Proof lineage
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arXiv:2607.17021v1
A statement-restricted graph for the proof-critical sources behind the homogeneous-dynamics thickness theorem, its weighted Diophantine application, and the paper's separately marked ancillary claims. Contextual citations and results reproved in full are excluded.
48 nodes · 53 proof-critical links
Open dependence graph →Uniform Distribution Theory 17(1), 89–104 (2022) · arXiv:2107.08084v1
A statement-restricted graph for the winning, hyperplane-absolute-winning, exponent-bound, and best-approximation claims in the published paper. Historical and metric-context citations that do not enter those proofs are excluded.
11 nodes · 13 proof-critical links
Open dependence graph →arXiv:2503.21180v3
A statement-restricted graph for the transference, second-moment, lacunary escaping-set, and product-dimension inputs used in the paper. Historical attributions, contextual literature, and arguments reproduced in the focal paper or an included source are excluded; published books are terminal.
7 nodes · 6 proof-critical links
Open dependence graph →arXiv:2504.02258v1
Statement-restricted proof-dependence graph for the submanifold genericity theorems, the multiplicative Dani correspondence, the full-measure half of the Khintchine-type theorem, and Proposition 1.7. It includes the uncited cusp-volume input required on p. 28, its independent Schmidt 1957 proof repair, and two source-use failures: BEG20 is applied after dropping its strong-spectral-gap hypothesis, and BG23 is cited with the wrong theorem number.
33 nodes · 38 proof-critical links
Open dependence graph →arXiv:2504.19329v4
A statement-restricted graph for the proved $D$-module, $q$-difference, and quantum-cluster assertions in arXiv:2504.19329v4. It follows every proof-critical non-book source recursively and stops at books or monographs. Historical antecedents that are reproved in the focal paper, definitional pointers, alternative citations, and the explicitly conjectural symplectic-singularity discussion are excluded.
15 nodes · 16 proof-critical links
Open dependence graph →arXiv:1501.01260v4
Bounded statement-restricted first-order citation graph for arXiv:1501.01260v4. It has one focal node and fourteen selected direct ancestor nodes. It is not a full recursive closure: selected source proof citations are unexpanded/unverified, and superseded Gies is excluded because the focal manuscript re-proves its lemma. It does not repair T1.1.
15 nodes · 23 proof-critical links
Open dependence graph →arXiv:2508.08440v2
A statement-restricted proof-dependence graph for arXiv:2508.08440v2. It covers the paper's main analytic continuation and convergence theorems, exact $q$-Bessel formulas, monotonicity and almost-everywhere singularity, the unit-circle Jones/Burau bounds, the mod-$m$ classification results, and the hypergeometric construction of $q$-complex numbers. Expansion follows only claims whose truth can affect one of those marked results and stops at published books. Historical citations, comparisons, conjectures, and alternate routes are documented separately as non-edges.
15 nodes · 15 proof-critical links
Open dependence graph →arXiv:2506.00941v2
Statement-restricted proof-dependence graph for the projection formula, the braid, graphical and natural-unit-interval interpretations, and the type-B signed-source-component extension. It traces the two independent projection sources and the signed-graph arrangement inputs to exact published claims, while stopping at published books and monographs. Context, comparison, historical citations and results reproved internally are excluded. The type-A and natural-unit-interval results check, but an exact rank-two counterexample disproves the type-B projection and coefficient branch.
14 nodes · 16 proof-critical links
Open dependence graph →arXiv:2603.24885v1
Statement-restricted proof-dependence graph for the contribution-one theorem and its tree-contribution consequences. The graph follows the exact face-bijection input through Bernardi's region bijection and its counting step. It excludes background, historical citations, independently sufficient alternative sources, and results reproved inside the focal paper.
5 nodes · 5 proof-critical links
Open dependence graph →arXiv:2603.24886v1
A statement-restricted dependence graph for the right-inverse construction, the bijectivity criterion for (m, epsilon)-arrangements, and the necessity theorem inside the transitive class. Only external results that can affect those marked claims are included. Historical comparisons, alternative proofs, and citations used only for context are excluded; expansion stops at published books and monographs.
4 nodes · 3 proof-critical links
Open dependence graph →Results in Mathematics 80(8), 236 (2025) · arXiv:2401.00614v3
A statement-restricted graph for arXiv:2401.00614v3. It traces the uniform-distribution, Schmidt-game, Hausdorff-dimension, and Borel-determinacy results actually imported by the v3 proofs. Theorem 2.6 is marked self-contained. Contextual citations, an earlier bound that is not used, and the different determinacy citation added only in the later journal version are excluded.
