏不能保证内部概率增加。
+
+所有组外半径相同，但平均范数并未强行匹配。若把每个向量都归一化为相同范数，径向差异会被直接抹去。固定半径组的浮点误差约1e-16；理论上全部在球面，图中仅为数值表示。
+
+这不证明好模型的参数更靠内部。用于模型时还需明确采样单元、中心、半径和缩放约定；参数可做保持功能的重参数化，原点距离不是模型质量不变量。
+
+文件：run.py为生成程序；measured-radii.npz为全部实测范数；interior-probabilities.csv保留理论与经验概率；三种图格式及SHA清单便于长期保存。没有读取、训练或改动任何模型。
+
+有限样本说明：每组20000个点不足以精确估计约1e-6的极小尾概率；极小概率和倍数以解析公式为准，不把零次命中当成严格零概率。

results/sparse-shell/audit.json

@@ -0,0 +1,41 @@
+{
+  "passed": true,
+  "ambient_dimension": 128,
+  "sparse_dimensions": 16,
+  "sample_count_per_distribution": 20000,
+  "seed": 20261111,
+  "outer_radius": 1,
+  "controls": {
+    "uniform_d_ball": {
+      "active_dimensions": 128,
+      "max_norm": 0.9999994375833048,
+      "mean_squared_norm": 0.9846031601593415,
+      "radial_dimension": 128
+    },
+    "uniform_k_ball": {
+      "active_dimensions": 16,
+      "max_norm": 0.999996278718437,
+      "mean_squared_norm": 0.8894302831319619,
+      "radial_dimension": 16
+    },
+    "sparse_same_radial": {
+      "active_dimensions": 16,
+      "max_norm": 0.9999995857902936,
+      "mean_squared_norm": 0.9847321079586993,
+      "radial_dimension": 128
+    },
+    "sparse_sphere": {
+      "active_dimensions": 16,
+      "max_norm": 1.0000000000000002,
+      "mean_squared_norm": 1.0,
+      "radial_dimension": null
+    }
+  },
+  "elapsed_seconds": 3.858549994882196,
+  "limitations": [
+    "synthetic distributions, not trained-network parameters",
+    "same outer radius, not same mean squared norm",
+    "changing angular support alone leaves the chosen radial law unchanged",
+    "fixed-radius shell is a deliberately structured counterexample"
+  ]
+}

results/sparse-shell/checksums.json

@@ -0,0 +1,9 @@
+{
+  "REPORT.md": "aca991f55ab215f637e310430eef267b0d2a327bb11be5ebdefb8a759fe5a4e5",
+  "audit.json": "a9c294b86872676ad2cd721d6e4ab819f01ecb43194829246d2505d2fa5a4555",
+  "interior-probabilities.csv": "e1aaddec9b4079d212a43c05b2b348e449bb6a120314225558ed4a7624de7f28",
+  "measured-radii.npz": "6a44587524196f4281f43753c87146344a38b72e141d7c06135385cd680b613c",
+  "sparse_shell_controls.pdf": "d2f749b107b391b58a5309ad14383c41716dc864c0e24d2b517cc1f8f6aa6da2",
+  "sparse_shell_controls.png": "7d4a6810838a7d377e0e68f19a952baa6784e3fa5a112396f5cbd65e61992eb1",
+  "sparse_shell_controls.svg": "b34aa1f561b316c06d0c620f594c1e3a64bcaa1ce2cb175d69946e867822b43d"
+}

results/sparse-shell/interior-probabilities.csv

@@ -0,0 +1,17 @@
+distribution,radius_fraction,theory_probability,empirical_probability,sample_count
+uniform_d_ball,0.8,3.940200619639476e-13,0.0,20000
+uniform_k_ball,0.8,0.028147497671065624,0.02645,20000
+sparse_same_radial,0.8,3.940200619639476e-13,0.0,20000
+sparse_sphere,0.8,0.0,0.0,20000
+uniform_d_ball,0.9,1.3900845237714517e-06,0.0,20000
+uniform_k_ball,0.9,0.18530201888518416,0.1821,20000
+sparse_same_radial,0.9,1.3900845237714517e-06,0.0,20000
+sparse_sphere,0.9,0.0,0.0,20000
+uniform_d_ball,0.95,0.001408061023535215,0.0012,20000
+uniform_k_ball,0.95,0.44012666865176536,0.4402,20000
+sparse_same_radial,0.95,0.001408061023535215,0.0011,20000
+sparse_sphere,0.95,0.0,0.0,20000
+uniform_d_ball,0.99,0.2762516676992083,0.27975,20000
+uniform_k_ball,0.99,0.8514577710948755,0.8515,20000
+sparse_same_radial,0.99,0.2762516676992083,0.27645,20000
+sparse_sphere,0.99,0.0,0.0,20000

