METRIC REALISATION AND CURVATURE AMPLITUDE IN AN OVERCONSTRAINED 21:16 REGGE LATTICE V1.0

METRIC REALISATION AND CURVATURE AMPLITUDE IN AN OVERCONSTRAINED 21:16 REGGE LATTICE Charles Emmanuel Levine Independent Researcher ABSTRACT A minimal metrological model for incidence–driven curvature begins with a stepped subdivision of a square domain refined by diagonal simplices. In its smallest nontrivial instance—a 4 × 4array of square cells—the refinement produces a canonical three–layer (1 , 1 , 3) boundary and forces twenty–one regions into a frame of sixteen. This fixed 21 : 16 overpacking leaves a combinatorial surplus that can be accommodated only by concentrating curvature at four symmetry–related hinges. The same boundary layering also partitions the Regge deficit at each hinge into three contributions in the exact ratio 1 : 1 : 3. A symmetric isosceles surrogate metric gives closed–form expressions for the associated lift and curvature amplitude, showing that although the metric is not uniquely determined, the pattern of curvature is fixed entirely by the boundary data. In this respect the configuration functions as a minimal metrological reference for incidence–driven curvature, providing a benchmark geometry in which the curvature response is uniquely determined by boundary constraints. A planar realisation of a minimally closed cycloidal horn–toroid carries the same (1 , 1 , 3) boundary pattern, establishing a direct geometric link to the stepped lattice. Viewed in this way, the 21 : 16 configuration acts as a discrete representative of the boundary invariant of the flattened horn–toroid, giving a minimal, boundary–set model for how curvature arises in an overconstrained simplicial complex. Keywords:Regge calculus; discrete curvature; angular deficit; combinatorial geometry; simplicial lattices; overconstrained meshes; metric embedding; curvature amplitude ©2025 Charles Emmanuel Levine. All rights reserved. 1 Contents 1 Introduction 3 2 Lattice structure and combinatorial overpacking 3 3 Regge deficit and curvature distribution 5 4 Regge calculus for the inverse subdivision pattern 7 4.1 Angular budget and positive deficit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2 Symmetry and localisation of curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Inverse hinge geometry and metric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.4 Inversecurvatureamplitude........................................ 8 5 Metric realisation and illustrative lift 9 5.1 Isoscelestrianglemodel .......................................... 9 5.2 Illustrative three–dimensional embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.3 Curvatureamplitude............................................ 10 6 Discussion 11 7 Conclusion 11 A Angle–lift relation without symmetry 12 B Boundary–Induced Curvature Determination 12 B.1 Boundaryconstraints............................................ 12 B.2 Boundary identification of curvature hinges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 B.3 Reconstruction of the deficit distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 B.4 Inverse subdivision pattern and deficit duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 B.5 The 1–1–3 boundary pulse per quadrant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 C Correspondence Between the Flattened Horn–Toroid Geometry and the 21:16 Regge Lattice 16 C.1 Flattened horn–toroid geometry and its boundary decomposition . . . . . . . . . . . . . . . . . . . 16 C.2 Justification of the boundary correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 C.3 Boundary structure of the stepped 4×4Reggelattice ......................... 17 C.4 Geometric correspondence of the two constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 D Reproducibility and Traceability 18 E Glossary of symbols 19 ©2025 Charles Emmanuel Levine. All rights reserved. 