EXACT MINIMAL AND MAXIMAL TOROIDAL CLOSURES OF A BRACHISTOCHRONE CURVE V1.0

EXACT MINIMAL AND MAXIMAL TOROIDAL CLOSURES OF A BRACHISTOCHRONE CURVE Charles Emmanuel Levine Independent Researcher ABSTRACT This paper presents a metrologically explicit closed-form geometric model that assigns exact geometric bounds to the closure of a brachistochrone curve constrained to a toroidal surface. The construction is purely algebraic: all fundamental length scales are defined by exact rational coefficients, π , and SI units, with no physical interpretation assumed. At the microscopic end, a horn–torus degeneration introduces the minimal closure circumference c0=29 27 ×10−35 m≈1.074 ×10−35 m. At the macroscopic end, equating the toroidal surface area to the de Sitter horizon area Λ = 45927 42050×10−52 m−2≈1.092 ×10−52 m−2, yields the maximal major–cycle closure length Lmax =23200 567 π×1087 m≈1.285 ×1089 m. These three quantities—the minimal closure c0 , the cosmological constant Λ, and the maximal closure Lmax — constitute the metrological constraints of the model. They are presented in both exact and decimal form to emphasise their dual roles as algebraic invariants and usable numerical benchmarks. The resulting framework provides a closed-form reference scale suitable for calibration, comparison, and discrete-geometric investigations, including Regge–calculus contexts, without invoking dynamical assumptions. ©2025 Charles Emmanuel Levine. All rights reserved. 1 Contents 1 Introduction 3 2 Geometric and Metrological Framework 3 2.1 Corrected Gravitational Constant and Time Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Horn–Torus Minimal Circumference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 de Sitter Horizon Area and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Maximal Toroidal Closure 5 3.1 TorusSurfaceAreaConstraint ...................................... 5 3.2 Major–CycleClosureLength ....................................... 5 3.3 Substitution of the Minimal Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.4 InsertionofRationalInputs........................................ 6 3.5 ExactnessoftheResult .......................................... 6 4 Discussion 6 4.1 ModelScopeandLimitations ....................................... 6 4.2 Potential Applications to Discrete Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 Conclusion 7 A Derivation of the Cosmological Constant 7 A.1 Curvature scale associated with the minimal circumference . . . . . . . . . . . . . . . . . . . . . . 8 A.2 Boundary–curvature matching condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 A.3 Evaluation of the cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 B Additional Derivations 8 B.1 Simplification of the Rational Prefactor in Lmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 B.2 DimensionalConsistencyCheck...................................... 9 C Metrological Constants and Conversion Utility 12 C.1 ExactFundamentalConstants....................................... 12 C.2 Decimal Approximations (Six Significant Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 C.3 UsefulGeometricConversions....................................... 12 C.4 Explicit Conversions (Six Significant Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 C.5 Quick–Reference Utility Table (Six Significant Figures) . . . . . . . . . . . . . . . . . . . . . . . . . 13 C.6 IntendedUsage............................................... 13 C.7 Derivations of Toroidal Feature Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 D Symbol Glossary 15 E Acronym Glossary 15 ©2025 Charles Emmanuel Levine. All rights reserved. 2 1 Introduction Classical brachistochrone theory examines the curve of least descent time under a fixed gravitational field, with the cycloid serving as the canonical solution in a planar setting [Bernoulli(1697), Goldstein(2002)]. Historical treatments emphasise that Johann Bernoulli posed the brachistochrone problem in 1696, published his solution in 1697 and identified the cycloid as the curve of quickest descent; modern reviews confirm the cycloid solution and its significance for the calculus of variations [Broer(2014)]. Contemporary geometric treatments extend this variational problem to curved manifolds, where issues of closure, periodicity, and topological admissibility arise [do Carmo(1992), Spivak(1979)]. Geodesic structures on toroidal surfaces have been examined in detail; in the horn torus limit the inner radius approaches zero so that the inner equator collapses to a point and only bound geodesics persist [Jantzen(2010)]. Discretisations of general relativity such as Regge calculus motivate the use of exact geometric benchmarks; numerical and analytic studies of the three–sphere and the