Prediction Limits and Koopman Closure of Geometry-Induced Soft State Abstractions

1 Introduction

• Independent-data prediction-error certificates can rule out a target RMSE for every admissible predictor when the lower bound exceeds that target, without fitting a prediction matrix on evaluation data.The conclusion applies only to the fixed representation, evaluation distribution, and predictor class; a zero certificate does not establish that an accurate predictor exists.Source: §1 · p. 2, §1.1 · p. 3

1.4 Contributions and scope

• Under exact deterministic closure, the coordinate-function span is Koopman-invariant, with a faithful finite-dimensional operator requiring redundant coordinate directions to be removed.Small prediction error and forecasting experiments do not establish exact closure.Source: §1 · p. 2, §1.2 · p. 3

1.2 Representation and prediction task

• The prediction target is the next soft-coordinate vector under a fixed joint evaluation law, and the representation and labels must be fixed independently of evaluation outcomes.Transitions may be deterministic or stochastic, and current and successor marginal distributions need not coincide.Source: §1.2 · p. 3, §2.1 · p. 6

1.3 KAHM-based coordinates in brief

• KAHM coordinates normalize positive affinities derived from class-specific reconstruction scores; a reconstruction-score advantage for the reference class controls deviation from its label.The general risk certificates require only a fixed measurable simplex-valued representation, not this particular coordinate construction.Source: §1 · p. 2, §1.3 · p. 3, §1.4 · p. 4

1.4 Contributions and scope

• In a four-state construction, the class-centered population bound equals the exact optimum, showing that close label agreement need not imply accurate forecasting.The paper also studies matrix-fitting update error separately from the prediction-risk certificates.Source: §1.1 · p. 2, §1.4 · p. 4
• The studies use 117,000 archived datasets from fixed four-state designs to compare certificates with ERM, and 64 independently sampled datasets to test a label-based certificate for one preselected representation.The repeated four-state datasets are not independently learned abstractions.Source: §1 · p. 2, §1.4 · p. 4

2.1 Evaluation law and a population lower bound

• The class-centered population bound subtracts the norm-limited benefit of within-class input variation from successor variation, capturing how much states with the same label evolve differently.It is at least as strong as the label-deviation relaxation, though neither lower bound is generally exact.Source: §1.1 · p. 3, §2.1 · pp. 7–8

2.2 A computable lower confidence bound

• The computable certificates adjust within-class successor dispersion and input-to-label disagreement for sampling uncertainty using independent evaluation pairs, rather than a fitted predictor’s training loss.The guarantee is conditional on representation construction and selection data; a large successor spread can still yield an inconclusive certificate when input-to-label discrepancy is large.Source: §2.2 · pp. 8–9
• At confidence 1−δ, the label-based bound covers every spectral-norm limit κ simultaneously and excludes tolerance ε for all admissible matrices only when Lκ,δ > ε.The bound concerns the population optimum, not the error of a fitted matrix; a zero bound does not establish that an accurate predictor exists.Source: §2.2 · p. 9

2.3 A class-centered certificate on the same evaluation pairs

• The class-centered certificate measures residual within-class input dispersion on the same independent pairs, but finite-sample occupancy corrections can make it weaker than the label-based bound.Its population inequality is stronger, and estimating class means requires no separate evaluation sample.Source: §2.3 · p. 10

2.4 When additional evaluation data make the certificate informative

• With a fixed representation, finite class count, and i.i.d. pairs, either certificate eventually exceeds a tolerance almost surely when its population limit is strictly above it.Singleton classes contribute zero empirical within-class variance, and more data cannot eliminate a gap between the structural bound and the actual optimum.Source: §2.4 · pp. 11–12

2.5 Comparison with certificates based on empirical risk minimization

• Structural certificates avoid matrix optimization, cost O(MC), and cover all κ simultaneously, whereas the constrained-ERM comparator requires optimization and lacks that simultaneous guarantee.Structural bounds can be conservative, and neither structural certificate nor the comparator can certify positive risk when a feasible matrix interpolates the evaluation sample.Source: §2.5 · pp. 12–13

