Create notebook_cell_3_theorem_2_output.txt

notebook_cell_3_theorem_2_output.txt

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+Loading Freckles v40 + CIFAR-10...
+Collecting spatial friction maps (full test set)...
+
+
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+
+Collected 10000 images, 2000 individual maps
+
+======================================================================
+  1. SPATIAL STRUCTURE — Do friction maps have spatial variance?
+======================================================================
+
+  Per-image spatial friction variance (across 256 patches):
+  Mode 0 (S₀): mean=0.0359 std=0.0196
+  Mode 1 (S₁): mean=0.0207 std=0.0143
+  Mode 2 (S₂): mean=0.3235 std=0.1607
+  Mode 3 (S₃): mean=0.0000 std=0.0000
+
+  Per-image spatial CV (std/mean):
+    Mode 0: CV mean=0.0145 median=0.0145 max=0.0300
+    Mode 1: CV mean=0.0109 median=0.0106 max=0.0272
+    Mode 2: CV mean=0.0294 median=0.0297 max=0.0533
+    Mode 3: CV mean=0.0000 median=0.0000 max=0.0000
+
+  VERDICT: SPATIALLY UNIFORM (mean CV = 0.0137)
+
+======================================================================
+  2. PER-CLASS SPATIAL PATTERNS — Do classes have different friction maps?
+======================================================================
+
+  Inter-class friction map L2 distances:
+              airpl   auto   bird    cat   deer    dog   frog  horse   ship  truck
+    airplane  0.000  5.791  5.016  6.309  6.924  5.982  7.640  5.202  3.186  4.733
+        auto  5.791  0.000  3.181  3.194  3.779  3.585  3.552  3.158  3.951  2.838
+        bird  5.016  3.181  0.000  2.093  2.065  2.115  2.931  1.535  4.403  4.120
+         cat  6.309  3.194  2.093  0.000  1.806  1.493  1.996  2.043  5.563  4.821
+        deer  6.924  3.779  2.065  1.806  0.000  2.135  1.485  2.355  6.168  5.364
+         dog  5.982  3.585  2.115  1.493  2.135  0.000  2.682  2.474  5.349  5.220
+        frog  7.640  3.552  2.931  1.996  1.485  2.682  0.000  3.063  6.488  5.457
+       horse  5.202  3.158  1.535  2.043  2.355  2.474  3.063  0.000  4.650  3.783
+        ship  3.186  3.951  4.403  5.563  6.168  5.349  6.488  4.650  0.000  3.010
+       truck  4.733  2.838  4.120  4.821  5.364  5.220  5.457  3.783  3.010  0.000
+
+  Mean inter-class distance: 3.8820
+  Mean cosine similarity:    0.999958
+  Min cosine similarity:     0.999881
+  VERDICT: NEARLY IDENTICAL PATTERNS
+
+======================================================================
+  3. CENTER vs EDGE — Do boundary patches have higher friction?
+======================================================================
+
+  Class        Center     Edge  Edge/Center
+  ----------------------------------------
+  airplane     12.155   12.126       0.9975
+  auto         12.203   12.219       1.0013
+  bird         12.193   12.177       0.9987
+  cat          12.206   12.212       1.0005
+  deer         12.207   12.198       0.9993
+  dog          12.179   12.200       1.0017
+  frog         12.201   12.222       1.0018
+  horse        12.214   12.183       0.9974
+  ship         12.147   12.166       1.0015
+  truck        12.210   12.185       0.9980
+
+======================================================================
+  4. PER-PATCH-POSITION DISCRIMINABILITY
+======================================================================
+
+  F-statistic (inter-class var / intra-class var) per mode:
+    Mode 0: mean=0.054267 max=0.113004 top 5% threshold=0.096192
+    Mode 1: mean=0.165008 max=0.262047 top 5% threshold=0.254798
+    Mode 2: mean=0.069616 max=0.170688 top 5% threshold=0.149573
+    Mode 3: mean=0.000000 max=0.000000 top 5% threshold=0.000000
+    Mode 0 best position: (1, 15) F=0.113004
+    Mode 1 best position: (1, 13) F=0.262047
+    Mode 2 best position: (0, 14) F=0.170688
+    Mode 3 best position: (0, 0) F=0.000000
+
+  Overall best discriminative patch position: 0, 14 F=0.133089
+  Overall mean F-statistic: 0.072223
+  VERDICT: POSITIONALLY DISCRIMINATIVE
+
+======================================================================
+  5. PER-MODE SPATIAL VARIANCE — Which mode has the most structure?
