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The table gives the six pairs and their recorded values; the parity criterion in the source explains when equality holds at every starting row.","math_labels":[],"tables":[{"caption":"Recorded mixed order-two examples","headers":["Steps","Baseline","Exact valuation","Exact difference"],"rows":[["(1, 1)",3,3,"72"],["(1, 2)",4,6,"-33728"],["(2, 2)",5,5,"60376672"],["(1, 4)",5,6,"-288354427328"],["(2, 4)",6,6,"3019098102225472"],["(3, 4)",5,10,"-60921822444127689728"]]}]},"metric":{"receipt":{"name":"metric","source":"papers/paper-ix/current/figures/metric.pdf","source_sha256":"cb9a43a9d24c76df095996e7304afe5d1cacc6611bf3f41f5c954568ec9ab16f","image":"assets/atlas/paper-ix/metric.png","image_sha256":"8cb0b9b4432a3d1f8d3ad46b62d3d4cfa7ee2c42a2db561eace4f55531624853","dimensions":[1366,799],"native_text":"0\n1\n2\n3\n4\n5\n6\nv\n2\n(\nH\n)\n02468\n10\n12\nv2(E(4, j) 4H − E(4, j) 0 )\nj = 0, B = 1\nj = 1, B = 1\nj = 2, B = 5\nj = 3, B = 3"},"alt":"First-difference valuation against step valuation","description":"Four lines have slope one, with original Euler scale offsets one, one, five and three for addresses zero through three. Addresses zero and one overlap. Points at step valuations zero through six illustrate the exact arithmetic. The line law is universal; these displayed points are finite illustrations.","math_labels":[{"line":279,"text":"$v_2(H)$"},{"line":279,"text":"$v_2(E_{4H}^{(4,j)}-E_0^{(4,j)})$"}],"tables":[]},"periods":{"receipt":{"name":"periods","source":"papers/paper-ix/current/figures/periods.pdf","source_sha256":"7dbb2f01a71bbcb247d4161ccf21680f95376ff71783c7b51c191ddaf1495662","image":"assets/atlas/paper-ix/periods.png","image_sha256":"eba51decbab54af16b82ab5163d00335828afbcc26052d87bec094a630937439","dimensions":[1354,790],"native_text":"1\n2\n3\n4\n5\n6\n7\n8\n9 10 11 12\np\nr\ne\nc\ni\ns\ni\no\nn\nr\ni\nn\nm\no\nd\nu\nl\nu\ns\n2\nr\n02468\n10\nlog2 (least row period)\nj = 0, 1\nj = 2\nj = 3"},"alt":"Binary logarithms of least row periods","description":"For precision exponents one through twelve, each staircase is zero until precision exceeds its address scale and then rises by one per bit. Addresses zero and one share a staircase. Height zero means row period one, not period zero. Euler degree periods are four times the row periods.","math_labels":[{"line":288,"text":"precision r in modulus $2^r$"},{"line":288,"text":"$\\log_2$(least row period)"}],"tables":[]},"residue_permutations":{"receipt":{"name":"residue_permutations","source":"papers/paper-ix/current/figures/residue_permutations.pdf","source_sha256":"dd736266d04997a7b86dfd54d81559ec8805a02a761829d1e3f58aa2ba0be290","image":"assets/atlas/paper-ix/residue_permutations.png","image_sha256":"183dda61638f0edabaacdd0bfdbd3c1bfdeccc0181e13b4d6d26d15e0e97168b","dimensions":[1537,1060],"native_text":"0 64 127\nrow n\n0\n64\n127\nscaled residue\nj = 0: 128 rows, 128 permitted residues\n0 64 127\nrow n\n0\n64\n127\nscaled residue\nj = 1: 128 rows, 128 permitted residues\n0 1 2 3 4 5 6 7\nrow n\n0\n1\n2\n3\n4\n5\n6\n7\nscaled residue\nj = 2: 8 rows, 8 permitted residues\n0 16 31\nrow n\n0\n16\n31\nscaled residue\nj = 3: 32 rows, 32 permitted residues"},"alt":"Four exact residue permutations modulo 256","description":"The four panels have periods 128, 128, 8 and 32. The horizontal coordinate is row index over one full period. The vertical coordinate subtracts the address factorial and divides by the known power-of-two scale before reducing