Geometry

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2025 - 2504(1) - 2506(3) - 2507(2) - 2511(1)

Recent submissions

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[7] ai.viXra.org:2511.0069 [pdf] submitted on 2025-11-21 11:45:38

Geometric -Ladder Universal Constants are Eigenvalues of a Circle

Authors: David Charles
Comments: 6 Pages. (Note by ai.viXra.org Admin: Part of the table is cutoff; please use standard citiation such as numerical APS style)

Using only the intrinsic geometry of the circle S1 and standard spectral theory of flattori T n = (S1)n, we derive three unavoidable mathematical constraints: angular dualityD = 180/π with reciprocal U = π/180, factorial escalation governed by 3! = 6 beyonddimension three, and transcendence-toll saturation with damping n!/π|n|−33|n|−3 for |n| ≥ 4.These constraints force a unique spectral ladder Tn on integer rungs n ∈ Z. Selectedeigenvalues (or simple rational multiples fixed by the Basel regularisation ζ(2) = π2/6)reproduce the dimensionless combinations underlying the Planck constant h, the speed oflight c, the fine-structure constant α, and the gravitational constant G to better than 10−10relative precision against CODATA 2022, without adjustable parameters. The fundamentalconstants of physics are therefore exact eigenvalues of the circle.
Category: Geometry

[6] ai.viXra.org:2507.0098 [pdf] submitted on 2025-07-21 17:52:52

Hodge Conjecture as a Theorem of Functional Geometry

Authors: Maxim Govorushkin
Comments: 72 Pages. 10.5281/zenodo.16279209

In this paper we develop a fully constructive proof of the Hodge Conjecture for rational classes of type ((p,p)) on a smooth projective complex manifold (M). Rather than relying on deep motivic or trans-cendental arguments, we introduce a new emph{functional—geometric} (FG) framework that isolates the key analytical—algebraic steps into four emph{primitive operations}:begin{enumerate} item textbf{FG-GAGA:} uniformly approximate local holomorphic FG—functions by polynomials (Q(y)) with explicit remainder estimates, and show that the zero—sets and radicals of the ideal are preserved (Lemma 1.1, Appendix A.2). item textbf{FG-Nullstellensatz:} represent (1) in a radical ideal via polynomial combinations of the (Q)’s, with a modern effective degree bound (deg g_ile (8d)^{2n+1}) (Lemma A.4). item textbf{FG-Resolution:} resolve singularities of the local complete—intersection cycle by a functorial sequence of FG—blow—ups, stopping in at most (Nle C'(n)(d+1)^n) steps and yielding a smooth strict transform with normal crossings (Lemma A.3.5). item textbf{FG-Algebraization:} assemble the smooth strict—transform CI—cycle into a global algebraic cycle (Z_{m alg}subsetPP^N) of controlled degree, matching the original fundamental class (Theorem 4.1, Appendix A.4).end{enumerate}Using these primitives we implement an eight—step pipeline:begin{itemize} item Construct a local analytic cycle from the positive current (T=w^p) via Siu—Demailly decomposition (Appendix B). item FG-GAGA approximates its defining holomorphic equations. item FG-Nullstellensatz provides a unity—decomposition in the ideal. item FG-Resolution produces a smooth polynomial CI—cycle of the same class. item Global Čech—gluing and Serre Vanishing ((H^1(M,OO(d))=0)) produce a single global system of equations. item Denominator cleaning via (N=mathrm{lcm}{mathrm{den}(a_sigma)}) yields an integral CI-cycle. item Phantom classes are ruled out by a meromorphic asymmetry functional (Phi) with Zariski—dense vanishing locus (Appendix D). item Analytic continuation on each irreducible component of the Hodge locus establishes no remaining obstructions.end{itemize}We prove that every rational ((p,p))—class (ain H^{2p}(M,Q)) is represented by a genuine algebraic (p)—cycle (Z) with ([Z]=a). The entire procedure is accompanied by explicit degree and complexity estimates, a Lean4 formalization of key lemmas (Cech—gluing, flat FG—connection), and a Colab notebook verifying FG—cycle computations on classical test—cases (Fermat—K3, Noether—Lefschetz surfaces). Our FG formalism thus provides a emph{flat}, algorithmic language for Hodge theory, bypassing standard motivic conjectures and opening the door to effective computation of Hodge classes in concrete geometric settings.
Category: Geometry

[5] ai.viXra.org:2507.0078 [pdf] submitted on 2025-07-14 21:17:11

Functional Coordinate System (Fcs)bottom-up Functional Geometry: from Dynamic Axes and SVD to Classical Riemann—Cartan Formalism

