Source-linked AI summary
Diagnosing Faults in Reinforcement Learning Simulators and World Models with Canonical Polynomial Invariants
Tesfay Zemuy Gebrekidan, Hadush Hailu Gebrerufael
TL;DR
The paper asks whether exact physical structure improves learned-dynamics prediction and develops exact polynomial-invariant diagnostics for RL simulators and world models. It finds little predictive benefit but strong fault-localisation value, while noting that attribution requires clean data and a trusted reference generator.
Problem
The paper tests whether enforcing physical structure improves learned-dynamics prediction, addressing limited evidence based on exact rather than approximate invariants.
Method
It recovers polynomial invariants from trajectories, canonicalises invariant ideals with reduced Gröbner bases, and uses screening plus parameter attribution for diagnosis.
Results
Across fifteen injected faults, screening localises every broken constraint without false alarms, and attribution recovers the responsible parameter for all seven parameter faults.
Takeaways & Limitations
Canonical invariants are valuable chiefly for localisation and equality-based diagnosis, not for improving prediction or residual-based shaping.
Takeaways & Limitations
Attribution fails between observation-noise levels σ = 10^-6 and 10^-5, and screening requires a correct reference generating set.
Abstract
from arXiv · showhide
A large literature builds physical structure into learned dynamics on the premise that models respecting the underlying physics predict better. We test that premise using exact polynomial invariants recovered from trajectories and canonicalised as reduced Gröbner bases over $\mathbb{Q}$. On Acrobot, exactness provides little benefit for prediction: a consistency regulariser reduces algebraic residual while leaving rollout fidelity essentially unchanged, and a shaping potential recovered from a system with a 100% mass error accelerates learning as effectively as the correct potential. Exact canonical invariants instead prove valuable for diagnosis. We develop two procedures: screening, which identifies the violated physical constraint, and attribution, which recovers the faulty invariant and identifies the responsible physical parameter. To enable this, we introduce normal-form deflation and quotient-space recovery. Across fifteen injected faults, screening localises every broken constraint with no false alarms, whereas observation-space baselines do not localise any; attribution recovers the responsible parameter on all seven parameter faults. Paired difference tests detect all faults, showing that the advantage is localisation rather than detection. Perturbing reference generators by $10^{-4}$ preserves 14--15/15 localisations, showing that screening does not require exactness, whereas ideal-equality decisions distinguish perturbations of only $10^{-12}$, showing that exactness is required for algebraic comparison. Applied to 350 release pairs across eleven RL environments, the diagnostic finds no evidence of changed simulator dynamics, instead revealing properties of the benchmark implementations themselves.
I. INTRODUCTION
The paper tests whether exact physical structure improves learned dynamics and finds little predictive benefit, but substantial diagnostic value in canonical invariants for locating simulator faults.
- Motivation: The paper challenges the premise that enforcing physical structure improves prediction by replacing approximate invariants with exact, canonical ones.The comparison targets a premise underlying conservation penalties, structured architectures, and projected kinematic constraints.
- Motivation: Exact canonical invariants make equality of invariant ideals decidable, enabling comparisons that approximate or learned invariants cannot settle.Reduced Gröbner bases are unique for a fixed monomial order, while approximate invariants mainly support residual-based uses.
- Contributions: The diagnostic separates screening, which identifies violated constraints, from attribution, which recovers the faulty invariant and responsible physical parameter.Its broader motivation is that aggregate validation errors and rollout divergences indicate inaccuracy without identifying the cause.
- Application: Across 350 release pairs in eleven RL environments, the diagnostic found no evidence of changed simulator dynamics but exposed implementation properties instead.Reported properties include integrator sensitivity, an observation inconsistency in Reacher, and long-horizon limits from chaotic dynamics.
- Invariant types: The approach distinguishes vanishing constraints, canonicalised as ideals, from conserved quantities, which form a subalgebra and require trajectory-difference recovery.Examples include rigid-link constraints, trigonometric identities, forward kinematics, loop closures, and energy-like quantities.
