The Polarized Hodge Energy Spectrum: a high-dimensional unifying object in Euclidean space——based on the Selberg trace Formula, Dual Invariants, Wulff Convex Geometry, the Global Langlands Correspondence and Hodge Theory
Year: 2026
Type: preprint
Abstract: The Polarized Hodge Energy Spectrum: a high-dimensional unifying object in Euclidean space Based on the Selberg Trace Formula, Dual Invariants, Wulff Convex Geometry, the Global Langlands Correspondence and Hodge Theory ZITAI QINORCID: 0009-0004-5467-0074November 8, 2026 MSC (2020): 52A20 (convex sets); 46B20 (normed linear spaces); 11M36 (spectral theory and Selberg trace formula); 11F70 (automorphic representations); 14D07 (Hodge structures); 11G40 (the Birch–Swinnerton-Dyer conjecture); 11R39 (the Langlands correspondence). Keywords: polarized Hodge energy spectrum; polar-body duality class; Wulff shape; support function; Minkowski functional; polar body; gauge; energy control; Banach–Mazur distance; L^p Mahler product; dual invariant; spectral zeta function; explicit formula; Tannaka duality; Langlands spectral correspondence; Artin conjecture; Birch–Swinnerton-Dyer spectral decomposition. Conventions of this document: the main text is plain text; mathematical formulas are written in LaTeX notation (inline as $...$, displayed as $$...$$). The paper itself is a 21-chapter manuscript of about 500 pages, organized in five parts, with 97 references, listed in full at the end of this document. 1 OVERVIEW This paper introduces the polarized Hodge energy spectrum, and reveals its status as a high-dimensional unifying object in Euclidean space. The object is defined on the polar-body duality class $$\mathbb K(\rho)=\{K(\rho),\,K(\rho)^\circ\}:$$ with a positive continuous directional trace $\rho$ on the sphere as input, the Wulff construction produces a compact convex body $K(\rho)$, whose polar body $K(\rho)^\circ$ together with it forms an unordered pair; $\mathbb K$ is identified with a polarized Hodge structure through a bijection, where "polarization" means polar-body duality, and in the Hodge case it coincides with Hodge polarization. The object lies in a deeper layer of the unified structure: the classical theorems of convex geometry, Hodge theory, the Langlands correspondence, Galois representations, and arithmetic geometry are organized through explicit formulas as invariants of one and the same object — they are projections of one and the same Hodge structure under different explicit formulas, and the three-level identity of objects, theorems and boundaries characterizes this layer. The families of invariants on $\mathbb K$ include the $L^p$ Mahler products $\mathcal Q_p(\mathbb K)=\|h_K\|_p\,\|g_K\|_p$, the spherical Laplace modulus-square spectrum $\{p_k\}(\mathbb K)$, the full complex-amplitude spectrum, and the moment-map recovered full spectrum. The primary structure is the two-sided energy control $$\rho_{\min}^2\,\|x\|_K^2\ \le\ \|x\|_2^2\ \le\ \rho_{\max}^2\,\|x\|_K^2,$$ with optimal constants; in the ellipsoid case the ratio is exactly $\sqrt{\kappa(A)}$. Main result I: the CY spectral Torelli theorem. It is proved that for a general odd-dimensional Calabi–Yau variety the spectral map determines the complex structure: the zero set is a proper closed real-analytic subset with empty interior, the map is non-stationary on an open dense subset, and the dimension of the spectral stratum equals exactly the dimension of the period domain; maximal rank holds unconditionally on product-attainable moduli spaces, and the general case reduces to a conditional form. Main result II: the Hodge-direction interface and computable closed forms. It is proved that Hodge–Riemann positivity is the well-definedness condition of the Weil ellipsoid construction, and the Kähler cone is its domain of definition; all provable subclasses of the standard conjectures enter the spectral language in theorem form; closed forms of the spectrum are given for the $n=2,3$ ellipsoid families and the Fermat quintic. Main result III: Langlands-direction faithfulness and the boundary of the Artin conjecture. It is proved that the directional trace is a faithful invariant of the category of Artin representations (determining the representation, preserving tensor products, induction and symmetric powers), i.e., the spectral form of Tannaka duality; the two-dimensional known side of the Artin conjecture is proved; the minimal coordinates and obstruction invariants of the remaining frontier in the $n\ge3$ stratum are located; the local layers and the known global layers of the Langlands correspondence are proved item by item in the spectral language. Main result IV: the visible/invisible decomposition of Birch–Swinnerton-Dyer. It is proved that the real period and the regulator of the Birch–Swinnerton-Dyer leading coefficient are completely spectralizable, and the Tamagawa numbers, torsion orders and the order of Ш lie in the spectrally invisible stratum; the remaining part is exactly the statement "the invisible stratum is uniquely determined by the spectral stratum"; the analytic rank is readable in all strata, and the algebraic rank