12 Symmetry Group Quotient Pose Multimodality Calibration

Symmetry, Ambiguity & Uncertainty

Why Can One Observation Have
Several Correct Poses?

A symmetric object's pose is an equivalence class under a proper symmetry group, rather than one arbitrary matrix. Occlusion, projection, and sensor failures add alternatives beyond global object symmetry. Separate these sources before choosing losses, distributions, confidence, and robot actions.

Open the Equivalent Pose Lab Start with quotient spaces

01 · Problem Contract

Define equivalent poses before training or evaluation

Symmetry is not a constant derived solely from shape. Instances, modalities, task costs, and evaluation tolerances change equivalence relations; give each layer independent fields and versions.

01

Object

Metric meshes, part identities, proper groups, axes/offsets, instance differences

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02

Appearance

Whether texture, text, materials, spectrum, and illumination break geometric symmetry

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03

Observation

Modes supported by visible surfaces, occlusion, projection, noise, and modality validity

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04

Task

Pose changes with equal grasping, insertion, placement, or control costs

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05

Protocol

Losses, metrics, correctness thresholds, confidence, calibration splits

Scope of This Topic This topic defines equivalent poses, multimodal posteriors, and calibration contracts. Topic 13 details ADD(-S), VSD, MSSD, MSPD, BOP Average Recall, matching, and aggregation. Every result needs symmetry rules, thresholds, and implementation versions.

02 · Five Distinct Layers

Geometry, appearance, observations, tasks, and representations define different equivalences

Calling every pair of plausible poses symmetric is a common error. These five layers differ in scope, evidence, and handling.

01

Geometric symmetry

Question: Which proper rigid motions preserve surfaces, part identities, and scale?
Scope: Object-level and view-independent; usually defined by CAD/scans and object semantics
Example: C₄ for a textureless square prism, axial SO(2) for a cylinder, SO(3) for an ideal sphere
Handling: Store symmetry metadata; train, match, and evaluate in the quotient space

02

Appearance symmetry

Question: Which changes preserve the observable object after considering texture, text, color, and material?
Scope: Depends on sensor spectrum and illumination; can be smaller than the geometric group
Example: A one-sided label usually reduces a geometric C₄ box to identity in visible light
Handling: Specify geometry-only or appearance-aware symmetry; automatic mesh analysis must not overwrite semantic definitions

03

Visible / observational ambiguity

Question: Which poses explain the current image, occlusion, projection, and noise?
Scope: Sample-level and view-dependent; not necessarily global object symmetry
Example: A hidden mug handle permits a 180° flip; frontal silhouettes can yield two solutions
Handling: Use multimodal posteriors, tracking, active views, or abstention; keep this out of global object metadata

04

Task equivalence

Question: Which visually distinct poses have equal cost for grasping, insertion, placement, or avoidance?
Scope: Defined jointly by actions, end effectors, contact, and safety constraints
Example: Any disc yaw may suit suction grasping, while a keyed assembly needs unique yaw
Handling: Merge or separate modes in downstream costs without rewriting perceptual facts

05

Representation equivalence

Question: Does one rotation have multiple numerical parameterizations?
Scope: Representation-level; independent of object shape
Example: Unit quaternions q and −q always represent one SO(3) element
Handling: Handle antipodal equivalence in quaternion losses/interpolation separately from object symmetry

Reflections are not ordinary 6D rotational symmetries SO(3) contains proper rotations with det(R)=+1. Reflections, mirrored instances, and handedness changes are improper O(3) transforms; matching silhouettes do not justify right-multiplying them into poses. Use instance/category semantics, shape correspondences, or a separate mirror variable.

03 · Pose as an Equivalence Class

A symmetric object's state lies in SE(3) / 𝒮, beyond one canonical matrix

Even after fixing the object frame, multiple object-to-camera transforms may align every semantically labeled surface. They are representatives of one pose.

