allaun
514bf8da72
- 68 problem records (67 unsolved + 1 solved boundary marker) - 10 alignment clusters spanning number theory, complexity, geometry, topology, analysis, logic, physics, and cosmology - 30×30 crossing matrix of known reductions, shared techniques, and analogies - RRC shape/axis tags consistent with existing rrc_equation_classification.md - JSON validated
63 KiB
Unsolved Hard Problems — RRC Manifold/Projection Survey
Date: 2026-06-20
Scope: A curated survey of well-known unsolved problems across mathematics, theoretical computer science, physics, and logic, tagged with Rainbow Raccoon Compiler (RRC) manifold/projection metadata.
Claim boundary: This document is a research artifact. It records RRC-style projections (shape class, status, axes, alignment fingerprint) and known interconnections. It does not claim any problem is solved unless explicitly marked ACCEPT.
Sources consulted: Clay Mathematics Institute Millennium Problems, Wolfram MathWorld, arXiv survey literature, and standard mathematical references.
Summary
- Total problem records: 68
- Unsolved problems surveyed: 67
- Solved boundary markers: 1
- Alignment clusters: 10
RRC axis schema
Projections use the 16-axis RRC manifold vector from 6-Documentation/docs/rainbow_raccoon_compiler_integration.md, restricted here to the most relevant axes:
semantic_entropygeometric_masscompression_pressuretopology_torsionresidual_riskproof_readinessscale_band_declarednegative_control_strengthprojection_declaredshape_closure
Axis values are dimensionless scores in [0,1] (Q0_16-compatible normalized units). Higher values indicate stronger presence of that axis.
RRC shape vocabulary
Shape classes are taken from 4-Infrastructure/shim/rrc_ray_tagger.py and 6-Documentation/docs/rainbow_raccoon_compiler_integration.md:
BurgersRGSolverCadForceProbeReceiptCognitiveLoadFieldComputeKernelReceiptErdosBoundConjectureHoldForUnlawfulOrUnderspecifiedShapeLanguageSetManifoldGraphLeanTheoremReceiptLogogramProjectionProjectableGeometryTopologySignalShapedRouteCompiler
Problems
Riemann Hypothesis {id:riemann_hypothesis}
Fields: Mathematics, Number Theory, Analysis
Statement. All non-trivial zeros of the Riemann zeta function ζ(s) have real part 1/2.
Why it is unsolved. No analytic proof forces the spectral projection of zeta zeros onto the critical line; random-matrix and numerical evidence are strong but non-rigorous.
Known reductions to: generalized_riemann_hypothesis, birch_swinnerton_dyer_conjecture, elliott_halberstam_conjecture
Known reductions from: generalized_riemann_hypothesis
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_01
Top axes: semantic_entropy=0.85, compression_pressure=0.80, scale_band_declared=0.90, negative_control_strength=0.70, projection_declared=0.95
Alignment fingerprint. Spectral line on the critical axis; projection is sharp, proof readiness is low.
Generalized Riemann Hypothesis {id:generalized_riemann_hypothesis}
Fields: Mathematics, Number Theory
Statement. All non-trivial zeros of Dirichlet L-functions and automorphic L-functions lie on the critical line.
Why it is unsolved. The family of L-functions lacks a universal positivity or monotonicity argument; GRH implies RH but is harder.
Known reductions to: riemann_hypothesis, elliott_halberstam_conjecture, fermat_catalan_conjecture
Known reductions from: riemann_hypothesis
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_01
Top axes: semantic_entropy=0.90, geometric_mass=0.65, compression_pressure=0.85, scale_band_declared=0.85, projection_declared=0.90
Alignment fingerprint. Family of spectral manifolds; projection declared but proof readiness even lower than RH.
P vs NP {id:p_vs_np}
Fields: Mathematics, Theoretical Computer Science, Logic
Statement. Is every language decidable by a nondeterministic polynomial-time Turing machine also decidable by a deterministic polynomial-time machine?
Why it is unsolved. Relativization, natural proofs, and algebrization barriers block diagonalization and algebraic techniques; no super-polynomial circuit lower bound for NP is known.
Known reductions to: np_intermediate_existence, graph_isomorphism_in_p, factoring_in_p, discrete_log_in_p, bpp_vs_p, exponential_time_hypothesis, unique_games_conjecture, bqp_vs_np, p_np_algebrization_barrier, matrix_rigidity, derandomization_polynomial_identity_testing
Known reductions from: np_intermediate_existence
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.80, compression_pressure=0.95, scale_band_declared=0.95, negative_control_strength=0.80, projection_declared=1.00
Alignment fingerprint. High-compression decision boundary; barriers create strong topology torsion.
Navier-Stokes existence and smoothness {id:navier_stokes_existence_smoothness}
Fields: Mathematics, Analysis, PDE
Statement. Do the 3D incompressible Navier-Stokes equations admit smooth solutions for all smooth initial data?
Why it is unsolved. Finite-time singularity (blow-up) has not been ruled out; weak solutions exist but uniqueness and regularity remain open.
Known reductions to: navier_stokes_blowup, turbulence_closure_problem, hilbert_sixteenth_problem
Known reductions from: navier_stokes_blowup
RRC shape: BurgersRGSolver
RRC status: CANDIDATE
Alignment cluster: cluster_03
Top axes: semantic_entropy=0.85, geometric_mass=0.75, compression_pressure=0.80, scale_band_declared=0.85, projection_declared=0.90
Alignment fingerprint. Geometric PDE mass concentrated; residual risk of blow-up dominates.
Yang-Mills existence and mass gap {id:yang_mills_mass_gap}
Fields: Mathematical Physics, Mathematics
Statement. Prove that quantum Yang-Mills theory exists in four dimensions and has a mass gap.
Why it is unsolved. Constructive quantum field theory in 4D is missing; mass gap is supported numerically and by lattice gauge theory but not proven rigorously.
Known reductions to: quantum_gravity, quantum_supremacy_verification
RRC shape: SignalShapedRouteCompiler
RRC status: CANDIDATE
Alignment cluster: cluster_03
Top axes: semantic_entropy=0.90, geometric_mass=0.60, compression_pressure=0.85, scale_band_declared=0.80, projection_declared=0.80
Alignment fingerprint. Quantum-field route with high semantic entropy; continuum limit projection underspecified.
Hodge Conjecture {id:hodge_conjecture}
Fields: Mathematics, Algebraic Geometry
Statement. Every Hodge class on a non-singular complex projective variety is a rational linear combination of classes of algebraic cycles.
Why it is unsolved. No general construction converts Hodge-theoretic data into algebraic cycles; known only in special cases.
Known reductions to: tate_conjecture, standard_conjectures, birch_swinnerton_dyer_conjecture
Known reductions from: tate_conjecture, standard_conjectures
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_01
Top axes: semantic_entropy=0.80, geometric_mass=0.90, compression_pressure=0.75, scale_band_declared=0.75, projection_declared=0.85
Alignment fingerprint. High geometric mass and topology torsion; algebraic-cycle witness missing.
