Finite-Sample Metric Non-Collapse for Geometrically Supervised Latent World Models in Control

Motivation and problem

• Prediction loss and latent variance alone permit collapse or folding, so deterministic control requires pointwise metric non-collapse and uniformly faithful latent dynamics.A constant encoder can achieve zero prediction loss, while globally spread representations may still identify distinct states or lose local resolution.Source: §1 · pp. 1–2

Geometric supervision

• The encoder-only local–global metric hinge penalizes infinitesimal directional collapse and separated-state overlap until prescribed margins are reached.The one-sided hinge is decoder-free, inverse-model-free, and independent of reconstruction objectives, using observable-state metrics and tangent directions during training.Source: §1 · p. 2

Planning transfer

• The representation and semiconjugacy certificates transfer modularly to finite-horizon trajectory, cost, learned-cost-head, and approximate-planner guarantees.Learned latent cost heads contribute an additional 2(Tεℓ+ εg) term, with compatible physical and terminal costs handled separately.Source: §1 · pp. 2–3

Main theorem

• Under observable-geometry, coverage, approximation, regularity, and optimization hypotheses, every approximate empirical minimizer is pointwise co-Lipschitz and uniformly approximately semiconjugate with high probability.A computable one-sided regularization half-line yields the selection property, while approximation, statistical, and training-optimization errors remain explicit.Source: §1 · pp. 2–3

Approximation theory

• Norm-constrained tensor-product B-splines constructively satisfy the finite-capacity hypotheses, and the interpolation exponent converting mean residuals to uniform error is sharp.These results establish the approximation component needed by the finite-sample theorem rather than treating capacity assumptions abstractly.Source: §1 · p. 3

Empirical validation

• Controlled experiments isolate collapse and folding, quantify the analytic certificate’s uniformity reserve, and demonstrate improved control after metric resolution is restored.A finite-net proposition provides a sharper model-specific a posteriori certificate, distinct from the theorem’s continuum guarantee.Source: §1 · p. 3

Coercivity and stability gaps

• The local–global hinge closes pure prediction’s geometric stability gap, forcing pointwise non-collapse and uniform semiconjugacy for every approximate minimizer under explicit certification hypotheses.This supplies a conditional lower stability estimate from empirical objective value to latent geometry and controlled dynamics.Source: §2 · p. 4

Quantitative injectivity and metric regularization

• Local directional control resolves singular-value loss, while separated-pair margins eliminate distant folds and yield a continuum co-Lipschitz certificate under C1,1 regularity and coverage.The result quantifies injectivity for every approximate empirical minimizer rather than only for an ideal population solution.Source: §2 · p. 4

Observability, abstraction, bisimulation, and semiconjugacy

• The observable factor retains control-relevant distinctions while simultaneously receiving a lower metric certificate and a uniform deterministic controlled-semiconjugacy estimate.This links abstraction-level behavioral preservation to latent geometric faithfulness.Source: §2 · p. 4

Empirical-to-uniform estimates

• Capacity control, Lipschitz budgets, and lower-Ahlfors coverage convert empirical estimates into pointwise continuum bounds with an explicit, sharp dimension-dependent L2-to-L∞ rate.The sharp exponent identifies the regularity-limited conversion rate rather than merely asserting uniform convergence.Source: §2 · p. 4

Simulation lemmas and planning transfer

• The deterministic transfer argument propagates sup-norm latent dynamics error η and compatible-cost Lipschitz constant c∗ into finite-horizon optimization guarantees over action sequences.The representation theorem supplies exactly these two constants for planning transfer.Source: §2 · p. 4

JEPA and learned world models

• Relative to modern JEPA and world-model theory [8] [9], the contribution is an objective-level finite-sample implication from geometric supervision to certified non-collapse, semiconjugacy, and optimizer transfer.The theorem addresses the subsequent consequence of a fixed geometrically supervised objective, not merely identifiability or prediction generalization.Source: §2 · p. 4