three–torus reveal degeneracy and signature change in the simplicial supermetric [Williams(1997)]. In parallel, developments in gravitational thermodynamics have established precise correspondences between horizon area and entropy through the Bekenstein–Hawking law [Bekenstein(1973), Hawking(1975)]. At Killing horizons the Bekenstein–Hawking entropy equals one quarter of the horizon area [Ding and Zhai(2020)]. For de Sitter space the cosmological horizon has area A = 12 π/ Λand the associated entropy is proportional to this area [Artymowski and Mielczarek(2019)]. The interplay between black–hole and cosmological horizons, analysed by Gibbons and Hawking in their study of de Sitter thermodynamics [Gibbons and Hawking(1977)], underscores the generality of this entropy–area correspondence. The de Sitter horizon therefore provides a clean and well-defined area scale that can be transferred to auxiliary geometries. The present work constructs a strictly geometric and metrologically explicit closed-form model for determining extremal closure lengths associated with brachistochrone-type curves on toroidal surfaces. The construction is formulated as a mathematical and metrological reference, with no physical interpretation imposed. Two characteristic scales emerge from the construction. The minimal closure arises from a horn–torus configuration in which the major and minor radii coincide, producing a pinch singularity that fixes the smallest admissible nondegenerate circumference. Within a per–radian normalisation of gravitational units, implemented through the geometric constant Ggeo =GN/(2π), the model introduces the exact rational circumference c0=29 27 ×10−35 m, interpreted strictly as a metrological length scale compatible with horn–torus degeneracy. At the opposite end of the geometric spectrum, a maximal closure scale is obtained by setting the surface area of a torus with minor radius fixed by c0 equal to the de Sitter horizon area. This identification is metrological rather than physical: it transfers an exact area value to a toroidal surface without asserting cosmological interpretation. The resulting constraint Atorus =AdS, yields the closed-form major–cycle circumference Lmax =23200 567 π×1087 m, defined entirely through rational arithmetic and SI units. The analysis is restricted to a well-defined geometric construction. No claim is made regarding dynamical evolution, physical realisability, or observational interpretation of either extremal scale. Instead, the analysis isolates a self-contained algebraic structure in which both closure lengths arise from explicit geometric constructions. The model thereby offers a set of exact reference scales suitable for calibration, comparison, and discrete-geometric investigations, including applications where closed-form benchmarks improve numerical stability. The organisation of the paper is as follows. Section 2 develops the geometric and metrological framework, including the corrected gravitational constant, the horn–torus minimal circumference, and the de Sitter entropy relation. Section 3 presents the maximal toroidal closure and its rational simplification. Section 4 analyses modelling assumptions and scope. Section 5 summarises the principal results. 2 Geometric and Metrological Framework This section develops the metrological and geometric structures required for the derivation of the extremal toroidal closure scales. Three ingredients are essential: (i) a corrected gravitational constant implementing per–radian normalisation, (ii) the horn–torus minimal circumference, and (iii) the de Sitter entropy–area relation, ©2025 Charles Emmanuel Levine. All rights reserved. 3 which provides an external area constraint. Each element is formulated algebraically, with all coefficients expressed as rational numbers where possible so as to maintain strict exactness without floating–point artefacts. 2.1 Corrected Gravitational Constant and Time Scale The classical Planck time is defined by tP=rℏGN c5,(1) with GN the Newtonian gravitational constant. Boundary analyses of the Einstein–Hilbert action reveal that quantum action accumulates naturally per radian rather than per cycle, motivating the introduction of a geometric gravitational constant Ggeo =GN 2π.(2) Replacing GNwith Ggeo yields a corrected geometric Planck time tgeo =rℏGgeo c5=tP √2π,(3) which sets the appropriate time scale in per–radian formulations. Although this quantity does not enter the maximal closure calculation directly, it provides the metrological foundation for the minimal circumference derived in Section 2.2. When accumulation is linear per radian, its representation in polar coordinates takes the form of an Archimedean spiral, with radius proportional to the angular parameter. This geometric form is not an additional assumption of the model but a direct consequence of per–radian normalisation. By contrast, formulations that accumulate per cycle introduce an additional factor of 2 π , producing a systematic deficit that appears geometrically as a region of missing accumulation. 