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No full external-proof or global computational approval is inferred.", "dependency_review_states": { "sections17-22": "independent_mathematical_snapshot_accepted_after_R1_R2; later_integration_delta_separate" } }, { "id": "appB.positive_trace", "labels": [ "app:conventions" ], "row_number": 24, "quantity": "Positive geometric trace", "status": "convention_pointer_checked", "statement": "$t_\\gamma=2\\cosh(\\ell_\\gamma/2)=|\\tr\\widetilde\\rho(\\gamma)|$ for a hyperbolic element; compatible lift signs are retained in trace identities; \\eqref{eq:figuretrace}.", "audit_note": "Positive geometric traces must not be substituted for globally signed character traces inside a skein identity.", "dependency_labels": [ "eq:figuretrace" ], "dependency_sources": [ "book/chapters/18-figure-eight-curves.tex" ], "dependencies": [ "book/claims/sections17-22.json#18.01" ], "inherited_statuses": { "book/claims/sections17-22.json#18.01": "derived_geometric" }, "independent_review_boundary": "The exact Sections17-22 mathematical snapshot and ledger are independently accepted after R1/R2; current computation-pointer/layout/metadata changes require separate integration review. No full external-proof or global computational approval is inferred.", "dependency_review_states": { "sections17-22": "independent_mathematical_snapshot_accepted_after_R1_R2; later_integration_delta_separate" } }, { "id": "appB.coefficient_basis", "labels": [ "app:conventions" ], "row_number": 25, "quantity": "Coefficient basis", "status": "convention_pointer_checked", "statement": "$T_C=\\prod_{\\alpha\\subset C}\\tr\\rho(\\alpha)$, $T_\\varnothing=1$, with unnegated traces and parallel multiplicities; \\eqref{eq:coeffbasis}.", "audit_note": "These functions are not pointwise positive on SU(2). 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their mathematical/source-statement reviews are inherited, not repeated in full." }, { "id": "appB.heat", "labels": [ "app:conventions" ], "row_number": 27, "quantity": "Coefficient heat transform", "status": "convention_pointer_checked", "statement": "$H_C(t)=e^{-t/4}(4\\pi t)^{-1/2}\\int e^{-L^2/(4t)}\\,\\dd\\sigma_C(L)$; \\eqref{eq:coeffheatdefinition}.", "audit_note": "The signed coefficient measure and the spectral 1/4 shift use the closed curvature-minus-one convention.", "dependency_labels": [ "eq:coeffheatdefinition" ], "dependency_sources": [ "book/chapters/23-trace-coefficients.tex" ], "dependencies": [ "book/claims/sections23-25.json#23.11" ], "inherited_statuses": { "book/claims/sections23-25.json#23.11": "derived_analytic" }, "independent_review_boundary": "Consistency with the exact earlier reviewed sources was checked; their mathematical/source-statement reviews are inherited, not repeated in full." }, { "id": "appB.laplace", "labels": [ "app:conventions" ], "row_number": 28, "quantity": "Coefficient Laplace parameter", "status": "convention_pointer_checked", "statement": "Kernel $e^{-sL}$; product parameter $v=s+1/2$; \\eqref{eq:coeffLaplacedefinition}, \\eqref{eq:coeffproductderivative}.", "audit_note": "The proved half-plane for nonempty coefficient transforms is Re(s)>1; formal coefficient extraction after continuation requires separate control.", "dependency_labels": [ "eq:coeffLaplacedefinition", "eq:coeffproductderivative" ], "dependency_sources": [ "book/chapters/23-trace-coefficients.tex" ], "dependencies": [ "book/claims/sections23-25.json#23.11", "book/claims/sections23-25.json#23.20" ], "inherited_statuses": { "book/claims/sections23-25.json#23.11": "derived_analytic", "book/claims/sections23-25.json#23.20": "derived_analytic" }, "independent_review_boundary": "Consistency with the exact earlier reviewed sources was checked; their mathematical/source-statement reviews are inherited, not repeated in full." }, { "id": "appB.total_laminations", "labels": [ "app:conventions" ], "row_number": 29, "quantity": "Total multicurve count", "status": "convention_pointer_checked", "statement": "All nonzero integral laminations, including disconnected and parallel supports; 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their mathematical/source-statement reviews are inherited, not repeated in full." }, { "id": "appB.support_variable", "labels": [ "app:conventions" ], "row_number": 31, "quantity": "Support variable $z$", "status": "convention_pointer_checked", "statement": "Exponent counts distinct geodesic supports, not parallel copies; \\eqref{eq:countsupportpartition}.", "audit_note": "A finite Fourier projection does not supply a uniform global coefficient extraction estimate.", "dependency_labels": [ "eq:countsupportpartition" ], "dependency_sources": [ "book/chapters/24-quantitative-counting.tex" ], "dependencies": [ "book/claims/sections23-25.json#24.11" ], "inherited_statuses": { "book/claims/sections23-25.json#24.11": "derived_analytic" }, "independent_review_boundary": "Consistency with the exact earlier reviewed sources was checked; their mathematical/source-statement reviews are inherited, not repeated in full." }, { "id": "appB.one_crossing", "labels": [ "app:conventions" ], "row_number": 32, "quantity": "One-crossing torus curves", "status": "convention_pointer_checked", "statement": "Primitive curves with one transverse double point; each simple class gives two N-type and one H-type labels, including equal-length multiplicity; \\eqref{eq:allfigureidentity}.", "audit_note": "Hyperelliptic and non-hyperelliptic types are both retained.", "dependency_labels": [ "eq:allfigureidentity" ], "dependency_sources": [ "book/chapters/18-figure-eight-curves.tex" ], "dependencies": [ "book/claims/sections17-22.json#18.09", "book/claims/sections17-22.json#18.05" ], "inherited_statuses": { "book/claims/sections17-22.json#18.09": "derived_from_external_statements", "book/claims/sections17-22.json#18.05": "derived_geometric" }, "independent_review_boundary": "The exact Sections17-22 mathematical snapshot and ledger are independently accepted after R1/R2; current computation-pointer/layout/metadata changes require separate integration review. 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