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The table gives rational evaluations, showing the mismatch introduced by the chosen transport and rule.","blocks":[{"kind":"text","nodes":[0,1,2]},{"kind":"table","caption":"Drawing grid 1 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2","Column 3"],"rows":[[3,4,5,6],[7,8,9,10],[11,12,13,14],[15,16,17,18]]},{"kind":"text","nodes":[19]}]}},"bilinear_vs_nonlinear":{"name":"bilinear_vs_nonlinear","plate":22,"source":"papers/paper-iii/current/manuscript/visual_graphics/bilinear_vs_nonlinear.tex","source_sha256":"acdc8acc7ded63fe7cbd2f8aad0e2e06db7973a517a2f0da0c8f2f832ffc6b40","image":"assets/atlas/paper-iii/bilinear_vs_nonlinear.png","image_sha256":"0c1acd3d83972fe309981591aa5f7eff95adcfd3b72950404af0e7e0c0bb1d3f","pdf_page":24,"crop_pdf_points":[55,188,557,377],"nodes":[{"text":"$ac=10$","source_text":"$ac=10$","position":[0.65,1.125],"line":3},{"text":"$ad=14$","source_text":"$ad=14$","position":[0.65,2.625],"line":5},{"text":"$bc=15$","source_text":"$bc=15$","position":[2.275,1.125],"line":7},{"text":"$bd=21$","source_text":"$bd=21$","position":[2.275,2.625],"line":9},{"text":"$(2+3)(5+7)=60$","source_text":"$(2+3)(5+7)=60$","position":[1.62,4.25],"line":10},{"text":"$a=2$","source_text":"$a=2$","position":[0.65,0.15],"line":11},{"text":"$b=3$","source_text":"$b=3$","position":[2.27,0.15],"line":12},{"text":"$c=5$","source_text":"$c=5$","position":[-0.25,1.1],"line":13},{"text":"$d=7$","source_text":"$d=7$","position":[-0.25,2.62],"line":14},{"text":"Diagonal pieces are not the whole product","source_text":"Diagonal pieces are not the whole product","position":[8.0,4.1],"line":16},{"text":"$ac+bd=10+21=31$","source_text":"$ac+bd=10+21=31$","position":[8.0,3.1],"line":17},{"text":"$ad+bc=14+15=29$","source_text":"$ad+bc=14+15=29$","position":[8.0,2.25],"line":18},{"text":"$(2+3)^2=25\\ne2^2+3^2=13$","source_text":"$(2+3)^2=25\\ne2^2+3^2=13$","position":[8.0,1.1],"line":19},{"text":"missing square term: $2\\cdot2\\cdot3=12$","source_text":"missing square term: $2\\cdot2\\cdot3=12$","position":[8.0,0.15],"line":20}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],"reading_aid":{"alt":"All four terms of a bilinear expansion","description":"The rectangle splits into four products with separately scaled horizontal and vertical units. 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Its final displacement supplies the named nonzero packet witness.","blocks":[{"kind":"text","nodes":[0]},{"kind":"table","caption":"Drawing grid 1 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2"],"rows":[["Row 1",1,2],["Row 2",3,4]]},{"kind":"text","nodes":[5]},{"kind":"table","caption":"Drawing grid 2 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2"],"rows":[["Row 1",6,7],["Row 2",8,9]]},{"kind":"text","nodes":[10]},{"kind":"table","caption":"Drawing grid 3 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2"],"rows":[["Row 1",11,12],["Row 2",13,14]]},{"kind":"text","nodes":[15]},{"kind":"table","caption":"Drawing grid 4 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2"],"rows":[["Row 1",16,17],["Row 2",18,19]]},{"kind":"text","nodes":[20,21,22]}]}}},"included_sources":{},"manuscript_references":{"1":[["Section 1","read/paper-iii/finite-coordinates.html#L5"],["Section 