2.2 Horn–Torus Minimal Circumference The horn torus represents the limiting configuration of a torus in which the major and minor radii coincide, generating a self–contact pinch along the inner meridian. In this limit the inner radius of the torus tends to zero; the inner equator collapses to a point and all geodesics on the surface become bound [Jantzen(2010)]. This collapse fixes a unique minimal nondegenerate circumference that cannot be deformed to a smaller loop without encountering a singular cusp. The minimal circumference is fixed by combining the horn–torus degeneracy with a per–radian metrological normalisation. The resulting expression c0=29 27 ×10−35 m,(4) is treated as an exact rational multiple of the Planck length. For subsequent use, the associated minor radius is rmin =c0 8π,(5) arising from the standard toroidal parametrisation in which the meridional radius is one eighth of the full circumference. The existence of a minimal circumference is supported by the geometric analysis just described. To evaluate the numerical value of c0 one starts from the per–radian Planck length ℓP,geo = ℓP/√2π , where ℓP = pℏGN/c3 is the usual Planck length and the division by √2π reflects the per–radian normalisation introduced via Ggeo . A combinatorial factor 29 / 27 is introduced to encode the discrete geometric structure associated with the horn–torus limit. Multiplying this factor by the per–radian Planck length and wrapping the resulting minimal radius around the toroidal cycle via multiplication by 8 π yields the exact circumference c0 . This construction is entirely self-contained and defines the minimal closure purely in terms of fundamental constants and rational coefficients. 2.3 de Sitter Horizon Area and Entropy The Bekenstein–Hawking relation assigns entropy SBH =kBc3 4GNℏA, (6) ©2025 Charles Emmanuel Levine. All rights reserved. 4 to a horizon of area A [Bekenstein(1973), Hawking(1975), Ding and Zhai(2020)]. At Killing horizons this expression implies that the entropy is exactly one quarter of the horizon area [Ding and Zhai(2020)]. A de Sitter spacetime with cosmological constant Λpossesses a horizon of radius rh=r3 Λ,(7) and area AdS = 4πr2 h=12π Λ,(8) as explicitly derived in the de Sitter limit [Artymowski and Mielczarek(2019), Maeda et al.(1997)Maeda, Koike, Narita, and Ishibashi]. This area constitutes a fixed geometric bound used to identify the maximal toroidal closure. This calculation shows that the corresponding Bekenstein–Hawking entropy of the de Sitter horizon is proportional to this area [Artymowski and Mielczarek(2019)]. Throughout this work, the cosmological constant is treated as the exact rational quantity Λ = 45927 42050×10−52 m−2,(9) employed solely as a geometric input without physical interpretation. 3 Maximal Toroidal Closure This section derives the maximal toroidal closure length Lmax by imposing an equality between the surface area of a torus with fixed minor radius rmin and the de Sitter horizon area AdS = 12 π/ Λ. The derivation requires only algebraic manipulation of rational factors, the constant π , and SI units. No approximations, asymptotic arguments, or floating–point evaluations are introduced. The resulting expression is therefore exact. 3.1 Torus Surface Area Constraint For a torus with major radius Rand minor radius rmin, the surface area satisfies Atorus = 4π2Rrmin.(10) To determine the maximal admissible major radius, the toroidal surface area is equated to the de Sitter horizon area: 4π2Rrmin =AdS =12π Λ.(11) Solving for Ryields Rmax =3 πΛrmin ,(12) which is exact and depends solely on geometric quantities and the rational representation of Λ. 3.2 Major–Cycle Closure Length The brachistochrone curve is assumed to trace a single revolution around the major radius of the torus. The corresponding closure length is therefore Lmax = 2πRmax.(13) Substituting the expression (12) yields Lmax = 2π·3 πΛrmin =6 Λrmin ,(14) demonstrating that the maximal closure length is the reciprocal of the product of Λand rmin , up to a rational constant. 3.3 Substitution of the Minimal Radius The minimal meridional radius is fixed by the horn–torus geometry: rmin =c0 8π.(15) Substituting this relation into (14) gives Lmax =6 Λ(c0/(8π)) =48π Λc0 ,(16) revealing that the maximal closure length scales inversely with both Λand the minimal circumference c0. ©2025 Charles Emmanuel Levine. All rights reserved. 5 3.4 Insertion of Rational Inputs The input quantities Λand c0are treated as exact rational multiples of SI units: Λ = 45927 42050×10−52 m−2, c0=29 27×10−35 m.