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

Motivation

• Offline goal-conditioned reinforcement learning must separate task structure from executor-specific subgoals because executor variation can degrade transfer when dynamics or embodiments change.The proposed object is recovered from static experience recorded by other executors, without new interaction.Source: §1 · p. 1

Task-level subgoals

• Topological necessities are carrier-relative, falsifiable stages whose route-conditioned order is shared by successful trajectories rather than tied to any executor’s parameters.They represent unavoidable separations and route choices, supporting transfer across executors or embodiments in fixed, isomorphic free spaces.Source: §1 · p. 2

Certified construction

• The framework recovers bottleneck order with H0 and route bifurcation with persistent H1 on a transport-weighted, rectified carrier, producing enumerable gates with shell-level certificates.Certified gates form a recursive hierarchy that reanalyzes inter-gate segments for persistent sub-necks.Source: §1 · p. 2

Evidence chain

• The method transfers frozen gates from PointMaze to Ant and Humanoid without retraining, reaching a 96.1 Humanoid aggregate and +36.0 on the multi-route task (p=1.4×10−5).On Kitchen, the unsupervised readout matches the supervised event-label upper bound at 96.5 vs. 94.3.Source: §1 · p. 2
• Across OGBench, the method saturates PointMaze at 100 ± 0, improves AntMaze giant by +22.9, and exceeds HIQL on Kitchen by +15.8/+12.6.The evidence chain also includes controlled mechanism interventions, four mechanism regimes, and zero-label Kitchen neck sufficiency.Source: §1 · p. 2

Offline GCRL and hierarchical subgoal discovery

• The paper contrasts executor-dependent subgoals with task-level stages recovered from other executors’ data and falsifiable under executor replacement.Classical methods use visitation, values, policies, or transition statistics; this work uses explicit carriers, admissible paths, and goal-relative bottlenecks.Source: §2 · p. 3

Geometric, graph-based, and topological structure in RL

• Unlike graph-based planners and BGSS, the method certifies gates by separation: every admissible path must cross the bottleneck.It selects gates from the persistence spectrum of a goal-relative geodesic-radial field, making topology the native language of the certificate.Source: §2 · p. 3

Planning, monitoring, and transfer across embodiments

• Unlike typical embodiment-transfer methods, the paper freezes a bottleneck set from one embodiment’s offline data and tests it on another without adaptation.This uses mechanism persistence to assess whether the gates capture task-level structural content rather than executor-specific behavior.Source: §2 · p. 3

3.1 CARRIER AND GOAL-RELATIVE GEOMETRY

• Successful trajectories define a data-induced carrier with transport-weighted geometry and a graph-geodesic radial coordinate measured relative to the goal.A filtered kNN graph suppresses nuisance-dominated dimensions and unsupported shortcuts; shell measurement uses a rectifying embedding, while planning retains graph structure.Source: §3.1 · p. 4

3.2 THE PERSISTENT BOTTLENECK READOUT

• Persistent H0 valleys in the shell-measure profile become enumerable separating gate regions under a data-adaptive persistence elbow, forming route-conditioned chains.The certificate concerns crossing a narrow region rather than visiting a particular state, and no gate is forced when the persistence spectrum lacks a positive registered gap.Source: §3.2 · p. 4, §3.1 · p. 4
• H0 captures bottleneck order while H1 detects route forks through holes and winding-number signatures, selecting route chains without route switches in evaluated discriminating cases.When signatures coincide, deployment falls back to the geometric rule; the representation adds an enumerable, certificate-carrying structure beyond scalar valley ranking.Source: §3.2 · p. 5

3.3 RECURSIVE TOPOLOGICAL GATE HIERARCHY

• Reapplying the certified readout within route-conditioned segments produces a recursive gate hierarchy whose levels retain shell-level certificates relative to restricted carriers.The same hierarchy fixes route topology while in-distribution waypoint interfaces make gates executable; neck shells also monitor drift, stalls, and completion inconsistencies through global re-anchoring.Source: §3.3 · pp. 5–6

4.1 SETUP AND PROTOCOL

• The experiments use fixed readout and gate selection across domains, matched interfaces and seeding protocols, and frozen run logs, with domain-specific executors and waypoint interfaces.AntMaze uses a retrained HIQL latent-action executor, Kitchen uses a shortcut-style diffusion executor, and Section 4.3 uses a unified BFS progress pointer.Source: §4.1 · p. 6