2 1 Introduction Curvature in discrete settings is described using the framework of Regge calculus[Regge(1961)], in which a manifold is replaced by a simplicial complex and curvature is concentrated at hinges. In two dimensions the hinges are vertices, and the associated Regge deficit measures the deviation of the sum of incident angles from 2 π . Because these deficits depend only on the incidence relations of the underlying mesh, the resulting curvature structure is determined by combinatorial data rather than by any choice of metric; this feature has made Regge calculus central to discrete differential geometry and numerical relativity[Williams and Tuckey(1992), Gentle(2002), Bobenko and Suris(2008), Bobenko and Springborn(2023)]. The present work examines a minimal configuration in which curvature arises solely from a boundary–driven combinatorial incompatibility. A 4 × 4lattice equipped with a diagonal subdivision rule generates twenty–one subdivision regions inside a frame containing sixteen square cells. This 21 : 16 overpacking produces a fixed combinatorial surplus that forces curvature to localise at four symmetry–related vertices and renders the curvature distribution uniquely determined by the stepped boundary. A direct analysis of the boundary shows that each quadrant decomposes into a canonical three–layer (1 , 1 , 3) structure, and this decomposition induces an exact three–term (1 : 1 : 3) partition of the Regge deficit at every curvature hinge. The curvature behaviour of the lattice is therefore encoded entirely by a single boundary invariant, while the metric geometry remains otherwise unconstrained. In this respect the configuration acts as a minimal metrological reference for incidence–driven curvature, providing a benchmark geometry in which the curvature response is fixed uniquely by boundary data. A planar realisation of a minimally closed cycloidal horn–toroid exhibits an identical (1 , 1 , 3) boundary decomposition. This coincidence permits a direct geometric correspondence between the stepped lattice and the flattened horn–toroid boundary, through which the 21 : 16 configuration may be regarded as a discrete realisation of the same underlying two–dimensional structure. In this sense the lattice serves as a minimal, fully boundary– determined reference model for curvature localisation, boundary–to–bulk determinacy, and the behaviour of overconstrained simplicial complexes. To the best of the author’s knowledge, no prior work has explicitly analyzed this 21 : 16 configuration or demonstrated how boundary layering alone determines curvature distribution in such a minimal lattice. This study therefore represents the first metrological treatment of this specific combinatorial overpacking. 2 Lattice structure and combinatorial overpacking Consider a square region partitioned into a 4 × 4grid of planar cells of side length a . The underlying vertex set is V={(i, j)|i, j ∈ {0,1,2,3,4}},(1) yielding sixteen quadrilateral subdivision regions in the coarse frame. A diagonal subdivision rule is applied within each quadrant to generate twenty–one distinct subdivision regions inside the 4 × 4subdivision frame, which contains sixteen square cells; the remaining regions are not planar cells but pieces created by the diagonal pattern. Although the refined pattern contains eighty–four subdivision regions across the full lattice, the curvature behaviour is governed by a minimal 21 : 16 subsystem whose subdivision region –to–cell mismatch provides the fundamental angular surplus. Here and throughout the article the term ‘subdivision region’ refers to one of the twenty–one regions generated within the full 4 × 4frame by the diagonal rule; the diagrams isolate a single quadrant for clarity. Figure 1 shows the subdivision applied to a single quadrant. ©2025 Charles Emmanuel Levine. All rights reserved. 3 x+ y+ Figure 1. Diagonal subdivision rule applied to the upper–right quadrant of the 4 × 4lattice. The heavy solid lines denote the boundaries of the quadrant and the subdivision lines within it. The dashed diagonals show the simplicial refinement applied after the diagonal subdivision; the 21 count refers to the subdivision regions generated within the full 4 × 4frame; the diagram shows the pattern restricted to one quadrant. When this subdivision is mirrored across the horizontal and vertical axes, the full 4 × 4lattice arises. Figure 2 shows the piecewise–linear boundary of the full lattice. Figure 3 overlays the complete set of diagonals on this boundary, revealing the full simplicial refinement used throughout this work. The bold horizontal and vertical lines mark the boundaries between quadrants. Figure 2. Piecewise–linear boundary of the full 4 × 4lattice obtained by mirroring the single–quadrant subdivision across horizontal and vertical axes. Only the boundary lines are shown here; diagonals are displayed in Figure 3. The heavy axes mark the quadrant boundaries. ©2025 Charles Emmanuel Levine. All rights reserved. 