System setup

• The model uses compact convex state and action domains, Lipschitz deterministic dynamics and observations, and the product state-action metric.The state-action dimension is d_m = d_s + d_a, and C^1,1 extensions are assumed wherever derivatives are used.Source: §3 · p. 5

Observable quotient

• When H is non-injective, observational equivalence identifies states with identical controlled observation histories, yielding the appropriate observable quotient.The equivalence relation is the deterministic controlled analogue of bisimulation and state abstraction.Source: §3 · p. 5
• The quotient uses a chain pseudometric with zero-cost jumps within equivalence classes, because a naive infimum metric may fail to define a metric.The dynamics and observation map descend to the quotient, followed by a further identification of zero-distance classes when necessary.Source: §3 · pp. 5–6

Regular observable factor

• Assumption A1 replaces the abstract quotient with a finite-dimensional compact convex Euclidean observable factor whose dynamics, observations, and costs factor through q.The induced observation map H_o is bi-Lipschitz onto its image and admits C^1,1 extensions; the deployed encoder uses current observations rather than histories.Source: §3 · p. 6

Metric convention

• The theory measures distances in the Euclidean observable-factor metric, which need not equal the quotient chain metric without an additional bi-Lipschitz comparison.Assumption A1 supplies the Euclidean coordinate, but translating guarantees back to d_obs requires explicit metric comparability.Source: §3 · pp. 6–7

Coverage

• Assumption A2 requires the training measure on the observable state-action space to be lower Ahlfors regular, ensuring local coverage at every sufficiently small radius.A lower density bound plus uniform interior volume provides this condition in common absolutely continuous settings.Source: §3 · p. 7

Model and regularity

• A positive local directional margin requires an injective encoder derivative, hence latent dimension dz ≥ ds; a smooth bi-Lipschitz coordinate embedding into the latent interior is available.The dimension condition is assumed throughout the metric results.Source: §4 · p. 7
• The model uses an encoder and action-conditioned latent transition map, with only the encoder image of observable states and the transition restriction on that image affecting the analysis.The latent space is constrained to a compact convex set with nonempty interior, and the metric results require an ambient C1,1 encoder representative.Source: §4 · pp. 7–8

Compatible latent costs

• Once ψ = Φ◦H is lower co-Lipschitz, Lipschitz stage and terminal costs admit latent extensions that agree on encoded states and satisfy exact compatibility with separate Lipschitz bounds.The construction uses McShane’s extension theorem [36], including metric-quotient handling when a weight vanishes.Source: §4 · pp. 8–9

Prediction objective

• The JEPA residual is R[Φ,F](s,a) = ψ(G(s,a)) − F(ψ(s),a), and uniform Lipschitz assumptions make it uniformly Lipschitz while defining the population prediction loss.The residual formulation identifies prediction error directly with the mismatch between encoded true successors and latent transitions.Source: §4 · p. 9

Empirical optimization

• For every λ > 0, the regularized empirical objective attains its infimum when the admissible class is compact and its prediction loss and encoder regularizer are lower semicontinuous.Uniform convergence topology applies in the Lipschitz-only setting, whereas the derivative-dependent metric regularizer uses C1 compactness at fixed capacity.Source: §4 · p. 10

• Pure forward JEPA admits collapsed zero-loss minimizers, so forward prediction alone cannot imply non-collapse or useful controlled semiconjugacy.The constant latent map has identically zero residual but no positive lower co-Lipschitz constant unless the state space is a singleton.Source: §5 · p. 10
• Covariance and spectral-spread penalties prevent only global second-moment collapse, not injectivity or a pointwise lower co-Lipschitz bound.These regularizers constrain coordinate variances or eigenvalues, but do not enforce pointwise geometric separation.Source: §5 · p. 11
• A non-injective latent map can have zero prediction residual and positive variance, disproving covariance or spectral spread as sufficient geometric guarantees.For the example ψ(s)=Cs^2, the residual is zero and variance is positive, yet ψ(s)=ψ(−s), so no positive lower co-Lipschitz constant exists.Source: §5 · pp. 11–12
• The local–global objective supplies directional and separated-pair information needed to establish a uniform co-Lipschitz certificate.It targets the precise pointwise lower geometry left unconstrained by forward prediction and second-moment spread.Source: §5 · p. 12