4.2 WHAT THE TOPOLOGY FINDS

• Controlled interventions show that topology detects causal load-bearing necks, recovers predicted reward bottlenecks, and tracks monotonic progress rather than random data artifacts.The strongest valley is 6.1–8.5× the elbow; swapping the fixed gate changes completion from 50/50 to 0/50 and back, while the predicted route returns +357 versus −192 geometrically.Source: §4.2 · p. 6

4.4 DEPLOYMENT VALUE

• Zero-label necks match supervised interface performance, while planning-level structure—not executor or interface substitutions alone—accounts for the deployment gains.Kitchen reaches 96.5±1.3 without labels versus 94.3±2.5 for supervised event bands; grafting HIQL’s high level drops Kitchen by −47.3/−44.3, and direct tactical targets reduce performance by −3.7.Source: §4.2 · p. 7, §4.4 · p. 8

4.3 THE ORDER IS TASK-LEVEL

• Frozen gates remain task-level under map and executor changes: 18/18 gates are bit-identical to layout perturbations, while surviving-data rediscovery restores deployment from 7.3±1.9 to 92.0±2.8.Independent Ant discovery remains actionable on Humanoid at 91.0, and executor replacement under the same waypoint contract reaches 95.1±0.6.Source: §4.3 · pp. 7–8
• 96.1 aggregate Humanoid success transfers a PointMaze-discovered frozen gate set across embodiments, exceeding the map-privileged direct reference by +10.8 and spectral by +18.5.On multi-route task t4, the method gains +36.0 over direct (94.8 vs. 58.8) and +43.2 over spectral, while support-flow deployment reaches 95.5.Source: Table 1 · p. 7, §4.3 · p. 8

4.4 DEPLOYMENT VALUE

• 100±0 on PointMaze, +22.9 on AntMaze giant, and +15.8/+12.6 on Kitchen establish the strongest matched benchmark gains in long-horizon, multi-route, and multi-stage tasks.PointMaze leads all sizes by +14 to +32; AntMaze medium is +0.5 over HIQL and large is +3.4, while Kitchen improves over HIQL by +15.8/+12.6.Source: §4.4 · p. 8, Table 2 · p. 8

4.5 RECURSION IS ALREADY OPERATIVE

• 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 H1 winding-number lock prevents route switching without measurable success cost.The PH-ant refinement cuts stall events by 64%; on t5 its gain is absent, indicating recursion is structurally conditional, and H1 runs show zero route switches.Source: §4.5 · p. 9, Table 3 · p. 9

• Topological necessities recover route-conditioned orders of unavoidable stages from offline data, with H0 shell certificates and recursive filtration producing a deployable gate hierarchy.Between certified gates, the same shell filtration is reapplied on each inter-gate segment.Source: §5 · p. 9
• Frozen PointMaze gates transfer to two dynamically distinct embodiments, attaining the highest aggregate on each; Ant-discovered gates remain actionable on Humanoid at 91.0%.The evidence is strongest on long-horizon, multi-route tasks, including comparisons against map-privileged direct references.Source: §5 · p. 9
• The carrier-relative construction extracts task bottlenecks rather than the source executor’s action patterns, supporting mechanism-change stability under fixed maze geometry.The claim is task-conditioned transferability across evaluated embodiments, not unrestricted invariance.Source: §5 · p. 9
• Future work targets fully online recursion, integration with skill- and option-discovery, and richer uses of higher-order readouts.These directions are identified in Appendix A.12.Source: §5 · p. 10

AI USE STATEMENT

• AI tools assisted with experiment coding and manuscript language polishing, while authors executed, debugged, verified, reviewed, and took responsibility for the final content.The authors also retained responsibility for the research problem, mathematical formulation, theorem statements and proofs, experimental design, and interpretation of results.Source: AI USE STATEMENT · p. 15

A.1 BASELINE PROVENANCE

• 90.9 is the highest protocol-matched AntMaze-giant point estimate, exceeding CFHRL by +22.9, HIQL by +25.9, and AQM by +2.9.The appendix also audits competing methods against four structural checks: enumerable must-pass subgoals, false-alarm control, executor replacement, and reproducible cross-embodiment gate sets.Source: §A.1 · p. 16

A.2 KITCHEN PIPELINE DETAILS

• The Kitchen pipeline builds a 21-object-joint graph, extracts completion events with 97.6% count-match, and plans along the microwave→kettle→light→slide subtask chain.The subtask lattice formalizes precedence and generalizes the chain representation to partially ordered tasks.Source: §A.2 · pp. 16–17