4 Figure 3. Complete 4 × 4lattice with all diagonals inserted according to the subdivision rule. Thin dashed lines represent the full diagonal pattern, while heavy lines denote the quadrant boundaries. There are twenty–one subdivision regions generated inside the 4 × 4subdivision frame, which contains sixteen square cells. Mirroring the quadrant produces 84 subdivision regions across the full lattice; the simplicial refinement introduces additional triangles not enumerated here. The combinatorial surplus arises from the 21:16 region–to–cell overpacking present in this mesh. The mismatch between the number of subdivision regions and the number of square cells produces a combinatorial surplus that manifests as an angular excess at certain vertices. The diagonal subdivision rule generates twenty–one distinct subdivision regions within the footprint of a 4×4 cell frame, which contains sixteen square cells. The resulting surplus is therefore σ=N△−N□= 5,(2) a property of the subdivision rule itself rather than an additive contribution from any geometric partition of the full lattice. This 21:16 overpacking defines the fundamental source of angular surplus in the assembled mesh. In a planar embedding, the sum of angles around each vertex must equal 2 π . Exceeding this limit forces some vertex to carry a negative Regge deficit, εv= 2π−X ∆∈T(v) θ∆<0,(3) where T ( v )is the set of simplices incident on v and θ∆ denotes the interior angle of simplex ∆at v . The quantity εvtherefore measures the angular surplus at v. Geometrically, this means that too many triangular regions meet at the vertex to lie flat in the plane; the excess angle prevents an isometric planar embedding, requiring the vertex to be lifted out of the plane and producing a saddle-like structure indicative of negative curvature. 3 Regge deficit and curvature distribution Regge calculus assigns curvature by accumulating deficits at hinges. In the 21 : 16 lattice, hinges correspond to vertices where multiple simplices meet. The total angular surplus arises from the global incidence structure of the assembled lattice and is not attributable to any individual region of the mesh. The construction exhibits four principal directions aligned with the coordinate axes, and the combinatorial structure identifies four interior vertices where the incidence pattern accumulates angular surplus. Although these vertices form a natural symmetry–related set, Regge calculus imposes no requirement that the surplus be divided equally among them; the actual distribution depends on the complete incidence pattern of the assembled lattice. Equal allocation is therefore a modelling choice rather than a combinatorial consequence. ©2025 Charles Emmanuel Levine. All rights reserved. 5 Let H = {η1, η2, η3, η4} denote the four hinges where the surplus is concentrated. If εηi is the deficit at hinge ηi, symmetry implies εη1=εη2=εη3=εη4, 4 X i=1 εηi=εtot,(4) where εtot < 0is the net angular surplus. The combinatorial structure fixes εtot , but Regge calculus does not determine a unique metric deformation. In particular, no specific diagonal length or out–of–plane lift is fixed by the deficit alone; the metric remains free unless an explicit length assignment is made. These four representative hinges lie at symmetry–related interior vertices along the principal axes; they form a minimal symmetric set when their local incidence patterns coincide to distribute the total deficit. In Figure 3, they correspond to the central crossing nodes where the quadrant’s subdivision pattern is most concentrated. In the full lattice shown in Figure 3, the curvature assignment depends on the incidence of all diagonals in the assembled mesh. Mirroring the quadrant introduces additional diagonals along the