Decoy and disclosure radii of invariant shape descriptors

DECOY AND DISCLOSURE RADII OF INVARIANT SHAPE DESCRIPTORS

arXiv:2610.08410v1 [cs.CV] 6 Oct 2026

T. SHASKA AND L. BESHAJ

Abstract. A recognizer that compares rotation-invariant descriptors sees a surface only up to the fiber of the descriptor. We measure this fiber by its radius in the orbit distance from the enrolled surface. A large radius admits decoys, that is, distant shapes that pass the matcher. A small radius discloses the enrolled shape to anyone who captures the stored value. For star-shaped surfaces truncated to spherical harmonics of degree at most L, with n coefficients, a descriptor of generic rank r has generic fibers of dimension n − 3 − r modulo rotations. The standard pool of band powers, even bispectra, and three invariants of the degree-three band therefore admits decoy families of dimension 5, 13, 20 at L = 4, 6, 8. Its rank first reaches n − 3 at L = 16, and a mirror decoy remains at every L. The odd bispectra remove the mirror decoy generically for L ≥ 4. Yet at fixed mean radius the same pool determines the enclosed volume exactly, and it does not determine whether a surface meets a clearance requirement. We certify two cases by exact and interval arithmetic. At L = 6 a decoy matches all 32 invariants to relative precision 2 · 10−18 at orbit distance at least 0.87 times the norm of the enrolled tuple. For the radar shape model of asteroid (101955) Bennu, the pool recovers the modeled volume, misses the handedness, and leaves the keep-out radius uncertain by more than 7 m.

1. Introduction Consider a recognizer that identifies three-dimensional objects from their geometry and must ignore the pose of the object. A standard way to do this is to replace the surface by a descriptor that is invariant under rotations, and to compare the descriptor with an enrolled library. The recognizer then sees only the descriptor. Two surfaces with the same descriptor are indistinguishable from that representation alone. The set of such surfaces is the fiber of the descriptor, and we measure it modulo rotations. We analyze this fiber from two adversarial perspectives: decoy construction and disclosure from a released descriptor. The first adversary builds decoys. A decoy is a surface that is not a rotated copy of an enrolled object but has the same descriptor, up to the tolerance of the recognizer. At zero tolerance a decoy lies in the exact fiber; at positive tolerance it lies in a thickened fiber. Large fibers therefore permit geometrically distinct accepted shapes. A fiber of positive dimension modulo rotations is a whole family of decoys; this ambiguity is intrinsic to the representation and exists independently of classifier training. Every classifier that acts on the descriptor is constant on its fibers, so no training removes them. The second adversary obtains the stored descriptor of an enrolled object. A captured descriptor is consistent with every surface in its fiber, so small fibers disclose the enrolled geometry. We measure both threats by one number, the radius of the fiber in the orbit distance, centered at the enrolled surface. For a fixed admissible set and tolerance, this radius, centered at the enrolled point, has a decoy interpretation and a disclosure interpretation. An observer who knows only the released value faces a different uncertainty, which is governed by the orbit diameter, or by the corresponding minimax radius, of the set of 2020 Mathematics Subject Classification. Primary 13A50, 68M25; Secondary 14P10, 68T45, 68U05. Key words and phrases. invariant theory, spherical harmonics, bispectrum, separating invariants, orbit spaces, shape recognition, decoys, disclosure, task certification. 1

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surfaces consistent with that value. The diameter lies between the radius and twice the radius (Prop. 3.3). This statement concerns the shape as a whole. It does not address the accuracy of a particular task, and it is not a statistical notion of privacy. The setting is that of [15, 16]. A star-shaped surface is given by its radial function on the sphere, truncated to spherical harmonics of degree at most L. Rotations act band by band on the coefficients, and descriptors are polynomial invariants of the coefficient tuple. A descriptor that separates orbits admits no exact decoys; conversely, an exact released value determines the enrolled orbit. On a compact semialgebraic admissible set, the radius of any descriptor at tolerance ε exceeds its exact radius by at most Cεα , with constants that may depend on the enrolled tuple. The stability inequality of [16] bounds the orbit distance by a Hölder power of the descriptor discrepancy. It was proved there to certify the descriptor. Read as a disclosure statement, it says that a released descriptor known to accuracy ε locates the enrolled surface within orbit distance Cεα . On the dipole and quadrupole bands the separation proof of [16] is constructive and yields an explicit reconstruction. A descriptor that does not separate admits decoys, and the constant-rank theorem measures how many. Let the descriptor have generic rank r on a coefficient space of dimension n. Modulo rotations, the part of its fiber in the free maximal-rank locus is a manifold of dimension n − 3 − r. If r = n − 3, that regular part is a finite union of orbits. If r < n − 3, the recognizer accepts a family of decoys of positive dimension near every generic enrolled surface. A descriptor with fewer than n − 3 components is always in the second case, whatever its components are. The standard spectral pool consists of the band powers of [10], the even bispectra of [9], and three invariants of the band-three sextic. It is the invariant block used in [15]. We compute its generic rank exactly. For 2 ≤ L ≤ 15 the rank equals the size of the pool, which is less than n − 3. At the cutoffs L = 4, 6, 8 the decoy families have dimension 5, 13, and 20. At L = 16 the rank reaches n − 3 for the first time, and the continuous ambiguity disappears on the free maximal-rank locus. A discrete ambiguity remains. The pool consists of true scalars, so a surface and its mirror image have the same pool. For every surface that is not a rotated copy of its mirror image, the mirror image is an exact decoy, at every L, and at L = 16 the finite ambiguity still contains the mirror pair. The odd bispectra are pseudoscalars. For L ≥ 4, adjoining them removes the mirror decoy for generic surfaces; the augmented pool first saturates at L = 10. The latter threshold also follows from the degree-three orbit-recovery theorem of [7]. A certified L = 6 benchmark complements the dimension count: an integer tuple and a rational tuple agree in all 32 invariants to relative precision 2 · 10−18 and have orbit distance at least 0.87 times the enrolled tuple norm. The rational comparison is exact, and the distance bounds are established by interval arithmetic through a validated covering criterion given in Sec. 7. On the quadrupole band, the decoy radius of the power can be computed exactly. The fiber of the quadrupole power ∥Q∥2 is a sphere. The Hoffman–Wielandt inequality of [8] turns the orbit distance on it into the distance between sorted spectra. The decoy radius√at Q is √ √ 2 2r sin(max{θ, π/3 − θ}/2), where r = ∥Q∥ and cos 3θ = 3 6 det Q/r3 . It lies between 6− r 2 and r, and adjoining the cubic invariant det Q makes it zero. With the power known exactly, the radius of (∥Q∥2 , det Q) at determinant tolerance ε grows linearly at matrices with distinct eigenvalues and like ε1/2 at matrices with a repeated eigenvalue. On the quadrupole band, the passage from the power spectrum to the bispectrum is the passage from decoys to disclosure. Whole-shape ambiguity does not settle task utility. We therefore introduce task intervals and prove an exact attribute-disclosure result: at a fixed constant radial term, the even pool determines enclosed volume. An explicit pair of positive radial surfaces has identical descriptors and volumes but opposite outcomes for a specified clearance requirement. The example can be checked by hand, so the conflict is exhibited for a stated task and is not merely inferred

DECOY AND DISCLOSURE RADII OF INVARIANT SHAPE DESCRIPTORS

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from a difference in shape. In Sec. 11 the same analysis is applied to the radar shape model of the near-Earth asteroid (101955) Bennu, with certified descriptor, distance, positivity, and task bounds for the rounded L = 6 model. The coefficient model assumes a fully observed closed surface that is star-shaped about a specified origin. A centroid convention is an additional constraint. It follows neither from positivity nor from omitting the constant harmonic, and Sec. 10.1 states it explicitly. Surfaces of higher genus and partial views are outside this model. The discussion of applications in Sec. 10.4 proposes a validation setting; it makes no claim about any deployed system. 2. Descriptors, fibers, and radii 3

Let R carry the Euclidean inner product ⟨x, y⟩ = x1 y1 + x2 y2 + x3 y3 and the norm |x| = ⟨x, x⟩1/2 . Let S 2 = {x ∈ R3 : |x| = 1} be the unit sphere, and let the special orthogonal group SO(3) be the group of real 3 × 3 matrices R with R⊤ R = I and det R = 1, where I is the identity matrix. Every R ∈ SO(3) maps S 2 onto S 2 . Let l ≥ 1 be an integer. A polynomial P ∈ R[x1 , x2 , x3 ] is harmonic if ∆P = 0, where ∆ = ∂12 + ∂22 + ∂32 . Let Hl be the real vector space of the functions S 2 → R that are restrictions of homogeneous harmonic polynomials of degree l. Its elements are the real spherical harmonics of degree l. The space Hl has dimension 2l + 1; see [16, Sec. 2.1] for further details. Let R ∈ SO(3) and f ∈ Hl . Define R · f : S 2 → R by (R · f )(p) = f (R−1 p),

p ∈ S2,

where R−1 p is the product of the matrix R−1 = R⊤ with the vector p. Let P be a homogeneous harmonic polynomial of degree l whose restriction to S 2 is f . Then R · f is the restriction of P ◦ R−1 . This polynomial is homogeneous of degree l, and it is harmonic because ∆ commutes with orthogonal changes of variables. Hence R · f ∈ Hl . The group SO(3) acts on Hl via SO(3) × Hl → Hl , (R, f ) 7→ R · f. Then I · f = f and (R1 R2 ) · f = R1 · (R2 · f ) for all R1 , R2 ∈ SO(3) and f ∈ Hl , and the action is linear in f . Hence the map ρl : SO(3) → GL(Hl ), defined by ρl (R)(f ) = R · f , is a group homomorphism, that is, a representation of SO(3) on Hl . By [16, Thm. 3.8], the complexification Hl ⊗ C is equivariantly isomorphic to the space of binary forms of degree 2l, which is an irreducible SL2 (C)-module. The group SO(3) acts on it through the covering SU(2) → SO(3), and SU(2) is Zariski dense in SL2 (C), so a subspace invariant under SO(3) is invariant under SL2 (C). Hence Hl ⊗ C is irreducible, and every SO(3)-equivariant linear map Hl → Hl is a real multiple of the identity. For l = ̸ l′ the representations on Hl and Hl′ are not isomorphic, because their dimensions 2l + 1 and 2l′ + 1 differ. Remark 2.1 (Julia reduction). Let l ≥ 1 and f ∈ Hl . Let Ff be the binary form of degree 2l that corresponds to f under the isomorphism above. Assume that Ff has 2l distinct roots. Let JFf be the Julia quadratic of Ff , and let ξ(JFf ) ∈ H3 be its Julia point, as defined in [4, Sec. 4.1] and [3]. Then ξ(JFf ) = j, where j = (0, 1). Indeed, put σ(z0 , z1 ) = (−z̄1 , z̄0 ). The Veronese map v of [16] satisfies v(σ(z)) = −v(z). Since f is real and homogeneous of degree l, we have Ff (σ(z)) = (−1)l Ff (z). Hence the set of roots of Ff is invariant under α 7→ −1/ᾱ. The Julia point is SL2 (C)-equivariant by [4, Thm. 3.4, Lem. 4.4]. The group SU(2) fixes j, so after a rotation we may assume that no root of Ff is at

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infinity. By [4, Thm. 4.1], the Julia point is determined by the roots, and conjugating the roots replaces (z, t) by (z̄, t). The map α 7→ −1/ᾱ extends to the isometry   −z t (z, t) 7→ , |z|2 + t2 |z|2 + t2 of H3 . Its only fixed point is j. The stabilizer of j in PSL2 (C) is PSU(2) ∼ = SO(3). Hence the reduction theory of binary forms in [3, 4] gives no normalization of pose on a single real band. Let H0 be the space of constant functions on S 2 . Let dω be the surface measure on S 2 and dµ = dω/(4π) the round probability measure. For real continuous functions f, g on S 2 put Z ⟨f, g⟩ = f g dµ. S2