to the period. Every allowed scaled residue occurs once. The tables retain every plotted row, Euler residue and scaled residue.","math_labels":[],"tables":[{"caption":"Address 0: recorded residues modulo 256","headers":["Row","Euler residue","Scaled residue"],"rows":[[0,1,0],[1,255,127],[2,69,34],[3,131,65],[4,233,116],[5,39,19],[6,109,54],[7,107,53],[8,81,40],[9,207,103],[10,21,10],[11,211,105],[12,57,28],[13,247,123],[14,61,30],[15,187,93],[16,161,80],[17,159,79],[18,229,114],[19,35,17],[20,137,68],[21,199,99],[22,13,6],[23,11,5],[24,241,120],[25,111,55],[26,181,90],[27,115,57],[28,217,108],[29,151,75],[30,221,110],[31,91,45],[32,65,32],[33,63,31],[34,133,66],[35,195,97],[36,41,20],[37,103,51],[38,173,86],[39,171,85],[40,145,72],[41,15,7],[42,85,42],[43,19,9],[44,121,60],[45,55,27],[46,125,62],[47,251,125],[48,225,112],[49,223,111],[50,37,18],[51,99,49],[52,201,100],[53,7,3],[54,77,38],[55,75,37],[56,49,24],[57,175,87],[58,245,122],[59,179,89],[60,25,12],[61,215,107],[62,29,14],[63,155,77],[64,129,64],[65,127,63],[66,197,98],[67,3,1],[68,105,52],[69,167,83],[70,237,118],[71,235,117],[72,209,104],[73,79,39],[74,149,74],[75,83,41],[76,185,92],[77,119,59],[78,189,94],[79,59,29],[80,33,16],[81,31,15],[82,101,50],[83,163,81],[84,9,4],[85,71,35],[86,141,70],[87,139,69],[88,113,56],[89,239,119],[90,53,26],[91,243,121],[92,89,44],[93,23,11],[94,93,46],[95,219,109],[96,193,96],[97,191,95],[98,5,2],[99,67,33],[100,169,84],[101,231,115],[102,45,22],[103,43,21],[104,17,8],[105,143,71],[106,213,106],[107,147,73],[108,249,124],[109,183,91],[110,253,126],[111,123,61],[112,97,48],[113,95,47],[114,165,82],[115,227,113],[116,73,36],[117,135,67],[118,205,102],[119,203,101],[120,177,88],[121,47,23],[122,117,58],[123,51,25],[124,153,76],[125,87,43],[126,157,78],[127,27,13]]},{"caption":"Address 1: recorded residues modulo 256","headers":["Row","Euler residue","Scaled residue"],"rows":[[0,1,0],[1,51,25],[2,253,126],[3,111,55],[4,25,12],[5,11,5],[6,213,106],[7,135,67],[8,177,88],[9,99,49],[10,45,22],[11,31,15],[12,201,100],[13,59,29],[14,5,2],[15,55,27],[16,97,48],[17,147,73],[18,93,46],[19,207,103],[20,121,60],[21,107,53],[22,53,26],[23,231,115],[24,17,8],[25,195,97],[26,141,70],[27,127,63],[28,41,20],[29,155,77],[30,101,50],[31,151,75],[32,193,96],[33,243,121],[34,189,94],[35,47,23],[36,217,108],[37,203,101],[38,149,74],[39,71,35],[40,113,56],[41,35,17],[42,237,118],[43,223,111],[44,137,68],[45,251,125],[46,197,98],[47,247,123],[48,33,16],[49,83,41],[50,29,14],[51,143,71],[52,57,28],[53,43,21],[54,245,122],[55,167,83],[56,209,104],[57,131,65],[58,77,38],[59,63,31],[60,233,116],[61,91,45],[62,37,18],[63,87,43],[64,129,64],[65,179,89],[66,125,62],[67,239,119],[68,153,76],[69,139,69],[70,85,42],[71,7,3],[72,49,24],[73,227,113],[74,173,86],[75,159,79],[76,73,36],[77,187,93],[78,133,66],[79,183,91],[80,225,112],[81,19,9],[82,221,110],[83,79,39],[84,249,124],[85,235,117],[86,181,90],[87,103,51],[88,145,72],[89,67,33],[90,13,6],[91,255,127],[92,169,84],[93,27,13],[94,229,114],[95,23,11],[96,65,32],[97,115,57],[98,61,30],[99,175,87],[100,89,44],[101,75,37],[102,21,10],[103,199,99],[104,241,120],[105,163,81],[106,109,54],[107,95,47],[108,9,4],[109,123,61],[110,69,34],[111,119,59],[112,161,80],[113,211,105],[114,157,78],[115,15,7],[116,185,92],[117,171,85],[118,117,58],[119,39,19],[120,81,40],[121,3,1],[122,205,102],[123,191,95],[124,105,52],[125,219,109],[126,165,82],[127,215,107]]},{"caption":"Address 2: recorded residues modulo 256","headers":["Row","Euler residue","Scaled