Where the sources disagree or the structure was guessed, ranked. 88 items · 2 warn · 86 info. In v1 these are paper-internal; Lean-vs-paper rows arrive in Phase 2.
| Severity | Kind | Node | Note | Source |
|---|---|---|---|---|
| warn | unresolved_ref | Interior term relies on the Schur-complement identity eq:schur-energy-identity and Green formula eq:calculus-green, equations rather than statements. | sections/energy.tex:200 | |
| warn | unresolved_ref | C^{2,alpha} and Schauder steps rest on external Gilbarg-Trudinger, Evans-Krylov and Caffarelli-Nirenberg-Spruck results. | sections/dirichlet.tex:24 | |
| info | inferred_dep | critique: The proof (transfer.tex:100-110) rests on , which transfer.tex:88-94 obtains by "Interpolating Theorem with the two bounds above". Introduction.tex:106-107 says the same: "Interpolation and summation over annuli yield Theorem ". | sections/introduction.tex:16 | |
| info | structure_uncertain | Appendix D (sections/planar.tex) gives an alternative proof of the n=2 case via a linear Dirichlet problem, reproving Proposition 5.1's inequalities for a planar energy; it has no statement of its own. | sections/introduction.tex:97 | |
| info | inferred_dep | critique: Prop-current's zero-divergence claim comes from the pushforward computation (current.tex:240-257), where "The first sum vanishes by Lemma ". Its proof (current.tex:329) identifies the column of , which is built from the tensor of that lemma. | sections/current.tex:313 | |
| info | inferred_dep | The proof relies on the Hessian calculus: "Green's formula ", the Schur energy identity, and the boundary restriction. These are in the Hessian calculus section (§3.1), which probably holds that lemma. | sections/energy.tex:200 | |
| info | inferred_dep | critique: The proof (transfer.tex:157) says "By , the j-th norm on the right is at most...". That estimate is the interpolation of Theorem (transfer.tex:88-94). | sections/transfer.tex:115 | |
| info | inferred_dep | "Both lower bounds" refers to this bound together with the earlier obstruction in the same section, Proposition 8.1 (the sharpness proposition). | sections/sharpness.tex:83 | |
| info | structure_uncertain | Proved in Appendix C (sections/weak.tex 15-124) without a proof environment. | sections/sharpness.tex:83 | |
| info | structure_uncertain | Statement carries two labels (cor:spherical-restriction and cor:cone-extension); references to either point to the same node. | sections/global-extension.tex:13 | |
| info | inferred_dep | critique: The lemma assumes . Its proof (kakeya.tex:166-167) explicitly says to "apply on the enclosing ball", so the dependency should be visible in the graph. | sections/kakeya.tex:133 | |
| info | inferred_dep | critique: The proof (consequences.tex:73-74) uses . Consequences.tex:34-35 introduces that estimate with "We rescale Theorem to a unit frequency annulus". | sections/consequences.tex:49 | |
| info | inferred_dep | critique: The proof (stein_square.tex:65-67) applies "the rescaled wave estimate ". That estimate is Theorem rescaled (consequences.tex:34-47). | sections/stein_square.tex:39 | |
| info | structure_uncertain | No proof environment; the proof is the prose of §B.3 (sections/angular.tex 276-404), concluding 'This proves Theorem B.1'. | sections/angular.tex:9 | |
| info | inferred_dep | "the unrestricted annular estimate gives both conclusions" when H>N^eta refers to the one-frequency-annulus estimate of Theorem 1.2 | sections/angular.tex:33 | |
| info | inferred_dep | critique: Angular.tex:235-236 reads "Finally and scaling give \|U(t)\Pi_N\|_{1\to\infty}\lesssim N^{\alpha+1}". That kernel expansion is established in the proof of Lemma (angular.tex:81-90). | sections/angular.tex:167 | |
| info | structure_uncertain | Proved in §§E.1-E.3 (sections/pencil.tex 50-299) without a proof environment; §E.4 is a separate lower-bound discussion, not part of the proof. | sections/pencil.tex:11 | |
| info | inferred_dep | outline: "Laplacian of X_0 equals wall flux minus terms" rests on claim-prop-energy-flux-1 | sections/energy.tex:250 | |
| info | inferred_dep | outline: "Laplacian of segment energy equals full determinant" rests on claim-prop-energy-flux-1 | sections/energy.tex:257 | |
| info | inferred_dep | outline: "Current and density bounds with wall Gaussian decay" rests on claim-prop-fixed-barrier-1 | sections/barrier.tex:66 | |
| info | inferred_dep | outline: "Hessian lower bound for comparison function rP_*" rests on claim-prop-fixed-barrier-1 | sections/barrier.tex:113 | |
| info | inferred_dep | outline: "Upper bound on boundary quotient u/r" rests on claim-prop-fixed-barrier-2 | sections/barrier.tex:129 | |
