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Topological Necessities: Mechanism-Invariant Strategic Subgoals for Cross-Embodiment Goal-Conditioned Control

Hao Shi, Xi Li

arXiv:2609.11014v1cs.LGcs.AIcs.RO

TL;DR

Offline goal-conditioned control needs subgoals that can be reused when the executor or embodiment changes, but common subgoals are tied to executor-specific fitted pipelines. This paper recovers route-conditioned unavoidable stages from a topology-aware carrier of successful offline trajectories, then deploys them as certified recursive gate hierarchies. Frozen gates transfer across evaluated embodiments, achieving the highest Humanoid aggregate of 96.1 and strong results across PointMaze, AntMaze, and Kitchen.

  • Problem

    Existing offline GCRL subgoals are often produced by executor-tied value functions, latent actions, or policy outputs, limiting task knowledge to the executor that produced it.

  • Method

    The paper builds a transport-weighted carrier from successful trajectories and uses H0 bottleneck order, H1 route bifurcation, and recursive certified gates to represent unavoidable stages.

  • Results

    Frozen PointMaze gates transfer without retraining to Ant and Humanoid, reaching the highest Humanoid aggregate at 96.1, while matching or exceeding strong baselines across PointMaze, AntMaze, and Kitchen.

  • Takeaways & Limitations

    The results support task-conditioned strategic subgoals that remain actionable across evaluated executor and embodiment changes when maze geometry is held fixed.

  • Takeaways & Limitations

    The claim is task-conditioned rather than unrestricted invariance: stability is supported across evaluated embodiments under fixed maze geometry, with fully online recursion left for future work.

Abstract

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Long-horizon goal-conditioned reinforcement learning delegates control to a high-level module that proposes subgoals, but existing subgoals are implicit byproducts of value functions or latent actions, tied to the executor that produced them. We study a different object: a route-conditioned order of unavoidable stages that every successful executor must traverse, recoverable from offline trajectories and belonging to none of them. Its defining properties are topological: an unskippable stage is a separating set that every admissible path must cross, and a loop in free space forces a route choice. We read the two by homology in dimensions 0 and 1 over a transport-weighted carrier built from successful trajectories, yielding an enumerable gate set with shell-level certificates; the certified gates are what we call topological necessities. Certified gates enter the decision loop as a recursive topological gate hierarchy. Under a fixed, isomorphic free space, the object survives executor replacement: gates frozen on PointMaze data transfer without retraining to Ant and Humanoid, attaining the highest Humanoid aggregate under a unified interface (96.1), with +36.0 over a map-privileged reference on the multi-route task (p=1.4e-5); the planner saturates PointMaze (100+/-0) and matches or exceeds the strongest baselines on AntMaze (giant +22.9) and Kitchen (+15.8/+12.6).

1 INTRODUCTION

The paper defines transferable subgoals as route-conditioned orders of unavoidable stages recovered from successful offline trajectories rather than from any executor. A certified carrier-based construction and recursive hierarchy support transfer across mechanisms and embodiments.

  • Contribution: The framework recovers carrier-relative topological necessities—unavoidable, route-conditioned stages—without defining subgoals through the downstream executor.The object is transmitted by successful trajectories and belongs to none of the executors that produced them.
  • Method: The method combines a transport-weighted carrier, rectification embedding, dimension-matched H0/H1 readouts, and recursive gate hierarchy.H0 reads bottleneck order, H1 reads route bifurcation, and certified gates are recursively densified for execution.
  • Evidence chain: The evidence chain includes mechanism interventions, four-regime invariance, zero-label Kitchen sufficiency, layout perturbations, and cross-embodiment frozen transfer.These tests evaluate whether strategic subgoals emerge structurally and survive executor replacement.

2 RELATED WORK

Related work spans offline GCRL and hierarchical subgoal discovery, graph-based and topological representations, and planning or transfer methods operating at adjacent layers. The paper positions its contribution around structural subgoals rather than executor-tied value, latent, or policy outputs.