Metric hinge design

• The encoder-only regularizer promotes both local directional resolution and global separation without requiring a decoder, inverse model, or hard architectural constraint.Its two complementary terms target infinitesimal and well-separated-state geometry.Source: §6 · p. 12

Training-information convention

• Geometric supervision uses observable-state distances and tangent directions during training, while deployment retains the observation–action encoder–transition interface.Certified state-estimation or pixel-based geometry is permitted when independently bounded errors are absorbed into effective margins.Source: §6 · p. 12

Metric penalties

• The local and global penalties are averaged hinge relaxations of infinitesimal metric and separated-state conditions, respectively.The global term samples state pairs conditioned on separation at least ρ, while the local term samples state–direction pairs.Source: §6 · pp. 12–13

Empirical objective

• The empirical objective combines prediction loss with λ times the metric regularizer, using mutually independent transition, unit-direction, and separated-pair sample families.The regularizer is formed from empirical local and separated-pair penalties with distinct sample sizes when needed.Source: §6 · pp. 13–14

Sampling geometry and interpolation

• Lower Ahlfors regularity of the direction and separated-pair sampling spaces enables Lipschitz–Ahlfors L2-to-L∞ interpolation.This interpolation converts average residual control into uniform control under the stated compactness, measure, and Lipschitz conditions.Source: §6 · p. 14

7.1. A smooth non-collapsed comparator and a concrete finite-capacity class.

• The comparator realizes exact latent observations and controlled dynamics through a smooth bi-Lipschitz embedding, range-enforcing map, and Lipschitz latent dynamics.The construction preserves Φ0 ◦ H = ψ0 and F0(ψ0(s), a) = ψ0(G(s, a)).Source: §7.1 · p. 16
• Norm-constrained tensor-product cubic B-spline classes satisfy finite-capacity C1,1 approximation with capacity-independent regularity budgets and compactness in C1.For sufficiently large W, approximants clear both geometric margins and yield zero population and empirical metric penalties; the admissible margin regime is nonempty.Source: §7.1 · p. 16

7.2. Uniform statistical control.

• Uniform C1,1 budgets make prediction and metric-loss integrand classes bounded and equi-Lipschitz, yielding finite covering numbers and simultaneous high-probability deviation control.For fixed capacity, the statistical error tends to zero with n; finite-parametric estimates also apply under polynomial capacity schedules.Source: §7.2 · pp. 17, 19, 20

7.3. Principal finite-sample representation theorem.

• The principal representation theorem fixes three interpolation moduli for estimates based on the classes GM(W), GU(W), and GP(W).This subsection begins the theorem’s interpolation-based argument but does not provide its later conclusions in the supplied passage.Source: §7.3 · p. 21
• With probability at least 1 −δ, every εtrain-approximate empirical minimizer satisfies metric non-collapse and controlled semiconjugacy with error ηW,n,δ(εtrain) = ΘM(β(W) + 2εstat,W(n,δ) + εtrain).The guarantee applies under the stated structural, approximation, and statistical hypotheses; when approximation, statistical, and training errors vanish, the error modulus tends to zero.Source: §7.3 · pp. 21–22
• The admissible regularization strengths form a nonempty one-sided half-line determined by β(W), statistical error, and training error, with λmin → 0 when all three vanish.Thus, every fixed λ > 0 is eventually admissible along a capacity sequence with vanishing errors.Source: §7.3 · p. 23

7.4. Sharpness of the interpolation exponent.

• No uniform L2-to-L∞ estimate on the stated Lipschitz class can improve the interpolation exponent beyond 1/(q + 2).The sharpness result is established for Y = [0, 1]^q with fixed L using a truncated-cone construction.Source: §7.4 · p. 23

