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Research-Stack/6-Documentation/docs/research/unsolved_hard_problems_rrc_alignments.json
allaun 514bf8da72 docs(research): RRC survey of 67 unsolved hard problems and alignment clusters 
- 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
2026-06-20 19:12:13 -05:00

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{
"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,
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"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,
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"proof_readiness": 0.25,
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"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,
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"residual_risk": 0.3,
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"scale_band_declared": 0.75,
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"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,
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"proof_readiness": 0.15,
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"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,
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"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,
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"compression_pressure": 0.9,
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"scale_band_declared": 0.5,
"negative_control_strength": 0.3,
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"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": {
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},
"top_axes": [
"semantic_entropy",
"geometric_mass",
"compression_pressure",
"topology_torsion",
"residual_risk"
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"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",
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"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"
}
],
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{
"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 regularity and singularities",
"description": "Existence, smoothness, and blow-up questions for nonlinear PDEs, plus related dynamical-systems bounds.",
"problems": [
"navier_stokes_existence_smoothness",
"navier_stokes_blowup",
"turbulence_closure_problem",
"hilbert_sixteenth_problem",
"yang_mills_mass_gap"
]
},
{
"id": "cluster_04",
"name": "Arithmetic and Diophantine structures",
"description": "Additive/multiplicative patterns in integers, exponential Diophantine equations, and prime distribution.",
"problems": [
"abc_conjecture",
"beal_conjecture",
"goldbach_conjecture",
"twin_prime_conjecture",
"collatz_conjecture",
"polignacs_conjecture",
"elliott_halberstam_conjecture",
"fermat_catalan_conjecture",
"schinzel_hypothesis_h",
"brocards_problem",
"pillai_conjecture",
"mersenne_prime_infinitude",
"perfect_numbers_odd_existence",
"eulers_constant_irrationality",
"rational_points_high_genus",
"generalized_riemann_hypothesis"
]
},
{
"id": "cluster_05",
"name": "Topology and geometry",
"description": "Manifold classification, asphericity, curvature, and quantum-topological invariants.",
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"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.",
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"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.",
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"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 points.",
"problems": [
"hodge_conjecture",
"tate_conjecture",
"standard_conjectures",
"birch_swinnerton_dyer_conjecture",
"rational_points_high_genus",
"algebraic_k_theory_integers",
"langlands_program",
"hilbert_twelfth_problem"
]
}
],
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],
"note": "Connection strength 0–1 based on known reductions, shared techniques, cluster co-membership, and analogies."
}
}
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