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Efficient Learning of Mechanism Designs and Compositionality

João Pedro Gandarela, Thiago Rios, Stefan Menzel, André Freitas

arXiv:2610.40005v1cs.AIcs.CE

TL;DR

Mechanism learning must capture coupled geometric constraints and the subassemblies and dependencies that joint-level representations leave implicit. The paper represents mechanisms with Assur groups, interfaces, and solve dependencies, and evaluates that representation against flattened inputs using the same MLP. In the evaluated low-data range, Assur-derived features yield a geometric-mean 1.8× sample-efficiency advantage, while transfer remains bounded by interface information and sensitivity.

  • Problem

    Mechanism motion depends on coupled geometric constraints, and joint-level graphs leave implicit the jointly solved subassemblies and their dependencies that may support learning from limited examples.

    Source: §1 · p. 2

  • Method

    The paper represents planar mechanisms through Assur groups, their interfaces, and solve dependencies, retaining geometry and evaluating matched mechanism-curve pairs with the same MLP used for flattened inputs.

    Source: §1 · p. 2, §5.1 · p. 7

  • Results

    1.8× geometric-mean sample efficiency over flattened adjacency-and-position inputs across budgets of 50 through 10,000 mechanisms.

    Source: §5.1 · p. 7

  • Takeaways & Limitations

    The theory identifies complete interface information, recurring local problems, and their influence on outputs as important to compositional reuse.

    Source: §7 · p. 9

    The sample-efficiency advantage is confined to the low-data regime and adjacency-plus-position baseline; at the full 72,013-example pool, adjacency is lower by 0.0014.

    Source: §5.1 · p. 7

Abstract

from arXiv · show

Learning mechanism behaviour requires a representation of how connected parts jointly constrain motion. We study whether exposing reusable constraint subassemblies improves prediction from limited data. Assur-KB represents planar mechanisms through irreducible modules, boundary interfaces, and solve dependencies while retaining source geometry. Our theoretical results characterize transfer through module interfaces and establish physical limits arising from missing interface information and accumulated sensitivity. To test the representation claim directly, the experimental design compares matched mechanism-curve pairs across training budgets within the same multilayer-perceptron family. This controlled comparison establishes the utility of the complete representation in the evaluated learning setting. Assur-derived features provide a geometric-mean $1.8\times$ advantage in sample efficiency over flattened adjacency and position inputs across budgets of 50-10,000 mechanisms. Separate experiments establish invariance under tested joint relabelings, higher valid-generation rates under dyadic composition, and a coarser structural index over 10.02 million mechanisms. In summary, the theory and experiments identify how physical constraint structure supports limited-data prediction and which interface conditions govern compositional reuse.

1 INTRODUCTION

The paper asks whether explicit constraint subassemblies provide useful reusable units for learning mechanism motion from limited data. It represents planar mechanisms with Assur groups, interfaces, and solve dependencies, and reports lower-data curve prediction error than flattened adjacency-and-position inputs.

1 INTRODUCTION

  • Mechanism trajectories depend on coupled geometric constraints, while geometry and assembly configuration can make identical connectivity produce different trajectories.The combinations of topology, geometry, and attachment patterns create a behaviour space that training examples only partially cover.Source: §1 · p. 2
  • Joint-level graphs leave implicit the subassemblies solved together and their dependencies, requiring learners to recover that organization for prediction.Making recurring constraint patterns explicit could reduce the data needed to learn structure-to-motion relationships.Source: §1 · p. 2
  • Assur groups are irreducible subassemblies solved from supplied boundary configurations, and their dependencies form a directed acyclic solve graph.The graph specifies how local constraint solutions combine to determine mechanism configuration and motion.Source: §1 · p. 2
  • The paper evaluates Assur-based representations at corpus scale for trajectory prediction, structural composition, and indexing.The authors describe this as the first large-scale evaluation of Assur-based representations for limited-data mechanism behaviour learning.Source: §1 · p. 2
  • At limited data, Assur-derived features reach comparable curve-prediction error with roughly half the training examples required by flattened adjacency-and-position inputs.The comparison uses the same multilayer perceptron for both representations.Source: §1 · p. 2

2 FROM PHYSICAL PARTS TO CONSTRAINT MODULES

The section frames mechanism learning as prediction of a normalized output trajectory from connected rigid links and joints. It motivates constraint modules as computational units that expose jointly solved internal joints and their boundary inputs.