7.5. A finite-net a posteriori certificate and use of the threshold.

• A validated finite-net certificate converts model-specific net maxima, Lipschitz bounds, and fill-distance remainders into uniform margins and semiconjugacy constants usable by the deterministic planning corollary.This route is typically sharper than the class-uniform a priori threshold, but sweeping λ and checking held-out stability becomes rigorous only after validating the Lipschitz and fill-distance remainders; it is not a monotonicity claim for nonconvex optimization.Source: §7.5 · pp. 23–24

Modular planning transfer

• Theorem 7.11 certificates yield deterministic trajectory, cost, and optimizer-transfer guarantees when combined with Lipschitz costs, a finite horizon, and a planner.The modular formulation makes the roles of the learned geometric and semiconjugacy constants explicit.Source: §8 · p. 24

Uniform planning transfer

• Under co-Lipschitz representation and compatible latent cost extensions, uniform finite-horizon planning transfer bounds cover true-versus-latent objectives and approximate latent optimization.Learned cost-head compatibility adds explicit stage and terminal discrepancies to the optimizer comparison.Source: §8 · pp. 24–25

Finite-sample consequence

• Under Theorem 7.11 and Lipschitz physical costs, every fixed-horizon approximately optimal latent action sequence inherits a certified finite-sample planning consequence.The result also provides a corresponding bound for compatible learned cost heads on the relevant compact latent domain.Source: §8 · p. 25

Horizon dependence

• Certified error envelopes depend on horizon: they grow linearly when latent dynamics are nonexpansive and geometrically in the worst case when the latent Lipschitz factor exceeds one.Trajectory-local or contractive structure can yield tighter realized errors than the worst-case envelope.Source: §8 · p. 25

Mean-to-uniform residual control

• Mean residual control alone cannot guarantee planning reliability, because a localized Lipschitz bump may have vanishing L2 norm while a deterministic optimizer visits its high-error region.Theorem 7.11 closes this mean-to-uniform gap through regularity and coverage; supplementary results extend the analysis to adversarial planning, Lyapunov, and reachability consequences.Source: §8 · p. 25

9. Numerical evidence

• The experiments use matched architectures, bounded latent ranges, fixed data and optimization budgets, adversarially refined finite sets, and repeated restarts to align diagnostics with certification.The nonlinear benchmark compares latent MPC with an exact-model reference, whose best of 6 restarts receives more total optimization effort.Source: §9 · p. 26, §9.3 · p. 27

9.1. Exact and trained folding obstructions.

• Exact folding examples achieve zero JEPA residual and full-rank covariance while remaining non-injective, motivating the local–global geometric penalty.The construction satisfies ψC(s) = ψC(−s), with latent variance 4C2/45 and covariance diag(4C2/45, D2/3).Source: §9.1 · p. 26
• In the trained folding stress test, the combined hinge is injective in all 12 plain-initialization runs, while upper control reduces non-injective outcomes from 9 to 3 under symmetry-biased pretraining.The study compares prediction, covariance, local-only, global-only, local–global, upper-penalized, and two-sided objectives using lower and upper geometric diagnostics.Source: §9.1 · p. 26, §9 · p. 26

9.2. Finite-capacity threshold diagnostic.

• The finite-capacity spline diagnostic identifies an intermediate regularization regime where local and global margins strengthen while uniform residuals remain controlled.The interpolation threshold is P ∗= 4.81×10−6, class-uniform statistical deviations range from 5.19 to 18.46, and the analytic bound retains a substantial uniformity reserve.Source: §9.2 · pp. 26–27, Figure 2 · p. 27

9.3. Nonlinear latent-MPC benchmark.

• Geometry-aware training substantially reduces paired control-cost differences relative to pure prediction, while regularized objectives remain competitively effective in the nonlinear latent-MPC benchmark.Pure prediction has bηeval = 0.01, but geometry-aware objectives raise the median chord constant from bcmin = 1.75×10−4; its paired cost difference is 1.41 at T = 5 and 1.55 at T = 20, versus 4.03–25.79 times the regularized-arm medians.Source: §9.3 · pp. 27–28, Table 3 · p. 28
• The ratio bηeval/bcmin correlates with worst-case paired cost difference more strongly than bηeval alone, providing a diagnostic gate between collapsed and non-collapsed models.The pooled Spearman correlation is 0.62 with interval [0.30, 0.80], versus −0.34 for bηeval alone; within geometry-aware models it is 0.42 with interval [0.04, 0.69].Source: §9.3 · p. 28