  • Offline GCRL and hierarchical subgoal discovery: Offline GCRL and hierarchical methods commonly derive high-level actions or subgoals from value functions, latent representations, options, managers, or skills.These approaches provide temporal abstraction but remain distinct from the paper’s carrier-relative topological object.
  • Geometric, graph-based, and topological structure in RL: Graph-based, geometric, and topological RL methods read transition structure through cuts, spectra, shortest paths, keypoints, quasimetrics, or learned subgoal sequences.The related-work landscape includes multiple structural representations for planning and navigation.
  • Planning, monitoring, and transfer across embodiments: Planning, monitoring, recovery, diffusion-policy, and embodiment-transfer methods address planning or execution layers adjacent to mechanism-invariant strategic subgoals.The paper distinguishes its structural task representation from approaches that transfer policies, representations, or correspondences.

3 READING THE DATA-INDUCED STRATEGY SPACE

The method builds a strategy-space carrier from successful trajectories, orders it by goal-relative geometry, and detects certified bottleneck gates through persistent shell-measure valleys. H0 captures gate order, H1 captures route forks, and recursion turns certified gates into an executable hierarchy.

  • 3.1 CARRIER AND GOAL-RELATIVE GEOMETRY: The carrier is the data-supported transition structure induced by successful trajectories, equipped with goal-relative geometry; admissible paths are those supported by this carrier.The framework studies this carrier instead of reconstructing the full environment manifold.
  • 3.1 CARRIER AND GOAL-RELATIVE GEOMETRY: A certified gate is a narrow separating neck region, while a gate representative is a data-supported state selected from that region for deployment.The certificate is defined at the region level rather than for an individual point.
  • 3.1 CARRIER AND GOAL-RELATIVE GEOMETRY: The transport-weighted graph and goal-relative geodesic coordinate reduce nuisance-motion distortion and order carrier states into radial shells.The radial coordinate is computed by graph geodesics, while rectification supports shell-measure estimation.
  • 3.2 THE PERSISTENT BOTTLENECK READOUT: Persistent valleys in the shell-measure profile survive an adaptive persistence threshold to form enumerable gate chains ordered by radial coordinate.Segments without a registered positive persistence gap receive no forced gate.
  • 3.2 THE PERSISTENT BOTTLENECK READOUT: H0 detects and ranks bottleneck order, whereas persistent H1 detects bounded holes and assigns route classes that scalar valleys cannot distinguish.Each subproblem is matched to the lowest-dimensional homology invariant that expresses it.
  • Recursive gate hierarchy: The same certified shell readout is recursively reapplied within route segments, producing a hierarchical chain whose gates retain shell-level certificates.The hierarchy can be deployed through in-distribution waypoints and the neck shells also support runtime monitoring and re-anchoring.

4 EXPERIMENTS

Experiments test whether certified gates capture task structure rather than executor-specific behavior and whether they transfer across embodiments. Across benchmarks and controlled interventions, the method achieves strong long-horizon performance, while recursive refinement improves deployment selectively.