Proper object symmetry group

\mathcal S_O=\left\{S=(R_S,t_S)\in SE(3)\ \middle|\ S(\mathcal M)=\mathcal M\right\},\qquad R_S^\top R_S=I,\ \det R_S=+1

𝓜 can include parts, texture, or task semantics alongside unlabeled points. Choosing invariant attributes chooses the symmetry group.

Equivalent pose orbit

[T]=T\mathcal S_O=\left\{TS\mid S\in\mathcal S_O\right\},\qquad \mathcal P_O \cong SE(3)/\mathcal S_O

This site maps object points into the camera with ᶜTₒ. Right-multiplying S selects an equivalent object-internal representative before applying the external pose.

Distance between quotient poses

d_{\mathcal P}\!\left([T_1],[T_2]\right)=\min_{S_1,S_2\in\mathcal S_O}d_{SE(3)}(T_1S_1,T_2S_2)\ \equiv\ \min_{S\in\mathcal S_O}d_{SE(3)}(T_1,T_2S)

The right-hand equivalent form requires suitable invariance of the base distance. Rotation geodesics, surfaces, projections, and task costs answer different questions.

Representative

One matrix is an interface choice

A numeric label can select a canonical representative without making it truer than other orbit elements. Output the symmetry rule/object version with it.

Canonical section

Global canonical choices usually have boundaries

Mapping C₄ yaw to [−45°,45°) jumps at ±45°. Tiny input changes can demand labels separated by 90°.

Gauge

Continuous symmetric directions cannot be recovered without evidence

An ideal cylinder's axial yaw is gauge. Choosing yaw=0 is a convention; low variance without identifying cues expresses false certainty.

Double cover

q and −q describe parameterization, not the object orbit

Quaternions lie on S³ and map 2-to-1 to SO(3). Handle antipodal equivalence before adding object-specific symmetry.

04 · Symmetry Groups & Metadata

From Cₙ to continuous axes: store full rigid actions, beyond angles

Discrete, continuous, and approximate symmetries require different numerical handling. An off-origin symmetry axis necessarily couples rotation and translation.

Family	Structure	Pose orbit	Engineering notes
Trivial {I}	Asymmetric objects	One local representative	Occlusion, projection, or weak evidence can still create observational alternatives
Cₙ	n discrete rotations around one axis	n representatives	Gears, square prisms; appearance or tasks may reduce n
Dₙ proper part	n principal-axis rotations + n perpendicular half-turns	2n proper rotations	Do not insert planar reflection matrices directly into SO(3)
T / O / I	Regular tetrahedron, cube/octahedron, icosahedron/dodecahedron	12 / 24 / 60 proper rotations	Finite noncommutative groups, beyond one yaw period
SO(2)	Continuous rotation around a fixed object axis	One continuous orbit	Axial surfaces of revolution; rotation about the axis is gauge
SO(3)	Ideal spheres or fully isotropic tasks	Full rotation orbit	Rotation is unidentifiable, while position may remain meaningful
Approximate / partial symmetry	Transformed surfaces are close but not identical	Depends on tolerance and observation resolution	State tolerance, sensor, and task; do not claim an exact group

Rotation about an offset axis

x'=R(x-o)+o=Rx+(I-R)o,\qquad S=\begin{bmatrix}R&(I-R)o\\0&1\end{bmatrix}

o is any object-frame point on the axis. If the origin is off-axis, t=(I−R)o is essential.