A.3 PUZZLE NEGATIVE CONTROL / REPRODUCIBILITY STATEMENT / A EXPERIMENTAL AND IMPLEMENTATION DETAILS

• The Rubik environments provide a negative control: puzzle-3x3 reaches 98.4±0.3 versus GCIQL’s 95±1, while flat-goal ablation collapses to 0.4%.The appendix map locates the benchmark provenance, protocol, sensitivity, transfer, and implementation materials, while the artifact archive provides code, checkpoints, logs, per-figure data, and checksums.Source: A EXPERIMENTAL AND IMPLEMENTATION… · p. 15, REPRODUCIBILITY STATEMENT · p. 15, §A.3 · p. 17

A.4 KITCHEN PH NECESSITY, SEMANTICS, AND RECOVERY

• The configuration measure finds bottlenecks for light and slide, whereas the action measure finds valleys for all four subtasks, with microwave valleys surviving 14/14 perturbation rounds.Configuration valleys indicate narrow object-state tubes; action valleys indicate phases where admissible arm actions collapse.Source: §A.4 · pp. 17–18
• On arbitrary-progress recovery, ph reaches 70.4 versus 69.1 for event supervision, indicating zero-supervision recovery matches supervision out of distribution under identical sampling conditions.Absolute performance is capped by executor and dataset dynamics mismatch rather than planner quality.Source: §A.4 · p. 18

A.5 RECOVERY, CUBE-BOUNDARY, AND PLANNER-AUDIT FIGURES

• The cube boundary is execution-limited: GCIQL scores 79.6±1.4 and 36.3±2.3 versus diffusion’s 5.3 and 2.0, while planner audits show 100% waypoint coverage and 90% pseudo-execution convergence.Supplying waypoints to an executor without waypoint training collapses performance from 79.6 to 12.8, exposing a training-semantic interface contract.Source: §A.5 · pp. 18–19, Figure A.3 · p. 20, Figure A.4 · p. 20

A.6 RECURSIVE TOPOLOGY ON KITCHEN: SUB-NECKS IN ARM SPACE

• All four Kitchen subtasks yield stable recursive sub-necks, with CV/top-persistence pairs of 0.228/1.114, 0.171/0.883, 0.223/1.028, and 0.310/1.633.The early valleys identify intra-segment grasp or contact commitments, while later valleys mark place or finish convergence, extending recursive topology beyond mazes.Source: §A.6 · p. 20

A.7 THE PH-WAYPOINT PROBE

• 84–94% completion from isolated PH gate coordinates shows that certified bottlenecks contain sufficient routing information, but sparse gates are strictly worse than dense lookahead waypoints.The support-flow interface retains the gate chain while densifying it into same-corridor, in-distribution waypoints, resolving the executor’s training-horizon mismatch.Source: §A.7 · pp. 21–22

A.8 EXECUTOR PLUGGABILITY AND INTERFACE ABLATIONS

• 95.1 ± 0.6 with HIQL versus 89.7 ± 2.5 with FQL confirms executor pluggability under the same waypoint contract, with differences attributed to executor capability.Both replace only the low-level executor while retaining the planning layer; HIQL’s score is not its full-stack result.Source: §A.8 · p. 22
• 86.0 ± 2.0 under aligned gate-target training shows distant gate attraction remains harmful, motivating stepwise support-flow targets that retain route structure while avoiding out-of-distribution sparse coordinates.This is 3.7 points below the same executor with lookahead targets, while task 2 drops 24.7 points; fresh re-anchoring is load-bearing.Source: §A.8 · p. 23

A.9 TWO GATE CALIBERS AND EFFICIENCY-BASED ROUTE SELECTION

• 95.5 overall with support-flow exceeds the map-shaped BFS interface’s 89.3 plain and 86.9 pruned scores, while the two fixed calibers remain selection-free and separate.Plain preserves full route coverage; pruning removes detour routes, and frame-level efficiency reweighting is rejected because t4 collapses to 0.0.Source: §A.9 · p. 24
• 96.7 versus 62.0 on long-chain task t4 shows the certified method’s largest margin under the larger baseline budget, while its score shifts only 89.3 → 86.9 across budgets.Baseline scores are budget-dominated, whereas certified gate counts are determined by detection; efficiency pruning uses route-level criteria rather than frame-level weights.Source: §A.9 · p. 24

A.10 LAYOUT PERTURBATION: THE CARRIER IS THE SOLE MEDIATOR

• 18/18 gate sets remain unchanged when the map changes but zero-coverage data does not, establishing that the carrier—not hidden layout information—determines discovery.Across t2/t4/t5, the frozen gate counts are 5/5, 9/9, and 4/4, with zero new or ghost gates.Source: §A.10 · pp. 24–25
• 92.0±2.8 after re-identification versus 7.3±1.9 with frozen gates demonstrates an +84.7 deployment benefit when a physical blockage changes the surviving trajectory carrier.The frozen executor enters the blocked corridor and times out, whereas re-identification rebuilds three gates from 116 surviving trajectories.Source: §A.10 · p. 25