residue"],"rows":[[0,2,0],[1,34,1],[2,194,6],[3,226,7],[4,130,4],[5,162,5],[6,66,2],[7,98,3]]},{"caption":"Address 3: recorded residues modulo 256","headers":["Row","Euler residue","Scaled residue"],"rows":[[0,6,0],[1,190,23],[2,118,14],[3,46,5],[4,230,28],[5,158,19],[6,86,10],[7,14,1],[8,198,24],[9,126,15],[10,54,6],[11,238,29],[12,166,20],[13,94,11],[14,22,2],[15,206,25],[16,134,16],[17,62,7],[18,246,30],[19,174,21],[20,102,12],[21,30,3],[22,214,26],[23,142,17],[24,70,8],[25,254,31],[26,182,22],[27,110,13],[28,38,4],[29,222,27],[30,150,18],[31,78,9]]}]},"isometry_matrices":{"receipt":{"name":"isometry_matrices","source":"papers/paper-ix/current/figures/isometry_matrices.pdf","source_sha256":"ffd44e12766614b49b9df251432aadc998e3220d1a7a943d3f90f35f4a5b5036","image":"assets/atlas/paper-ix/isometry_matrices.png","image_sha256":"5bca0665a6ed7c67aed94144a6547008ca43332fe7cd22888cd9e293edd898c0","dimensions":[1258,641],"native_text":"0\n1\n2\n3\n4\n5\n6\n7\nrow m\n01234567\nrow n\ni\nn\np\nu\nt\n:\nv\n2\n(\nn\n−\nm\n)\n0\n1\n2\n3\n4\n5\n6\n7\nrow m\n01234567\nrow n\no\nu\nt\np\nu\nt\n:\nv\n2\n(\ny\nn\n−\ny\nm\n)\n012\nshared low-bit precision"},"alt":"Input and output low-bit precision matrices for address two","description":"Both eight-by-eight matrices show the valuation of a difference between two rows. The right matrix uses the scaled output residues from the address-two permutation. Their colors agree at every off-diagonal position. Blank diagonals mean the infinite valuation of zero. The recorded input-output pairs below identify all eight rows without interpreting the blank diagonal as a finite number.","math_labels":[{"line":327,"text":"input: $v_2(n-m)$"},{"line":327,"text":"output: $v_2(y_n-y_m)$"}],"tables":[{"caption":"Address 2: recorded residues modulo 256","headers":["Row","Euler residue","Scaled residue"],"rows":[[0,2,0],[1,34,1],[2,194,6],[3,226,7],[4,130,4],[5,162,5],[6,66,2],[7,98,3]]}]},"rank_scales":{"receipt":{"name":"rank_scales","source":"papers/paper-ix/current/figures/rank_scales.pdf","source_sha256":"0eb6b331e0e576f683b97238ea5bddb7eb3ed7722d8928d896c15d67adae5b39","image":"assets/atlas/paper-ix/rank_scales.png","image_sha256":"7678a2ef0e24202847b5e9de7ccac230826b05d543167634c3543539cc647f60","dimensions":[1355,739],"native_text":"0 1 2 3 4 5 6 7\naddress j\nN = 2\nN = 4\nN = 8\n1 2\n1 1 4 2\n1 1 1 1 4 3 2 2\nnormalized scale bN, j\n0 1 2 3 4 5 6 7\naddress j\n1 2\n1 1 5 3\n1 1 2 2 7 6 6 6\nEuler scale BN, j"},"alt":"Normalized and Euler scales at ranks two, four and eight","description":"Two arrays place section rank on the vertical axis and address zero through seven on the horizontal axis. Blank cells are outside the permitted address range. The left entries are normalized scales; the right entries add the valuation of the address factorial. These proved arrays do not extend the same metric law to every higher-rank address.","math_labels":[{"line":337,"text":"normalized scale $b_{N,j}$"},{"line":337,"text":"Euler scale $B_{N,j}$"}],"tables":[]},"analytic_slopes":{"receipt":{"name":"analytic_slopes","source":"papers/paper-ix/current/figures/analytic_slopes.pdf","source_sha256":"fa4875d4b6b8582ca04e4830441080ab3ffe4cd9d116fb5c1f5da20df8e018ad","image":"assets/atlas/paper-ix/analytic_slopes.png","image_sha256":"42d8e13c9730747c9d23f6718d6bf4c8e7e4adc44643266ecf5e32c84772405d","dimensions":[1369,803],"native_text":"1\n4\n8 16 32 48 64\nTaylor degree d\n0\n20\n40\n60\n80\n100\n120\n140\n160\ndyadic valuation of the degree-d coefficient\nj = 