The Wall Was a Coordinate: Solved Fields Collapse to Short Generators in the Fold Basis

One axiom. One operation. Zero free parameters. Solved games and logical fields are stored as exponentially large tables — the tablebase wall. This paper shows the wall is a bad coordinate. The fold’s characters χ_s(n) = (−1)^popcount(s AND n) are the natural basis. A rule ‘bit b is zero’ is exactly a fold projector (χ₀ ± χ_{2^b})/2 of two characters. A solved field that is a product of bit-rules is a product of projectors, and its generator is DERIVED by the group law χ_a·χ_b = χ_{a XOR b} — forced from the rule, not measured, each coefficient a dyadic fold value, reconstruction exact. The subtraction game is two projectors; Nim’s generator has size 2^(bits) regardless of heap count — a theorem, not an observation. The exponential wall was an artifact of the position basis; in the fold basis the answer is short. Machine-checked; reproduces from one command. A standalone result within the Smithian Fold Theory of Everything (SFTOE). Full corpus, code, and the run-it-yourself VERIFY.md protocol: https://github.com/MettaMazza/Smithian-Fold-Theory