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Every fixed finite digit window eventually reads only zeros.","source_text":"Every $a_k$ is nonzero. Every fixed finite digit window eventually reads only zeros.","position":[7.0,-0.15],"line":101}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55],"reading_aid":{"alt":"A finite condition that is not preserved at the limit","description":"Rows are successive powers of three, each with one nonzero digit. That digit moves beyond any fixed finite window, illustrated by the dashed box. Every displayed integer is nonzero, but the sequence converges to zero. 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Passing to absolute values must retain the real sign.","source_text":"The ideal norm is positive. Passing to absolute values must retain the real sign.","position":[7.0,-0.15],"line":30}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11],"reading_aid":{"alt":"Signed field norms versus positive ideal norms","description":"The exact examples retain both the sign of the element norm and the corresponding ideal data. 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Every nonzero element is a unit.","source_text":"All 25 elements are shown. Every nonzero element is a unit.","position":[7.0,-0.1],"line":68}],"node_order":[0,31,32,33,34,35,26,1,2,3,4,5,27,6,7,8,9,10,28,11,12,13,14,15,29,16,17,18,19,20,30,21,22,23,24,25,36,37,38,39,40],"reading_aid":{"alt":"A finite-field norm table from multiplication determinants","description":"Rows and columns give coordinates in the quadratic quotient algebra over the field with five elements. Each cell is the exact norm residue. The displayed multiplication matrix explains how its determinant computes that norm.","blocks":[{"kind":"text","nodes":[0]},{"kind":"table","caption":"Drawing grid 1 · rows top to bottom, columns left to right","headers":["Row / column",31,32,33,34,35],"rows":[[26,1,2,3,4,5],[27,6,7,8,9,10],[28,11,12,13,14,15],[29,16,17,18,19,20],[30,21,22,23,24,25]]},{"kind":"text","nodes":[36,37,38,39,40]}]}},"23_mean_cancellation":{"name":"23_mean_cancellation","plate":23,"source":"papers/paper-x/current/visual_appendix/graphics/23_mean_cancellation.tex","source_sha256":"86f7c6ffab5d4a47319dda811b4ab4ac8ad3c8a96cb3e42abe4e1f7a9a3dcd7a","image":"assets/atlas/paper-x/23_mean_cancellation.png","image_sha256":"e4572828faa3ac730ca39c43466ef44886cda1b69eee90a2aacffa882107226c","pdf_page":26,"crop_pdf_points":[74,197,525,365],"nodes":[{"text":"$(0,1)$","source_text":"$(0,1)$","position":[3.84,2.45],"line":5},{"text":"$(4,0)$","source_text":"$(4,0)$","position":[3.359655,3.609655],"line":8},{"text":"$(0,3)$","source_text":"$(0,3)$","position":[2.2,4.09],"line":11},{"text":"$(2,0)$","source_text":"$(2,0)$","position":[1.040345,3.609655],"line":14},{"text":"$(0,4)$","source_text":"$(0,4)$","position":[0.56,2.45],"line":17},{"text":"$(1,0)$","source_text":"$(1,0)$","position":[1.040345,1.290345],"line":20},{"text":"$(0,2)$","source_text":"$(0,2)$","position":[2.2,0.81],"line":23},{"text":"$(3,0)$","source_text":"$(3,0)$","position":[3.359655,1.290345],"line":26},{"text":"sum $=0$","source_text":"sum $=0$","position":[2.2,2.45],"line":27},{"text":"$(1,1)$","source_text":"$(1,1)$","position":[8.49,2.45],"line":30},{"text":"$(4,2)$","source_text":"$(4,2)$","position":[8.009655,3.609655],"line":33},{"text":"$(3,3)$","source_text":"$(3,3)$","position":[6.85,4.09],"line":36},{"text":"$(2,1)$","source_text":"$(2,1)$","position":[5.690345,3.609655],"line":39},{"text":"$(4,4)$","source_text":"$(4,4)$","position":[5.21,2.45],"line":42},{"text":"$(1,3)$","source_text":"$(1,3)$","position":[5.690345,1.290345],"line":45},{"text":"$(2,2)$","source_text":"$(2,2)$","position":[6.85,0.81],"line":48},{"text":"$(3,4)$","source_text":"$(3,4)$","position":[8.009655,1.290345],"line":51},{"text":"sum $=0$","source_text":"sum $=0$","position":[6.85,2.45],"line":52},{"text":"$(1,2)$","source_text":"$(1,2)$","position":[13.14,2.45],"line":55},{"text":"$(3,2)$","source_text":"$(3,2)$","position":[12.659655,3.609655],"line":58},{"text":"$(3,1)$","source_text":"$(3,1)$","position":[11.5,4.09],"line":61},{"text":"$(4,1)$","source_text":"$(4,1)$","position":[10.340345,3.609655],"line":64},{"text":"$(4,3)$","source_text":"$(4,3)$","position":[9.86,2.45],"line":67},{"text":"$(2,3)$","source_text":"$(2,3)$","position":[10.340345,1.290345],"line":70},{"text":"$(2,4)$","source_text":"$(2,4)$","position":[11.5,0.81],"line":73},{"text":"$(1,4)$","source_text":"$(1,4)$","position":[12.659655,1.290345],"line":76},{"text":"sum $=0$","source_text":"sum $=0$","position":[11.5,2.45],"line":77},{"text":"Multiply by $v=2u$. Its norm is $2$, so $\\chi(N(vz))=-\\chi(N(z))$.","source_text":"Multiply by $v=2u$. Its norm is $2$, so $\\chi(N(vz))=-\\chi(N(z))$.","position":[7.0,0.1],"line":78},{"text":"Filled node: $+1$. Open node: $-1$. The omitted zero element has weight $0$.","source_text":"Filled node: $+1$. Open node: $-1$. The omitted zero element has weight $0$.","position":[7.0,-0.4],"line":79}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],"reading_aid":{"alt":"Three nonzero orbits with alternating character weights","description":"Multiplication by the displayed unit cycles through each eight-element orbit. Filled and open vertices have weights plus and minus one. Each orbit sums to zero; the omitted zero element has weight zero. 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Each constraint belongs to this explicit finite model.","blocks":[{"kind":"table","caption":"Drawing grid 1 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2","Column 3","Column 4","Column 5","Column 6"],"rows":[["Row 1",0,1,2,3,4,5],["Row 2",6,7,8,9,10,11],["Row 3",12,13,14,15,16,17],["Row 4",18,19,20,21,22,23],["Row 5",24,25,26,27,28,29]]},{"kind":"text","nodes":[30,31,32,33,34,35]}]}},"26_boundary_count":{"name":"26_boundary_count","plate":26,"source":"papers/paper-x/current/visual_appendix/graphics/26_boundary_count.tex","source_sha256":"f6ae9da5859e30abbd6c3c84ae8dff27a5e9c395b6381583dbeb2e970cd88494","image":"assets/atlas/paper-x/26_boundary_count.png","image_sha256":"16b9db117e1f6154f0f90f3d471210797a2db3f67a1cbf3bb0e44be67c5803d9","pdf_page":29,"crop_pdf_points":[106,182,463,358],"nodes":[{"text":"$[0,12)^2\\cap\\mathbb Z^2$","source_text":"$[0,12)^2\\cap\\mathbb Z^2$","position":[3.0,4.4],"line":286},{"text":"four full $5\\times5$ cells","source_text":"four full $5\\times5$ cells","position":[9.8,3.75],"line":287},{"text":"$100$ bulk points: sum $0$","source_text":"$100$ bulk points: sum $0$","position":[9.8,2.85],"line":288},{"text":"$44$ boundary points","source_text":"$44$ boundary points","position":[9.8,1.95],"line":289},{"text":"actual total $S_{12}=1$","source_text":"actual total $S_{12}=1$","position":[9.8,1.05],"line":290},{"text":"$|S_L|\\le L^2-(L-(L\\bmod5))^2$ for this square-lattice model.","source_text":"$|S_L|\\le L^2-(L-(L\\bmod5))^2$ for this square-lattice model.","position":[7.0,-0.1],"line":291}],"node_order":[0,1,2,3,4,5],"reading_aid":{"alt":"Complete residue tiles and the remaining boundary strips","description":"Four complete five-by-five tiles fill a ten-by-ten square inside the displayed twelve-by-twelve box. 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Earlier scalar/Mellin checkpoints are historical."},{"number":2,"title":"An atlas of the objects","category":"NOTATION BEFORE PICTURES","locator":"Main text: Definitions 1.1, 5.1, 6.1, 7.1--7.2, and 8.1."},{"number":3,"title":"Higher digits cannot change the root","category":"RESIDUAL PARTITION","locator":"Main text: Theorems 2.1--2.2 and Corollary 2.3."},{"number":4,"title":"A branch fixes digits permanently","category":"INVERSE-LIMIT RECONSTRUCTION","locator":"Main text: Theorem 3.2 and the compatible-digit example in §3."},{"number":5,"title":"A smooth chart has independent choices","category":"ACTUAL FINITE POINTS","locator":"Main text: Proposition 3.4 and Theorem 6.2."},{"number":6,"title":"Weights delay visibility","category":"WEIGHTED DYADIC CHARTS","locator":"Main text: Proposition 5.3, Theorem 6.3, and §12.2."},{"number":7,"title":"Many approximate roots, one limit","category":"SINGULAR POWER SEEDS","locator":"Main text: Theorem 6.4 and Corollary 6.6."},{"number":8,"title":"One equation, several local outcomes","category":"A SCALAR COMPANION MODEL","locator":"Main text: Theorem 8.2, Corollary 8.3, and Example 8.4."},{"number":9,"title":"Why finiteness enters compactness","category":"AN INFINITE-LEVEL COUNTERMODEL","locator":"Main text: Theorem 3.3 and its finite-branching hypotheses."