0, 1: exact coefficient valuation\nj = 2: coefficient lower bound\nj = 2: exact along powers of two\nj = 3: proved lower bound"},"alt":"Positive-degree Taylor coefficient valuation bounds","description":"The degree axis runs from one through sixty-four. Addresses zero and one have exact valuation equal to degree. Address three has that same proved lower bound. The address-two lower envelope grows more steeply; powers of two from four onward mark exact values. Degree zero is excluded. The source gives an exact radius for address two and a lower bound for address three.","math_labels":[],"tables":[]},"worked/reciprocal_rows":{"receipt":{"name":"worked/reciprocal_rows","source":"papers/paper-ix/current/figures/worked/reciprocal_rows.pdf","source_sha256":"7c223666e70b5032d94a7f626258d80e969e3a890c8736768225ab7af510436a","image":"assets/atlas/paper-ix/worked-reciprocal_rows.png","image_sha256":"6c339e854405d82d53c057ac482248fce494b911c238f51f60afeaa43fdbe099","dimensions":[1428,456],"native_text":"address j C(0) C(1) C(2) 0 1 -1 69 1 1 -1/5 121/45 2 1 -1/15 13/45 3 1 -1/35 59/1155"},"alt":"Exact reciprocal coefficients at rank four","description":"The four rows correspond to addresses zero through three. Columns give the normalized reciprocal at rows zero, one and two as exact rational values. They are retained from the worked-example result record.","math_labels":[],"tables":[{"caption":"Exact reciprocal rows","headers":["Address","C(0)","C(1)","C(2)"],"rows":[[0,"1","-1","69"],[1,"1","-1/5","121/45"],[2,"1","-1/15","13/45"],[3,"1","-1/35","59/1155"]]}]},"worked/normalization_loss":{"receipt":{"name":"worked/normalization_loss","source":"papers/paper-ix/current/figures/worked/normalization_loss.pdf","source_sha256":"ffd636f54b9d0a85dfac5ec2b8ef656c7b3ab33174574a2e762600ee8bb7e115","image":"assets/atlas/paper-ix/worked-normalization_loss.png","image_sha256":"6e842a6cb72fb20d6843c02ac2a235b4dd60dea2a913b47ed3aa6d2b5256cbd1","dimensions":[1427,563],"native_text":"normalized difference original Euler difference\n01\n3-adic valuation\n64/455\n384/455"},"alt":"A factorial changes the three-adic difference valuation","description":"The normalized difference has value 64/455 and valuation zero. Multiplication by the address factorial gives original Euler difference 384/455 and valuation one. The dashed level is one. This example concerns width six and address three.","math_labels":[],"tables":[{"caption":"Retained normalization witness","headers":["Quantity","Exact value"],"rows":[["C1","-1/84"],["C2","703/5460"],["difference","64/455"],["original_difference","384/455"]]}]},"worked/odd_prime_sign":{"receipt":{"name":"worked/odd_prime_sign","source":"papers/paper-ix/current/figures/worked/odd_prime_sign.pdf","source_sha256":"beeeb1cb7215af2b15b7304bcdb7bef35a23ab36767744f0432af03e14d23ef9","image":"assets/atlas/paper-ix/worked-odd_prime_sign.png","image_sha256":"8bfaa17c506d868ac2a986166f226f59effab511944ab533d61af0b1bffffab8","dimensions":[1428,456],"native_text":"row n C at width 3, address 0 (-1)^n C(n)\n0 1 1\n1 -1 1\n2 19 19"},"alt":"Untwisted and sign-twisted coefficients at width three","description":"Rows zero, one and two compare the original normalized coefficients with their alternating-sign twist. The untwisted first difference is minus two, a three-adic unit. The twisted first difference is zero, with infinite valuation. The next twisted difference is eighteen.","math_labels":[],"tables":[{"caption":"Exact sign correction","headers":["Row","Untwisted