| {"id": "transport_minimum", "title": "Transport-optimal exact completion", "proof_source": "source/holonomy_completion_proof.tex", "admissible_category": "exact_parallel_abelian", "error_norm": "not_applicable", "claim": "Minimum auxiliary rank equals the dimension of the holonomy-saturated curvature space.", "essential_assumptions": "Fixed potential, flat coefficient action, parallel boundary, adjacent-arity abelian repair.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Theorem 2.1"} | |
| {"id": "residual_quotient", "title": "Residual quotient without a parallel complement", "proof_source": "source/holonomy_completion_proof.tex", "admissible_category": "exact_parallel_abelian", "error_norm": "not_applicable", "claim": "The canonical residual quotient inclusion is a quasi-isomorphism.", "essential_assumptions": "Rank-minimal repair; finite base-form filtration; a smooth complement suffices.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Theorem 2.2"} | |
| {"id": "compact_overhead", "title": "Unbounded global rank from a local defect line", "proof_source": "source/holonomy_completion_proof.tex", "admissible_category": "exact_parallel_abelian", "error_norm": "uniform_curvature", "claim": "Fixed closed five-dimensional bases support arbitrarily large exact-compatible repair rank.", "essential_assumptions": "The same global potential and curvature are retained as transport varies within each constructed family.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Sections 2 and 5"} | |
| {"id": "unitary_spectral_frontier", "title": "Unitary rank-error law", "proof_source": "source/unitary_error_frontier.tex", "admissible_category": "parallel_unitary_subbundles", "error_norm": "L2_curvature_squared", "claim": "Isotypic reduced positive matrices determine the exact rank-error budget frontier.", "essential_assumptions": "Unitary flat transport; invariant subbundles; integer costs are irreducible factor dimensions.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Theorem 5.1"} | |
| {"id": "variable_boundary_obstruction", "title": "Global compatibility obstruction for variable boundaries", "proof_source": "source/global_compatibility_proof.tex", "admissible_category": "arbitrary_smooth_isometric_boundary", "error_norm": "uniform_Hilbert_Schmidt_compatibility", "claim": "Normal compatibility is at least s sqrt(r(d-r)/d) divided by sqrt(2 sum loop_lengths_squared).", "essential_assumptions": "Flat Hermitian original bundle; unitary auxiliary connection; isometric boundary; selected holonomy commutator gap.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Theorem 6.1"} | |
| {"id": "linear_rank_reduction_scale", "title": "Matching-order compatibility transition", "proof_source": "source/global_compatibility_proof.tex", "admissible_category": "arbitrary_smooth_isometric_boundary", "error_norm": "uniform_curvature_and_compatibility", "claim": "The least compatibility budget for any rank reduction has order absolute(epsilon).", "essential_assumptions": "Explicit fixed closed connected-sum five-manifold; curvature tolerance below one; constants depend on fixed dimension and geometry; flat auxiliary line upper bound.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Theorem 6.3"} | |
| {"id": "rank_one_frontier", "title": "Complete constant-boundary rank-one frontier", "proof_source": "source/rank_one_frontier_proof.tex", "admissible_category": "constant_isometric_line_and_scalar_connection", "error_norm": "uniform_curvature_and_compatibility", "claim": "A compact scalar minimization with an exact SDP dual determines the frontier; rational spectral data admit exact-rational global additive certificates.", "essential_assumptions": "Hermitian generator and unit seed; secular-root specialization requires simple spectrum and positive seed weights. Exact-rational certificate execution assumes rational spectral data.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Theorem 7.1 and Corollary 7.2"} | |
| {"id": "two_level_frontier", "title": "Closed two-level curvature-compatibility formula", "proof_source": "source/two_level_frontier_fragment.tex", "admissible_category": "constant_isometric_line_and_scalar_connection", "error_norm": "uniform_curvature_and_compatibility", "claim": "For balanced two-level seeds, normalized minimum compatibility is Delta/2 times positive_part(1-2 tau_squared).", "essential_assumptions": "Two distinct Hermitian eigenvalues; the external connection scale is factored out.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Theorem 7.3"} | |