Birch and Swinnerton-Dyer conjecture {id:birch_swinnerton_dyer_conjecture}
Fields: Mathematics, Number Theory, Algebraic Geometry
Statement. The rank of the group of rational points of an elliptic curve equals the order of vanishing of its L-function at s=1.
Why it is unsolved. The Shafarevich-Tate group and Selmer groups are not controlled; only partial results (parity, rank ≤ 1) are known.
Known reductions to: tate_conjecture, rational_points_high_genus
Known reductions from: hodge_conjecture, tate_conjecture
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_01
Top axes: semantic_entropy=0.75, geometric_mass=0.85, compression_pressure=0.80, scale_band_declared=0.80, projection_declared=0.85
Alignment fingerprint. L-function/geometry bridge; low proof readiness due to Tate-Shafarevich torsion.
Smooth 4D Poincaré conjecture {id:smooth_4d_poincare_conjecture}
Fields: Mathematics, Topology
Statement. Every smooth closed 4-manifold homotopy equivalent to S⁴ is diffeomorphic to S⁴.
Why it is unsolved. Topological and smooth categories diverge in dimension 4; exotic smooth structures prevent a simple recognition theorem.
Known reductions to: generalized_poincare_conjecture_smooth
Known reductions from: generalized_poincare_conjecture_smooth
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_05
Top axes: semantic_entropy=0.70, geometric_mass=0.95, topology_torsion=0.95, scale_band_declared=0.75, projection_declared=0.85
Alignment fingerprint. Dimension-4 smooth topology torsion peak; projection declared but no constructive witness.
Generalized Poincaré conjecture (smooth category) {id:generalized_poincare_conjecture_smooth}
Fields: Mathematics, Topology
Statement. Every closed smooth n-manifold homotopy equivalent to Sⁿ is diffeomorphic to Sⁿ for n ≥ 4.
Why it is unsolved. The topological version is settled except for the smooth 4D case; the smooth category lacks a general classification.
Known reductions to: smooth_4d_poincare_conjecture
Known reductions from: smooth_4d_poincare_conjecture
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_05
Top axes: semantic_entropy=0.75, geometric_mass=0.90, compression_pressure=0.75, topology_torsion=0.90, projection_declared=0.80
Alignment fingerprint. Smooth-topology equivalence across dimensions; 4D torsion dominates.
abc conjecture {id:abc_conjecture}
Fields: Mathematics, Number Theory
Statement. For every ε > 0 there are only finitely many coprime positive integer triples a + b = c with c > rad(abc)^{1+ε}.
Why it is unsolved. The interplay between additive and multiplicative structure of integers is not captured by existing Diophantine tools; Mochizuki's claimed proof remains contested.
Known reductions to: beal_conjecture, fermat_catalan_conjecture, brocards_problem, pillai_conjecture
Known reductions from: beal_conjecture, fermat_catalan_conjecture
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.75, compression_pressure=0.80, scale_band_declared=0.75, negative_control_strength=0.60, projection_declared=0.90
Alignment fingerprint. Diophantine logogram with sharp projection; proof readiness stalled.
Goldbach conjecture {id:goldbach_conjecture}
Fields: Mathematics, Number Theory
Statement. Every even integer greater than 2 is the sum of two primes.
Why it is unsolved. Additive structure of primes lacks a sieve/inclusion-exclusion argument that closes at all scales; verified computationally to very large bounds.
Known reductions to: twin_prime_conjecture, polignacs_conjecture
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.60, compression_pressure=0.70, scale_band_declared=0.85, negative_control_strength=0.70, projection_declared=0.95
Alignment fingerprint. Simple logogram with strong computational witness; proof closure missing.
Twin prime conjecture {id:twin_prime_conjecture}
Fields: Mathematics, Number Theory
Statement. There are infinitely many primes p such that p + 2 is also prime.
Why it is unsolved. Sieve methods cannot yet distinguish consecutive prime gaps at bounded distance; Zhang and Maynard produced bounded gaps but not gap 2.
Known reductions to: polignacs_conjecture, elliott_halberstam_conjecture, schinzel_hypothesis_h
Known reductions from: polignacs_conjecture
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.65, compression_pressure=0.75, scale_band_declared=0.80, negative_control_strength=0.65, projection_declared=0.90
Alignment fingerprint. Bounded-gap sieve route; residual risk from parity of sieves.
Collatz conjecture (3n+1 problem) {id:collatz_conjecture}
Fields: Mathematics, Number Theory, Dynamical Systems
Statement. Iterating the 3n+1 map always reaches 1 for every positive integer seed.
Why it is unsolved. No invariant controls the combined expand/contract dynamics across all scales; verified empirically to huge bounds.
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.55, compression_pressure=0.70, scale_band_declared=0.80, negative_control_strength=0.60, projection_declared=0.85
Alignment fingerprint. Simple iterative logogram with dynamical-systems torsion.
Beal conjecture {id:beal_conjecture}
Fields: Mathematics, Number Theory
Statement. If A^x + B^y = C^z with positive integers and x,y,z > 2, then A, B, C share a common prime factor.
Why it is unsolved. Generalizes Fermat's last theorem; arbitrary-exponent Diophantine methods are insufficient; abc conjecture would imply it.
Known reductions to: abc_conjecture
Known reductions from: abc_conjecture
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.65, compression_pressure=0.70, scale_band_declared=0.75, negative_control_strength=0.55, projection_declared=0.85
Alignment fingerprint. Exponential Diophantine logogram; reduction to abc creates one-way dependency.
Unique Games Conjecture {id:unique_games_conjecture}
Fields: Theoretical Computer Science, Mathematics
Statement. For every ε,δ > 0 it is NP-hard to distinguish (1−δ)-satisfiable from ε-satisfiable Unique Games instances.
Why it is unsolved. Resists sum-of-squares and SDP integrality-gap attacks; equivalent to many optimal hardness-of-approximation results.
Known reductions to: p_vs_np, small_set_expansion_conjecture
Known reductions from: p_vs_np, small_set_expansion_conjecture
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.80, compression_pressure=0.85, scale_band_declared=0.80, negative_control_strength=0.65, projection_declared=0.85
Alignment fingerprint. Hardness-of-approximation router; high compression pressure from SDP gaps.
Existence of NP-intermediate problems {id:np_intermediate_existence}
Fields: Theoretical Computer Science, Logic
Statement. Does there exist a problem in NP that is neither in P nor NP-complete?
Why it is unsolved. Ladner's theorem gives such problems conditionally on P ≠ NP; unconditional existence is exactly as hard as separating P from NP.
Known reductions to: p_vs_np
Known reductions from: p_vs_np
RRC shape: CognitiveLoadField
RRC status: HOLD
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.70, compression_pressure=0.80, residual_risk=0.55, scale_band_declared=0.70, projection_declared=0.75
Alignment fingerprint. Conditional on P vs NP; shape closure blocked by complexity boundary.
Graph isomorphism in P? {id:graph_isomorphism_in_p}
Fields: Theoretical Computer Science, Mathematics
Statement. Can graph isomorphism be decided in deterministic polynomial time?