The angle brackets thus denote the Euclidean product for vectors and this product for functions; the arguments determine which is meant. The measure µ is invariant under SO(3), so ⟨R · f, R · g⟩ = ⟨f, g⟩ for every R ∈ SO(3). The spaces H0 , H1 , H2 , . . . are pairwise orthogonal for this product; see [16, Sec. 2.1]. We call Hl the band of degree l. A closed surface Σ ⊂ R3 \ {0} is star-shaped about the origin if every ray {tp : t > 0} with p ∈ S 2 meets Σ in exactly one point. Let Σ be such a surface, and for p ∈ S 2 let ρ(p) > 0 be the unique number with ρ(p) p ∈ Σ. The function ρ : S 2 → R is the radial function of Σ, and Σ = {ρ(p) p : p ∈ S 2 }. We assume that ρ is continuous. Conversely, every positive continuous function ρ : S 2 → R is the radial function of the star-shaped surface {ρ(p) p : p ∈ S 2 }. This surface is the boundary of the solid {tp : p ∈ S 2 , 0 ≤ t ≤ ρ(p)}, which contains the segment from the origin to each of its points. Let L ≥ 1. For 0 ≤ l ≤ L let fl be the orthogonal projection of ρ onto Hl . The truncation of ρ at degree L is ρL = f0 + f1 + · · · + fL . The band H0 is omitted from the coefficient space below because SO(3) acts trivially on it. Put UL =

L M

Hl ,

n = dim UL =

l=1

L X (2l + 1) = L2 + 2L. l=1

We write c = (f1 , . . . , fL ) for a point of UL . The point c built from the projections of ρ is the coefficient tuple of Σ at degree L. The group SO(3) acts on UL diagonally by R · c = (R · f1 , . . . , R · fL ). Rotating the surface by R ∈ SO(3) replaces ρ by R · ρ, where (R · ρ)(p) = ρ(R−1 p). This action preserves each Hl and the product ⟨·, ·⟩, so it commutes with the orthogonal projections. Hence the rotation replaces each fl by R · fl , and the coefficient tuple c by R · c. When comparing physical radial solids, the constant f0 must be fixed or retained separately; throughout the volume and clearance statements it is a known constant ρ0 . Let c ∈ UL . The orbit of c is the set SO(3) · c = {R · c : R ∈ SO(3)}. Two points of UL lie in the same orbit if and only if one is obtained from the other by a rotation. The orbits partition UL , and the set of orbits is denoted UL /SO(3). An inner product on UL is SO(3)-invariant if ⟨R · c, R · c′ ⟩ = ⟨c, c′ ⟩ for all R ∈ SO(3) and c, c′ ∈ UL . The bands are irreducible, their complexifications are irreducible, and distinct bands are not isomorphic. By Schur’s lemma, every SO(3)-invariant inner product on UL therefore has the form L X ⟨c, c′ ⟩ = wl ⟨fl , fl′ ⟩, l=1

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where ⟨·, ·⟩ on Hl is the product defined above and wl > 0. We fix such an inner product and its norm ∥·∥. Let c, c′ ∈ UL . The function R 7→ ∥c − R · c′ ∥ is continuous on the compact group SO(3), so it attains its minimum. The orbit distance is δ(c, c′ ) =

min ∥c − R · c′ ∥. R∈SO(3)

It is a metric on UL /SO(3), and δ(c, c′ ) = 0 if and only if c′ ∈ SO(3) · c. It is continuous and semialgebraic on UL × UL . These statements are [16, Thm. 7.3], whose proof uses only the invariance of the norm and the compactness of SO(3). The space UL is a real vector space of dimension n. A map φ : UL → R is a polynomial if it is a polynomial in the coordinates of UL with respect to some basis. A change of basis is linear, so this property does not depend on the basis. A map Φ = (Φ1 , . . . , ΦN ) : UL → RN is a polynomial map if each component Φj is a polynomial. A map Φ : UL → RN is SO(3)invariant if Φ(R · c) = Φ(c) for every R ∈ SO(3) and every c ∈ UL . Equivalently, Φ is constant on every orbit, so it induces a map UL /SO(3) → RN . Definition 2.2. A descriptor is an SO(3)-invariant polynomial map Φ : UL → RN . Let c ∈ UL . The fiber of Φ through c is the set Φ−1 (Φ(c)) of all points x of UL with Φ(x) = Φ(c). The descriptor Φ separates orbits if every fiber of Φ is a single orbit. Every fiber contains the orbit of each of its points, so it is a union of orbits. The descriptor Φ separates orbits if and only if Φ(c) = Φ(c′ ) implies c′ ∈ SO(3) · c. We use the Euclidean norm |·| on RN . Definition 2.3. Let Φ : UL → RN be a descriptor, let K ⊂ UL be compact, let c ∈ K, and let ε ≥ 0. The radius of Φ at c with tolerance ε is ε rΦ (c; K) = max{δ(c, c′ ) : c′ ∈ K, |Φ(c′ ) − Φ(c)| ≤ ε}. 0 We write rΦ (c; K) = rΦ (c; K).

The maximum exists. The set over which it is taken is closed in K, hence compact, and it contains c. The function δ is continuous. The radius is nondecreasing in ε. By the properties of δ, rΦ (c; K) = 0 if and only if the fiber of Φ through c meets K only in SO(3) · c. The radius has two interpretations. In the first, an adversary builds decoys against a recognizer. The recognizer holds Φ(c) for the coefficient tuple c of an enrolled surface and accepts c′ when |Φ(c′ ) − Φ(c)| ≤ ε. ε A decoy for c is an accepted point c′ ∈ / SO(3) · c. Then rΦ (c; K) is the largest orbit distance from c of a decoy in K, and it is zero if there is none. We call it the decoy radius. In the second interpretation, an adversary has obtained Φ(c) and knows that c ∈ K. Every point of the fiber of Φ through c that lies in K is consistent with this information, and rΦ (c; K) is the largest orbit distance from c among them. A large radius serves the first adversary. A small radius serves the second. Fig. 1 shows two surfaces whose band powers coincide. Each is a decoy for the other under the descriptor P = (P1 , P2 ) of Sec. 5. For comparison across components of different degrees, fix a nonsingular matrix W ∈ RN ×N and use the discrepancy |W (Φ(c′ ) − Φ(c))|. This is equivalent to replacing Φ by the descriptor

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W Φ. It leaves exact fibers and Jacobian ranks unchanged, but in general it changes positivetolerance radii. The matrix must be fixed when the tolerance is specified. In particular, a componentwise relative error corresponds to W = diag(1/|Φ1 (c)|, . . . , 1/|ΦN (c)|), which depends on c and is undefined when a component vanishes; it requires a positive absolute floor ηj , that is, the entries 1/ max{|Φj (c)|, ηj }. Definition 2.4. Let Φ : UL → RN be a descriptor, let W ∈ RN ×N be nonsingular, let K ⊂ UL be compact, let y ∈ RN , and let ε ≥ 0. The observation-consistent set is ε FΦ,W (y; K) = {x ∈ K : |W (Φ(x) − y)| ≤ ε}.

When this set is nonempty, its orbit diameter is ε DΦ,W (y; K) =

max

ε x,x′ ∈FΦ,W (y;K)

δ(x, x′ ).

ε Let c ∈ K. If y = Φ(c), the largest distance from c in this set is rW Φ (c; K). If both

|W (y − Φ(c))| ≤ ε

and

ε x ∈ FΦ,W (y; K),

then |W (Φ(x) − Φ(c))| ≤ 2ε by the triangle inequality. An empty consistent set signals incompatibility among the observation, the error budget, and K; it is not a certificate about an enrolled shape. All information statements below are conditional on the specified K. A catalog or a geometric prior can make the part of a fiber that lies in K much smaller than the whole fiber. 3. Separating descriptors disclose Separating descriptors exist on every UL . Real forms of the generators of the invariant ring separate the SO(3)-orbits by [16, Thm. 4.1]. Recovery of an orbit from the values of invariants, estimated from noisy samples, is the subject of [1]. Let I be a separating descriptor and let K ⊂ UL be compact and semialgebraic. By [16, Thm. 7.3(iii)] there are constants CK > 0 and αK ∈ (0, 1] such that δ(c, c′ ) ≤ CK |I(c) − I(c′ )|αK ,

c, c′ ∈ K.

The proof is the Lojasiewicz inequality for the continuous semialgebraic functions δ(c, c′ ) and |I(c) − I(c′ )| on K × K, which have the same zero set. It applies verbatim on UL with any invariant inner product. In [16] the inequality certifies the descriptor. The same argument bounds the positive-tolerance radius of every descriptor by its exact radius plus a Hölder error term. Proposition 3.1. Let Φ : UL → RN be a descriptor, let K ⊂ UL be compact and semialgebraic, and let c ∈ K. Then there are C > 0 and α ∈ (0, 1] such that ε rΦ (c; K) ≤ rΦ (c; K) + Cεα

for every ε ≥ 0. Proof. Let F be the set of x ∈ K with Φ(x) = Φ(c). It is compact and semialgebraic, and it contains c. The functions d(x) = miny∈F ∥x − y∥ and h(x) = |Φ(x) − Φ(c)| are continuous and semialgebraic on K, with the same zero set F . The Lojasiewicz inequality in [5] gives C > 0 and α ∈ (0, 1] with d(x) ≤ Ch(x)α on K. Let c′ ∈ K with h(c′ ) ≤ ε, and choose x ∈ F with ∥c′ − x∥ = d(c′ ). Then δ(c, c′ ) ≤ δ(c, x) + δ(x, c′ ) ≤ rΦ (c; K) + ∥x − c′ ∥ ≤ rΦ (c; K) + Cεα . Taking the maximum over c′ proves the claim.

