Nodal Chamber Equipartition, Critical-Point Index Dichotomy, and Weighted Prime-Simplex Dilogarithm Integrals for Riemann's Xi-Function
Year: 2026
Type: preprint
Abstract: Studies the Cauchy-Riemann nodal geometry and Prime-Number-Theorem simplex integrals associated with Riemann's completed zeta function Xi(t - i*d) = R(d,t) + i*I(d,t) (d >= 0) under the canonical transverse Wronskian normalization W(d,t) = (1/2)*d_d|Xi(t-id)|^2 = R*d_d R + I*d_d I = d*W_tilde(d,t). (1) Places the local nodal-branch dichotomy within the critical-point index framework of Craven, Csordas, and Smith (1987) and Ki and Kim (2000), certifies Case-I critical-line Laguerre defects c_1^{DH}(tau_{false,m}) < 0 across the first five off-critical zeros of the self-dual Davenport-Heilbronn function Xi_DH, and details the global obstacles of 1-parameter heat-flow saddle collisions, unbounded hairpin components (requiring Phragmen-Lindelof and Euler-product control), and Speiser-level difficulty of Xi' != 0 on d > 0. (2) Proves the unconditional 50%-50% Green-Dirichlet-Neumann chamber equipartition int R*d_d R dt == int I*d_d I dt == iint_{Omega_k} |Xi'|^2 dd dt = (1/2)*W_{Omega_k}(d_0) > 0 and Wronskian-slope reciprocity. (3) Derives exact closed-form Euler dilogarithm Li_2(2*theta) formulas contrasting factorized simplex weights (eta_fac^{(2)}(1/6) = 0.676891 > 1/2) against the true additive semiprime Wronskian weight (1-u-v)^2, which collapses to eta_add^{(2)}(1/6) = 0.150604 < 1/2 with polynomial deficit (1/8)*(1-2*tau)^2*(3-4*tau) > 0.
Author: Missiaen, Thijs
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225814028
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