https://kenan.works/ Things I cannot stop thinking about, mathematically. en Wed, 22 Jul 2026 12:00:00 +0000 https://kenan.works/posts/to-beat-one-over-log-the-coefficients-must-get-wild/ https://kenan.works/posts/to-beat-one-over-log-the-coefficients-must-get-wild/ Wed, 22 Jul 2026 12:00:00 +0000 A fixed self-similar mixture of exponential Nyman scales has a sharp 1/log N packet floor, and any bounded-variation adaptive mixture inherits the same order. Faster suppression forces coefficient variation to diverge. https://kenan.works/posts/fourteen-endpoint-shifts-of-xi/ https://kenan.works/posts/fourteen-endpoint-shifts-of-xi/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/degree-seven-is-the-sharp-karlin-threshold/ https://kenan.works/posts/degree-seven-is-the-sharp-karlin-threshold/ Wed, 22 Jul 2026 12:00:00 +0000 Every signed centered consecutive-derivative determinant is nonnegative for real-rooted polynomials through degree six, and degree seven is the first possible failure. https://kenan.works/posts/froissart-shells-are-missing-recurrence-modes/ https://kenan.works/posts/froissart-shells-are-missing-recurrence-modes/ Wed, 22 Jul 2026 12:00:00 +0000 Block-Hankel minimal realizations turn arbitrary mixed and repeated Froissart shells into missing denominator modes, while contour counts and all-channel Hermite jets make noisy removal fail closed. https://kenan.works/posts/one-extra-pf-order-strictifies-every-nonforced-minor/ https://kenan.works/posts/one-extra-pf-order-strictifies-every-nonforced-minor/ Wed, 22 Jul 2026 12:00:00 +0000 A positive nonpolynomial entire PF_(p+1) sequence has every structurally possible order-p Toeplitz minor strictly positive, with no shift cutoff. https://kenan.works/posts/one-positive-zero-deflation-detects-every-mixed-shell/ https://kenan.works/posts/one-positive-zero-deflation-detects-every-mixed-shell/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/odd-even-feedback-is-a-p-matrix/ https://kenan.works/posts/odd-even-feedback-is-a-p-matrix/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/power-sum-quotients-have-compressed-tangent-cones/ https://kenan.works/posts/power-sum-quotients-have-compressed-tangent-cones/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/first-theta-shell-is-a-generalized-bell-sturm-chain/ https://kenan.works/posts/first-theta-shell-is-a-generalized-bell-sturm-chain/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/chj-perron-weights-have-a-sharp-phase-transition/ https://kenan.works/posts/chj-perron-weights-have-a-sharp-phase-transition/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/rosati-polarization-gives-all-order-positive-kernels/ https://kenan.works/posts/rosati-polarization-gives-all-order-positive-kernels/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/the-480-boundary-layer/ https://kenan.works/posts/the-480-boundary-layer/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/two-scales-for-dirichlet-l-functions/ https://kenan.works/posts/two-scales-for-dirichlet-l-functions/ Wed, 22 Jul 2026 12:00:00 +0000 The global pair (10.295195, 61.272145) and the sharper large-height pair (9.93082, 48.07157) solve genuinely different optimization problems. https://kenan.works/posts/endpoint-flat-xi-approximants/ https://kenan.works/posts/endpoint-flat-xi-approximants/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/four-rows-of-cup-positivity/ https://kenan.works/posts/four-rows-of-cup-positivity/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/two-prime-counting-table-corrections/ https://kenan.works/posts/two-prime-counting-table-corrections/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/a-prime-with-any-large-digit-sum/ https://kenan.works/posts/a-prime-with-any-large-digit-sum/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/fifteen-shifts-of-xi/ https://kenan.works/posts/fifteen-shifts-of-xi/ Wed, 22 Jul 2026 12:00:00 +0000 Rectangular Jacobi-Trudi duality turns an unbounded-rank Toeplitz problem into fifteen fixed-size reciprocal-Xi determinants, whose first poles eventually force positivity. https://kenan.works/posts/the-full-reciprocal-xi-ensemble/ https://kenan.works/posts/the-full-reciprocal-xi-ensemble/ Wed, 22 Jul 2026 12:00:00 +0000 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. https://kenan.works/posts/debruijn-newman-stops-at-four/ https://kenan.works/posts/debruijn-newman-stops-at-four/ Wed, 22 Jul 2026 12:00:00 +0000 A complete confluent flag, used twice through ECT collocation, proves every de Bruijn-Newman translation minor through order four is positive. An equally spaced order-five minor is negative.