□

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The constants depend on c. For a separating descriptor rΦ (c; K) = 0, and the inequality of [16] makes the constants uniform on K. The next proposition gives the corresponding bounds under both interpretations. Proposition 3.2. Let I : UL → RN be a separating descriptor. Let K ⊂ UL be compact and semialgebraic, and let CK and αK be as above. Then the following hold. (1) For every c ∈ K and every ε ≥ 0, rIε (c; K) ≤ CK εαK . In particular rI (c; K) = 0. (2) Let c ∈ K, let ε ≥ 0, and let y ∈ RN satisfy |y − I(c)| ≤ ε. Then every c′ ∈ K with |I(c′ ) − y| ≤ ε satisfies δ(c, c′ ) ≤ CK (2ε)αK . Proof. Let c′ ∈ K with |I(c′ ) − I(c)| ≤ ε. The displayed inequality gives δ(c, c′ ) ≤ CK εαK . Taking the maximum over c′ proves (1). For ε = 0 the bound is zero. In (2) the triangle inequality gives |I(c′ ) − I(c)| ≤ 2ε, and (1) applies with 2ε. □ Part (1) says that a separating descriptor admits no decoys at tolerance zero, and only decoys within CK εαK at tolerance ε. Part (2) gives the disclosure interpretation: if a released value y is known with error ε, then the enrolled tuple lies within orbit distance CK (2ε)αK of every point of K whose descriptor lies within ε of y. That set of points contains c, and membership in it is decided by evaluating I. The differential of I annihilates the orbit directions, so its rank never exceeds n − 3 on the free locus. Where the Jacobian of I restricted to a slice transverse to the orbit has full rank n − 3 on the principal stratum, the exponent can be taken equal to one locally [16, Thm. 7.3(iv)]. On U2 the disclosure is explicit. Identify H1 with R3 and H2 with the traceless symmetric 3 × 3 matrices by f1 (x) = ⟨a, x⟩ and f2 (x) = x⊤ Qx. Then R · (a, Q) = (Ra, RQR⊤ ). By [16, Thm. 4.9] the invariants |a|2 ,

tr Q2 ,

det Q,

a⊤ Qa,

a⊤ Q2 a,

det(a, Qa, Q2 a)

separate the SO(3)-orbits in U2 . The proof in [16] is constructive and yields an explicit reconstruction. The eigenvalues of Q are the roots of t3 − 12 (tr Q2 ) t − det Q. Let a1 , a2 , a3 be the coordinates of a in a positively oriented orthonormal eigenbasis of Q, with eigenvalues λ1 , λ2 , λ3 . Then Y det(a, Qa, Q2 a) = a1 a2 a3 (λj − λi ). i<j

When the eigenvalues are distinct, the squares a2i solve the Vandermonde system i λki a2i = a⊤ Qk a for k = 0, 1, 2, and the displayed identity fixes the sign of a1 a2 a3 . Thus, on the locus P

where Q has distinct eigenvalues, the six numbers determine (a, Q) up to rotation by one cubic equation and one linear system. The cases of a repeated eigenvalue are treated separately in the proof of [16, Thm. 4.9], and separation holds on all of U2 . ε Proposition 3.3 (Reconstruction from a released value). Let F = FΦ,W (y; K) be nonempty ε and put D = DΦ,W (y; K). Define the deterministic worst-case reconstruction error by

R∗ (y; K) = min max δ(b c, x). b c∈K

x∈F

∗

Then D/2 ≤ R (y; K) ≤ D. For every c ∈ F, if rc = maxx∈F δ(c, x), then rc ≤ D ≤ 2rc . Proof. Compactness and continuity give all maxima. The function of b c being minimized is continuous, since its change is at most δ(b c, b c′ ) ≤ ∥b c−b c′ ∥, so the minimum also exists. For any x, x′ ∈ F and b c ∈ K, the triangle inequality gives δ(x, x′ ) ≤ δ(x, b c) + δ(b c, x′ ). Taking maxima proves the lower bound. Taking b c to be any point of F proves the upper bound and rc ≤ D. The same triangle inequality through c gives D ≤ 2rc . □

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This is an information bound. It provides neither a reconstruction algorithm nor a probabilistic privacy guarantee: it assumes no distribution on K and gives no bound on the computational effort needed to find a consistent point. A large shape diameter need not hide a particular attribute; Secs. 8 and 9 make that distinction explicit. 4. Fibers of descriptors Let L ≥ 2. For c ∈ UL let (a, Q) correspond to (f1 , f2 ) as in Sec. 3, and put ΩL = {c ∈ UL : det(a, Qa, Q2 a) ̸= 0}. Lemma 4.1. Let L ≥ 2. The set ΩL is open, dense, and SO(3)-invariant, and SO(3) acts freely on it. Proof. For R ∈ SO(3), det(Ra, RQR⊤ Ra, RQ2 R⊤ Ra) = det R · det(a, Qa, Q2 a) = det(a, Qa, Q2 a). So ΩL is invariant. The polynomial det(a, Qa, Q2 a) is not zero: for a = (1, 1, 1) and Q = diag(1, 0, −1) it equals −2. So ΩL is the complement of the zero set of a nonzero polynomial, and it is open and dense.Q Let c ∈ ΩL , and let λi and ai be as in Sec. 3. The identity det(a, Qa, Q2 a) = a1 a2 a3 i<j (λj − λi ) shows that the λi are distinct and that every ai is nonzero. Let R · c = c. Then RQR⊤ = Q, so R preserves each eigenline of Q. In the eigenbasis R is diagonal with entries ±1. Since Ra = a and every ai is nonzero, all entries equal 1. Hence R = I. □ Theorem 4.2. Let L ≥ 2 and let Φ : UL → RN be a descriptor. Let r be the maximum over UL of the rank of dΦ, and let U ′ = {c ∈ ΩL : rank dΦc = r}. Then the following hold. (1) The set U ′ is open, dense, and SO(3)-invariant, and r ≤ min{N, n − 3}. (2) For every c ∈ U ′ the set Mc = Φ−1 (Φ(c)) ∩ U ′ is an SO(3)-invariant embedded submanifold of UL of dimension n − r, and Mc /SO(3) is a smooth manifold of dimension n − 3 − r. (3) If r < n − 3, then rΦ (c; K) > 0 for every c ∈ U ′ and every compact neighborhood K of c. (4) If r = n − 3, then Mc is a finite union of SO(3)-orbits for every c ∈ U ′ . Proof. (1) If r = 0, then Φ is constant, the set where dΦ has rank r is UL , and U ′ = ΩL . Assume r > 0. The Jacobian of Φ has rank at most r everywhere. So the set where it has rank r is the complement of the common zero set of its r × r minors. These minors are polynomials, and one of them is not identically zero. Hence this set is a nonempty Zariski open subset of UL . By Lem. 4.1 so is ΩL . Their intersection U ′ is nonempty because UL is irreducible, so it is open and dense. Invariance of Φ gives dΦR·c ◦ R = dΦc . So the rank of dΦ is constant on orbits, and U ′ is invariant. Clearly r ≤ N . Let c ∈ U ′ . The action is free at c, so the orbit SO(3) · c is an embedded submanifold of dimension 3. The map Φ is constant on it, so its tangent space at c lies in the kernel of dΦc . Hence r ≤ n − 3. (2) The map Φ has constant rank r on the open set U ′ . By the constant-rank level set theorem in [11], Mc is a properly embedded submanifold of U ′ of dimension n − r. It is invariant because Φ and U ′ are. The compact group SO(3) acts smoothly and freely on Mc . By the quotient manifold theorem in [11], Mc /SO(3) is a smooth manifold of dimension n − r − 3. (3) Let r < n − 3, and let K be a compact neighborhood of c. Choose η > 0 such that the closed ball of radius η about c lies in K. The intersection O of Mc with the open ball of radius η about c is a nonempty open subset of Mc . It has dimension n − r > 3. The orbit SO(3) · c is a submanifold of dimension 3, so it has measure zero in Mc and does not contain O. Choose c′ ∈ O outside SO(3) · c. Then c′ ∈ K, Φ(c′ ) = Φ(c), and δ(c, c′ ) > 0. Hence rΦ (c; K) ≥ δ(c, c′ ) > 0.

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(4) Let r = n − 3. Then Mc is a manifold of dimension 3. Every orbit in Mc is a compact embedded submanifold of dimension 3, so it is open and closed in Mc . Since SO(3) is connected, the orbits in Mc are its connected components. The set U ′ is semialgebraic: it is the intersection of ΩL with the set where the sum of the squares of the r × r minors is nonzero. So Mc is semialgebraic, and it has finitely many connected components; see [5]. Hence Mc is a finite union of orbits. □ We call U ′ the free maximal-rank locus of Φ. For c ∈ U ′ we call Mc the regular fiber of Φ through c. Corollary 4.3. Let L ≥ 2 and let Φ : UL → RN be a descriptor with N < n − 3. Then Φ does not separate orbits. Moreover, there is an open dense set U ′ ⊂ UL such that rΦ (c; K) > 0 for every c ∈ U ′ and every compact neighborhood K of c. Proof. By Thm. 4.2, r ≤ N < n − 3. Part (3) of Thm. 4.2 gives the second claim. It also gives a point of a fiber outside the orbit of c, which proves the first claim. □ Only the number of components of Φ enters the corollary; the components themselves play no role. A recognizer built on fewer than n − 3 invariants admits decoys near every generic enrolled surface, at every tolerance. Corollary 4.4 (Combined releases). Let L ≥ 2, let Φ1 , . . . , Φm be descriptors on UL , and let Ψ = (Φ1 , . . . , Φm ). Then m \ Φ−1 Ψ−1 (Ψ(c)) = j (Φj (c)). j=1

On the open dense free maximal-rank locus for Ψ, the regular fiber modulo rotations has dimension n − 3 − rΨ , where   d(Φ1 )c rΨ = max rank  ...  . c∈UL d(Φm )c For separately specified observation error budgets, the jointly consistent set is the intersection of the individual consistent sets, and when nonempty, its diameter cannot exceed any individual diameter. Proof. Equality of the concatenated values is equivalent to equality of each component value, and the displayed matrix is dΨc . Apply Thm. 4.2. The last assertion follows from set inclusion and from the monotonicity of the maximum that defines the diameter. □ Thus separate releases about the same tuple must be audited jointly. Their ranks need not add, because their differentials can be dependent. The conclusion applies to releases about one underlying tuple. It does not apply to measurements of unrelated objects. 5. Band powers 2

The band powers Pl (c) = ∥fl ∥ form the descriptor P = (P1 , . . . , PL ) of [10]. The fiber of P through c is a product of spheres. It contains (R1 · f1 , . . . , RL · fL ) for all R1 , . . . , RL ∈ SO(3). The gradient of Pl is a nonzero multiple of fl , so P has rank L wherever every fl is nonzero. By Thm. 4.2, for L ≥ 2 the fiber of P through a generic point is, modulo rotations, a manifold of dimension L2 + L − 3. On H2 we use the matrix Q of Sec. 3 and the Frobenius norm ∥Q∥. The invariant inner product on H2 is unique up to a positive factor. Any other choice multiplies all radii on H2 by one constant. In the next two statements the definitions of Sec. 2 are applied with H2 in place

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of UL ; the orbit distance on H2 is the restriction of the orbit distance on U2 to the pairs with a = a′ = 0. Proposition 5.1. Let r > 0 and let Q ∈ H2 with ∥Q∥ = r. Let θ ∈ [0, π/3] satisfy cos 3θ = √ 3 6 det Q/r3 . Let K ⊂ H2 be compact and contain the sphere {Q′ : ∥Q′ ∥ = r}. Then rP2 (Q; K) = 2r sin √

max{θ, π/3 − θ} . 2

√

2 In particular 6− r ≤ rP2 (Q; K) ≤ r. The lower bound is attained if and only if det Q = 0. 2 The upper bound is attained if and only if Q has a repeated eigenvalue.

Proof. For φ ∈ R put µ(φ) = r The identity φ′ ). Hence

p
2/3 cos φ, cos(2π/3 − φ), cos(2π/3 + φ) .