IV. SCALABLE INVARIANT RECOVERY
The recovery pipeline removes algebraically redundant polynomial content before extracting invariant directions, making the result shorter and more auditable.
- IV. SCALABLE INVARIANT RECOVERY: The pipeline compares reference and test trajectories, constructs polynomial differences, deflates known content, projects into quotient space, and extracts a recovered invariant.Screening then uses generator residual tests, while attribution fits the recovered invariant to physical parameters.
- IV. SCALABLE INVARIANT RECOVERY: Normal-form deflation removes the entire filtered space of redundant invariant multiples, rather than only projecting out the known generator span.On Acrobot, trivial multiples comprise nearly the entire degree-3 nullspace.
B. Tolerance-Free Recovery via Quotient Spaces
Quotient-space recovery replaces tolerance-based nullspace dimension selection with exact algebraic projection and a dimensionless spectral-gap criterion.
- B. Tolerance-Free Recovery via Quotient Spaces: Quotient-space recovery computes an exact rational normal-form basis, restricts the difference matrix to it, and extracts the smallest-singular-value direction without estimating nullspace dimension.This removes the units-dependent absolute tolerance used by numerical nullspace methods.
- B. Tolerance-Free Recovery via Quotient Spaces: Deflation resolves noisy-data failure by removing redundant multiples before recovery; without projection, the recovered direction is entirely trivial, whereas with projection it aligns with energy at 1.0000.The method uses a fixed spectral-gap threshold of 0.1 and refuses systems with no gap.
- Canonical comparison: Canonical reduced Gröbner bases support exact ideal comparison: identical ideals reduce to identical bases, while changed ideals are distinguished by two-way normal-form reduction.The experiments associate parameter and observation faults with changed ideals, but integration and step-size faults with preserved ideals.
- Canonical comparison: Canonicalisation alone does not guarantee minimality, physical correctness, or complete dynamics coverage; diagnostic capability comes from combining it with screening and attribution.It is unnecessary for residual-based consistency regularisation and shaping, and required only for equality-based diagnosis.
VI. DIAGNOSTIC PROCEDURE
The diagnostic compares normalized invariant residuals from a reference system and a system under test, then screens violated constraints and attributes parameter faults when fitting is decisive.
- VI. DIAGNOSTIC PROCEDURE: The input is a known reference generating set plus trajectories from another simulator release, a suspected faulty simulator, or a learned world model.The same procedure applies across all three settings.
- VI. DIAGNOSTIC PROCEDURE: Each generator’s residual is normalized by its natural data scale, using trajectory standard deviation for conserved quantities rather than distance to zero.This prevents degree and unit scale from making otherwise comparable generators appear differently violated.
- VI. DIAGNOSTIC PROCEDURE: Screening splits both trajectories into 32 contiguous blocks and applies Bonferroni-corrected one-sided Mann–Whitney U tests against reference variability.Contiguous blocks preserve the effects of conservation drift.
- VI. DIAGNOSTIC PROCEDURE: Attribution fits a symbolic parameterized template modulo arbitrary scale and additive gauge, freeing one parameter at a time while holding the others to specification.A wrong hypothesis produces a large residual because templates contain more monomials than parameters.
- VI. DIAGNOSTIC PROCEDURE: The procedure returns parameter fault, consistent with specification, or not explicable as a parameter fault, refusing cases without decisive recovery or fit evidence.This prevents a naive best-fit choice from labeling healthy systems or structurally mismatched faults as parameter errors.
C. Downstream Applications
The diagnostic evaluates known simulator faults using invariant residuals, separating constraint screening from parameter attribution. It localises all fifteen injected faults in noiseless settings, remains useful under noise, and identifies all seven faulty parameters when attribution is applicable.
- Diagnostic design: Screening uses reference invariant residuals to identify violated physical constraints, whereas attribution recovers faulty invariants and responsible parameters.The evaluation uses fifteen fixed faults with known constraint loci and parameter values, plus two healthy controls.
