The Spiral Geometry — Global Geometric Structure of the Universe
Year: 2026
Type: preprint
Abstract: THE SPIRAL GEOMETRY — GLOBAL GEOMETRIC STRUCTURE OF THE UNIVERSE DUAL GEOMETRIC ORIENTATION, OPPOSITE MOTIONS, ALTERNATING TORSION AND A DYNAMIC FORMULATION OF GRAVITY Overview This preprint presents The Spiral Geometry — Global Geometric Structure of the Universe, a theoretical framework that formulates a unified geometric architecture based on circular structure, dual orientation, fundamental axes, temporal scale, logarithmic spiral geometry, hierarchical alternation, relative angular motion, radial acceleration, gravitational transduction, and Newtonian mass–structure closure. The architecture begins with a complete circular structure of 360∘360^\circ, divided into two opposite semicircular orientations of 180∘180^\circ, represented by the states s=±1s=\pm1. Two fundamental perpendicular lines, HH and VV, organize the structure. The HH-axis represents time, while the VV-axis represents the gravitational axis. Their intersection defines the equilibrium point H∩V=12.H\cap V=\frac{1}{2}. Time is treated as the primary scalar variable of geometric evolution. The fundamental temporal scale is A(t)=a0eHt.A(t)=a_0e^{Ht}. The temporal variable generates the scale through which the geometry develops. From this scale, the fundamental logarithmic spiral is defined by r(t,θ)=A(t)eθ.r(t,\theta)=A(t)e^\theta. The architecture admits two strictly opposite orientations, r+(t,θ)=A(t)eθ,r−(t,θ)=A(t)e−θ.r_+(t,\theta)=A(t)e^\theta, \qquad r_-(t,\theta)=A(t)e^{-\theta}. Their product and ratio are r+r−=A2(t),r+r−=e2θ.r_+r_-=A^2(t), \qquad \frac{r_+}{r_-}=e^{2\theta}. The two orientations are therefore treated as opposite states of the same underlying geometric structure. Hierarchical Alternation and Torsion The spiral structure is extended through hierarchical levels whose orientations alternate according to sN=−sN−1,s_N=-s_{N-1}, or equivalently, sN=(−1)N.s_N=(-1)^N. The four principal levels are organized as Atom, Planet Earth, Galaxy, and Cosmos/External Structure, with s0=−1,s1=+1,s2=−1,s3=+1.s_0=-1,\qquad s_1=+1,\qquad s_2=-1,\qquad s_3=+1. Each level possesses the angular velocity ωN=sNω∗,\omega_N=s_N\omega^*, where ω∗\omega^* is the fundamental angular magnitude. At every interface between adjacent levels of opposite orientation, the relative angular velocity has constant magnitude ∣ωrel∣=2ω∗.|\omega_{\mathrm{rel}}|=2\omega^*. This constant relative motion constitutes the fundamental kinematic element of the proposed gravitational transduction. Gravitational Transduction For an interface with characteristic radius RNR_N, the radial acceleration associated with the relative angular motion is gN=−RNωrel2.g_N=-R_N\omega_{\mathrm{rel}}^2. Since ∣ωrel∣=2ω∗,|\omega_{\mathrm{rel}}|=2\omega^*, the structural gravitational field becomes gN=−4(ω∗)2RN.g_N=-4(\omega^*)^2R_N. Defining K=4(ω∗)2,K=4(\omega^*)^2, the structural field can be written as gN=−KRN.g_N=-KR_N. Within the proposed architecture, gravity is interpreted as the radial manifestation, or kinematic transduction, of the relative angular motion generated at interfaces between adjacent hierarchical levels of opposite orientation. Newtonian Closure and Mass–Structure Relation The structural gravitational field is closed against the classical Newtonian expression gN=−GMNRN2.g_N=-\frac{GM_N}{R_N^2}. Equating the two forms gives 4(ω∗)2RN=GMNRN2,4(\omega^*)^2R_N = \frac{GM_N}{R_N^2}, and therefore MN=4(ω∗)2GRN3.M_N=\frac{4(\omega^*)^2}{G}R_N^3. The architecture thus produces a cubic mass–radius relation, MN=CMRN3,M_N=C_MR_N^3, with CM=4(ω∗)2G.C_M=\frac{4(\omega^*)^2}{G}. This relation constitutes the algebraic mass–structure closure of the proposed geometric model. Space-Time and Metric Formulation The geometric architecture is formulated within a single four-dimensional pseudo-Riemannian state, ds2=c2dτ2=gμνdxμdxν,ds^2=c^2d\tau^2=g_{\mu\nu}dx^\mu dx^\nu, with xμ=(ct,x,y,z).x^\mu=(ct,x,y,z). The temporal and spatial components are therefore treated as components of the same geometric state. In the nonrelativistic weak-field limit, the spatial geodesic equation yields ai=−c22∂ig00.a_i=-\frac{c^2}{2}\partial_i g_{00}. With g00=1+2Φc2,g_{00}=1+\frac{2\Phi}{c^2}, the classical relation follows: a=−∇Φ.