On the Nature of Nature is the title of my book. The framework developed through that work is the Shadow Framework. GPPVerify2 is the Lean 4 repository where its mathematical claims are being separated into four categories that matter: proved, conditional, numerically supported, and open.
sorry does not mean “the whole theory is proved.” The repository also counts explicit axioms and deliberately parked True := trivial stubs. Those are open mathematics, not proofs. The distinction is enforced precisely because a clean compiler result by itself can otherwise be misleading.What changed since the original version of this page
The first version overstated the status by saying every core theorem had been verified. That is no longer how I describe the project. The formalization has grown enormously, and the audit has become stricter at the same time. Some old assumptions were proved. Some were retired because they were circular. Some attractive ideas were falsified. Others were decomposed into smaller statements until the exact missing theorem became visible.
The result is better than a blanket claim of completion: we now have a substantial kernel-checked core and a much sharper map of what remains.
Examples of genuinely closed mathematics
The relevant Haar self-duality layer and the functional-equation machinery are formalized without custom axioms in their clean modules.
The chart transition used in the mass construction satisfies τ∘τ = −id exactly. An earlier axiom-based version was replaced by a proof.
Orthogonal complement sends k-planes to (n−k)-planes and restricts to a self-map exactly when n = 2k, including Gr(2,4).
RH is reduced rigorously to finite positivity of the reflection pairing. The former “arithmetic admissibility” axiom—which restated RH—was deleted.
The Mellin transform of the sinh kernel is proved to produce the Γζ factor, with exact special values replacing earlier numerical checks.
The Euler product, prime occupation law, Bose–Einstein form, and related thermodynamic identities have been formalized in their convergence domain.
Sewing L disjoint pairs of extra legs onto the relevant cubic tree raises the cycle rank by exactly L. The remaining celestial analytic identity is isolated separately.
The pseudo-isothermal profile’s forward projection and inverse reconstruction are exact Lean theorems, not fitted numerical identities.
The critical line is appearing structurally
One of the cleanest recent results concerns the half-density dilation character
For nontrivial positive scale, the critical line Re(s)=1/2 is exactly the locus on which this character has unit modulus. The shadow involution s ↦ 1−s sends it to its reciprocal:
That does not prove RH. It does show that the ubiquitous one-half is not being inserted by hand at this layer: it is the center of the unitary half-density representation.
What is not proved
The Riemann Hypothesis itself is still open in GPPVerify2. So are substantial parts of the Yang–Mills, Wightman, twistor-googly, Majorana, and arithmetic-geometry towers. The decisive RH gap is no longer “why one-half?”; it is the global analytic positivity/transport bridge that forces the actual arithmetic zeros onto the unitary locus.
The same discipline applies elsewhere. If a physical theorem depends on a geometric object that has not yet been built in Lean, the correct next move is not to leave the consequence as an eternal stub. It is to formalize the object and prove the prerequisite.
Three numbers that must always be reported together
This accounting rule was adopted after the repository itself exposed how easy it is to make a formalization sound more complete than it is. That lesson is now part of the method.
The next phase
GPPVerify2 is moving beyond isolated consequences. The next serious phase is foundational: genuinely formalize the geometry and analysis that the Shadow construction is supposed to live on—Grassmannians, Plücker coordinates and charts, the relevant bundles and dualities, celestial kinematics, measure structures, representation-theoretic objects, and then the physical constructions built on top of them.
That is how the stub count should fall: not by deleting placeholders, weakening statements, or renaming assumptions, but by proving the mathematics underneath them.
Follow the formalization live
The proof tree is public. The live blueprint exposes theorem dependencies; the source repository exposes the actual Lean files, history, CI, and audit scripts.