From 68db1911a41ebeed57f274ee900aae82a5ae6f45 Mon Sep 17 00:00:00 2001 From: allaun Date: Sat, 20 Jun 2026 19:12:13 -0500 Subject: [PATCH] docs(research): RRC survey of 67 unsolved hard problems and alignment clusters MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - 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 --- ...unsolved_hard_problems_rrc_alignments.json | 3982 +++++++++++++++++ .../unsolved_hard_problems_rrc_survey.md | 1480 ++++++ 2 files changed, 5462 insertions(+) create mode 100644 6-Documentation/docs/research/unsolved_hard_problems_rrc_alignments.json create mode 100644 6-Documentation/docs/research/unsolved_hard_problems_rrc_survey.md diff --git a/6-Documentation/docs/research/unsolved_hard_problems_rrc_alignments.json b/6-Documentation/docs/research/unsolved_hard_problems_rrc_alignments.json new file mode 100644 index 00000000..65b18efb --- /dev/null +++ b/6-Documentation/docs/research/unsolved_hard_problems_rrc_alignments.json @@ -0,0 +1,3982 @@ +{ + "schema": "unsolved_hard_problems_rrc_alignments_v1", + "generated_at": "2026-06-20", + "claim_boundary": "survey/projection-only; no proofs or solutions claimed", + "total_problems": 68, + "unsolved_count": 67, + "solved_boundary_count": 1, + "axis_schema": [ + "semantic_entropy", + "geometric_mass", + "compression_pressure", + "topology_torsion", + "residual_risk", + "proof_readiness", + "scale_band_declared", + "negative_control_strength", + "projection_declared", + "shape_closure" + ], + "problems": [ + { + "id": "riemann_hypothesis", + "name": "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_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", + "rrc_axes": { + "semantic_entropy": 0.85, + "geometric_mass": 0.6, + "compression_pressure": 0.8, + "topology_torsion": 0.45, + "residual_risk": 0.3, + "proof_readiness": 0.25, + "scale_band_declared": 0.9, + "negative_control_strength": 0.7, + "projection_declared": 0.95, + "shape_closure": 0.5 + }, + "top_axes": [ + "projection_declared", + "scale_band_declared", + "semantic_entropy", + "compression_pressure", + "negative_control_strength" + ], + "alignment_fingerprint": "Spectral line on the critical axis; projection is sharp, proof readiness is low.", + "alignment_cluster": "cluster_01" + }, + { + "id": "generalized_riemann_hypothesis", + "name": "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_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", + "rrc_axes": { + "semantic_entropy": 0.9, + "geometric_mass": 0.65, + "compression_pressure": 0.85, + "topology_torsion": 0.5, + "residual_risk": 0.35, + "proof_readiness": 0.2, + "scale_band_declared": 0.85, + "negative_control_strength": 0.65, + "projection_declared": 0.9, + "shape_closure": 0.45 + }, + "top_axes": [ + "semantic_entropy", + "projection_declared", + "compression_pressure", + "scale_band_declared", + "geometric_mass" + ], + "alignment_fingerprint": "Family of spectral manifolds; projection declared but proof readiness even lower than RH.", + "alignment_cluster": "cluster_01" + }, + { + "id": "p_vs_np", + "name": "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_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", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.3, + "compression_pressure": 0.95, + "topology_torsion": 0.7, + "residual_risk": 0.5, + "proof_readiness": 0.15, + "scale_band_declared": 0.95, + "negative_control_strength": 0.8, + "projection_declared": 1.0, + "shape_closure": 0.55 + }, + "top_axes": [ + "projection_declared", + "compression_pressure", + "scale_band_declared", + "semantic_entropy", + "negative_control_strength" + ], + "alignment_fingerprint": "High-compression decision boundary; barriers create strong topology torsion.", + "alignment_cluster": "cluster_02" + }, + { + "id": "navier_stokes_existence_smoothness", + "name": "Navier-Stokes existence and smoothness", + "fields": [ + "Mathematics", + "Analysis", + "PDE" + ], + "statement": "Do the 3D incompressible Navier-Stokes equations admit smooth solutions for all smooth initial data?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.85, + "geometric_mass": 0.75, + "compression_pressure": 0.8, + "topology_torsion": 0.65, + "residual_risk": 0.55, + "proof_readiness": 0.2, + "scale_band_declared": 0.85, + "negative_control_strength": 0.6, + "projection_declared": 0.9, + "shape_closure": 0.4 + }, + "top_axes": [ + "projection_declared", + "semantic_entropy", + "scale_band_declared", + "compression_pressure", + "geometric_mass" + ], + "alignment_fingerprint": "Geometric PDE mass concentrated; residual risk of blow-up dominates.", + "alignment_cluster": "cluster_03" + }, + { + "id": "yang_mills_mass_gap", + "name": "Yang-Mills existence and mass gap", + "fields": [ + "Mathematical Physics", + "Mathematics" + ], + "statement": "Prove that quantum Yang-Mills theory exists in four dimensions and has a mass gap.", + "why_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" + ], + "known_reductions_from": [], + "rrc_shape": "SignalShapedRouteCompiler", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.9, + "geometric_mass": 0.6, + "compression_pressure": 0.85, + "topology_torsion": 0.55, + "residual_risk": 0.4, + "proof_readiness": 0.15, + "scale_band_declared": 0.8, + "negative_control_strength": 0.55, + "projection_declared": 0.8, + "shape_closure": 0.35 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "scale_band_declared", + "projection_declared", + "geometric_mass" + ], + "alignment_fingerprint": "Quantum-field route with high semantic entropy; continuum limit projection underspecified.", + "alignment_cluster": "cluster_03" + }, + { + "id": "hodge_conjecture", + "name": "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_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", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.9, + "compression_pressure": 0.75, + "topology_torsion": 0.7, + "residual_risk": 0.3, + "proof_readiness": 0.2, + "scale_band_declared": 0.75, + "negative_control_strength": 0.55, + "projection_declared": 0.85, + "shape_closure": 0.4 + }, + "top_axes": [ + "geometric_mass", + "projection_declared", + "semantic_entropy", + "compression_pressure", + "scale_band_declared" + ], + "alignment_fingerprint": "High geometric mass and topology torsion; algebraic-cycle witness missing.", + "alignment_cluster": "cluster_01" + }, + { + "id": "birch_swinnerton_dyer_conjecture", + "name": "Birch and 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_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", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.85, + "compression_pressure": 0.8, + "topology_torsion": 0.6, + "residual_risk": 0.35, + "proof_readiness": 0.25, + "scale_band_declared": 0.8, + "negative_control_strength": 0.6, + "projection_declared": 0.85, + "shape_closure": 0.45 + }, + "top_axes": [ + "geometric_mass", + "projection_declared", + "compression_pressure", + "scale_band_declared", + "semantic_entropy" + ], + "alignment_fingerprint": "L-function/geometry bridge; low proof readiness due to Tate-Shafarevich torsion.", + "alignment_cluster": "cluster_01" + }, + { + "id": "smooth_4d_poincare_conjecture", + "name": "Smooth 4D Poincaré conjecture", + "fields": [ + "Mathematics", + "Topology" + ], + "statement": "Every smooth closed 4-manifold homotopy equivalent to S⁴ is diffeomorphic to S⁴.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.7, + "geometric_mass": 0.95, + "compression_pressure": 0.7, + "topology_torsion": 0.95, + "residual_risk": 0.45, + "proof_readiness": 0.2, + "scale_band_declared": 0.75, + "negative_control_strength": 0.55, + "projection_declared": 0.85, + "shape_closure": 0.35 + }, + "top_axes": [ + "geometric_mass", + "topology_torsion", + "projection_declared", + "scale_band_declared", + "semantic_entropy" + ], + "alignment_fingerprint": "Dimension-4 smooth topology torsion peak; projection declared but no constructive witness.", + "alignment_cluster": "cluster_05" + }, + { + "id": "generalized_poincare_conjecture_smooth", + "name": "Generalized Poincaré conjecture (smooth category)", + "fields": [ + "Mathematics", + "Topology" + ], + "statement": "Every closed smooth n-manifold homotopy equivalent to Sⁿ is diffeomorphic to Sⁿ for n ≥ 4.