+======================================================================
+  Mode 0: map_mean=12.5844 map_var=0.000355 map_cv=0.0015
+  Mode 1: map_mean=12.4741 map_var=0.000701 map_cv=0.0021
+  Mode 2: map_mean=18.6994 map_var=0.004943 map_cv=0.0038
+  Mode 3: map_mean=5.0000 map_var=0.000000 map_cv=0.0000
+
+======================================================================
+  6. SAMPLE FRICTION MAPS — Individual images
+======================================================================
+
+  Class      Img     Mean      Std      Max  Entropy  HotSpots
+  -------------------------------------------------------
+  airplane     0    12.11     0.22    19.60    1.000         0
+  airplane     1    12.09     0.11    19.89    1.000         0
+  auto         0    12.35     0.26    20.45    1.000         0
+  auto         1    12.15     0.24    20.03    1.000         0
+  bird         0    12.18     0.23    20.62    1.000         0
+  bird         1    12.30     0.26    20.28    1.000         0
+  cat          0    12.19     0.20    19.87    1.000         0
+  cat          1    12.07     0.18    20.44    1.000         0
+  deer         0    12.30     0.09    19.88    1.000         0
+  deer         1    12.26     0.24    20.43    1.000         0
+  dog          0    12.26     0.23    20.28    1.000         0
+  dog          1    12.19     0.30    20.55    1.000         0
+  frog         0    12.18     0.19    20.18    1.000         0
+  frog         1    12.23     0.17    20.43    1.000         0
+  horse        0    12.38     0.34    20.69    1.000         0
+  horse        1    12.15     0.21    19.89    1.000         0
+  ship         0    12.16     0.29    20.76    1.000         0
+  ship         1    12.17     0.23    20.06    1.000         0
+  truck        0    12.17     0.26    20.30    1.000         0
+  truck        1    12.21     0.23    20.33    1.000         0
+
+======================================================================
+  7. LINEAR PROBE — Can flattened friction maps classify?
+======================================================================
+
+  Features: flattened friction map (1024 dims)
+  Train: 1600, Test: 400
+  Train accuracy: 92.6%
+  Test accuracy:  24.3%
+  Chance:          10.0%
+
+  Class         Acc
+  ------------------
+  airplane   32.6% ██████
+  auto       24.4% ████
+  bird       20.6% ████
+  cat        15.7% ███
+  deer       10.3% ██
+  dog        21.9% ████
+  frog       37.8% ███████
+  horse      22.9% ████
+  ship       24.1% ████
+  truck      27.8% █████
+
+  VERDICT: DISCRIMINATIVE spatial friction signal
+
+======================================================================
+  8. SETTLE MAP — Spatial convergence patterns
+======================================================================
+  Settle map linear probe:
+  Test accuracy: 10.5%
+  VERDICT: NOT DISCRIMINATIVE
+
+======================================================================
+  9. COMBINED CONDUIT — All evidence stacked
+======================================================================
+
+  Collecting eigenvalue spatial maps...
+
+  Linear probe comparison (all use same train/test split):
+
+  Eigenvalues (S) spatial        dims= 1024  test_acc=21.0%
+  Friction spatial               dims= 1024  test_acc=24.3%
+  Settle spatial                 dims= 1024  test_acc=10.5%
+  Combined (S+fric+settle)       dims= 3072  test_acc=22.0%
+
+  Chance: 10.0%
+  VERDICT: Combined vs eigenvalues-only lift = +1.0 percentage points
+
+======================================================================
+  SPATIAL FRICTION ANALYSIS — SUMMARY
+======================================================================
+  1. Spatial structure within images:   CV = 0.0137
+  2. Inter-class pattern distance:      cos_min = 0.999881
+  3. Center vs edge asymmetry:          (see table above)
+  4. Per-position F-statistic:          max = 0.133089
+  5. Friction map linear probe:         24.3%
+  6. Settle map linear probe:           10.5%
+  7. Eigenvalue map linear probe:       21.0%
+  8. Combined conduit linear probe:     22.0%
+  9. Conduit lift over eigenvalues:     +1.0pp