Graphs Have Structure: Declarative Manifolds, 3D Z-Order Monoid Roll-ups, and Zero-Simulation Visual Analytics in a Graph DBMS

arXiv:2611.01879v1 [cs.DB] 1 Nov 2026

Graphs Have Structure: Declarative Manifolds, 3D Z-Order Monoid Roll-ups, and Zero-Simulation Visual Analytics in a Graph DBMS Madhulatha Mandarapu

Sandeep Kunkunuru

Samyama.ai madhulatha@samyama.ai

Samyama.ai sandeep@samyama.ai

Abstract—Every real-world graph is embedded in an ambient structural manifold: proteins fold in Euclidean space (R3 ), telecom networks sit on a planetary surface (S2 ×R), and ontologies and calendars form discrete subsumption posets ((V, ⊑)). Yet property graph databases treat node coordinates as opaque payload scalars, leave spatial proximity edges unindexed, and force visualizers into O(V 2 ) client-side force-directed simulations that destroy true geometry and truncate hierarchy roll-ups. We elevate ambient manifold structure to a firstclass primitive in Samyama Graph. First, we introduce a declarative and auto-detected structural convention contract (samyama.structure/1) persisted in .sgsnap headers that binds node labels to Euclidean 3D, GeoSpherical, Causal Spacetime, or Poset Hierarchy manifolds. Second, we show that both discrete posets (via Order-Embedded Hierarchy [tin, tout] DFS intervals) and continuous 3D manifolds (via a 63-bit 3D Morton Z-order Structural-Geometric Index, SGI) reduce regional aggregation to 1D rank-interval monoid folds answered directly from index-resident Fenwick trees (SUM, COUNT) and Segment Trees (MIN, MAX). At 100,000 3D nodes, SGI builds in 28.14 ms and answers Mortoninterval SUM and MAX roll-ups in 0.048 µs and 0.104 µs— 8,345.1× and 3,824.4× faster than property scans. Third, we introduce 3D voxel-hashed virtual contact edges weighted by Topological–Geometric Stress σ(u, v) = dgraph (u, v)/dmanifold (u, v) to isolate long-range tertiary folds, plus manifold-guided A∗ shortest-path search that cuts explored states by up to 3.80× over Dijkstra. Fourth, we replace client layout simulation with servercomputed 3D Cone-Tree/Ribbon coordinates and a non-WebGL 2.5D Isometric projection with painter’sorder depth sorting and sub-millisecond Level-of-Detail (LoD) macro-cuts. Index Terms—graph databases, spatial indexing, Morton Z-order curve, hierarchy indexing, monoid rollup, protein structure graphs, visual analytics

I. Introduction Property graph databases model data as a combinatorial pair G = (V, E) adorned with key-value property maps. In this abstraction, a graph has adjacency but no intrinsic geometry: the coordinates (x, y, z) of an amino acid residue in a Protein Data Bank (PDB) structure [1], the (lat, lon, alt) of a cellular base station, and the taxonomic depth of a concept in the NCBI Taxonomy [2] or

GeoNames [3] are stored as uninterpreted floating-point or integer payload properties. Treating ambient structure as opaque payload creates four systemic failures across the query engine and its visual analytics stack: 1) Regional roll-ups degenerate into linear scans. Asking for the sum, minimum, or maximum of a measure (such as crystallographic B-factor, cell traffic volume, or descendant taxon count) over a 3D spatial region or taxonomic subtree forces the query engine to scan candidate nodes and dereference property storage, rather than answering the aggregate from an index. 2) Implicit spatial adjacency is either unindexed or quadratic. In structural biology and wireless infrastructure, the most consequential interactions are non-bonded contacts—pairs of nodes (u, v) that are far apart along the covalent or routing backbone (dG (u, v) ≫ 1) but lie within a physical cutoff radius ϵ in ambient space (dM (u, v) ≤ ϵ). Materializing all pairs within ϵ across multiple cutoffs bloats edge storage, while computing them on the fly without a 3D index costs O(V 2 ). 3) Graph algorithms run blind to the manifold. Pathfinding (shortestPath) and community detection execute uninformed combinatorial searches (BFS or Dijkstra) that expand wavefronts radially in hop space, ignoring admissible geometric lower bounds available from the node coordinates. 4) Visualizers fabricate O(V 2 ) hairballs and truncate hierarchy roll-ups. When a browser client such as Samyama Insight fetches a subgraph of 250 to 2,000 nodes, it runs an iterative force-directed layout [4] that destroys true 3D molecular or geospatial geometry. Worse, when rendering a hierarchy from a client-loaded subgraph, client-side aggregation can only sum over the loaded subset, misreporting the size and roll-up of every collapsed ancestor near the root. a) The thesis.: Every real-world graph is embedded in an ambient structural manifold M—discrete or

continuous—and declaring that manifold to the database unifies indexing, regional roll-up, structure-guided algorithms, and zero-simulation visual analytics under a single abstraction.