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.9, + "compression_pressure": 0.75, + "topology_torsion": 0.9, + "residual_risk": 0.45, + "proof_readiness": 0.18, + "scale_band_declared": 0.7, + "negative_control_strength": 0.5, + "projection_declared": 0.8, + "shape_closure": 0.3 + }, + "top_axes": [ + "geometric_mass", + "topology_torsion", + "projection_declared", + "semantic_entropy", + "compression_pressure" + ], + "alignment_fingerprint": "Smooth-topology equivalence across dimensions; 4D torsion dominates.", + "alignment_cluster": "cluster_05" + }, + { + "id": "abc_conjecture", + "name": "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_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", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.3, + "compression_pressure": 0.8, + "topology_torsion": 0.35, + "residual_risk": 0.4, + "proof_readiness": 0.25, + "scale_band_declared": 0.75, + "negative_control_strength": 0.6, + "projection_declared": 0.9, + "shape_closure": 0.5 + }, + "top_axes": [ + "projection_declared", + "compression_pressure", + "semantic_entropy", + "scale_band_declared", + "negative_control_strength" + ], + "alignment_fingerprint": "Diophantine logogram with sharp projection; proof readiness stalled.", + "alignment_cluster": "cluster_04" + }, + { + "id": "goldbach_conjecture", + "name": "Goldbach conjecture", + "fields": [ + "Mathematics", + "Number Theory" + ], + "statement": "Every even integer greater than 2 is the sum of two primes.", + "why_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" + ], + "known_reductions_from": [], + "rrc_shape": "LogogramProjection", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.6, + "geometric_mass": 0.2, + "compression_pressure": 0.7, + "topology_torsion": 0.2, + "residual_risk": 0.3, + "proof_readiness": 0.3, + "scale_band_declared": 0.85, + "negative_control_strength": 0.7, + "projection_declared": 0.95, + "shape_closure": 0.55 + }, + "top_axes": [ + "projection_declared", + "scale_band_declared", + "compression_pressure", + "negative_control_strength", + "semantic_entropy" + ], + "alignment_fingerprint": "Simple logogram with strong computational witness; proof closure missing.", + "alignment_cluster": "cluster_04" + }, + { + "id": "twin_prime_conjecture", + "name": "Twin prime conjecture", + "fields": [ + "Mathematics", + "Number Theory" + ], + "statement": "There are infinitely many primes p such that p + 2 is also prime.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.65, + "geometric_mass": 0.25, + "compression_pressure": 0.75, + "topology_torsion": 0.25, + "residual_risk": 0.35, + "proof_readiness": 0.25, + "scale_band_declared": 0.8, + "negative_control_strength": 0.65, + "projection_declared": 0.9, + "shape_closure": 0.45 + }, + "top_axes": [ + "projection_declared", + "scale_band_declared", + "compression_pressure", + "semantic_entropy", + "negative_control_strength" + ], + "alignment_fingerprint": "Bounded-gap sieve route; residual risk from parity of sieves.", + "alignment_cluster": "cluster_04" + }, + { + "id": "collatz_conjecture", + "name": "Collatz conjecture (3n+1 problem)", + "fields": [ + "Mathematics", + "Number Theory", + "Dynamical Systems" + ], + "statement": "Iterating the 3n+1 map always reaches 1 for every positive integer seed.", + "why_unsolved": "No invariant controls the combined expand/contract dynamics across all scales; verified empirically to huge bounds.", + "known_reductions_to": [], + "known_reductions_from": [], + "rrc_shape": "LogogramProjection", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.55, + "geometric_mass": 0.35, + "compression_pressure": 0.7, + "topology_torsion": 0.45, + "residual_risk": 0.4, + "proof_readiness": 0.2, + "scale_band_declared": 0.8, + "negative_control_strength": 0.6, + "projection_declared": 0.85, + "shape_closure": 0.4 + }, + "top_axes": [ + "projection_declared", + "scale_band_declared", + "compression_pressure", + "negative_control_strength", + "semantic_entropy" + ], + "alignment_fingerprint": "Simple iterative logogram with dynamical-systems torsion.", + "alignment_cluster": "cluster_04" + }, + { + "id": "beal_conjecture", + "name": "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_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", + "rrc_axes": { + "semantic_entropy": 0.65, + "geometric_mass": 0.25, + "compression_pressure": 0.7, + "topology_torsion": 0.25, + "residual_risk": 0.3, + "proof_readiness": 0.2, + "scale_band_declared": 0.75, + "negative_control_strength": 0.55, + "projection_declared": 0.85, + "shape_closure": 0.4 + }, + "top_axes": [ + "projection_declared", + "scale_band_declared", + "compression_pressure", + "semantic_entropy", + "negative_control_strength" + ], + "alignment_fingerprint": "Exponential Diophantine logogram; reduction to abc creates one-way dependency.", + "alignment_cluster": "cluster_04" + }, + { + "id": "unique_games_conjecture", + "name": "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_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", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.3, + "compression_pressure": 0.85, + "topology_torsion": 0.6, + "residual_risk": 0.45, + "proof_readiness": 0.25, + "scale_band_declared": 0.8, + "negative_control_strength": 0.65, + "projection_declared": 0.85, + "shape_closure": 0.5 + }, + "top_axes": [ + "compression_pressure", + "projection_declared", + "semantic_entropy", + "scale_band_declared", + "negative_control_strength" + ], + "alignment_fingerprint": "Hardness-of-approximation router; high compression pressure from SDP gaps.", + "alignment_cluster": "cluster_02" + }, + { + "id": "np_intermediate_existence", + "name": "Existence of NP-intermediate problems", + "fields": [ + "Theoretical Computer Science", + "Logic" + ], + "statement": "Does there exist a problem in NP that is neither in P nor NP-complete?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.7, + "geometric_mass": 0.2, + "compression_pressure": 0.8, + "topology_torsion": 0.5, + "residual_risk": 0.55, + "proof_readiness": 0.15, + "scale_band_declared": 0.7, + "negative_control_strength": 0.5, + "projection_declared": 0.75, + "shape_closure": 0.4 + }, + "top_axes": [ + "compression_pressure", + "projection_declared", + "semantic_entropy", + "scale_band_declared", + "residual_risk" + ], + "alignment_fingerprint": "Conditional on P vs NP; shape closure blocked by complexity boundary.", + "alignment_cluster": "cluster_02" + }, + { + "id": "graph_isomorphism_in_p", + "name": "Graph isomorphism in P?", + "fields": [ + "Theoretical Computer Science", + "Mathematics" + ], + "statement": "Can graph isomorphism be decided in deterministic polynomial time?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.7, + "geometric_mass": 0.35, + "compression_pressure": 0.75, + "topology_torsion": 0.5, + "residual_risk": 0.45, + "proof_readiness": 0.3, + "scale_band_declared": 0.75, + "negative_control_strength": 0.6, + "projection_declared": 0.85, + "shape_closure": 0.5 + }, + "top_axes": [ + "projection_declared", + "compression_pressure", + "scale_band_declared", + "semantic_entropy", + "negative_control_strength" + ], + "alignment_fingerprint": "Symmetry-classification route; quasipolynomial witness improves readiness.", + "alignment_cluster": "cluster_02" + }, + { + "id": "factoring_in_p", + "name": "Integer factorization in P", + "fields": [ + "Theoretical Computer Science", + "Number Theory", + "Cryptography" + ], + "statement": "Can integer factorization be solved in deterministic polynomial time?