2 FROM PHYSICAL PARTS TO CONSTRAINT MODULES

  • A mechanism’s output trajectory is determined by solving geometric constraints from an input, but identical connectivity can yield different trajectories under different geometry or assembly configurations.The experiment predicts the normalized, arc-length-resampled shape of the resulting curve.Source: §2 · p. 2
  • A dyad groups two bars that must be solved jointly to locate an unknown joint, with an assembly branch selecting between two circle-intersection solutions.Reusing the dyad means reusing a conditional constraint relationship; different boundary positions and lengths generally produce different motions.Source: §2 · p. 3
  • An Assur group is generically rigid and independent with its boundary joints pinned, and has no proper subassembly with those properties.The paper uses the Servatius et al. (2010) definition.Source: §2 · p. 3
  • Boundary joints are a module’s inputs, while its internal joints are unknowns solved together; irreducibility sets the granularity when it cannot be split into sequentially solvable units.Larger groups can contain several coupled internal joints.Source: §2 · p. 3
  • Dependencies between groups form a directed acyclic graph whose topological traversal gives a dependency-respecting solve order.The graph exposes constraint-computation organization while preserving mechanism geometry.Source: §2 · p. 3
  • The nominal mobility count assumes independent constraints, so special geometries or redundant constraints can change actual mobility.An Assur decomposition describes generic structure, whereas real assembly solutions and full-cycle continuation are geometric questions.Source: §2 · p. 3

3 A COMPOSITIONAL REPRESENTATION OF MECHANISMS

The representation converts source mechanisms into records of irreducible groups, interfaces, dependencies, and verification statuses while preserving recoverable source information. Source-specific proposal oracles feed a shared structural acceptance test; the resulting representation supports learning and structural operations.

3 A COMPOSITIONAL REPRESENTATION OF MECHANISMS

  • The record encodes each mechanism as irreducible groups with boundaries, owned links, tracers, assembly branches, and statuses, while ownership induces an acyclic solve graph and preserves recoverability.The shared acceptance test distinguishes pinned rigidity and minimality; the record keeps structural status separate from geometric validation.Source: §3 · p. 3, Figure 2 · p. 4

3.1 MODULES, INTERFACES, AND DEPENDENCIES

  • The shared peeling loop grows a solved set from ground and the driven joint, while each source’s oracle proposes the next links and tracers are held out then reattached to their owning group.The build encountered 153,124 link-incidence records with no module below the ground group.Source: §3.1 · p. 4
  • Joint-adjacency proposals yield classical dyads, whereas link-incidence proposals can reach n-ary bodies, prismatic pairs, and boundary arity above two.Figure 3 illustrates a ternary link fixing two joints in one module before a remaining dyad closes at D.Source: §3.1 · p. 4, Figure 3 · p. 5

3.2 STRUCTURAL TESTS AND GEOMETRIC INFORMATION

  • The pebble-game acceptance test requires pinned rigidity and minimality; mobility counts alone cannot establish both conditions.Minimality checking is exponential in internal-joint count and is recorded unchecked beyond twelve; no built module reaches that size, while modules touching prismatic pairs fall outside the bar-joint test.Source: §3.2 · p. 5

3.3 WHAT THE RECORD STORES

  • Each record stores the realized assembly branch and distinguishes passed, failed, or unevaluated statuses rather than treating structural acceptance as motion validation.The schema reports boundary arity rather than inferring classical Assur order without a one-to-one boundary-pair convention.Source: Figure 2 · p. 4, §3.3 · p. 5

3.4 USING THE REPRESENTATION FOR LEARNING

  • The representation’s learning experiment uses fixed-vector summaries of modules, interfaces, dependencies, branch, and geometry with the same three-layer MLP as the flattened adjacency-and-position baseline.This controls for architecture so the comparison isolates the input representation.Source: §3.4 · p. 6
  • The representation also supports attaching a dyad to compatible boundary joints for candidate construction and retrieving related structures through a coarse module-and-dependency fingerprint.These uses motivate composition and indexing experiments, while curve prediction remains the central learning test.Source: §3.4 · p. 6

