Riemann and Hodge Research Program — Current Repository Map
Evidence-scoped map of the independent research program connecting Riemann zeta structures, foliated dynamics, spectral geometry, and Hodge-theoretic formalization.
Riemann and Hodge Research Program
This page records the current public-facing map of the mathematical research repository dazzling-shannon. It is a research program and repository inventory, not a claim of a completed proof of the Riemann Hypothesis.
Scope
The repository studies a connected set of constructions around:
- the completed Riemann zeta function, its critical-line symmetry, and Riemann–Siegel coordinates;
- foliated and solenoidal dynamical models, including transfer operators, natural extensions, and projective limits;
- spectral and noncommutative-geometric structures, including Dirac-type operators, Fredholm modules, Cuntz representations, and trace formulas;
- Hodge-theoretic structures, including leafwise Dolbeault ideas, Gauss–Manin/Picard–Fuchs systems, period matrices, and -adic Hodge-inspired modules;
- explicit falsification and obstruction work intended to identify which local or structural properties do not by themselves imply the global Riemann Hypothesis.
Repository topology
The source organizes work by strata and campaigns. Relevant current modules include:
| Area | Repository evidence |
|---|---|
| Riemann symmetry and rotating frames | RiemannMirror.lean, RotatingFrameCalculus.lean, XiFullStripParity.lean |
| Solenoid and noncommutative geometry | SolenoidDiscreteModel.lean, SolenoidHolonomyGroupoid.lean, SolenoidMultiscaleWavelets.lean |
| Spectral/cohomological structures | AdelicMeyerTrace.lean, NoncommutativeFredholmKHomology.lean, CuntzAlgebraRepresentation.lean |
| Hodge and period structures | GaussManinPicardFuchs.lean, PadicHodgeTateCurve.lean, MatchboxDolbeaultCohomology.lean |
| Falsification boundary | research/claim_ledger.md, research/handoffs/CLOSED_divisor_current_and_logarithmic_cover_1788043851_REPORT.md, Chapter 27 of the unified report |
Hodge line of inquiry
The Hodge-related work is not a single theorem. It is a cluster of formal and computational investigations:
- A Legendre-family model connects Gauss–Manin transport to a hypergeometric Picard–Fuchs operator.
- A Tate-curve model records a period matrix and filtration relations in a -adic setting.
- Leafwise complex structures are used to state Hodge-star, Dolbeault, and decomposition identities on a matchbox/solenoid model.
- These structures are studied as possible components of a spectral-geometric framework, not as an established bridge to the zeros of .
The repository distinguishes kernel-checked Lean statements, exact identities, numerical evidence, conditional results, hypotheses, and open proof obligations. Those categories must remain visible when the work is summarized elsewhere.
Riemann line of inquiry
The Riemann-related work develops local and structural mechanisms around the critical line. Examples include the mirror involution , rotating-frame derivatives, symplectic lifts, logarithmic potentials, divisor currents, logarithmic covers, and spectral/trace constructions.
The research value is not limited to positive results. The repository also records countermodels and failed routes. In particular, local harmonicity, reflection symmetry, integer residues, or a local non-ramification statement do not, by themselves, exclude off-line zeros.
Current status boundary
The current accepted checkpoint identifies a historical, unreviewed baseline: reviewed_git_head is null, accepted_delta_claim_ids is empty, and no last accepted reviewer verdict is recorded. The repository therefore supports saying that these constructions and formalization artifacts exist and that some are mechanically checked within their stated assumptions. It does not support presenting the overall campaign as an accepted proof of the Riemann Hypothesis.
The exact open bridge is the construction of a global arithmetic-geometric space whose dynamics, spectrum, prime data, and regularized measure satisfy all required identifications simultaneously. That bridge remains an active research program.
Related pages
Sources
dazzling-shannon/GEMINI.md, repository state and epistemic rules.dazzling-shannon/goals/active_goal.json, current goal state.dazzling-shannon/research/checkpoints/current_checkpoint.json, accepted-checkpoint fields.dazzling-shannon/research/claim_ledger.md, claim classifications and evidence pointers.dazzling-shannon/research/handoffs/MASTER_OVERNIGHT_RESEARCH_REPORT_EPOCHS_1_19.md.dazzling-shannon/research/handoffs/MASTER_MATHEMATICAL_ODYSSEY_ALL_EPOCHS_1_19_UNIFIED.md.