Beyond RH: Formalizing Shadow from Gr(2,4) Up

A large part of GPPVerify2 is currently “top-down”: we formalized consequences because they were concrete enough to attack. That produced real theorems. It also exposed the next job. We have to go back down and build the mathematical objects those consequences are supposed to come from.

This is how the stub count should fall. Not by deleting stubs. Not by changing their statements. Not by replacing an open theorem with a weaker theorem under the same name. We retire a stub only when the mathematics it depended on has actually been formalized far enough to prove it.

What we already have around the Grassmannian

The current repository is not empty at the foundation. Several important pieces are already genuine Lean theorems.

Orthogonal self-dualityGrassmannianSelfDuality.lean proves the finite-dimensional subspace statement Gr(k,n) ↔ Gr(n−k,n), with self-duality exactly at n=2k.

Plücker relationHolographicChain.lean defines the 2×2 minors of a pair of vectors in ℂ⁴ and proves the Gr(2,4) Plücker quadric identity exactly.

Chart transitionGrassmannianMass.lean proves the explicit transition map squares to −id rather than assuming it.

Jacobian algebraGrassmannianJacobian.lean proves the corrected fourth-power polynomial identity for the cleared Jacobian numerator.

Hodge star witnessesThe six-dimensional ∧²ℂ⁴ matrix, its involution, trace, and explicit self-dual/anti-self-dual eigenvectors are formalized.

Minkowski determinantThe 2×2 Hermitian representation of a spacetime vector has determinant t²−x²−y²−z² as an actual matrix theorem.

Those are meaningful results. But they are still pieces around the geometric object. We have not yet built a full reusable Grassmannian layer with the level of structure the Shadow program ultimately asks of it.

What “genuinely formalize Gr(2,4)” should mean

The object

Bundle the Grassmannian itself as the type of 2-dimensional subspaces of a fixed 4-dimensional vector space, rather than repeatedly passing raw submodules plus separate finrank hypotheses.

Plücker geometry

Construct the exterior-square/Plücker map as a map of the bundled object; prove well-definedness under change of basis; identify its image with the decomposable locus satisfying the quadratic Plücker relation.

Charts and atlas

Define the standard big cells, their coordinate matrices, overlaps and transition functions. Then our explicit τ is a theorem about an actual chart overlap, not a free-floating rational map.

Tangent and differential structure

Formalize the tangent model Hom(Λ,V/Λ) or an equivalent chart derivative; connect the Jacobian computations to the differential of the transition map and prove coordinate-independence of any claimed invariant.

Tautological and quotient bundles

Build the canonical rank-2 subbundle and quotient bundle. Many Shadow statements about fields, helicities and dual sectors should ultimately be statements about sections, duals and induced representations of these bundles.

Hodge and duality structures

Lift the current explicit 6×6 Hodge-star calculation into exterior-algebra language. Prove the ± eigenspace decomposition intrinsically, then connect orthogonal-complement duality to the middle-dimensional geometry.

Group actions and measures

Formalize the relevant GL/SL action, stabilizer description and homogeneous-space structure. Only then should claims about invariant or Haar-induced measures on the geometry be treated as native theorems.

Celestial representation layer

Connect the geometric action to the Mellin/principal-series characters, shadow involution, celestial weights and the actual representation spaces used later in the physics.

Why this retires stubs

A stub is often not missing because its final line is hard. It is missing because Lean does not yet possess the objects that make the line meaningful. Once those objects exist, whole families of downstream statements can become ordinary lemmas.

geometry → bundles → representations → measures → spectral statements → physics

For example, a “Penrose correspondence” stub should not be attacked by asserting a correspondence theorem in isolation. We need the twistor/celestial spaces, maps and bundle data required to state the correspondence precisely. A Wightman stub about covariance or spectrum should eventually depend on a genuine representation and measure-theoretic construction, not a symbolic stand-in. A mass-gap claim should sit downstream of an actual gauge-field/configuration-space formalization, not only of a finite arithmetic ratio.

There are other foundations to build too

Adèles and idèlesMathlib does not hand us the full idèle class group in the form needed by the RH program. We have started building the p-adic and diagonal-embedding infrastructure ourselves.

Celestial amplitudesThe graph sewing theorem is closed, but the analytic tree-to-loop extraction still needs actual meromorphic celestial objects, residues, dispersion reconstruction and completeness.

Quantum field theoryWightman-level claims require distributions, covariance representations, spectrum conditions and locality on explicitly constructed fields—not only the final implication.

Gauge theoryThe Yang–Mills folder contains real arithmetic/spectral subresults but not a formal construction satisfying the Millennium problem. That is a separate, large program.

Arithmetic geometryBSD/Hasse–Weil claims need elliptic curves, local factors, cohomological or analytic inputs and Tamagawa/Selmer infrastructure at the level actually used.

Standard Model structureCounting identities and Koide-phase algebra are useful, but gauge representations, anomaly cancellation, mass operators and symmetry breaking must be constructed if the derivation is to become foundational.

The standard going forward

The target is not “a repository with no red labels.” The target is a dependency tree in which every green node means what a mathematician or physicist thinks it means. If a result needs an assumption, it stays conditional. If a concept is absent, we build it. If the statement is false, we change the theory rather than the proof checker.

That is slower than translating formulas into Lean. It is also the only route that makes the formalization scientifically valuable.

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Beyond RH:Formalizing Shadow from Gr(2,4) Up