# 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_entropy`
- `geometric_mass`
- `compression_pressure`
- `topology_torsion`
- `residual_risk`
- `proof_readiness`
- `scale_band_declared`
- `negative_control_strength`
- `projection_declared`
- `shape_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`:

- `BurgersRGSolver`
- `CadForceProbeReceipt`
- `CognitiveLoadField`
- `ComputeKernelReceipt`
- `ErdosBoundConjecture`
- `HoldForUnlawfulOrUnderspecifiedShape`
- `LanguageSetManifoldGraph`
- `LeanTheoremReceipt`
- `LogogramProjection`
- `ProjectableGeometryTopology`
- `SignalShapedRouteCompiler`

## 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 Hypothesis
- `generalized_riemann_hypothesis` — Generalized Riemann Hypothesis
- `birch_swinnerton_dyer_conjecture` — Birch and Swinnerton-Dyer conjecture
- `hodge_conjecture` — Hodge Conjecture
- `tate_conjecture` — Tate conjecture
- `standard_conjectures` — Grothendieck's standard conjectures
- `langlands_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 NP
- `np_intermediate_existence` — Existence of NP-intermediate problems
- `graph_isomorphism_in_p` — Graph isomorphism in P?
- `factoring_in_p` — Integer factorization in P
- `discrete_log_in_p` — Discrete logarithm in P
- `bpp_vs_p` — BPP vs P (derandomization)
- `exponential_time_hypothesis` — Exponential Time Hypothesis
- `strong_exponential_time_hypothesis` — Strong Exponential Time Hypothesis
- `unique_games_conjecture` — Unique Games Conjecture
- `small_set_expansion_conjecture` — Small-Set Expansion conjecture
- `quantum_pcp_conjecture` — Quantum PCP conjecture
- `matrix_rigidity` — Matrix rigidity
- `derandomization_polynomial_identity_testing` — Derandomization of Polynomial Identity Testing
- `bqp_vs_np` — BQP vs NP
- `p_np_algebrization_barrier` — Algebrization barrier
- `cap_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 smoothness
- `navier_stokes_blowup` — Navier-Stokes finite-time blow-up
- `turbulence_closure_problem` — Turbulence closure problem
- `hilbert_sixteenth_problem` — Hilbert's 16th problem
- `yang_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 conjecture
- `beal_conjecture` — Beal conjecture
- `goldbach_conjecture` — Goldbach conjecture
- `twin_prime_conjecture` — Twin prime conjecture
- `collatz_conjecture` — Collatz conjecture (3n+1 problem)
- `polignacs_conjecture` — Polignac's conjecture
- `elliott_halberstam_conjecture` — Elliott-Halberstam conjecture
- `fermat_catalan_conjecture` — Fermat-Catalan conjecture
- `schinzel_hypothesis_h` — Schinzel's Hypothesis H
- `brocards_problem` — Brocard's problem
- `pillai_conjecture` — Pillai's conjecture
- `mersenne_prime_infinitude` — Infinitude of Mersenne primes
- `perfect_numbers_odd_existence` — Odd perfect numbers
- `eulers_constant_irrationality` — Irrationality of Euler's constant
- `rational_points_high_genus` — Rational points on higher-genus varieties
- `generalized_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 conjecture
- `generalized_poincare_conjecture_smooth` — Generalized Poincaré conjecture (smooth category)
- `volume_conjecture` — Volume conjecture
- `novikov_conjecture` — Novikov conjecture
- `borel_conjecture` — Borel conjecture
- `hopf_conjecture` — Hopf conjecture (S²×S²)
- `hodge_conjecture` — Hodge Conjecture
- `tate_conjecture` — Tate conjecture
- `standard_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 Hypothesis
- `consistency_of_zfc` — Consistency of ZFC
- `singular_cardinal_hypothesis` — Singular Cardinal Hypothesis
- `p_vs_np` — P vs NP
- `p_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 verification
- `black_hole_information_paradox` — Black hole information paradox
- `bqp_vs_np` — BQP vs NP
- `quantum_pcp_conjecture` — Quantum PCP conjecture
- `quantum_gravity` — Quantum gravity
- `yang_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 identity
- `cosmological_constant_problem` — Cosmological constant problem
- `dark_energy_equation_of_state` — Dark energy equation of state
- `baryon_asymmetry_problem` — Baryon asymmetry of the universe
- `cosmological_inflation_origin` — Origin of cosmic inflation
- `origin_of_magnetic_fields` — Origin of cosmic magnetic fields
- `quantum_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 smoothness
- `navier_stokes_blowup` — Navier-Stokes finite-time blow-up
- `turbulence_closure_problem` — Turbulence closure problem
- `yang_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 Conjecture
- `tate_conjecture` — Tate conjecture
- `standard_conjectures` — Grothendieck's standard conjectures
- `birch_swinnerton_dyer_conjecture` — Birch and Swinnerton-Dyer conjecture
- `rational_points_high_genus` — Rational points on higher-genus varieties
- `algebraic_k_theory_integers` — Algebraic K-theory of the integers
- `langlands_program` — Langlands program
- `hilbert_twelfth_problem` — Hilbert's 12th problem

## Notable interconnection patterns

1. **Zeta/L-function cluster** is the densest mathematical hub: RH, GRH, BSD, and the arithmetic implications of EH form a high-weight subnetwork.
2. **P vs NP** is the dominant complexity hub, with reductions to derandomization, fine-grained hardness, hardness of approximation, and algebraic barriers.
3. **Navier-Stokes / turbulence / Yang-Mills** form a PDE-singularity cluster where regularity questions and constructive QFT share regularity DNA.
4. **Smooth 4D Poincaré / generalized Poincaré** are tightly coupled and isolated from the number-theory clusters except through Langlands/motives.
5. **Cosmology and dark sectors** are the most underspecified region: weak `projection_declared`, weak `scale_band_declared`, and high `residual_risk`.
6. **Logic/foundations** (CH, ZFC consistency) sits at the meta-limit: high `semantic_entropy` and `residual_risk`, near-zero `proof_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.