3.3 WHAT THE RECORD STORES

  • The build contains 12,986,754 mechanism records, with indexing on 10,020,000 records and learning on 88,910 matched mechanism-curve pairs.Initial assembly and full-cycle motion remain undecided build-wide because decomposition alone does not resolve geometry.Source: §3.3 · p. 5, §3.4 · p. 6

4 WHEN CONSTRAINT MODULES SUPPORT PREDICTION

The theory characterizes when local module predictors can transfer across assemblies and how interface information and downstream sensitivity constrain that reuse. The results establish that local coverage and controlled propagation matter, while missing distinctions and geometric configurations can limit prediction.

4 WHEN CONSTRAINT MODULES SUPPORT PREDICTION

  • Local prediction error transfers across assemblies under the theorem’s domain and Lipschitz assumptions, with a bound combining familiar-input error, distribution displacement, residual variation, and downstream influence.The bound concerns composed predictors; experiments instead test representation benefit within a global MLP.Source: §4 · p. 6
  • Two unit-bar dyad chains with identical dependency graphs and local Jacobian singular values can have endpoint sensitivities of 1 and 2m, while tangencies can reduce regularity to exponent 2−m after m stages.These constructions show that interface geometry matters alongside modular structure.Source: §4 · p. 6
  • Exchanging which boundary port receives each dyad length can change output shape even after 128-point arc-length sampling, and continuous features need enough dimensions to retain independently observable variations.Assur component size and undirected graph width do not bound one another.Source: §4 · p. 6
  • Figure 4 presents mean symmetric Chamfer distance over ten stratified fold units and marks the data-axis gain and corrected-test significance sizes.Its caption describes solve-graph features as worth about a factor of two in training data.Source: Figure 4 · p. 7

5 EXPERIMENTAL EVALUATION

The experiments test whether explicit constraint structure improves trajectory learning and supports structural composition, indexing, and invariance. Assur-derived features improve sample efficiency in the evaluated low-data regime, while the advantage closes at larger budgets.

5 EXPERIMENTAL EVALUATION

  • The evaluation first tests the central hypothesis that making constraint structure explicit improves prediction from limited data, then examines other representation properties.The learning comparison uses 88,910 matched pairs, ten joint-count-stratified folds, and symmetric Chamfer distance on normalized, 128-point curves.Source: §5 · p. 7, §5.1 · p. 7

5.1 LEARNING OUTPUT TRAJECTORIES WITH FEWER EXAMPLES

  • A geometric-mean 1.8× sample-efficiency advantage holds from 50 through 10,000 training mechanisms, but adjacency is lower at 40,000 and 72,013 examples.The comparison uses matched mechanism-curve pairs and the same three-layer MLP for both representations.Source: §5.1 · p. 7

5.2 CONSTRUCTIVE SAMPLING IMPROVES VALIDITY WITHIN THE TESTED FAMILY

  • Composition has higher overall validity at every evaluated size, while a 10,000-sample coverage control rebuilds every valid adjacency-sampled mechanism.The matched adjacency sampler has rounded validity rates at most 0.001 in the empirical branch; composition reaches 100% coverage of valid adjacency samples.Source: §5.2 · pp. 7–8, Figure 5 · p. 8

5.3 GROUPING MECHANISMS BY COARSE CONSTRAINT STRUCTURE

  • Across 10,020,000 mechanisms, solve-graph fingerprints yield 908,186 distinct structures versus 1,173,078 bare joint-graph hashes, a 1.29× coarsening.At n = 5,000, only the condensation-DAG feature block clearly contributes; the branch block costs performance, while six others fall within fold-to-fold spread.Source: §5.3 · p. 8

5.4 RELABELLING JOINTS LEAVES THE SOLVE GRAPH IDENTICAL AND THE BASELINE UNRECOGNIZABLE

  • Across 6,000 relabeling trials, the fingerprint and Assur features match their original values exactly, while the adjacency-plus-position baseline is unchanged in 0.68% of trials.The N = 4 residue reflects identity permutations or graph automorphisms; canonical dyad ordering avoids index-based tie breaking.Source: §5.4 · p. 8

6 RELATED WORK

Related work spans mechanism datasets, validity-aware generation, decomposition and rigidity, and compositional learning. This paper distinguishes its structural abstraction from learned instance embeddings and measures its contribution using a fixed network architecture.