2.6 Interpreting and using the certificate

• For a prespecified representation, evaluation law, norm limit, and RMSE tolerance, compare the corrected lower bound with ε: exceedance excludes accuracy, while zero is inconclusive.Under stochastic dynamics, realized-successor variation can contribute to prediction error even when the conditional mean evolves linearly.Source: §2.6 · p. 13

3 KAHM Reconstruction Scores and Koopman Operator Preliminaries

• The section sets up KAHM reconstruction scores and operator-theoretic conventions for interpreting exact coordinate evolution; neither preliminary is required to prove the general certificate.Source: §3 · pp. 13–14

3.1 Projection and function-space conventions

• An RKHS is a Hilbert space of functions where inner products with kernel sections reproduce function evaluation.The section also specifies the inner-product convention used for complex-valued RKHSs.Source: §3.1 · p. 14

3.2 Kernel Affine Hull Machine (KAHM)

• A KAHM represents a query as an affine combination of reference samples, using normalized reconstruction weights that may be signed.The final soft class weights are nonnegative, unlike these reconstruction coefficients.Source: §3.2 · p. 14

3.3 KAHM reconstruction score

• The reconstruction score combines Euclidean and angular discrepancies; a lower score indicates a better match under that combination.The two components can trade off, and the score is not generally invariant under state-coordinate transformations.Source: §3.3 · p. 15

3.4 Koopman Operators and Reproducing-Kernel Hilbert Spaces

• When composition with the state map preserves the RKHS, the Koopman operator is invariant on that space; the paper’s adjoint statements assume it is defined and bounded there.The finite-dimensional analysis requires removing redundant coefficient directions for a faithful operator representation.Source: §3.4 · pp. 15–16

4.1 Data, reference classes, and transition samples

• The analysis fixes reference classes and datasets; deterministic operator results assume a measurable one-step map that stays in the state space, while stochastic transitions yield a conditional-mean equation.Evaluation transitions are independent of representation construction, selection, and model fitting; the separate spectral dataset's independence alone does not guarantee a statistical result for its diagnostics.Source: §4.1 · p. 17

4.2 Construction of the soft abstraction

• KAHM soft coordinates normalize positive, exponentiated affinities so better relative reconstruction by a class yields a larger coordinate; the offset and exponent tune positivity and contrast.The coordinates express relative reconstruction quality, not posterior class probabilities; implemented score extensions and normalization thresholds must be fixed as part of the representation.Source: §4.2 · pp. 18–19

4.3 Finite-rank kernel and spectral correspondence

• Under exact closure, removing redundant coefficient directions yields a reduced matrix representing Koopman action on the soft-coordinate function space, with corresponding eigenfunction and adjoint-eigenfunction conditions.Different coefficient vectors can represent the same function, so the RKHS adjoint and spectral representation use the feature-vector span.Source: §4.3 · pp. 19–21
• Finite-sample adjoint-residual calculations represent the stated RKHS residual under exact closure, but can depend on the chosen weights and sample when that implication fails.A zero relative residual establishes the adjoint eigenrelation only for a nonzero represented function with positive denominator; it is undefined when the weighted successor vector is zero.Source: §4.3 · pp. 21–22