C(n)","Sign-twisted C(n)"],"rows":[[0,"1","1"],[1,"-1","1"],[2,"19","19"]]}]},"worked/newton_cutoff":{"receipt":{"name":"worked/newton_cutoff","source":"papers/paper-ix/current/figures/worked/newton_cutoff.pdf","source_sha256":"91712583281f1b6738195cd36126cf3990d6587b22ea32ead1fad415e5fc0e85","image":"assets/atlas/paper-ix/worked-newton_cutoff.png","image_sha256":"47231688d1bc993da4339e02dca9e8c464ed8ffa32b23a01939c510acce76331","dimensions":[1427,584],"native_text":"1\n2\n3\n4\n5\n8 12 16\nNewton order k\n05\n10\n15\n20\n25\n30\nv2 of the Newton coefficient\nprecision cutoff: 8 bits"},"alt":"Newton coefficient valuations and the eight-bit cutoff","description":"Points show the recorded Newton valuations for orders one through sixteen at rank four, address zero. The dashed line is eight bits. Orders five and above lie at or above it. The source uses the coefficient theorem for the uniform truncation and separately records finite reconstruction checks through row thirty-two.","math_labels":[],"tables":[{"caption":"Recorded Newton valuations","headers":["Order","Valuation"],"rows":[[1,1],[2,3],[3,4],[4,7],[5,8],[6,10],[7,11],[8,15],[9,16],[10,18],[11,19],[12,22],[13,23],[14,25],[15,26],[16,31]]}]},"worked/smooth_tower":{"receipt":{"name":"worked/smooth_tower","source":"papers/paper-ix/current/figures/worked/smooth_tower.pdf","source_sha256":"47d4bb1e6dfc283b771b5b0c259d5783e0ae2f39dc9eaf0affd23ab843bad603","image":"assets/atlas/paper-ix/worked-smooth_tower.png","image_sha256":"a6a79e2a3a3e00b8cdca9d04e3baa28159c4a0f352160d827e9b023ee15addd9","dimensions":[1428,457],"native_text":"L 3^L current x next digit next x\n1 3 1 1 4\n2 9 4 1 13\n3 27 13 0 13\n4 81 13 2 175\n5 243 175 0 175"},"alt":"Five successive compatible three-adic root corrections","description":"Rows give the first five stages for a root of x squared minus seven, starting at one modulo three. Each row shows the precision modulus, current representative, correcting digit and next representative. A zero correcting digit preserves compatibility and does not stop the construction.","math_labels":[],"tables":[{"caption":"First five recorded root corrections","headers":["Stage","Modulus","Current representative","Digit","Next representative"],"rows":[[1,3,1,1,4],[2,9,4,1,13],[3,27,13,0,13],[4,81,13,2,175],[5,243,175,0,175]]}]},"worked/singular_block":{"receipt":{"name":"worked/singular_block","source":"papers/paper-ix/current/figures/worked/singular_block.pdf","source_sha256":"411fcd20f289f100a819f0a4a0abdbb96eb3ecae6441c5bf1839c17e814f4b3a","image":"assets/atlas/paper-ix/worked-singular_block.png","image_sha256":"a354b8678cb3609a384e12081debb9c9eed2a3f9b222cbc7e9cd6f9bcc70d937","dimensions":[1428,456],"native_text":"digit t candidate y y^2 - 3 residue mod 9\n0 0 -3 6\n1 3 6 6\n2 6 33 6"},"alt":"All three candidate digits fail to lift the singular root","description":"For x squared minus three, the old root class is zero modulo three. The candidates zero, three and six all leave residue six modulo nine. The table displays every possible next digit and the exact residual. The source also distinguishes singularity with a zero obstruction, where lifts can exist.","math_labels":[],"tables":[{"caption":"Recorded residues for the three candidate digits","headers":["Digit","Residual modulo 