Case	Recommended metadata	Required checks
Discrete symmetry centered at the object origin	Store full 4×4 rigid matrices S; equivalent camera poses are ᶜTₒS	Every matrix must be a proper rigid transform with det(R)=+1
Symmetry axis offset from the object origin	Rotation around a point o includes translation t=(I−R)o	R alone moves the surface incorrectly and corrupts evaluation distances
Continuous axial symmetry	Store object-frame axis and a point/offset; evaluate analytically or discretize per protocol	Step size is a numerical approximation tied to maximum surface-displacement tolerance
Texture or instance variation breaks geometric symmetry	Keep the geometric group and version appearance-aware subgroups / instance rules	Category CAD, physical stickers, and test images may imply different groups
Part and task semantics	Version equivalence rules with tasks/affordances rather than universal object truth	Gripper width, contact surfaces, and assembly direction change the task quotient space

Versioned Symmetry Contract · Example Schema

object_id: gear_07
asset_version: sha256(mesh + semantic_parts)
frame: object_model_v3
geometry_symmetry:
  discrete: [S_0, S_1, ...]     # complete 4×4 SE(3)
  continuous: [{axis_o, point_o, tolerance}]
appearance_subgroup:
  sensor_band: visible
  texture_version: print_v2
task_equivalence:
  task_id: parallel_jaw_pick_v4
  action_rule_version: 17
validation:
  surface_max_deviation_mm: 0.04
  det_rotation: +1
  reviewed_instances: [...]

05 · Observation-Conditioned Ambiguity

An asymmetric object can still be unidentifiable in one frame

Identifiability depends on the observation model and evidence. More similarly distributed training data cannot recover information hidden by occlusion, projection collapse, or sensor loss.

Observation equivalence set

\mathcal A(x;\epsilon)=\left\{T\in SE(3)\ \middle|\ D\!\left(h(T,\eta),x\right)\le\epsilon\ \text{for some nuisance }\eta\right\}

h is the imaging/sensor model; η includes lighting, occlusion, and noise. The global symmetry orbit remains in the ideal observational equivalence set; additional modes may exist only for the current x.

Source	Why alternatives arise	Diagnostic evidence	Handling
Self-occlusion / external occlusion	Hidden asymmetric parts let different poses render the same visible region	Visible masks, amodal support, intermode render residuals	Change views, retain modes across frames, output top-K
Projection and silhouettes	3D differences collapse along viewing rays; frontal, weak-perspective, or nearly planar structures yield alternatives	Multiview baseline, depth/normals, oblique silhouettes	Add oblique views, metric depth, or structural priors
Repeated parts / local patches	Local descriptors match multiple locations in periodic structures	Correspondence-vote mode counts, part identity	Group-aware matching, global verification, multimodel RANSAC
Weak texture / lighting changes	Blur, overexposure, shadows, or low resolution remove unique appearance cues	cue visibility, MTF / exposure, appearance ablation	Improve capture or explicitly use a geometric subgroup
Mirrors / transparency / depth holes	Mismatched observation models make intensity/depth unreliable surface evidence	sensor confidence, multi-view consistency, OOD score	Sensor fallback, polarization/touch/silhouettes, rejection
Cropping / truncation	Key parts leave the image; scale and center compensate for one another	Full bboxes, truncation, proposal-boundary stress tests	Widen ROIs, use full-image context, multiple frames, or active fallback
Category and instance variation	Canonical frames, proportions, or local structures differ across instances	instance metadata, shape posterior, scale residual	Joint shape–scale–pose distributions; label disagreement is not simply noise
Annotation / canonical conflicts	Annotators or datasets choose different orbit representatives	quotient-aware inter-annotator distance	Save equivalence sets/rule versions and marginalize during training

Global?

The whole object and all required attributes remain invariant

Yes → quotient pose; new views cannot break true symmetry.

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Visible?

Is the current asymmetric cue observed validly?

No → view-induced modes; inspect occlusion, truncation, and validity.

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Separating evidence?

Do expected intermode observation differences exceed noise?

No → retain the posterior; avoid forcing an average pose.

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Action?

Do modes still have different task costs?

Different → acquire / top-K / abstain; equal → action quotient.

06 · Supervision, Loss & Matching

Supervise equivalence sets instead of the annotator's arbitrary representative

Parameter regression, surface matching, correspondences, and probability densities must specify finite/continuous groups, visible ambiguity, and representation double cover.