A.11 E1 AND E4 PROTOCOL DETAILS

• The E1 neck is detected across all four mechanism regimes, and swapping only corridor executors yields 50/50 before blocking, 0/50 for the blocked left route, and 50/50 for the right route.B and BLK coincide bit-for-bit because absent mechanisms are indistinguishable to the trajectory filtration.Source: §A.11 · pp. 25–26, Figure A.8 · p. 26, Table A.3 · p. 27
• Gate-segmented monitoring complements stall-window monitoring by providing zero pure false alarms on task 5 without material recovery loss, but sparse chains and post-final-gate states remain blind spots.The deployed stack therefore retains the stall window and uses gates as a precision layer rather than a replacement.Source: §A.11 · pp. 26–27, Table A.4 · p. 27

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

• The cubical filtration constructs nested coverage offsets and finite dimension-one birth–death intervals, providing a multiscale existence test distinct from fixed-scale hole localization.The implementation uses one-cell dilation, h = 0.5 maze units before rescaling, a 30,000-cell budget, and GUDHI 3.13.0; an annulus recovers one long finite bar [0, 4.596].Source: §A.12 · p. 27
• AntMaze-medium tasks 2, 4, and 5 each contain a dominant finite H1 bar, with task-4 routes passing opposite sides of the localized central obstacle.Top bars are [0, 2.062] (lifetime 2.062), [0, 2.236] (lifetime 2.236), and [0, 2.500] (lifetime 2.500); task 4 also has [1.118, 2.062] (lifetime 0.944).Source: §A.12 · pp. 27–28, Figure A.10 · p. 28
• The dominant route-forking bar survives 70% trajectory subsampling in 3/3 runs for both tasks 2 and 5, although fixed-scale localization is stable in only 2/3 seeds.The route-forking interpretation holds in all three seeds, while one localization shifts to another real obstacle block.Source: §A.12 · p. 28
• A winding-number route lock yields zero route switches on discriminating task t4, versus 17 for PH-pt and 26 for PH-ant geometric locks, with comparable success rates.The t4 route signatures are [0] and [−1]; non-discriminating tasks remain within ±1.4 points of geometric-lock baselines, so deployment retains the zero-cost geometric lock.Source: §A.12 · pp. 28–29

A.13 DETECTOR-EQUIVALENCE AND CARRIER CONTROLS

• On the same carrier, topographic prominence exactly reproduces the H0 reference on all five tasks: 18/18 gates match with no spurious detections.Every valley and persistence value, including the deepest-valley convention, agrees to machine precision.Source: §A.13 · p. 29
• Replacing the carrier with raw observation coordinates matches only 11/18 gates and misses structurally important gates rather than merely displacing them.Missed task-4 gates drift 3.3–3.6 units, task-5 detections are 4.6–6.7 units away, and task 3 fragments three gates into seven detections.Source: §A.13 · pp. 29–30
• The raw carrier fails because prominence collapses, PCA directions capture gait rather than navigation, and Euclidean progress inflates within-shell dispersion by 2.6–4.6×.Raw-valley prominence is ≤0.657 (median 0.211), xy loadings are ≤0.052, and reference shell dispersion is 1.6–3.5 units.Source: §A.13 · p. 29
• Together, the controls show that detector replaceability is carrier-local, whereas transport-weighted carrier construction supplies the structure and guarantees that scalar profiles cannot express.The H1 route-forking analysis establishes that higher-homology structure lies outside any one-dimensional prominence profile.Source: §A.12 · p. 27, §A.13 · p. 30

A.14 ABLATIONS AND SENSITIVITY

• Parameter selection used pre-benchmark grid scans for kNN degree k, lookahead ℓ, and waypoint horizon, while monitoring constants were fixed mechanistically and validated by perturbation protocols.No parameter in either tier was tuned on benchmark outcomes.Source: §A.14 · p. 30, Table A.5 · p. 30

A.15 IMPLEMENTATION DETAILS

• Implementation handles rank-deficient shells in their effective 3–4-dimensional subspaces, reports median shell coordinates, and uses real support-flow trajectory points for control waypoints.Median reporting can leave L-shaped shells outside free space, but wall containment does not enter the control loop.Source: §A.15 · p. 30
• The offline AntMaze-giant graph with 25k nodes and k=12 builds in ≈2 s on one CPU core and is amortized across goals and episodes.The planar Delaunay dual avoids computing a 29-dimensional dual, using the effective projection for tractability.Source: §A.15 · p. 31