P2

3 ′ 2 k=0 cos(x + 2πk/3) cos(y + 2πk/3) = 2 cos(x − y) gives ⟨µ(φ), µ(φ )⟩ = r cos(φ −

|φ − φ′ | , |φ − φ′ | ≤ π. 2 So µ traces the circle of radius r in the plane of vectors with coordinate sum zero, at unit angular speed. The coordinates of µ(φ) are in decreasing order exactly on the arc φ ∈ [0, π/3]. So the spectra, in decreasing order, of the matrices Q′ ∈ H2 with ∥Q′ ∥ = r are exactly the vectors µ(φ) with φ ∈ [0, π/3]. Let λ(A) denote the spectrum of a symmetric matrix A in decreasing order. By the Hoffman– Wielandt inequality in [8] and the rearrangement inequality, ∥Q − U Q′ U ⊤ ∥ ≥ ∥λ(Q) − λ(Q′ )∥ for every U ∈ O(3). Equality holds when U carries an orthonormal eigenbasis of Q′ to one of Q with the eigenvalues in the same order. Replacing U by −U does not change U Q′ U ⊤ , so the minimum over SO(3) equals the minimum over O(3). Hence δ(Q, Q′ ) = ∥λ(Q) − λ(Q′ )∥. Q2 Write λ(Q) = µ(θ′ ) with θ′ ∈ [0, π/3]. The identity k=0 cos(θ′ + 2πk/3) = 41 cos 3θ′ gives ∥µ(φ) − µ(φ′ )∥ = 2r sin

det Q = r3 ′

 2 3/2 cos 3θ′ 3

′

4

=

r3 cos 3θ′ √ . 3 6

′

Since 3θ ∈ [0, π], this determines θ , and θ = θ. The fiber of P2 through Q is the sphere {∥Q′ ∥ = r}, which lies in K. Therefore rP2 (Q; K) =

max 2r sin φ∈[0,π/3]

|θ − φ| max{θ, π/3 − θ} = 2r sin , 2 2

because sin is increasing on [0, π/6]. The quantity max{θ, π/3 − θ} lies in [π/6, π/3]. It equals π/6 exactly when θ = π/6, that is, when det Q = 0. It equals π/3 exactly when θ ∈ {0, π/3}, that √ √ 6− 2 is, when two eigenvalues of Q coincide. The values 2r sin(π/12) = r and 2r sin(π/6) = r 2 complete the proof. □ Adjoining det Q to P2 gives a separating descriptor on H2 . The numbers tr Q2 and det Q determine the spectrum of Q, and symmetric matrices with the same spectrum are conjugate by a rotation. So the radius of (P2 , det Q) vanishes at every point. The invariant cubic forms on H2 are the multiples of det Q. So det Q is, up to a nonzero factor, the bispectrum of the quadrupole with itself in the sense of [9]. At positive tolerance the radius of the separating pair is again explicit. Knowledge of the power is side information, recorded by the set K.

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Corollary 5.2. Let r > 0, let Sr = {Q′ ∈ H2 : √ ∥Q′ ∥ = r}, and let I = (P2 , det). Let Q ∈ Sr , let θ be as in Prop. 5.1, let ε ≥ 0, and put η = 3 6 ε/r3 and
φ− = 31 arccos min{1, cos 3θ + η} ,
φ+ = 13 arccos max{−1, cos 3θ − η} . Then

max{θ − φ− , φ+ − θ} . 2 √ r 2η rη 2 ε As ε → 0, rIε (Q; Sr ) = 3 sin + O(η 3/2 ) if 3θ + O(η ) if 0 < θ < π/3, and rI (Q; Sr ) = 3 θ ∈ {0, π/3}. rIε (Q; Sr ) = 2r sin

as in the proof of Prop. 5.1. The Proof. For Q′ ∈ Sr write λ(Q′ ) = µ(φ) with φ ∈ [0, π/3], √ power is constant on Sr , and det Q′ = r3 cos 3φ/(3 6). So Q′ is accepted if and only if |cos 3φ − cos 3θ| ≤ η. Since cos 3φ decreases on [0, π/3], this holds if and only if φ ∈ [φ− , φ+ ]. By the proof of Prop. 5.1, δ(Q, Q′ ) = 2r sin(|θ − φ|/2). Its maximum over [φ− , φ+ ] is the stated value. For 0 < θ < π/3 the derivative of arccos at cos 3θ is −1/ sin 3θ, which gives φ± = √ θ ± η/(3 sin 3θ) + O(η 2 ). For θ = 0 we have φ− = 0 and φ+ = 13 arccos(1 − η) = 13 2η + O(η 3/2 ). The case θ = π/3 is symmetric. □ The maximum over the fixed-power sphere is also explicit:  
1 arccos 1 − min{η, 2} . (1) max rIε (Q; Sr ) = 2r sin Q∈Sr 6 To see this, for 0 ≤ a ≤ b ≤ π the identity cos a − cos b = 2 sin((a + b)/2) sin((b − a)/2) gives cos a − cos b ≥ 1 − cos(b − a). Apply this with a, b the ordered angles 3θ, 3φ; equality is attained when one angle is an endpoint. Thus the uniform exponent 1/2 is attained and cannot be increased. At a fixed spectrum with 0 < θ < π/3 the radius is linear to first order. These statements assume exact knowledge of the power and do not describe joint noise in both invariants. Proposition 5.3. Let L = 2 and use on U2 the invariant norm ∥(a, Q)∥2 = |a|2 + ∥Q∥2 . Let r1 , r2 > 0, let F = {(a, Q) : |a| = r1 , ∥Q∥ = r2 } be the corresponding fiber of P = (P1 , P2 ), and put   r2 c± = r1 e1 , ± √ diag(2, −1, −1) . 6 Then c± ∈ F and  4 2r2 + r2 − r1 , r2 ≤ 3r2 , 1 2 1 2 3r22 δ(c+ , c− )2 =  2 2 4r2 , r1 ≥ 3r22 . Consequently rP (c; K) ≥ 12 δ(c+ , c− ) for every c ∈ F and every compact K ⊃ F . √ Proof. Put s = r2 / 6 and E = 3e1 e⊤ 1 − I = diag(2, −1, −1), so c± = (r1 e1 , ±sE). Both points lie in F because ∥E∥2 = 6. Let R ∈ SO(3), put u = Re1 , and put t = ⟨u, e1 ⟩. Then ∥r1 e1 − R(r1 e1 )∥2 = 2r12 (1 − t),
⊤ sE − R(−sE)R⊤ = s 3e1 e⊤ 1 + 3uu − 2I . ⊤ The symmetric matrix e1 e⊤ 1 + uu has eigenvalues 1 + t, 1 − t, and 0. So the second matrix has eigenvalues s(1 + 3t), s(1 − 3t), and −2s, and its squared norm is s2 (6 + 18t2 ) = r22 (1 + 3t2 ). Hence ∥c+ − R · c− ∥2 = g(t), g(t) = 2r12 (1 − t) + r22 (1 + 3t2 ).

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Every t ∈ [−1, 1] occurs. The function g is convex, with minimum at t0 = r12 /(3r22 ) > 0. If t0 ≤ 1, then δ(c+ , c− )2 = g(t0 ) = 2r12 + r22 − r14 /(3r22 ). If t0 ≥ 1, then g is decreasing on [−1, 1], and δ(c+ , c− )2 = g(1) = 4r22 . For c ∈ F the triangle inequality gives max{δ(c, c+ ), δ(c, c− )} ≥ 1 □ 2 δ(c+ , c− ), and both c± lie in F ⊂ K. √ √ For the norm w1 |a|2 + w2 ∥Q∥2 , replace r1 and r2 by w1 r1 and w2 r2 . The map (a, Q) 7→ √ ( w1 a, w2 Q) is an equivariant isometry onto U2 with the norm of Prop. 5.3, and it maps fibers of P to fibers of P . √

c¡

c+

Figure 1. The surfaces c+ (left) and c− (right) of Prop. 5.3, drawn as radial surfaces ρ0 + f1 + f2 with ρ0 = 1, r1 = 0.25, and r2 = 0.35, where f1 (p) = ⟨a, p⟩ and f2 (p) = p⊤ Qp. Their band powers coincide. Their orbit distance is δ(c+ , c− ) ≈ 0.487.

6. The spectral pool For binary forms F and G of degrees m and m′ in (z0 , z1 ), the transvectant of order k is   k ∂kF ∂kG (m − k)! (m′ − k)! X i k (F, G)k = (−1) , i ∂z0k−i ∂z1i ∂z0i ∂z1k−i m! m′ ! i=0 as in [16, Eq. (18)]. By [16, Thm. 3.8], restriction to the null cone gives an SL2 (C)-equivariant linear isomorphism Ψl from Hl ⊗ C onto the binary forms of degree 2l. Write Fl = Ψl (fl ) and A(L) = {(l1 , l2 , l3 ) ∈ Z3 : 1 ≤ l1 ≤ l2 ≤ l3 ≤ L, l3 ≤ l1 + l2 , l1 + l2 + l3 even}, A− (L) = {(l1 , l2 , l3 ) ∈ Z3 : 1 ≤ l1 < l2 < l3 ≤ L, l3 ≤ l1 + l2 , l1 + l2 + l3 odd}. For L ≥ 2 the pool ΦL has the following components. (1) The powers Pl = (Fl , Fl )2l for 1 ≤ l ≤ L.
(2) The bispectra Bl1 l2 l3 = (Fl1 , Fl2 )l1 +l2 −l3 , Fl3 2l3 for (l1 , l2 , l3 ) ∈ A(L). (3) For L ≥ 3, the invariants J4 , J6 , J10 of the sextic F3 . They are obtained from its Igusa–Clebsch invariants by [16, Eqs. (20)–(21)], J2 = I2 /8, J4 = (4J22 − I4 )/96, J6 = (8J23 − 160J2 J4 − I6 )/576, and J10 = I10 /4096.

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The number of components is N (2) = 4 and N (L) = L + |A(L)| + 3 for L ≥ 3. The odd bispectra Bl1 l2 l3 with (l1 , l2 , l3 ) ∈ A− (L) are given by the same formula. In the symmetric 3j normalization, an invariant trilinear contraction changes by (−1)l1 +l2 +l3 under interchange of two factors; another nonzero normalization gives the same vanishing criterion. Thus an odd bispectrum with two equal bands vanishes when evaluated on the same band vector twice; equivalently, after reordering the contraction, (F, F )k = 0 for odd k. The pool Φ+ L consists of ΦL and the odd bispectra divided by i, and it has N + (L) = N (L) + |A− (L)| components. We + call ΦL the spectral pool, or the even pool when it is contrasted with Φ+ L , and we call ΦL the augmented pool. The powers are the power spectrum of [10] and the Bl1 l2 l3 are bispectra in the sense of [9]. The pool is the radial invariant block of [15, Sec. 5.2]. There the power of band l is ∥fl ∥2 . Its complexification is a nonzero multiple of Pl , because an invariant quadratic form on the irreducible module of binary forms of degree 2l is unique upP to a scalar. Let dl be the degree of a component in Fl . For every component of ΦL the sum l l dl is even, and for every odd bispectrum it is odd. So the components of ΦL are real on real harmonics and the odd bispectra are purely imaginary by [16, Lem. 3.6(iii)]. Hence ΦL and Φ+ L are descriptors on UL . The block of [15] does not contain the odd bispectra. L 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

n n − 3 N (L) r(L) n − 3 − r(L) N + (L) r+ (L) n − 3 − r+ (L) 8 15 24 35 48 63 80 99 120 143 168 195 224 255 288

5 12 21 32 45 60 77 96 117 140 165 192 221 252 285

4 10 16 22 32 42 57 72 93 114 142 170 206 242 287

4 10 16 22 32 42 57 72 93 114 142 170 206 242 285

1 2 5 10 13 18 20 24 24 26 23 22 15 10 0

4 10 17 24 37 50 71 92 123 154 197 240 297 354 427

4 10 17 24 37 50 71 92 117 140 165 192 221 252 285

1 2 4 8 8 10 6 4 0 0 0 0 0 0 0

Table 1. Sizes, generic ranks, and generic fiber dimensions of the pools ΦL and Φ+ L.

Theorem 6.1. Let 2 ≤ L ≤ 16, and let r(L) and r+ (L) be the generic ranks of ΦL and Φ+ L. Then r(L) = N (L) < n − 3 for L ≤ 15, and r(16) = n − 3. Moreover r+ (L) = N + (L) < n − 3 ′ for L ≤ 9, and r+ (L) = n − 3 for 10 ≤ L ≤ 16. Let Φ be ΦL or Φ+ L , and let U and Mc be as in Thm. 4.2. Then the following hold. (1) If the generic rank of Φ is less than n − 3 and c ∈ U ′ , then Mc /SO(3) is a smooth manifold of positive dimension, listed in Tab. 1, and rΦ (c; K) > 0 for every compact neighborhood K of c. (2) If the generic rank of Φ equals n−3 and c ∈ U ′ , then Mc is a finite union of SO(3)-orbits. Proof. By Thm. 4.2, the generic rank is at most the number of components and at most n − 3. The counts in Tab. 1 give N (L) < n − 3 for L ≤ 15 and N (16) > n − 3, and N + (L) < n − 3 for L ≤ 9 and N + (L) > n − 3 for L ≥ 10. It remains to bound the ranks from below.