Authors: Maxim Govorushkin
Comments: 81 Pages. 10.5281/zenodo.15837404

This paper introduces and studies in detail Functional Geometry (FG) — an independent "bottom-up"formalism, in which instead of a predetermined metric tensor, the primary ones are the dynamic matrices of local "axes"Xij(t) and their integral synchronization condition. FG specifies its own affine connection, torsionand curvature forms through the Maurer-Cartan formula w = X−1(t)dX(t), without a preliminary choice of metric. At the same time, it uniquely reproduces the classical Riemannian (or Riemannian—Cartanian, if torsion is taken into account) structure on a smooth manifold M: it reconstructs the metric tensor, Christoffelsymbols, and curvature tensor, proving their invariance under smooth transformations.In the case of an asymmetric matrix X(t), the smooth singular value decomposition (SVD) ensures the inclusion of torsion and the continuity of the constructions. Thus,FG is not just an extension of the classical formalism, but an independent geometrywith its own internal structure, strictly equivalent to Riemannian geometry.
Category: Geometry

[4] ai.viXra.org:2506.0102 [pdf] submitted on 2025-06-23 20:43:13

The Known Circle: A Physical Thought Experiment Challenging Symbolic Incompleteness

Authors: Christopher R. Parks
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)

The Known Circle proposes that a perfect physical circle, formed and measured in an idealized system, can yield an exact, finite area through proportional mass — without invoking p as an infinite symbolic constant. By treating p as an emergent product of measurable systems, the paper challenges the assumption that circular area must remain symbolically incomplete. This is not a numerical approximation, but a conceptual challenge to the structure of mathematical limits andtheir dependence on symbolic infinity.
Category: Geometry

[3] ai.viXra.org:2506.0064 [pdf] submitted on 2025-06-16 02:08:34

Two-Sided Symmetry and Holonic Maps: From Koestler’s Holarchy to Intuitionist Geometry and Archetypal Resonance

Authors: Stephen P. Smith
Comments: 11 Pages. (Note by ai.viXra.org Admin: Please cite and list sceintific references)

This paper explores Arthur Koestler’s concept of Janus-faced holons within a dynamic holarchy, integrating insights from CPT symmetry, intuitionist mathematics, Michael Schneider’s generative geometry, and Karl Friston’s free energy principle. It critiques static holonic diagrams and proposes a more resonant, fractal, and bilaterally symmetrical mapping rooted in archetypal forms. Drawing from sacred geometry, musical structures, and biological patterns, the essay argues that reality unfolds through intuitive construction and pre-existing mathematical orders. Holonic development, like nature itself, reflects a deep, two-sided balance—unifying form, transformation, and perception in a cosmological vision where cognition participates in creation.
Category: Geometry

[2] ai.viXra.org:2506.0016 [pdf] submitted on 2025-06-04 22:14:30

Observer-Centric Internal Spherical Geometry

Authors: Larry Whitaker
Comments: 9 Pages. (Note by ai.viXra.org Admin: Please cite listed sceintific references)

We present a practical mathematical framework for interior spherical geometry that en-ables complete sphere characterization and three-dimensional positioning from an observer-centric perspective. The methodology addresses fundamental limitations in traditionalspherical geometry by eliminating dependence on external reference points and establish-ing the observer as the coordinate system origin. Our approach integrates perpendicularchord measurement techniques with computational algorithms to provide accurate interiorpositioning for practical applications. The framework introduces measurement protocols forgeometric calculations, establishes cone-based coordinate systems with scalable precision,and addresses measurement uncertainty propagation. Applications include navigation systems for enclosed environments, geometric modeling of spherical structures, and scientificinstrumentation requiring interior perspective calculations.
Category: Geometry

[1] ai.viXra.org:2504.0071 [pdf] submitted on 2025-04-19 22:56:37

Study by Vector Projection of a Construction Made with a Rule and Compass to Understand Squaring the Circle

Authors: José Rodrigo Alejandro Martinez Diaz
Comments: (Improper cover page stripped by ai.viXra.org Admin)

The conditions of the problem establish the definition of a right triangle whose angle subtends the opposite cathetus with a value equal to alpha = arctan (1/ sqrt{pi}). Starting from this main condition, a series of steps are proposed to compare the precision between two idealized methods that describe the construction of a circle and a square linked to the triangle mentioned above. The first approximation is made taking into account "valid" compass and ruler constructions to obtain such geometric figures. The second construction is based on the lengths obtained by intersecting the previously mentioned geometric figures by performing the vector analysis. In both cases, the areas of the circle and square obtained are compared to find out if they are equal.
Category: Geometry

Replacements of recent Submissions

[1] ai.viXra.org:2506.0064 [pdf] replaced on 2025-06-20 06:48:42

Two-Sided Symmetry and Holonic Maps: From Koestler’s Holarchy to Intuitionist Geometry and Archetypal Resonance

Authors: Stephen P. Smith
Comments: 13 Pages.