A.17 THEORETICAL CLOSURE AND THE FAILURE BOUNDARY

• On t5, extending the budget converts 19 of 39 near-miss failures (+5.2 overall), whereas t4 changes by −2.0 inside noise, identifying budget and executor failures respectively.Audited logs show no structure-layer failures; executor deadlocks occur in transition zones and goal corners, while gate chains carry no observed failure.Source: §A.17 · p. 31

A.16 THE δ-CLOSURE: THEORETICAL PREDICTION VERSUS MEASURED VALUE

• δpred = 0.0511 versus δemp = 0.0483 yields a 1.06× ratio, outperforming competing models that deviate by 3.62×, 1.37×, and 1.14×.The three-state Voronoi factorization is the only candidate within the 1.06× discrimination band, making the closure discriminative rather than curve-fitting.Source: §A.16 · p. 31, Figure A.12 · p. 32

A.17 THEORETICAL CLOSURE AND THE FAILURE BOUNDARY

• The final scope is limited to static, low-dimensional-state tasks and frozen-field enumerability, with unresolved T4 calibration and executor-side AntMaze-giant falls.Gates closer than filter resolution merge without being missed, and Kitchen dataset–environment mismatch caps arbitrary-progress recovery for all methods equally.Source: §A.17 · p. 32

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

• The protocol uses successful PointMaze trajectories, α-paired evaluation, and a single confirmatory t4 comparison; t4 versus direct is +36.0 with p=1.4 × 10−5.Other aggregate and per-task contrasts are descriptive, with paired differences and 95% confidence intervals; standard deviations use uniformly defined population conventions.Source: §A.18 · pp. 32–33, Table A.7 · p. 35
• 95.5 overall is achieved by the final FQL-xy executor with the support-flow interface, exceeding BFS-interface performance while removing deployment-side map access.The identical frozen gate set yields per-task scores 91.3/96.7/96.7/97.3; support-flow is the recursive hierarchy expressed at waypoint resolution.Source: §A.18 · pp. 33–34
• The frozen gates are executor-invariant, but scores progress from 0 to 24.5–30 to 95.5 across behavioral-cloning, GCIQL, and FQL-xy executors.This ablation localizes the dominant performance constraint to the low-level executor rather than the unchanged strategic gate set.Source: §A.18 · pp. 33–34
• 84.7 overall for Ant-discovered gates versus 89.3 for PointMaze-discovered gates reveals a 4.6 pp gap concentrated on t4, attributed to coarse MVEE resolution of sub-4-unit necks.The strategic chain remains actionable after embodiment change; the shortfall is attributed to estimator-side gait-density coarsening rather than mechanism dependence.Source: §A.18 · p. 34

B PROOFS

• Appendix B supplies full proofs of Theorems T1–T6, including graph consistency, cascading gates, coordinate stability, calibration, and δ-predictability results.The appendix maps each theorem to its proof section and records operating-point constants and numerical witnesses.Source: B PROOFS · p. 36

B.1 PRELIMINARIES

• The preliminaries define metrics, Lipschitz maps, intrinsic submanifold distances, and graph geodesic distance as the mathematical foundations for the construction.These definitions establish the distance notions used by the transport-weighted graph and subsequent consistency arguments.Source: §B.1 · p. 36

B.2 FORMAL SETUP

• The formal setup builds a transport-weighted kNN manifold graph, computes goal-distance radial progress, and trains a 16-dimensional rectification embedding using radial and isometry objectives.The graph removes low-density wall-crossing edges, while the η-weighted metric stretches high-η dimensions and compresses low-η dimensions.Source: §B.2 · p. 37
• The formal guarantees assume graph connectivity and sufficient successful-trajectory sampling, while Theorem T1 specifically does not rely on the sampling assumption.Sampling requires density bounded below and shrinking local kNN edge lengths; frozen dimensions are quotiented into an effective subspace.Source: §B.2 · p. 37

Theorem T1

• Under Definitions 8–9, dG is an extended metric, becomes genuine under Assumption 1, r is 1-Lipschitz, and shortest paths satisfy r(v) = r(u) + dG(v, u).The characteristic equality means r decreases at full rate along shortest-path descent.Source: §B.3 · p. 38

