GPPVerify2 has crossed an important threshold. It is no longer a collection of toy encodings attached to a manuscript. It contains real analysis, linear algebra, measure theory, spectral reductions, finite-dimensional operator theory, number-theoretic identities, and geometry developed far enough that the proof assistant has started changing the research itself.
The strongest closed pieces
The chart transition τ used in the mass construction satisfies τ² = −id exactly. The earlier axiom-based version is gone.
Orthogonal complement realizes Gr(k,n) ↔ Gr(n−k,n), with self-duality precisely at n=2k. Gr(2,4) is the canonical middle-dimensional case.
Core Haar self-duality and the zeta functional-equation machinery are cleanly formalized in their corresponding modules.
The nontrivial-zero statement is equivalent to positivity of the reflection pairing on every finite subset. This is a rigorous reduction, not RH itself.
The first Li coefficient is handled unconditionally, including a proof of λ₁>0. It is a genuine positive result on the zeta side.
The sinh/Planck kernels are connected to Γ(s)ζ(s) by actual integration theorems, replacing old high-precision numerics with exact analysis.
The prime Euler factors produce exact occupation laws and a Bose–Einstein form in the convergence domain; the thermodynamic interpretation now has formal algebra under it.
The L-pair sewing cycle-rank theorem is proved for the graph-combinatorial layer. The difficult celestial analytic sewing identity remains explicitly separate.
A cleaner reason for one-half
The half-density dilation character is
At a nontrivial positive scale, its unitary locus is exactly Re(s)=1/2. Under the shadow map s↦1−s, the centered exponent changes sign, so
This theorem contains no claim about zeta zeros. That is precisely why it matters. It isolates the representation-theoretic origin of the one-half independently of the arithmetic step that RH still requires.
What Lean killed
Formalization has been useful not only because it confirms results, but because it has forced retractions. A supposed “arithmetic admissibility” input was recognized as RH in disguise and removed. A proposed Jacobian complex-structure identity was false and replaced by the correct period-four polynomial identity. A global residue-positivity hope for a finite Weil-parity object failed a concrete stress test. Historical scalar/graviton-box numerics were found inconsistent with their own formulas and were not carried forward as evidence.
That is exactly what this machinery is for. A theory that cannot survive its own falsification attempts is not worth formalizing.
The exact RH frontier
The repo now proves a strong finite positivity criterion:
The zero-side algebra is therefore not the mystery. The hard step is to derive the required global positivity from an independent analytic/operator construction without importing RH through a disguised hypothesis. Work on support ladders, Cauchy positive-type kernels, truncated transport, heat traces, Suzuki–Herglotz structures, and Weil-parity operators all attacks versions of that bridge.
What “verified” means here
As of the August 24 merged build, the combined tree completed 3,325 Lean jobs successfully with zero executable sorry and 13 explicit custom axioms. The current main branch remains CI-green. Those numbers are useful only when the stub ledger is reported beside them, because a stub can compile while proving nothing. GPPVerify2 now says that plainly.
Why this matters beyond RH
RH has driven a lot of the recent work because it is brutally sensitive to circularity. But the same verification discipline is now being applied across the rest of Shadow: the Grassmannian geometry, celestial holography, quantum gravity, Standard Model structure, Yang–Mills, cosmology, and number theory.
The next upgrade is not “more consequences.” It is deeper foundations. If the framework says the physics comes from Gr(2,4), then Lean should eventually know what that Grassmannian is—not merely a handful of coordinate identities extracted from it.