is readable for $r\le1$. Main result V: the surrounding theorem group of the conjectures. The standard conjectures A, B, D and the Hodge conjecture are proved on very general families and on one complete geometric family; the dimension-independent chain on the Artin side, the four Birch–Swinnerton-Dyer spectral-stratum items, the three Langlands-correspondence threshold items at $n=4$, and the $\mathcal Q$-faithfulness system are established. Main result VI: the equivariant Tannaka structure theorem of the Hodge energy spectrum. The two groupoidifications of the polar-body duality class — a contractible two-object groupoid and a one-object quotient groupoid with classifying space $B\mathbb Z/2$ — form a double cover; the spectral invariants stratify accordingly, with $\mathcal Q_p$ and $\{p_k\}$ reading out the two strata of the groupoid; the three faithfulness theorems (directional trace, Hodge realization, étale realization) are unified as three fibers of one and the same motivic Tannaka fiber functor, the faithfulness of the Hodge realization being equivalent to standard conjecture B; the Chow-stratum gap and the $g^2$ deficit are the two images of $\ker(\omega)$. Other results: the boundary of spectral faithfulness; the direction-resolved stationary-phase constant; the Wulff-shape realization of the main term and the error term of the Weyl law; the term-by-term boundary of the Selberg trace formula and the Mellin transform; the extremal characterization of two-dimensional lattices; the two-sided mutual control of the Hodge spectral gap and $\mathcal Q_\infty$; the $E_6$ spectrum of the Fermat cubic surface and the $E_8$ fiber-block spectral closed forms; the error-bounded $\mathcal Q$ spectrum of the Möbius projection convex hull. All theorems of the paper are explicit assemblies resting on unconditional classical theorems; the novelty of the new theorems lies in the explicit forms and the exact constants. The remaining parts of the standard conjectures, the Artin conjecture, the Birch–Swinnerton-Dyer conjecture and the global Langlands correspondence are marked item by item in the paper as "remaining part" and are never mixed with the proved theorems. 2 COMPLETE EXPOSITION OF THE IDEAS 2.1 The input and the Wulff construction The input is a positive continuous directional trace $\rho$ on the unit sphere $S^{n-1}$ — a spherical function carrying infinitely many degrees of freedom. In applications it arises from support-function evaluation (convex geometry), surface energy densities (crystallography, Wulff), periods (Hodge theory) and Radon transforms (integral geometry). The Wulff construction $$K(\rho)=\{x\in\mathbb R^n:\langle x,u\rangle\le\rho(u)\ \ \forall u\in S^{n-1}\}$$ turns $\rho$ into an intersection of closed half-spaces. Theorem 2.3 (Minkowski support-function duality) states that sublinear functions and compact convex sets correspond one-to-one. Theorem 2.17 proves the well-definedness and the information preservation of the construction: $K(\rho)$ is a compact convex set containing the origin in its interior, sandwiched between the two balls $\overline B(0,\rho_{\min})$ and $\overline B(0,\rho_{\max})$; the support function $h_{K(\rho)}$ is the greatest sublinear minorant (sublinear envelope) of the homogeneous extension of $\rho$. When $\rho$ is sublinear, traces and convex bodies correspond one-to-one and no directional information is lost; otherwise the difference is exactly the convexification loss, quantified by $\|\rho-h_{K(\rho)}\|_\infty$. On the polar side, $$K(\rho)^\circ=\overline{\mathrm{conv}}\{u/\rho(u):u\in S^{n-1}\},$$ an identity that follows row by row from the exchange of polar with intersection (Theorem 2.12) and is the geometric root of all duality conclusions of the paper. 2.2 The object and its identification with a polarized Hodge structure The unordered pair $$\mathbb K(\rho)=\{K(\rho),\,K(\rho)^\circ\}$$ is the polar-body duality class. The reason for taking the unordered pair is that $\mathcal Q_p(K)=\|h_K\|_p\|g_K\|_p$ is invariant under the interchange $K\leftrightarrow K^\circ$, so dual invariance is part of the definition on $\mathbb K$ rather than a property requiring proof. The class is identified with a polarized Hodge structure through a bijection: in weight 1 (Proposition 5.38), $C\mapsto B=Q(C\cdot,\cdot)$ is a bijection from $Q$-compatible complex structures to the positive-definite symplectic matrices, with inverse $C=Q^{-1}B$; in general weight (Theorem 14.38), the Weil form $B=\operatorname{Re}H$ determines the complex structure block by block through the same formula. The double meaning of "polarization" is deliberate: polar-body duality $K\leftrightarrow K^\circ$ coincides in the Hodge case with Hodge polarization, because $B=\operatorname{Re}H$ is exactly the real part of the Hodge polarization Hermitian form. 2.3 The primary structure: two-sided energy control Through the Minkowski functional the convex body gives the gauge $\|\cdot\|_K$, and the gauge gives the Finsler energy $E_K=\tfrac12\|\cdot\|_K^2$. Theorem 3.39
Author: 秦, 子泰
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225573806