4.4 Exact Linear Closure and Koopman-Invariant Coordinate Functions

• Exact deterministic closure requires one matrix to propagate coordinates at every state in the stated domain, whereas zero population prediction risk establishes equality only almost surely under the evaluation distribution.The empirical adjoint-residual diagnostic has its RKHS interpretation only under exact closure; otherwise it measures agreement between sampled weighted successor and current sums.Source: §4.4 · pp. 22–23
• R1 specifies population prediction risk; R2 measures finite-sample error and simplex feasibility, while R3 diagnostics do not independently certify Koopman invariance.Population interpretation of R2 requires a separate statistical guarantee; R3 tolerances concern squared ratios.Source: §4.4 · p. 24
• As the exact normalized-affinity parameter ω approaches zero, coordinates approach C−11_C and identity prediction has vanishing error, so selection also considers coordinate variation.The forecasting problem is formulated for the resulting fixed map.Source: §4.4 · p. 24
• The protocol selects among fixed reference constructions and soft maps, fits each closure matrix on fitting data, then holds the selected map and matrix fixed.Selection may favor coordinate variation among candidates with similar prediction errors.Source: §4.4 · p. 24
• The contribution is the resulting coordinate-prediction bounds and their conditional operator interpretation, not the original KAHM construction.An arbitrary fitted matrix need not preserve the simplex; the experimental identity-initialized NLMS procedure preserves sums of one in exact arithmetic, but components can remain negative.Source: §4.4 · p. 25

5 Reconstruction Scores, Prediction Bounds, and Matrix Fitting

• The section distinguishes representation agreement with reference labels, successor variation within reference classes, and the accuracy of a particular fitted prediction matrix.Reconstruction-score margins bound label deviation under stated normalization assumptions, while transition dynamics and fitting affect the other two quantities.Source: §5 · p. 25

5.1 Class-conditional successor means and matrix fitting

• The class-mean matrix has mean successor vectors as rows and minimizes hard-label prediction error, but may not be optimal for soft coordinates; NLMS preserves unit row sums, though entries can become negative.The NLMS recursion also applies to arbitrary transition sequences, including dependent samples.Source: §5.1 · p. 26

5.2 Finite-update NLMS error analysis

• Upper bounds for a fixed fitted matrix combine label deviation, within-class successor variation, and prediction discrepancy from the class-mean matrix, or replace that discrepancy with a hypothetical-update term.The update analysis is a parameter-error identity, not a computable lower confidence bound on the population optimum; rates with training epochs require separate analysis.Source: §5.2 · pp. 27–28

5.3 Deviation from Reference Labels and Coordinate Second Moments

• Squared deviation from one-hot reference labels measures fidelity, not hard-classification error; its prediction effect is smaller when class-mean successors are closer together.Weight transferred between classes with identical mean successors does not change the class-mean prediction; within-class successor variation and matrix-dependent terms remain.Source: §5.3 · pp. 28, 30

5.4 Reconstruction-score margins and prediction-error bounds

• A positive reconstruction-score margin means the labelled class has greater affinity than competitors and bounds deviation from reference labels, but does not directly constrain successor variation or fitted-matrix error.The margin-based contribution to prediction bounds is multiplied by the largest distance between class-mean successors; full bounds also require successor-variation and matrix-dependent terms.Source: §5.4 · pp. 30, 32

5.5 Minimum prediction error and successor variation

• A lower certificate above an error tolerance rules out that accuracy, while attainability requires an upper bound from the class-mean matrix when it satisfies the norm limit.The four-state example isolates predictive obstruction by holding coordinates fixed while changing successor behavior.Source: §5.5 · p. 32
• If σ_res > ε + κ p2Ageom, every predictor with spectral norm at most κ exceeds error tolerance ε; a converse sufficient condition guarantees attainment when κ ≥ ∥B*∥2.These bounds concern one fixed representation, need not determine the exact minimum, and rely on the norm limit to prevent amplification of small within-label coordinate differences.Source: §5.5 · p. 33
• 15/32 is the exact optimal RMSE in the four-state KAHM example, while the population centered bound is exact and assignment error tends to zero as coordinates approach hard labels.For the illustrative balanced count vector, the 95% pointwise label, centered, fixed-prototype, and ERM bounds are 0.303065, 0.128038, 0.336172, and 0.329043, respectively; these are deterministic calculations, not sampled campaign replicates.Source: §5.5 · p. 34
• For stochastic successors, one-step prediction error separates error in predicting the conditional mean successor coordinates from random variation around that mean.Successor variation includes both differences in conditional mean successors across same-class states and randomness at a fixed state; the spectral identities retain deterministic-transition assumptions.Source: §5.5 · p. 35