II. The Manifold-Embedded Property Graph Model A. Formal Definition A Manifold-Embedded Property Graph is a tuple

Building on our prior work on Order-Embedded Hierarchy indexing (OEH) for subsumption posets [5], we observe a fundamental algebraic duality between discrete hierarchies and continuous 3D space. In a tree or near-tree poset, a depth-first numbering maps every subtree to a contiguous 1D rank interval [tin(u), tout(u)]. In continuous 3D space, quantizing (x, y, z) into a 63-bit 3D Morton Z-order code [6] maps octree cells and Morton ranges into contiguous 1D rank intervals [rank(mlo ), rank(mhi )]. Consequently, both discrete posets and continuous 3D manifolds can be aggregated by the exact same indexresident monoid engines: a Fenwick tree [7] for invertible group operations (SUM, COUNT) and a Segment Tree for idempotent semilattice operations (MIN, MAX).

G = (V, E, λ, M, πM , µ)

b) Contributions.: We design, implement, and evaluate this architecture inside the Rust-based Samyama Graph engine [8]:

C1 — Declarative & Auto-Detected Structural Conventions (§II): We introduce the samyama.structure/1 specification persisted in .sgsnap snapshot headers, together with a schema-free property-signature detector that binds node labels to four manifold classes: Euclidean3d, GeoSpherical, HierarchyPoset, and CausalSpacetime. • C2 — Unified 1D Rank-Interval Monoid Indexing via OEH and SGI (§III): We pair OEH (extended with a 4-way structural probe including NearTree exception lists and Segment Tree MIN/MAX roll-ups) with the Structural-Geometric Index (SGI), a 63-bit 3D Morton Z-order index with index-resident Fenwick and Segment Tree roll-ups. • C3 — Structure-Guided Graph Operators (§IV): We implement 3D voxel-hashed virtual contact extraction, Topological–Geometric Stress σ(u, v) = dG (u, v)/dM (u, v) to isolate long-range tertiary folds and routing detours, and manifold-guided A∗ shortest-path search [9]. • C4 — Zero-Simulation 3D & Non-WebGL 2.5D Isometric Visual Cues (§V): We expose engine-level HTTP endpoints (/api/hierarchies and /api/structure/cues) that return authoritative whole-graph roll-ups, 3D Cone-Tree [10] and polymer ribbon coordinates, and a deterministic nonWebGL 2.5D Isometric projection with painter’salgorithm depth sorting and sub-millisecond Level-ofDetail (LoD) macro-cuts.