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The coefficients of the forms Fl are complex linear coordinates on UL ⊗C. In these coordinates the components of ΦL , and the odd bispectra up to the constant i, are polynomials with rational coefficients, and they define the complexifications. A minor of the Jacobian that vanishes on UL vanishes on UL ⊗ C, because a polynomial that vanishes on a real form of a complex vector space is zero. So each generic rank equals that of the complexification. It is at least the rank of the Jacobian at any point of UL ⊗ C. Put i = (F3 , F3 )4 and ∆ = (i, i)2 . In this paragraph i and ∆ denote these covariants, and not the imaginary unit and the Laplacian. In the rows of the Jacobian we replace J4 , J6 , J10 by the invariants (i, i)4 , (i, ∆)4 , and the discriminant of F3 , of degrees 4, 6, 10. Together with P3 , these generate the same algebra as P3 , J4 , J6 , J10 . Indeed, the Clebsch invariants (F3 , F3 )6 , (i, i)4 , (i, ∆)4 and the Clebsch invariant of degree ten generate the algebra of invariants of even degree of the sextic. The invariants I2 , I4 , I6 , I10 are polynomials in them, and I10 is the discriminant up to a nonzero factor, with a nonzero coefficient on the degree-ten generator. Each set is therefore a polynomial function of the other. By the chain rule, each Jacobian row space is contained in the other at every point; hence the two row spaces coincide. We evaluate the Jacobian at a point whose coordinates in the monomial bases are integers in [−9, 9]. The coordinates are produced, band by band and coefficient by coefficient, by the linear congruential generator x 7→ (1664525 x + 1013904223) mod 232 , started at x = 1, with coordinate (⌊x/216 ⌋ mod 19) − 9. The same point serves for every L. The entries of the Jacobian are rational numbers. Their denominators divide products of factorials of integers at most 2L ≤ 32. So they are prime to the prime p = 231 − 1. The rank of the reduction modulo p is at most the rank over Q. At this point the reductions have the ranks listed in Tab. 1. This gives the lower bounds. Parts (1) and (2) follow from Thm. 4.2. □ The accompanying program rank witness.py performs this computation exactly modulo p for 2 ≤ L ≤ 16. For each L and each pool it prints the evaluation point, the labels of a maximal linearly independent set of Jacobian rows, the pivot columns of that row set, and the determinant of the resulting square minor modulo p. A nonzero determinant certifies the rank. For L ≤ 15 the minor of ΦL has size N (L), and for L = 16 it has size 285. The starting value of the generator is a parameter of the program; the starting values 2, 3 and 7 were also tested and produced the same ranks. At the cutoffs L ∈ {4, 6, 8} of [15, Sec. 7.3], the fiber dimensions are 5, 13, 20 for ΦL and 4, 8, 6 for Φ+ L. We compare these ranks with known orbit-recovery results. For one spherical function including the constant band, [7, Thm. 4.1 and Cor. 4.2] proves finite generic recovery from invariants of degree at most three for every L ≥ 10. On H0 ⊕ UL the invariants of degree at most three are polynomials in f0 , the powers, and the even and odd bispectra. Finite generic recovery gives their map the generic rank (L + 1)2 − 3. Fixing f0 lowers the rank by at most one, so the powers and bispectra have generic rank at least n − 3 on UL ; adjoining the sextic invariants cannot lower it. In particular, r+ (L) = n − 3 holds for every L ≥ 10, and the saturation of Φ+ L at L = 10 is not a new generic-recovery threshold. The comparison made here concerns the even pool ΦL , its radii and mirror ambiguity, and the information retained for particular tasks. For R spherical shells, [6, Cor. 5.1] gives recovery under L ≥ 3 and R ≥ L + 2, while [2, Thm. 2.2] gives generic separation from degree-at-most-three invariants for R ≥ 3. The radial function considered here has one copy of each harmonic band, whereas those results require several independently sampled shells. The odd bispectra change sign under reflection, and ΦL does not change. Let τ : UL → UL be τ (f1 , . . . , fL ) = (−f1 , f2 , . . . , (−1)L fL ), so that τ replaces the radial function by p 7→ ρ(−p). Put χ(c) = det(a, Qa, Q2 a), as in Sec. 4. Proposition 6.2. Let L ≥ 2. Then the following hold.

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15

(1) The map τ commutes with the action of SO(3). Moreover ΦL ◦ τ = ΦL , B ◦ τ = −B for every odd bispectrum B, and χ ◦ τ = −χ. (2) For every c ∈ ΩL , the point τ c lies in the fiber of ΦL through c and not in SO(3) · c. In particular ΦL does not separate orbits. (3) Let U ′ and Mc be as in Thm. 4.2 for Φ = ΦL . Then τ (U ′ ) = U ′ and τ (Mc ) = Mc for every c ∈ U ′ . For L = 16 and c ∈ U ′ , the number of orbits in Mc is even, and at least two. Ψl is linear, so τ Proof. (1) For R ∈ SO(3), (R · τ f )(p) = f (−R−1 p) = (τ (R · f ))(p). The map P l multiplies Fl by (−1) . A component of degree dl in Fl is multiplied by (−1) l l dl , which is 1 for the components of ΦL and −1 for the odd bispectra. The map τ fixes Q and negates a, so χ(τ c) = det(−a, −Qa, −Q2 a) = −χ(c). (2) By (1), ΦL (τ c) = ΦL (c). The function χ is constant on orbits by the proof of Lem. 4.1, and χ(τ c) = −χ(c) ̸= χ(c) for c ∈ ΩL . Hence τ c ∈ / SO(3) · c. The set ΩL is not empty, so ΦL does not separate orbits. (3) By (1), τ is linear with ΦL ◦ τ = ΦL , so d(ΦL )τ c ◦ τ = d(ΦL )c , and the rank is the same at c and τ c. By (1) also τ (ΩL ) = ΩL . Hence τ (U ′ ) = U ′ and τ (Mc ) = Mc . By (1), τ permutes the orbits in Mc , and τ 2 = 1. No orbit is fixed, because χ is constant and nonzero on each orbit in Mc and changes sign under τ . For L = 16 the orbits in Mc are finite in number by Thm. 6.1, so their number is even. It is at least two, since Mc contains c and τ c. □ The point τ c matches every component of ΦL exactly. It is a distinct mirror decoy precisely when τ c ∈ / SO(3) · c, equivalently when the surface has no orientation-reversing orthogonal symmetry about the chosen origin. Such a symmetry need not be reflection in a plane. On ΩL the two rotation classes are always distinct. At L = 16, continuous ambiguity disappears on the free maximal-rank locus, but the mirror pair remains. The proposition neither counts all other finite ambiguities nor asserts that singular parts of a fiber have been classified. At L = 2 the pool consists, up to nonzero factors, of |a|2 , tr Q2 , a⊤ Qa, and det Q. Its fibers have dimension one. By [16, Thm. 4.9], adjoining a⊤ Q2 a and χ = det(a, Qa, Q2 a) gives a separating descriptor. These two invariants have degrees four and six, and every bispectrum is cubic. The second is odd under τ . For the two adversaries of Sec. 2 these results read as follows. For 2 ≤ L ≤ 15 and every c ∈ U ′ , ε the decoy radius rΦ (c; K) is positive for every tolerance ε and every compact neighborhood K L of c. A recognizer built on ΦL accepts, near every generic enrolled surface, a family of decoys of the dimension listed in Tab. 1, and it accepts τ c whenever τ c ∈ / SO(3) · c, at every L. For 2 ≤ L ≤ 15, a captured value of ΦL at a point of U ′ is consistent with a family of rotation classes of positive dimension, and that family contains the mirror class. For L = 16, within U ′ , a captured value is consistent with finitely many rotation classes, including at least the mirror pair; additional finite ambiguities are not excluded. For L ≥ 4 the odd bispectrum B234 is a nonzero trilinear invariant, so it is nonzero on an open dense subset and distinguishes c from τ c there. There are no odd bispectra in these pools for L = 2, 3. Neither the addition of odd bispectra nor full generic rank proves global separation of all orbits.

7. An explicit finite-tolerance benchmark In this section wl = 1 for all l, so ∥·∥ is the L2 norm for the round probability measure. For θ ∈ R3 let R(θ) ∈ SO(3) be the rotation with rotation vector θ. The closed ball of radius π is mapped onto SO(3).

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Lemma 7.1. Let c, c′ ∈ UL with c′ = (f1′ , . . . , fL′ ), and put K′ =

L X

l2 ∥fl′ ∥2

1/2

.

l=1

Then the following hold. (1) |∥c − R(θ1 ) · c′ ∥ − ∥c − R(θ2 ) · c′ ∥| ≤ K ′ |θ1 − θ2 | for all θ1 , θ2 ∈ R3 . (2) For R1 , R2 ∈ SO(3) let ϕ(R1 , R2 ) ∈ [0, π] be the rotation angle of R1−1 R2 . Then |∥c − R1 · c′ ∥ − ∥c − R2 · c′ ∥| ≤ K ′ ϕ(R1 , R2 ). Proof. (2) Write R1−1 R2 = R(ϕw) with a unit vector w and ϕ = ϕ(R1 , R2 ). On Hl the generator of the rotations about a unit axis is skew, with eigenvalues im for |m| ≤ l. So the curve t 7→ R(tϕw) · c′ has speed at most ϕK ′ , and ∥R1 · c′ − R2 · c′ ∥ = ∥c′ − R1−1 R2 · c′ ∥ ≤ K ′ ϕ. The triangle inequality completes the proof. d where ω (1) Put θ(t) = θ1 + t(θ2 − θ1 ) and Rt = R(θ(t)) for t ∈ [0, 1]. Then Ṙt = Rt ω(t), b 3 b is the skew matrix of ω ∈ R . The derivative of the exponential map of SO(3) at θ is exp(θ) composed with a normal operator on R3 whose eigenvalues are 1 and (1 − e∓i|θ| )/(±i|θ|). These have modulus at most 1, so |ω(t)| ≤ |θ2 − θ1 |. On Hl the generator of the rotations about a unit axis is skew, with eigenvalues im for |m| ≤ l. Hence L X 1/2
d l2 ∥fl′ ∥2 Rt · c′ ≤ |ω(t)| ≤ K ′ |θ2 − θ1 |, dt l=1

because Rt acts isometrically. Integration gives ∥R(θ1 ) · c′ − R(θ2 ) · c′ ∥ ≤ K ′ |θ1 − θ2 |, and the triangle inequality completes the proof. □ A second parametrization avoids trigonometric functions. For a unit quaternion q = (q0 , q1 , q2 , q3 ) let R(q) ∈ SO(3) be the rotation it represents; the entries of R(q) are quadratic forms in q, and R(q) = R(−q). For u ∈ R3 and k ∈ {0, 1, 2, 3} let Sk (u) = R(q), where q is the unit quaternion whose k-th coordinate is (1 + |u|2 )−1/2 and whose remaining three coordinates, in order, are the coordinates of (1 + |u|2 )−1/2 u. The entries of Sk (u) are rational functions of u with denominator 1 + |u|2 . Lemma 7.2. Every rotation is Sk (u) for some k and some u ∈ [−1, 1]3 . For u, u′ ∈ [−1, 1]3 √ ′ with |u − u | < 2,
π ϕ Sk (u), Sk (u′ ) ≤ √ |u − u′ |. 2 Proof. Let qk be a coordinate of largest absolute value of a unit quaternion q; then qk ̸= 0. Replacing q by −q, we may take qk > 0; then q = qk (1, u) up to the position of the coordinate 1, with |uj | = |qj |/qk ≤ 1. This proves the first claim. Let q and q ′ be the unit quaternions of Sk (u) and Sk (u′ ). The map v 7→ v/|v| has differential of norm at most 1/|v|, so on √ the segment from (1, u) to (1, u′ ), where |v| ≥ 1, it is 1-Lipschitz; hence |q − q ′ | ≤ |u − u′ | < 2. Put ψ = arccos⟨q, q ′ ⟩. From |q − q ′ |2 = 2 − 2⟨q, q ′ ⟩ < 2 we get ⟨q, q ′ ⟩ > 0 and ψ ∈ [0, π/2). The rotation R(q)−1 R(q ′ ) is represented by the quaternion q̄q ′ , whose real part is√⟨q, q ′ ⟩ = cos ψ, so its rotation angle is 2ψ. Since sin is concave on [0, π/2], sin(ψ/2) ≥ (2 2/π)(ψ/2) for √ ψ/2 ∈ [0, π/4]. Hence |q − q ′ | = 2 sin(ψ/2) ≥ ( 2/π)(2ψ), which is the claim. □ Corollary 7.3 (Validated rotation covering). Let c, c′ ∈ UL , let K ′ be as in Lem. 7.1, and let K ≥ K ′ . Then the following hold.