axes, and these extra connections change the valences of vertices near the quadrant boundaries. The diagonals inserted by mirroring contribute to the hinge incidence on an equal footing with those present before reflection; the curvature is determined by the complete network of diagonals rather than by a preferred subset. Nevertheless, the incidence patterns within each mirrored quadrant are equivalent up to symmetry, so the distribution of curvature can still be analysed on a per–quadrant basis provided that the full set of diagonals is taken into account. x+ y+ Figure 4. Schematic reconstruction of the inverse subdivision pattern in Quadrant I based on adjacency relations determined by the subdivision rule. The diagram includes only the principal boundary and diagonal segments used in the inverse construction; secondary diagonals used solely for simplicial refinement are omitted; the figure represents adjacency structure only and is not a metric embedding. ©2025 Charles Emmanuel Levine. All rights reserved. 6 Figure 5. Inverse 4×4subdivision lattice (boundary only, no diagonals). 4 Figure 6. Full inverse 4 × 4lattice obtained by mirroring the Quadrant I inverse pattern across both axes. The sign pairs (+ , +),( −, +),( −,− )and (+ ,− )indicate the orientations used for reflection. Although the mirrored quadrants share the same incidence lists, the figure is schematic and not intended as an exact metric embedding. 4 Regge calculus for the inverse subdivision pattern The inverse subdivision pattern shown in Figures 4–6 reverses the adjacency relations that generate the 21 : 16 region–to–cell mismatch in the original lattice. Instead of overpacking subdivision regions into the fixed quadrant footprint, the inverse construction removes the interior diagonals responsible for producing the angular surplus. Because Regge curvature depends solely on the incidence of simplices, this inverse configuration produces a complementary curvature structure whose behaviour can be characterised explicitly. 4.1 Angular budget and positive deficit In the original 21 : 16 lattice, the number of triangular regions incident on certain vertices exceeds the planar value, producing angle sums X ∆∈T(v) θ∆>2π, ©2025 Charles Emmanuel Levine. All rights reserved. 7 and thus a negative deficit (angular surplus). The inverse lattice, by contrast, reduces the valence of each vertex relative to the overpacked case. Let Ninv △ ( v ) denote the number of simplices incident on v in the inverse configuration. A reduction in valence tends to lower the total angle sum around v , but whether the sum falls below the planar value 2 π depends on the actual incidence of simplices and their internal angles. If the reduced valence is sufficiently small, the angle sum satisfies X ∆∈T(v) θ∆<2π, and the resulting deficit εv= 2π−X ∆∈T(v) θ∆ is positive. Explicit enumeration of the incidence in the underpacked configuration considered here shows that the angle sums at the four symmetry–related hinge vertices are indeed below 2 π , yielding a positive Regge deficit. Thus the inverse pattern realises the curvature–dual of the 21 : 16 angular surplus configuration, but the sign change is conditional on the detailed incidence pattern rather than guaranteed by a mere reduction in valence. 4.2 Symmetry and localisation of curvature The polarity rules (+ , +),( −, +),( −,− ),(+ ,− )used to mirror the Quadrant I inverse pattern preserve the same fourfold symmetry as the direct subdivision. Therefore curvature remains localised at four symmetry–related hinge vertices: εη1=εη2=εη3=εη4, 4 X i=1 εηi=εinv tot >0. The total positive deficit εinv tot is determined by the combinatorial underpacking: eliminating the five extra subdivision regions of the overpacked case reduces the valence of certain vertices. Because the incident angles of the constituent triangles remain unchanged, any reduction in valence lowers the total angle sum around those vertices relative to the planar 2 π budget, producing a positive deficit. Whether this sum falls below 2 π depends on the detailed incidence pattern; in the inverse configuration considered here the underpacking is sufficient to yield a positive Regge deficit at the same four hinges as in the overpacked lattice. A positive deficit occurs only when the reduced incidence pattern lowers the total interior angle strictly below 2 π ; valence reduction alone is not sufficient. 