•

where (V, E, λ) is a labeled property graph, (M, dM ) is an ambient metric or partially ordered manifold, πM : VM ⊆ V → M is a partial embedding mapping nodes of a target label to points in M, and µ : VM → M maps nodes to elements of a commutative monoid (M, ⊕, 0). Unlike a purely combinatorial graph where distance is solely shortest-path hop count dG (u, v), every embedded pair u, v ∈ VM possesses two distances: 1) Topological distance dG (u, v), the shortest path length along explicit edges E; and 2) Manifold distance dM (πM (u), πM (v)), evaluated directly in the ambient space M. The interplay between dG and dM governs both structural indexing and structure-guided graph algorithms. B. Canonical Manifold Classes Figure 1 illustrates how Samyama Graph binds node labels and edge types to four canonical manifold classes (ManifoldSpec in src/index/structure/mod.rs): a) 1. Euclidean3d (M = R3 ).: Nodes carry three floating-point coordinate properties (x, y, z) with a physical unit (e.g., angstrom for biomolecular structures or m for LiDAR/CAD digital twins), an optional monotonic sequence property s(u) (such as residue_seq along a polypeptide chain), a chain grouping property (such as chain_id), and a default spatial contact cutoff ϵ > 0. The metric is Euclidean L2 : dR3 (u, v) = ∥πM (u) − πM (v)∥2 . b) 2. GeoSpherical (M = S2 × R).: Nodes carry geodetic latitude and longitude (ϕ, λ) in degrees (default CRS EPSG:4326) and an optional altitude h in meters. The ambient distance combines great-circle Haversine distance dhav ((ϕu , λu ), (ϕv , λv )) on Earth radius R⊕ = 6,371,000 m with vertical separation ∆h: p dS2 ×R (u, v) = dhav (u, v)2 + (hu − hv )2 . For 3D rendering and voxel hashing, (ϕ, λ, h) is projected to Earth-Centered, Earth-Fixed (ECEF) Cartesian coordinates (r cos ϕ cos λ, r cos ϕ sin λ, r sin ϕ) with r = R0 + αh. c) 3. HierarchyPoset (M = (V, ⊑)).: Generated by a transitive covering relation R⊑ ⊆ E (e.g., IS_A, PART_OF, SUBCLASS_OF) maintained by an OehIndex. Here the structural predicates are subsumption (u ⊑ v), Lowest Common Ancestor (LCA(u, v)), and poset depth depth(u). d) 4. CausalSpacetime (M = R2 × T).: Nodes carry spatial coordinates (x, y) and a timestamp t, supporting causal cone queries where tu ≤ tv and ∥(xu − xv , yu − yv )∥2 ≤ cmax (tv − tu ).

.sgsnap Snapshot Header & Auto-Probe (detect_conventions)

Discrete Poset (OEH) Tree / NearTree / Chain DFS Interval [tin(u), tout(u)]

1D RankInterval Monoids Fenwick: SUM, COUNT (O(log n)) SegmentTree: MIN, MAX (O(log n))

Continuous 3D (SGI) Euclidean3d / GeoSpherical 63-bit Morton [rank(mlo ), rank(mhi )]

StructureGuided Operators 3D Voxel Contacts + Stress σ(u, v) Manifold-Guided A∗ Pathfinding

Zero-Simulation Visual Cues 3D Cone-Tree / Ribbon Coords Non-WebGL 2.5D Isometric + LoD

Fig. 1. Architecture of Manifold-Embedded Property Graphs in Samyama Graph. Both discrete subsumption posets (OEH) and continuous 3D manifolds (SGI) linearize structural regions into 1D rank intervals backed by the same Fenwick and Segment Tree monoid engines, powering structure-guided operators and zero-simulation 3D and 2.5D visual analytics.

C. Declarative Persistence in .sgsnap & Auto-Detection To ensure that datasets ship with their structural semantics intact, we extend the .sgsnap binary snapshot header (SnapshotHeader) with an additive structures: Vec<StructuralConvention> field. When empty, the field is omitted via skip_serializing_if = "Vec::is_empty", preserving byte-for-byte backward compatibility with existing snapshots. When a graph is loaded without explicit conventions, detect_conventions(&GraphStore) performs a bounded sample probe (k ≤ 32 nodes per label) checking: (i) all active OehIndex instances → HierarchyPoset; (ii) coordinate triples (x, y, z) or (ca_x, ca_y, ca_z) plus sequence properties (residue_seq, seq_id) → Euclidean3d with preset protein_ribbon_3d (default cutoff ϵ = 8.0 Å); and (iii) bounded latitude/longitude pairs (ϕ ∈ [−90◦ , 90◦ ], λ ∈ [−180◦ , 180◦ ]) → GeoSpherical. III. Unified 1D Rank-Interval Monoid Indexing A. Rank-Interval Monoid Duality The central architectural insight of this work is that discrete subsumption hierarchies and continuous 3D spatial manifolds share an identical aggregation substrate once their regions are linearized into 1D rank intervals. Proposition 1 (Rank-Interval Monoid Unification): Let π : VM → {0, . . . , n − 1} be a bijection ordering the n = |VM | embedded nodes along a 1D curve (DFS preorder rank for posets; 63-bit 3D Morton Z-order rank for continuous 3D space), and let µ : {0, . . . , n − 1} → M be a node measure. For any structural region Q ⊆ M −1 whose image π(VM ∩ πM (Q)) decomposes into k disjoint Fk contiguous rank intervals i=1 [li , ri ]: 1) If (M, ⊕, ⊖, 0) is an abelian group L (COUNT, SUM), a Fenwick tree of n words evaluates u∈Q µ(u) in O(k log n) time and updates a single node’s measure in O(log n) time. 2) If (M, ⊕, 0) is an idempotent commutative monoid without inversesL (MIN, MAX), a flat 2n-word Segment Tree evaluates u∈Q µ(u) in O(k log n) time and updates a single node’s measure in O(log n) time.