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.7, + "geometric_mass": 0.25, + "compression_pressure": 0.8, + "topology_torsion": 0.45, + "residual_risk": 0.4, + "proof_readiness": 0.2, + "scale_band_declared": 0.8, + "negative_control_strength": 0.65, + "projection_declared": 0.85, + "shape_closure": 0.45 + }, + "top_axes": [ + "projection_declared", + "compression_pressure", + "scale_band_declared", + "semantic_entropy", + "negative_control_strength" + ], + "alignment_fingerprint": "Cryptographic hardness router; quantum route swappable but classical proof missing.", + "alignment_cluster": "cluster_02" + }, + { + "id": "discrete_log_in_p", + "name": "Discrete logarithm in P", + "fields": [ + "Theoretical Computer Science", + "Number Theory", + "Cryptography" + ], + "statement": "Can the discrete logarithm problem be solved in deterministic polynomial time?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.7, + "geometric_mass": 0.3, + "compression_pressure": 0.8, + "topology_torsion": 0.4, + "residual_risk": 0.4, + "proof_readiness": 0.2, + "scale_band_declared": 0.8, + "negative_control_strength": 0.6, + "projection_declared": 0.85, + "shape_closure": 0.45 + }, + "top_axes": [ + "projection_declared", + "compression_pressure", + "scale_band_declared", + "semantic_entropy", + "negative_control_strength" + ], + "alignment_fingerprint": "Algebraic-group route; closely coupled to factoring.", + "alignment_cluster": "cluster_02" + }, + { + "id": "bpp_vs_p", + "name": "BPP vs P (derandomization)", + "fields": [ + "Theoretical Computer Science", + "Mathematics" + ], + "statement": "Does every polynomial-time randomized algorithm have a deterministic polynomial-time simulation?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.25, + "compression_pressure": 0.8, + "topology_torsion": 0.55, + "residual_risk": 0.45, + "proof_readiness": 0.2, + "scale_band_declared": 0.75, + "negative_control_strength": 0.55, + "projection_declared": 0.8, + "shape_closure": 0.4 + }, + "top_axes": [ + "compression_pressure", + "projection_declared", + "semantic_entropy", + "scale_band_declared", + "topology_torsion" + ], + "alignment_fingerprint": "Pseudorandomness compression gate; blocked by circuit lower bounds.", + "alignment_cluster": "cluster_02" + }, + { + "id": "exponential_time_hypothesis", + "name": "Exponential Time Hypothesis", + "fields": [ + "Theoretical Computer Science" + ], + "statement": "3-SAT cannot be solved in time 2^{o(n)}.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.2, + "compression_pressure": 0.85, + "topology_torsion": 0.55, + "residual_risk": 0.4, + "proof_readiness": 0.25, + "scale_band_declared": 0.8, + "negative_control_strength": 0.65, + "projection_declared": 0.85, + "shape_closure": 0.5 + }, + "top_axes": [ + "compression_pressure", + "projection_declared", + "scale_band_declared", + "semantic_entropy", + "negative_control_strength" + ], + "alignment_fingerprint": "Fine-grained complexity anchor; high compression pressure.", + "alignment_cluster": "cluster_02" + }, + { + "id": "strong_exponential_time_hypothesis", + "name": "Strong Exponential Time Hypothesis", + "fields": [ + "Theoretical Computer Science" + ], + "statement": "CNF-SAT requires time 2^{(1−ε)n} for some ε > 0.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.2, + "compression_pressure": 0.85, + "topology_torsion": 0.6, + "residual_risk": 0.45, + "proof_readiness": 0.2, + "scale_band_declared": 0.75, + "negative_control_strength": 0.6, + "projection_declared": 0.8, + "shape_closure": 0.45 + }, + "top_axes": [ + "compression_pressure", + "semantic_entropy", + "projection_declared", + "scale_band_declared", + "topology_torsion" + ], + "alignment_fingerprint": "Tight SAT lower-bound route; topology torsion from exact constants.", + "alignment_cluster": "cluster_02" + }, + { + "id": "continuum_hypothesis", + "name": "Continuum Hypothesis", + "fields": [ + "Mathematics", + "Logic", + "Set Theory" + ], + "statement": "Is there a set whose cardinality is strictly between ℵ₀ and 2^{ℵ₀}?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.9, + "geometric_mass": 0.2, + "compression_pressure": 0.7, + "topology_torsion": 0.5, + "residual_risk": 0.8, + "proof_readiness": 0.05, + "scale_band_declared": 0.6, + "negative_control_strength": 0.4, + "projection_declared": 0.7, + "shape_closure": 0.2 + }, + "top_axes": [ + "semantic_entropy", + "residual_risk", + "compression_pressure", + "projection_declared", + "scale_band_declared" + ], + "alignment_fingerprint": "Axiomatic boundary; projection underdetermined by ZFC.", + "alignment_cluster": "cluster_06" + }, + { + "id": "consistency_of_zfc", + "name": "Consistency of ZFC", + "fields": [ + "Mathematics", + "Logic" + ], + "statement": "Are the Zermelo-Fraenkel axioms with Choice consistent?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.95, + "geometric_mass": 0.1, + "compression_pressure": 0.75, + "topology_torsion": 0.55, + "residual_risk": 0.9, + "proof_readiness": 0.02, + "scale_band_declared": 0.5, + "negative_control_strength": 0.3, + "projection_declared": 0.6, + "shape_closure": 0.15 + }, + "top_axes": [ + "semantic_entropy", + "residual_risk", + "compression_pressure", + "projection_declared", + "topology_torsion" + ], + "alignment_fingerprint": "Meta-mathematical limit point; negative controls extremely weak.", + "alignment_cluster": "cluster_06" + }, + { + "id": "hilbert_sixth_problem", + "name": "Hilbert's 6th problem", + "fields": [ + "Mathematics", + "Physics", + "Logic" + ], + "statement": "Axiomatize all of physics in a mathematically rigorous way.", + "why_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" + ], + "known_reductions_from": [], + "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", + "rrc_status": "HOLD", + "rrc_axes": { + "semantic_entropy": 0.95, + "geometric_mass": 0.4, + "compression_pressure": 0.9, + "topology_torsion": 0.6, + "residual_risk": 0.7, + "proof_readiness": 0.05, + "scale_band_declared": 0.3, + "negative_control_strength": 0.2, + "projection_declared": 0.35, + "shape_closure": 0.1 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "residual_risk", + "topology_torsion", + "geometric_mass" + ], + "alignment_fingerprint": "Underspecified universal axiomatization; projection and scale band weak.", + "alignment_cluster": "unclustered" + }, + { + "id": "hilbert_sixteenth_problem", + "name": "Hilbert's 16th problem", + "fields": [ + "Mathematics", + "Analysis", + "Dynamical Systems" + ], + "statement": "Bound the number of limit cycles for planar polynomial vector fields.", + "why_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" + ], + "known_reductions_from": [], + "rrc_shape": "ProjectableGeometryTopology", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.75, + "compression_pressure": 0.75, + "topology_torsion": 0.65, + "residual_risk": 0.45, + "proof_readiness": 0.2, + "scale_band_declared": 0.7, + "negative_control_strength": 0.5, + "projection_declared": 0.75, + "shape_closure": 0.35 + }, + "top_axes": [ + "semantic_entropy", + "geometric_mass", + "compression_pressure", + "projection_declared", + "scale_band_declared" + ], + "alignment_fingerprint": "Planar topology-dynamics; projection declared but scale band diffuse.", + "alignment_cluster": "cluster_03" + }, + { + "id": "hilbert_twelfth_problem", + "name": "Hilbert's 12th problem", + "fields": [ + "Mathematics", + "Number Theory", + "Algebra" + ], + "statement": "Construct all abelian extensions of arbitrary algebraic number fields (Kronecker's Jugendtraum).", + "why_unsolved": "Solved for Q and imaginary quadratic fields; general base fields lack explicit class-field-theory generators.", + "known_reductions_to": [ + "langlands_program" + ], + "known_reductions_from": [], + "rrc_shape": "ProjectableGeometryTopology", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.85, + "geometric_mass": 0.7, + "compression_pressure": 0.75, + "topology_torsion": 0.5, + "residual_risk": 0.35, + "proof_readiness": 0.2, + "scale_band_declared": 0.65, + "negative_control_strength": 0.45, + "projection_declared": 0.7, + "shape_closure": 0.3 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "geometric_mass", + "projection_declared", + "scale_band_declared" + ], + "alignment_fingerprint": "Non-abelian class-field route; projection geometry over number fields.", + "alignment_cluster": "cluster_10" + }, + { + "id": "navier_stokes_blowup", + "name": "Navier-Stokes finite-time blow-up", + "fields": [ + "Mathematics", + "Analysis" + ], + "statement": "Does there exist a finite-time singularity for 3D Navier-Stokes with smooth initial data?