  • Controlled interventions: Controlled mechanism interventions identify the same load-bearing neck, and fixing that gate changes completion from 50/50 to 0/50 and back to 50/50 across executor swaps.The intervention is presented as evidence that the located gate is causal rather than correlational.
  • Cross-embodiment transfer: The frozen PointMaze-discovered gate set transfers without retraining to Humanoid under a shared BFS interface, while Ant-discovered gates remain actionable on Humanoid.The transfer evaluation uses frozen gate sets and a unified interface.
  • Task-level structure: Map perturbations leave all 18 gates bit-identical, whereas blocking a detour or neck removes only the corresponding route skeleton.This supports the carrier as the mediator of the detected structure.
  • Main benchmark: 100±0 on all PointMaze sizes, with AntMaze gains up to +22.9 over CFHRL and Kitchen gains of +15.8/+12.6 over HIQL.The strongest gains occur on long-horizon, multi-route, and multi-stage tasks.
  • Runtime monitoring: Gate-segmented monitoring reduces pure false alarms from 3.2/1.46 to 0.92/0.00 per episode.The certified neck shells therefore serve as runtime monitors in addition to planning landmarks.
  • Recursive hierarchy: Recursive sub-necks raise PH-pt aggregate success from 91.1 to 96.1 and PH-ant t4 success from 70.0 to 96.0, while the t5 PH-ant gain is absent.The refinement removes transition-zone stalls but is not uniformly additive when the bottleneck is execution-layer.
  • Route selection: The H1 winding-number route lock preserves success within the noise band and records zero route switches across every task seed.On t4, success is 94.8 versus 96.4 with p=0.46.

5 CONCLUSION

The paper frames topological necessities as route-conditioned stage orders recoverable from offline trajectories and independent of any executor. Evidence is strongest for transfer across evaluated embodiments in fixed maze geometry, while broader invariance remains outside the claim.

  • Conclusion: Topological necessities are route-conditioned orders of unavoidable stages transmitted by successful trajectories and belonging to none of the executors.H0 shell persistence certifies bottleneck gates, while recursion makes the certificate deployable as a hierarchy.
  • Conclusion: Frozen gates transfer to two dynamically distinct embodiments and achieve the highest aggregate on each, while Ant-discovered gates remain actionable on Humanoid at 91.0%.The construction is presented as extracting task bottleneck structure rather than source-executor action patterns.
  • Scope: The claim is task-conditioned transferability: bottleneck structure remains stable across evaluated embodiments when maze geometry is held fixed.This fixed-geometry condition bounds the scope of the invariance claim.
  • Future work: Future work includes fully online recursion, integration with skill and option discovery, and richer uses of higher-order readouts.

AI USE STATEMENT

AI tools assisted experiment-code development and manuscript language polishing. The authors state that the experiments and scientific content were produced and verified by them.

  • AI use: AI tools were used for collaborative experiment-code development and language assistance during manuscript preparation.

REPRODUCIBILITY STATEMENT

The paper supports reproducibility through released artifacts, explicit benchmark and pipeline documentation, and targeted audits that separate planning, execution, topology, and interface effects.

  • Artifacts: The released archive provides code, checkpoints, logs, per-figure data, frozen algorithm stacks, experiment scripts, executor checkpoints, and binary checksums.The artifact README also documents the theorem-to-code registry and excluded-large-file policy.
  • Documentation: Benchmark provenance and appendix documentation cover baseline audits, Kitchen pipelines, cube boundaries, planner audits, recovery figures, and recursive sub-necks.The appendix map identifies these materials across Sections A.1–A.6.
  • Kitchen pipeline: The Kitchen pipeline defines completion events as dense-count increments and reconstructs per-subtask completion flags from offline signature-dimension bands with 97.6% count agreement.The planner consumes an extracted subtask order; on Kitchen partial/mixed, that order is a total chain represented by the subtask lattice.
  • Ablations: The zero-event-label ablation reaches 96.5±1.3 with PH necks versus 94.3±2.5 for supervised event bands, while strong low-level control yields 94.0±0.5 without decomposition.This bounds the interpretation: the result demonstrates sufficiency under the stated pipeline and a near-saturated executor, not universal planner superiority.
  • Attribution and robustness: Planner and executor audits attribute the cube boundary to executor semantics: flat-goal GCIQL reaches 79.6±1.4/36.3±2.3, whereas the diffusion executor reaches 5.3/2.0.Waypoint supply covers 100% of paired starts, pseudo-execution converges at 90%, and goal-retrieval execution reaches 74.4; mismatched waypoint semantics collapse performance from 79.6 to 12.8.