Why it is unsolved. Babai gave a quasipolynomial-time algorithm, but group-theoretic obstacles remain; GI is not known to be NP-complete.
Known reductions to: p_vs_np
Known reductions from: p_vs_np
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.70, compression_pressure=0.75, scale_band_declared=0.75, negative_control_strength=0.60, projection_declared=0.85
Alignment fingerprint. Symmetry-classification route; quasipolynomial witness improves readiness.
Integer factorization in P {id:factoring_in_p}
Fields: Theoretical Computer Science, Number Theory, Cryptography
Statement. Can integer factorization be solved in deterministic polynomial time?
Why it is unsolved. No polynomial-time classical algorithm is known; Shor's algorithm uses quantum resources.
Known reductions to: p_vs_np, discrete_log_in_p
Known reductions from: p_vs_np, discrete_log_in_p
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.70, compression_pressure=0.80, scale_band_declared=0.80, negative_control_strength=0.65, projection_declared=0.85
Alignment fingerprint. Cryptographic hardness router; quantum route swappable but classical proof missing.
Discrete logarithm in P {id:discrete_log_in_p}
Fields: Theoretical Computer Science, Number Theory, Cryptography
Statement. Can the discrete logarithm problem be solved in deterministic polynomial time?
Why it is unsolved. Number-field-sieve algorithms are subexponential but not polynomial; reductions closely couple factoring and discrete log.
Known reductions to: p_vs_np, factoring_in_p
Known reductions from: p_vs_np, factoring_in_p
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.70, compression_pressure=0.80, scale_band_declared=0.80, negative_control_strength=0.60, projection_declared=0.85
Alignment fingerprint. Algebraic-group route; closely coupled to factoring.
BPP vs P (derandomization) {id:bpp_vs_p}
Fields: Theoretical Computer Science, Mathematics
Statement. Does every polynomial-time randomized algorithm have a deterministic polynomial-time simulation?
Why it is unsolved. Hardness-vs-randomness links derandomization to circuit lower bounds; no explicit pseudorandom generator covers all of BPP.
Known reductions to: p_vs_np, exponential_time_hypothesis, matrix_rigidity, derandomization_polynomial_identity_testing
Known reductions from: p_vs_np
RRC shape: CognitiveLoadField
RRC status: HOLD
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.75, compression_pressure=0.80, topology_torsion=0.55, scale_band_declared=0.75, projection_declared=0.80
Alignment fingerprint. Pseudorandomness compression gate; blocked by circuit lower bounds.
Exponential Time Hypothesis {id:exponential_time_hypothesis}
Fields: Theoretical Computer Science
Statement. 3-SAT cannot be solved in time 2^{o(n)}.
Why it is unsolved. Strongly supported by algorithmic experience but unproven; a large web of conditional lower bounds depends on it.
Known reductions to: p_vs_np, strong_exponential_time_hypothesis, unique_games_conjecture
Known reductions from: p_vs_np, strong_exponential_time_hypothesis
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.75, compression_pressure=0.85, scale_band_declared=0.80, negative_control_strength=0.65, projection_declared=0.85
Alignment fingerprint. Fine-grained complexity anchor; high compression pressure.
Strong Exponential Time Hypothesis {id:strong_exponential_time_hypothesis}
Fields: Theoretical Computer Science
Statement. CNF-SAT requires time 2^{(1−ε)n} for some ε > 0.
Why it is unsolved. Stronger than ETH; underpins many tight lower bounds but remains unproven.
Known reductions to: exponential_time_hypothesis, p_vs_np
Known reductions from: exponential_time_hypothesis, p_vs_np
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.80, compression_pressure=0.85, topology_torsion=0.60, scale_band_declared=0.75, projection_declared=0.80
Alignment fingerprint. Tight SAT lower-bound route; topology torsion from exact constants.
Continuum Hypothesis {id:continuum_hypothesis}
Fields: Mathematics, Logic, Set Theory
Statement. Is there a set whose cardinality is strictly between ℵ₀ and 2^{ℵ₀}?
Why it is unsolved. Independent of ZFC by Gödel and Cohen; cannot be resolved within standard axioms without adopting new axioms.
Known reductions to: consistency_of_zfc, singular_cardinal_hypothesis
Known reductions from: consistency_of_zfc
RRC shape: LanguageSetManifoldGraph
RRC status: HOLD
Alignment cluster: cluster_06
Top axes: semantic_entropy=0.90, compression_pressure=0.70, residual_risk=0.80, scale_band_declared=0.60, projection_declared=0.70
Alignment fingerprint. Axiomatic boundary; projection underdetermined by ZFC.
Consistency of ZFC {id:consistency_of_zfc}
Fields: Mathematics, Logic
Statement. Are the Zermelo-Fraenkel axioms with Choice consistent?
Why it is unsolved. Gödel's second incompleteness theorem shows ZFC cannot prove its own consistency unless it is inconsistent.
Known reductions to: continuum_hypothesis, singular_cardinal_hypothesis
Known reductions from: continuum_hypothesis
RRC shape: LanguageSetManifoldGraph
RRC status: HOLD
Alignment cluster: cluster_06
Top axes: semantic_entropy=0.95, compression_pressure=0.75, topology_torsion=0.55, residual_risk=0.90, projection_declared=0.60
Alignment fingerprint. Meta-mathematical limit point; negative controls extremely weak.
Hilbert's 6th problem {id:hilbert_sixth_problem}
Fields: Mathematics, Physics, Logic
Statement. Axiomatize all of physics in a mathematically rigorous way.
Why it is unsolved. Scope is open-ended; physics contains effective theories, emergent phenomena, and the measurement problem, none fully axiomatized.
Known reductions to: quantum_gravity, measurement_problem, cosmological_constant_problem
RRC shape: HoldForUnlawfulOrUnderspecifiedShape
RRC status: HOLD
Alignment cluster: unclustered
Top axes: semantic_entropy=0.95, geometric_mass=0.40, compression_pressure=0.90, topology_torsion=0.60, residual_risk=0.70
Alignment fingerprint. Underspecified universal axiomatization; projection and scale band weak.
Hilbert's 16th problem {id:hilbert_sixteenth_problem}
Fields: Mathematics, Analysis, Dynamical Systems
Statement. Bound the number of limit cycles for planar polynomial vector fields.
Why it is unsolved. Hilbert number H(n) is unknown even for n=2; tied to bifurcation theory and o-minimality.
Known reductions to: navier_stokes_existence_smoothness
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_03
Top axes: semantic_entropy=0.80, geometric_mass=0.75, compression_pressure=0.75, scale_band_declared=0.70, projection_declared=0.75
Alignment fingerprint. Planar topology-dynamics; projection declared but scale band diffuse.
Hilbert's 12th problem {id:hilbert_twelfth_problem}
Fields: Mathematics, Number Theory, Algebra
Statement. Construct all abelian extensions of arbitrary algebraic number fields (Kronecker's Jugendtraum).