Finite-group symmetry-aware rotation loss

\mathcal L_R(\hat R,R^\star)=\min_{S\in\mathcal S_R}\left\|\log\!\left((R^\star S)^\top\hat R\right)\right\|_2

This is rotational geodesic min-over-group. Offset translations in S also require translation coupling in full SE(3)/surface losses.

Marginal likelihood over equivalent truth

\mathcal L_{\mathrm{NLL}}=-\log p([T^\star]\mid x)=-\log\!\left(\sum_{S\in\mathcal S_O}p(T^\star S\mid x)\right)

Finite groups use sums; continuous groups require integration or quotient-space density. Measures and normalization are essential.

Parameter regression + min-over-group

Matrices, quaternions, continuous 6D rotations, or se(3) residuals

Minimize geodesic / parameter-consistent losses over finite symmetry representatives

Strength: Simple and compatible with existing regressors
Trap: Discontinuous canonical-cell boundaries; Euclidean parameter distance may differ from surface/task distance

Surface matching / ADD-S style

Transformed model point sets

Compare nearest surfaces or apply explicit symmetry transforms first

Strength: Directly reflects model geometry and scale
Trap: Nearest points may overaccept repeated local structure; density and invisible surfaces affect results

Symmetry-aware correspondence

2D–3D / 3D–3D correspondences, object coordinates, or keypoints

Treat equivalent surface points as one target set, or predict fragments/regions before composition

Strength: Avoid contradictory canonical labels upstream
Trap: Conflating global symmetry with view-induced ambiguity in correspondence sets

Likelihood marginalization

p(T|x), or discrete / continuous pose densities

Sum or integrate over equivalent ground truth rather than rewarding one representative

Strength: Preserves modes and probability mass for NLL and decisions
Trap: Coarse continuous-group discretization; unspecified base measure / normalization

Multiple-choice / best-of-M

M pose hypotheses + scores

Cover the ground-truth orbit with at least one hypothesis; train mode scores/diversity

Strength: Easy integration with PnP, tracking, and verification
Trap: Oracle best-of-M alone hides unusable scores, collapsed modes, and K-dependent cost

Task-aware expected risk

Action, grasp, or control outcomes

Expected downstream costs, worst-case modes, or chance constraints under the posterior

Strength: Aligns visual uncertainty with actual failure costs
Trap: Task equivalence does not replace general pose reporting; wrong costs can create false safety

A minimum can also hide failures Min-over-group handles known equivalent truth, not incorrect metadata, new occlusion modes, ranking, or collapse. Oracle-only best-of-K training can always contain a close candidate while consistently selecting the wrong one in deployment.

07 · Interactive Lab

Distinguish equivalent orbits, view-induced modes, and false confidence

The first experiment changes geometric/appearance/task equivalence. The second creates modes through occlusion of an asymmetric object. The third compares top-K, NLL, entropy, epistemic signals, and calibration.

ground-truth representative equivalent orbit estimate One physical state can have multiple transform representatives

Symmetry group

—

—Proper rotations only; mirrored shapes are not automatically equivalent 6D poses

Naïve angle error

|wrap(ŷ − y)|

—Single-representative error ignoring object equivalence

Quotient-pose error

min over symmetry orbit

——

Equivalence verdict

—

——

Canonicalization risk

—

—Canonical representatives are interface choices, not added observations

Quaternions q and −q do not describe object symmetry. They form the quaternion double cover of SO(3): every object and rotation has both numerical representatives. Object C₂, C₄, and continuous axial symmetries come from proper object groups. Handle these separately in losses, interpolation, deduplication, and annotation.

08 · Pose Distributions

Point estimates, local covariances, and multimodal posteriors have different capacities

Choose distributions/interfaces after distinguishing small unimodal errors, finite separated modes, and continuous symmetry orbits.

Local SE(3) perturbation contract

T=\operatorname{Exp}(\hat\xi)\bar T\ \text{(left)}\quad\text{or}\quad T=\bar T\operatorname{Exp}(\hat\xi)\ \text{(right)},\qquad \xi\sim\mathcal N(0,\Sigma_{6\times6})

Σ requires a tangent point, left/right convention, and frame; record meter/radian scaling. Distant modes need mixtures rather than one enlarged ellipsoid.