In Samyama Graph, Fenwick and SegmentTree (src/index/hierarchy/oeh.rs) are implemented once and shared directly between OehIndex and SgiIndex. B. Discrete Posets: OEH with 4-Way Structural Probe Our prototype OEH work [5] introduced nested-set intervals for strict trees and Dilworth chain decomposition [11], [12] for low-width DAGs. However, real-world geographical and biomedical hierarchies such as GeoNames [3] are near-trees: over 99% of nodes have a single parent, while a small fraction of multi-parent exception edges (e.g., dual administrative containment) violate the strict tree invariant. In the Rust engine (ADR-035), OehIndex::build executes a 4-way structural probe: 1) Tree (→ NestedSet): Every non-root node has indegree ≤ 1. A DFS assigns [tin(u), tout(u)]. Both u ⊑ v and subtree roll-up reduce to a single interval (k = 1). 2) NearTree (→ NestedSet + sorted exception list): When the fraction of multi-parent nodes is small (≤ 2%), OEH extracts a primary spanning tree and records the remaining multi-parent edges in a compact exception list. This indexes all 544,093 GeoNames nodes without dropping a single multiparent edge. 3) LowWidthDag (→ Chain): When poset width w ≤ √ 8 n, OEH constructs a path partition with perchain reachability offsets and per-chain suffix monoid arrays (k = w). 4) HighWidthDag (→ Declined): Above the width cap, OEH declines index materialization and defers to graph traversal. Furthermore, the Cypher query planner rewrites hierarchy subsumption and aggregation patterns into index operators (HierarchyRollup, HierarchyDescendantScan), achieving a 23.4× speedup on selective cross-hierarchy conjunctions across the 112-query HIER benchmark suite.

C. Continuous 3D Manifolds: The 63-Bit Morton SGI Index To bring the same index-resident monoid capability to 3D continuous manifolds, we introduce the Structural-Geometric Index (SGI, SgiIndex in src/index/structure/mod.rs). a) 63-bit 3D Morton encoding.: Given the 3D axisaligned bounding box B = [bmin , bmax ] of the embedded nodes, each coordinate ca ∈ [bmin,a , bmax,a ] along axis a ∈ {0, 1, 2} is normalized and quantized to a 21-bit unsigned integer:   1 ca − bmin,a 21 · (2 − 1) + . qa (ca ) = max(bmax,a − bmin,a , 10−12 ) 2 A branchless bitwise dilator (part1by2_21) spreads the 21 bits of each axis using five magic-number shift-and-mask steps so that bit i moves to bit 3i, interleaving (qx , qy , qz ) into a 63-bit Morton code: morton(qx , qy , qz ) = S(qx ) ∨ (S(qy ) ≪ 1) ∨ (S(qz ) ≪ 2), where S(·) = part1by2_21(·). Its exact inverse (morton_decode_3d) compacts every third bit back in O(1) bitwise operations. b) Index construction and Morton-range roll-ups.: SgiIndex::build extracts the coordinates and optional measure µ(u) of each candidate node, computes its 63bit Morton code, and sorts the point array in ascending (morton, node_id) order. Over the sorted ranks {0, . . . , n− 1}, it constructs: • fenwick_sum: a Fenwick tree over µ(points[r]), and • seg_min, seg_max: two SegmentTree instances over µ(points[r]). For any contiguous Morton interval [mlo , mhi ] (such as any octree cell at depth d ∈ {0, . . . , 21} where the top 3d bits are fixed), rollup_morton_range locates the rank boundaries [l, r] via two binary searches (partition_point) and answers COUNT in O(1) via r − l + 1, SUM in O(log n) via fenwick_sum.range(l, r), and MIN/MAX in O(log n) via seg_min.range(l, r) and seg_max.range(l, r)— touching zero node or property records. For arbitrary axisaligned 3D query boxes that span multiple Z-order runs, rollup_bbox narrows the scan to the Morton interval [morton(qmin ), morton(qmax )] and filters the candidate slice against the exact box predicates. IV. Structure-Guided Graph Analytics Once πM and SgiIndex are resident in the engine, graph algorithms no longer need to operate blind to ambient geometry. A. 3D Voxel-Hashed Virtual Contact Edges In biomolecular graphs (where explicit edges E represent covalent peptide bonds) and spatial infrastructure graphs (where E represents fiber or road links), analysts frequently query implicit proximity edges:  Eϵ = {u, v} ⊆ VM {u, v} ∈ / E ∧ 0 < dM (u, v) ≤ ϵ .

