\begin{equation} \mathrm {Valid}(r,d) = \mathbf {1}\left \[ \max \_{(p_i,t_i),(p_j,t_j)\in r} \frac {\\p_i-p_j\\}{\|t_i-t_j\|+\epsilon } \le v\_{\max }(d) \right \], \label {eq:valid} \end{equation}
with stricter checks when acceleration, curvature, or S-KBM state is available. A query is therefore a budgeted planning problem over typed tools:
\begin{equation} \min \_{\mathcal {G}}\\\mathrm {Cost}(\mathcal {G};q) \quad \text {s.t.}\quad \forall r\in \mathcal {R}(\mathcal {G},q),\\ \mathrm {Valid}(r,d)=1,\quad \mathrm {Budget}(\mathcal {G})\preceq b, \label {eq:budgeted-plan} \end{equation}
where \\\mathcal {G}\\ is the PlanGraph emitted by the planner. The LLM is used to propose \\\mathcal {G}\\ and summarize its results; geometry, validation, caching, and budget accounting are deterministic.
PlanGraph semantics: A PlanGraph is a finite directed acyclic graph \\\mathcal {G}=(V,E)\\ whose node \\v\in V\\ has type \\\tau (v)\\, dependency set \\\mathrm {deps}(v)\\, JSON-serializable inputs, output schema, expected token cost, expected tool cost, and confidence prior. Topological layers \\\mathcal {L}\_1,\ldots ,\mathcal {L}\_m\\ define concurrency: nodes within a layer may run in parallel, but no node runs before all dependencies finish. Invalid plans fail before execution if a node type is unknown, a dependency points outside the graph, the graph has a cycle, or expected costs exceed the request budget. This is the main difference between GeoTrace-Agent and a free-form ReAct loop: the model is allowed to plan, but the orchestrator owns the type system.
\begin{equation} \mathcal {E}\_t = \left \\ p : \frac {\\p - A\\}{v\_{\max }} \le t-t_A \\\wedge \\ \frac {\\p - B\\}{v\_{\max }} \le t_B-t \right \\, \quad t\in \[t_A,t_B\], \label {eq:ellipse} \end{equation}
which we approximate by the inscribed ellipse with foci \\A, B\\ and semi-major axis \\a(t) = \tfrac {1}{2} v\_{\max } (T)\\. Distances are evaluated in a local equirectangular projection centered on the anchor midpoint so they are Euclidean to first order; for continental-scale anchors we switch to a UTM zone via pyproj. The MOBR is the orthogonal bounding rectangle of the boundary ellipse; the dynamic region merge (DRM) of overlapping ellipses uses an R\*-tree index and a maximal-union sweep, recovering the STAGD-DRM behavior of Sharma et al. \[[37](#Xsharma2024tist)\]. The two-prism intersection intersect(P, Q) samples \\n\\ time slices in the overlap window and unions per-slice ellipse intersections; this is the operation that powers TGARD.
### 4.5 STAGD-DRM and TGARD / DC-TGARD agents
The GapDetectorAgent walks a trajectory, finds gaps where consecutive samples are farther apart than a coverage threshold, indexes each gap’s prism MOBR in an R\*-tree, unions overlapping bounding boxes, and scores each cluster with the Abnormal Gap Measure
\begin{equation} \mathrm {AGM}(g) = \lambda \\ (1 - p\_{\text {phys}}(g)) + (1-\lambda )\\ p\_{\text {data}}(g), \label {eq:agm} \end{equation}
where \\p\_{\text {phys}} = \min (1, v\_{\max } / v\_{\text {required}})\\ and \\p\_{\text {data}}\\ is a calibrated tail probability over the Pi-DPM \[[38](#Xsharma2025geoanomalies)\] reconstruction error.
The RendezvousFinderAgent pairwise-intersects prisms; in the dual-convergence variant DC-TGARD \[[35](#Xsharma2022sigspatial)\] it walks the time interval from both ends, pruning slices whose intersection becomes empty, and records the tightened time window \\\[\\t_0 + (t_1-t_0)\\\ell ,\\ t_0 + (t_1-t_0)\\h\\\]\\. The DC variant is provably correct and complete and is empirically faster in expectation when overlaps are short.
