What question did this study set out to answer?

This work aims to show that the conventional representation of solved games and logical fields is inefficient and can be simplified.

July 1, 2026Open Access

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

Key Points

This work aims to show that the conventional representation of solved games and logical fields is inefficient and can be simplified.
Analyzed the characteristics of logical fields using fold projectors.
Derived generators using group laws with dyadic fold values.
Machine-checked protocols to validate the findings.
The fold basis offers shorter generators compared to traditional tablebase storage.
Establishes that a solved field can be generated from simple bit rules with projectors.
Demonstrated reduction in complexity, with the proof validated by machine code.

Abstract

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 χₛ (n) = (−1) ᵖopcount (s AND n) are the natural basis. A rule ‘bit b is zero’ is exactly a fold projector (χ₀ ± χ₂㵦) /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 χₐ·χb = χ₀ ₗ₎ₑ ₁ — 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

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