Rather than materializing Eϵ on disk for a fixed ϵ or executing an O(n2 ) all-pairs distance loop, SgiIndex::contact_edges constructs a 3D spatial hash grid with voxel side length ℓ = ϵ. Each point u is hashed
into cell c(u) = ⌊xu /ℓ⌋, ⌊yu /ℓ⌋, ⌊zu /ℓ⌋ ∈ Z3 . Testing only the 3 × 3 × 3 = 27 neighboring voxel buckets around each point extracts all non-bonded contacts within ϵ in O(n + |Eϵ |) expected time for bounded-density manifolds (such as folded proteins where steric exclusion caps the 3 number of residues per Å ). B. Topological–Geometric Stress σ(u, v) Not all spatial contacts are equally informative: two residues at sequence positions i and i + 2 are close in 3D space simply because they are separated by two covalent bonds. The contacts that govern tertiary protein folding, allosteric communication, and network vulnerability are those where topological distance is large relative to ambient manifold distance. We define the Topological–Geometric Stress of a pair (u, v) as: dG (u, v) , (1) σ(u, v) = max(dM (u, v), δ0 ) where dG (u, v) is evaluated via a bounded bidirectional/BFS hop search (capped at hmax = 12 hops in bounded_hop_distance to guarantee O(1) per-contact work on bounded-degree graphs) and δ0 = 0.1. Sorting Eϵ in descending order of σ(u, v) immediately isolates longrange tertiary contacts (β-sheet hairpins, disulfide bridges, and domain-domain interfaces) at the top of the result set without requiring any domain-specific secondary-structure caller. C. Manifold-Guided A∗ Shortest Path When edge weights represent physical segment lengths w(u, v) ≥ dM (u, v), the ambient manifold distance h(u) = dM (πM (u), πM (t)) to a target node t is both admissible (h(u) ≤ dG (u, t)) and consistent (h(u) ≤ w(u, v) + h(v) by the triangle inequality on M) [9]. SgiIndex::astar_shortest_path prioritizes open frontier states by f (u) = g(u) + dM (u, t), where dM is Euclidean distance for Euclidean3d and 3D Haversinealtitude distance for GeoSpherical. Alongside the exact optimal path and its hop count, the operator returns the path stretch ratio σpath (s, t) = dG (s, t)/dM (s, t) and the exact ratio of states explored by A∗ versus uninformed Dijkstra. V. Zero-Simulation 3D & Non-WebGL 2.5D Visual Analytics A major bottleneck in visual graph analytics tools (including Samyama Insight at graph.samyama.cloud) is reliance on client-side force-directed layout simulations. Force-directed placement costs O(V 2 ) (or O(V log V ) with Barnes–Hut approximation) per tick across hundreds of

ticks, scrambles true 3D coordinates into arbitrary rotational hairballs, and requires WebGL shaders to remain responsive. Moreover, when a user explores a large hierarchy by loading a 250-node neighborhood into the browser, client-side hierarchy roll-ups only see the loaded nodes, displaying misleading partial aggregates. We resolve both issues by moving structural layout and Level-of-Detail (LoD) culling into the database engine via two low-latency HTTP endpoints: POST /api/hierarchies/:name/cues and POST /api/structure/cues. A. 3D Cone-Tree Coordinates for OEH Hierarchies For any OehIndex, visual_cues maps each node’s nested-set interval [tin(u), tout(u)] and poset depth d(u) ∈ {0, . . . , Dmax } into deterministic 3D Cone-Tree [10] coordinates in O(1) arithmetic operations per node: tin(u) + tout(u) , (2) θ(u) = 2π · 2n   tout(u) − tin(u) r(u) = R0 1 − 0.55 , (3) n
(xu , yu , zu ) = r(u) cos θ(u), −∆y d(u), r(u) sin θ(u) . (4) Because subtree intervals [tin(u), tout(u)] are disjoint for sibling subtrees at every depth, angular sectors never overlap, and every node’s payload carries its authoritative whole-poset descendant_count and index-resident rollups (sum, min, max, count) regardless of how few nodes the browser renders.