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.8, + "compression_pressure": 0.8, + "topology_torsion": 0.7, + "residual_risk": 0.75, + "proof_readiness": 0.1, + "scale_band_declared": 0.7, + "negative_control_strength": 0.45, + "projection_declared": 0.75, + "shape_closure": 0.25 + }, + "top_axes": [ + "geometric_mass", + "compression_pressure", + "semantic_entropy", + "residual_risk", + "projection_declared" + ], + "alignment_fingerprint": "Singularity endpoint; high residual risk, weak negative controls.", + "alignment_cluster": "cluster_03" + }, + { + "id": "turbulence_closure_problem", + "name": "Turbulence closure problem", + "fields": [ + "Physics", + "Applied Mathematics" + ], + "statement": "Derive a closed finite set of equations for the statistics of turbulent flows.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.9, + "geometric_mass": 0.7, + "compression_pressure": 0.85, + "topology_torsion": 0.6, + "residual_risk": 0.55, + "proof_readiness": 0.1, + "scale_band_declared": 0.55, + "negative_control_strength": 0.35, + "projection_declared": 0.6, + "shape_closure": 0.2 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "geometric_mass", + "topology_torsion", + "projection_declared" + ], + "alignment_fingerprint": "Multi-scale PDE route; closure gap prevents projection.", + "alignment_cluster": "cluster_03" + }, + { + "id": "quantum_supremacy_verification", + "name": "Quantum supremacy verification", + "fields": [ + "Theoretical Computer Science", + "Physics" + ], + "statement": "Can a quantum computation be certified as infeasible for any classical computer?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.85, + "geometric_mass": 0.3, + "compression_pressure": 0.8, + "topology_torsion": 0.55, + "residual_risk": 0.6, + "proof_readiness": 0.15, + "scale_band_declared": 0.7, + "negative_control_strength": 0.5, + "projection_declared": 0.65, + "shape_closure": 0.35 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "scale_band_declared", + "projection_declared", + "residual_risk" + ], + "alignment_fingerprint": "Hardware-software verification gap; compute receipt incomplete.", + "alignment_cluster": "cluster_07" + }, + { + "id": "black_hole_information_paradox", + "name": "Black hole information paradox", + "fields": [ + "Physics", + "Quantum Gravity" + ], + "statement": "Is information preserved during black hole evaporation?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.9, + "geometric_mass": 0.75, + "compression_pressure": 0.8, + "topology_torsion": 0.7, + "residual_risk": 0.65, + "proof_readiness": 0.1, + "scale_band_declared": 0.6, + "negative_control_strength": 0.35, + "projection_declared": 0.55, + "shape_closure": 0.2 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "geometric_mass", + "topology_torsion", + "residual_risk" + ], + "alignment_fingerprint": "Geometry-quantum boundary; high topology torsion, projection underspecified.", + "alignment_cluster": "cluster_07" + }, + { + "id": "dark_matter_identity", + "name": "Dark matter identity", + "fields": [ + "Physics", + "Cosmology" + ], + "statement": "What is the particle or gravitational nature of dark matter?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.85, + "geometric_mass": 0.5, + "compression_pressure": 0.75, + "topology_torsion": 0.4, + "residual_risk": 0.7, + "proof_readiness": 0.1, + "scale_band_declared": 0.55, + "negative_control_strength": 0.3, + "projection_declared": 0.45, + "shape_closure": 0.15 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "residual_risk", + "scale_band_declared", + "geometric_mass" + ], + "alignment_fingerprint": "Dark-sector force probe; projection and scale band weak.", + "alignment_cluster": "cluster_08" + }, + { + "id": "cosmological_constant_problem", + "name": "Cosmological constant problem", + "fields": [ + "Physics", + "Cosmology" + ], + "statement": "Why is the observed vacuum energy density ~120 orders of magnitude smaller than naive QFT predictions?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.9, + "geometric_mass": 0.45, + "compression_pressure": 0.85, + "topology_torsion": 0.45, + "residual_risk": 0.75, + "proof_readiness": 0.05, + "scale_band_declared": 0.5, + "negative_control_strength": 0.25, + "projection_declared": 0.4, + "shape_closure": 0.1 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "residual_risk", + "scale_band_declared", + "geometric_mass" + ], + "alignment_fingerprint": "Force-probe hierarchy mismatch; projection almost absent.", + "alignment_cluster": "cluster_08" + }, + { + "id": "baryon_asymmetry_problem", + "name": "Baryon asymmetry of the universe", + "fields": [ + "Physics", + "Cosmology" + ], + "statement": "Why is the universe made of matter rather than equal amounts of matter and antimatter?", + "why_unsolved": "Sakharov conditions are known, but no Standard Model source produces the observed asymmetry.", + "known_reductions_to": [ + "dark_matter_identity", + "cosmological_inflation_origin" + ], + "known_reductions_from": [], + "rrc_shape": "CadForceProbeReceipt", + "rrc_status": "HOLD", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.4, + "compression_pressure": 0.7, + "topology_torsion": 0.3, + "residual_risk": 0.6, + "proof_readiness": 0.15, + "scale_band_declared": 0.55, + "negative_control_strength": 0.35, + "projection_declared": 0.5, + "shape_closure": 0.25 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "residual_risk", + "scale_band_declared", + "projection_declared" + ], + "alignment_fingerprint": "Force-probe asymmetry; projection diffuse.", + "alignment_cluster": "cluster_08" + }, + { + "id": "quantum_gravity", + "name": "Quantum gravity", + "fields": [ + "Physics", + "Mathematics" + ], + "statement": "Reconcile general relativity and quantum mechanics into a single consistent theory.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.95, + "geometric_mass": 0.85, + "compression_pressure": 0.9, + "topology_torsion": 0.85, + "residual_risk": 0.75, + "proof_readiness": 0.05, + "scale_band_declared": 0.45, + "negative_control_strength": 0.2, + "projection_declared": 0.4, + "shape_closure": 0.1 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "geometric_mass", + "topology_torsion", + "residual_risk" + ], + "alignment_fingerprint": "Ultimate geometry-quantum projection; all axes extreme, projection weakest.", + "alignment_cluster": "cluster_07" + }, + { + "id": "measurement_problem", + "name": "Quantum measurement problem", + "fields": [ + "Physics", + "Foundations" + ], + "statement": "Explain the apparent collapse of the quantum wavefunction upon measurement.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.9, + "geometric_mass": 0.3, + "compression_pressure": 0.75, + "topology_torsion": 0.5, + "residual_risk": 0.65, + "proof_readiness": 0.05, + "scale_band_declared": 0.35, + "negative_control_strength": 0.2, + "projection_declared": 0.3, + "shape_closure": 0.1 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "residual_risk", + "topology_torsion", + "scale_band_declared" + ], + "alignment_fingerprint": "Interpretational hold; projection and scale band severely underspecified.", + "alignment_cluster": "unclustered" + }, + { + "id": "tate_conjecture", + "name": "Tate conjecture", + "fields": [ + "Mathematics", + "Algebraic Geometry" + ], + "statement": "Algebraic cycles generate the l-adic cohomology classes invariant under Galois action.