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17

(1) Let B be a finite collection of boxes covering [−π, π]3 . For B ∈ B let θB be its center, let hB bound the distance from the center to every point of the box, and let ℓB ≤ ∥c − R(θB ) · c′ ∥ be a validated lower bound. Then  δ(c, c′ ) ≥ max 0, min(ℓB − KhB ) . B∈B

(2) For k = 0, 1, 2, 3 let Bk be a√finite collection of boxes covering [−1, 1]3 . For B ∈ Bk let uB be its center, let hB < 2 bound the distance from the center to every point of the box, and let ℓB ≤ ∥c − Sk (uB ) · c′ ∥ be a validated lower bound. Then n  o π δ(c, c′ ) ≥ max 0, min min ℓB − √ KhB . k B∈Bk 2 ′ Any validated upper bound on ∥c − R0 · c ∥ at a single rotation R0 is an upper bound on δ(c, c′ ). Proof. (1) For θ ∈ B, Lem. 7.1(1) gives ∥c − R(θ) · c′ ∥ ≥ ℓB − KhB . The boxes cover all rotation vectors of length at most π, which represent every rotation. Taking the minimum over all boxes gives the stated bound. (2) For u ∈ B ∈ Bk , Lem. 7.1(2) and Lem. 7.2 give √ ∥c − Sk (u) · c′ ∥ ≥ ℓB − (π/ 2)KhB , and by Lem. 7.2 the four families cover SO(3). The upper bound follows by evaluating the function at one rotation. □ For the computation we parametrize the null cone of C3 by (x, y, z) = (z02 − z12 , i(z02 + −2z0 z1 ) and take Fl to be the restriction of fl . Two parametrizations differ by a linear substitution in (z0 , z1 ). Under it every component of ΦL is multiplied by a nonzero constant, because transvectants are relative invariants. Relative differences of components do not depend on the choice. We fix bases of H1 , . . . , H6 consisting of harmonic polynomials with integer coefficients, listed in the data file sh-183-benchmark.json that accompanies this paper together with the programs used below. The enrolled tuple is z12 ),

f1 = (−2, −3, 0),

f2 = (−3, 3, 3, 3, 2),

f4 = (3, 0, −1, −3, 1, −4, −4, −4, 4),

f3 = (−1, −3, 3, −4, 2, 2, −4),

f5 = (−4, 2, −1, 2, −4, 4, −1, 3, 3, 4, −1),

f6 = (1, −1, −1, 3, 0, −4, 2, 4, −3, −2, 0, −3, 1), in coordinates with respect to these bases, and ∥c∥2 = 1633361/45045, so ∥c∥ < 6.0217. The decoy c′ has coordinates in 10−20 Z; they are listed in the data file. Example 7.4 (Certified L = 6 benchmark). The tuples c, c′ ∈ U6 above satisfy the following. (1) For every component φ of Φ6 , |φ(c′ )−φ(c)| ≤ 2·10−18 |φ(c)|. In particular |φ(c′ )−φ(c)| ≤ 10−14 |φ(c)|. (2) 21/4 ≤ δ(c, c′ ) ≤ 5.3643. In particular δ(c, c′ ) ≥ 0.87 ∥c∥. (3) 4.5 ≤ δ(c, τ c) ≤ 4.8048 and 4.3 ≤ δ(τ c, c′ ) ≤ 4.6035. (4) Let ε = 10−14 |Φ6 (c)|, and let K ⊂ U6 be compact and contain c and c′ . Then ε rΦ (c; K) ≥ 21/4. 6 (5) sc > −20 and sc′ > −20 on S 2 , where sc = f1 + · · · + f6 for c = (f1 , . . . , f6 ) and sc′ = f1′ + · · · + f6′ for c′ = (f1′ , . . . , f6′ ). Computation. (1) The components are evaluated at c and c′ in exact arithmetic over Q(i), with the invariants J4 , J6 , J10 computed from the Clebsch covariants of F3 . All 32 values are real and nonzero, and the largest relative difference is 1.54 · 10−18 . (2) The norms are computed from the exact Gram matrices of the bases. The upper bound is the exact value of ∥c − R0 · c′ ∥ at the rational rotation R0 = R(q0 ) of the quaternion q0 = (1427/905074, 290689/653469, −292467/330422, 1), normalized; ∥c − R0 · c′ ∥2 is a rational number smaller than 28.77556 < 5.36432 . The lower bound is Cor. 7.3(2). Each cube [−1, 1]3 is divided into 83 boxes. For a box B the value ∥c − Sk (uB ) · c′ ∥2 = ∥c∥2 + ∥c′ ∥2 − 2⟨c, Sk (uB ) · c′ ⟩

18

T. SHASKA AND L. BESHAJ

is enclosed by evaluating ⟨c, Sk (uB ) · c′ ⟩ as an explicit polynomial in the nine entries of Sk (uB ), in interval arithmetic, in IEEE double precision with rounding to nearest and with outward rounding by one unit in the last place after√every operation. With K = 22.806 > K ′ and hB the half-diagonal of B, a box with ℓB − (π/ 2)KhB ≥ 21/4 is discarded, and every other box is divided into eight. All boxes are discarded after at most eight levels, with 990 624 center evaluations in total. The ratio follows from (21/4)2 > 0.872 · 1633361/45045. (3) The same procedure, applied to the pairs (c, τ c) and (τ c, c′ ) with the thresholds 9/2 and 43/10, discards all boxes after 249 744 and 221 200 evaluations. The upper bounds come from the rational quaternions listed in the data file. (4) By (1), |Φ6 (c′ ) − Φ6 (c)| ≤ ε. By (2), c′ ∈ / SO(3) · c and δ(c, c′ ) ≥ 21/4. P (5) Write p = v/|v| with v on a face of the cube [−1, 1]3 , so that sc (p) = l fl (v)/|v|l . Each face is covered by boxes, and sc is enclosed on each box by interval arithmetic as in (2), with boxes subdivided until the lower bound exceeds −20. The distance bounds distinguish c′ from both c and τ c. The mirror match itself is an exact algebraic identity by Prop. 6.2. The rational inequality in part (1) is a finite-tolerance match: it does not establish that c′ belongs to the exact fiber or to a particular connected component of that fiber. We found c′ by numerical continuation along the descriptor level set, moving in the quotient tangent space of Thm. 6.1, which has dimension 13 at a free maximal-rank point, with Gauss– Newton correction of the descriptor residual, followed by Newton iteration in 60-digit arithmetic and rounding of the coordinates to 10−20 Z. The continuation is not part of the certificate; parts (1)–(5) are verified from the final rational pair alone. Part (1) is a componentwise relative check. No component of Φ6 (c) vanishes, so it is the discrepancy of Sec. 2 with W the diagonal matrix of the reciprocals 1/|φ(c)| and with the maximum norm in place of the Euclidean norm. A matcher on Φ6 with componentwise relative tolerance 10−14 therefore accepts c′ , and the Euclidean tolerance ε of part (4) is a consequence. The band powers of c and c′ agree to the same precision, so their energies agree to that precision in every band. By (5), the radial functions 24 + sc and 24 + sc′ exceed 4 on the sphere. Fig. 2 shows both as radial surfaces with constant band f0 = 24. Remark 7.5 (Certification status). Part (1) is established in exact rational arithmetic. Parts (2), (3) and (5) are established by interval arithmetic in IEEE double precision in round-to-nearest mode, with every elementary result widened by one unit in the last place in each direction; this covers the rounding error of correctly rounded addition, subtraction, multiplication, division √ and square root. The constant√π/ 2 is enclosed from above by the double-precision quotient√of the nearest doubles to π and 2, increased by two units in the last place; this exceeds π/ 2 because the nearest double to π lies below π by less than 1.3 · 10−16 and the nearest double to √ √ 2 lies above 2 by less than 10−16 . The entries of Sk (uB ) have absolute value at most 1 and all coefficients are bounded, so no overflow occurs, and results in the subnormal range remain correctly rounded. The certificate depends on the correctness of this floating-point arithmetic and of the interval routines in the accompanying program certify.py, and on nothing else. The exact value of ∥c − R0 · c′ ∥2 , the rational quaternions, the box counts and the Lipschitz constants are recorded by the program. The factor 0.87 is normalized by the nonconstant coefficient tuple. For the rendered model with ρ0 = 24, p ∥ρc ∥L2 (dµ) = 576 + 1633361/45045 ≈ 24.744. The distance interval [5.25, 5.3643] corresponds to approximately 0.2122–0.2168 of that full radial norm. Both normalizations are radial L2 quantities. They are neither Hausdorff distances nor violations of a dimensional tolerance.

DECOY AND DISCLOSURE RADII OF INVARIANT SHAPE DESCRIPTORS

19

Figure 2. The enrolled surface c (left) and the decoy c′ (right), drawn with f0 = 24; c′ is rotated by the rotation R0 of Ex. 7.4(2). All 32 components of Φ6 agree within relative tolerance 2 · 10−18 ; see Rem. 7.5. 8. Task-specific certification and disclosure A shape collision is not necessarily an error for every task. An inspection quantity can be constant on a fiber that contains many noncongruent shapes. Conversely, a shape descriptor can fail to determine geometry while revealing an attribute exactly. The distinction can be stated without choosing a classifier. Definition 8.1. Let T : UL → R be a continuous rotation-invariant task quantity. For a ε nonempty set F = FΦ,W (y; K), define T− (y) = min T (x), T+ (y) = max T (x), x∈F x∈F ε AΦ,W,T (y; K) = T+ (y) − T− (y). We call [T− (y), T+ (y)] the task interval and AεΦ,W,T (y; K) its width. The interval is an enclosure of the attainable task values; these values need not fill it when the consistent set is disconnected. For a requirement T ≤ b, an established upper bound tup ≥ T+ (y) with tup ≤ b certifies compliance. An established lower bound tlow ≤ T− (y) with tlow > b certifies noncompliance. To demonstrate ambiguity one needs two feasible shapes with validated task values on opposite sides of b. A loose enclosing interval that crosses b does not establish such a pair. Proposition 8.2 (Task information). With the notation of Def. 8.1, the following hold. (1) The smallest worst-case error of a scalar estimate based on y is T+ (y) − T− (y) . 2 The midpoint of the task interval attains this value. (2) If |T (x) − T (x′ )| ≤ MT δ(x, x′ ) for all x, x′ ∈ K, then min max|t − T (x)| = t∈R x∈F

ε AεΦ,W,T (y; K) ≤ MT DΦ,W (y; K).