TABLE I SGI 63-bit 3D Morton index build time, index-resident monoid roll-up latency vs. property scan, and LoD-2000 visual cue latency across 3D node scales. Nodes Build Fenwick SegTree N (ms) SUM (µs) MAX (µs) 1,000 10,000 100,000

0.26 3.38 28.14

0.027 0.036 0.048

Scan (µs)

Speedup LoD (SUM / MAX) (ms)

0.065 0.74 27.3× / 11.4× 0.36 0.087 28.28 774.8× / 325.4× 2.85 0.104 398.23 8345.1× / 3824.4× 27.78

TABLE II 3D voxel-hashed non-bonded contact extraction (ϵ = 6.0 Å) and Topological–Geometric Stress (σ ≥ 1.5) on folded polypeptide backbones. Res. Contacts High-Stress Max Voxel+BFS Pairwise N (dM ≤ 6 Å) (σ ≥ 1.5) σ(u, v) (ms) (ms) 500 2,000 5,000

642 3,318 8,670

618 3,222 8,430

2.50 2.50 2.50

1.60 8.54 22.28

0.14 1.71 11.30

VI. Experimental Evaluation We evaluate all four components inside Samyama Graph (Rust 1.85, release profile -O3) on a Linux x86_64 workstation. Every number reported in Tables I– IV is generated by the reproducible benchmark suite in tests/structural_conventions_and_cues.rs (standalone reproducibility harness at https: //github.com/samyama-ai/structural-manifold-index; DBMS integration in https://github.com/samyama-ai/ samyama-graph/pull/2038) and committed in results/benchmark_results.json. A. SGI 3D Morton Roll-up vs. Property Scan

B. Non-WebGL 2.5D Isometric Projection & Painter’s Order To support accurate 3D structural visualization in restricted enterprise browsers, headless exports, and 2D HTML5 Canvas contexts without WebGL, the engine precomputes an orthographic 2.5D isometric projection at azimuth α = 45◦ and elevation β = 30◦ over normalized bounding-box coordinates (x̂, ŷ, ẑ) ∈ [−1, 1]3 : iso_x = (x̂ − ŷ) cos 45◦ ,

(5)

◦

◦

(6)

◦

◦

(7)

iso_y = (x̂ + ŷ) sin 30 − ẑ cos 30 , zcam = (x̂ + ŷ) cos 30 + ẑ sin 30 .

The engine normalizes zcam to depth_norm ∈ [0, 1], assigns each node a back-to-front painter’s-algorithm draw rank (draw_order_rank), and computes a depth-fog opacity opacity = clamp(1.0−0.6 depth_norm, 0.3, 1.0). A 2D canvas renderer simply iterates nodes in draw_order_rank order and draws each node at (iso_x, iso_y) with its precomputed depth-fog opacity—achieving true 3D occlusion and perspective cues in a single O(V ) pass with zero clientside matrix math or physics ticks.

Table I evaluates SgiIndex construction time, Mortoninterval monoid roll-up latency (Fenwick SUM and SegmentTree MAX over a 50% Morton-code range [mN/4 , m3N/4 ]), linear property-scan latency over the same Morton range, and LoD-2000 visual cue generation across N ∈ {1,000, 10,000, 100,000} 3D folded-helix residue nodes. Because Fenwick::range and SegmentTree::range execute O(log N ) array lookups over contiguous Morton ranks, roll-up latency grows from 0.027 µs at N = 1,000 to only 0.048 µs at N = 100,000 (SUM), and from 0.065 µs to 0.104 µs (MAX). At N = 100,000, index-resident Morton roll-ups are 8,345.1× (SUM) and 3,824.4× (MAX) faster than scanning the node properties in the interval. B. 3D Voxel Contact Extraction & Topological–Geometric Stress Table II evaluates SgiIndex::contact_edges (ϵ = 6.0 Å) on folded antiparallel β-hairpin polypeptide chains of N ∈ {500, 2,000, 5,000} residues (with turns every 50 residues bringing sequence-distant strands within 4.8– 5.2 Å).

TABLE III Manifold-guided A∗ vs. uninformed Dijkstra across 3D spatial lattice graphs. Nodes Hops N 400 1,296 2,500

Path Direct Stretch States Red. Dist Dist σpath (A∗ / Dijk.)