### 4.6 Kinematic validator (S-KBM gate)
Every region returned to the user is forced through a ValidatorAgent that recomputes a worst-case required speed across the region’s centroid and time window, compares against the per-domain envelope (vessel: \\25\\ kt, vehicle: \\130\\ km/h, pedestrian: \\2\\ m/s, UAV: \\30\\ m/s), and raises KinematicViolation on infeasibility. The validator is a hard invariant: a region that does not pass is never returned. This is the single most important difference between an LLM-only system and ours.
### 4.7 MCP server and A2A protocol
The prism kernel is exposed as a Model Context Protocol \[[3](#Xanthropic2024mcp)\] server speaking JSON-RPC 2.0 over stdio with three tools: prism.compute, prism.intersect, prism.merge_dynamic. Any MCP-aware client, IDE plugin, or sibling agent can invoke them directly. The orchestrator additionally exposes a JSON-RPC 2.0 Agent-to-Agent endpoint at /a2a/jsonrpc and advertises a capability card at /a2a/.well-known/capabilities; outbound A2AClient calls cache cards for 60 seconds and propagate the OpenTelemetry trace identifier as a traceparent header.
Prism geometry in local coordinates: For a pair of anchors \\A,B\\ we project latitude/longitude to a local tangent plane centered at the midpoint. Let \\c=\\B-A\\/2\\, \\a(t)=v\_{\max }(t_B-t_A)/2\\, and \\b(t)=\sqrt {a(t)^2-c^2}\\ when the anchor pair is reachable. The ellipse at time \\t\\ can be written as
\begin{equation} \frac {(u^\top (p-m))^2}{a(t)^2}+ \frac {(v^\top (p-m))^2}{b(t)^2} \le 1, \label {eq:ellipse-canonical} \end{equation}
where \\m=(A+B)/2\\, \\u=(B-A)/\\B-A\\\\, and \\v\\ is the perpendicular unit vector. The MOBR is obtained by projecting the ellipse axes onto the coordinate axes, giving half-widths
\begin{equation} h_x=\sqrt {a(t)^2 u_x^2+b(t)^2 v_x^2},\qquad h_y=\sqrt {a(t)^2 u_y^2+b(t)^2 v_y^2}. \label {eq:mobr} \end{equation}
These bounds are cheap enough for repeated tool calls and tight enough for R-tree pruning. Exact polygon intersections are only computed after MOBR filtering. This mirrors the database principle that a coarse index should prune before the expensive exact predicate runs.
Require: trajectory samples \(s_{1:n}\), domain speed cap \(v_{\max }\), gap threshold \(\Delta t_{\min }\) 1: initialize R-tree index \(\mathcal {I}\) and candidate list \(\mathcal {C}\)
2: for \(i=1\) to \(n-1\) do
3: if \(t_{i+1}-t_i > \Delta t_{\min }\) then
4: compute prism \(P_i=\mathrm {Prism}(s_i,s_{i+1},v_{\max })\)
5: insert MOBR\((P_i)\) into \(\mathcal {I}\) and append \(P_i\) to \(\mathcal {C}\)
6: end if
7: end for
8: merge overlapping candidates with dynamic region merge to obtain clusters \(\mathcal {M}\)
9: for cluster \(m\in \mathcal {M}\) do
10: compute \(p_{\mathrm {phys}}(m)\) from required speed and \(p_{\mathrm {data}}(m)\) from Pi-DPM tail score
11: score \(\mathrm {AGM}(m)=\lambda (1-p_{\mathrm {phys}}(m))+(1-\lambda )p_{\mathrm {data}}(m)\)
12: end for
13: return clusters sorted by AGM, each with geometry, time window, and validator status
2: for \(i=1\) to \(n-1\) do
3: if \(t_{i+1}-t_i > \Delta t_{\min }\) then
4: compute prism \(P_i=\mathrm {Prism}(s_i,s_{i+1},v_{\max })\)
5: insert MOBR\((P_i)\) into \(\mathcal {I}\) and append \(P_i\) to \(\mathcal {C}\)
6: end if
7: end for
8: merge overlapping candidates with dynamic region merge to obtain clusters \(\mathcal {M}\)
9: for cluster \(m\in \mathcal {M}\) do
10: compute \(p_{\mathrm {phys}}(m)\) from required speed and \(p_{\mathrm {data}}(m)\) from Pi-DPM tail score
11: score \(\mathrm {AGM}(m)=\lambda (1-p_{\mathrm {phys}}(m))+(1-\lambda )p_{\mathrm {data}}(m)\)
12: end for
13: return clusters sorted by AGM, each with geometry, time window, and validator status
\begin{equation} \sum \_{v\in V\_{\mathrm {done}}} c_v \preceq b \label {eq:cost} \end{equation}
after every layer. If the next layer would exceed budget, the orchestrator returns a partial answer with terminated_by_budget=true, completed regions, and missing-node diagnostics. This behavior is preferable to silent truncation because the caller can decide whether to reissue the query with a larger budget.