20

T. SHASKA AND L. BESHAJ

In particular, for y = Φ(c), ε max|T (x) − T (c)| ≤ MT rW Φ (c; K). x∈F

(3) At exact tolerance, every task interval is a singleton if and only if there is a map g : Φ(K) → R such that T |K = g ◦ Φ|K . In that case g is continuous. Proof. For (1), both endpoints are attained, so every estimate has error at least half their difference. Every attainable value lies between them, so their midpoint has error at most half their difference. For (2), apply the assumed inequality to a maximizing pair and, for the second assertion, to c and a feasible point. For (3), singleton intervals say exactly that T is constant on each fiber in K, making g(Φ(x)) = T (x) well-defined. Conversely, such a factorization makes the intervals singletons. The continuous surjection Φ|K from a compact space onto the Hausdorff space Φ(K) is a quotient map, so continuity of T |K gives continuity of g. □ For exact descriptor collisions, every deterministic predictor h ◦ Φ takes the same value at both shapes. Thus if the required task values differ, no such predictor can be correct at both. At positive tolerance, closeness of two descriptors alone does not imply that an arbitrary predictor assigns them the same label. The corresponding impossibility statement concerns a common observation y: if both shapes lie in its consistent set, no decision based only on that observation can be guaranteed correct for both under the stated error model. A local test uses the same Jacobian calculation as the fiber theorem. Proposition 8.3 (Differential test for task ambiguity). Let L ≥ 2, let Φ : UL → RN be a descriptor, and let U ′ and Mc be as in Thm. 4.2. Let c ∈ U ′ , and let T : UL → R be a C 1 rotation-invariant function. If rank d(Φ, T )c > rank dΦc , then every neighborhood of c in the regular fiber Mc contains points c− and c+ with T (c− ) < T (c) < T (c+ ). The same conclusion holds inside an admissible set K whenever K contains a neighborhood of c in that fiber. Proof. The rank inequality is equivalent to the existence of v ∈ ker dΦc with dTc (v) ̸= 0. By the constant-rank theorem, ker dΦc is the tangent space to the regular fiber. Choose a C 1 curve in that fiber through c with derivative v. The derivative of T along the curve is nonzero at c, so sufficiently small parameters of opposite signs give the two inequalities. □ The rank test detects first-order variation only. It does not measure the size of the variation, and at a single point it is sufficient but not necessary. It also cannot exclude a jump between disconnected components. For example, let c lie in the free maximal-rank locus of Φ16 . The differential of every invariant annihilates the three-dimensional tangent space of the orbit, so adjoining χ does not increase the rank at c. Yet χ takes opposite signs at c and τ c, which Φ16 does not distinguish. Full local rank therefore does not imply that a task quantity is determined globally. 9. Volume disclosed by the even spectral pool Fix a known constant radial term ρ0 > 0 and use the round probability measure dµ = dω/(4π) on S 2 . Put L X sc (p) = fl (p), ρc (p) = ρ0 + sc (p), l=1

DECOY AND DISCLOSURE RADII OF INVARIANT SHAPE DESCRIPTORS

21

and suppose ρc > 0. The enclosed radial solid is {tp : p ∈ S 2 , 0 ≤ t ≤ ρc (p)}. Its volume is Z 4π (2) V (c) = ρc (p)3 dµ(p). 3 S2 R 3 For a positive continuous radial function ρ on S 2 we write V (ρ) = 4π 3 S 2 ρ dµ, so that V (c) = V (ρc ). This is the volume enclosed by the modeled surface. For a hollow object it is neither the material volume nor a measure of mass. Theorem 9.1 (Exact volume disclosure). For L ≥ 2 and fixed ρ0 , there is an explicitly computable vector β(ρ0 ) such that every c ∈ UL with ρc > 0 satisfies (3)

V (c) =

4π 3 ρ + β(ρ0 )⊤ ΦL (c). 3 0

The entries of β corresponding to J4 , J6 , J10 are zero. The same identity holds for Φ+ L with zero coefficients on the odd bispectra. Consequently, at fixed ρ0 , every exact fiber intersected with the set {c : ρc > 0} has zero volume-task width, irrespective of its shape dimension or diameter. Proof. Every fl has mean zero. Orthogonality gives Z Z L X X ∥fl ∥2L2 (dµ) + fi fj fk dµ. ρ3c dµ = ρ30 + 3ρ0 l

i,j,k=1

For each l, the quadratic integral is a nonzero fixed multiple al Pl . A triple integral vanishes when the sum of the degrees is odd, by parity, or when the triangle condition fails. For an ordered even triangle triple, the invariant trilinear space is one-dimensional: in the Clebsch– Gordan decomposition, Hk occurs with multiplicity one in Hi ⊗ Hj . The triple integral and the iterated transvectant are proportional. The Gaunt formula shows that the former is nonzero for these triples; see [12, Eq. (34.3.5) and §34.3(vii)]. Thus there are nonzero constants bijk with R fi fj fk dµ = bijk Bijk . Collecting the ordered triples gives   L X X 4π  3 (4) V (c) = ρ0 + 3ρ0 al Pl (c) + mijk bijk Bijk (c) , 3 l=1

(i,j,k)∈A(L)

where mijk = 1, 3, 6 according as all three indices, exactly two indices, or no indices coincide. This gives the stated vector and shows that the other invariants are unnecessary for volume. □ 9.1. Normalization and exact coefficient computation. The constants must match the descriptor normalization; norm powers cannot be substituted directly for the transvectant powers. For the null-cone parametrization of Sec. 7, let Pl be the Legendre polynomial with Pl (1) = 1. The solid harmonic extending Pl (pz ) restricts to   2l l l ql (z0 , z1 ) = (−1)l z z . l 0 1 Indeed only the leading term in Pl survives restriction to x2 + y 2 + z 2 = 0, and z = −2z0 z1 . Consequently al = (5) bijk =

1
, (2l + 1)(−1)l 2ll Gijk
, (qi , qj )i+j−k , qk 2k

Gijk =

1 2

Z 1 Pi (t)Pj (t)Pk (t) dt. −1

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T. SHASKA AND L. BESHAJ

The denominator is nonzero for (i, j, k) ∈ A(L). With s = (i + j + k)/2, the zonal Gaunt coefficient is (s!)2 (2s − 2i)!(2s − 2j)!(2s − 2k)! Gijk = , (s − i)!2 (s − j)!2 (s − k)!2 (2s + 1)! by [12, Eq. (34.3.5) and §34.3(vii)]. Thus the calibration uses rational arithmetic only. For example, 1 a1 = − , 6 1 , b112 = 30

a2 =

1 , 30

b222 = −

1 . 105

The accompanying program volume from pool.py implements these formulas, evaluates volume as an exact rational multiple of π when the supplied descriptor is rational, and propagates componentwise absolute error bounds. It does not test realizability of an arbitrary descriptor or validity of a supplied measurement-error bound. 9.2. Descriptor and radial-model error. Corollary 9.2. Let L ≥ 2, let c ∈ UL with ρc > 0, let W be a nonsingular matrix, and put 3 ⊤ Vb (y) = 4π 3 ρ0 + β(ρ0 ) y. If |W (ΦL (c) − y)| ≤ ε, then (6)

|V (c) − Vb (y)| ≤ |W −⊤ β(ρ0 )| ε.

If Pinstead each component satisfies |ΦL,j (c) − yj | ≤ ej , the right-hand side may be replaced by j |βj |ej . Suppose also that the actual radial function ρ and the modeled radial function ρc satisfy 0 < ρ, ρc ≤ ρmax and ∥ρ − ρc ∥L2 (dµ) ≤ γ. Then (7)

|V (ρ) − Vb (y)| ≤ |W −⊤ β(ρ0 )| ε + 4πρ2max γ.

The componentwise version holds with the same additional term. Proof. Subtract Eq. (3) from the definition of Vb and apply Cauchy–Schwarz to (W −⊤ β)⊤ W (ΦL (c) − y). The componentwise estimate is the triangle inequality. For a, b ∈ [0, ρmax ] we have |a3 − b3 | ≤ 3ρ2max |a − b|. Therefore Eq. (2) and the fact that µ is a probability measure give |V (ρ) − V (c)| ≤ 4πρ2max ∥ρ − ρc ∥L1 (dµ) ≤ 4πρ2max γ. Another triangle inequality proves Eq. (7).

□

An uncertain constant term must be accounted for: its contribution can be included in the radial error γ or bounded separately. Similarly, a descriptor of a truncated tuple determines the modeled volume. It determines the physical volume only up to an estimate of the truncation and acquisition errors.

DECOY AND DISCLOSURE RADII OF INVARIANT SHAPE DESCRIPTORS

23

9.3. An exact clearance test with identical volume. The following example exhibits a task conflict in closed form, without a numerical search. The reference origin is fixed; no centroid normalization is imposed. Proposition 9.3. Let p = (x, y, z) ∈ S 2 and define ρA (p) = 3 + 23 x + 13 y + 23 z + x2 − z 2 , ρB (p) = 3 + y + x2 − z 2 . Both radial functions are at least 1. Their coefficient tuples cA , cB ∈ U2 , taken with ρ0 = 3, have exactly the same even pool Φ2 , and their enclosed volumes are both 216π/5. Nevertheless, for Tout (c) = maxp∈S 2 ρc (p), 17 9 14 < < ≤ Tout (cA ). 4 2 3 Thus the same exact descriptor supports incompatible decisions for containment in the origincentered ball of radius 9/2, while it determines the enclosed volume exactly. Tout (cB ) =

Proof. Both have Q = diag(1, 0, −1), with aA = (2/3, 1/3, 2/3) and aB = (0, 1, 0). Then |aA |2 = |aB |2 = 1,

tr Q2 = 2,

det Q = 0,

⊤ a⊤ A QaA = aB QaB = 0.

These are the four invariants of Φ2 , up to its fixed nonzero normalization factors. In the null-cone convention of Sec. 7, the ordered values (P1 , P2 , B112 , B222 ) are (−2, 8, 0, 0) for both tuples. Also |⟨a, p⟩| ≤ 1 for a ∈ {aA , aB } and |p⊤ Qp| ≤ 1, so each radius is at least 3 − 1 − 1 = 1. Applying Eq. (4) with the constants in Eq. (5) gives    4π 4 1 216π V = + . 27 + 9 = 3 3 15 5 For ρB , use x2 − z 2 ≤ 1 − y 2 to obtain 17 ρB ≤ 4 + y − y 2 ≤ . 4 √ Equality is attained at y = 1/2, z = 0, x = 3/2. For ρA , evaluation at p = (1, 0, 0) gives 14/3. The displayed strict inequalities follow. □ The two objects cannot be congruent even under O(3), since Tout is invariant under that group and their values differ. The failure is therefore not solely a handedness ambiguity. At the common descriptor value the task interval for any admissible K containing both tuples has width at least 14/3 − 17/4 = 5/12. By Prop. 8.2, every scalar estimator based on that value has worst-case outer-radius error at least 5/24 on these two objects. No comparable error is forced for volume, whose exact task width is zero. 10. Admissible shapes and a representation audit 10.1. Origin, scale, and the feasible domain. The set K records geometric side information for both recognition and disclosure. One concrete model, with fixed ρ0 , is (8)

K = {c ∈ UL : ∥c∥ ≤ M,

ρmin ≤ ρ0 + sc (p) ≤ ρmax for all p ∈ S 2 },

where 0 < ρmin < ρmax . It is compact and rotation-invariant. It is semialgebraic, because each harmonic is the restriction of a polynomial and the quantified conditions on p · p = 1 can be eliminated over the reals; see [5]. Positivity of this kind is an assumption on the truncated model: the truncation of a positive function need not be positive.