Scope

• Training requires observable-state distances and tangent directions with positive effective margins, whereas deployment remains observation- and action-conditioned.Simulator state, proprioception, certified state estimators, and error-controlled pixel geometry estimators can supply the required supervision.Source: §10 · p. 29

Certification assumptions

• The spline construction makes the regularity and approximation assumptions non-vacuous, while coverage and optimization assumptions become actionable sampling and margin-monitoring requirements.Source: §10 · p. 29

Certification limits

• The analytic threshold is a class-uniform sufficient condition, not the smallest empirically useful regularization weight, and its reserve motivates class-specific or finite-net certification.Experiment B reveals reserve in the sufficient threshold.Source: §10 · p. 29

Empirical outlook

• Several regularized objectives restore non-collapse and achieve competitive control, while the local–global hinge uniquely aligns with the pointwise theorem.Experiment C supports the broader geometric mechanism and distinguishes the proposed objective by theorem alignment.Source: §10 · p. 29

Outlook

• A natural next direction is extending the objective-to-certificate principle to partially observed stochastic systems and independently certified visual metrics.Source: §10 · p. 29

11. Conclusion

• The local–global hinge supplies pointwise stability, while every approximate empirical minimizer in the certified regime is co-Lipschitz and uniformly approximately semiconjugate to controlled dynamics.These certificates connect empirical prediction with deterministic planning.Source: §11 · pp. 29–30
• The modular planning result transfers the learned certificates to trajectory, finite-horizon cost, learned-cost-head, and optimizer bounds.Source: §11 · p. 30
• Constructive approximation theory, sharp interpolation, finite-net certification, and controlled experiments complete the learning-to-control chain.Together, they show how empirical objectives select metrically faithful dynamics models and support reliable finite-horizon planning without conflating analytic certification with finite-model evidence.Source: §11 · p. 30

S1. Exact realizers and detailed finite-capacity spline construction

• Norm-constrained tensor-product splines satisfy the finite-capacity requirements through range enforcement, compactness, coefficient counts, margin clearing, and a nonempty geometric parameter regime.These checks provide the explicit approximation framework used by the main theorem.Source: S1. Exact realizers and detailed… · p. 34

S1.1. Exact and smooth non-collapsed realizers.

• The construction satisfies Φ0(H(s)) = ψ0(s), F0(ψ0(s), a) = ψ0(G(s, a)), and R[Φ0, F0] ≡ 0, yielding an exact non-collapsed realization in admissible hypothesis classes containing these maps.Source: S1.1. Exact and smooth non-collapsed… · p. 34
• Quantitative co-Lipschitz observable embeddings admit uniformly sized C1,1 local inverse charts with bi-Lipschitz constants 1/2 and 3/2.The chart constants depend only on the co-Lipschitz bound, the fixed-neighborhood C1,1 budget, and a finite covering.Source: S1.1. Exact and smooth non-collapsed… · pp. 35–36

0 A0 is invertible and Ps0 is the Moore–Penrose

• F_s0 is a global C1 diffeomorphism onto the open set O_s0, satisfying 1/2|s−s′| ≤ |F_s0(s)−F_s0(s′)| ≤ 3/2|s−s′|.The inverse representative Γ_s0 has the claimed quantitative C1,1 bounds and recovers (f|Ω)^−1 on the relevant observations.Source: §0 · pp. 36–37
• The construction provides C1,1 exact representatives Φ0 and F0 with Φ0(H(s)) = ψ0(s), F0(ψ0(s),a) = ψ0(G(s,a)), and zero residual.The encoder maps into KZ and satisfies Φ0 ∈ C1,1(X; KZ).Source: §0 · pp. 37–39
• Assumption A4 requires finite-capacity classes with W-independent Lipschitz and C1,1 budgets and approximation error δ(W) → 0 in C1.For sufficiently large W, representatives (ΦW, FW) satisfy the stated combined C1 approximation bound δ(W).Source: §0 · p. 39

S1.2. Finite-capacity spline approximation.