Mechanism corpora and retrieval

  • Mechanism corpora include simulated planar linkages and higher-arity or slider families; this work adds a structural abstraction above those datasets.LINKS and the Purwar Lab dataset are named as corpus sources.Source: §6 · pp. 8–9

Keeping the discrete part valid

  • Existing mechanism generators constrain topology through libraries or grammars, or generate freely and filter; this work decomposes given mechanisms into an acceptance test.The construction rule is applied to corpus records rather than only to graphs built by a generator.Source: §6 · p. 9

Decomposition and rigidity

  • Prior decomposition and rigidity work establishes structural questions about modules, while whether learners benefit from being given those modules remains open.The paper compares its structural baseline with Weisfeiler-Lehman hashing of the bare joint graph.Source: §6 · p. 9

Compositional structure and data efficiency

  • Assur decomposition abstracts mechanisms into irreducible modules, interfaces, and solve order, and the paper measures the resulting low-data benefit and conditions for reuse.The related work contrasts this structural abstraction with learned embeddings that abstract instances by similarity.Source: §6 · p. 9

7 CONCLUSION

The conclusion frames solve graphs as a representation that makes irreducible modules, interfaces, and local-solve dependencies explicit without changing the mechanism. The accompanying tutorial develops the constraint, rigidity, and dataset concepts behind that representation.

7 CONCLUSION

  • Solve graphs expose irreducible modules, typed interfaces, and dataflow order, making structural validity and solve order available before simulation while motion remains emergent.The conclusion connects this representation to a measurable low-data advantage and a more abstract structural search space.Source: §7 · p. 9

B.1 MECHANISMS AS CONSTRAINT SYSTEMS

  • A planar rigid body has three degrees of freedom, and each revolute joint removes two relative freedoms; the resulting mobility count is structural, not a geometry or full-cycle certificate.Under generic regularity and independence assumptions, F = 1 means one generalized coordinate locally drives the others.Source: §B.1 · p. 12

B.2 THE 4-BAR LINKAGE

  • For the 4-bar linkage, the driver determines one dyad boundary joint while the other is fixed; the dyad’s internal joint is solved from those boundary positions.The tutorial distinguishes the one-DOF mechanism as a whole from the pinned-rigid dyad sub-chain.Source: Figure 6 · p. 13, §B.3 · p. 13

B.3 GENERIC MOBILITY AND MINIMAL PINNED COMPONENTS

  • A zero-mobility count is necessary but not sufficient for generic isostaticity; the tutorial keeps counting, decomposition, geometric assembly, and full-cycle motion as separate checks.The count can hold even when constraints are dependent or a particular geometry fails to assemble.Source: §B.3 · p. 13

B.4 THE PEBBLE GAME

  • The pebble game inserts bars when four endpoint freedoms can be gathered, using path reversal to find available freedoms and detect redundant constraints.In the pinned test, exactly three free pebbles indicate pinned rigidity; the method is described as affordable on the modules considered.Source: §B.4 · pp. 13–14

B.5 ASSUR GROUPS: DEFINITION

  • An Assur group is a minimal pinned-isostatic component; contracting groups into nodes yields a solve graph whose traversals give dependency-respecting solve orders.A dyad is the simplest non-trivial group, with two links, one internal joint, and two external joints.Source: §B.5 · p. 14

B.6 A HIGHER-CLASS EXAMPLE

  • In a higher-class mechanism, the Assur view isolates three mutually coupled internal joints in G2, while the remaining groups factor into small local solves.The flat adjacency view does not identify the coupled joints or name the dyad.Source: §B.6 · p. 15, Figure 9 · p. 15

B.7 THE LINKS DATASET IN DETAIL

  • The LINKS source represents each mechanism by an adjacency matrix, initial joint positions, and grounded-joint flags; the driver is fixed by convention to the link between joints 0 and 1.The cited upstream release contains ten million mechanisms, while the published dataset reports one hundred million.Source: §B.7 · p. 16

B.8 THE PURWAR LAB DATASET IN DETAIL

The section details the source dataset and experiment summaries, then describes the learning-curve comparison, feature representation, and relabelling protocol. It also explains why geometric sampling can reduce compositional validity despite structural admissibility.