Why it is unsolved. Solved for Q and imaginary quadratic fields; general base fields lack explicit class-field-theory generators.
Known reductions to: langlands_program
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_10
Top axes: semantic_entropy=0.85, geometric_mass=0.70, compression_pressure=0.75, scale_band_declared=0.65, projection_declared=0.70
Alignment fingerprint. Non-abelian class-field route; projection geometry over number fields.
Navier-Stokes finite-time blow-up {id:navier_stokes_blowup}
Fields: Mathematics, Analysis
Statement. Does there exist a finite-time singularity for 3D Navier-Stokes with smooth initial data?
Why it is unsolved. Candidate blow-up constructions have been proposed and debated; rigorous confirmation or refutation is absent.
Known reductions to: navier_stokes_existence_smoothness
Known reductions from: navier_stokes_existence_smoothness
RRC shape: BurgersRGSolver
RRC status: HOLD
Alignment cluster: cluster_03
Top axes: semantic_entropy=0.75, geometric_mass=0.80, compression_pressure=0.80, residual_risk=0.75, projection_declared=0.75
Alignment fingerprint. Singularity endpoint; high residual risk, weak negative controls.
Turbulence closure problem {id:turbulence_closure_problem}
Fields: Physics, Applied Mathematics
Statement. Derive a closed finite set of equations for the statistics of turbulent flows.
Why it is unsolved. The moment hierarchy is infinite and scale interactions span many orders; no universal closure exists.
Known reductions to: navier_stokes_existence_smoothness, navier_stokes_blowup
Known reductions from: navier_stokes_blowup
RRC shape: BurgersRGSolver
RRC status: HOLD
Alignment cluster: cluster_03
Top axes: semantic_entropy=0.90, geometric_mass=0.70, compression_pressure=0.85, topology_torsion=0.60, projection_declared=0.60
Alignment fingerprint. Multi-scale PDE route; closure gap prevents projection.
Quantum supremacy verification {id:quantum_supremacy_verification}
Fields: Theoretical Computer Science, Physics
Statement. Can a quantum computation be certified as infeasible for any classical computer?
Why it is unsolved. Verification of sampling tasks is hard; cross-entropy benchmarking gives statistical evidence, not proof.
Known reductions to: p_vs_np, bpp_vs_p, bqp_vs_np
Known reductions from: p_vs_np
RRC shape: ComputeKernelReceipt
RRC status: HOLD
Alignment cluster: cluster_07
Top axes: semantic_entropy=0.85, compression_pressure=0.80, residual_risk=0.60, scale_band_declared=0.70, projection_declared=0.65
Alignment fingerprint. Hardware-software verification gap; compute receipt incomplete.
Black hole information paradox {id:black_hole_information_paradox}
Fields: Physics, Quantum Gravity
Statement. Is information preserved during black hole evaporation?
Why it is unsolved. Tension between general relativity (no-hair) and quantum mechanics (unitarity); no consensus mechanism.
Known reductions to: quantum_gravity, cosmic_censorship_conjecture
Known reductions from: quantum_gravity
RRC shape: ProjectableGeometryTopology
RRC status: HOLD
Alignment cluster: cluster_07
Top axes: semantic_entropy=0.90, geometric_mass=0.75, compression_pressure=0.80, topology_torsion=0.70, residual_risk=0.65
Alignment fingerprint. Geometry-quantum boundary; high topology torsion, projection underspecified.
Dark matter identity {id:dark_matter_identity}
Fields: Physics, Cosmology
Statement. What is the particle or gravitational nature of dark matter?
Why it is unsolved. No non-gravitational detection; candidates (WIMPs, axions, primordial black holes) remain hypothetical.
Known reductions to: cosmological_constant_problem, baryon_asymmetry_problem, quantum_gravity
Known reductions from: cosmological_constant_problem
RRC shape: CadForceProbeReceipt
RRC status: HOLD
Alignment cluster: cluster_08
Top axes: semantic_entropy=0.85, geometric_mass=0.50, compression_pressure=0.75, residual_risk=0.70, scale_band_declared=0.55
Alignment fingerprint. Dark-sector force probe; projection and scale band weak.
Cosmological constant problem {id:cosmological_constant_problem}
Fields: Physics, Cosmology
Statement. Why is the observed vacuum energy density ~120 orders of magnitude smaller than naive QFT predictions?
Why it is unsolved. No known cancellation mechanism for quantum corrections; anthropic explanations are not predictive.
Known reductions to: dark_matter_identity, quantum_gravity, dark_energy_equation_of_state, cosmological_inflation_origin
Known reductions from: dark_matter_identity
RRC shape: CadForceProbeReceipt
RRC status: HOLD
Alignment cluster: cluster_08
Top axes: semantic_entropy=0.90, geometric_mass=0.45, compression_pressure=0.85, residual_risk=0.75, scale_band_declared=0.50
Alignment fingerprint. Force-probe hierarchy mismatch; projection almost absent.
Baryon asymmetry of the universe {id:baryon_asymmetry_problem}
Fields: Physics, Cosmology
Statement. Why is the universe made of matter rather than equal amounts of matter and antimatter?
Why it is unsolved. Sakharov conditions are known, but no Standard Model source produces the observed asymmetry.
Known reductions to: dark_matter_identity, cosmological_inflation_origin
RRC shape: CadForceProbeReceipt
RRC status: HOLD
Alignment cluster: cluster_08
Top axes: semantic_entropy=0.80, compression_pressure=0.70, residual_risk=0.60, scale_band_declared=0.55, projection_declared=0.50
Alignment fingerprint. Force-probe asymmetry; projection diffuse.
Quantum gravity {id:quantum_gravity}
Fields: Physics, Mathematics
Statement. Reconcile general relativity and quantum mechanics into a single consistent theory.
Why it is unsolved. Non-renormalizability of GR, background independence, and the measurement problem block direct quantization.
Known reductions to: black_hole_information_paradox, cosmological_constant_problem, yang_mills_mass_gap, measurement_problem, cosmic_censorship_conjecture
Known reductions from: black_hole_information_paradox
RRC shape: ProjectableGeometryTopology
RRC status: HOLD
Alignment cluster: cluster_07
Top axes: semantic_entropy=0.95, geometric_mass=0.85, compression_pressure=0.90, topology_torsion=0.85, residual_risk=0.75
Alignment fingerprint. Ultimate geometry-quantum projection; all axes extreme, projection weakest.
Quantum measurement problem {id:measurement_problem}
Fields: Physics, Foundations
Statement. Explain the apparent collapse of the quantum wavefunction upon measurement.
Why it is unsolved. Interpretational gap between unitary evolution and observed outcomes; no universally accepted resolution.
Known reductions to: quantum_gravity, hilbert_sixth_problem
Known reductions from: quantum_gravity
RRC shape: HoldForUnlawfulOrUnderspecifiedShape
RRC status: HOLD
Alignment cluster: unclustered
Top axes: semantic_entropy=0.90, compression_pressure=0.75, topology_torsion=0.50, residual_risk=0.65, scale_band_declared=0.35
Alignment fingerprint. Interpretational hold; projection and scale band severely underspecified.