Output	Representation	Suitable use	Limits
Point + score	One transform and a potentially unnormalized score	Low latency and simple interfaces	Cannot express distant modes; scores are not inherently success probabilities
Top-K hypotheses	K transforms, weights, and equivalence deduplication rules	Suitable for PnP / templates / tracking / verification	Report K, coverage, ranking, and downstream costs
Local mean + 6×6 covariance	A local ellipsoid with specified left/right perturbations, frame, and tangent point	Unimodal small errors, filtering, optimization	Cannot span distant symmetry modes; scale rotation/translation units appropriately
Bingham on S³	antipodal-symmetric quaternion orientation distribution	Respects q/−q and models directional axes/anisotropy	One Bingham does not represent an arbitrary multimodal SE(3) posterior
Matrix Fisher on SO(3)	Exponential family defined directly on rotation matrices	No coordinate singularities; structured local uncertainty	A finite parameter family cannot represent every complex/continuous multimodal density
Mixture / particles	Multiple local distributions or weighted samples	Explicitly retains symmetry, occlusion, and association modes	Protocols must specify component alignment, deduplication, normalization, and compute budgets
Implicit density / flow	Queryable relative / normalized density for any R or T	Arbitrary SO(3) modes and higher-order structure	More complex sampling, normalization, training measures, and calibration

Aleatoric / data uncertainty

Noise or irreducible ambiguity under a given observation mechanism

Conditional spread for occluded, blurred, or weak-return samples under one model

Additional views/better sensors may reduce it; retraining alone may not

Epistemic / model uncertainty

Unknowns from missing training coverage, parameters, or model families

ensemble / posterior disagreement, density support, OOD stress test

Add relevant data/models/priors; one softmax entropy does not identify this alone

Symmetry multiplicity

Several numerical representatives of one physical pose

Object symmetry group and quotient orbit

Merge into equivalence classes instead of counting as task risk/error

Observation ambiguity

Current evidence supports physically distinct quotient poses

Multiview / modality likelihoods and mode separability

Propagate modes, sense actively, or reject; avoid Euclidean averaging

Label / convention uncertainty

Inconsistent frames, units, timing, or canonical choices

annotation provenance, round trip, inter-annotator quotient distance

Fix data contracts rather than teaching models to absorb the error

Downstream risk

Failure probability induced jointly by posterior and action cost

expected cost, chance constraint, collision / grasp simulation

Can disagree with pose entropy; compute for the specific task

09 · Confidence & Calibration

Define correctness before judging probability reliability

Pose scores, detector scores, render residuals, densities, and robot success probabilities are distinct. Bind calibration to symmetry-aware events, domains, and versions.

Reliability and expected calibration error

\operatorname{acc}(B_m)=\frac1{|B_m|}\sum_{i\in B_m}\mathbf1[E_i],\quad \operatorname{conf}(B_m)=\frac1{|B_m|}\sum_{i\in B_m}\hat p_i,\quad \operatorname{ECE}=\sum_m\frac{|B_m|}{n}\left|\operatorname{acc}(B_m)-\operatorname{conf}(B_m)\right|

Define Eᵢ, for example: at least one predicted pose satisfies MSSD<τ under object symmetry. Changing τ, K, metric, instance group, or task changes the event.