A.8 EXECUTOR PLUGGABILITY AND INTERFACE ABLATIONS

The ablations separate executor capability from the frozen planning interface and test whether certified gates remain valid under mechanism and layout changes. Results show pluggable execution, harmful fixed gate attraction, carrier-mediated gate changes, and severe deployment costs when blocked layouts are not re-identified.

  • Executor pluggability: 95.1 ± 0.6 versus 89.7 ± 2.5 shows that executor class changes performance while both consume the same waypoint contract.The planning layer and interface remain fixed; the score difference is attributed to executor capability.
  • Interface ablations: 86.0 ± 2.0 overall, −3.7 versus the lookahead contract, shows that gates used as distant attractor targets remain harmful after contract alignment.The finding motivates recursive sub-neck detection and support-flow densification rather than fixed gate targeting.
  • Gate calibers: 95.5 under support-flow exceeds the map-shaped BFS interface scores of 89.3 plain and 86.9 pruned, using one fixed selection-free pipeline.The two gate calibers are not combined into per-task best results; plain preserves fuller route coverage while pruning removes detours.
  • Layout perturbation: 18/18 gates remain unchanged when the map changes but the successful-trajectory carrier does not, while blocking either route causes the corresponding gate skeleton to disappear.Detour blocking deletes 143 trajectories and removes 0/5 detour gates; direct-neck blocking removes 0/4 direct gates.
  • Layout perturbation: 92.0±2.8 after re-identification versus 7.3±1.9 with frozen gates demonstrates the deployment cost of retaining paths through a newly blocked corridor.The net benefit of re-identification is +84.7, with 93% of frozen episodes timing out.

A.12 MULTISCALE PERSISTENT-H1: ROUTE-FORKING OBSTACLES

Persistent H1 captures route-forking holes that the one-dimensional radial filtration cannot represent, using cubical complexes built from trajectory-induced coverage. The route signature is stable under subsampling and improves route locking, while carrier controls show that the carrier—not merely the detector—is essential.

  • Construction: The cubical construction rasterizes trajectory coverage, dilates the occupied mask, and filters cells by Euclidean distance to coverage, producing nested offsets for persistent H1.The resulting hole structure is computed from behavior-induced coverage rather than map access.
  • Multiscale persistent-H1: The dominant finite H1 bars are [0, 2.062] on task 2, [0, 2.500] on task 5, and [0, 2.236] on task 4, identifying multiscale route-forking structure.Task 4 also has a secondary bar [1.118, 2.062] with lifetime 0.944; direct and detour trajectories pass the localized hole on opposite sides.
  • Stability: The finite route-forking bar persists in 3/3 subsampled runs for both task 2 and task 5, although localization is less rigid than existence.The result supports stability of the topological signal under substantial trajectory removal.
  • Decision-loop integration: Zero route switches on task 4 with the topological lock contrasts with 17 for the geometric lock on PH-pt and 26 on PH-ant.The topological branch uses winding signatures for discriminating routes and falls back to the geometric branch on non-discriminating tasks.
  • Carrier controls: The detector control reproduces all 18 reference gates exactly, whereas the raw-coordinate carrier matches only 11/18 and misses deep task-4 and task-5 gates.Off-carrier failures reflect collapsed prominence, gait-dominated PCA directions, and inflated spatial dispersion rather than simple gate displacement.
  • Interpretation: H0 expresses bottleneck order, while H1 expresses route bifurcation, so scalar prominence cannot replace the higher-dimensional route-forking readout.The contribution is located in the carrier construction and dimension-matched invariants, not solely in the final detector.

A.14 ABLATIONS AND SENSITIVITY

The appendix validates parameter choices, implementation costs, theoretical closure, and failure boundaries through sensitivity scans, audits, and perturbation analyses.