shell_study/run.py

@@ -0,0 +1,49 @@
+"""Synthetic geometric controls; no model weights, training or GPU."""
+import argparse,json,csv,hashlib,time
+from pathlib import Path
+import numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+
+def main(root):
+    root=Path(root);root.mkdir(parents=True,exist_ok=False);started=time.monotonic();rng=np.random.default_rng(20261111);d,k,n=128,16,20000
+    # Construct actual vectors, then measure norms. Inactive coordinates are zero.
+    radii={};checks={}
+    for name,active,radial_dim in [('uniform_d_ball',d,d),('uniform_k_ball',k,k),('sparse_same_radial',k,d),('sparse_sphere',k,None)]:
+        direction=rng.normal(size=(n,active));direction/=np.linalg.norm(direction,axis=1,keepdims=True)
+        radius=np.ones(n) if radial_dim is None else rng.uniform(size=n)**(1/radial_dim)
+        vectors=np.zeros((n,d));vectors[:,:active]=direction*radius[:,None]
+        measured=np.linalg.norm(vectors,axis=1);np.testing.assert_allclose(measured,radius,atol=1e-14)
+        checks[name]={'active_dimensions':active,'max_norm':float(measured.max()),'mean_squared_norm':float(np.mean(measured**2)),'radial_dimension':radial_dim}
+        radii[name]=measured;del vectors,direction
+    np.savez_compressed(root/'measured-radii.npz',**radii)
+    rows=[]
+    for a in [.8,.9,.95,.99]:
+        for name,x in radii.items():
+            m=checks[name]['radial_dimension'];theory=0. if m is None else a**m;empirical=float(np.mean(x<=a))
+            # Six standard errors plus finite-sample allowance; diagnostic, not a model-quality claim.
+            assert abs(empirical-theory)<=6*np.sqrt(theory*(1-theory)/n)+6/n
+            rows.append({'distribution':name,'radius_fraction':a,'theory_probability':theory,'empirical_probability':empirical,'sample_count':n})
+    with (root/'interior-probabilities.csv').open('w') as f:
+        w=csv.DictWriter(f,fieldnames=list(rows[0]));w.writeheader();w.writerows(rows)
+    fig,axs=plt.subplots(1,2,figsize=(12,4.5),layout='constrained');labels={'uniform_d_ball':'Uniform 128D ball','uniform_k_ball':'Uniform 16D ball embedded in 128D','sparse_same_radial':'16-sparse, same radii as 128D ball law','sparse_sphere':'16-sparse, fixed radius 1'}
+    styles=['-','-','--',':']
+    for (name,x),style in zip(radii.items(),styles):
+        xx=np.sort(x);yy=np.arange(1,n+1)/n;axs[0].plot(xx,yy,style,label=labels[name]);axs[1].plot(xx,yy,style,label=labels[name])
+    axs[0].set(xlim=(.65,1.005),ylim=(0,1),xlabel='Measured norm / common radius R',ylabel='Interior probability: P(norm <= r)',title='Empirical radial CDF')
+    axs[1].set(xlim=(.97,1.001),ylim=(0,1),xlabel='Measured norm / R',title='Near-boundary detail')
+    for ax in axs:ax.grid(alpha=.2)
+    axs[0].legend(fontsize=8,loc='upper left');fig.suptitle('Same ambient dimension and outer radius; sparsity alone does not fix radial probability')
+    for ext in ['png','pdf','svg']:fig.savefig(root/('sparse_shell_controls.'+ext),dpi=180)
+    proof={'passed':True,'ambient_dimension':d,'sparse_dimensions':k,'sample_count_per_distribution':n,'seed':20261111,'outer_radius':1,'controls':checks,'elapsed_seconds':time.monotonic()-started,'limitations':['synthetic distributions, not trained-network parameters','same outer radius, not same mean squared norm','changing angular support alone leaves the chosen radial law unchanged','fixed-radius shell is a deliberately structured counterexample']}
+    (root/'audit.json').write_text(json.dumps(proof,indent=2)+'\n')
+    lines=['# 稀疏分布与高维壳：条件定理及反例','', '在同一外半径R下，若X在d维球内均匀分布，P(||X||<=aR)=a^d。若X均匀分布于嵌入该空间的k维子空间球，则概率为a^k。0<a<1、k<d时严格有a^k>a^d。','', '因此用户猜想在上述模型下成立。k仍很大时也有薄壳，只是由有效维度k决定，而非环境维度d。','', '| r/R | 128维均匀球（理论） | 16维子空间均匀球（理论） | 内部概率倍数 |','|---|---:|---:|---:|']
+    for a in [.8,.9,.95,.99]:lines.append(f'| {a} | {a**d:.6g} | {a**k:.6g} | {a**(k-d):.6g} |')
+    lines+=['', '模拟构造4组实际128维向量，每组20000个，再测量其范数：全维均匀球、16维子空间均匀球、同样16稀疏但采用原128维球半径分布、以及全部固定在半径R的16稀疏点。前两者验证条件定理；后两者说明仅凭稀疏不能保证内部概率增加。','', '所有组外半径相同，但平均范数并未强行匹配。若把每个向量都归一化为相同范数，径向差异会被直接抹去。固定半径组的浮点误差约1e-16；理论上全部在球面，图中仅为数值表示。','', '这不证明好模型的参数更靠内部。用于模型时还需明确采样单元、中心、半径和缩放约定；参数可做保持功能的重参数化，原点距离不是模型质量不变量。','', '文件：run.py为生成程序；measured-radii.npz为全部实测范数；interior-probabilities.csv保留理论与经验概率；三种图格式及SHA清单便于长期保存。没有读取、训练或改动任何模型。']
+    lines+=['', '有限样本说明：每组20000个点不足以精确估计约1e-6的极小尾概率；极小概率和倍数以解析公式为准，不把零次命中当成严格零概率。']
+    (root/'REPORT.md').write_text('\n'.join(lines)+'\n')
+    (root/'checksums.json').write_text(json.dumps({p.name:hashlib.sha256(p.read_bytes()).hexdigest() for p in sorted(root.iterdir()) if p.is_file()},indent=2)+'\n')
+    print((root/'REPORT.md').read_text(),end='')
+if __name__=='__main__':
+    p=argparse.ArgumentParser();p.add_argument('--out',required=True);a=p.parse_args();main(a.out)