MathOverflow 513954, revision 2: singular signed downdates, fixed coordinates, and a sharp polynomial sphere bound

MathOverflow 513954, revision 2 has affirmative answers: we give a self-contained proof of the inertia formula previously proved by user1551, and prove the continuous-family assertion. For any real symmetric A and nonzero Delta=x^T A x, the matrix A-(Ax)(Ax)^T/Delta loses one eigenvalue of the pivot sign and gains one zero eigenvalue, with kernel ker A direct-sum span{x}. For continuous diagonal families with a positive-semidefinite modification, connectedness fixes the pivot sign and, in the negative branch, one negative coordinate. The conclusions extend to countably punctured simplices, and a block congruence removes any nondegenerate subspace. For the linked sphere question, the increasing odd polynomial h(t)=(3t-t^3)/2 has squared ratio infimum 16(17-9sqrt(3))/23 over all dimensions, strictly below the two-coordinate infimum 80/81.