4.3 Inverse hinge geometry and metric interpretation Let ninv denote the number of congruent triangles meeting at a hinge in the inverse subdivision pattern. The interior angle at the hinge must satisfy θinv =2π−εinv η ninv . Because εinv η> 0, the hinge angle is strictly smaller than in the direct 21 : 16 lattice. A symmetric isosceles embedding may again be constructed by placing the hinge at height hinv relative to the boundary vertices. Repeating the law–of–cosines derivation gives hinv2=a21−cosh2π−εinv η ninv i 21 + cosh2π−εinv η ninv i. Because the deficit is positive, the metric realisation is typically concave relative to the direct (convex) lift. 4.4 Inverse curvature amplitude Define the dimensionless curvature amplitude for the inverse configuration by Ainv =hinv a. The amplitude increases as the hinge angle θinv decreases and becomes unbounded only in the formal limit θinv → 0 (equivalently, when εinv η→ 2 π ). By contrast, as εinv η→ 0the hinge angle tends to 2 π/ninv and the amplitude approaches a finite value, just as in the overpacked case. Larger positive deficits correspond to hinge angles closer to 2 π/ninv and therefore to smaller amplitudes. The inverse subdivision pattern thus realises the positive–curvature branch of the same geometric family and functions as the dual configuration to the overpacked 21:16 lattice. ©2025 Charles Emmanuel Levine. All rights reserved. 8 5 Metric realisation and illustrative lift The combinatorial analysis above fixes the existence and distribution of the Regge deficit but leaves the metric geometry unspecified: Regge calculus constrains only the angle sums, while the edge lengths remain free unless prescribed. A symmetric ansatz based on four congruent isosceles triangles meeting at a lifted vertex provides an explicit metric configuration that realises a non–zero deficit. Although the 21 : 16 lattice contains a more complex star of simplices at each hinge with valence greater than four, the four–triangle isosceles model serves as an illustrative surrogate that captures the qualitative metric freedom and yields closed–form relations among the deficit, the lattice spacing, and the out–of–plane lift. The construction is one admissible embedding compatible with the angular data rather than a unique realisation. 5.1 Isosceles triangle model This construction is a valence–four surrogate used solely to illustrate the metric–deficit relation; the full 21 : 16 lattice has a higher hinge valence and a more complex star of simplices. Consider four identical isosceles triangles meeting at a central hinge. Each triangle has two equal edges of length s connecting the hinge to boundary vertices and a base edge of length d lying in the plane. Denote the interior angle of each triangle at the hinge by θ . Because four such triangles meet around the hinge, the local deficit εhsatisfies 4θ= 2π−εh,=⇒θ=2π−εh 4.(5) The law of cosines applied to an isosceles triangle with equal sides sand base dgives cos θ= 1 −d2 2s2.(6) To relate s to the underlying lattice spacing and the out–of–plane lift, choose coordinates so that the central hinge is located at  0 , 0 , h and the four boundary vertices lie at ±a/ 2 ,±a/ 2 , 0  . The distance from the hinge to any boundary vertex is then s2=a2 2+h2,(7) and the base of each triangle is the diagonal of the square, d = √2a . Substituting d = √2a and s2 = a2/ 2 + h2 into Eq.(6) and rearranging with Eq.(7) yields a constraint relating the interior angle θ , the lattice spacing a , and the lift h:a2 2+h2=d2 21−cos θ=2a2 21−cos θ=a2 1−cos θ.(8) Solving for hgives h2=a2 1−cos θ−a2 2=a21 + cos θ 21−cos θ.(9) The right–hand side of Eq. (9) is non–negative whenever − 1 <cos θ < 1, ensuring the existence of a real lift. Combining Eq. (5) with Eq. (9) expresses the lift in terms of the deficit: h(εh)=av u u u t 1 + cosh2π−εh/4i 21−cosh2π−εh/4i.