Cache correctness: The exact cache is safe when the key includes tool name, normalized arguments, code version, and domain config. The semantic cache is riskier because near-duplicate queries can differ in ways that matter. GeoTrace-Agent therefore uses semantic hits only for high-level retrieval and summary reuse; deterministic geometry results require exact keys. This distinction prevents a dangerous failure mode: reusing a prism from a near but not identical anchor pair. A semantic cache hit can save text tokens, but it cannot replace a geometric predicate.
Human-in-the-loop queue: The validator does not force every uncertain case into a binary answer. When confidence falls below the configured threshold, the response marks hitl_required=true and writes a Postgres row containing the question, PlanGraph, regions, validator outputs, and trace id. The Streamlit console lets a reviewer accept, reject, or annotate the answer. The review event is later exportable as a preference triple for Pi-GRPO. This is an important design connection: GeoTrace-Agent creates structured supervision from ambiguous operational cases instead of discarding them as failures.
## 5 Experiments
Golden dataset. We curated three anchor questions (g-001 rendezvous, g-002 gap audit, g-003 prism only) and replay them through the orchestrator. Each item carries an expected block (region count bounds, allowed methods, required validator pass) that the offline evaluator checks. Results are written under evaluation/eval_results/\| Metric | p50 latency (s) | p95 latency (s) | tokens / query | cost USD / query |
| Full pipeline | 1.9 | 3.4 | 4,210 | 0.034 |
| Without semantic cache (ablation) | 2.6 | 4.5 | 6,980 | 0.054 |
| Without tool dedup (ablation) | 2.1 | 3.7 | 4,980 | 0.039 |
Check |
Evidence captured |
Expected status |
Prism geometry |
ellipse/MOBR area, reachability, impossible-anchor rejection |
pass |
Planner schema |
PlanGraph type validation, dependency validation, cycle rejection |
pass |
Orchestrator |
topo-layer execution, budget stop, stage trace serialization |
pass |
Semantic cache |
exact hit, miss, and near-hit behavior remain separated |
pass |
Validator |
infeasible speed raises KinematicViolation |
pass |
API integration |
/healthz, /v1/query, A2A capability card serialize |
pass |
Case |
Primary route |
Required checks |
Expected result |
g-001 |
prism intersection \(\rightarrow \) rendezvous \(\rightarrow \) validator |
at least one candidate region, all regions feasible |
pass |
g-002 |
gap detector \(\rightarrow \) DRM merge \(\rightarrow \) AGM score |
abnormal gap score and time window present |
pass |
g-003 |
direct prism kernel |
geometry returned without unnecessary agents |
pass |
| Configuration | schema fail \(\downarrow \) | tokens/query \(\downarrow \) | p95 latency \(\downarrow \) | infeasible regions \(\downarrow \) |
| Free-form agent baseline | 0.18 | 7,400 | 5.8 | 0.12 |
| Typed PlanGraph only | 0.04 | 6,100 | 5.0 | 0.08 |
| + exact cache / tool dedup | 0.04 | 5,200 | 3.9 | 0.08 |
| + semantic cache | 0.04 | 4,200 | 3.4 | 0.08 |
| + S-KBM validator | 0.04 | 4,250 | 3.5 | 0.00 |
Ablation |
Primary metric shift |
Likely diagnosis |
Free-form planner output |
schema failures and replay failures increase |
model is doing planning in prose |
No topo parallelism |
p95 latency increases |
independent tool calls serialized |
No tool dedup |
duplicate tool calls per query increase |
agents converge on same request |
No exact cache |
repeated prism latency increases |
deterministic calls recomputed |
No semantic cache |
tokens/query and cost increase |
near-duplicate text not reused |
No validator |
infeasible regions can be returned |
correctness boundary removed |
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