- Fault screening: 15/15 faults are detected and localised in both noiseless regimes, with no false alarms on either healthy control.A paired two-sample test also detects 15/15, showing the diagnostic’s advantage is localisation rather than detection.
- Fault screening: At σ = 10^-3, screening detects 12/15 faults per trajectory and still localises every fault it detects.Observation-space baselines do not name constraints, while residual-space testing can reach 14/15 localisation.
- Parameter attribution: Attribution identifies the correct parameter on all 7 parameter faults with relative error 1.0 × 10^-7.It rejects parameter explanations when none applies and returns specification-consistent results on both healthy controls at σ = 0.
- Operating limits: Screening tolerates larger perturbations and noise than attribution, whose conserved-quantity branch fails between σ = 10^-6 and 10^-5.Level-1 screening remains useful at noise three orders of magnitude higher, while level-2 attribution is limited to float64, noise-free simulator or learned-model data.
B. World-Model Consistency Regularisation
On Acrobot, enforcing exact invariant consistency does not improve long-horizon world-model fidelity, while discovered-energy shaping accelerates learning. The shaping benefit persists under substantial mass-parameter error, indicating different practical value for prediction and control.
- Consistency regularisation: No regularisation weight improves divergence horizon, with the largest estimate only −0.1 steps out of 45.9.At λ = 100, the horizon loses 37 steps, despite the algebraic residual falling to 0.79 of its unregularised value.
- Consistency regularisation: The algebraic residual can fall while rollout fidelity does not improve, because both exact and approximate generators supply the same residual-based training signal.Validation error correlates more strongly with long-horizon fidelity than algebraic residual, with Spearman correlations of 0.93 and 0.77 respectively.
- Reward shaping: Shaping from the discovered energy improves AUC from −147.8 to −91.3, with the gain appearing in learning speed rather than final return.Energy-derived conditions converge to final returns within about one standard deviation of one another.
- Faulty shaping potentials: A 100% mass error leaves shaping AUC between −90.8 and −88.6, close to the correct potential’s −91.3 and better than no shaping.Only the 300% error degrades performance to −100.5, while still outperforming no shaping.
- Implications: Learning can absorb a parameter error without reporting it, shifting the practical risk to hardware transfer and benchmark comparisons.The application analysis reports two shipped-code defects that do not change the reported return.
D. Auditing Shipped Releases
The release audit finds no changed simulator dynamics across 350 release pairs, while exposing implementation-level inconsistencies, integrator sensitivity, and chaos-related limits that source comparison would not reveal.
- All 275 classic-control pairs were bitwise identical, while 75 MuJoCo pairs differed by at most 9.1 × 10−15 in four non-chaotic environments, indicating unchanged dynamics.The reported differences are consistent with a recompiled physics library reordering floating-point operations.
- At ∆t = 0.2, Acrobot’s RK4 relative energy drift was 3.55×10−3, exceeding the violation from a one-percent LINK_LENGTH_1 error and limiting energy-based screening.Reducing the integration step lowers the drift floor and the smallest localisable parameter error.
- On InvertedDoublePendulum, thirteen release pairs separated from 2.2×10−16 initially to clipped values over time, so exact comparisons must use first-step separation.The growth rate reflects amplified rounding differences rather than changed dynamics.
- Attribution assumes a single fault and can return a confident partial answer on two-fault systems, while the method also requires clean data and a correct reference generating set.The paper reports attribution failure between σ = 10−6 and 10−5 of observation noise.
- Block-level false-alarm rates reached 10.2% against a nominal 1%, whereas trajectory-level permutation tests remained consistent with nominal levels.Blocks from one trajectory are not independent replicates for the rank test.
IX. CONCLUSION
The paper concludes that exact canonical invariants are more valuable for diagnosing simulator faults than for improving learned-dynamics prediction. The diagnostic localises broken constraints and faulty parameters, and its application to benchmark releases finds implementation properties rather than changed dynamics.