Tate conjecture {id:tate_conjecture}
Fields: Mathematics, Algebraic Geometry
Statement. Algebraic cycles generate the l-adic cohomology classes invariant under Galois action.
Why it is unsolved. Relates arithmetic and geometry; known in special cases but a full proof is missing.
Known reductions to: hodge_conjecture, standard_conjectures, birch_swinnerton_dyer_conjecture
Known reductions from: hodge_conjecture, standard_conjectures
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_01
Top axes: semantic_entropy=0.80, geometric_mass=0.90, compression_pressure=0.75, topology_torsion=0.70, projection_declared=0.80
Alignment fingerprint. Arithmetic-geometric topology; close cousin of Hodge.
Grothendieck's standard conjectures {id:standard_conjectures}
Fields: Mathematics, Algebraic Geometry
Statement. Standard conjectures on algebraic cycles, including Lefschetz and Hodge standard.
Why it is unsolved. Would imply the Weil and Tate conjectures but remain unproven; no general approach exists.
Known reductions to: hodge_conjecture, tate_conjecture
Known reductions from: hodge_conjecture, tate_conjecture
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_01
Top axes: semantic_entropy=0.85, geometric_mass=0.95, compression_pressure=0.80, topology_torsion=0.75, projection_declared=0.75
Alignment fingerprint. Motivic topology anchor; high geometric mass and torsion.
Polignac's conjecture {id:polignacs_conjecture}
Fields: Mathematics, Number Theory
Statement. For every even integer 2k there are infinitely many prime gaps of size 2k.
Why it is unsolved. Generalizes the twin prime conjecture; sieve parity problem blocks even bounded gaps.
Known reductions to: twin_prime_conjecture, elliott_halberstam_conjecture, schinzel_hypothesis_h
Known reductions from: twin_prime_conjecture
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.65, compression_pressure=0.70, scale_band_declared=0.75, negative_control_strength=0.60, projection_declared=0.85
Alignment fingerprint. Generalized prime-gap logogram; reduction tree rooted in twin prime.
Elliott-Halberstam conjecture {id:elliott_halberstam_conjecture}
Fields: Mathematics, Number Theory
Statement. Primes in arithmetic progressions are distributed as uniformly as GRH predicts up to a factor.
Why it is unsolved. Strong sieve input; would imply bounded prime gaps and related results.
Known reductions to: twin_prime_conjecture, polignacs_conjecture, generalized_riemann_hypothesis
Known reductions from: generalized_riemann_hypothesis
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.80, compression_pressure=0.80, scale_band_declared=0.70, negative_control_strength=0.50, projection_declared=0.80
Alignment fingerprint. Distribution hypothesis; high compression pressure via GRH-like uniformity.
Fermat-Catalan conjecture {id:fermat_catalan_conjecture}
Fields: Mathematics, Number Theory
Statement. Only finitely many perfect powers differ by 1.
Why it is unsolved. Special cases (Catalan's theorem, Fermat-Catalan conjecture) are solved or limited; abc would settle the general case.
Known reductions to: abc_conjecture
Known reductions from: abc_conjecture
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.65, compression_pressure=0.65, scale_band_declared=0.70, negative_control_strength=0.55, projection_declared=0.80
Alignment fingerprint. Power-difference logogram; conditional on abc closure.
Schinzel's Hypothesis H {id:schinzel_hypothesis_h}
Fields: Mathematics, Number Theory
Statement. Every admissible finite set of integer polynomials simultaneously takes prime values infinitely often.
Why it is unsolved. Generalizes twin primes, Green-Tao, and Dickson's conjecture; sieve obstructions are not overcome.
Known reductions to: twin_prime_conjecture, polignacs_conjecture, elliott_halberstam_conjecture
Known reductions from: twin_prime_conjecture
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.75, compression_pressure=0.75, scale_band_declared=0.70, negative_control_strength=0.50, projection_declared=0.80
Alignment fingerprint. Polynomial-prime sieve bottleneck; broad implications.
Brocard's problem {id:brocards_problem}
Fields: Mathematics, Number Theory
Statement. Are there finitely many integer solutions to n! + 1 = m² beyond n = 4,5,7?
Why it is unsolved. Exponential Diophantine equation with factorial; abc heuristics suggest a finite list.
Known reductions to: abc_conjecture
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.55, compression_pressure=0.60, scale_band_declared=0.70, negative_control_strength=0.55, projection_declared=0.80
Alignment fingerprint. Sparse factorial logogram; finite-list expectation.
Pillai's conjecture {id:pillai_conjecture}
Fields: Mathematics, Number Theory
Statement. For fixed positive integers A,B, the equation Ax^m − By^n = k has finitely many solutions for each k.
Why it is unsolved. Catalan's theorem is the k=1 case; the general case needs effective Diophantine bounds.
Known reductions to: abc_conjecture, brocards_problem
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.65, compression_pressure=0.65, scale_band_declared=0.70, negative_control_strength=0.55, projection_declared=0.80
Alignment fingerprint. Exponential Diophantine family; dependent on abc.
Infinitude of Mersenne primes {id:mersenne_prime_infinitude}
Fields: Mathematics, Number Theory
Statement. Are there infinitely many Mersenne primes?
Why it is unsolved. No proof exists; heuristic predictions are strong but no lower-bound theorem is known.
Known reductions to: perfect_numbers_odd_existence
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.60, compression_pressure=0.60, scale_band_declared=0.80, negative_control_strength=0.50, projection_declared=0.75
Alignment fingerprint. Sparse exponential prime logogram; scale band strong but proof absent.
Odd perfect numbers {id:perfect_numbers_odd_existence}
Fields: Mathematics, Number Theory
Statement. Does an odd perfect number exist?
Why it is unsolved. No example and no impossibility proof; many restrictions on size and form are known.
Known reductions to: mersenne_prime_infinitude
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.60, compression_pressure=0.65, scale_band_declared=0.80, negative_control_strength=0.60, projection_declared=0.80
Alignment fingerprint. Existence/impossibility logogram; negative controls provide partial bounds.
Rational points on higher-genus varieties {id:rational_points_high_genus}
Fields: Mathematics, Number Theory, Algebraic Geometry
Statement. Characterize and bound rational points on curves and varieties of general type (effective Faltings).
Why it is unsolved. Faltings' theorem is non-effective; uniform bounds (Bombieri-Lang) remain conjectural.
Known reductions to: abc_conjecture, birch_swinnerton_dyer_conjecture
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.75, geometric_mass=0.80, compression_pressure=0.75, scale_band_declared=0.65, projection_declared=0.75
Alignment fingerprint. Arithmetic geometry projection; effective methods missing.
Algebraic K-theory of the integers {id:algebraic_k_theory_integers}
Fields: Mathematics, Algebra, Number Theory
Statement. Compute the algebraic K-groups K_n(Z) for all n.
Why it is unsolved. Known for many n but no complete pattern; relates to Bernoulli numbers and motivic cohomology.