| {"id": "fourth_moment_law", "title": "Fourth-moment sensitivity at perfect capture", "proof_source": "source/two_level_frontier_fragment.tex", "admissible_category": "rank_one_capture_variance", "error_norm": "variance_at_capture_constraint", "claim": "The first-order improvement at small curvature error is governed by the square root of fourth central moment minus variance squared.", "essential_assumptions": "Any Hermitian matrix and unit seed; a zero linear term is included.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Proposition 7.4"} | |
| {"id": "compatible_rounding", "title": "Dimension-preserving invariant-projector rounding", "proof_source": "source/general_pareto_proof.tex", "admissible_category": "finite_unitary_holonomy_generators", "error_norm": "Hilbert_Schmidt_commutator", "claim": "Below the commutator-gap threshold, conditional-expectation spectral rounding preserves rank and controls curvature loss.", "essential_assumptions": "Positive least singular value on the commutant orthogonal complement; scalar commutant-complement-zero case treated separately.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Theorem 8.1"} | |
| {"id": "all_rank_joint_budget", "title": "Exact marked flat all-rank joint frontier", "proof_source": "source/general_pareto_proof.tex", "admissible_category": "marked_constant_boundary_and_separated_connections", "error_norm": "L2_curvature_and_compatibility", "claim": "Simultaneous optima are the curvature spectral tail and epsilon_squared*r*(d-r)/d.", "essential_assumptions": "Fixed marked T5 family with constant isometric boundary and separated flat connection forms; not unrestricted variable boundaries.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Theorem 8.2"} | |
| {"id": "nonlinear_relative_completion", "title": "Finite nonlinear relative Weil completion", "proof_source": "source/noncentral_relative_cone.tex", "admissible_category": "fixed_ground_nonlinear_coefficients", "error_norm": "not_an_approximation_budget", "claim": "One shifted generator per coefficient generator gives an exact square-zero differential and a quasi-isomorphism to the ground.", "essential_assumptions": "Finite graded vector bundle; coefficient degrees at least two; fixed ground CE differential. Existence upper bound using the standard Weil mechanism; not nonlinear minimum rank.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Theorem 9.1 and Proposition 9.2"} | |
| {"id": "jordan_resolution", "title": "Sharp fixed-loop Jordan resolution laws", "proof_source": "source/resolution_frontier_proof.tex", "admissible_category": "fixed_seed_orbit_observations", "error_norm": "declared_matrix_noise", "claim": "Jordan probe singular scales determine deterministic certification radii and Gaussian consistency thresholds.", "essential_assumptions": "Fixed loop counts and input seed; geometrically consecutive-rank alternatives are truncated Jordan chains; longer loops change the resources.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Section 10"} | |
| {"id": "cyclic_singular_hierarchy", "title": "Cyclic singular hierarchy beyond nilpotence", "proof_source": "source/cyclic_resolution_extension.tex", "admissible_category": "fixed_cyclic_generator_and_seed", "error_norm": "singular_value_scales", "claim": "The jth singular value of the fixed-loop orbit matrix has order absolute(epsilon)^(j-1).", "essential_assumptions": "Fixed cyclic pair and dimension; real-Gaussian full-rank certification applies to real means; complex noise needs its own convention.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Theorem 11.1"} | |
| {"id": "algebraic_infinitesimal_scales", "title": "Algebraic infinitesimal precision hierarchy", "proof_source": "source/AELYNTHIEL_Holonomy_Completion.tex", "admissible_category": "finite_algebraic_scalar_extension", "error_norm": "ordered_coefficient_scales", "claim": "Polynomial Jordan-orbit, finite determinant, and Gramian identities admit suitable real-closed scalar extension.", "essential_assumptions": "Finite algebraic coefficient statements only; smooth geometry, general analytic exponentials, and Gaussian probability are not transferred.", "proof_status": "proved_in_manuscript", "historical_priority": "unconfirmed", "manuscript_location": "Section 12"} | |