Corollaries

• The latent gradient, graph Lipschitz inequality, and geodesic equality are all theorems of the single object dG, converting Eikonal supervision from regression targets into an identity.The latent element satisfies ∥∇_z r∥ = 1 almost everywhere, while the state-space and characteristic elements follow from T1.Source: §B.3 · pp. 38–39
• The goal set is uniquely minimal: r(u) is nonnegative and equals zero exactly when u belongs to Vg.This follows from metric non-negativity and positive definiteness.Source: §B.3 · p. 39
• The radial labels and edge weights share the same generating object dG, eliminating objective-level conflict between the two supervision signals.Both loss terms vanish simultaneously under an isometric embedding with nodes on spheres matching their radial labels; exactness may fail, yielding an MMSE compromise.Source: §B.3 · p. 39

Three-state dynamics

• Anchor switching produces an exhaustive three-state process: plateaus with Δt = 0, forward moves with Δt < 0, and bounces with Δt > 0.The plateau atom arises from discrete anchoring, while forward and bounce states correspond to descending or backtracking relative to the shortest-path tree.Source: §B.3.1 · pp. 39–40
• The relapse tail decomposes into a rollout-dependent bounce event rate and a graph-only magnitude prior, whose conditional magnitude rate is 0.615 versus prior 0.662.Their approximately independent mechanisms separate whether anchor switching occurs from how far the adjacent-node switch moves in r.Source: §B.3.1 · pp. 39–40

Numerical validation

• 0.0511 predicted δ versus 0.0483 measured δ yields a 1.06 prediction-to-measurement ratio, but this is a same-sample consistency check rather than held-out validation.The event rate and empirical tail were measured on the same 21 successful rollouts; cross-task held-out validation remains future work.Source: §B.3.1 · p. 40

Mechanistic attribution

• All observed relapses are benign Voronoi-neighbor switches inside corridors, with zero wall-crossing failures and no pathological component.The conditional readings attribute 96.5% to benign corridor switching and report a zero count for the defined wall-crossing failure event.Source: §B.3.1 · p. 40

Model comparison

• Only the three-state Voronoi-prior decomposition achieves 1.0× agreement, whereas comparison models miss the plateau atom or misrepresent discrete bounces as continuous skewness.Across 21 trajectories, median δemp is 0.049 and the median prediction/measurement ratio is 1.09.Source: §B.3.1 · pp. 40–41

Theorem T2

• (1−ε)d_M(x,y) ≤ d_G_P(x,y) ≤ (1+ε)d_M(x,y) uniformly with probability tending to one under the stated sampling and neighborhood-scale conditions.The theorem assumes a compact smooth manifold, k = c_k log P, and a neighborhood scale inside the BdSL window.Source: §B.4 · p. 41

Corollary 4

• The resulting radial labels satisfy |r_P(u)−d_M(u,G)| ≤ εd_M(u,G)+o_P(1), making them consistent geodesic-distance estimates and unbiased Lyapunov readings.This contrasts with systematically biased time-to-go labels caused by policy wandering.Source: §B.4 · p. 41

Proof strategy

• The proof transfers BdSL convergence through metric equivalence, density-filter fidelity, wall-crossing soundness, and multi-source Dijkstra’s set-distance reduction.The filters retain intra-manifold neighbor edges and delete wall-crossing chord shortcuts, preserving the graph conditions needed for consistency.Source: §B.4 · pp. 41–42

Empirical audit

• On five maze tasks, filtered graphs remained connected and deleted zero or only a handful of edges, consistent with an asymptotically idle filtering regime.Finite-sample experiments use fixed neighborhoods: k = 12 for AntMaze/PointMaze and k = 16 for Kitchen, rather than asserting the asymptotic condition exactly.Source: §B.4 · pp. 41, 43

Remark 6: Robustness

• Kneedle separation is empirically robust but theoretically conservative: failures can raise the threshold and lose recall, not admit false positives, while larger persistence gaps improve localization.Across multiple 2D and 29D task suites, the elbow separated true gates from spurious valleys in every reported run.Source: §B.5 · p. 45

Theorem 4 (T3)

• Persistent-depth separation makes the elbow threshold recover the true-gate set with recall and precision under the theorem’s geometric and distributional conditions.True waists generate deep valleys, while shape-switch and noise valleys remain shallow; the kneedle threshold falls in the persistence gap, with elbow-tied marginal cases registered rather than counted.Source: §B.5 · pp. 44–46
• The complementary JS-divergence channel detects shape-switching events, and its union with measure-valley detection forms a complete topological event map.JS peaks precede measure valleys at corners, with spacing determined by corner geometry, because the channels respond to distributional abruptness versus cross-sectional narrowness.Source: §B.5 · pp. 44, 46