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.9, + "compression_pressure": 0.75, + "topology_torsion": 0.7, + "residual_risk": 0.35, + "proof_readiness": 0.2, + "scale_band_declared": 0.7, + "negative_control_strength": 0.5, + "projection_declared": 0.8, + "shape_closure": 0.4 + }, + "top_axes": [ + "geometric_mass", + "semantic_entropy", + "projection_declared", + "compression_pressure", + "topology_torsion" + ], + "alignment_fingerprint": "Arithmetic-geometric topology; close cousin of Hodge.", + "alignment_cluster": "cluster_01" + }, + { + "id": "standard_conjectures", + "name": "Grothendieck's standard conjectures", + "fields": [ + "Mathematics", + "Algebraic Geometry" + ], + "statement": "Standard conjectures on algebraic cycles, including Lefschetz and Hodge standard.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.85, + "geometric_mass": 0.95, + "compression_pressure": 0.8, + "topology_torsion": 0.75, + "residual_risk": 0.4, + "proof_readiness": 0.15, + "scale_band_declared": 0.65, + "negative_control_strength": 0.45, + "projection_declared": 0.75, + "shape_closure": 0.3 + }, + "top_axes": [ + "geometric_mass", + "semantic_entropy", + "compression_pressure", + "topology_torsion", + "projection_declared" + ], + "alignment_fingerprint": "Motivic topology anchor; high geometric mass and torsion.", + "alignment_cluster": "cluster_01" + }, + { + "id": "polignacs_conjecture", + "name": "Polignac's conjecture", + "fields": [ + "Mathematics", + "Number Theory" + ], + "statement": "For every even integer 2k there are infinitely many prime gaps of size 2k.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.65, + "geometric_mass": 0.2, + "compression_pressure": 0.7, + "topology_torsion": 0.25, + "residual_risk": 0.35, + "proof_readiness": 0.2, + "scale_band_declared": 0.75, + "negative_control_strength": 0.6, + "projection_declared": 0.85, + "shape_closure": 0.45 + }, + "top_axes": [ + "projection_declared", + "scale_band_declared", + "compression_pressure", + "semantic_entropy", + "negative_control_strength" + ], + "alignment_fingerprint": "Generalized prime-gap logogram; reduction tree rooted in twin prime.", + "alignment_cluster": "cluster_04" + }, + { + "id": "elliott_halberstam_conjecture", + "name": "Elliott-Halberstam conjecture", + "fields": [ + "Mathematics", + "Number Theory" + ], + "statement": "Primes in arithmetic progressions are distributed as uniformly as GRH predicts up to a factor.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.25, + "compression_pressure": 0.8, + "topology_torsion": 0.3, + "residual_risk": 0.4, + "proof_readiness": 0.15, + "scale_band_declared": 0.7, + "negative_control_strength": 0.5, + "projection_declared": 0.8, + "shape_closure": 0.4 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "projection_declared", + "scale_band_declared", + "negative_control_strength" + ], + "alignment_fingerprint": "Distribution hypothesis; high compression pressure via GRH-like uniformity.", + "alignment_cluster": "cluster_04" + }, + { + "id": "fermat_catalan_conjecture", + "name": "Fermat-Catalan conjecture", + "fields": [ + "Mathematics", + "Number Theory" + ], + "statement": "Only finitely many perfect powers differ by 1.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.65, + "geometric_mass": 0.2, + "compression_pressure": 0.65, + "topology_torsion": 0.2, + "residual_risk": 0.3, + "proof_readiness": 0.2, + "scale_band_declared": 0.7, + "negative_control_strength": 0.55, + "projection_declared": 0.8, + "shape_closure": 0.45 + }, + "top_axes": [ + "projection_declared", + "scale_band_declared", + "semantic_entropy", + "compression_pressure", + "negative_control_strength" + ], + "alignment_fingerprint": "Power-difference logogram; conditional on abc closure.", + "alignment_cluster": "cluster_04" + }, + { + "id": "schinzel_hypothesis_h", + "name": "Schinzel's Hypothesis H", + "fields": [ + "Mathematics", + "Number Theory" + ], + "statement": "Every admissible finite set of integer polynomials simultaneously takes prime values infinitely often.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.2, + "compression_pressure": 0.75, + "topology_torsion": 0.25, + "residual_risk": 0.4, + "proof_readiness": 0.15, + "scale_band_declared": 0.7, + "negative_control_strength": 0.5, + "projection_declared": 0.8, + "shape_closure": 0.4 + }, + "top_axes": [ + "projection_declared", + "semantic_entropy", + "compression_pressure", + "scale_band_declared", + "negative_control_strength" + ], + "alignment_fingerprint": "Polynomial-prime sieve bottleneck; broad implications.", + "alignment_cluster": "cluster_04" + }, + { + "id": "brocards_problem", + "name": "Brocard's problem", + "fields": [ + "Mathematics", + "Number Theory" + ], + "statement": "Are there finitely many integer solutions to n! + 1 = m² beyond n = 4,5,7?", + "why_unsolved": "Exponential Diophantine equation with factorial; abc heuristics suggest a finite list.", + "known_reductions_to": [ + "abc_conjecture" + ], + "known_reductions_from": [], + "rrc_shape": "LogogramProjection", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.55, + "geometric_mass": 0.15, + "compression_pressure": 0.6, + "topology_torsion": 0.15, + "residual_risk": 0.35, + "proof_readiness": 0.2, + "scale_band_declared": 0.7, + "negative_control_strength": 0.55, + "projection_declared": 0.8, + "shape_closure": 0.45 + }, + "top_axes": [ + "projection_declared", + "scale_band_declared", + "compression_pressure", + "semantic_entropy", + "negative_control_strength" + ], + "alignment_fingerprint": "Sparse factorial logogram; finite-list expectation.", + "alignment_cluster": "cluster_04" + }, + { + "id": "pillai_conjecture", + "name": "Pillai's 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_unsolved": "Catalan's theorem is the k=1 case; the general case needs effective Diophantine bounds.", + "known_reductions_to": [ + "abc_conjecture", + "brocards_problem" + ], + "known_reductions_from": [], + "rrc_shape": "LogogramProjection", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.65, + "geometric_mass": 0.2, + "compression_pressure": 0.65, + "topology_torsion": 0.2, + "residual_risk": 0.3, + "proof_readiness": 0.2, + "scale_band_declared": 0.7, + "negative_control_strength": 0.55, + "projection_declared": 0.8, + "shape_closure": 0.45 + }, + "top_axes": [ + "projection_declared", + "scale_band_declared", + "semantic_entropy", + "compression_pressure", + "negative_control_strength" + ], + "alignment_fingerprint": "Exponential Diophantine family; dependent on abc.", + "alignment_cluster": "cluster_04" + }, + { + "id": "mersenne_prime_infinitude", + "name": "Infinitude of Mersenne primes", + "fields": [ + "Mathematics", + "Number Theory" + ], + "statement": "Are there infinitely many Mersenne primes?", + "why_unsolved": "No proof exists; heuristic predictions are strong but no lower-bound theorem is known.", + "known_reductions_to": [ + "perfect_numbers_odd_existence" + ], + "known_reductions_from": [], + "rrc_shape": "LogogramProjection", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.6, + "geometric_mass": 0.15, + "compression_pressure": 0.6, + "topology_torsion": 0.15, + "residual_risk": 0.45, + "proof_readiness": 0.15, + "scale_band_declared": 0.8, + "negative_control_strength": 0.5, + "projection_declared": 0.75, + "shape_closure": 0.35 + }, + "top_axes": [ + "scale_band_declared", + "projection_declared", + "semantic_entropy", + "compression_pressure", + "negative_control_strength" + ], + "alignment_fingerprint": "Sparse exponential prime logogram; scale band strong but proof absent.", + "alignment_cluster": "cluster_04" + }, + { + "id": "perfect_numbers_odd_existence", + "name": "Odd perfect numbers", + "fields": [ + "Mathematics", + "Number Theory" + ], + "statement": "Does an odd perfect number exist?", + "why_unsolved": "No example and no impossibility proof; many restrictions on size and form are known.", + "known_reductions_to": [ + "mersenne_prime_infinitude" + ], + "known_reductions_from": [], + "rrc_shape": "LogogramProjection", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.6, + "geometric_mass": 