• Range-constrained tensor-product cubic B-splines achieve C1 error O(h) while retaining uniform C1,1 bounds, with coefficient capacity W_h ≍ h^−d.The construction uses affine reproduction, local stability, and a range-enforcing map that is the identity near the target image.Source: S1.2. Finite-capacity spline… · pp. 39–41
• Localized range-constrained spline approximation preserves a uniform C1,1 budget when the target image lies in a neighborhood where the range map is the identity.The resulting bound is uniform over sufficiently small mesh widths h.Source: S1.2. Finite-capacity spline… · p. 41
• Norm-constrained tensor-product spline classes satisfy Assumption A4 with uniform C1,1 budgets, range constraints into KZ, C1 approximation, compactness, and O(dzW) scalar coefficients.The construction uses localized spline approximation and fixed range-enforcing maps; all requirements of Assumption A4 are thereby met.Source: S1.2. Finite-capacity spline… · pp. 43, 45, 46

S1.3. Margin-clearing competitors and a nonempty parameter regime.

• For sufficiently large capacity W, the approximating pair is a margin-clearing competitor with directional margin κ and separated-state margin α.Specifically, the encoded map satisfies the directional lower bound on U and the separated-point lower bound on Pρ.Source: S1.3. Margin-clearing competitors and… · p. 46
• These competitors make the empirical local, global, and metric penalties exactly zero for every sample.The result follows after choosing W sufficiently large so approximation errors fall below the prescribed margins.Source: S1.3. Margin-clearing competitors and… · pp. 46–47
• A nonempty parameter regime exists for compact nontrivial S: any κ>0, ρ<diam(S), and α∈(0,α0(ρ)) satisfying the stated bounds is valid.The regime is supported by the bi-Lipschitz embedding, positive co-Lipschitz constant, and nonempty off-diagonal pair set Pρ.Source: S1.3. Margin-clearing competitors and… · pp. 48–49
• The proof establishes a nonempty admissible parameter regime with 0 < κ < κ0 and 0 < α < α0(ρ), alongside a ρ satisfying B_loc,ρ ≤ κ/4.These choices are made after showing α0(ρ) is positive.Source: S1.3. Margin-clearing competitors and… · p. 49

Coefficient stability

• Spline dictionaries have coefficient stability constants depending only on degree, dimension, and boundary patterns, not mesh width, with componentwise extension to vector-valued splines.The proof uses bounded local overlap, finitely many reference-cell patterns, and local linear independence of active basis functions.Source: S2. Architecture-specific covering… · pp. 50–51

Parameter-Lipschitz bound

• The norm-constrained cubic B-spline architecture satisfies the finite-parametric deviation hypothesis for all three integrand classes G_M(W), G_U(W), and G_P(W).The resulting finite-parametric estimate therefore applies to the spline architecture of Proposition S1.6.Source: S2. Architecture-specific covering… · p. 51
• Range-enforcing composition with fixed C^{1,1} maps preserves uniform value and derivative parameter bounds, while identity near H(S) retains the encoder scaling in the integrand estimates.The relevant derivative norms and derivative Lipschitz constants are bounded independently of mesh width and capacity level.Source: S2. Architecture-specific covering… · pp. 52–53
• The combined spline coefficient set lies in a cube with radius R independent of W, so the deviation corollary may take R_W = R.This uniform radius follows from coefficient stability and the raw norm constraints.Source: S2. Architecture-specific covering… · p. 53
• The three integrand classes are uniformly Lipschitz in the combined encoder-transition parameter vector, and their covering numbers grow at most polynomially in W.The bounds combine residual boundedness, the transition Lipschitz budget, spline parameter-value estimates, and the capacity-dependent parameter dimension.Source: S2. Architecture-specific covering… · pp. 53–54