(10) Equation (10) is the central result of this subsection. It relates the lift directly to the interior angle at the hinge and, through Eq. (5), to the deficit. The lift decreases smoothly as the interior angle increases and becomes unbounded only in the extreme limit θ→ 0, where the four triangles flatten completely against one another. In contrast, as εh→ 0the interior angle tends to π/ 2and the lift approaches the finite value h = a/√2 . Thus there is no divergence of the lift as the deficit vanishes; the divergence arises only for deficits tending toward 2 π , which lies outside the regime relevant to this four–triangle surrogate. The surrogate model does not admit a real configuration with h = 0. From Eq. (9), setting h = 0 requires 1 + cos θ = 0, implying cos θ = − 1and θ = π . This angle cannot occur for four congruent isosceles triangles with base d = √2a and s2 = a2/ 2 + h2 at h = 0, since the geometry cannot close. Consequently, the apparent “transition angle” is an artefact of extending the formula beyond its geometric domain. The admissible range for the surrogate is θ∈ [ π/ 3 , 2 π/ 3], corresponding to deficits − 2 π/ 3 < εh< 2 π/ 3, and values outside this interval require additional triangles or asymmetric configurations. The divergence behaviour in the surrogate therefore does not represent the behaviour of the actual 21:16 lattice, whose hinge valence exceeds four. ©2025 Charles Emmanuel Levine. All rights reserved. 9 and, by the foregoing correspondence, one has |F(k) η|=Nk, k = 0,1,2. Under the symmetric isosceles metric assignment used for the surrogate model, all simplices incident on η are congruent and share the same interior hinge angle θ . Consequently each triangle in Fη contributes the same amount δ=θ−2π n to the angular surplus at η (where n is the hinge valence in the surrogate), and the Regge deficit can be written as εη= 2π−X ∆∈Fη θ=−X ∆∈Fη δ=− 2 X k=0 X ∆∈F(k) η δ. Define ε(k) η=−P∆∈F(k) ηδ. Because δis the same for each simplex, ε(k) η=−|F(k) η|δ=−Nkδ, k = 0,1,2. Taking absolute values and using (N0, N1, N2) = (1,1,3) gives ε(0) η:ε(1) η:ε(2) η=N0:N1:N2=1:1:3. Thus the deficit at the quadrant hinge admits a three–term decomposition whose magnitudes are in the fixed ratio 1:1:3, and the associated curvature pulse Πquad(η)has the claimed internal structure. C Correspondence Between the Flattened Horn–Toroid Geometry and the 21:16 Regge Lattice This section establishes that the stepped 4 × 4boundary used in the 21:16 lattice realises the same three–layer (1 , 1 , 3) structural pattern that appears in the planar representation of a flattened horn–toroid geometry. The comparison is entirely geometric and relies only on the combinatorial structure of the boundaries. C.1 Flattened horn–toroid geometry and its boundary decomposition Consider a closed planar curve formed by a periodic cycloid with minimal closure: a least–circumference toroidal loop formed from twelve congruent cycloidal arches arranged around a square fundamental region. The boundary of this region admits a decomposition into three concentric, axis–aligned layers: 1. an inner layer, consisting of a single segment generated by the minimal approach of the cycloid; 2. an intermediate layer, consisting of a single segment arising from the first outward advance of the periodic cycloid; 3. an outer layer, consisting of three segments that complete the closure of one quadrant of the cycloidal toroid. Let ( L0, L1, L2 )denote the number of boundary segments in these layers within one quadrant of the flattened toroid. Direct enumeration gives (L0, L1, L2) = (1,1,3), a pattern that is repeated symmetrically in each of the four quadrants of the figure. This (1 , 1 , 3) triple is a structural invariant of the flattened horn–toroid boundary and reflects the three distinct stages of advance required to achieve minimal cycloidal closure. C.2 Justification of the boundary correspondence The correspondence with the 21 : 16 lattice is based solely on the combinatorial structure of the boundary. In the stepped 4 × 4subdivision frame, the boundary admits a decomposition into three concentric, axis–aligned rings at radii r0=a 2, r1=3a 2, r2=5a 2, ©2025 Charles Emmanuel Levine. All rights reserved. 