0.2, + "compression_pressure": 0.65, + "topology_torsion": 0.2, + "residual_risk": 0.5, + "proof_readiness": 0.2, + "scale_band_declared": 0.8, + "negative_control_strength": 0.6, + "projection_declared": 0.8, + "shape_closure": 0.4 + }, + "top_axes": [ + "scale_band_declared", + "projection_declared", + "compression_pressure", + "semantic_entropy", + "negative_control_strength" + ], + "alignment_fingerprint": "Existence/impossibility logogram; negative controls provide partial bounds.", + "alignment_cluster": "cluster_04" + }, + { + "id": "rational_points_high_genus", + "name": "Rational points on higher-genus varieties", + "fields": [ + "Mathematics", + "Number Theory", + "Algebraic Geometry" + ], + "statement": "Characterize and bound rational points on curves and varieties of general type (effective Faltings).", + "why_unsolved": "Faltings' theorem is non-effective; uniform bounds (Bombieri-Lang) remain conjectural.", + "known_reductions_to": [ + "abc_conjecture", + "birch_swinnerton_dyer_conjecture" + ], + "known_reductions_from": [], + "rrc_shape": "ProjectableGeometryTopology", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.8, + "compression_pressure": 0.75, + "topology_torsion": 0.55, + "residual_risk": 0.4, + "proof_readiness": 0.2, + "scale_band_declared": 0.65, + "negative_control_strength": 0.5, + "projection_declared": 0.75, + "shape_closure": 0.35 + }, + "top_axes": [ + "geometric_mass", + "semantic_entropy", + "compression_pressure", + "projection_declared", + "scale_band_declared" + ], + "alignment_fingerprint": "Arithmetic geometry projection; effective methods missing.", + "alignment_cluster": "cluster_04" + }, + { + "id": "algebraic_k_theory_integers", + "name": "Algebraic K-theory of the integers", + "fields": [ + "Mathematics", + "Algebra", + "Number Theory" + ], + "statement": "Compute the algebraic K-groups K_n(Z) for all n.", + "why_unsolved": "Known for many n but no complete pattern; relates to Bernoulli numbers and motivic cohomology.", + "known_reductions_to": [ + "standard_conjectures", + "langlands_program" + ], + "known_reductions_from": [], + "rrc_shape": "ProjectableGeometryTopology", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.75, + "compression_pressure": 0.7, + "topology_torsion": 0.6, + "residual_risk": 0.3, + "proof_readiness": 0.25, + "scale_band_declared": 0.6, + "negative_control_strength": 0.45, + "projection_declared": 0.7, + "shape_closure": 0.35 + }, + "top_axes": [ + "semantic_entropy", + "geometric_mass", + "compression_pressure", + "projection_declared", + "topology_torsion" + ], + "alignment_fingerprint": "Homotopy-number theory bridge; projection geometry over Z.", + "alignment_cluster": "cluster_10" + }, + { + "id": "novikov_conjecture", + "name": "Novikov conjecture", + "fields": [ + "Mathematics", + "Topology", + "Geometry" + ], + "statement": "Higher signatures of compact oriented manifolds are oriented homotopy invariants.", + "why_unsolved": "Proven for large classes but not in full generality; connects index theory and C*-algebras.", + "known_reductions_to": [ + "borel_conjecture" + ], + "known_reductions_from": [], + "rrc_shape": "ProjectableGeometryTopology", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.85, + "compression_pressure": 0.7, + "topology_torsion": 0.75, + "residual_risk": 0.3, + "proof_readiness": 0.25, + "scale_band_declared": 0.65, + "negative_control_strength": 0.5, + "projection_declared": 0.75, + "shape_closure": 0.4 + }, + "top_axes": [ + "geometric_mass", + "semantic_entropy", + "topology_torsion", + "projection_declared", + "compression_pressure" + ], + "alignment_fingerprint": "Surgery-theory topology; high topology torsion.", + "alignment_cluster": "cluster_05" + }, + { + "id": "borel_conjecture", + "name": "Borel conjecture", + "fields": [ + "Mathematics", + "Topology" + ], + "statement": "Aspherical closed manifolds are determined up to homeomorphism by their fundamental group.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.7, + "geometric_mass": 0.85, + "compression_pressure": 0.65, + "topology_torsion": 0.8, + "residual_risk": 0.35, + "proof_readiness": 0.2, + "scale_band_declared": 0.65, + "negative_control_strength": 0.5, + "projection_declared": 0.75, + "shape_closure": 0.35 + }, + "top_axes": [ + "geometric_mass", + "topology_torsion", + "projection_declared", + "semantic_entropy", + "compression_pressure" + ], + "alignment_fingerprint": "Aspherical topology; torsion from rigidity.", + "alignment_cluster": "cluster_05" + }, + { + "id": "volume_conjecture", + "name": "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_unsolved": "Connects quantum topology and hyperbolic geometry; known for many knots but general proof open.", + "known_reductions_to": [ + "quantum_gravity" + ], + "known_reductions_from": [], + "rrc_shape": "ProjectableGeometryTopology", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.85, + "compression_pressure": 0.75, + "topology_torsion": 0.75, + "residual_risk": 0.35, + "proof_readiness": 0.2, + "scale_band_declared": 0.65, + "negative_control_strength": 0.45, + "projection_declared": 0.75, + "shape_closure": 0.35 + }, + "top_axes": [ + "geometric_mass", + "semantic_entropy", + "compression_pressure", + "topology_torsion", + "projection_declared" + ], + "alignment_fingerprint": "Quantum-geometric bridge; high topology torsion.", + "alignment_cluster": "cluster_05" + }, + { + "id": "hopf_conjecture", + "name": "Hopf conjecture (S²×S²)", + "fields": [ + "Mathematics", + "Geometry" + ], + "statement": "There is no Riemannian metric of positive sectional curvature on S² × S².", + "why_unsolved": "Few examples of positive curvature exist; topological obstructions in product manifolds are subtle.", + "known_reductions_to": [], + "known_reductions_from": [], + "rrc_shape": "ProjectableGeometryTopology", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.7, + "geometric_mass": 0.9, + "compression_pressure": 0.65, + "topology_torsion": 0.8, + "residual_risk": 0.4, + "proof_readiness": 0.2, + "scale_band_declared": 0.65, + "negative_control_strength": 0.5, + "projection_declared": 0.75, + "shape_closure": 0.35 + }, + "top_axes": [ + "geometric_mass", + "topology_torsion", + "projection_declared", + "semantic_entropy", + "compression_pressure" + ], + "alignment_fingerprint": "Product-manifold curvature torsion; projection declared.", + "alignment_cluster": "cluster_05" + }, + { + "id": "poincare_conjecture_3d_solved", + "name": "3D Poincaré conjecture (solved boundary)", + "fields": [ + "Mathematics", + "Topology" + ], + "statement": "Every simply connected closed 3-manifold is homeomorphic to S³.", + "why_unsolved": "Solved by Grigori Perelman (2002–2003) using Ricci flow with surgery; included as a solved RRC boundary marker.", + "known_reductions_to": [], + "known_reductions_from": [ + "smooth_4d_poincare_conjecture", + "generalized_poincare_conjecture_smooth" + ], + "rrc_shape": "LeanTheoremReceipt", + "rrc_status": "ACCEPT", + "rrc_axes": { + "semantic_entropy": 0.4, + "geometric_mass": 0.9, + "compression_pressure": 0.3, + "topology_torsion": 0.85, + "residual_risk": 0.05, + "proof_readiness": 1.0, + "scale_band_declared": 1.0, + "negative_control_strength": 1.0, + "projection_declared": 1.0, + "shape_closure": 1.0 + }, + "top_axes": [ + "proof_readiness", + "scale_band_declared", + "negative_control_strength", + "projection_declared", + "shape_closure" + ], + "alignment_fingerprint": "Solved theorem receipt; demonstrates the ACCEPT boundary for manifold topology.", + "alignment_cluster": "unclustered" + }, + { + "id": "singular_cardinal_hypothesis", + "name": "Singular Cardinal Hypothesis", + "fields": [ + "Mathematics", + "Logic", + "Set Theory" + ], + "statement": "Does 2^κ = κ⁺ hold for every singular strong-limit cardinal κ?