UNIFIED DIFFERENTIAL ALGEBRAIC THEORY OF DENSE SPHERE PACKING: FROM FIRST PRINCIPLES TO HIGH-DIMENSIONAL CONSTRUCTIONS
Fuente:
Zenodo
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Recurso digital |
| Lingua: | inglese |
| Pubblicazione: |
Zenodo
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866901745578278912 |
|---|---|
| author | liu, shifa |
| author_facet | liu, shifa |
| contents | <p>This paper develops a systematic differential algebraic framework for the study of dense sphere packings, unifying the Euclidean, non-Euclidean (spherical and hyperbolic), and quantum state space settings. We construct a packing differential closure KPack by adjoining a generic solution of a differential ideal encoding all packing constraints, within a differentially closed field whose existence is guaranteed by model theory. We prove the existence of geometric differential closures with constant fields isomorphic to R (Theorem 2.3), providing a rigorous semantic foundation for transferring geometric existence statements via the Tarski transfer principle. This closure provides a complete algebraic context for studying all geometric configurations satisfying contact constraints, critical points of density functions, and associated symmetry generators. Within this framework, we prove the existence of locally optimal packing configurations in a fixed container and establish a differential algebraic characterization of periodic packing density as a function of lattice basis vectors (Theorem 2.4),with a complete proof using differential algebra quotient ring constructions. Based on linear programming methods, we derive a precise asymptotic upper bound for the kissing number.We extend this framework to quantum state space (Bloch sphere packing), hyperbolic space Hn, and sphere Sn, establishing the corresponding closures KQPack, KHPack, and KSPack. Using known results on spherical packings, we prove a lower bound theorem for the minimum fidelity of maximally entangled state packings with an explicit constant expressed in terms of Gamma functions. We prove the existence and algebraicity of optimal auxiliary functions for spherical packing bounds (Theorem 3.2), showing they are finite Gegenbauer series with algebraic coefficients. By introducing a time variable, we construct the dynamical extension closure KDynPack = KPack⟨t, ∂t⟩ and derive the mode-coupling equations and their aging scaling law form within the mode-coupling approximation. We provide a complete self-contained derivation of the aging exponent equation (5.10) from first principles using asymptotic analysis and Mellin transform techniques (Theorem 5.2). In computational complexity, we prove that the complexity class DAR−NP in the Blum-Shub-Smale model is exactly equal to the existential theory of the reals ∃R, and establish the ∃R-completeness of the decision problem for high-density packings via constructive polynomial-time reduction from planar 3-coloring.We formulate and partially prove the Algebraicity Conjecture for optimal packings (Conjecture 8.1), establishing its equivalence within the differential framework (Theorem 8.2) and providing low-dimensional verification (Theorem 8.3), while demonstrating the possibility of transcendental optimal lattices for periodic problems (Theorem 8.4). This paper not only incorporates classical results from two-dimensional hexagonal packings to high-dimensional E8 and Leech lattices as special cases within a unified formulation but also, through a series of rigorously proven new theorems, provides a solid and profound theoretical foundation for understanding dense packing problems from first principles.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18644567 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | UNIFIED DIFFERENTIAL ALGEBRAIC THEORY OF DENSE SPHERE PACKING: FROM FIRST PRINCIPLES TO HIGH-DIMENSIONAL CONSTRUCTIONS liu, shifa differential algebraic framework,dense sphere packings,critical points,density functions,Mellin transform techniques, <p>This paper develops a systematic differential algebraic framework for the study of dense sphere packings, unifying the Euclidean, non-Euclidean (spherical and hyperbolic), and quantum state space settings. We construct a packing differential closure KPack by adjoining a generic solution of a differential ideal encoding all packing constraints, within a differentially closed field whose existence is guaranteed by model theory. We prove the existence of geometric differential closures with constant fields isomorphic to R (Theorem 2.3), providing a rigorous semantic foundation for transferring geometric existence statements via the Tarski transfer principle. This closure provides a complete algebraic context for studying all geometric configurations satisfying contact constraints, critical points of density functions, and associated symmetry generators. Within this framework, we prove the existence of locally optimal packing configurations in a fixed container and establish a differential algebraic characterization of periodic packing density as a function of lattice basis vectors (Theorem 2.4),with a complete proof using differential algebra quotient ring constructions. Based on linear programming methods, we derive a precise asymptotic upper bound for the kissing number.We extend this framework to quantum state space (Bloch sphere packing), hyperbolic space Hn, and sphere Sn, establishing the corresponding closures KQPack, KHPack, and KSPack. Using known results on spherical packings, we prove a lower bound theorem for the minimum fidelity of maximally entangled state packings with an explicit constant expressed in terms of Gamma functions. We prove the existence and algebraicity of optimal auxiliary functions for spherical packing bounds (Theorem 3.2), showing they are finite Gegenbauer series with algebraic coefficients. By introducing a time variable, we construct the dynamical extension closure KDynPack = KPack⟨t, ∂t⟩ and derive the mode-coupling equations and their aging scaling law form within the mode-coupling approximation. We provide a complete self-contained derivation of the aging exponent equation (5.10) from first principles using asymptotic analysis and Mellin transform techniques (Theorem 5.2). In computational complexity, we prove that the complexity class DAR−NP in the Blum-Shub-Smale model is exactly equal to the existential theory of the reals ∃R, and establish the ∃R-completeness of the decision problem for high-density packings via constructive polynomial-time reduction from planar 3-coloring.We formulate and partially prove the Algebraicity Conjecture for optimal packings (Conjecture 8.1), establishing its equivalence within the differential framework (Theorem 8.2) and providing low-dimensional verification (Theorem 8.3), while demonstrating the possibility of transcendental optimal lattices for periodic problems (Theorem 8.4). This paper not only incorporates classical results from two-dimensional hexagonal packings to high-dimensional E8 and Leech lattices as special cases within a unified formulation but also, through a series of rigorously proven new theorems, provides a solid and profound theoretical foundation for understanding dense packing problems from first principles.</p> |
| title | UNIFIED DIFFERENTIAL ALGEBRAIC THEORY OF DENSE SPHERE PACKING: FROM FIRST PRINCIPLES TO HIGH-DIMENSIONAL CONSTRUCTIONS |
| topic | differential algebraic framework,dense sphere packings,critical points,density functions,Mellin transform techniques, |
| url | https://doi.org/10.5281/zenodo.18644567 |