- IX. CONCLUSION: Exact canonical invariants support diagnostics: screening identifies the broken physical constraint, while attribution identifies the faulty invariant and responsible constant.The instrument operates on trajectories from simulators, releases, or learned world models.
- IX. CONCLUSION: For prediction, enforcing physical structure did not improve rollout fidelity, even when consistency regularisation increased one-step error by a factor of sixty.A shaping potential recovered from a system with a 100% mass error performed as well as the correct potential.
- IX. CONCLUSION: Canonicity provides decidable equality and reproducibility, but not minimality, sufficiency, or safety.The substitution ablation further shows that screening continues through a coefficient perturbation four decades larger than numerical recovery error.
- IX. CONCLUSION: Across 350 shipped release pairs, the diagnostic reports no changed simulator dynamics and instead reveals three benchmark implementation properties.The accompanying release analysis is presented as exposing findings that source comparison would not surface.
- IX. CONCLUSION: The implementation is reproducible through released experiment code, scripts mapping results to claims, and checks that fail when Appendix I claims drift from the text.The repository covers every experiment and the numerical results in Section VII and Appendix H.
A Extended Related Work
The extended related work distinguishes this diagnostic from approximate invariant discovery, system identification, dynamic invariant detection, and safety verification. It also details the algebraic and numerical design choices needed to recover and compare physically meaningful relations from trajectories.
- A Extended Related Work: Unlike approximate discovery methods, reduced Gröbner bases make ideal membership and equality decidable through exact normal-form reduction.Approximate border bases reduce membership to a threshold, whereas the exact predicate is NFG(q) = 0 in rational arithmetic.
- A Extended Related Work: Dynamic invariant detection uses fixed template relations over program variables, whereas this work recovers physical polynomial constraints from executions for simulator diagnosis.The supplied related-work passage identifies dynamic invariant detection as the closest precedent but distinguishes its hypothesis class from the paper’s approach.
- A Extended Related Work: Normal-form reduction is an exact algebraic analogue of deflation: it removes trivial multiples while simultaneously producing a canonical representation.This contrasts with neural or projection-based approaches that remove redundancy approximately.
- A Extended Related Work: Learned latent invariants cannot be inspected, passed to a solver, or compared for equality, unlike the paper’s exact rational canonical polynomials.This is the central distinction from contrastive invariant-enforcement methods such as ConCerNet.
- A Extended Related Work: The paper complements, rather than replaces, dynamics randomisation: randomisation broadens policy tolerance, while screening and attribution identify whether a specific parameter is wrong and what it is.The comparison is framed as two objectives that can compose.
- A Extended Related Work: The diagnostic differs from system identification because screening first tests whether the assumed physical structure is correct, while attribution runs only when a parameter explains the fault.Identification instead presupposes a correct parametric dynamics model and estimates constants within it.
- B.3 Balance Laws for Actuated Systems: The transition dictionary uses monomial differences and action-dependent terms, while balance-law recovery must tolerate quadrature drift and therefore requires a tolerance above that floor.The trapezoidal discretisation has an O(∆t3) defect, and the dictionary is smaller than the alternative joint-space construction.
C Deflation, Arity, Lifting and Conditioning
Normal-form deflation removes algebraically trivial nullspace content exactly, while recovery quality remains limited by dictionary growth, noise, conditioning, and integrator drift. The diagnostic then uses these recovered invariants for exact comparisons across declared, recovered, and installed simulator specifications.
- C.2 Arity and lifting: Dictionary sizes remain manageable for Reacher: M is 78, 364, and 1365 for (n, d) = (11, 2), (11, 3), and (11, 4), respectively.Support search scales with M, while sparse support selection is combinatorial and nullspace estimation requires N ≫ M.