Tool	Measurement / procedure	Reporting and limits
Correctness event	Define a symmetry-aware threshold event, such as MSSD<τ, task success, or top-K orbit coverage	Different events need separate calibration; one confidence cannot support every claim
Reliability diagram / ECE	Bin confidence and compare empirical correctness; ECE is a weighted bin gap	Report bins, counts, and class/visibility strata; one ECE hides local miscalibration
NLL / proper scoring	Rewards probability assigned to the correct equivalence set and penalizes false confidence	Sum finite groups; continuous groups need a relative measure/tolerance or quotient density
Brier score	Compare probability with outcome for an explicit binary success event	Reveals overconfidence beyond accuracy, but still depends on the event definition
Coverage / set size	Does a prediction set / top-K contain equivalent ground truth, and how large is it?	Report coverage with K, angular/surface thresholds, and symmetry rules
Risk–coverage / selective pose	Reject low-confidence samples and plot retained coverage versus risk	Choose deployment thresholds by failure cost and capacity, beyond the most attractive operating point
Temperature scaling	Scale logits on validation data, usually preserving ranking	Can reduce global over/underconfidence, but cannot create missing modes or fix OOD/symmetry metadata
Shift / OOD audit	Retest calibration by instance, scene, occlusion, sensor, and time	In-distribution calibration does not guarantee deployment reliability; version it and trigger fallback

Every Deployed Confidence Field Should Carry

target eventmetric + thresholdsymmetry versionK / set rule calibration splitdomain / sensorcalibrator versionvalidity / OOD flags

10 · Temporal Inference & Robotics

Time resolves view ambiguity without creating globally erased information

Track in quotient space while retaining necessary mode identity. Robot decisions then merge, verify, or reject modes using contact, collision, reachability, and safety costs.

01

Orbit-aware association

Gate using the minimum distance from track predictions to equivalent representatives, avoiding false jumps for one physical pose.

02

Symmetry unwrapping

For continuous display/control, select the representative nearest the previous state while retaining quotient pose. Unwrapping chooses temporal gauge, not recovered truth.

03

Mode persistence

Under occlusion, preserve weighted modes rather than resetting to an average pose; merge/prune when identifying cues return.

04

Motion can break ambiguity

Motion supplies parallax/new surfaces to resolve view-induced modes, but cannot break true global geometric-plus-appearance symmetry.

05

Filter manifold contract

EKF / factor-graph tangent covariances need left/right perturbations, reference frames, rotation/translation scaling, and relinearization rules.

06

Data association uncertainty

Identity, pose modes, and occlusion are coupled; fixing the wrong identity before pose filtering produces confident wrong tracks.

Task	Visual facts	Task equivalence	Decision interface
Parallel-jaw grasping	The object may have C₄ symmetry	Jaw contacts are usually equivalent only under 180°; collisions/cables may break that equivalence	Map pose modes to grasp sets and merge action-equivalent modes by collision/force closure
Suction grasping a disc	Continuous unidentifiability of yaw about the normal	Central suction is approximately insensitive to yaw	Estimate position and surface normal, declare yaw gauge, and avoid fabricated angular precision
Insertion / assembly	Approximately axially symmetric housing	Notches, keys, or pins make final roll unique with tight tolerances	Task symmetry is smaller than visual geometry; inspect key features actively or search by touch
Placement / orientation	Multiple poses explain the current RGB-D	Centers of mass, supports, labels, and reachability induce different costs	Evaluate stable placement / reachability per mode; output expected risk and safe fallback
Visual servoing	Canonical representatives can jump across cells	Controllers need continuous errors; jumps can trigger large actions	Compute quotient-space errors and unwrap temporally; include velocity limits and mode switching in the state machine
Active sensing	Two modes differ only in handle visibility	Reduce collision / grasp risk within limited time	Choose reachable, collision-free views maximizing expected intermode observation differences, beyond visible-pixel count

Posterior-aware robot decision

a^\star=\arg\min_{a\in\mathcal A}\ \mathbb E_{[T]\sim p([T]\mid x)}\!\left[C(a,[T])\right]\qquad\text{subject to}\qquad \Pr\!\left(C_{\mathrm{unsafe}}(a,[T])>0\right)\le\delta

Perception expresses a posterior in object quotient space; action costs determine further mode equivalence. Safety-sensitive tasks can use chance constraints, worst-case modes, or active observations.

