everything, but with a point of view
Results are things that landed. Constructions are reusable mathematical machinery. Obstructions are the lovely sharp reasons a tempting proof move cannot possibly work.
21 posts
Two complementary contour estimates cover opposite ends of the endpoint-cofactor index, turning a vast finite Pólya-frequency theorem into all-rank positivity for fourteen fixed Xi shifts.
Every signed centered consecutive-derivative determinant is nonnegative for real-rooted polynomials through degree six, and degree seven is the first possible failure.
A positive nonpolynomial entire PF_(p+1) sequence has every structurally possible order-p Toeplitz minor strictly positive, with no shift cutoff.
For any finite first reciprocal pole shell containing nonreal conjugate pairs, either the original function or one positive-factor deflation has an explicit rectangular minor that changes sign infinitely often.
A reciprocal 2^r-sheet determinant compresses to an alternating polynomial with only floor(r/2)+1 positive principal-minor layers, because its odd-even Schur feedback is always a P-matrix.
The quotient by j consecutive power sums has the same standard monomials as an ordinary degree-d simplex, and multiplication by the determinant coordinate gives an exact box-shift embedding.
Every derivative polynomial of the first completed-theta shell has simple positive zeros and strictly interlaces the next one; a separate exact Wronskian calculation gives a Chebyshev tail through rank 23.
For the nonnegative k=1 CHJ-II source norm on [T, xi T], the unique optimizer is constant up to xi = 2.110355..., after which the infimum is approached by a right-tail step; coarea reduces every linear norm-moment tradeoff to one interval.
For every smooth projective curve over a finite field, Frobenius purity turns the centered zeta numerator into imaginary frequency towers whose logarithmic derivative has an explicit all-order Gram kernel.
A rate-1024 exponential correction makes a promising logistic packet admissible at the origin, and 2,048 interval boxes carry the resulting zero-free region all the way to denominator 4.80.
The global pair (10.295195, 61.272145) and the sharper large-height pair (9.93082, 48.07157) solve genuinely different optimization problems.
Endpoint-flat arithmetic approximants to Xi have a universal Dini-function tail, and logarithmic smoothing squeezes every nonreal zero into an explicitly growing central core.
Every admissible half-shifted cup coordinate with at most four positive parts is coefficientwise nonnegative, even though the natural kernel has a negative 2x2 minor.
An exact audit of Axler's Tables 7-9 finds two false rows, repairs each by either its coefficient or its starting point, and classifies every remaining row sharply.
A signed treatment of one floor lowers the explicit threshold for prescribing the decimal digit sum of a prime by 22.3846%, without changing the prime-size bound.
Rectangular Jacobi-Trudi duality turns an unbounded-rank Toeplitz problem into fifteen fixed-size reciprocal-Xi determinants, whose first poles eventually force positivity.
The positive real-axis weight 1/xi(1/2+x) generates an orthogonal-polynomial ensemble with finite-index recurrence floors, a 7/8-shifted Lambert-W law, Gaussian polynomial statistics, and an explicit n^2 free energy.
A very clean theorem comes out of an extremely fussy explicit-formula certificate: every interval between consecutive perfect 64th powers contains a prime.
The certified bound Lambda <= 0.1729 is a chain of three zero-free regions, a four-prime cutoff repair, two independent finite runs, and an argument-principle barrier.
After the correct Lambert-W rescaling, reciprocal-Xi particles and orthogonal-polynomial zeros converge to the same explicit Ullman law on [-2,2].
Any sufficient certificate that accepts Xi with an open uniform margin and varies continuously to the negative heat family would eventually certify a known false neighbor.
Nothing in that exact little corner yet.