- C.1 Gr¨obner deflation: Normal-form reduction removes the entire filtered ideal component, unlike orthogonal projection, which removes only one coefficient-space direction per generator.For Acrobot, sequential projection can remove one direction in total, whereas normal-form reduction eliminates redundant multiples by ideal membership.
- C.1 Gr¨obner deflation: At Δt = 0.2, the nullspace count collapses to 1 but the surviving direction lies 62% outside span{E, 1}, so residual validation—not dimension counting—decides recovery.Acrobot energy is not recoverable at the default timestep because the integrator drift floor requires a loose tolerance that admits a spurious direction.
- C.3 Conditioning and units: Nondimensionalisation determines whether rational snap-rounding can represent coefficients, making units a recovery condition rather than a cosmetic preprocessing choice.Reacher’s 0.11 coefficient is unavailable at Qmax = 16 until lengths are rescaled, after which 1 and 11/10 are admissible.
- D Canonical Forms: Extended Analysis: Declared, recovered, and installed specifications support different comparison roles: parameter faults move the ideal, observation changes transform it, and integrator or timestep changes leave algebraic constraints unchanged.On seven parameter faults, recovered attribution reproduces the declared ideal exactly; installed Acrobot and Reacher parameters match the reference ideal.
- D Canonical Forms: Extended Analysis: The screen tolerates four decades more coefficient error than clean-data recovery incurs, while equality decisions distinguish perturbations as small as 10^-12.Across 184 healthy controls, no tested reference perturbation caused a false alarm; equality testing is therefore much more sensitive than screening.
E The Two-Level Diagnostic: Construction
The diagnostic constructs scale-normalised residual tests around exact invariant recovery, using quotient-space deflation to isolate informative directions and calibrated floors to control false alarms. Screening is reliable at the trajectory level, while block dependence and observation precision constrain interpretation.
- Construction: Normal-form deflation removes trivial algebraic directions before recovery, changing Acrobot’s energy alignment from 0.0000 without projection to 1.0000 with it.Without projection, the least-varying direction lies entirely in the 14-dimensional trivial component.
- Screening and calibration: The screen ranks scale-relative residuals, but an absolute residual floor is required because rank tests can flag machine-precision differences as faults.Float32 observations floor unit-norm identities near 1.7 × 10^-8, and shipped MuJoCo observations near 10^-8.
- Screening and calibration: Removing the floor raises Acrobot’s false-alarm rate from zero in 500 null comparisons to 10.2%, while violations below the floor become invisible.A Reacher violation at 6 × 10^-9 was found only by the exact stage.
- Screening and calibration: The reported p-value saturates at the test’s resolution limit, so residual ratios—not p-values—distinguish fault magnitudes such as 0.1% gravity and 30% mass errors.With 32 blocks, the smallest attainable one-sided p is 3.26 × 10^-12.
- Screening and calibration: At trajectory level, the test remains consistent with nominal error rates, whereas block-level Acrobot testing exceeds nominal in three of four settings, reaching 10.2% versus a 1% nominal level.Blocks share trajectories, initial conditions, energy levels, and drift rates, so treating them as independent makes p-values too small.
E.3 Attribution: Construction and Verdict Logic
Attribution fits recovered invariant directions to parameterised templates after screening, while separate fit and margin tests distinguish identifiable parameter faults from faults that no parameter can explain. The downstream constructions use recovered invariants for consistency regularisation and potential-based shaping, with different requirements from diagnosis.
- Attribution: A wrong parameter hypothesis leaves residual mismatch because invariant templates contain more monomials than parameters, making cross-hypothesis comparison meaningful after dimensionless normalisation.Recovered directions have arbitrary scale, and conserved quantities also have arbitrary additive constants.
- Attribution: Attribution names a physical parameter only when one hypothesis fits absolutely and beats the runner-up by a margin; otherwise it reports non-identifiability or not explicable as a parameter fault.The separate tests prevent selecting a merely least-bad hypothesis or confusing indistinguishable parameters with a valid attribution.