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.85, + "geometric_mass": 0.15, + "compression_pressure": 0.7, + "topology_torsion": 0.45, + "residual_risk": 0.75, + "proof_readiness": 0.05, + "scale_band_declared": 0.55, + "negative_control_strength": 0.35, + "projection_declared": 0.6, + "shape_closure": 0.2 + }, + "top_axes": [ + "semantic_entropy", + "residual_risk", + "compression_pressure", + "projection_declared", + "scale_band_declared" + ], + "alignment_fingerprint": "Set-theoretic higher cardinal; axiomatic risk high.", + "alignment_cluster": "cluster_06" + }, + { + "id": "p_np_algebrization_barrier", + "name": "Algebrization barrier", + "fields": [ + "Theoretical Computer Science", + "Logic" + ], + "statement": "A meta-barrier showing that many known techniques cannot separate P and NP.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.2, + "compression_pressure": 0.75, + "topology_torsion": 0.7, + "residual_risk": 0.5, + "proof_readiness": 0.1, + "scale_band_declared": 0.7, + "negative_control_strength": 0.75, + "projection_declared": 0.75, + "shape_closure": 0.3 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "negative_control_strength", + "projection_declared", + "topology_torsion" + ], + "alignment_fingerprint": "Complexity barrier node; high topology torsion from negative controls.", + "alignment_cluster": "cluster_02" + }, + { + "id": "small_set_expansion_conjecture", + "name": "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_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", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.3, + "compression_pressure": 0.8, + "topology_torsion": 0.55, + "residual_risk": 0.45, + "proof_readiness": 0.2, + "scale_band_declared": 0.7, + "negative_control_strength": 0.6, + "projection_declared": 0.8, + "shape_closure": 0.45 + }, + "top_axes": [ + "compression_pressure", + "projection_declared", + "semantic_entropy", + "scale_band_declared", + "negative_control_strength" + ], + "alignment_fingerprint": "Graph expansion route; tightly coupled to UGC.", + "alignment_cluster": "cluster_02" + }, + { + "id": "quantum_pcp_conjecture", + "name": "Quantum PCP conjecture", + "fields": [ + "Theoretical Computer Science", + "Physics" + ], + "statement": "Approximating the ground-state energy of local Hamiltonians is QMA-hard.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.85, + "geometric_mass": 0.3, + "compression_pressure": 0.85, + "topology_torsion": 0.6, + "residual_risk": 0.4, + "proof_readiness": 0.15, + "scale_band_declared": 0.7, + "negative_control_strength": 0.55, + "projection_declared": 0.75, + "shape_closure": 0.35 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "projection_declared", + "scale_band_declared", + "topology_torsion" + ], + "alignment_fingerprint": "Quantum complexity hardness router.", + "alignment_cluster": "cluster_02" + }, + { + "id": "sunflower_conjecture", + "name": "Erdős-Rado sunflower conjecture", + "fields": [ + "Mathematics", + "Combinatorics", + "Theoretical Computer Science" + ], + "statement": "Bound the size of set systems with restricted pairwise intersections (sunflowers).", + "why_unsolved": "Lower-bound constructions are limited; recent upper-bound improvements still leave a gap.", + "known_reductions_to": [ + "cap_set_problem", + "matrix_rigidity" + ], + "known_reductions_from": [], + "rrc_shape": "ErdosBoundConjecture", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.7, + "geometric_mass": 0.35, + "compression_pressure": 0.7, + "topology_torsion": 0.4, + "residual_risk": 0.35, + "proof_readiness": 0.25, + "scale_band_declared": 0.7, + "negative_control_strength": 0.6, + "projection_declared": 0.8, + "shape_closure": 0.45 + }, + "top_axes": [ + "projection_declared", + "semantic_entropy", + "compression_pressure", + "scale_band_declared", + "negative_control_strength" + ], + "alignment_fingerprint": "Combinatorial sunflower bound; Erdős-style projection.", + "alignment_cluster": "cluster_02" + }, + { + "id": "cap_set_problem", + "name": "Cap set problem (exact growth)", + "fields": [ + "Mathematics", + "Combinatorics" + ], + "statement": "Determine the maximum size of a cap set in F_3^n.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.65, + "geometric_mass": 0.3, + "compression_pressure": 0.65, + "topology_torsion": 0.35, + "residual_risk": 0.3, + "proof_readiness": 0.3, + "scale_band_declared": 0.75, + "negative_control_strength": 0.65, + "projection_declared": 0.8, + "shape_closure": 0.5 + }, + "top_axes": [ + "projection_declared", + "scale_band_declared", + "semantic_entropy", + "compression_pressure", + "negative_control_strength" + ], + "alignment_fingerprint": "Polynomial-method route; recent progress but closure not tight.", + "alignment_cluster": "cluster_02" + }, + { + "id": "matrix_rigidity", + "name": "Matrix rigidity", + "fields": [ + "Theoretical Computer Science", + "Mathematics" + ], + "statement": "Are high-rank matrices far from low-rank matrices under bounded-entry changes?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.4, + "compression_pressure": 0.75, + "topology_torsion": 0.5, + "residual_risk": 0.45, + "proof_readiness": 0.2, + "scale_band_declared": 0.7, + "negative_control_strength": 0.55, + "projection_declared": 0.8, + "shape_closure": 0.4 + }, + "top_axes": [ + "projection_declared", + "semantic_entropy", + "compression_pressure", + "scale_band_declared", + "negative_control_strength" + ], + "alignment_fingerprint": "Linear-algebraic complexity gate; recent rigidity results create residual risk.", + "alignment_cluster": "cluster_02" + }, + { + "id": "derandomization_polynomial_identity_testing", + "name": "Derandomization of Polynomial Identity Testing", + "fields": [ + "Theoretical Computer Science", + "Mathematics" + ], + "statement": "Find explicit hitting sets for polynomial identity testing or prove PIT is in P.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.25, + "compression_pressure": 0.8, + "topology_torsion": 0.55, + "residual_risk": 0.4, + "proof_readiness": 0.2, + "scale_band_declared": 0.75, + "negative_control_strength": 0.6, + "projection_declared": 0.85, + "shape_closure": 0.45 + }, + "top_axes": [ + "projection_declared", + "compression_pressure", + "semantic_entropy", + "scale_band_declared", + "negative_control_strength" + ], + "alignment_fingerprint": "Algebraic derandomization node; reduction to circuit lower bounds.", + "alignment_cluster": "cluster_02" + }, + { + "id": "eulers_constant_irrationality", + "name": "Irrationality of Euler's constant", + "fields": [ + "Mathematics", + "Number Theory" + ], + "statement": "Is the Euler-Mascheroni constant γ irrational (or transcendental)?", + "why_unsolved": "No proof of irrationality exists; standard Diophantine methods do not apply.", + "known_reductions_to": [], + "known_reductions_from": [], + "rrc_shape": "LogogramProjection", + "rrc_status": "CANDIDATE", + "rrc_axes": { + "semantic_entropy": 0.55, + "geometric_mass": 0.1, + "compression_pressure": 0.6, + "topology_torsion": 0.1, + "residual_risk": 0.4, + "proof_readiness": 0.15, + "scale_band_declared": 0.75, + "negative_control_strength": 0.5, + "projection_declared": 0.8, + "shape_closure": 0.4 + }, + "top_axes": [ + "projection_declared", + "scale_band_declared", + "compression_pressure", + "semantic_entropy", + "negative_control_strength" + ], + "alignment_fingerprint": "Analytic constant logogram; projection clear, proof tools absent.", + "alignment_cluster": "cluster_04" + }, + { + "id": "dark_energy_equation_of_state", + "name": "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_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", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.35, + "compression_pressure": 0.7, + "topology_torsion": 0.3, + "residual_risk": 0.6, + "proof_readiness": 0.15, + "scale_band_declared": 0.55, + "negative_control_strength": 0.35, + "projection_declared": 0.5, + "shape_closure": 0.25 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "residual_risk", + "scale_band_declared", + "projection_declared" + ], + "alignment_fingerprint": "Dark-energy force probe; w parameter underspecified.", + "alignment_cluster": "cluster_08" + }, + { + "id": "bqp_vs_np", + "name": "BQP vs NP", + "fields": [ + "Theoretical Computer Science", + "Physics" + ], + "statement": "Can every efficient quantum computation be verified classically in nondeterministic polynomial time?