{ "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; 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