| info | inferred_dep | outline: "Upper bound on boundary quotient u/r" rests on claim-prop-fixed-barrier-3 | sections/barrier.tex:129 | |
| info | inferred_dep | outline: "Energy upper bound and wall flux bound" rests on claim-prop-fixed-barrier-1 | sections/barrier.tex:148 | |
| info | inferred_dep | outline: "Energy upper bound and wall flux bound" rests on claim-prop-fixed-barrier-2 | sections/barrier.tex:148 | |
| info | inferred_dep | outline: "Energy upper bound and wall flux bound" rests on claim-prop-fixed-barrier-4 | sections/barrier.tex:148 | |
| info | inferred_dep | outline: "Interior density lower bound by Gaussian inversion" rests on prop:current | sections/barrier.tex:170 | |
| info | inferred_dep | outline: "Interior density lower bound by Gaussian inversion" rests on claim-prop-fixed-barrier-2 | sections/barrier.tex:170 | |
| info | inferred_dep | outline: "Interior energy lower bound via ball quadratic" rests on claim-prop-fixed-barrier-6 | sections/barrier.tex:202 | |
| info | inferred_dep | outline: "Interior energy lower bound via ball quadratic" rests on lem:fiber-geometry | sections/barrier.tex:202 | |
| info | inferred_dep | outline: "Choose gamma, K; divide to get potential" rests on claim-prop-fixed-barrier-5 | sections/barrier.tex:233 | |
| info | inferred_dep | outline: "Choose gamma, K; divide to get potential" rests on claim-prop-fixed-barrier-7 | sections/barrier.tex:233 | |
| info | inferred_dep | outline: "Outward passage bounds band mass" rests on claim-lem-cylinder-1 | sections/propagation.tex:63 | |
| info | inferred_dep | outline: "Harnack lower bound on unit circle" rests on claim-lem-cylinder-2 | sections/propagation.tex:79 | |
| info | inferred_dep | outline: "Return passage to inner radius" rests on claim-lem-cylinder-1 | sections/propagation.tex:94 | |
| info | inferred_dep | outline: "Return passage to inner radius" rests on claim-lem-cylinder-3 | sections/propagation.tex:94 | |
| info | inferred_dep | outline: "Logarithmic volume bound for neighborhoods" rests on claim-cor-quant-kakeya-1 | sections/kakeya.tex:203 | |
| info | inferred_dep | outline: "Infinite gauge Hausdorff measure via Keich grouping" rests on claim-cor-quant-kakeya-1 | sections/kakeya.tex:211 | |
| info | inferred_dep | outline: "Shell bounds summed over radius intervals" rests on claim-thm-stein-square-2 | sections/stein_square.tex:204 | |
| info | inferred_dep | outline: "Limiting argument and extension to Lp" rests on claim-thm-stein-square-3 | sections/stein_square.tex:233 | |
| info | inferred_dep | outline: "Limiting argument and extension to Lp" rests on claim-thm-stein-square-4 | sections/stein_square.tex:233 | |
| info | inferred_dep | outline: "Global mixed paraboloid extension by epsilon removal" rests on claim-cor-parabolic-1 | sections/parabolic.tex:72 | |
| info | inferred_dep | outline: "Near contribution via Fresnel formula; critical estimate" rests on claim-cor-parabolic-1 | sections/parabolic.tex:100 | |
| info | inferred_dep | outline: "Near contribution via Fresnel formula; critical estimate" rests on claim-cor-parabolic-3 | sections/parabolic.tex:100 | |
| info | inferred_dep | outline: "Mixed-norm Besov local smoothing" rests on claim-cor-parabolic-2 | sections/parabolic.tex:131 | |
| info | inferred_dep | outline: "Barriers, gradient bound, Hessian reduced to boundary" rests on claim-thm-dirichlet-1 | sections/dirichlet.tex:92 | |
| info | inferred_dep | outline: "Mixed boundary derivatives via quaternionic rigid motion" rests on claim-thm-dirichlet-2 | sections/dirichlet.tex:112 | |
| info | inferred_dep | outline: "Mixed boundary derivatives via quaternionic rigid motion" rests on lem:hessian-calculus | sections/dirichlet.tex:112 | |
| info | inferred_dep | outline: "Double normal derivative via Schur; full C2 bound" rests on claim-thm-dirichlet-3 | sections/dirichlet.tex:150 | |
| info | inferred_dep | outline: "Double normal derivative via Schur; full C2 bound" rests on lem:hessian-calculus | sections/dirichlet.tex:150 | |
| info | inferred_dep | outline: "Evans-Krylov C^{2,alpha} estimate" rests on claim-thm-dirichlet-4 | sections/dirichlet.tex:171 | |
| info | inferred_dep | outline: "Closedness and smoothness up to boundary" rests on claim-thm-dirichlet-5 | sections/dirichlet.tex:196 | |
| info | inferred_dep | outline: "Descent to three slots and parameter dependence" rests on claim-thm-dirichlet-1 | sections/dirichlet.tex:203 | |
| info | inferred_dep | outline: "Descent to three slots and parameter dependence" rests on claim-thm-dirichlet-6 | sections/dirichlet.tex:203 | |
| info | inferred_dep | outline: "Shell profile estimate independent of radius" rests on claim-lem-angular-ball-1 | sections/angular.tex:103 | |
