Shadow Theory / Source Lift
Recovering Yang–Mills
through Shadow Theory.
A construction is a chain of operations. Shadow Theory makes that chain explicit: which objects a readout recovers, what it discards, and why the next operation remains valid. The flagship develops this recovery across the Wilson Yang–Mills construction.
One construction. Every recovery made explicit.
The flagship constructs a native support, its literal preparation and a realized family that carries configurations, measures, effective quantities and physical fields. Reverse sections establish coverage. Frame fibres identify redundant information. Operation identities establish compatibility. Residual estimates quantify departure from exact recovery. The complete Yang–Mills calculations remain in the reading edition.
Jeremy Rodgers originated Shadow Theory, the source–readout architecture and the Source Lift protocol, and directed this work with AI assistance. OpenAI is credited for the underlying Lean Yang–Mills construction. The second paper presents the mathematical construction in conventional language, with its own complete derivations.
Shadow Theory and the Wilson Yang–Mills Construction: A Source–Readout Reconstruction
Shadow Theory constructs a connected realization of the Wilson Yang–Mills argument: its configurations, measures, effective quantities, physical fields and operator correlations, with explicit recovery maps and compatible operations throughout.
91 reading chapters →A Wilson Yang–Mills Construction: A Natural-Language Mathematical Exposition
Follow the mathematical construction from finite Wilson measures through calibrated continuum limits, physical fields and Wightman reconstruction to operator products and the centered trace. A standalone companion to the Shadow Theory reconstruction.
72 reading chapters →
From finite laws to a common quantum-field hierarchy.
The argument passes through quadratic kernels, nonlinear group geometry, marked integration, coupling calibration and continuum limits. Its physical labels, phases, operator products, stress and centered trace are recovered on compatible witnesses. The full fixed-period mesh limit and the outer volume extraction retain their different quantifiers.
The group, faithful representation, four-dimensional Wilson law and analytic hypotheses are the inputs of this realization. The centered trace asymptotic and comparison exclusions have their stated domains. Neither edition asserts a positive mass gap.
OpenAI’s Yang–Mills construction, reconstructed through Shadow Theory.
Readers arriving from the formal construction can follow the source–readout reconstruction here, or start with the mathematical companion. The following public materials give the upstream context and the assessment of the October construction:
- OpenAI’s announcement and academic collaboration ↗
- The October OpenAI Lean source distribution ↗
- The academic group’s assessment and verification materials ↗
The group’s assessment of 10 November describes the October submission as a construction of a relativistic quantum field theory satisfying the Wightman axioms from four-dimensional Wilson lattice gauge theory for simple compact gauge groups. It leaves uniqueness of the infinite-volume limit and a positive mass gap open.
The papers use a supplied expanded mathematical translation. These links identify the related public formal project; this website does not assert an independently checked revision-level correspondence or Lean certification of the reconstruction.
Explore the recovery interfaces in the Atlas → · Continue into the Einstein reconstruction →