Known reductions to: standard_conjectures, langlands_program
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_10
Top axes: semantic_entropy=0.80, geometric_mass=0.75, compression_pressure=0.70, topology_torsion=0.60, projection_declared=0.70
Alignment fingerprint. Homotopy-number theory bridge; projection geometry over Z.
Novikov conjecture {id:novikov_conjecture}
Fields: Mathematics, Topology, Geometry
Statement. Higher signatures of compact oriented manifolds are oriented homotopy invariants.
Why it is unsolved. Proven for large classes but not in full generality; connects index theory and C*-algebras.
Known reductions to: borel_conjecture
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_05
Top axes: semantic_entropy=0.75, geometric_mass=0.85, compression_pressure=0.70, topology_torsion=0.75, projection_declared=0.75
Alignment fingerprint. Surgery-theory topology; high topology torsion.
Borel conjecture {id:borel_conjecture}
Fields: Mathematics, Topology
Statement. Aspherical closed manifolds are determined up to homeomorphism by their fundamental group.
Why it is unsolved. Proven in many cases; general proof is blocked by torsion and non-positive-curvature issues.
Known reductions to: novikov_conjecture
Known reductions from: novikov_conjecture
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_05
Top axes: semantic_entropy=0.70, geometric_mass=0.85, compression_pressure=0.65, topology_torsion=0.80, projection_declared=0.75
Alignment fingerprint. Aspherical topology; torsion from rigidity.
Volume conjecture {id:volume_conjecture}
Fields: Mathematics, Topology, Physics
Statement. The hyperbolic volume of a knot complement equals the asymptotic growth rate of the colored Jones polynomial.
Why it is unsolved. Connects quantum topology and hyperbolic geometry; known for many knots but general proof open.
Known reductions to: quantum_gravity
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_05
Top axes: semantic_entropy=0.80, geometric_mass=0.85, compression_pressure=0.75, topology_torsion=0.75, projection_declared=0.75
Alignment fingerprint. Quantum-geometric bridge; high topology torsion.
Hopf conjecture (S²×S²) {id:hopf_conjecture}
Fields: Mathematics, Geometry
Statement. There is no Riemannian metric of positive sectional curvature on S² × S².
Why it is unsolved. Few examples of positive curvature exist; topological obstructions in product manifolds are subtle.
RRC shape: ProjectableGeometryTopology
RRC status: CANDIDATE
Alignment cluster: cluster_05
Top axes: semantic_entropy=0.70, geometric_mass=0.90, compression_pressure=0.65, topology_torsion=0.80, projection_declared=0.75
Alignment fingerprint. Product-manifold curvature torsion; projection declared.
3D Poincaré conjecture (solved boundary) {id:poincare_conjecture_3d_solved}
Fields: Mathematics, Topology
Statement. Every simply connected closed 3-manifold is homeomorphic to S³.
Why it is unsolved. Solved by Grigori Perelman (2002–2003) using Ricci flow with surgery; included as a solved RRC boundary marker.
Known reductions from: smooth_4d_poincare_conjecture, generalized_poincare_conjecture_smooth
RRC shape: LeanTheoremReceipt
RRC status: ACCEPT
Alignment cluster: unclustered
Top axes: proof_readiness=1.00, scale_band_declared=1.00, negative_control_strength=1.00, projection_declared=1.00, shape_closure=1.00
Alignment fingerprint. Solved theorem receipt; demonstrates the ACCEPT boundary for manifold topology.
Singular Cardinal Hypothesis {id:singular_cardinal_hypothesis}
Fields: Mathematics, Logic, Set Theory
Statement. Does 2^κ = κ⁺ hold for every singular strong-limit cardinal κ?
Why it is unsolved. Independent of ZFC; sensitive to large-cardinal assumptions.
Known reductions to: continuum_hypothesis, consistency_of_zfc
Known reductions from: continuum_hypothesis
RRC shape: LanguageSetManifoldGraph
RRC status: HOLD
Alignment cluster: cluster_06
Top axes: semantic_entropy=0.85, compression_pressure=0.70, residual_risk=0.75, scale_band_declared=0.55, projection_declared=0.60
Alignment fingerprint. Set-theoretic higher cardinal; axiomatic risk high.
Algebrization barrier {id:p_np_algebrization_barrier}
Fields: Theoretical Computer Science, Logic
Statement. A meta-barrier showing that many known techniques cannot separate P and NP.
Why it is unsolved. Any resolution of P vs NP must use non-algebrizing, non-relativizing, and non-naturalizing arguments.
Known reductions to: p_vs_np
Known reductions from: p_vs_np
RRC shape: CognitiveLoadField
RRC status: HOLD
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.80, compression_pressure=0.75, topology_torsion=0.70, negative_control_strength=0.75, projection_declared=0.75
Alignment fingerprint. Complexity barrier node; high topology torsion from negative controls.
Small-Set Expansion conjecture {id:small_set_expansion_conjecture}
Fields: Theoretical Computer Science, Mathematics
Statement. It is NP-hard to distinguish small-set expanding graphs from those with sparse small cuts.
Why it is unsolved. Equivalent to the Unique Games Conjecture in some regimes and resists sum-of-squares lower bounds.
Known reductions to: unique_games_conjecture, p_vs_np
Known reductions from: unique_games_conjecture
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.75, compression_pressure=0.80, scale_band_declared=0.70, negative_control_strength=0.60, projection_declared=0.80
Alignment fingerprint. Graph expansion route; tightly coupled to UGC.
Quantum PCP conjecture {id:quantum_pcp_conjecture}
Fields: Theoretical Computer Science, Physics
Statement. Approximating the ground-state energy of local Hamiltonians is QMA-hard.
Why it is unsolved. Quantum analogue of the PCP theorem; would have broad implications for quantum complexity.
Known reductions to: p_vs_np, unique_games_conjecture, quantum_supremacy_verification, bqp_vs_np
Known reductions from: p_vs_np
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.85, compression_pressure=0.85, topology_torsion=0.60, scale_band_declared=0.70, projection_declared=0.75
Alignment fingerprint. Quantum complexity hardness router.
Erdős-Rado sunflower conjecture {id:sunflower_conjecture}
Fields: Mathematics, Combinatorics, Theoretical Computer Science
Statement. Bound the size of set systems with restricted pairwise intersections (sunflowers).
Why it is unsolved. Lower-bound constructions are limited; recent upper-bound improvements still leave a gap.
Known reductions to: cap_set_problem, matrix_rigidity
RRC shape: ErdosBoundConjecture
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.70, compression_pressure=0.70, scale_band_declared=0.70, negative_control_strength=0.60, projection_declared=0.80
Alignment fingerprint. Combinatorial sunflower bound; Erdős-style projection.
Cap set problem (exact growth) {id:cap_set_problem}
Fields: Mathematics, Combinatorics
Statement. Determine the maximum size of a cap set in F_3^n.
Why it is unsolved. The polynomial method gave strong upper bounds, but matching lower bounds and exact growth remain open; ties to sunflower questions.