- Downstream constructions: Consistency regularisation penalises predicted departures from vanishing constraints or conserved quantities, with generator scaling compensating for differing magnitudes.Canonicalisation does not justify uniform weights because a reduced basis can contain algebraic consequences of other basis elements.
- Downstream constructions: Potential-based shaping uses Φ(s) = −|E(s) − E⋆| rather than an invariant residual, because conservation alone supplies no task signal; the resulting shaping preserves the set of optimal policies.On swing-up, the recovered potential reproduces energy-pumping shaping that control engineers hand-design.
- Diagnostic scope: A zero consistency penalty is sound for predictions on the constraint variety but is not sufficient for dynamic validity unless the invariant variety and local reachable trajectory manifold coincide.This local sufficiency condition must be tested rather than inferred from canonicalisation.
F.2 Localisation Guarantee of Screening
Screening localises a violated reference invariant because positive residual identifies a reachable test state outside that constraint’s variety, independently of sampling reparameterisation. The guarantee is local and depends on rank and trajectory-manifold estimates, with Acrobot and Reacher providing exact-invariant examples.
- Localisation guarantee: If a reference generator vanishes on system A but has positive residual on system B, screening identifies that specific violated constraint independently of how B’s reachable states are sampled.The implemented screen is a rank test on scale-normalised block residuals rather than the exact predicate ρ > 0.
- Local validity: The guarantee is local and sampling-dependent: the implicit-function argument applies only where rank is checked, and unsampled regions may have lower rank.For Acrobot, rank JG = 3 matches n − dtraj = 6 − 3; dropping energy reduces rank to 2 and leaves necessity without sufficiency.
- Environment scope: Acrobot’s exact degree-3 energy and unit-norm structure support the worked diagnostic, while Reacher supplies five exact degree-at-most-two invariants in an eleven-dimensional observation.Reacher’s kinematic identities require the standard link lengths and nondimensionalisation for recovery.
- Environment scope: The recovery construction scales with invariant arity and degree rather than state dimension, as illustrated by linear moiety conservation in reaction networks.No reaction-network experiments are run; the example delimits the analytic class rather than adding results.
- Negative controls: Actuated, dissipative, contact-rich Hopper and HalfCheetah are negative controls where the correct discovery output is no invariant, and the procedure rejects spurious trajectory correlations through validation.No result in the paper extends to these control environments as targets.
G.4 The fault catalogue
The fault catalogue separates detection from localisation: paired difference tests detect every injected fault, but only invariant-residual features recover the violated physical constraint. Attribution correctly names parameter faults and refuses structural or otherwise non-parameter explanations.
- Detection and localisation: Paired difference tests detect every injected fault, while the screen localises all 15 faults and observation-space baselines localise none.The comparison isolates localisation as the screen’s advantage rather than detection.
- Detection and localisation: Residual features substantially improve baseline localisation, with KS reaching 14/15 and MMD rising from 7/15 on raw states to 14/15 on residuals, still below the screen’s 15/15.The residual vector contains one coordinate per reference generator, enabling constraint-level readout.
- Attribution: On all seven parameter faults, attribution names the correct parameter, with worst relative errors of 1.0 × 10^-7 unpaired and 1.7 × 10^-7 paired.The Reacher ℓ2 = 0.13 fault is returned as ℓ2/ℓ1 = 1.3000 against the specified 1.1.
- Attribution: For seven faults unexplainable by parameter changes, attribution names no parameter and returns not explicable as a parameter fault on every case.These include dropped Coriolis terms, unmodelled torque, integrator and timestep changes, scaled cosine, and transposed observations.
- Verdict logic: The verdict logic distinguishes spectral-gap failure from failed parameter fit, with six non-parameter faults rejected at the gap stage and scaled cos θ1 rejected after direction recovery.Healthy controls at σ = 0 are returned as consistent with specification; unconditional argmin attribution would falsely name faults on correct software.