12 157.28 20 270.42 27 369.41

130.77 242.28 340.73

1.203 1.116 1.084

186 / 400 2.15× 420 / 1,296 3.09× 658 / 2,500 3.80×

TABLE IV OEH whole-poset vs. LoD-250 visual cue latency and authoritative root descendant counts. Nodes Auth. Root Client-250 Full Cues LoD-250 N Descendants Apparent (ms) (ms) 781 3,906 5,461

781 3,906 5,461

250 250 250

0.17 0.72 0.99

0.055 0.080 0.089

In all three structures, the 3D voxel grid recovers 100% of the exact pairwise non-bonded contacts (642, 3,318, and 8,670 contacts, respectively) while simultaneously computing 12-hop bounded-BFS graph distance and Topological–Geometric Stress σ(u, v). At N = 5,000 residues, voxel hashing plus per-contact BFS stress completes in 22.28 ms (scaling linearly with contact count, whereas naive O(N 2 ) distance-only scanning grows 80.7× from 0.14 ms to 11.30 ms across a 10× increase in N ). More importantly, ranking contacts by σ(u, v) ≥ 1.5 automatically separates the 8,430 inter-strand tertiary contacts (max σ = 2.50) from the 240 local turn contacts without any external secondary-structure annotation. C. Manifold-Guided A∗ vs. Uninformed Dijkstra Table III compares SgiIndex::astar_shortest_path against uninformed Dijkstra between opposite corners of 3D spatial lattice graphs of 400 (10 × 10 × 4), 1,296 (18 × 18 × 4), and 2,500 (25 × 25 × 4) waypoints. Because the Euclidean heuristic is admissible and consistent, A∗ returns the exact shortest path in every graph while reducing the number of explored states from 400 → 186 (2.15×), 1,296 → 420 (3.09×), and 2,500 → 658 (3.80×), with the state-space reduction growing monotonically with graph diameter. D. OEH Whole-Poset Visual Cues & LoD Truncation Finally, Table IV evaluates OehIndex::visual_cues on balanced taxonomic trees of 781, 3,906, and 5,461 nodes when the client requests either the full poset or a Levelof-Detail macro-cut of lod_budget = 250 nodes. Without engine-level OEH cues, a client that loads 250 nodes under-reports the root’s subtree size as 250 (an error of 95.4% at N = 5,461). In contrast, OehIndex::visual_cues with lod_budget = 250 returns the top 250 macro-nodes in 0.089 ms while preserving the exact whole-poset descendant count (5,461) and indexresident monoid roll-ups on every returned node.

VII. Related Work and Limitations a) Related work.: Order-embedding of trees via DFS intervals goes back to Grust’s XPath accelerator [13] and SAP HANA’s hierarchy indexing [14], [15], while chain decomposition for poset reachability goes back to Dilworth [11] and Jagadish [12], complemented by 2-hop labeling [16]. Our prior OEH work [5] unified nested-set and chain encodings with index-resident Fenwick roll-ups on static posets. Space-filling Morton Z-order curves [6] are widely used in multidimensional point access methods and point-cloud engines, while spatial network community detection [17] and 3D Cone Trees [10] have been studied in network science and information visualization. To our knowledge, Samyama Graph is the first property graph DBMS to unify discrete poset intervals (OEH) and continuous 3D Morton intervals (SGI) under a shared indexresident monoid engine (Fenwick + SegmentTree) while serving structure-guided operators and zero-simulation 3D and 2.5D visual cues directly from the database kernel. b) Limitations.: We state three limitations of the current implementation explicitly: 1) Z-order multi-run boundary refinement for arbitrary 3D boxes. While any single Morton interval [mlo , mhi ] (such as an octree cell) is answered in O(log n) purely from the Fenwick or Segment Tree, an arbitrary axis-aligned 3D bounding box that straddles high-bit Z-order quadrant boundaries spans multiple non-contiguous Morton runs. Our current rollup_bbox implementation scans the bounding Morton range [l, r] and applies exact point-in-box filtering; decomposing a query box into O(log span) maximal Z-order sub-intervals (BIGMIN/LITMAX) so that arbitrary 3D boxes also execute purely via tree lookups is future work (samyama-graph#2036). 2) Static vs. streaming 3D coordinate maintenance. While OehIndex supports O(log n) in-place measure updates (update_measure), SgiIndex currently rebuilds its sorted Morton array when node coordinates (x, y, z) mutate. Supporting dynamic octree/Morton block splits for high-frequency streaming telemetry is future work. 3) Bounded-hop stress approximation. To prevent all-pairs shortest-path blowup during interactive visual cue requests, contact_edges caps its percontact BFS hop search at hmax = 12 hops. Contacts separated by more than 12 covalent bonds receive a conservative lower-bound stress numerator of 12; exact backbone sequence differences |s(u)−s(v)| can be substituted when sequence_prop is present. VIII. Conclusion Real-world graphs are not shapeless combinatorial hairballs: they inhabit discrete taxonomic posets and continuous 3D geometric manifolds. By elevating structural conventions (samyama.structure/1) into the snapshot

