Add NC1 README

NC1/README.md

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+# NC¹ Groups (Non-Solvable)
+
+This directory contains all non-solvable permutation groups from our dataset. Computing compositions in these groups is NC¹-complete, meaning it requires at least logarithmic-depth circuits. This is beyond the computational power of TC⁰ models like Transformers and SSMs, leading to the "Illusion of State" phenomenon.
+
+## Groups in this category:
+
+### Symmetric Groups (n ≥ 5)
+- S5 (order 120) - First non-solvable symmetric group
+- S6 (order 720)
+- S7 (order 5,040)
+- S8 (order 40,320)
+- S9 (order 362,880)
+
+### Alternating Groups (n ≥ 5)
+- A5 (order 60) - The smallest non-solvable group
+- A6 (order 360)
+- A7 (order 2,520)
+- A8 (order 20,160)
+- A9 (order 181,440)
+
+### Projective Special Linear Groups
+- PSL(2,5) (order 60) - Isomorphic to A5
+- PSL(2,7) (order 168)
+
+### Sporadic Simple Groups
+- M11 (order 7,920) - Mathieu group
+- M12 (order 95,040) - Mathieu group
+
+## Key Property
+These groups are all non-solvable (simple or containing simple subgroups), placing them in NC¹-complete. The boundary at A5/S5 marks the theoretical limit of what TC⁰ models can solve.