Lemma 8: Mechanism invariance

• Certified gates are necessary for every successful trajectory across execution mechanisms that share the same free space and provide sufficient successful-trajectory coverage.The gates are defined by geometry rather than by any particular executor; mechanism-dependent estimator factors may affect only marginal, elbow-tied survivors.Source: §B.5 · pp. 46–48

Lemma 9: Radial-coordinate stability

• An O(εT2) radial-coordinate perturbation changes persistence by at most 4τ and shifts a valley by at most 4τ/s, leaving bottleneck detection stable when the separation gap dominates.The result follows from sup-norm control of cross-sectional measures, non-expansive detrending, and bottleneck stability of the 1D filtration.Source: §B.5 · p. 48

Remark 9: Cascading gates

• Cascading gates remain detectable, but gates closer than roughly 2–3 filter bins can merge into one composite valley, preserving bottleneck recall while degrading localization resolution.When spacing exceeds the filter scale, valleys are detected separately; otherwise the composite valley localizes near its center and the separation condition becomes tighter.Source: §B.5 · pp. 48–49

Empirical anchors

• All four ground-truth gates were detected with persistence 0.69–1.35 above elbow τ = 0.056, while one borderline shallow valley survived at the elbow.The extra survivor reflects a marginal violation of the bimodal-separation heuristic and does not affect four-gate recall.Source: §B.5 · p. 49

Object-side audit

• 20 of 20 testable gates satisfy Lemma 4’s lower bound, while the maximum subthreshold depth 0.530 remains below Lemma 5’s 1.067 upper bound.The lower bound is informative for 2 of 20 gates; the remaining valid cases are sometimes vacuous because the right-hand side is negative.Source: §B.5 · p. 49
• The main-term constant is met by 3/20 gates under max-shoulder depths but by 0/20 under conservative min-flank depths, indicating distributional rather than universal separation.For multi-candidate tasks, the elbow lies at the upper edge between the largest subthreshold candidate and the smallest surviving valley.Source: §B.5 · p. 50

Marginal instances: adjudication

• Two elbow-tied survivors were adjudicated as marginal false positives because all object-side depth estimates were negative, with size-3 smoothing able to manufacture shallow valleys.The cases were t5b21, persistence 0.056 = τ, and t3b34, persistence 0.557 = τ.Source: §B.5 · p. 50

Field dependence (seed axis)

• Across two fresh retrains, all five task-2 gates re-localized within 0.17/0.27 median deviation and task-5 core gates re-localized at 6/8 and 8/8, despite elbow drift up to ≈13×.The theorem asserts correctness given the field, not invariance across fields; task-3/4 gate sets were not consumed downstream and were excluded from reproducibility comparisons.Source: §B.5 · p. 50

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

• With sufficient effective sample size, any threshold inside the spectral window recovers dψ with probability at least 1−η, and ˆmeff converges at O(Peff^-1/2).Dependence changes the convergence scale through covariance mass rather than invalidating the dimension-recovery identity.Source: §B.6 · pp. 51–52
• T4 equates the Jacobian spectral gap with the embedding’s effective dimension, with significant eigenvalues scaling as c^2σ_i^2 and the remaining m−dψ eigenvalues exactly zero.Under approximate isometry, dψ is the number of directions along which ψ varies over the data support.Source: §B.6 · pp. 51–52
• Empirically, centered Jacobian correlations have length d⋆≈1.65, while the spectral-gap position meff=3 is unchanged under independent cross-trajectory sampling.The measured effective sample size is Peff∼O(10^2–10^3), and sampling correlation changes Peff but not spectral shape or gap position.Source: §B.6 · p. 52, Figure B.1 · p. 53
• T4 closes structural dimension recovery but leaves exact noise-edge calibration under short-range mixing unresolved, so the remaining gap concerns λ+ rather than the recovery theorem.The MP calibration is empirically reproduced under the independent benchmark, while edge-rigidity transfer remains future work.Source: §B.6 · pp. 52–53

B.7 THEOREM T6: A DISTRIBUTION-MATCHING BOUND FOR THE SUBGOAL INTERFACE

• T6 bounds cascade execution error by training consistency plus Lδ times the Wasserstein distance between training and planner-induced subgoal distributions.The bound addresses whether the low-level policy receives subgoals from the distribution on which its conditional policy was trained.Source: §B.7 · p. 53

B.7.1 SETUP

• The setup defines µ from coherent future frames, µpath from path-aligned conditioning, and νr from each graph-based frame-selection rule and terminal rule.Training subgoals are reachable future states whose velocity and posture components follow the true dynamics.Source: §B.7.1 · p. 53
• The interface guarantee assumes per-slice µ-consistency, Lδ-Lipschitz execution in the subgoal, and non-degenerate path coverage, with candidate frame count ≥4 as the finite-sample criterion.Outside the training support, execution error can be arbitrarily large, making the transport term essential.Source: §B.7.1 · p. 54