  • Sensitivity: The operating point (ℓ, k) = (2.0, 12) lies at the peak of a scan that is unimodal in ℓ and plateaued in k.
  • Sensitivity: The waypoint-horizon criterion selects the shortest horizon preserving frame-level tracking, with Kitchen press success 84 at w→5 matching w=25.
  • Implementation: The largest AntMaze-giant graph builds offline in approximately 2 seconds on one CPU core and is amortized across goals and episodes.
  • Theoretical closure: 1.06× closure matches the measured δemp = 0.0483, while competing models deviate by 1.14× to 3.62×.The three-state Voronoi factorization is the closest candidate in the closure comparison.
  • Failure boundary: Residual failures separate into route-geometry budget exhaustion on t5 and executor deadlocks on t4, with no audited structure-layer failures.

A.18 CROSS-EMBODIMENT TRANSFER: PROTOCOL, EXECUTOR HISTORY, AND ERROR BARS (FOR TAB. 1)

Cross-embodiment evaluation freezes gates discovered from PointMaze while varying deployment interfaces, executor classes, statistical protocols, and reference methods.

  • Deployment interface: The main benchmark expresses recursive topology through support-flow waypoint subdivision rather than introducing a separate mechanism.
  • Statistical protocol: Comparisons use α-paired evaluation and per-seed success rates, with paired t-tests over five seed pairs for per-task contrasts.
  • Transfer and executor history: PointMaze-discovered gates transfer to Humanoid and Ant through frozen interfaces, while executor-class ablations localize performance constraints to low-level execution.
  • Transfer and executor history: PH-pt reaches 89.3 on the Humanoid plain-caliber harness versus 84.7 for PH-ant, a 4.6-point gap concentrated on t4.
  • Error bars: Table A.7 reports success percentages as mean ± population standard deviation across Ant and Humanoid panels with differing seed counts.

B.3 THEOREM T1: THE DISCRETE EIKONAL TRIPLET

Theorem T1 establishes a discrete Eikonal triplet for graph geodesic distance: metric structure, 1-Lipschitz radial distance, and exact decrease along shortest paths.

  • Metric structure: Theorem T1 proves that dG is an extended metric, becomes genuine under Assumption 1, and defines the graph geometry used by the method.
  • Discrete Eikonal inequality: The radial coordinate r is 1-Lipschitz with respect to dG, giving |r(u) − r(v)| ≤ dG(u, v).
  • Characteristic property: Along a shortest path from v toward the goal, r(v) = r(u) + dG(v, u), so radial distance decreases at full edge-weight rate.
  • Interpretation: The framework turns the Eikonal constraint from a regression target into an intrinsic identity of the graph-geodesic object.
  • Embedding compatibility: The radial and edge-length supervision signals derive from the same dG, so an isometric embedding can satisfy both loss terms simultaneously.

B.3.1 THE STATISTICAL LYAPUNOV PROPERTY AND THE PREDICTABILITY OF THE ONE-STEP RELAPSE PROBABILITY

The statistical Lyapunov analysis models anchored rollout progress as a three-state process and predicts relapse probability from rollout switching and graph-structural magnitude factors.

  • Three-state dynamics: Anchored increments comprise plateau, forward, and bounce states, with the bounce state being the sole source of positive relapse increments.
  • Graph-structural predictability: The relapse probability factors into bounce-event frequency and a Voronoi/Delaunay graph magnitude prior, whose conditional rate differs from the prior by approximately 7%.
  • Closure: The three-state model predicts δpred = 0.0511 versus δemp = 0.0483, achieving a 1.06 ratio on the measured threshold.
  • Scope: The closure is a same-sample consistency check rather than a held-out prediction, leaving cross-task validation for future work.
  • Model comparison: Only the three-state decomposition reaches 1.0×-level agreement, while alternative models miss the anchoring atom, skewness, or discrete bounce mechanism.

