Equation 677 Database

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Magma cf1a5c5aa45b…

magma cf1a5c5aa45b
Size
847
Isomorphism class hash
cf1a5c5aa45bf2636038d85e6e46fdcb96a0d896714e233ddff0af3f96cf09df
Satisfies Equation 255
yes
Right-cancellative
no
Idempotent
no
Fiber matrix
symmetric: no · normal: no · rank: 127 (nullity 720) what is this?
Submitted by
dwrensha
Submitted at
2026-09-05 02:08:14
Display reorder
846,725,723,724,720,721,722,840,672,712,662,692,682,702,834,622,602,612,642,632,652,842,674,714,664,694,684,704,845,673,713,663,693,683,703,844,671,711,660,691,681,701,831,625,605,615,645,635,655,835,626,606,616,646,636,656,832,623,603,613,643,633,653,841,670,710,661,690,680,700,833,624,604,614,644,634,654,839,676,716,666,696,686,706,779,317,417,442,367,342,392,726,296,299,295,298,297,108,792,323,424,449,374,348,399,781,314,414,439,364,339,389,791,324,423,448,373,349,398,742,119,117,118,115,120,116,764,182,178,180,186,184,175,753,141,139,140,143,142,144,788,311,411,436,361,336,386,730,127,131,132,129,128,130,826,620,600,610,641,631,651,743,82,80,81,78,83,79,804,555,558,559,556,586,557,765,29,24,27,33,31,35,750,46,44,45,42,47,43,754,54,58,59,56,55,57,818,501,504,500,502,575,503,814,464,462,463,460,567,461,806,531,534,530,532,581,533,761,68,66,67,70,69,71,819,480,483,484,481,571,482,838,679,719,669,699,689,709,784,313,413,438,363,338,388,759,97,101,96,99,98,100,777,320,420,445,370,345,395,797,305,405,430,355,330,380,786,301,401,425,351,326,376,733,236,240,235,238,237,239,749,181,177,179,185,183,176,740,11,9,10,7,6,8,798,308,408,433,358,333,383,771,89,87,88,85,84,86,837,678,718,668,698,688,708,795,309,409,434,359,334,384,744,220,218,219,222,221,217,790,322,422,447,372,347,397,776,300,400,426,350,325,375,783,318,418,443,368,343,393,766,234,232,233,230,229,231,731,198,196,197,194,193,195,772,17,15,16,13,12,14,796,304,404,429,354,329,379,727,206,210,205,208,207,209,843,675,715,665,695,685,705,787,303,403,428,353,328,378,768,264,262,263,260,259,261,800,312,412,437,362,337,387,799,307,407,432,357,332,382,778,315,415,440,365,340,390,728,250,248,249,252,251,247,755,163,167,168,165,164,166,745,155,153,154,151,156,152,785,319,419,444,369,344,394,735,272,276,271,274,273,275,829,628,608,618,648,638,658,773,280,278,279,282,281,277,809,546,549,545,547,584,548,752,148,146,147,150,149,145,763,162,160,161,158,157,159,762,257,255,256,253,258,254,801,529,527,528,525,580,526,824,486,489,485,487,572,488,825,514,512,513,510,577,511,775,265,269,270,267,266,268,812,465,468,469,466,568,467,827,627,607,617,647,637,657,758,199,203,204,201,200,202,821,472,470,471,473,569,474,732,18,22,23,20,19,21,739,191,189,190,187,192,188,770,227,225,226,223,228,224,807,544,542,543,540,583,541,805,450,453,454,451,565,452,815,564,562,563,560,587,561,748,213,211,212,215,214,216,820,499,497,498,495,574,496,828,621,601,611,640,630,650,767,90,94,95,92,91,93,823,493,491,492,494,573,490,741,292,290,291,294,293,289,751,170,174,169,172,171,173,734,243,241,242,245,244,246,811,457,455,456,458,566,459,822,479,477,478,475,570,476,802,553,551,552,554,585,550,760,283,287,288,285,284,286,808,516,519,515,517,578,518,836,677,717,667,697,687,707,793,306,406,431,356,331,381,738,61,65,60,63,62,64,782,316,416,441,366,341,391,789,302,402,427,352,327,377,794,310,410,435,360,335,385,757,53,51,52,49,48,50,737,39,37,38,41,40,36,774,4,2,3,1,5,0,780,321,421,446,371,346,396,747,75,73,74,77,76,72,830,629,609,619,649,639,659,729,126,124,125,122,121,123,813,508,506,507,509,576,505,756,134,138,133,136,135,137,769,28,25,26,32,30,34,746,112,110,111,114,113,109,816,588,596,599,592,590,594,810,523,521,522,524,579,520,817,589,597,598,593,591,595,736,104,102,103,106,105,107,803,538,536,537,539,582,535 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

A NON-IDEMPOTENT Eq677 magma with a NONNORMAL fiber matrix, built using a LINEAR base and AFFINE operations in EVERY fiber block. Order847=121*7. It still satisfies Eq255, so it is not a counterexample to the main implication. Exactly121 elements are idempotent;726 are not. Carrier F11^2 x F7. Write base vectors i=(i1,i2), j=(j1,j2), and define (i,s)*(j,t) = (6j-5i, H_ij(s,t)). Select the two coordinate axes in F11^2. If i,j are distinct and lie on the same selected axis, let delta be j-i in that axis coordinate. Set H_ij(s,t)=t when delta is a nonzero square in F11, and H_ij(s,t)=-s-t otherwise. For every other pair, including i=j, set H_ij(s,t)=4s+3t. Fiber arithmetic is modulo7. Every fiber block is affine: its (left,right) coefficients are (0,1),(-1,-1), or(4,3). No nonlinear exceptional core is used. Why Eq677 holds: for distinct base inputs, all four pairs in its evaluation remain on the same affine line and have distinct entries. On a selected axis the fiber trace is A,A,B,B or B,B,A,A, where A(s,t)=t and B(s,t)=-s-t; both chains return the required input. On any other line the four operations are C(s,t)=4s+3t, a standard seven-element Eq677 law. Equal base inputs also stay within a C fiber. Fiber matrix explanation: let K be the121-by121 matrix formed by embedding the Paley tournament matrix B[z,j]=1 iff j-z is a nonzero nonsquare on each of the two axes. The axes share only the origin, and K has no diagonal entries. Put P=7I7-J7. Then F = J847 + K tensor P, FF^T-F^TF = 7*(KK^T-K^TK) tensor P. K is the sum of two individually normal tournament matrices, but is nonnormal because the axes intersect. For the points (0,1) and(1,0), the KK^T entry is1 and the K^TK entry is0. This gives nonnormality directly. K has rank21, hence F has rank127. Corresponding rows and columns of F still have matching entry multisets. Pack (i1,i2,s) as7*(11*i1+i2)+s. In these construction labels, 1*1=0, and (FF^T)[7,77]=889 whereas(F^TF)[7,77]=847. Canonical server labels can differ. All717409 Eq677 instances, Eq255, the idempotent count, and both witnesses were independently checked with a separate scalar verifier. The full block-matrix and commutator identities were also checked exactly. Constructed and verified with OpenAI Codex, 2026-09-04. This shows that affine fiber operations over a linear base do not by themselves force normality of the fiber matrix.

last edited by dwrensha at 2026-09-05 02:09:34 · history