Hilbert's 23 Problems — Status & Cascade Catalog

David Hilbert presented 23 problems at the International Congress of Mathematicians, Paris, 8 August 1900. Most have been resolved. The genuinely-open ones (plus some considered "resolved by independence" or partially solved) get cascade-research subdirectories here.

Source: Wikipedia — Hilbert's problems

Full status table

#	Title (short)	Status	Cascade dir
1	Continuum hypothesis	Resolved by independence (Gödel 1940, Cohen 1963 — proved independent of ZFC). Some philosophical debate remains.	—
2	Consistency of arithmetic	Partially resolved (Gödel 1931 — cannot be proved within arithmetic itself; Gentzen 1936 gave proof using transfinite induction).	—
3	Equal-volume polyhedra dissection	Solved — Dehn 1900 (negative answer via Dehn invariant).	—
4	Foundation of geometry	Resolved as too vague; partially addressed by Hamel 1903 and successors.	—
5	Lie groups without smoothness	Solved — Gleason, Montgomery, Zippin 1952.	—
6	Mathematical treatment of axioms of physics	Open / ongoing — partially addressed (Kolmogorov axioms for probability 1933; quantum-mechanics axiomatization; QFT formalization remains incomplete).	hilbert_06_axiomatize_physics/
7	Irrationality / transcendence of a^b	Solved — Gelfond-Schneider 1934.	—
8	Riemann hypothesis, Goldbach, twin primes	OPEN — the headliner. Three major sub-questions.	hilbert_08_riemann_hypothesis/, hilbert_08_goldbach_conjecture/, hilbert_08_twin_prime/
9	Reciprocity in algebraic number fields	Solved — Artin reciprocity 1927; Hasse and others.	—
10	Decidability of Diophantine equations	Solved — Matiyasevich 1970 (negative: undecidable).	—
11	Quadratic forms with algebraic coefficients	Solved — Hasse-Minkowski local-global principle.	—
12	Kronecker's Jugendtraum (abelian extensions)	Partially solved / open — solved for ℚ (Kronecker-Weber) and imaginary quadratic (complex multiplication); open for general number fields.	hilbert_12_kronecker_jugendtraum/
13	7th-degree solvability via 2-variable functions	Solved — Kolmogorov-Arnold 1957 (continuous version); algebraic version remains technically debated.	—
14	Finite generation of certain rings of invariants	Solved — Nagata 1959 (negative counterexample).	—
15	Schubert calculus rigor	Solved — Severi school, modern intersection theory.	—
16	Topology of real algebraic curves + planar polynomial vector field limit cycles	PARTIALLY OPEN — (16a) topology of real algebraic curves partially solved; (16b) bound on number of limit cycles of polynomial vector field is OPEN. Also Smale's 16th problem.	hilbert_16_limit_cycles/
17	Positive rationals as sums of squares	Solved — Artin 1927.	—
18	Polyhedra space-filling + Kepler conjecture	Solved — Bieberbach (18a); Hales 1998 / formal proof 2014 (18b Kepler).	—
19	Solutions of Lagrangians always analytic	Solved — De Giorgi 1956, Nash 1957.	—
20	Boundary value problems	Solved — variational methods.	—
21	Existence of linear differential equations with given monodromy	Solved with refinement — Plemelj 1908 (with gap); Bolibruch 1989 counterexample; refined affirmative for various settings.	—
22	Uniformization of analytic relations	Solved — Koebe 1907, Poincaré (uniformization theorem).	—
23	Calculus of variations	Resolved as research direction — not a single problem; whole field developed.	—

Cascade-research priority

These are the open Hilbert problems with directories below:

Problem	Cascade tractability	Priority for autonomous dispatch
#8 Goldbach	High — Class J (primes) + Class I (cyclic Z/2nZ for even-sum structure) + Class L (Goldbach partition graph spectrum)	First demo — most tractable for cascade prototyping
#8 Twin Prime	High — Class J + Class I + Class K (asymptotic-DoF for prime gap distribution)	Second
#8 Riemann Hypothesis	High but heavy — Class L (Hilbert-Pólya: zeros as eigenvalues of self-adjoint operator) + Class M (HDC encoding of zeta function functional equation)	Third
#16 Limit Cycles	Moderate — Class L on phase-space Laplacian + Class C orientation of vector field	Fourth
#12 Kronecker Jugendtraum	Moderate — Class I (cyclic-group / class-field generation) + Class N (rational/algebraic approximation)	Fifth
#6 Axiomatize Physics	Low (vague) — Class H (self-introspection: which class operators are needed?) + Class A-N enumeration	Sixth

How each subdirectory is organized

Per the top-level REPORT_TEMPLATE.md:

hilbert_NN_short_name/
├── REPORT.md             ← uniform problem report (10 sections)
├── descriptor.toml       ← AMSC source descriptor (when cascade is dispatched)
├── schema.json           ← per-record JSON Schema
├── *.ndjson              ← cascade-output records
└── generate_catalog.py   ← reproducible cascade generator

REPORT.md is created at scaffold time (with status: open / cascade dispatched: awaiting dispatch). The other four files are created when the cascade is actually dispatched and results are recorded.