The Geometry of Correction

The cost to correct an error is not proportional to its magnitude but to its de- scription length—the complexity of the transform mapping the erroneous state to truth. Errors with high symmetry (antipodal errors) require minimal description: one bit can flip an entire inverted structure. Errors without symmetry (fractal er- rors) require unbounded description: each deviation must be specified individu- ally. This asymmetry has profound consequences: systematic wrongness is cheap to correct while random wrongness is ruinous. We formalize this distinction using Kolmogorov complexity, connect it to the round-trip fairness conditions of idiom translation, and show that the preservation of structure under error is equivalent to the preservation of witnesses under transformation. The framework provides a complexity-theoretic foundation for why guards and invariants matter: they ensure errors remain antipodal rather than fractal