5.6 Independent evaluation and finite-sample bounds

• Independent i.i.d. evaluation pairs yield conditional 1−δ confidence bounds with fitted objects held fixed, including a compact closure-risk bound that combines assignment-loss control with population class means and matrix-dependent terms.Auxiliary bound parameters may be selected after observing the assignment-loss estimate because the confidence event holds simultaneously for their positive values.Source: §5.6 · p. 36
• Independent transition pairs also give a computable prediction-risk upper bound by combining soft-coordinate deviation, successor variation around estimated class means, and disagreement between the fitted matrix and the class-mean matrix.The fitted representation, predictor, and class-mean estimates must be fixed before evaluation.Source: §5.6 · pp. 36–37
• The Koopman candidate residual bound controls an eigenfunction’s absolute mean-squared composition residual, but relative-error assessment requires positive L2 norm, and operator boundedness or spectral proximity needs separate conditions.For implemented reconstruction scores, a geometric margin bound must account for uniform fallback use; thresholds or representations selected after evaluation require fresh data or simultaneous guarantees.Source: §5.6 · pp. 38–39

6 Evaluating Prediction-Error Certificates

• The 117,000 controlled datasets were repeated samples from fixed designs, whereas the 64 Van der Pol datasets evaluated one representation selected before independent evaluation.The Van der Pol study could not assess certificate-to-optimum gaps because its population optimum was unknown, and it did not numerically test centered or prototype variants on the learned coordinates.Source: §6 · p. 40, §6.4 · p. 44

6.1 Quantities, trial units, and interpretation

• The certificate bounds the minimum population RMSE over norm-constrained matrices, while empirical RMSE-minus-bound differences describe one fitted predictor and do not establish the bound’s coverage.A positive certificate excludes a target error ε only when it exceeds ε; a zero certificate is inconclusive.Source: §6.1 · p. 40

6.2 Controlled validation with a known population optimum

• The controlled four-state family has class-centered population bound equal to the optimum, but at p = 4/5 the label-based certificate is zero although the optimum is 0.30.At p = 31/32, the optimum is 15/32 and the limiting certificate is 13/32, a persistent difference of 1/16.Source: §6.2 · p. 41, Figure 1 · p. 42
• Across 117,000 fixed-design datasets, no certificate exceeded the known optimum and no tolerance exclusion was erroneous; zero observed violations do not establish zero violation probability.The centered finite-sample correction remained conservative despite its exact population limit, while the ERM-based bound became tighter than the fixed-prototype bound at larger sample sizes.Source: §6.2 · pp. 41–42, Table 2 · p. 42

6.3 Pilot analysis and selection of the dynamical evaluation

• At M = 2048, pilot bounds were nonpositive at the fitted norms, even after omitting confidence corrections, so increasing sample size alone was unlikely to make those certificates informative there.Exploratory results motivated selecting the Van der Pol representation with ω = 128 for evaluation on fresh independent pairs; the resulting study concerns that selected map, not newly trained representations.Source: §6.3 · p. 43

6.4 Fixed representation and sampling of independent transition pairs

• Each evaluation pair came from a newly started trajectory, yielding independent pairs under matched-stochastic or deterministic Van der Pol dynamics; 32 replicates per law each contained 16,384 pairs.The fixed representation and fitted matrix were used under both laws.Source: §6.4 · p. 44

6.5 Confirmatory results across norm limits

• At the fitted matrix’s spectral norm, all 32 replicates under each Van der Pol transition law had positive certificates; median certificates were 0.084727 under matched stochastic and 0.084269 under deterministic evaluation.At larger norm limits, positivity weakened: at κ = 2, 11/32 matched stochastic and 8/32 deterministic replicates were positive, and at twice the fitted norm all certificates were zero.Source: §6.5 · pp. 44–45, Table 4 · p. 45, Table 5 · p. 45