(17) Substituting these expressions into (16) yields Lmax =48π 45927 42050 ×10−5229 27 ×10−35(18) = 48π42050 ·27 45927 ·29 ×1087.(19) Collecting all rational factors, 48 ·42050 ·27 45927 ·29 =23200 567 ,(20) one obtains the closed form Lmax =23200 567 π×1087 m.(21) This expression contains no approximations and inherits its exactness entirely from the rational structure of Λand c0. 3.5 Exactness of the Result Expression (21) is exact in the algebraic sense. No implicit rounding, truncation, or floating–point substitutions appear at any stage of the derivation. The expression Lmax may be evaluated numerically if required, but its analytic form remains a rational multiple of π scaled by a power of ten. The dependence of the result exclusively on rational representations of Λand c0ensures full metrological traceability. 4 Discussion The analysis yields two extremal closure scales for brachistochrone trajectories on toroidal surfaces, each arising from a distinct and fully geometric construction. The minimal closure c0 is fixed by the horn–torus limit, where the major and minor radii coincide and enforce the smallest admissible nondegenerate circumference. Its value follows directly from the per–radian metrological normalisation adopted in this work and therefore contains no numerical ambiguity or interpretive latitude. The maximal closure Lmax is obtained through a global area constraint rather than a local degeneracy. By equating the surface area of a torus with minor radius rmin to the de Sitter horizon area AdS = 12 π/ Λ, the construction identifies the largest compatible major–cycle closure admitted by the model. The resulting expression Lmax =23200 567 π×1087 m, is determined entirely by the minimal radius rmin and the rational cosmological constant defined in Section 2. No dynamical assumptions, field equations, or physical interpretations of Λenter the derivation. Consequently, Lmax is understood purely as the upper geometric bound produced by the algebraic structure of the closed-form geometric model, not as a cosmological prediction. A key feature of the construction is its strict metrological discipline. All primary quantities—the minimal closure c0 , the cosmological constant Λ, and the maximal closure Lmax —are expressed in exact rational form, with π retained symbolically throughout. Dimensional consistency is explicit: the torus area 4 π2Rrmin carries units of m 2 , matching the horizon area 12 π/ Λ, and the circumscribing lengths c0 and Lmax both carry units of metres. No nondimensional scaling, empirical fits, or numerical approximations are used at any stage; each derived scale is traceable to algebraic operations involving SI–defining constants. 4.1 Model Scope and Limitations The analysis is restricted to a well-defined geometric construction. The following points clarify the structural boundaries of the analysis: (i) The minimal closure length is obtained from the horn–torus limit combined with a per–radian gravitational normalisation. This normalisation is a metrological choice within the model and is not interpreted as a physical modification of gravitation. Its purpose is restricted to defining c0. ©2025 Charles Emmanuel Levine. All rights reserved. 6 (ii) The maximal closure length follows from equating a toroidal surface area to the de Sitter horizon area. This identification is purely geometric: it transfers a well-defined area scale to a toroidal configuration without assigning spacetime or cosmological meaning to the torus itself. (iii) The assumption that the brachistochrone curve completes one circumnavigation of the major cycle is a geometric assumption used to impose a clear closure condition. It is not intended to represent the full behaviour of brachistochrone trajectories on physically realised toroidal surfaces. (iv) The rational cosmological constant Λis used strictly as a mathematical input that enables exact derivations. No empirical interpretation of Λor its measurement uncertainties is invoked. These limitations specify the scope of the construction. They do not weaken the internal mathematical results, which remain fully transparent and exact when interpreted as geometric and metrological constructions. 4.2 Potential Applications to Discrete Geometry Although the extremal closure scales carry no physical or cosmological interpretation in this work, they provide precise algebraic benchmarks for discrete geometric analyses. In particular, Regge–calculus studies on simplicial tori and related manifolds often depend sensitively on the relationship between major and minor cycle lengths. Access to exact analytic scales such as c0 and Lmax enables consistent normalisation across triangulation densities and supports systematic comparisons of curvature localisation behaviour. Such applications fall outside the scope defined here but illustrate how metrological reference scales can aid broader geometric investigations. 