B.8 THE PURWAR LAB DATASET IN DETAIL

  • The build streams 2,966,754 mechanisms: 602,079 four-bar, 397,525 six-bar, and 1,967,150 eight-bar examples.The source dataset reports approximately three million single-DOF planar four-, six-, and eight-bar mechanisms with open and closed coupler curves.Source: §B.8 · p. 16
  • Each record includes a link-incidence matrix, joint-type labels, and one coupler curve; the grounded joints and driver pair come from the source topology table.No joint-coordinate array is released; the pose is decoded from the source filename and accepted only if the traced point maps onto the stored curve.Source: §B.8 · p. 16

C.1 COMPOSITIONAL VALIDITY

  • Extra stationary pivots can make sampled lengths fail to close a dyad, reducing empirical compositional validity while leaving structural admissibility intact.The decline is attributed to the geometry-drawing rule; the oracle records geometric validity separately from three structural conditions.Source: §C.1 · p. 16

C FULL BENCHMARK TABLES

  • Table 3 summarizes adjacency hashes and solve-graph fingerprints, while Tables 5 and 6 define the feature-ablation and relabelling-invariance summaries without supplying their cell values here.Table 5 covers feature ablations at train n = 5,000; Table 6 reports unchanged-trial fractions over 300 mechanisms and 20 permutations each.Source: Table 3 · p. 17, Table 5 · p. 18, Table 6 · p. 18

C.3 CURVE-REGRESSION LEARNING CURVE

  • At 40,000 examples, the −0.0004 difference lies within fold noise; at the full pool, adjacency is better by 0.0014, with all ten folds favoring it.The corrected paired interval at 40,000 examples is [−0.0009, +0.0001], with one-sided probability 0.068 for adjacency-better; the full-pool probability is 6.6 × 10−6.Source: §C.3 · p. 17, Table 4 · p. 17

C.4 THE DEFAULT FEATURE VECTOR

  • The default representation comprises eight pose-, scale-, and joint-number-independent blocks totaling 86 numbers, including group counts, solve-DAG structure, geometry, branches, and curve features.The structural-only sub-block contains 13 numbers from order counts and DAG-shape and gated DAG-branching features.Source: §C.4 · p. 17

C.5 ASSUR-FEATURE ABLATION

The ablation identifies the condensation-DAG block as the main contributor to the default feature vector. The surrounding analysis frames reusable composition in terms of module interfaces, local prediction error, and its propagation through a solve graph.

C.5 ASSUR-FEATURE ABLATION

  • The condensation-DAG block is the only clear contributor at +0.00219 ± 0.00057 Chamfer, while removing the branch block lowers error by 0.00209 ± 0.00061.The remaining six default blocks sit within fold noise; the structural sub-block alone scores 0.10375 versus 0.10576 for the full default vector.Source: §C.5 · p. 18

C.6 RELABELLING INVARIANCE

  • The relabelling test decomposes mechanisms before and after a random joint-index permutation and measures the fraction of trials with bitwise-unchanged descriptors.The motor pair and coupler joint are remapped by the same permutation.Source: §C.6 · p. 18

D CONDITIONS AND BOUNDARIES FOR PREDICTIVE COMPOSITION

  • A dyad’s reusable conditional rule depends on boundary positions, bar lengths, and assembly branch, and passes its solution to later modules.The accompanying figure illustrates that port correspondence and motion-transmission directions matter beyond local structure alone.Source: D CONDITIONS AND BOUNDARIES FOR… · p. 19, Figure 10 · p. 19

D.1 GENERALIZATION THROUGH MODULE INTERFACES

  • Composition separates local prediction error with correct inputs from additional error caused by imperfect parent outputs, with local coverage and solve-graph propagation governing the two sources.Hybrid inputs replace predicted parent outputs one at a time, attributing each incoming error to a dependency; the exact solver need not be defined on those hybrid inputs.Source: §D.1 · p. 19, §D.1.1 · p. 20

D.1.1 SPACES, DOMAINS, AND ERROR PROPAGATION

  • The propagation bound weights each module’s local error by downstream path influence, with contributions from multiple paths adding in the bound.The path weight includes the direct readout contribution; the framework allows module outputs to be paths, not only single configurations.Source: §D.1.1 · pp. 19–20

D.1.2 TRANSPORT FROM TRAINING INTERFACES TO NEW ASSEMBLIES

The transfer analysis relates module error on a new assembly’s interface distribution to error under a training reference distribution. It gives alternative distribution-shift conditions and accounts for physical reconstruction when normalized module charts are used.