B.4 THEOREM T2: GEODESIC CONSISTENCY OF THE TRANSPORT-WEIGHTED MANIFOLD GRAPH

The transport-weighted filtered graph consistently approximates intrinsic manifold geodesics, enabling unbiased radial labels for goal-reaching. Its guarantees rely on retaining intra-manifold neighbors while removing wall-crossing shortcuts under explicit sampling and geometry assumptions.

  • Theorem and consequence: (1 ± ε)d_M(x, y) bounds d_G_P(x, y) uniformly over graph vertices with probability tending to one.The consistency result transfers to set distances and supports graph-based geodesic estimation.
  • Consequence: The resulting radial labels consistently estimate true geodesic distance to the goal, unlike time-to-go labels that are biased by policy wandering.This establishes the geometric basis for the later shell-profile and gate-detection pipeline.
  • Proof strategy: The proof extends Euclidean kNN convergence through a constant-coefficient transport metric, then transfers pointwise graph-distance bounds to multi-source goal distances.The transport metric is bi-Lipschitz equivalent to the Euclidean metric, preserving consistency up to equivalence constants.
  • Assumptions: Under density, wall-separation, and local-regularity assumptions, kNN neighborhoods become sufficiently local for the graph-consistency argument.The sampling support is decomposed into a compact free domain and excluded wall neighborhoods with positive density on the free domain.
  • Filter fidelity: The two filters retain candidate edges whose chords lie in the free domain and delete sufficiently short chords crossing wall slabs.For sufficiently large samples, the filtered graph coincides with the free-domain intrinsic kNN graph with high probability.

B.5 THEOREM T3: A PERSISTENT-HOMOLOGY CHARACTERIZATION OF BOTTLENECKS

Persistent homology identifies bottlenecks as persistent valleys in a goal-relative shell-measure profile, while the theorem certifies their necessity as separating gate regions. Detection is mechanism-invariant only for shared free space with adequate coverage, and estimator-induced marginal gates remain a boundary.

  • Persistence selection: The elbow separates shallow spurious valleys from deeper true gates, but kneedle failure can conservatively miss small gates rather than admit false positives.The separation is reliable when the persistence gap dominates within-segment variation; experiments separated true and spurious valleys in every reported run.
  • Bottleneck characterization: Persistent valleys above the data-adaptive elbow correspond to certified narrow necks that every admissible path from the outer region to the goal must cross.The certificate combines valley persistence, shell localization, and a separation argument based on continuity of the radial coordinate.
  • Topological decomposition: The H0 readout orders separating gates, while H1 captures bounded holes that force route bifurcation.The framework assigns each structural question the lowest homology dimension that expresses it.
  • Mechanism invariance: For a fixed task manifold, certified gates are must-pass across mechanisms because their separating shells are determined by geometry rather than executor identity.The invariance claim excludes elbow-tied marginal instances, which may vary by mechanism and remain registered but uncounted.
  • Scope: Mechanism invariance requires shared free space and successful-trajectory coverage, while changing passages or omitting a mechanism from offline data invalidates the corresponding guarantee.The detector cannot distinguish a mechanism absent from the data from one absent from the world.

B.6 THEOREM T4: THE JACOBIAN SPECTRAL GAP AND THE EFFECTIVE DIMENSION

The Jacobian Gram spectrum separates signal directions from sampling-dependent noise, yielding an effective dimension estimate whose convergence depends on effective sample size. Empirical diagnostics support the residual-correlation model, while exact noise-edge calibration remains open.