7.6 What the experiments establish about predictive abstractions

• The experiments assess fitted predictors rather than the best population error over a norm-bounded class, and neither forecasting results nor certificates establish exact Koopman closure.Forecasting performance depends strongly on target coordinates and the tested system and horizon.Source: §7 · p. 46, §7.6 · pp. 51–52

7.1 Protocol, comparison targets, and error measures

• The forecasting protocol separates map construction, matrix fitting, selection, and independent certificate evaluation, while distinguishing shared-target comparisons from each method's own-coordinate task.The centered score compares prediction error with always predicting the evaluation-target mean; it is descriptive, and differs from the coordinate RMSE used by certificates.Source: §7.1 · pp. 46–47

7.2 Coordinate variation and the choice of prediction target

• For the reported configurations, KAHM affinities have the smallest 200-step relative squared error among the listed coordinate constructions, but each construction predicts its own target.The comparison therefore measures representation-specific predictability, not performance on a shared prediction task.Source: §7.2 · pp. 48–49, Table 7 · p. 48

7.3 Learning from additional trajectories

• At eight training trajectories, mean R2 across five horizons is 0.9021 for the noise-selected configuration, 0.8891 for the clean-selected configuration, and 0.9077 for the reference; ordering varies across training counts.These three-fit comparisons predict each map's own coordinates under the specified local initialization laws.Source: §7.3 · p. 49, Table 8 · p. 49

7.4 Comparing predictors on the same abstract target

• At 200 steps, degree-three EDMD reaches mean R2 = 0.9774 versus −0.3155 for direct coordinate prediction on Duffing, while direct prediction reaches 0.7724 versus 0.4075 for EDMD on Van der Pol.Both comparisons use a common output target within each system; each summarizes three evaluation trajectories for one fitted model per method.Source: §7.4 · p. 50, Table 9 · p. 50

7.5 Prediction under fixed randomized behavior policies

• MountainCar and Acrobot have positive mean one-step R2, but all three fixed-policy tasks have negative mean R2 50 against the evaluation-mean predictor.On Acrobot's common target, direct coordinate prediction has the smallest mean 50-step relative squared error, although all four predictors have negative mean R2 50.Source: §7.5 · p. 51, Table 10 · p. 51, Table 11 · p. 51
• The MountainCar figure colors visited states by their largest soft coordinate, with nominal coordinate indices that carry no quantitative meaning.The visualization does not measure class separation or predictive sufficiency.Source: §7.5 · p. 51, Figure 2 · p. 52

8.1 What the lower confidence bound establishes

• A certificate above the chosen tolerance rules out that population RMSE for every permitted matrix while the representation, evaluation law, and norm limit remain fixed.The bound does not identify which alternative representation, nonlinear predictor, or control policy would succeed.Source: §8.1 · pp. 52–53

8.2 Using a certificate to assess a representation

• Use independent transitions after fixing the representation, partition, evaluation law, tolerance, and matrix-norm limit; a zero or sub-tolerance bound is inconclusive.Coordinate-variation summaries can help identify nearly constant representations that may otherwise appear easy to forecast.Source: §8.2 · p. 53

8.3 Sampling uncertainty and the gap in the population bound

• In 117,000 four-state evaluations, 49,271 original label-based bounds were zero; at p = 31/32, the label-based population bound is 13/32 against an exact optimum of 15/32.The centered population bound equals the optimum when within-class input variation vanishes, but population identities do not establish finite-sample tightness.Source: §8.3 · p. 53
• All 64 independent Van der Pol evaluations had positive label-based certificates at the fitted matrix norm, but positivity was specific to that norm-constrained class.The representation was selected using earlier exploratory results, before its independent certificate evaluations.Source: §8.3 · p. 53

8.4 Interpreting the forecasting experiments

• Forecasting outcomes depend on the representation, task, predictor, and horizon; the experiments show no uniform method superiority or control benefit.Polynomial EDMD scores higher on long-horizon Duffing, while direct coordinate prediction scores higher for the selected Van der Pol target.Source: §8.4 · pp. 53–54