4","read/paper-iii/finite-coordinates.html#L372"],["Section 5","read/paper-iii/connections-and-obstructions.html#L1"]],"2":[["Definition 2.1","read/paper-iii/finite-coordinates.html#L64"],["Definition 5.5","read/paper-iii/connections-and-obstructions.html#L116"],["Definition 7.4","read/paper-iii/connections-and-obstructions.html#L469"]],"3":[["Lemma 2.2","read/paper-iii/finite-coordinates.html#L96"],["Theorem 2.3","read/paper-iii/finite-coordinates.html#L103"],["Theorem 2.4","read/paper-iii/finite-coordinates.html#L121"]],"4":[["Theorem 3.8","read/paper-iii/finite-coordinates.html#L285"],["Corollary 3.9","read/paper-iii/finite-coordinates.html#L320"]],"5":[["Lemma 3.7","read/paper-iii/finite-coordinates.html#L254"],["Section 9","read/paper-iii/connections-and-obstructions.html#L661"]],"6":[["Corollary 3.9","read/paper-iii/finite-coordinates.html#L320"],["Corollary 3.10","read/paper-iii/finite-coordinates.html#L341"]],"7":[["Corollary 3.11","read/paper-iii/finite-coordinates.html#L351"]],"8":[["Proposition 8.1","read/paper-iii/connections-and-obstructions.html#L607"]],"9":[["Theorem 4.1","read/paper-iii/finite-coordinates.html#L386"],["Theorem 4.2","read/paper-iii/finite-coordinates.html#L407"]],"10":[["Proposition 10.1","read/paper-iii/connections-and-obstructions.html#L711"]],"11":[["Definition 5.1","read/paper-iii/connections-and-obstructions.html#L10"],["Definition 5.5","read/paper-iii/connections-and-obstructions.html#L116"]],"12":[["Theorem 5.10","read/paper-iii/connections-and-obstructions.html#L207"],["Section 9","read/paper-iii/connections-and-obstructions.html#L661"]],"13":[["Definition 5.9","read/paper-iii/connections-and-obstructions.html#L189"],["Definition 7.4","read/paper-iii/connections-and-obstructions.html#L469"]],"14":[["Theorem 6.2","read/paper-iii/connections-and-obstructions.html#L325"]],"15":[["Theorem 6.3","read/paper-iii/connections-and-obstructions.html#L346"]],"16":[["Theorem 7.5","read/paper-iii/connections-and-obstructions.html#L490"],["Corollary 7.6","read/paper-iii/connections-and-obstructions.html#L549"]],"17":[["Theorem 7.5","read/paper-iii/connections-and-obstructions.html#L490"]],"18":[["Proposition 7.7","read/paper-iii/connections-and-obstructions.html#L573"]],"19":[["Theorem 7.2","read/paper-iii/connections-and-obstructions.html#L419"]],"20":[["Definition 11.1","read/paper-iii/connections-and-obstructions.html#L768"],["Theorem 11.2","read/paper-iii/connections-and-obstructions.html#L793"]],"21":[["Proposition 10.1","read/paper-iii/connections-and-obstructions.html#L711"],["Formalization target 11.7","read/paper-iii/connections-and-obstructions.html#L900"],["Theorem 20.2","read/paper-iii/nonlinear-coordinates.html#L748"]],"22":[["Formalization target 11.6","read/paper-iii/connections-and-obstructions.html#L868"],["Theorem 18.3","read/paper-iii/nonlinear-coordinates.html#L517"]],"23":[["Theorem 5.10","read/paper-iii/connections-and-obstructions.html#L207"],["Theorem 6.2","read/paper-iii/connections-and-obstructions.html#L325"],["Corollary 7.6","read/paper-iii/connections-and-obstructions.html#L549"]],"24":[["Section 12","read/paper-iii/connections-and-obstructions.html#L919"]]},"new_proof_review":false,"new_lean_execution":false,"new_mathematical_example_execution":false,"new_tex_build":false,"edition_checks":{}}