Part IThe geometric question
Arithmetic positivity structures
Sufficient primitive intersection and cohomological structures, with the exact comparison to the Weil pairing made explicit.
YonedaAI · Structural arithmetic geometry
The structure
behind positivity.
Which maps connect arithmetic geometry, Weil positivity, polarisation, and spectral realization?
The Arithmetic Positivity Program identifies sufficient structures and the comparisons they need. The arithmetic construction and the Riemann Hypothesis remain open.
Primitive intersection space
A geometric sign theorem on a specified space
Arithmetic comparison
The exact map and the correct pairing
Weil positivity
The implication is proved; its arithmetic inputs remain to be constructed
A sufficient structure is not yet an arithmetic instance.
Follow the maps.
Each route has a source structure, a comparison, and a property to preserve.
Source structureRequired bridgeWhat it establishes
Primitive intersection
Exact signed comparison
An independently proved negative pairing gives Weil positivity when an arithmetic cycle map pulls it back with the correct sign.
Cohomological purity
Positive pairing and realization
A weight identity locates represented eigenvalues. A concrete identification with the actual zeta zeros is a separate obligation.
Finite positive models
Vanishing comparison defect
Positivity survives a pointwise limit on common tests. The finite models must approximate the entire arithmetic form.
Ground-state transforms
Selection and holomorphic limit
Known proxy convergence must be joined to the actual ground states, with the hypotheses of the real-zero theorem verified.
Where the proof stops.
The general transfer mechanisms are accessible. The arithmetic comparisons carry the unresolved content.
A Hilbert-space factorization of the exact Weil form is itself equivalent to positivity. Calling it universal does not supply an independent proof.
Inspect the formal comparisonProved in Lean
Positivity transfer, primitive intersection comparisons, radical quotient descent, weighted transform estimates, and explicit counterexamples.
Known finite results
Exact prime sums at fixed support, convergent smooth zero sums, localized operator realizations, and particular proxy limits.
Remaining arithmetic work
Construct the geometric object and its exact pairing, or prove the finite-model comparison and sign control on every test. The classical Weil formula is not yet formalized in this Lean library.
Three complementary questions.
The geometry, the approximations, and the network of bridges between them.
Part IIThe limiting question
Finite Weil approximations
What finite prime formulas, spectral sums, positive models, and ground-state approximations actually establish.
Part IIIThe maps between them
Structural arithmetic geometry
Universal positive realizations, direct bridge maps, counterexamples, and the remaining arithmetic obligations.
Primary sources. Explicit limits.
The program distinguishes established comparisons from conjectural arithmetic constructions.
Recent research
- Connes, Consani, Moscovici: Zeta spectral triples
The proxy limit and the two remaining ground-state steps.
- Suzuki: Weil’s quadratic form via the screw function
A concrete localized operator comparison, with global positivity still open.
- Connes and van Suijlekom: Quadratic forms and real zeros
The precise hypotheses behind the real-zero transfer.
Formal evidence
- Quotients and counterexamples
Descent preserves a supplied sign. Spectral purity alone does not supply an invariant metric.
- Approximation and weighted transforms
Common tests, finite Gram limits, and bounds on complex strips.
- Source and circularity audit
Context revisions, claim scopes, and the missing arithmetic instances.