8.5 Exact closure and spectral interpretation

• Under exact deterministic closure, coordinate functions span a finite-dimensional Koopman-invariant space; removing redundant coefficient vectors gives its reduced matrix representation.The adjoint depends on the specified RKHS inner product, and the operator statements apply only when closure holds on the full stated domain.Source: §8.5 · p. 54

8.6 Limitations and further evaluation

• Confidence guarantees require fixed representation and class maps plus specified independent evaluation units; selection, optional stopping, and distribution shift need new samples or extended guarantees.Under stochastic transitions, the bound concerns realized-successor RMSE, including transition noise, and does not alone rule out exact evolution of the conditional mean.Source: §8.6 · p. 54
• The four-state certificate ordering cannot be generalized to learned high-dimensional maps, and the ERM baseline can be tighter for larger samples.The four-state study uses known low-dimensional representations, whereas the Van der Pol result concerns one representation selected using exploratory data.Source: §8.6 · p. 54

9 Conclusion

• Independent transition data yield lower confidence bounds on the minimum population RMSE for all linear predictors within a specified spectral-norm limit; exceeding tolerance rules out that accuracy.A zero bound leaves open whether an accurate predictor exists.Source: §9 · p. 55
• The class-centered population bound equals the optimum in a four-state example, and one Van der Pol map has positive label-based certificates in all 64 evaluations at the fitted norm.The Van der Pol optimum remains unknown; the four-state reanalysis uses 117,000 archived datasets to compare certificates with an ERM-based comparator.Source: §9 · p. 55
• Exact deterministic propagation by one matrix gives the nonredundant coordinate-function span a Koopman-invariant and reproducing-kernel interpretation, distinct from finite-data error certification.The certificates can rule out a target accuracy for a fixed representation, evaluation law, and predictor class before further fitting.Source: §9 · p. 55

Statements and Declarations

• The research was supported by Austrian federal ministries and the State of Upper Austria through SCCH’s INTEGRATE competence center, an FFG COMET programme grant.The listed grant number is 892418.Source: Statements and Declarations · p. 56
• The authors declare no competing interests; ethics approval and participant consent were not applicable because the work involved mathematical analysis and computer simulations.ChatGPT assisted with literature searches, language revision, and experimental-reporting and provenance descriptions.Source: Statements and Declarations · p. 56
• Mohit Kumar developed the mathematical theory, while Somayeh Kargaran implemented the experiments and maintained experimental provenance records.Source: Statements and Declarations · p. 56
• The archives include 117,000 four-state replicates and verification records, forecasting datasets, and the fixed Van der Pol representation with training and verification records.Confirmatory Van der Pol materials include 64 replicate summaries and numerical-verification records; full evaluation arrays are retained separately from the compact public release.Source: Statements and Declarations · p. 56
• Experimental source code, configurations, verifiers, and forecasting code are archived with reproduction instructions, dependency specifications, and reference-environment records.The source code is distributed under the MIT License.Source: Statements and Declarations · p. 56

A KAHM Construction Details

• KAHMs combine an encoding matrix, Gaussian kernel, regularized regression functions, and normalized affine reconstruction weights.Source: A KAHM Construction Details · p. 57
• The encoding matrix uses leading sample-covariance eigenvectors, retaining at most min(20, n, N − 1, rX) directions and removing low-range coordinates when multiple directions remain.Every retained eigenvalue exceeds machine epsilon.Source: A KAHM Construction Details · p. 57
• Each class’s encoded reference samples must be pairwise distinct and have positive-definite sample covariance for the construction to be admissible.Source: A KAHM Construction Details · p. 57
• The encoding and Gaussian-kernel covariances use denominator N − 1, and the reference values are stored in single precision while covariance and matrix calculations use double precision.For the two encoded points −1 and 1, this convention gives covariance 2.Source: A KAHM Construction Details · pp. 57–58
• For each reference sample, a kernel-RKHS function fits target 1 at that encoded point and 0 at the others using regularized least squares.The regularized solution balances target agreement against an RKHS-norm penalty.Source: A KAHM Construction Details · p. 58
• The regularization parameter is computed through fixed-point iterations that approximate a unique fixed point, with the implementation stopping at 10^-3 absolute change or 100 iterations.The reconstruction reuses the inverse from the final completed iteration before its final parameter update; its matrix solve depends only on reference data.Source: A KAHM Construction Details · p. 58