16 and, when restricted to a single quadrant, the number of boundary segments intersecting each ring satisfies (N0, N1, N2) = (1,1,3). This (1 , 1 , 3) triple coincides exactly with the layer counts ( L0, L1, L2 )obtained for the flattened horn–toroid boundary in the preceding subsection. Moreover, in both constructions the three layers are axis–aligned, partition one quadrant of a square fundamental region, and are related by the same dihedral symmetry of order eight. Within a given quadrant, the adjacency graph whose vertices are the endpoints of boundary segments and whose edges are the segments themselves is isomorphic in the two cases after identifying corresponding layers. In particular, there is a layer–preserving bijection between the sets of segments {inner, intermediate, outer segments of the cycloidal boundary} ←→ {inner, intermediate, outer steps of the 4 × 4 boundary}, which matches the intersection pattern and the (1 , 1 , 3) counts. In this sense the flattened horn–toroid boundary provides a smooth realisation of the same three–layer (1 , 1 , 3) boundary structure that appears in the stepped 21 : 16 Regge lattice. The cycloidal curve is therefore used here as a continuous analogue of the stepped boundary: the two boundaries share the same quadrant–wise layering, segment counts, and symmetry, and differ only in whether the layers are represented by straight segments (stepped lattice) or by rounded cycloidal arcs (flattened toroid). C.3 Boundary structure of the stepped 4×4Regge lattice Earlier sections established that the stepped boundary of the 4 × 4lattice admits an analogous three–layer decomposition when restricted to a single quadrant. Let the three axis–aligned rings be located at radii r0=a 2, r1=3a 2, r2=5a 2, and let Nk denote the number of stepped boundary segments intersecting the ring at radius rk within one quadrant. A direct count from the figure shows (N0, N1, N2) = (1,1,3), with N0 + N1 + N2 =5= σ , the subdivision surplus in the 4×4 frame. These three layers form the boundary bloom of the Regge lattice. Furthermore, the Quadrant Pulse Theorem proved above decomposes the Regge deficit at the hinge η on the quadrant axis into three contributions εη=ε(0) η+ε(1) η+ε(2) η, with magnitudes in the explicit ratio ε(0) η:ε(1) η:ε(2) η=1:1:3. Thus the Regge curvature pulse inherits the same (1,1,3) triple realised combinatorially on the boundary. C.4 Geometric correspondence of the two constructions Because both the flattened horn–toroid boundary and the stepped 4 × 4Regge boundary admit the same (1 , 1 , 3) decomposition into inner, intermediate, and outer layers per quadrant, the two structures are naturally comparable. The horn–toroid geometry provides a smooth, continuous model whose boundary layering reflects the minimal closure of the cycloidal toroid, while the Regge lattice furnishes a discrete, piecewise–linear model whose boundary layering drives the bulk curvature distribution. The correspondence may be summarised as follows: 1. The three–layer boundary structure is identical in both constructions: (1,1,3) per quadrant. 2. The boundary determines the bulk behaviour in each case: the horn–toroid boundary determines the cycloidal closure, and the Regge boundary determines curvature localisation. 3. The bulk response is partitioned into the same (1 , 1 , 3) triple: cycloidal advance in the smooth case and curvature pulse in the Regge case. 4. The two geometries therefore represent continuous and discrete realisations of the same underlying two–dimensional structure. In this sense the stepped 21:16 Regge lattice may be viewed as a discrete approximation of the boundary geometry of a flattened horn–toroid, with the (1,1,3) pattern serving as the invariant that links the two representations. ©2025 Charles Emmanuel Levine. All rights reserved. 