H.3 World-Model Consistency Experiments
World-model experiments show that algebraic consistency can reduce residuals without improving rollout fidelity, while recovered shaping potentials can accelerate learning despite severe parameter faults. The broader audit finds no changed simulator dynamics across release pairs, but exposes implementation-dependent limits from chaos, integration, observation precision, and step size.
- H.3 World-Model Consistency Experiments: At λ = 100, one-step error is 64 times its unregularised value while the algebraic residual remains 0.79, so stronger consistency regularisation does not reconcile local accuracy with 100-step rollout fidelity.The collapse in performance persists across divergence thresholds, whereas the smallest weights change sign.
- H.4 Shaping: protocol, controls and full curves: The discovered Acrobot potential is recovered rather than supplied, with relative coefficient error 2.8 × 10^-8, and its comparison against faulty potentials uses gauges fitted within each system.Faulty configurations regress against their own analytic energies, while arbitrary random polynomials require a different gauge treatment.
- H.4 Shaping: protocol, controls and full curves: A 100% mass-error shaping condition remains better than no shaping and is statistically separable from the correct potential, showing that effective shaping need not identify the correct dynamics.The condition is separable from the correct potential at p = 0.016 and from no shaping at p = 0.0079.
- H.5 The release audit: matrix, protocol and findings: In InvertedDoublePendulum, separations grow from 2.2 × 10^-16 to 6.4 × 10^-2 through chaotic amplification, so exact release comparisons must be interpreted at the first step or over shorter windows.Reacher remains flat at approximately machine precision, while loosening thresholds would sacrifice the certainty of exact comparison.
- H.5 The release audit: matrix, protocol and findings: Screening localises Acrobot’s omitted Coriolis term and 30% second-link mass fault to energy alone, with residual ratios 4.21 and 11.56 while unit-norm identities remain clean.At the shipped step size, simulator drift sets the diagnostic floor and reducing the step moves the floor and detection threshold together.
I Verified Environment and Algebraic Properties
The verification suite guards both diagnostic implementation and environment-specific algebraic behavior. It confirms the Acrobot model choice, exact energy representation, discretization behavior, Gröbner-basis redundancy handling, and limits imposed by coefficient representability.
- I Verified Environment and Algebraic Properties: The verification driver runs 145 checks and fails when claims drift beyond a 5% relative tolerance, providing executable guards for reported numerical and algebraic properties.The checks cover diagnostic logic, environment behavior, shaping quantities, chaotic growth, and table values.
- I Verified Environment and Algebraic Properties: Only Gymnasium’s book Acrobot variant conserves the derived energy; the nips variant omits a Coriolis term and exhibits a step-size-independent drift plateau near 1.7 × 10^-4.Leaving the variant unspecified would make subsequent drift and balance measurements uninterpretable.
- I Verified Environment and Algebraic Properties: The symbolic Acrobot energy agrees with mechanical energy to 1.14 × 10^-13, while passive relative drift decreases from 6.51 × 10^-4 at Δt = 0.2 to 2.20 × 10^-10 at Δt = 0.01 with convergence order 4.95.The measured drift therefore sets the resolution of diagnostics based on conserved quantities.
- I Verified Environment and Algebraic Properties: The power-balance residual scales as O(Δt^2), with residual/Δt^2 ratios of 0.99, 1.06, and 1.53 across the tested step sizes.This confirms the expected first-order identity under discretisation.
- I Verified Environment and Algebraic Properties: Reduced Gröbner bases expose generator redundancy: three basis elements represent a two-generator ideal, while exact normal-form deflation leaves a one-dimensional energy direction.Sequential projection instead removes only one direction overall and can fail because generator coefficient vectors are non-orthogonal.
- I Verified Environment and Algebraic Properties: Coefficient representability depends on scaling: metre-unit coefficients with denominator 100 exceed Qmax = 16, whereas rescaling lengths makes both coefficients admissible and recoverable.The example shows that exact invariant recovery is constrained by the chosen rational representation.