6 nodes · 6 proof-critical links
Open dependence graph →arXiv:2603.25988v1
A statement-restricted graph for the total-density criteria, their uniform-approximation consequences, the convex-hull criterion, and the separately marked optimality branch. Context, historical attributions, alternative proof sources, and arguments reproduced inside the focal paper are excluded.
4 nodes · 3 proof-critical links
Open dependence graph →Mathematika 72(2), e70089 (2026) · arXiv:2508.01912v1
A statement-restricted graph for the zero–full measure theorem, its weighted transference inputs, and the separately marked second-moment and geometry-of-numbers branches. Contextual citations, analogies, alternative routes, and sources whose needed result is reproved in the focal paper are excluded.
18 nodes · 19 proof-critical links
Open dependence graph →arXiv:2412.05858v1
A statement-restricted proof-dependence graph for arXiv:2412.05858v1. It covers Theorems 1–5, Corollary 1, and Lemmas 1–4; recursively follows every proof-critical non-book result; and stops only at disclosed published books or edited/seminar volumes. Corollary 1 is verified only for finite positive critical limsup: its endpoint assertions at zero and infinity do not follow from Theorem 3 and formula (5). The Dani 1981 branch retains the Garland–Raghunathan article and Kazhdan–Margulis input. The latter is expanded through the exact Borel–Tits, Borel, and Rosenlicht articles, preserving an insufficient Rosenlicht citation alongside its Hartshorne repair, and recursively through the Weil and Lang–Weil article chain rather than replacing historical sources with retrospective books.
39 nodes · 55 proof-critical links
Open dependence graph →arXiv:2608.24401v2
A full recursive graph for the proof-critical sources behind arXiv:2608.24401v2: the ambient and fixed-sequence HAW theorems, curve and line restrictions, dynamical formulations, and the two active appendix examples. Contextual citations and the inactive source block are excluded; exact-source limitations and terminal books are shown on the affected branches.
67 nodes · 83 proof-critical links
Open dependence graph →arXiv:2605.16028v1
Full statement-restricted recursive proof-dependence graph for arXiv:2605.16028v1. Theorem 3.1 and its main enumerative corollaries are proved locally and independently verified. The equivalence-class family in Theorem 4.5 is not correct under the literal Section 2 digraph definitions: an arc-set cocycle cannot contain a 0-way edge, while reciprocal arcs of one 2-way nonloop edge form a digraph 2-cycle. Replacing those notions by the standard underlying-edge-simple potential cocycle/cycle notions makes Definition 4.4 meaningful and verifies the intended theorem. The chosen proof chain for Fact 4.10 uses Backman's graph-specific, self-contained cut/cycle decomposition; the independently sufficient BBY and Ding routes were inspected but are excluded as alternative proofs rather than drawn as parallel mandatory parents. Flajolet–Sedgewick is the terminal book source for Corollary 3.7. Historical, terminology, comparison, and locally reproved citations are excluded. Example 3.4 additionally needs X and Y nonempty, and two informal specializations need G connected.
3 nodes · 2 proof-critical links
Open dependence graph →arXiv:2506.19060v1
A statement-restricted recursive graph for the matrix reformulation of Waldschmidt's conjecture, the uniform inhomogeneous BA/VWA criteria, the abelian-variety exponent bound, the rank-one theorem, and ancillary isogeny invariance in Fishman–Lambert–Merrill–Simmons. The graph distinguishes externally rescued statements from defects in the posted proofs: a false local-minima lemma, a missing cutoff that makes the displayed exceptional set false, an invalid reverse winning-game proof, an isogeny-inequality sign error, and several repairable omissions.
18 nodes · 22 proof-critical links
Open dependence graph →arXiv:2609.36411v1
Full statement-restricted proof-dependence graph for arXiv:2609.36411v1. The fixed-matrix zero-one theorem, successive-minima formulation, Lagrange and bad-target classifications, Hausdorff consequences, scalar comparison, and the full quasi-independence chain were checked. The proof-critical external branches close at Henk's lattice estimates and their terminal geometry-of-numbers background, Dani's published correspondence theorems, the general Beresnevich-Velani mass transference theorem, the exact Fuchs-Kim and Beresnevich-Haynes-Velani scalar estimates, and three terminal books for Mahler compactness, generalized Borel-Cantelli, and Hausdorff-Cantelli. Historical, motivational, comparison-only, and results fully reproved in the focal manuscript are excluded.
10 nodes · 9 proof-critical links
Open dependence graph →arXiv:2609.36412v1
A statement-restricted recursive graph for the proof-critical inputs behind the pressure formula, the no-dimension-drop theorem, the Hausdorff-measure zero-full law, the sharp escape-of-mass construction, and the transference applications. Contextual comparisons, historical attributions, and results merely recovered by the focal paper are excluded. Books are terminal sources.
12 nodes · 11 proof-critical links
Open dependence graph →