D.1.2 TRANSPORT FROM TRAINING INTERFACES TO NEW ASSEMBLIES

  • Theorem 2 bounds local error under a new interface distribution using training-distribution error plus a Wasserstein shift penalty when the residual is Lipschitz.The residual is the population error of the learned module, not its empirical training error; Lipschitz module maps suffice for the stated residual condition.Source: §D.1.2 · p. 21
  • A density-ratio bound on the new interface distribution provides an alternative transfer condition that requires no residual-Lipschitz assumption.The condition is absolute continuity with respect to the training law and a finite upper bound on the density ratio.Source: §D.1.2 · p. 22
  • If the transfer-bound right-hand side is uniformly at most B across a graph family, the RMS error is at most B throughout that family.For random graphs with conditional bound B_G, the unconditional RMS error is at most (E_G B_G^2)^(1/2); graph count does not enter separately.Source: §D.1.2 · p. 22
  • For normalized dyad charts, a physical scale bound d_max converts normalized solver error into physical error, but edge gains must still bound the complete physical predictor.For C1 path norms, reconstruction derivatives must also be included; d_max alone is insufficient.Source: §D.1.2 · p. 22

D.1.3 UNIFORM TRAJECTORY ERROR AND NORMALIZED POINT CLOUDS

The section connects module-level error bounds to normalized, arc-length-resampled curve comparisons, then shows how interface directions and repeated tangencies constrain global sensitivity and regularity. These constructions demonstrate that identical local module properties need not yield identical whole-mechanism behavior.

D.1.3 UNIFORM TRAJECTORY ERROR AND NORMALIZED POINT CLOUDS

  • With curve scale floors, shared-phase samples admit a normalized squared Chamfer bound from trajectory error; independent arc-length resampling additionally requires a positive speed floor and derivative-error control.The stated arc-length comparison uses E_arc = E0 + 2(b/a)^2 E1, and the normalized squared Chamfer bound is E_arc^2/s0^2 when both sampled clouds have RMS scale at least s0.Source: §D.1.3 · p. 23

D.2 LOCAL REGULARITY DOES NOT BOUND GLOBAL CONFIGURATION SENSITIVITY

  • Two chains with the same graph, unit bar lengths, and local Jacobian singular values can have constant versus exponentially different endpoint sensitivity because their interface response directions differ.The theorem concerns the configuration map from driving angle to endpoint joint; its relation to a curve observation also depends on readout and sampling.Source: §D.2 · pp. 24–25, A chain with amplifying endpoint… · p. 26

A chain with amplifying endpoint sensitivity. Again set q∗

  • The amplifying-chain construction has an analytic assembly branch on an interval around its reference configuration for each finite depth, without requiring a uniform interval width as depth grows.The implicit-function argument establishes existence for each finite m; the width may shrink with m.Source: A chain with amplifying endpoint… · p. 26

D.3 REPEATED TANGENCIES REDUCE CONFIGURATION REGULARITY

  • Repeated tangencies produce endpoint response scaling with exponent 2−m, so the endpoint map is not Hölder continuous for any exponent β > 2−m at the singular pose.Across mechanisms with arbitrarily many dyads, no positive depth-independent Hölder exponent holds at all singular assemblies.Source: §D.3 · pp. 27–28
  • The paper distinguishes amplification on regular branches from loss of smoothness at singular limits, both arising with simple Assur modules.Thus module type alone does not determine the regularity available to a learner; the geometric regime also matters.Source: §D.3 · pp. 26–28

D.4 WHY BOUNDARY-PORT ASSIGNMENTS MATTER

Boundary-port assignments can distinguish mechanisms with otherwise matching module structure, creating an error floor when representations discard them. More generally, continuous features need enough coordinates to preserve observable variation, while lossless decompositions do not improve unrestricted learners’ achievable risks.