B.7.2 THEOREM AND PROOF

• The interface-loss bound is ε_π + L_δ W_1(ν, µ_path), combining conditional path error with the frame-selection distribution’s Wasserstein mismatch.The proof uses Kantorovich–Rubinstein duality and an optimal coupling between ν_r(· | s) and µ_path(· | s).Source: §B.7.2 · p. 54
• The success-rate bound decreases monotonically with interface Wasserstein distance: an H-step rollout’s failure probability is at most H C_dyn(ε_π + L_δ W_1)/R.The result assumes execution error accumulates linearly through dynamics, with drift constant C_dyn and goal-ball radius R.Source: §B.7.2 · p. 54

B.7.3 TRANSPORT COSTS OF THE FIVE FRAME-SELECTION RULES

• Loose has W_1 ≤ r because direction-aligned real frames incur only positional smearing and remain on the path distribution’s support.This is the lowest-cost rule in the stated ordering.Source: §B.7.3 · p. 55
• Synth has W_1 ≥ Cξ because replacing positions with node coordinates moves joint frames off the dynamical coherence shell, making synth costlier than loose.The mismatch is a first-order position–velocity correlation term in corridor geometry.Source: §B.7.3 · p. 55
• Retarget incurs dynamic-freezing mismatch but improves near terminal goals as path displacement approaches zero, matching its measured terminal advantage.Retarget freezes the current posture instead of preserving the path’s posture in motion.Source: §B.7.3 · p. 55
• Off has the largest cost, W_1 ≈ C′v̄, because arbitrary frames randomize velocity and posture relative to the path direction.The cost corresponds to the expected chord length in a random direction.Source: §B.7.3 · p. 55
• Stand sets velocity components identically to zero, leaving the motion slice of µ and explaining terminal stalling when policies interpret the subgoal as already reached.This is a slice mismatch rather than ordinary positional displacement.Source: §B.7.3 · p. 55
• The transport ordering W_1(loose) < W_1(synth) ≲ W_1(retarget) < W_1(off) opposes success rates 90 > 77 ≳ 67 > 62, consistent with monotonicity.The constants depend on corridor geometry, including width and velocity scale, and are not made explicit.Source: §B.7.3 · p. 55

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

• The convergence proof treats anchoring failures as measurable probability panchor and controls path-level execution loss with Azuma–Hoeffding concentration under boundedness.It therefore does not exclude near-wall wall-crossing anchoring by assumption or require pointwise high-probability execution-loss tails.Source: §B.8 · pp. 55–56
• Theorem T5 guarantees terminal arrival with high probability when the path-averaged net progress π̄ is positive, separating planning, execution accumulation, and terminal-docking contributions.The result assumes the rollout remains within the λ-tubular neighborhood of data support and conditions on graph-construction failure probability εplan.Source: §B.8 · pp. 56–57

B.9 OPERATING POINT AND THE APPLICABILITY OF THE ASYMPTOTIC BOUNDS

• k = 12 makes the asymptotic filter bound vacuous: non-vacuity requires k ≥178, so εplan uses the measured zero-deletion audit and its confidence bound.The experimental graph had 6 × 10^4 nodes, 0/420292 deleted edges, and 100% connectivity; the per-edge Clopper–Pearson upper bound was 7.1 × 10^-6.Source: §B.8 · p. 57, §B.9 · p. 59

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

• 100 ± 0 PointMaze scores reflect near-zero execution noise and zero planning-layer failures, whereas AntMaze’s residual gap is attributed to execution accumulation.This benchmark attribution instantiates Theorem T5’s separation between planning and low-level execution effects.Source: §B.8 · p. 58

B.9 OPERATING POINT AND THE APPLICABILITY OF THE ASYMPTOTIC BOUNDS

• 0.103 median net progress per control step, with all 12 rollouts reaching the goal, verifies the positive operating-point margin required by Theorem T5.The measured r-traces use the same graph-distance units as the lookahead ℓ.Source: §B.8 · pp. 57–58, §B.9 · p. 59

B.10 NOTATION TABLE

• The notation table defines the 29-dimensional state space, transport-weighted metric dη, manifold graph G, radial progress coordinate r, subgoal distribution νr, execution loss δ(s,w), and W1.These quantities organize the graph, geometry, subgoal, and interface-distance terms used by the convergence analysis.Source: §B.10 · p. 60