S3. Secondary deterministic-control diagnostics

• Uniform residuals amplify through rollouts and running costs: C(T, LF) is bounded, linear, or geometric as LF is below, equal to, or above 1, while neutral-case costs scale as O(ηT^2).Table S1 reports the cumulative rollout factor ηC(T, LF) for η = 10^-2; at LF = 1, terminal-state error is O(ηT) but undiscounted running-cost error is generally O(ηT^2).Source: Table S1 · p. 54, S3. Secondary deterministic-control… · p. 54
• The exponent 1/(q + 2) in Lipschitz–Ahlfors L2-to-L∞ interpolation is optimal: truncated-cone residuals can have L2 error of order h^(q+2) while pointwise error remains order h.No uniform estimate with exponent α > 1/(q + 2) can hold under Lipschitz-only regularity.Source: S3. Secondary deterministic-control… · pp. 54–56
• Nearly equal L2 one-step errors can yield different planning outcomes: concentrated Model A error near the decision boundary produces approximately 60% more mean excess closed-loop cost than Model B.The adversarial-planner diagnostic shows that an optimizer can select trajectories through small-measure regions where the surrogate is optimistic, whereas uniform-error control provides the required worst-case guarantee.Source: S3. Secondary deterministic-control… · pp. 57–58, Table S3 · p. 58

S3.3. Why metric non-collapse is the additional ingredient.

• Metric non-collapse and dynamical accuracy play complementary roles: c∗ controls cost regularity and state recovery from latent codes, while η controls horizon-accumulated dynamics error and supports Lyapunov and reachability transfer.The resulting certificates include practical Lyapunov estimates and latent-space Hausdorff reachable-set bounds; state-space reachable-set transfer additionally requires a decoder or quantified projection.Source: S3.3. Why metric non-collapse is the… · pp. 58–59

S3.5. Summary.

• The summary identifies explicit c∗ and ηW,n,δ(εtrain) as the quantities enabling uniform cost-transfer, latent-optimizer, state-recovery, Lyapunov, and reachable-set guarantees while separating εtrain from ξplan.Learned-cost errors are further quantified through εℓ and εg, and the planning guarantee assumes the regularization strength lies in the principal theorem’s admissible range.Source: S3.5. Summary · p. 59

S4. Numerical protocols, summarized records, and supporting-material availability.

• The experiments use a common compact latent box, fixed DM, DU, DP arrays, matched update counts, upper diagnostics, and explicit restart accounting to align numerical evidence with theorem assumptions.The protocol reports objective components, gradient norms, sampled curves, and terminal optimization quality; complete raw records and verification documents are retained for provision upon request.Source: S4. Numerical protocols, summarized… · pp. 60–61, Table S4 · p. 61

S4.1. Experiment A: folding, full-rank covariance, and the local/global ablation.

• Experiment A analytically showed that zero prediction residual and positive full-rank latent covariance can coexist with non-injective encoders, while trained ablations separated local and global geometric roles.The scalar and two-dimensional constructions use fixed separated-pair arrays and exactly verify folding despite cleared variance or minimum-eigenvalue penalties.Source: S4. Numerical protocols, summarized… · p. 61, S4.1. Experiment A: folding, full-rank… · p. 62
• Under ordinary initialization, the local–global hinge produced 0 non-injective outcomes among 12 seeds, with median bcmin = 0.90, bcmax = 1.31, and bκgeom = 1.48.The upper directional term reduced expansive outcomes from 9 to 3 and reduced bcmax from 20.78 to 2.55 while preserving favorable conditioning.Source: S4.1. Experiment A: folding, full-rank… · p. 66

S4.2. Experiment B: a theorem-motivated finite-capacity proxy.