D.4 WHY BOUNDARY-PORT ASSIGNMENTS MATTER

  • Swapping which boundary port receives each dyad length changes the output curve, even after similarity, reflection, traversal, and 128-point sampling invariances are removed.If the representation merges the mechanisms, any shared prediction incurs nonzero error; complete link incidence distinguishes them.Source: §D.4 · pp. 28, 30
  • The port-assignment numerical floor is specific to chord normalization and Hausdorff loss, although the curve distinction holds under similarity-based normalization and traversal changes.The obstruction concerns representations discarding port assignments; complete link incidence preserves the distinction.Source: §D.4 · p. 29

D.5 THE DIMENSION REQUIRED TO RETAIN OBSERVABLE BEHAVIOUR

  • A continuous representation with fewer than q coordinates must incur worst-case error at least cr when outputs remain separated by 2cr across a q-dimensional family.A full-column-rank observable measurement supplies this separation locally, with lower bound σr/(2L).Source: §D.5 · pp. 30–31

D.6 LOSSLESS STRUCTURE EXPOSES COMPUTATION WITHOUT ADDING INFORMATION

  • Lossless, recoverable encodings and raw records have identical achievable prediction risks for unrestricted learners, every sample size, loss, and joint law.The representation comparison therefore tests whether explicit organization helps the fixed learner use data, not whether the encoding adds information.Source: §D.6 · p. 31

D.7 CONSTRAINT MODULES AND GRAPH WIDTH MEASURE DIFFERENT STRUCTURE

Assur component size and undirected treewidth capture different aspects of mechanism structure: neither bounds the other, even with two boundary joints. The census also shows complementary source coverage, with LINKS spanning larger mechanisms and Purwar Lab supplying structural variety.

D.7 CONSTRAINT MODULES AND GRAPH WIDTH MEASURE DIFFERENT STRUCTURE

  • A dyadic grid can have growing treewidth while each conditional solve remains small; conversely, a cycle with two pins can form one large Assur component despite bounded treewidth.The supplied feasible grid branch evaluates in O(r^2) scalar operations under the stated regularity conditions.Source: §D.7 · pp. 32–33
  • κ(Gr) = 1 and p(Gr) = 2 can coexist with tw(Gr) ≥ r, while κ(Hn) = n and p(Hn) = 2 coexist with tw(Hn) ≤ 4.Thus bounded component size does not imply bounded treewidth, nor do bounded treewidth and interface arity imply bounded component size.Source: §D.7 · p. 32

E CENSUS OF THE BUILT KNOWLEDGE BASE

E.2 MECHANISM-LEVEL STRUCTURE

  • LINKS spans 6–20 joints and solve-graph depth up to 14 but contains dyads alone; Purwar Lab reaches 11 joints and depth 4 while supplying higher interface arity, n-ary bodies, and prismatic pairs.The 153,124 PRPR records have a ground group and no module, and are marked as incomplete topology.Source: §E.2 · pp. 34–35

E.3 MODULES

The module census shows a stark contrast between the dyad-only LINKS population and Purwar Lab’s broader module structures. Purwar Lab’s higher-arity bodies and prismatic pairs also make typed link incidence consequential.

E.3 MODULES

  • 92,514,069 LINKS groups are all dyads, while a quarter of Purwar Lab modules have interface arity above two and its modules span 33 shape keys.Purwar Lab modules reach seven internal joints and six bodies; the classical dyad is only its fourth most common module at 10.3%.Source: §E.3 · pp. 35–36, Table 10 · p. 35
  • Table 12 covers the ten most frequent Purwar Lab shape keys, together accounting for 90.1% of its Assur groups.The shape key records module order, internal- and boundary-joint counts, and sorted body arities.Source: Table 12 · p. 36, §E.3 · p. 36

E.4 BODIES AND JOINTS

  • 50.1% of Purwar Lab bodies are ternary or quaternary, and every mechanism contains at least one; flattened pairwise adjacency would make each indistinguishable from a ring of bars.Prismatic pairs are 2.6% of joint instances but appear in 14.8% of Purwar Lab mechanisms.Source: §E.4 · p. 36, Table 13 · p. 36
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