From e1514f49522541c425611dc916d653caa01c987a Mon Sep 17 00:00:00 2001 From: Codex Bot Date: Wed, 11 Nov 2026 23:21:53 +0800 Subject: [PATCH] Verify sparse-ball shell probabilities with radial controls --- docs/FIGURE_GALLERY.md | 6 + results/sparse-shell/REPORT.md | 22 + results/sparse-shell/audit.json | 41 + results/sparse-shell/checksums.json | 9 + .../sparse-shell/interior-probabilities.csv | 17 + results/sparse-shell/measured-radii.npz | Bin 0 -> 435403 bytes .../sparse-shell/sparse_shell_controls.pdf | Bin 0 -> 44839 bytes .../sparse-shell/sparse_shell_controls.png | Bin 0 -> 164395 bytes .../sparse-shell/sparse_shell_controls.svg | 4392 +++++++++++++++++ shell_study/run.py | 49 + 10 files changed, 4536 insertions(+) create mode 100644 results/sparse-shell/REPORT.md create mode 100644 results/sparse-shell/audit.json create mode 100644 results/sparse-shell/checksums.json create mode 100644 results/sparse-shell/interior-probabilities.csv create mode 100644 results/sparse-shell/measured-radii.npz create mode 100644 results/sparse-shell/sparse_shell_controls.pdf create mode 100644 results/sparse-shell/sparse_shell_controls.png create mode 100644 results/sparse-shell/sparse_shell_controls.svg create mode 100644 shell_study/run.py diff --git a/docs/FIGURE_GALLERY.md b/docs/FIGURE_GALLERY.md index 9e80d14..9a8077a 100644 --- a/docs/FIGURE_GALLERY.md +++ b/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) diff --git a/results/sparse-shell/REPORT.md b/results/sparse-shell/REPORT.md new file mode 100644 index 0000000..d93af84 --- /dev/null +++ b/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。0a^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。 diff --git a/results/sparse-shell/audit.json b/results/sparse-shell/audit.json new file mode 100644 index 0000000..c51056d --- /dev/null +++ b/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" + ] +} diff --git a/results/sparse-shell/checksums.json b/results/sparse-shell/checksums.json new file mode 100644 index 0000000..6b566d8 --- /dev/null +++ b/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" +} diff --git a/results/sparse-shell/interior-probabilities.csv b/results/sparse-shell/interior-probabilities.csv new file mode 100644 index 0000000..a796650 --- /dev/null +++ b/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 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