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.2, + "compression_pressure": 0.8, + "topology_torsion": 0.55, + "residual_risk": 0.45, + "proof_readiness": 0.15, + "scale_band_declared": 0.7, + "negative_control_strength": 0.55, + "projection_declared": 0.8, + "shape_closure": 0.35 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "projection_declared", + "scale_band_declared", + "topology_torsion" + ], + "alignment_fingerprint": "Quantum-classical verification boundary.", + "alignment_cluster": "cluster_02" + }, + { + "id": "cosmological_inflation_origin", + "name": "Origin of cosmic inflation", + "fields": [ + "Physics", + "Cosmology" + ], + "statement": "What is the physical origin and detailed mechanism of cosmic inflation?", + "why_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" + ], + "known_reductions_from": [], + "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", + "rrc_status": "HOLD", + "rrc_axes": { + "semantic_entropy": 0.85, + "geometric_mass": 0.4, + "compression_pressure": 0.75, + "topology_torsion": 0.4, + "residual_risk": 0.65, + "proof_readiness": 0.1, + "scale_band_declared": 0.4, + "negative_control_strength": 0.25, + "projection_declared": 0.35, + "shape_closure": 0.15 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "residual_risk", + "geometric_mass", + "topology_torsion" + ], + "alignment_fingerprint": "Model-degeneracy hold; projection weak.", + "alignment_cluster": "cluster_08" + }, + { + "id": "langlands_program", + "name": "Langlands program", + "fields": [ + "Mathematics", + "Number Theory", + "Representation Theory" + ], + "statement": "A broad web of conjectures connecting number theory, automorphic forms, and algebraic geometry.", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.95, + "geometric_mass": 0.9, + "compression_pressure": 0.9, + "topology_torsion": 0.75, + "residual_risk": 0.5, + "proof_readiness": 0.1, + "scale_band_declared": 0.5, + "negative_control_strength": 0.3, + "projection_declared": 0.6, + "shape_closure": 0.2 + }, + "top_axes": [ + "semantic_entropy", + "geometric_mass", + "compression_pressure", + "topology_torsion", + "projection_declared" + ], + "alignment_fingerprint": "Vast correspondence manifold; projection declared but closure diffuse.", + "alignment_cluster": "cluster_01" + }, + { + "id": "cosmic_censorship_conjecture", + "name": "Cosmic censorship conjecture", + "fields": [ + "Physics", + "General Relativity" + ], + "statement": "Do naked singularities form from generic initial data?", + "why_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", + "rrc_axes": { + "semantic_entropy": 0.8, + "geometric_mass": 0.8, + "compression_pressure": 0.75, + "topology_torsion": 0.7, + "residual_risk": 0.6, + "proof_readiness": 0.15, + "scale_band_declared": 0.55, + "negative_control_strength": 0.35, + "projection_declared": 0.6, + "shape_closure": 0.25 + }, + "top_axes": [ + "semantic_entropy", + "geometric_mass", + "compression_pressure", + "topology_torsion", + "residual_risk" + ], + "alignment_fingerprint": "GR singularity censorship; geometry-quantum boundary.", + "alignment_cluster": "unclustered" + }, + { + "id": "origin_of_magnetic_fields", + "name": "Origin of cosmic magnetic fields", + "fields": [ + "Physics", + "Astrophysics" + ], + "statement": "Explain the origin and amplification of large-scale cosmic magnetic fields.", + "why_unsolved": "Dynamo theory is incomplete and primordial seeds are poorly constrained.", + "known_reductions_to": [ + "baryon_asymmetry_problem", + "cosmological_inflation_origin" + ], + "known_reductions_from": [], + "rrc_shape": "CadForceProbeReceipt", + "rrc_status": "HOLD", + "rrc_axes": { + "semantic_entropy": 0.75, + "geometric_mass": 0.4, + "compression_pressure": 0.65, + "topology_torsion": 0.3, + "residual_risk": 0.55, + "proof_readiness": 0.15, + "scale_band_declared": 0.5, + "negative_control_strength": 0.3, + "projection_declared": 0.5, + "shape_closure": 0.25 + }, + "top_axes": [ + "semantic_entropy", + "compression_pressure", + "residual_risk", + "scale_band_declared", + "projection_declared" + ], + "alignment_fingerprint": "Astrophysical force probe; multi-scale amplification gap.", + "alignment_cluster": "cluster_08" + } + ], + "alignment_clusters": [ + { + "id": "cluster_01", + "name": "Millennium, L-functions, and motives", + "description": "Problems anchored in zeta/L-functions, algebraic cycles, and arithmetic geometry. Riemann Hypothesis is the central spectral axis.", + "problems": [ + "riemann_hypothesis", + "generalized_riemann_hypothesis", + "birch_swinnerton_dyer_conjecture", + "hodge_conjecture", + "tate_conjecture", + "standard_conjectures", + "langlands_program" + ] + }, + { + "id": "cluster_02", + "name": "Computational complexity core", + "description": "P vs NP and its satellites: hardness of approximation, fine-grained complexity, derandomization, and algebraic barriers.", + "problems": [ + "p_vs_np", + "np_intermediate_existence", + "graph_isomorphism_in_p", + "factoring_in_p", + "discrete_log_in_p", + "bpp_vs_p", + "exponential_time_hypothesis", + "strong_exponential_time_hypothesis", + "unique_games_conjecture", + "small_set_expansion_conjecture", + "quantum_pcp_conjecture", + "matrix_rigidity", + "derandomization_polynomial_identity_testing", + "bqp_vs_np", + "p_np_algebrization_barrier", + "cap_set_problem", + "sunflower_conjecture" + ] + }, + { + "id": "cluster_03", + "name": "PDE 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+ ] + }, + { + "id": "cluster_05", + "name": "Topology and geometry", + "description": "Manifold classification, asphericity, curvature, and quantum-topological invariants.", + "problems": [ + "smooth_4d_poincare_conjecture", + "generalized_poincare_conjecture_smooth", + "volume_conjecture", + "novikov_conjecture", + "borel_conjecture", + "hopf_conjecture", + "hodge_conjecture", + "tate_conjecture", + "standard_conjectures" + ] + }, + { + "id": "cluster_06", + "name": "Logic and foundations", + "description": "Independence, consistency, and meta-mathematical limits of standard axiom systems.", + "problems": [ + "continuum_hypothesis", + "consistency_of_zfc", + "singular_cardinal_hypothesis", + "p_vs_np", + "p_np_algebrization_barrier" + ] + }, + { + "id": "cluster_07", + "name": "Quantum and information", + "description": "Quantum computation, verification, and quantum-gravity information puzzles.", + "problems": [ + "quantum_supremacy_verification", + "black_hole_information_paradox", + "bqp_vs_np", + "quantum_pcp_conjecture", + "quantum_gravity", + "yang_mills_mass_gap" + ] + }, + { + "id": "cluster_08", + "name": "Cosmology and dark sectors", + "description": "Dark matter, dark energy, vacuum energy, baryon asymmetry, and large-scale structure origins.", + "problems": [ + "dark_matter_identity", + "cosmological_constant_problem", + "dark_energy_equation_of_state", + "baryon_asymmetry_problem", + "cosmological_inflation_origin", + "origin_of_magnetic_fields", + "quantum_gravity" + ] + }, + { + "id": "cluster_09", + "name": "Fluid and field-theoretic singularities", + "description": "Turbulence, Navier-Stokes singularities, and constructive quantum field theory.", + "problems": [ + "navier_stokes_existence_smoothness", + "navier_stokes_blowup", + "turbulence_closure_problem", + "yang_mills_mass_gap" + ] + }, + { + "id": "cluster_10", + "name": "Algebraic geometry and motives", + "description": "Cycles, K-theory, Langlands duality, and the arithmetic of rational 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00000000..19888300 --- /dev/null +++ b/6-Documentation/docs/research/unsolved_hard_problems_rrc_survey.md @@ -0,0 +1,1480 @@ +# 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.