17 D Reproducibility and Traceability This section presents a fully worked numerical example to demonstrate the traceability and evaluability of the closed-form lift relation in Eq. (10). All steps are shown in detail, with no skipped transformations. A Regge deficit is chosen εh=−π 2 which lies within the valid range − 2 π/ 3 < εh< 0for a fourfold hinge. The lattice spacing is set to a = 1 (unitless) for simplicity. Substituting this value into Eq. (10): h(εh)=av u u u u u t 1 + cos 2π−εh 4 21−cos 2π−εh 4 =v u u u u u t 1 + cos 2π+π/2 4 21−cos 2π+π/2 4 (16) =v u u u u u t 1 + cos 5π 8 21−cos 5π 8 (17) Now evaluate the cosine: cos(5π/8) = cos(112.5◦)≈ −0.38268343 Plug this into the expression: h=s1+(−0.38268343) 2 (1 −(−0.38268343)) =r0.61731657 2·1.38268343 =r0.61731657 2.76536686 =√0.22323135 ≈0.472873 Thus, for a= 1 and εh=−π/2, the lift is h≈0.472873,A=h a≈0.472873 This value can be verified independently using a scientific calculator or symbolic computation software. It confirms that the analytic formula for h behaves as expected and yields a concrete geometric embedding consistent with the prescribed Regge deficit. The corresponding hinge angle is: θ=2π−εh 4=2π+π/2 4=5π 8≈112.5◦ which is consistent with the fact that each isosceles triangle must subtend an angle of 112 . 5 ◦ at the lifted vertex to accumulate a total angle sum of 4 θ = 450 ◦ = 5 π/ 2, which implies a negative deficit of εh = 2 π− 5 π/ 2 = −π/ 2as prescribed. This example validates the model quantitatively and illustrates how the lift amplitude depends on the chosen angular deficit. All numerical values are rounded to appropriate significant digits for clarity. As the results are derived from exact trigonometric identities and algebraic manipulations, rounding errors are negligible and do not affect geometric consistency. ©2025 Charles Emmanuel Levine. All rights reserved. 18 E Glossary of symbols Table 1: Key symbols and definitions used throughout this paper. Symbol Meaning Units θInterior angle of an isosceles triangle at the hinge. Radians ADimensionless curvature amplitude, defined as h/a. Dimensionless Ainv Curvature amplitude in the inverse configuration. Dimensionless N□Number of square cells in the 4×4subdivision frame (N□= 16). – εRegge deficit at a vertex, ε= 2π−Piθi. Radians εhDeficit assigned to a single hinge in the symmetric surrogate model. Radians P∆∈T(v)Sum of incident triangle angles around a vertex. Radians EEdge set of the 4×4lattice. – ηNotional hinge label (indexing curvature–carrying vertices). – εinv Positive deficit at a hinge in the inverse subdivision pattern. Radians εtot Total angular surplus summed over curvature–carrying hinges. Radians N△ Number of subdivision regions generated by the diagonal rule in the 4 × 4frame (N△= 21). – s Length of the equal edges from the hinge to the boundary vertices in the isosceles surrogate model. Same units as a hOut–of–plane lift of the central vertex in the illustrative embedding. Same units as a ninv Hinge valence in the inverse subdivision pattern. – aSide length of a square cell in the underlying lattice. Arbitrary length dBase length of each isosceles triangle (d=√2a). Same units as a σCombinatorial surplus of the subdivision rule, σ=N△−N□= 5. – ∆Triangular simplex incident on a vertex. – VVertex set of the 4×4lattice, V={(i, j)|i, j ∈ {0,1,2,3,4}}. – Acronyms 2D Two–Dimensional 3D Three–Dimensional DDG Discrete Differential Geometry GR General Relativity PL Piecewise Linear Acknowledgements The development of Regge calculus, discrete differential geometry, and combinatorial curvature has been the work of many researchers. The author acknowledges the foundational contributions of Tullio Regge and subsequent scholars in both mathematics and physics whose insights into angular deficits, simplicial curvature, and overconstrained triangulations have provided the framework for the present study. Published analyses of simplicial overpacking and rigidity, particularly those emphasising minimal examples and visual constructions, form the essential background on which this paper relies. REFERENCES T. Regge, Il Nuovo Cimento 19, 558 (1961). R. M. Williams and P. A. Tuckey, Classical and Quantum Gravity 9, 1409 (1992). A. Gentle, General Relativity and Gravitation 34, 1701 (2002). A. I. Bobenko and Y. B. Suris, Discrete Differential Geometry: Integrable Structure, Graduate Studies in Mathematics, Vol. 98 (American Mathematical Society, 2008). ©2025 Charles Emmanuel Levine. All rights reserved. 19 A. I. Bobenko and B. A. Springborn, Principles of Discrete Differential Geometry (AMS / EMS Press, 2023). H. Gluck and W. Pan, Experimental Mathematics 26, 321 (2017). U. Brehm and W. Kühnel, Topology 26, 465 (1987). ©2025 Charles Emmanuel Levine. All rights reserved. 20