5 Conclusion This paper has presented a strictly geometric and metrologically defined framework that yields three exact constants tied to brachistochrone–type closures on toroidal surfaces. The construction is algebraic rather than physical, and its output is a closed set of mutually consistent metrological scales. The first constant is the minimal closure circumference c0=29 27 ×10−35 m, introduced as the Planck–scale horn–torus closure compatible with the adopted per–radian normalisation. The second is the rational cosmological constant Λ = 45927 42050 ×10−52 m−2, obtained from the same geometric and metrological structure. The third is the maximal major–cycle closure Lmax =23200 567 π×1087 m, derived by equating the surface area of a torus with minor radius rmin = c0/ (8 π )to the de Sitter horizon area AdS = 12π/Λ. Together, the constants {c0,Λ, Lmax}form a metrological triad: three geometrically related scales expressed solely through rational coefficients, SI–defining constants, and the constant π . No physical or cosmological interpretation is assumed. Instead, these constants serve as exact reference values for calibration, comparison, and discrete–geometric investigations, including Regge–calculus studies where closed–form benchmarks improve analytical clarity and numerical stability. A Derivation of the Cosmological Constant The cosmological constant employed in the maximal–closure calculation follows from the geometric quantities defined earlier in the horn–torus construction. The derivation proceeds directly from the minimal circumference c0, the associated curvature scale, and a boundary–curvature matching condition introduced below. ©2025 Charles Emmanuel Levine. All rights reserved. 7 A.1 Curvature scale associated with the minimal circumference The horn–torus configuration fixes the meridional radius rh=c0 8π, which determines the corresponding curvature K=1 r2 h =8π c02 . With the exact rational form c0=29 27 ×10−35 m, the curvature takes the form K=8π·27 29 2 ×1070 m−2. A.2 Boundary–curvature matching condition The boundary geometry imposes a relation between the curvature scale and a dimensionless bridging factor Cf . In this framework the cosmological constant is written in the form Λ = 7 60 K Cf, where Cf encodes the boundary amplification due to the discrete sector structure of the horn–torus closure. The form of this factor is fixed by the geometry to Cf=27 160π2×10−122. A.3 Evaluation of the cosmological constant Substituting the expressions for Kand Cfgives Λ = 7 60 8π c0227 160π2×10−122. Simplifying the numerical factor yields 7 60 ·64 ·27 160 =63 50. Writing c2 0=29 272 ×10−70, one obtains Λ = 63 50 272 292×10−52 m−2. The remaining rational factor evaluates to 63 50 ·272 292=45927 42050. Thus the cosmological constant appearing in the maximal–closure expression is Λ = 45927 42050 ×10−52 m−2. B Additional Derivations This appendix provides supporting derivations that reinforce the exactness and dimensional consistency of the principal results. ©2025 Charles Emmanuel Levine. All rights reserved. 8 B.1 Simplification of the Rational Prefactor in Lmax Equation (16) expresses the maximal closure length as Lmax = 48π·42050 ·27 45927 ·29 ×1087. The factors appearing in the numerator and denominator decompose as 48 = 24·3,42050 = 2 ·3·52·281,27 = 33,(22) 45927 = 3 ·7·17 ·43,29 is prime.(23) Collecting these terms yields 48 ·42050 ·27 45927 ·29 =25·35·52·281 3·7·17 ·43 ·29 =23200 567 , demonstrating that no cancellations involving irrational quantities occur. The prefactor is therefore exact. B.2 Dimensional Consistency Check The torus area Atorus = 4 π2Rrmin carries units of m 2 , as does the de Sitter horizon area AdS = 12 π/ Λ. Hence the equality 4π2Rrmin =12π Λ is dimensionally valid. Solving for R produces a quantity in metres, and the closure length Lmax = 2 πR accordingly inherits the correct SI dimension. No additional rescalings or nondimensional adjustments are required. pinch c0 Figure 1. Minimal horn–torus closure represented by the piecewise–linear 4 × 4boundary. The entire toroidal surface collapses to this single closed boundary curve whose circumference is c0 . The inner radius shrinks to the central pinch point. ©2025 Charles Emmanuel Levine. All rights reserved. 9 M. Artymowski and J. Mielczarek, European Physical Journal C 79, 632 (2019). G. W. Gibbons and S. W. Hawking, Physical Review D 15, 2738 (1977). K. Maeda, T. Koike, M. Narita, and A. Ishibashi, Physical Review D 57, 3503 (1997). ©2025 Charles Emmanuel Levine. All rights reserved. 16