},{"number":10,"title":"See cycles, boundaries, and classes","category":"A TWO-PACKET COCHAIN COMPLEX","locator":"Main text: Theorem 4.2 and the finite-complex example."},{"number":11,"title":"An arbitrary cut need not descend","category":"THE CHAIN-MAP CONDITION","locator":"Main text: Theorem 4.2 and the scope of the cohomology splitting."},{"number":12,"title":"How many starting seeds are necessary?","category":"DIRECTED TRANSPORT","locator":"Main text: Theorem 7.3 and the seven-root example."},{"number":13,"title":"A transport must work at every depth","category":"ONE REVERSIBLE EXAMPLE","locator":"Main text: Definition 7.1 and the reduction-compatibility requirement."},{"number":14,"title":"Common coefficients, different residues","category":"THE ARITHMETIC SPINE","locator":"Main text: Definition 8.1 and Theorem 8.2."},{"number":15,"title":"A finite trace matrix can separate types","category":"CHARACTERS BEFORE ARITHMETIC PRIMES","locator":"Main text: Theorem 9.1; retained scalar/Mellin continuation: arithmetic-system separation."},{"number":16,"title":"A solution is not yet a summand","category":"INTEGRAL, NONNEGATIVE, WITHIN CAPACITY","locator":"Main text: Theorem 9.2 and its multiplicity-recovery example."},{"number":17,"title":"Separation can consume a digit","category":"EXACT PRECISION LOSS","locator":"Main text: The finite trace-recovery discussion in §9."},{"number":18,"title":"Pairwise occupancy can still fail jointly","category":"EXACT JOINT PROJECTORS","locator":"Main text: Theorem 10.1 and the finite occupancy models."},{"number":19,"title":"The hypotheses do real work","category":"TRACE AND SEMISIMPLICITY","locator":"Main text: Theorem 10.1 and the checked false controls following it."},{"number":20,"title":"A condition must survive the limit","category":"CLOSEDNESS IS NOT DECORATION","locator":"Main text: Theorems 8.2 and 11.1; the inverse-limit closure hypotheses."},{"number":21,"title":"The sign of a norm is arithmetic data","category":"ELEMENTS VERSUS IDEALS","locator":"Main text: §9, arithmetic residue-counting continuation; signed-norm comparison."},{"number":22,"title":"A norm is a determinant you can compute","category":"THE QUOTIENT ALGEBRA $\\FF_{25}$","locator":"Main text: §9; retained scalar/Mellin continuation: actual quotient determinant norms."},{"number":23,"title":"Permutation forces a zero mean","category":"A NONSCALAR UNIT WITNESS","locator":"Main text: §9; retained scalar/Mellin continuation: norm-residue lattice mean cancellation."},{"number":24,"title":"A nontrivial character may still be exceptional","category":"THE EXACT SCALAR TEST","locator":"Main text: §9; scalar/Mellin checkpoint: scalar-power and nonscalar-witness boundaries."},{"number":25,"title":"Avoid finitely many primes constructively","category":"A CRT MODEL FOR REPRESENTATIVE CHOICE","locator":"Main text: §9; retained scalar/Mellin continuation: finite-prime avoidance and automatic representatives."},{"number":26,"title":"Cancellation leaves a boundary problem","category":"A FULL LATTICE-BOX CALCULATION","locator":"Main text: §9; retained scalar/Mellin continuation: additive residue counts and norm-character bounds."},{"number":27,"title":"What a power-saving bound buys analytically","category":"A MELLIN ENDPOINT MODEL","locator":"Main text: §9; retained scalar/Mellin continuation: summatory measurability and first-order Mellin control."},{"number":28,"title":"Finite splitting data are not a density theorem","category":"THE REMAINING FROBENIUS STEP","locator":"Main text: Theorem 9.1. Latest continuation: GeneralArithmeticCoverage and ArithmeticSystemSeparation."},{"number":29,"title":"Changing coefficients changes the problem","category":"SCALAR ABSENCE VERSUS REPRESENTATION EXISTENCE","locator":"Main text: Example 8.4 and the declared coefficient categories in §§8--9."},{"number":30,"title":"Read the evidence at its actual scope","category":"FINAL READER'S CHECKLIST","locator":"Main text: Appendix D. Formal guide: statement register, dependency sources, and replay boundary."}],"manuscript_targets":"Stable source labels reconstructed with the Paper X manuscript reader","scope":"Historical 17/20, checked conditional 20/20, and later unpromoted 19/20 contracts remain distinct"}}