CAS I: A Geometric Coding Theorem

This paper establishes a direct analogue of the classical Coding Theorem in the setting of symmetry groups. We consider computable bijections on the set of binary strings and define the symmetry prior of a string x as the probability that a randomly chosen symmetry from a given group G has x as its unique fixed point. We show that for any fix-retractable symmetry group G, a group admitting a computable section that selects an isolating symmetry for every string, the symmetry prior is a universal lower semi-computable semi-measure. In this case, the Geometric Coding Theorem holds. This result is a coding-theoretic restatement of a theorem of Trejo, Kreinovich and Longpr'e, who showed that the complexity of describing a string by a symmetry with that string as its unique fixed point equals its Kolmogorov complexity. Our contribution is to recast it in terms of algorithmic probability and to treat the symmetry group as a parameter, identifying fix-retractability as exactly the condition under which symmetry complexity collapses onto Kolmogorov complexity.