9"],"rows":[[0,6],[1,6],[2,6]]}]},"worked/shared_obstruction":{"receipt":{"name":"worked/shared_obstruction","source":"papers/paper-ix/current/figures/worked/shared_obstruction.pdf","source_sha256":"6e61a13b3fce9736a5f30dafb9c00e88810ac8da29f1424335e1a29b2c87f1b5","image":"assets/atlas/paper-ix/worked-shared_obstruction.png","image_sha256":"0f12e7c92cb63b5e4f8c0bd248e74339cf15830088164a00a8c71e7e8a96cafa","dimensions":[1168,605],"native_text":"0\n1\n2\nfirst residual mod 3\n012\nsecond residual mod 3\nt = 0\nt = 1\nt = 2\nrequired residual\navailable shared digits"},"alt":"Available shared residuals miss the required zero vector","description":"The required residual is marked at the origin. The three available shared digits give residual pairs (0, 2), (1, 0) and (2, 1). Each coordinate can vanish separately, but no shared digit makes both vanish. The axes are the first and second residuals modulo three.","math_labels":[],"tables":[{"caption":"Available shared corrections","headers":["Digit","First residual","Second residual"],"rows":[[0,0,2],[1,1,0],[2,2,1]]}]},"worked/weighted_motion":{"receipt":{"name":"worked/weighted_motion","source":"papers/paper-ix/current/figures/worked/weighted_motion.pdf","source_sha256":"480217dbd580ce1777cb36a464f1d50a3802f02cbc2d16633bb9a9fb9539778a","image":"assets/atlas/paper-ix/worked-weighted_motion.png","image_sha256":"7aee56a7947ae765038b8e39e0f1a3d0dff15b8fc93a0d675d1feedafe4f780d","dimensions":[1427,562],"native_text":"2\n3\n4\n5\n6\n7\nA in 3x + 3^A\n1234567\n3-adic valuation\ncoefficient precision\nroot displacement precision"},"alt":"One digit of precision is lost in root displacement","description":"For the displayed exponents two through seven, coefficient precision is the exponent, while root displacement precision is one less. Both lines rise with slope one. The derivative has valuation one, and the source derives the loss exactly from the two linear equations.","math_labels":[],"tables":[{"caption":"Recorded root displacement","headers":["Exponent","Root","Coefficient valuation","Displacement valuation"],"rows":[[2,-3,2,1],[3,-9,3,2],[4,-27,4,3],[5,-81,5,4],[6,-243,6,5],[7,-729,7,6]]}]}},"macros":{"N":"\\mathbb N","Z":"\\mathbb Z","Q":"\\mathbb Q","F":"\\mathbb F","Zp":"\\mathbb Z_p","Qp":"\\mathbb Q_p","tightlist":"\\setlength{\\itemsep}{0pt}\\setlength{\\parskip}{0pt}","ZZ":"\\mathbb Z","QQ":"\\mathbb Q","FF":"\\mathbb F","NN":"\\mathbb N","Wp":"\\mathcal W_p","Hp":"\\mathcal H(\\Z_p)","ordp":"\\operatorname{ord}_p","ord":"\\operatorname{ord}","sech":"\\operatorname{sech}","EE":"\\mathcal E","PP":"\\mathcal P","Hcal":"\\mathcal H","Ecal":"\\mathcal E","Ccal":"\\mathcal C","Pcal":"\\mathcal P","Scal":"\\mathcal S","Ocal":"\\mathcal O","Rcal":"\\mathcal R","Vcal":"\\mathcal V","Wcal":"\\mathcal 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address","url":"read/paper-ix/euler-packets.html#L56"}],"2":[{"key":"thm:arith-nishibuchi","label":"Theorem 3.6","number":"3.6","kind":"theorem","title":"Nishibuchi congruence","url":"read/paper-ix/euler-packets.html#L166"}],"3":[{"key":"cor:all-dyadic-final","label":"Corollary 4.5","number":"4.5","kind":"corollary","title":"Threshold-free dyadic congruence","url":"read/paper-ix/strengthening-threshold-free-reciprocal-transfer-at-prime-power-section-orders.html#L224"}],"4":[{"key":"thm:all-prime-power-transfer","label":"Theorem 4.4","number":"4.4","kind":"theorem","title":"All-row prime-power transfer","url":"read/paper-ix/strengthening-threshold-free-reciprocal-transfer-at-prime-power-section-orders.html#L170"}],"5":[{"key":"cor:final-euler-congruence","label":"Corollary 9.1","number":"9.1","kind":"corollary","title":"Final common-step 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