| info | inferred_dep | outline: "Summation over shells and remainder kernel" rests on claim-lem-angular-ball-2 | sections/angular.tex:139 | |
| info | inferred_dep | outline: "Littlewood-Paley setup and small-source removal" rests on lem:angular-envelope | sections/angular.tex:278 | |
| info | inferred_dep | outline: "Selected low-pass via envelope and vector maximal" rests on lem:angular-envelope | sections/angular.tex:308 | |
| info | inferred_dep | outline: "Selected low-pass via envelope and vector maximal" rests on claim-thm-angular-1 | sections/angular.tex:308 | |
| info | inferred_dep | outline: "Strong and weak bounds for selected pieces" rests on claim-thm-angular-3 | sections/angular.tex:337 | |
| info | inferred_dep | outline: "Strong and weak bounds for selected pieces" rests on prop:angular-band | sections/angular.tex:337 | |
| info | inferred_dep | outline: "Strong bound for unselected angular bands" rests on thm:main | sections/angular.tex:358 | |
| info | inferred_dep | outline: "Strong bound for unselected angular bands" rests on lem:angular-ball | sections/angular.tex:358 | |
| info | inferred_dep | outline: "Strong bound for unselected angular bands" rests on claim-thm-angular-2 | sections/angular.tex:358 | |
| info | inferred_dep | outline: "Reassembly by synthesis operator and interpolation" rests on claim-thm-angular-4 | sections/angular.tex:389 | |
| info | inferred_dep | outline: "Reassembly by synthesis operator and interpolation" rests on claim-thm-angular-5 | sections/angular.tex:389 | |
| info | inferred_dep | outline: "Reassembly by synthesis operator and interpolation" rests on claim-thm-angular-2 | sections/angular.tex:389 | |
| info | inferred_dep | outline: "Modulated packets transport along tubes" rests on claim-prop-weak-obstruction-1 | sections/weak.tex:62 | |
| info | inferred_dep | outline: "Random signs give weak-norm lower bound" rests on claim-prop-weak-obstruction-1 | sections/weak.tex:102 | |
| info | inferred_dep | outline: "Random signs give weak-norm lower bound" rests on claim-prop-weak-obstruction-2 | sections/weak.tex:102 | |
| info | inferred_dep | outline: "Fiber chart and geometric bounds" rests on claim-thm-pencil-1 | sections/pencil.tex:80 | |
| info | inferred_dep | outline: "Fiber chart and geometric bounds" rests on lem:hessian-calculus | sections/pencil.tex:80 | |
| info | inferred_dep | outline: "Conserved current and Gaussian multiplier symbols" rests on lem:current-flow | sections/pencil.tex:135 | |
| info | inferred_dep | outline: "Conserved current and Gaussian multiplier symbols" rests on claim-thm-pencil-2 | sections/pencil.tex:135 | |
| info | inferred_dep | outline: "Determinant energy and balance on pencil fibers" rests on claim-thm-pencil-3 | sections/pencil.tex:190 | |
| info | inferred_dep | outline: "Determinant energy and balance on pencil fibers" rests on thm:dirichlet | sections/pencil.tex:190 | |
| info | inferred_dep | outline: "Determinant energy and balance on pencil fibers" rests on lem:cofactor-divergence | sections/pencil.tex:190 | |
| info | inferred_dep | outline: "Determinant energy and balance on pencil fibers" rests on prop:energy-flux | sections/pencil.tex:190 | |
| info | inferred_dep | outline: "Determinant energy and balance on pencil fibers" rests on lem:energy-comparison | sections/pencil.tex:190 | |
| info | inferred_dep | outline: "Wall decay and interior inversion estimates" rests on claim-thm-pencil-3 | sections/pencil.tex:213 | |
| info | inferred_dep | outline: "Wall decay and interior inversion estimates" rests on prop:fixed-barrier | sections/pencil.tex:213 | |
| info | inferred_dep | outline: "Energy bounds, potential inequality, torus cap estimate" rests on claim-thm-pencil-4 | sections/pencil.tex:249 | |
| info | inferred_dep | outline: "Energy bounds, potential inequality, torus cap estimate" rests on claim-thm-pencil-5 | sections/pencil.tex:249 | |
| info | inferred_dep | outline: "Energy bounds, potential inequality, torus cap estimate" rests on prop:fixed-barrier | sections/pencil.tex:249 | |
| info | inferred_dep | outline: "Energy bounds, potential inequality, torus cap estimate" rests on lem:cylinder | sections/pencil.tex:249 | |
| info | inferred_dep | outline: "Energy bounds, potential inequality, torus cap estimate" rests on prop:periodic-cap | sections/pencil.tex:249 | |
| info | inferred_dep | outline: "Remove period by scaling; smallest eigenvalue case" rests on claim-thm-pencil-6 | sections/pencil.tex:283 | |
| info | inferred_dep | outline: "Remove period by scaling; smallest eigenvalue case" rests on thm:main | sections/pencil.tex:283 |