format and unifying discrete poset intervals (OEH) with 63-bit 3D Morton intervals (SGI) over shared Fenwick and Segment Tree monoid engines, Samyama Graph accelerates regional roll-ups by up to 8,345.1×, isolates tertiary 3D structural contacts via Topological–Geometric Stress, prunes A∗ shortest-path search by up to 3.80×, and delivers sub-millisecond 3D and non-WebGL 2.5D visual cues without client-side simulation. References [1] H. M. Berman, J. Westbrook, Z. Feng, G. Gilliland, T. N. Bhat, H. Weissig, I. N. Shindyalov, and P. E. Bourne, “The Protein Data Bank,” Nucleic Acids Research, vol. 28, no. 1, pp. 235–242, 2000. [2] “NCBI taxonomy database,” https://ftp.ncbi.nih.gov/pub/ taxonomy/, 2026. [3] “GeoNames geographical database,” https://www.geonames. org, 2026, cC BY 4.0. [4] T. M. J. Fruchterman and E. M. Reingold, “Graph drawing by force-directed placement,” Software: Practice and Experience, vol. 21, no. 11, pp. 1129–1164, 1991. [5] M. Mandarapu and S. Kunkunuru, “One index for subsumption and roll-up across time, geography, and ontology,” arXiv preprint arXiv:2606.24677, 2026. [6] G. M. Morton, “A computer oriented geodetic data base and a new technique in file sequencing,” IBM Ltd., Ottawa, Tech. Rep., 1966. [7] P. M. Fenwick, “A new data structure for cumulative frequency tables,” Software: Practice and Experience, vol. 24, no. 3, pp. 327–336, 1994. [8] M. Mandarapu and S. Kunkunuru, “Samyama graph: A multitenant property graph engine with vector, hierarchy, and snapshot primitives,” arXiv preprint, 2026. [9] P. E. Hart, N. J. Nilsson, and B. Raphael, “A formal basis for the heuristic determination of minimum cost paths,” IEEE Transactions on Systems Science and Cybernetics, vol. 4, no. 2, pp. 100–107, 1968. [10] G. G. Robertson, J. D. Mackinlay, and S. K. Card, “Cone trees: Animated 3D visualizations of hierarchical information,” in Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI), 1991, pp. 189–194. [11] R. P. Dilworth, “A decomposition theorem for partially ordered sets,” Annals of Mathematics, vol. 51, no. 1, 1950. [12] H. V. Jagadish, “A compression technique to materialize transitive closure,” ACM Transactions on Database Systems, vol. 15, no. 4, 1990. [13] T. Grust, “Accelerating XPath location steps,” in Proceedings of the ACM SIGMOD International Conference on Management of Data, 2002. [14] R. Brunel, J. Finis, G. Franz, N. May, A. Kemper, T. Neumann, and F. Faerber, “Supporting hierarchical data in SAP HANA,” in IEEE International Conference on Data Engineering (ICDE), 2015. [15] R. Brunel, N. May, and A. Kemper, “Index-assisted hierarchical computations in main-memory RDBMS,” Proceedings of the VLDB Endowment, vol. 9, no. 12, 2016. [16] E. Cohen, E. Halperin, H. Kaplan, and U. Zwick, “Reachability and distance queries via 2-hop labels,” SIAM Journal on Computing, vol. 32, no. 5, 2003. [17] P. Expert, T. S. Evans, V. D. Blondel, and R. Lambiotte, “Uncovering space-independent communities in spatial networks,” Proceedings of the National Academy of Sciences (PNAS), vol. 108, no. 19, pp. 7663–7668, 2011.

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