• Experiment B calibrates the principal representation theorem using coefficient-box cubic B-spline and bounded MLP classes, with restarts measuring optimization sensitivity under fixed capacity and sampling.The spline class is a finite-capacity computational proxy, while Proposition S1.6 supplies the separate mesh-uniform C1,1 construction required by the theorem.Source: S4.2. Experiment B: a… · p. 66
• The experiments freeze nested sample arrays and matched planning budgets, report dense-grid and paired-cost diagnostics, and retain an exact-rational certificate with λcert = 53/500 for provision upon request.Experiment C uses a nonlinear embedding with singular values 0.20 and 1.18, avoiding raw access to θ and ω.Source: S4.2. Experiment B: a… · pp. 67–68, S4.3. Experiment C: nonlinear planning… · pp. 69–70
• Experiment B exhibits three stable regimes: λ = 0 gives residual 10^-8 but local defect κ2 and lower chord ratio 0, while positive λ forms geometric margins.At n = 2048 and λ = 0.03, dense-grid local defect is 0, lower chord ratio is 0.68, and evaluation residual is 0.02.Source: S4.2. Experiment B: a… · p. 67
• At λ = 0.3, 1 of 2 seeds clears both sampled margins while retaining bcmin = 0.88 and bηeval = 0.03, with sampled clearance appearing at n = 2048.Large λ becomes optimization-dominated, with restart choice affecting lower chord ratio and residual.Source: S4.2. Experiment B: a… · pp. 67–68
• The plug-in threshold diagnostic finds the sufficient condition far from met, with εstat/P∗ exceeding 10^5 throughout and uniform covering and Lipschitz–Ahlfors terms dominating improvement opportunities.It is a calibration diagnostic rather than a certified evaluation of Proposition S1.6.Source: S4.2. Experiment B: a… · p. 68, Table S10 · p. 69

S4.3. Experiment C: nonlinear planning benchmark.

• Pure prediction achieves the smallest uniform residual, 0.01, yet collapses metrically and incurs median paired cost differences of 1.41 at T = 5 and 1.55 at T = 20.Its median bcmin is 1.75×10^-4, while regularized arms perform 4.03–25.79 times better on corresponding medians.Source: S4.3. Experiment C: nonlinear planning… · p. 71
• The geometry-normalized ratio predicts control outcomes better than residual alone: worst-case paired-cost correlation is 0.62 versus −0.34 for bηeval alone.The bootstrap interval for the ratio correlation is [0.30, 0.80], while removing pure prediction yields correlation 0.42 with interval [0.04, 0.69].Source: S4.3. Experiment C: nonlinear planning… · p. 71
• Covariance spread, the local–global hinge, and the two-sided baseline achieve similarly favorable planning performance, whereas global-only regularization is weaker at short horizons.Each regularized mechanism beats the demanding exact-model reference on a non-negligible fraction of initial states.Source: S4.3. Experiment C: nonlinear planning… · p. 71
• Experiment C separates geometry, residual, cost-head, optimizer, and horizon effects rather than treating the planning proxy as a certified suboptimality bound.The exact-model reference uses six times the per-restart optimization effort, and Beval excludes planner-tolerance terms and is not a certified bound.Source: Table S12 · p. 73, S4.3. Experiment C: nonlinear planning… · pp. 73–74, Table S13 · p. 74

S4.4. Discussion.

• Experiment B identifies the uniform covering estimate as the dominant plug-in term at the studied sample sizes.The study uses fixed samples, a norm-constrained finite-capacity class, near-global restart diagnostics, and explicit evaluation of selected threshold ingredients; sharper class-specific complexity bounds and posterior finite-net validation are proposed as routes to a larger certified regime.Source: S4.4. Discussion · p. 75

S4.3. Experiment C: nonlinear planning benchmark.

• Geometry-aware objectives restore latent resolution and strongly improve planning over the residual-only reference on the true pendulum.Comparable performance across several regularizers supports the robustness of the geometric principle, while the local–global hinge additionally supplies a direct deterministic certificate and folding diagnostic.Source: S4.4. Discussion · p. 75

S4.4. Discussion.

• The numerical campaign retains optimization and uniformity quantities separately, but a continuum certificate still requires the validated remainders from the main article’s finite-net proposition.Retained records include objective curves, terminal gradients, restart spreads, finite-set suprema, adversarial refinements, and measured Lipschitz budgets.Source: S4.4. Discussion · p. 75