Known reductions to: sunflower_conjecture
Known reductions from: sunflower_conjecture
RRC shape: ErdosBoundConjecture
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.65, compression_pressure=0.65, scale_band_declared=0.75, negative_control_strength=0.65, projection_declared=0.80
Alignment fingerprint. Polynomial-method route; recent progress but closure not tight.
Matrix rigidity {id:matrix_rigidity}
Fields: Theoretical Computer Science, Mathematics
Statement. Are high-rank matrices far from low-rank matrices under bounded-entry changes?
Why it is unsolved. Valiant's program links rigidity to circuit lower bounds; recent constructions challenge expected bounds.
Known reductions to: p_vs_np, sunflower_conjecture, derandomization_polynomial_identity_testing
Known reductions from: p_vs_np
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.75, compression_pressure=0.75, scale_band_declared=0.70, negative_control_strength=0.55, projection_declared=0.80
Alignment fingerprint. Linear-algebraic complexity gate; recent rigidity results create residual risk.
Derandomization of Polynomial Identity Testing {id:derandomization_polynomial_identity_testing}
Fields: Theoretical Computer Science, Mathematics
Statement. Find explicit hitting sets for polynomial identity testing or prove PIT is in P.
Why it is unsolved. Randomized algorithms are known; deterministic derandomization implies circuit lower bounds.
Known reductions to: p_vs_np, bpp_vs_p, matrix_rigidity
Known reductions from: p_vs_np
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.75, compression_pressure=0.80, scale_band_declared=0.75, negative_control_strength=0.60, projection_declared=0.85
Alignment fingerprint. Algebraic derandomization node; reduction to circuit lower bounds.
Irrationality of Euler's constant {id:eulers_constant_irrationality}
Fields: Mathematics, Number Theory
Statement. Is the Euler-Mascheroni constant γ irrational (or transcendental)?
Why it is unsolved. No proof of irrationality exists; standard Diophantine methods do not apply.
RRC shape: LogogramProjection
RRC status: CANDIDATE
Alignment cluster: cluster_04
Top axes: semantic_entropy=0.55, compression_pressure=0.60, scale_band_declared=0.75, negative_control_strength=0.50, projection_declared=0.80
Alignment fingerprint. Analytic constant logogram; projection clear, proof tools absent.
Dark energy equation of state {id:dark_energy_equation_of_state}
Fields: Physics, Cosmology
Statement. Determine whether dark energy is a cosmological constant (w = −1 exactly) or a dynamical field.
Why it is unsolved. Observational degeneracy and theoretical quintessence models are not observationally distinguished.
Known reductions to: cosmological_constant_problem
Known reductions from: cosmological_constant_problem
RRC shape: CadForceProbeReceipt
RRC status: HOLD
Alignment cluster: cluster_08
Top axes: semantic_entropy=0.80, compression_pressure=0.70, residual_risk=0.60, scale_band_declared=0.55, projection_declared=0.50
Alignment fingerprint. Dark-energy force probe; w parameter underspecified.
BQP vs NP {id:bqp_vs_np}
Fields: Theoretical Computer Science, Physics
Statement. Can every efficient quantum computation be verified classically in nondeterministic polynomial time?
Why it is unsolved. No oracle separation fully resolves the inclusion; quantum proofs (QMA) form a larger class.
Known reductions to: p_vs_np, quantum_supremacy_verification, quantum_pcp_conjecture
Known reductions from: p_vs_np
RRC shape: CognitiveLoadField
RRC status: CANDIDATE
Alignment cluster: cluster_02
Top axes: semantic_entropy=0.80, compression_pressure=0.80, topology_torsion=0.55, scale_band_declared=0.70, projection_declared=0.80
Alignment fingerprint. Quantum-classical verification boundary.
Origin of cosmic inflation {id:cosmological_inflation_origin}
Fields: Physics, Cosmology
Statement. What is the physical origin and detailed mechanism of cosmic inflation?
Why it is unsolved. Many models predict similar observables; Planck data constrain but do not select a unique mechanism.
Known reductions to: cosmological_constant_problem, baryon_asymmetry_problem, quantum_gravity
RRC shape: HoldForUnlawfulOrUnderspecifiedShape
RRC status: HOLD
Alignment cluster: cluster_08
Top axes: semantic_entropy=0.85, geometric_mass=0.40, compression_pressure=0.75, topology_torsion=0.40, residual_risk=0.65
Alignment fingerprint. Model-degeneracy hold; projection weak.
Langlands program {id:langlands_program}
Fields: Mathematics, Number Theory, Representation Theory
Statement. A broad web of conjectures connecting number theory, automorphic forms, and algebraic geometry.
Why it is unsolved. Partial results exist (functoriality for some cases) but the full program is far from complete.
Known reductions to: hilbert_twelfth_problem, riemann_hypothesis, generalized_riemann_hypothesis, standard_conjectures, birch_swinnerton_dyer_conjecture
Known reductions from: hilbert_twelfth_problem
RRC shape: ProjectableGeometryTopology
RRC status: HOLD
Alignment cluster: cluster_01
Top axes: semantic_entropy=0.95, geometric_mass=0.90, compression_pressure=0.90, topology_torsion=0.75, projection_declared=0.60
Alignment fingerprint. Vast correspondence manifold; projection declared but closure diffuse.
Cosmic censorship conjecture {id:cosmic_censorship_conjecture}
Fields: Physics, General Relativity
Statement. Do naked singularities form from generic initial data?
Why it is unsolved. Counterexamples exist in special cases; no general theorem for generic matter and symmetry.
Known reductions to: black_hole_information_paradox, quantum_gravity
Known reductions from: black_hole_information_paradox
RRC shape: ProjectableGeometryTopology
RRC status: HOLD
Alignment cluster: unclustered
Top axes: semantic_entropy=0.80, geometric_mass=0.80, compression_pressure=0.75, topology_torsion=0.70, residual_risk=0.60
Alignment fingerprint. GR singularity censorship; geometry-quantum boundary.
Origin of cosmic magnetic fields {id:origin_of_magnetic_fields}
Fields: Physics, Astrophysics
Statement. Explain the origin and amplification of large-scale cosmic magnetic fields.
Why it is unsolved. Dynamo theory is incomplete and primordial seeds are poorly constrained.
Known reductions to: baryon_asymmetry_problem, cosmological_inflation_origin
RRC shape: CadForceProbeReceipt
RRC status: HOLD
Alignment cluster: cluster_08
Top axes: semantic_entropy=0.75, compression_pressure=0.65, residual_risk=0.55, scale_band_declared=0.50, projection_declared=0.50
Alignment fingerprint. Astrophysical force probe; multi-scale amplification gap.
Alignment clusters
Millennium, L-functions, and motives {id:cluster_01}
Problems anchored in zeta/L-functions, algebraic cycles, and arithmetic geometry. Riemann Hypothesis is the central spectral axis.