B Proof of Proposition 3.2

• For every RKHS function, the reproducing property and adjoint relation equate inner products with kernel sections at the transformed and original inputs.Since the functions belong to the RKHS and have identical inner products against every member, their difference is zero.Source: B Proof of Proposition 3.2 · p. 59

• The orthogonal component of w contributes zero to every function value, while the real-space and complex inner-product identities yield the norm formula.The represented function is zero exactly when the projected coefficient is zero.Source: C Proof of Proposition 4.7 · p. 59

• Because QΦ is real, its transpose is its conjugate transpose, and uniqueness of orthogonal projection gives (57).Source: D Proof of Proposition 4.8 · p. 59

• Exact closure makes VΦ invariant under B⊤, preserving redundant coefficient directions so equivalent coefficients remain equivalent after multiplication by B.Composition with F maps the represented function space into itself.Source: E Proof of Proposition 4.9 · pp. 59–60
• When Bw = λw and fw is nonzero, the represented function fw is a Koopman eigenfunction with eigenvalue λ.Source: E Proof of Proposition 4.9 · p. 60
• The reduced map ew ↦ fQΦ ew is an isometry, and its Koopman eigenrelation holds exactly when ew is nonzero and eBΦ ew = λ ew.Source: E Proof of Proposition 4.9 · p. 60
• In the orthonormal coefficient basis, a nonzero function fQΦ ew is an adjoint Koopman eigenfunction when its coefficient satisfies the corresponding adjoint eigenrelation.Injectivity of the coefficient-to-function map also gives the converse.Source: E Proof of Proposition 4.9 · p. 61

F Proof of Proposition 5.1

• The proof of Proposition 5.1 derives a residual bound by converting a quadratic inequality into a root bound and bounding its coefficients over the simplex.It first establishes the change in squared parameter distance and uses that Φ(x) lies in the simplex.Source: F Proof of Proposition 5.1 · pp. 62–63

G Proof of Proposition 5.2

• For Proposition 5.2, the direct comparison bounds residual error by the sum of the assignment, estimation, and within-class successor-variation terms.The inequalities hold for any fixed fitted matrix, including one obtained from dependent training transitions.Source: G Proof of Proposition 5.2 · pp. 64–65
• A second Proposition 5.2 bound uses the parameter-error change from Proposition 5.1 to compare the fitted predictor with the class-mean matrix.Source: G Proof of Proposition 5.2 · pp. 64–65
• Class-mean distances or KAHM reconstruction-score margins can bound the assignment contribution, while successor variation and predictor-error terms enter separately.Source: G Proof of Proposition 5.2 · p. 65

H Concentration Details and Independent-Trajectory Evaluation

• Reference-label and reconstruction-margin losses yield confidence upper bounds on label deviation and geometric loss, with the geometric bound no tighter than the label-based bound.The margin thresholds and representations must be fixed before evaluation unless a simultaneous guarantee covers their selection.Source: H Concentration Details and… · p. 66
• With J independent trajectories, the confidence correction uses J, and trajectory-weighted class means and losses target a law that weights trajectories equally and states within each trajectory uniformly.Within-trajectory dependence is unrestricted; a state in a short trajectory receives more weight than a state in a long one.Source: H Concentration Details and… · pp. 66–67

I Infimum identity used in Result 5.12

• The infimum identity is obtained by minimizing the stated expression over positive t, with boundary limits covering cases where either nonnegative term vanishes.When both terms vanish, every positive t attains the infimum.Source: I Infimum identity used in Result 5.12 · p. 67