11 · Failure Modes & Release Gates

Do not disguise multiple solutions as one confident answer

These checks cover group validity, metadata, canonical discontinuities, distributions, confidence semantics, and leakage.

Failure modes	Symptoms	Detection	Fix
Treating q/−q as object symmetry	Artificial bimodal labels or quaternion-loss jumps for arbitrary objects	Check R(q)=R(−q), then audit object symmetry separately	Use \|qᵀq̂\| or rotation geodesics for the double cover
Encoding reflections as rotations	det(R)=−1 and flipped handedness; meshes may overlap but the transform is outside SO(3)	Assert RᵀR=I and det(R)=+1 for every symmetry matrix	Retain proper rotations; model mirrored-instance semantics separately
Missing axis offsets	The model drifts when rotated about an off-origin axis	Check S(M)=M on sampled surfaces and t=(I−R)o	Store full 4×4 transforms or axis+point
Excessive canonicalization	Labels/predictions jump 90° / 180° between representatives near boundaries	Perturb inputs around cell boundaries and plot output continuity	Marginalize/minimize over groups during training; expose orbit rules at the interface
Averaging modes	Arithmetic averages of valid rotations can give a wrong pose or nonrotation matrix	Check whether the mean lies in a high-probability mode / SO(3)	Output mixtures/top-K; define the decision meaning of any Fréchet mean
Incorrect symmetry groups	Models/evaluation ignore stickers, notches, or instance differences	Audit geometry, appearance, instance, and task layers separately	Version subgroups/evidence and manually review approximate symmetries
Substituting entropy for epistemic uncertainty	Symmetric samples are labeled OOD while incorrect sharp modes appear reliable	Combine symmetry, OOD, false-confidence, and weak-evidence stress sets	Model multiplicity, aleatoric/epistemic uncertainty, and calibration separately
Reporting oracle top-K alone	Ground truth is inside K, but actual ranking sends the robot a wrong pose	Report top-1, top-K, NLL, score calibration, and downstream selection together	Train comparable weights and validate the verifier
One covariance spanning multiple modes	Ellipsoids cross low-probability regions and updates converge to nonexistent intermediate poses	Plot mode separation and tangent validity radii	Use per-mode covariance with mixture weights / particles
Inconsistent calibration events	A rotation-threshold score is interpreted as grasp success probability	Register target event, threshold, split, and version for each score	Recalibrate for the task event or report only the original meaning
Coarse continuous-group discretization	Metrics/NLL change strongly with symmetry sample counts	Refine sampling and check maximum surface displacement and convergence	Minimize analytically or choose justified steps from diameter/threshold
Evaluation leakage	Tuning temperatures, rejection thresholds, or symmetry tolerances on test data	Audit calibration / validation / test provenance	Fix the protocol and fit on independent calibration splits

Release Gate · Before Outputting Pose and Confidence

Are geometric, appearance, instance, and task symmetries separated and versioned?
Are matrices rigid, proper, and surface-invariant, with complete axis offsets?
Is q/−q equivalence handled separately from object symmetry?
Are supervision, outputs, and tracking stress-tested around canonical boundaries?
Are additional view-conditioned modes preserved outside global object metadata?
Do continuous-group integration/discretization converge with steps tied to surface tolerance?
Are top-K results accompanied by K, deduplication, ranking, coverage, NLL, and downstream selection?
Does every covariance specify tangent point, left/right perturbation, frame, and units?
Is confidence tied to events, metrics, thresholds, symmetry, and calibration versions?
Are calibration, rejection thresholds, and task costs selected only on independent validation/calibration data?
Are symmetry, occlusion, false peaks, OOD, sensor failures, and label conflicts tested separately?
Does the robot check collision, contact, reachability, and fallback for all retained modes?

Primary Sources on Symmetric Pose, Probability, Uncertainty, and CalibrationExpand equivalence-class pose, EPOS, Implicit-PDF, Bingham, uncertainty, calibration, and BOP sources