Member problems:
riemann_hypothesis— Riemann Hypothesisgeneralized_riemann_hypothesis— Generalized Riemann Hypothesisbirch_swinnerton_dyer_conjecture— Birch and Swinnerton-Dyer conjecturehodge_conjecture— Hodge Conjecturetate_conjecture— Tate conjecturestandard_conjectures— Grothendieck's standard conjectureslanglands_program— Langlands program
Computational complexity core {id:cluster_02}
P vs NP and its satellites: hardness of approximation, fine-grained complexity, derandomization, and algebraic barriers.
Member problems:
p_vs_np— P vs NPnp_intermediate_existence— Existence of NP-intermediate problemsgraph_isomorphism_in_p— Graph isomorphism in P?factoring_in_p— Integer factorization in Pdiscrete_log_in_p— Discrete logarithm in Pbpp_vs_p— BPP vs P (derandomization)exponential_time_hypothesis— Exponential Time Hypothesisstrong_exponential_time_hypothesis— Strong Exponential Time Hypothesisunique_games_conjecture— Unique Games Conjecturesmall_set_expansion_conjecture— Small-Set Expansion conjecturequantum_pcp_conjecture— Quantum PCP conjecturematrix_rigidity— Matrix rigidityderandomization_polynomial_identity_testing— Derandomization of Polynomial Identity Testingbqp_vs_np— BQP vs NPp_np_algebrization_barrier— Algebrization barriercap_set_problem— Cap set problem (exact growth)sunflower_conjecture— Erdős-Rado sunflower conjecture
PDE regularity and singularities {id:cluster_03}
Existence, smoothness, and blow-up questions for nonlinear PDEs, plus related dynamical-systems bounds.
Member problems:
navier_stokes_existence_smoothness— Navier-Stokes existence and smoothnessnavier_stokes_blowup— Navier-Stokes finite-time blow-upturbulence_closure_problem— Turbulence closure problemhilbert_sixteenth_problem— Hilbert's 16th problemyang_mills_mass_gap— Yang-Mills existence and mass gap
Arithmetic and Diophantine structures {id:cluster_04}
Additive/multiplicative patterns in integers, exponential Diophantine equations, and prime distribution.
Member problems:
abc_conjecture— abc conjecturebeal_conjecture— Beal conjecturegoldbach_conjecture— Goldbach conjecturetwin_prime_conjecture— Twin prime conjecturecollatz_conjecture— Collatz conjecture (3n+1 problem)polignacs_conjecture— Polignac's conjectureelliott_halberstam_conjecture— Elliott-Halberstam conjecturefermat_catalan_conjecture— Fermat-Catalan conjectureschinzel_hypothesis_h— Schinzel's Hypothesis Hbrocards_problem— Brocard's problempillai_conjecture— Pillai's conjecturemersenne_prime_infinitude— Infinitude of Mersenne primesperfect_numbers_odd_existence— Odd perfect numberseulers_constant_irrationality— Irrationality of Euler's constantrational_points_high_genus— Rational points on higher-genus varietiesgeneralized_riemann_hypothesis— Generalized Riemann Hypothesis
Topology and geometry {id:cluster_05}
Manifold classification, asphericity, curvature, and quantum-topological invariants.
Member problems:
smooth_4d_poincare_conjecture— Smooth 4D Poincaré conjecturegeneralized_poincare_conjecture_smooth— Generalized Poincaré conjecture (smooth category)volume_conjecture— Volume conjecturenovikov_conjecture— Novikov conjectureborel_conjecture— Borel conjecturehopf_conjecture— Hopf conjecture (S²×S²)hodge_conjecture— Hodge Conjecturetate_conjecture— Tate conjecturestandard_conjectures— Grothendieck's standard conjectures
Logic and foundations {id:cluster_06}
Independence, consistency, and meta-mathematical limits of standard axiom systems.
Member problems:
continuum_hypothesis— Continuum Hypothesisconsistency_of_zfc— Consistency of ZFCsingular_cardinal_hypothesis— Singular Cardinal Hypothesisp_vs_np— P vs NPp_np_algebrization_barrier— Algebrization barrier
Quantum and information {id:cluster_07}
Quantum computation, verification, and quantum-gravity information puzzles.
Member problems:
quantum_supremacy_verification— Quantum supremacy verificationblack_hole_information_paradox— Black hole information paradoxbqp_vs_np— BQP vs NPquantum_pcp_conjecture— Quantum PCP conjecturequantum_gravity— Quantum gravityyang_mills_mass_gap— Yang-Mills existence and mass gap
Cosmology and dark sectors {id:cluster_08}
Dark matter, dark energy, vacuum energy, baryon asymmetry, and large-scale structure origins.
Member problems:
dark_matter_identity— Dark matter identitycosmological_constant_problem— Cosmological constant problemdark_energy_equation_of_state— Dark energy equation of statebaryon_asymmetry_problem— Baryon asymmetry of the universecosmological_inflation_origin— Origin of cosmic inflationorigin_of_magnetic_fields— Origin of cosmic magnetic fieldsquantum_gravity— Quantum gravity
Fluid and field-theoretic singularities {id:cluster_09}
Turbulence, Navier-Stokes singularities, and constructive quantum field theory.
Member problems:
navier_stokes_existence_smoothness— Navier-Stokes existence and smoothnessnavier_stokes_blowup— Navier-Stokes finite-time blow-upturbulence_closure_problem— Turbulence closure problemyang_mills_mass_gap— Yang-Mills existence and mass gap
Algebraic geometry and motives {id:cluster_10}
Cycles, K-theory, Langlands duality, and the arithmetic of rational points.
Member problems:
hodge_conjecture— Hodge Conjecturetate_conjecture— Tate conjecturestandard_conjectures— Grothendieck's standard conjecturesbirch_swinnerton_dyer_conjecture— Birch and Swinnerton-Dyer conjecturerational_points_high_genus— Rational points on higher-genus varietiesalgebraic_k_theory_integers— Algebraic K-theory of the integerslanglands_program— Langlands programhilbert_twelfth_problem— Hilbert's 12th problem
Notable interconnection patterns
- Zeta/L-function cluster is the densest mathematical hub: RH, GRH, BSD, and the arithmetic implications of EH form a high-weight subnetwork.
- P vs NP is the dominant complexity hub, with reductions to derandomization, fine-grained hardness, hardness of approximation, and algebraic barriers.
- Navier-Stokes / turbulence / Yang-Mills form a PDE-singularity cluster where regularity questions and constructive QFT share regularity DNA.
- Smooth 4D Poincaré / generalized Poincaré are tightly coupled and isolated from the number-theory clusters except through Langlands/motives.
- Cosmology and dark sectors are the most underspecified region: weak
projection_declared, weakscale_band_declared, and highresidual_risk. - Logic/foundations (CH, ZFC consistency) sits at the meta-limit: high
semantic_entropyandresidual_risk, near-zeroproof_readiness.
References
- Clay Mathematics Institute. Millennium Problems. https://www.claymath.org/millennium-problems/
- Weisstein, Eric W. MathWorld—A Wolfram Web Resource. https://mathworld.wolfram.com/
- arXiv.org survey literature on the individual problems listed above.