\mathbf a=-\nabla\Phi. For the Newtonian potential Φ=−GMr,\Phi=-\frac{GM}{r}, the corresponding gravitational acceleration is recovered as a=−GMr2r^.\mathbf a=-\frac{GM}{r^2}\hat{\mathbf r}. The metric route and the spiral-kinematic route are therefore represented as two formulations within the same geometric state: the metric formulation connects the temporal component g00g_{00} with spatial acceleration, while the spiral formulation connects relative angular motion with radial gravitational acceleration. Unified Structural Chain The complete architecture can be summarized as 360∘→H/V→A(t)→rs→sN→ωN→ωrel→gN→MN.360^\circ \rightarrow H/V \rightarrow A(t) \rightarrow r_s \rightarrow s_N \rightarrow \omega_N \rightarrow \omega_{\mathrm{rel}} \rightarrow g_N \rightarrow M_N. The circular structure provides the fundamental duality; the HH-axis establishes time; the temporal scale establishes geometric evolution; the logarithmic spiral provides the geometric form; alternating orientation establishes the hierarchical direction; the interface generates relative angular motion; kinematic transduction produces radial acceleration; and Newtonian closure produces the mass–structure relation. The architecture consequently places time and gravity within the same geometric structure, with gravity represented as the spatial manifestation of temporal structure in spiral motion. Empirical Closure and Predictive Program The Addendum defines the next stage of the framework through four principal points for empirical and theoretical development. 1. Determination of the Master Constant ω∗\omega^*. The fundamental angular magnitude is to be determined from the structural closure relation using the Earth level as the reference scale. The resulting value is then maintained across the hierarchical interfaces of the model. 2. Cubic-Law Test and Effective Interface Mass. The relation MN=CMRN3M_N=C_MR_N^3 is to be examined across hierarchical scales using the proposed Effective Interface Mass, defined as the mass associated with the structure participating in gravitational transduction between adjacent levels. 3. Strong-Relativistic Limit — The Spiral Horizon. The relative transduction velocity is defined by v=∣ωrel∣R=2ω∗R.v=|\omega_{\mathrm{rel}}|R=2\omega^*R. The limiting condition v=cv=c defines the proposed kinematic Spiral Horizon and establishes the strong-relativistic boundary of the spiral kinematics. 4. Identification of HH — Expansion of the Cosmos. The temporal scale A(t)=a0eHtA(t)=a_0e^{Ht} is examined in relation to the cosmological expansion parameter H0H_0, establishing the proposed connection between the temporal scale of the architecture and large-scale cosmic expansion. Complementary Predictive Validation The Addendum further identifies a complementary validation stage associated with the continuation of the empirical program. This stage includes the determination of the exact numerical value of the fundamental constant from the Earth scale, the construction of a fractal representation of the mass–radius structure, and a blind cross-scale prediction in which the Earth-derived value of ω∗\omega^* is maintained without recalibration and compared with independently measured quantities at other hierarchical scales. The blind cross-scale prediction is therefore part of the complementary validation program and is not a fifth principal point of the Addendum. Scope of the Work The preprint presents the mathematical and geometric architecture of the Spiral Geometry together with its gravitational transduction mechanism, Newtonian closure, metric formulation, hierarchical structure, and defined program for empirical validation. Its central proposition is that circular duality, temporal scale, spiral orientation, hierarchical alternation, interface dynamics, radial acceleration, and gravitational structure can be represented as interconnected components of a single geometric framework. The resulting architecture establishes a continuous structural chain from 360∘360^\circ circular duality through temporal generation and spiral geometry to gravitational acceleration and mass–structure closure, providing the basis for the proposed geometric formulation of the universe.
Author: Vicente da Silva, Cláudio
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204910784
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