  • Empirical diagnostics: The measured centered Jacobian fluctuation decays over d⋆≈1.65, while the spectral gap remains at m_eff = 3 under subsampling and an independent cross-trajectory benchmark.The within-trajectory sampling interval is 5 steps versus an autocorrelation time of 80 steps.
  • Residual dependence: Dependence enters convergence through a Bartlett-type effective sample-size rescaling derived from covariance mass rather than assumed independently.The theorem uses bounded Jacobians and pairwise covariance decay under the sampling design.
  • Population identity: The Jacobian Gram matrix has exactly d_ψ nonzero population eigenvalues, so its rank equals the number of directions along which the embedding varies.The null space is the orthogonal complement of the span of Jacobian images.
  • Finite-sample recovery: With a nonempty signal–noise spectral window, the estimated effective dimension satisfies P( m̂_eff = d_ψ ) ≥ 1 − η and converges at O(P_eff^-1/2).The threshold is chosen between the smallest signal eigenvalue and the noise fluctuation bound.
  • Limitation: The framework proves signal–noise separability but leaves exact noise-bulk upper-edge calibration to future work under mixing.The remaining gap concerns rigorous edge-rigidity results for the Marchenko–Pastur boundary and related thresholds.

B.7.2 THEOREM AND PROOF

The interface theorem bounds low-level execution error by the mismatch between selected subgoal distributions and path-aligned training conditionings. Consequently, rollout success decreases monotonically with this Wasserstein mismatch under linear error accumulation.

  • Interface theorem: The one-step interface loss is bounded by ε_π + L_δW_1(ν, μ_path), separating training risk from subgoal-distribution mismatch.The bound follows by coupling the selected subgoal distribution to the ideal path-conditioned distribution.
  • Rollout consequence: After H-step linear error accumulation, the probability of ending outside the goal ball is at most H C_dyn(ε_π + L_δW_1)/R.The result uses the standard state-drift structure for absorbing-wall problems.
  • Monotonicity: Success rate therefore decreases monotonically with the interface Wasserstein distance relative to the ideal path-aligned interface.The relationship is an affine upper bound, and its constants are not claimed to be tight.
  • Terminal behavior: Retargeting is less mismatched near the terminal region because path displacement vanishes, reducing its dynamic-freezing error.Its posture mismatch is smaller at the terminal than midway along the path.
  • Rule comparison: Loose frame selection has the smallest estimated transport cost, followed by synthesis and retargeting, with arbitrary-node frames worst.The ordering W1(loose) < W1(synth) ≲ W1(retarget) < W1(off) matches success rates 90 > 77 ≳ 67 > 62.

B.8 THEOREM T5: CONVERGENCE OF SHORTEST-PATH-TREE PLANNING

The theorem establishes convergence of shortest-path-tree planning by combining one-step progress, path-level martingale concentration, and graph-legality conditions. It decomposes success into planning, execution-accumulation, and terminal-docking factors, with empirical measurements supporting the required conditions.

  • Concentration: Path-level martingale concentration controls execution deviations using bounded martingale differences and Azuma–Hoeffding rather than pointwise high-probability bounds.The anchoring-loss sequence is handled analogously through conditional bounds on failure and normal steps.
  • Planner construction: The planner anchors each state to its nearest graph node, follows a shortest-path-tree parent chain to a lookahead waypoint, and executes toward that waypoint.The manifold radial function measures distance to the goal, while εT2 captures graph–manifold bridging error.
  • One-step progress: One-step progress follows from Lipschitz distance bounds, graph–manifold bridging, waypoint radial decrease, and bounded execution deviation.The resulting chain reduces radial distance by the waypoint drop while accumulating εT2 and δt error terms.
  • Theorem T5: Theorem T5 guarantees terminal-zone arrival when average net progress is positive, subject to graph-construction failure, concentration, and terminal-docking conditions.The rollout must remain within the λ-tubular neighborhood of the data support, and the arrival probability combines planning and path-level concentration terms.
  • Empirical validation: Anchoring failure was observed zero times in 48029 planning steps, yielding panchor < 6.2 × 10−5 and a measured failure-cost term of 0.Together with the other measured constants, these observations support the theorem’s conditions and make average net progress a system-health indicator.
  • Empirical implications: 100 ± 0 scores on the three pointmaze sizes are attributed to near-zero execution noise and zero planning-layer failures.The residual AntMaze gap is attributed to low-level gait noise rather than the planning layer.
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