J.1 Purpose of the reported studies

• The studies address four questions: certificate detection, coordinate variation, common-output prediction comparisons, and prediction changes with horizon under a fixed policy.Reported outcomes include favorable, unfavorable, and inconclusive results.Source: §J.1 · p. 67, Table 12 · p. 68

J.2 State generation, reconstruction, and fitting

• KAHM class scores use the minimum over internal reconstruction models, while an absolute-threshold fallback can replace nonuniform coordinates with a uniform vector.Changing ω can alter both relative weights and which states trigger the fallback; claims for a changed map require reevaluating dependent quantities.Source: §J.2 · p. 69

J.3 Hyperparameter selection and repeated-data units

• The expanded Duffing search selected (10, 2) by average relative squared error across five horizons; Van der Pol selection chose (25, 6) without added observation noise and (25, 4) across noise levels, unlike certificate map (25, 128).Source: §J.3 · p. 70

J.4 Fixed-policy episode protocols

• The control studies use fixed policies and episode protocols, but overlapping reset-seed ranges make the three fits dependent, and no untouched final test set is provided.The reported CartPole and MountainCar configurations are exploratory fixed-configuration diagnostics.Source: §J.4 · p. 71

J.5 Numerical scope and source records

• The campaign validates the lower certificate empirically; fitted-predictor upper bounds, finite-update NLMS inequalities, and exact-closure results remain analytical contributions.Released records connect reported values with stored outputs and evaluation datasets.Source: §J.5 · p. 72

J.6 Detailed implementation conventions and remaining limits

• Forecasting segments used for selection are not independent final-test estimates, and the reported control summaries do not count uniform-coordinate fallback use.Without fallback counts, changes cannot be attributed solely to more concentrated relative affinities.Source: §J.6 · pp. 72–73

K Certificate Evaluation: Records and Verification Scope

• 117,000 four-state count records reproduce 49,271 zero certificates, with no certificate exceeding its analytical optimum.Reaggregation of 64 dynamical result records reproduces the displayed certificate summaries.Source: K Certificate Evaluation: Records and… · p. 73

K.1 Controlled four-state records

• The four-state archive contains 117 settings with 1,000 replicate datasets each, and the reanalysis reproduces 49,271 zero label-based certificates.Its maximum absolute discrepancy from archived label-based bounds is 1.11 × 10^-16.Source: §K.1 · p. 74

K.2 Dynamical protocol and random streams

• The confirmatory Van der Pol protocol reuses stored centers and reconstruction models, refits only the prediction matrix at ω = 128, and fixes the protocol before confirmatory sampling.A separate implementation reproduced the matrix and NLMS update history before evaluation.Source: §K.2 · pp. 74–75

K.3 Independent numerical verification of dynamical results

• The numerical verifier recomputes evaluation statistics for all 64 scheduled datasets from stored physical-state pairs and the fixed coordinate map.The summary generator runs numerical checks before computing reported tables.Source: §K.3 · p. 75

K.4 Confirmatory evaluation records

• The Van der Pol archive provides the protocol, run metadata, 64 primary replicate rows, and a numerical audit; access to full evaluation arrays is described in the Data availability statement.Source: §K.4 · p. 76

L.1 Effects of simplex projection on prediction

• Per-step vector projection lowers Van der Pol E200 from 0.180 to 0.153, while row projection substantially increases long-horizon error in these runs.Prediction accuracy depends on the projection rule as well as output feasibility.Source: §L.1 · p. 76, Table 15 · p. 76

L.2 Interpreting the empirical spectral diagnostic

• All four reported fits have numerical training rank C; Duffing median ρ is 5.88 × 10^-2 for NLMS and 6.44 × 10^-4 for ridge least squares.Both Van der Pol medians are below 2 × 10^-4, while one-step prediction scores are similar within each system.Source: §L.2 · p. 77
• Rounded spectral radii of 1.000 do not establish stability of long rollouts, and aggregate held-out residuals do not establish pointwise eigenrelations or deterministic exact closure.The sampled feature basis may also differ from the full feature span.Source: §L.2 · pp. 77–78