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A declared reference origin and a centroid convention are different assumptions. For the enclosed solid, the volume centroid is at the origin precisely when Z p ρc (p)4 dµ(p) = 0, S2

R since its unnormalized first moment is 14 pρ4c dω. A surface-area centroid imposes a different condition. Recentering changes the radial function and generally its harmonic coefficients. The exact example of Prop. 9.3 uses the declared origin; it is not a counterexample to an unspecified recentered pipeline. More generally, if equality constraints restrict coefficients to a smooth rotation-invariant manifold M of dimension d on which the action is free, the local regular-fiber dimension is
d − 3 − rank dΦc |Tc M , where the restricted rank is assumed constant in a neighborhood. This follows from the same constant-rank and quotient-manifold argument as Thm. 4.2. The ambient count n − 3 − r does not carry over after such a normalization. Physical size must also be retained when the task depends on dimensions; removing H0 for orbit analysis does not make it irrelevant to inspection. 10.2. A computable geometric bound. Proposition 10.1. For the weighted norm of Sec. 2, put !1/2 L X 2l + 1 ΓL,w = . wl l=1

Then ∥sc ∥∞ ≤ ΓL,w ∥c∥ for every c ∈ UL . At a fixed constant term, the outer-radius task satisfies |Tout (c) − Tout (c′ )| ≤ ΓL,w δ(c, c′ ) √ for all c, c′ ∈ UL . For wl = 1, ΓL,w = L2 + 2L. Proof. Choose an orthonormal basis of each band for L2 (dµ). The addition theorem makes the sum of its squared values at any point equal to 2l + 1. Weighted Cauchy–Schwarz therefore gives |sc (p)| ≤ ΓL,w ∥c∥. Apply this to c − R · c′ and use invariance of the maximum under rotations. Minimizing over R proves the task bound. □ In particular, ρ0 − ΓL,w M ≥ ρmin is a simple sufficient positivity condition on a coefficient ball. It is not necessary. The task bound yields conservative clearance certificates from an upper bound on the shape radius; optimizing the task itself can be substantially tighter. 10.3. An error budget for the actual representation. A numerical matching tolerance is not a sensor specification. Suppose c and a fitted tuple b c belong to a convex bounded region on which ∥W dΦx ∥op ≤ MΦ,W , with the domain norm fixed as in Sec. 2. If ∥b c − c∥ ≤ σ and the stored value satisfies |W (y − Φ(b c))| ≤ η, integration along the line segment gives |W (y − Φ(c))| ≤ MΦ,W σ + η. Thus ε = MΦ,W σ + η is a sufficient descriptor-error budget under these assumptions. Truncation error in a geometric quantity must still be propagated separately, as in Cor. 9.2. If the acquisition bounds hold only with a stated probability, the resulting certificate holds with that probability and is not unconditional.

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25

10.4. Use in independent evaluation. The framework is intended primarily for evaluators who select or test geometric representations, not for operators who apply a radius to an unspecified field observation. The input is one documented representation, one admissible shape model, a calibrated error budget, and a task requirement. The output is a task certificate, a verified conflicting pair, or an unresolved case. A failure to find a conflicting shape is not a certificate of its absence. Similarly, a numerical distance computed at one aligning rotation bounds the orbit distance from above only; Cor. 7.3 gives the lower bound needed to certify geometric separation. One such setting is an independent validation study of scan-based identification or inspection of equipment or calibration objects, for instance in defense or aerospace programs. Such a study is relevant only if the representation under test actually discards geometry in the manner modeled here. A comparison against complete geometry does not automatically inherit a collision of ΦL . A focused validation would first reproduce the exact quadrupole formulas and Prop. 9.3. It would then use a controlled collection of fully observed radial objects with independently measured reference geometry, evaluate band powers, the even pool, and a task-augmented representation under the same acquisition and preprocessing conditions, and report task-error bounds, conflicting decisions, the fraction of unresolved cases, computation time, and descriptor storage. The high-band benchmark of Sec. 7 is a certified calibration case for such a comparison, and Sec. 11 applies the same audit to a space object. For shared records, the same audit specifies what is released: at fixed known ρ0 , the modeled enclosed volume is recoverable by Thm. 9.1, and multiple records about one tuple must be assessed jointly by Cor. 4.4. An application need not demand complete shape reconstruction. It may be better served by retaining one additional task measurement than by enlarging a global invariant pool. That decision depends on the required task and on which attributes may be disclosed.

11. A case study: asteroid (101955) Bennu Rotation-invariant descriptors are natural for a tumbling object of unknown attitude. We apply the framework to the radar shape model of [13] for the near-Earth asteroid (101955) Bennu. It is archived in the Planetary Data System as the file data/101955bennu.tab of the bundle urn:nasa:pds:ast-bennu.radar.shape-model, version 1.1; see [14]. The model is a closed triangulated surface with 1348 vertices and 2692 facets. Its origin is the center of mass, and its axes are the principal axes. We use meters. The polyhedral volume of the model is Vmesh = 6.22656 · 107 m3 , and its equivalent-sphere diameter is 491.8 m. The facets are oriented so that the signed volume is positive. Every facet has outward normal ν with ⟨ν, a⟩ > 0 at its vertices a; the smallest value of ⟨ν, a⟩/(|ν||a|) is 0.667. So the origin lies in the open inner half-space of every facet plane, that is, in the interior of the kernel of the polyhedron, and the surface is star-shaped about the origin. As an independent check, the radial projection maps every facet onto a positively oriented spherical triangle, and the solid angles of these triangles sum to 4π; so the projection has degree one, and a regular direction is covered exactly once. The radial function ρ is positive and Lipschitz. The vertices of the model lie at distances between 224.2 m and 289.8 m from the center of mass. Since |x| is convex, ρ ≤ 289.8 m on S 2 . The constant term ρ0 and the bands f1 , . . . , fL approximate the orthogonal projections of ρ for the round probability measure; they are computed using a product Gauss rule with 200 × 400 nodes, without a validated quadrature bound, since ρ is only piecewise smooth. For L = 6 the coordinates of f1 , . . . , f6 in the bases of Sec. 7 are rounded to 10−4 m, and ρ0 is rounded to

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245.2830 m. The rounded tuple is denoted c ∈ U6 ; it is listed in the data file sh-183-bennu.json. All statements about L = 6 concern c and its radial model ρc = ρ0 + sc . We have ∥c∥ < 11.52 m. L 2 3 4 5 6 7 8 9 10 11 12

components

Vpool (km3 )

|Vpool − VL |/Vmesh

(Vpool − Vmesh )/Vmesh

γL (m)

−15

4 0.061969 1.1 · 10 −0.477 % 9.71 7 0.062009 8.4 · 10−16 −0.412 % 9.01 13 0.062186 3.6 · 10−16 −0.127 % 4.97 19 0.062204 2.4 · 10−16 −0.098 % 4.32 29 0.062227 < 10−16 −0.063 % 3.45 39 0.062232 3.6 · 10−16 −0.054 % 3.18 54 0.062248 3.6 · 10−16 −0.029 % 2.31 69 0.062250 3.6 · 10−16 −0.026 % 2.17 90 0.062253 1.2 · 10−16 −0.021 % 1.95 111 0.062254 3.6 · 10−16 −0.019 % 1.88 139 0.062256 4.8 · 10−16 −0.016 % 1.73 Table 2. Volume of Bennu from the even pool. Vpool is the right-hand side of Eq. (4) evaluated on the powers and even bispectra; VL is the volume of the truncated radial model; γL = ∥ρ − ρ0 − f1 − · · · − fL ∥L2 (dµ) is the truncation error of the radial function. The column “components” counts the powers and even bispectra.

Example 11.1 (Bennu). Let c ∈ U6 be the tuple of Bennu above, let Tout (x) = ρ0 + maxS 2 sx be the keep-out radius about the fixed reference origin, the center of mass of the radar model, and let V (x) be the volume of the radial model ρ0 + sx . There are tuples d+ , d− ∈ U6 with coordinates in 10−20 Z such that the following hold. (1) Φ6 (τ c) = Φ6 (c). The five odd bispectra of c are nonzero, and each changes sign under τ . Moreover 5 ≤ δ(c, τ c) ≤ 5.5138 m. (2) For 2 ≤ L ≤ 12, the volume obtained from the even pool by Thm. 9.1 equals the volume of the truncated model. In floating-point arithmetic the two agree to relative precision 2 · 10−15 , and Vpool differs from Vmesh by the amounts in Tab. 2. This part is numerical and is not certified. (3) For every component φ of Φ6 , |φ(d± ) − φ(c)| ≤ 4 · 10−19 |φ(c)|, and |V (d± ) − V (c)| ≤ 2 · 10−16 m3 . (4) The keep-out radii satisfy 272.161 ≤ Tout (d− ) ≤ 272.172, 274.321 ≤ Tout (c) ≤ 274.332, 279.588 ≤ Tout (d+ ) ≤ 279.599, in meters. In particular d+ and d− lie neither in O(3) · c nor in each other’s O(3)-orbit. (5) 4 ≤ δ(c, d+ ) ≤ 4.2930 m. (6) sc , sd+ and sd− exceed −40 m on S 2 . Hence each of the radial functions ρ0 + sc , ρ0 + sd+ and ρ0 + sd− exceeds 205 m in every direction. Computation. (1) The values of Φ6 at c and τ c are computed in exact arithmetic over Q(i) and coincide; this is also Prop. 6.2(1). The odd bispectra B234 , B245 , B256 , B346 , B456 are computed exactly at c and τ c. The distance bounds are obtained as in Ex. 7.4(2), with 64 176 boxes, and the upper bound at a rational rotation listed in the data file.

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27

(2) The identity Eq. (4) is exact. In floating-point arithmetic both sides agree to the relative accuracy in Tab. 2. The last two columns of the table are numerical: they combine the truncation error of the radial model with the quadrature error of the projection, and neither is enclosed by a validated bound. Cor. 9.2 with ρmax = 289.8 m and γ = γ6 bounds the volume error at L = 6 by 5.8 % of Vmesh ; the actual error is 0.063 %. (3) The tuples d± are found by numerical continuation along the level set of Φ6 through c, R in the direction of the projection onto the quotient tangent space of the gradient of ± (ρ0 +
1/24 sx )24 dµ , followed by Newton iteration in 60-digit arithmetic and rounding. The relative differences of the components are computed exactly, and are at most 3.7 · 10−19 . The volume bound is the componentwise estimate of Cor. 9.2, computed exactly. (4) Each lower bound is the exact value of ρ0 + sx at a rational point of S 2 of the form (2u, 2v, ±(1 − u2 − v 2 ))/(1 + u2 + v 2 ) with u, v ∈ Q. Each upper bound is established by the interval cover of Ex. 7.4(5) applied to −sx . The function Tout is invariant under O(3), which gives the last assertion. (5) As in Ex. 7.4(2), with 43 392 boxes. (6) As in Ex. 7.4(5).

Figure 3. Bennu: the radar shape model of [13], its L = 6 model c, the mirror image τ c, and the decoy d+ , labeled in each panel. The last two are rotated to their numerically best alignment with c. All four are drawn at the same scale. The color shows the distance from the origin in meters; see the color bar. Ex. 11.1 has the following implications for the two adversarial perspectives and for the keep-out task. The even pool at L = 6 determines the modeled volume of Bennu, and hence its mass for a given bulk density. It does not determine the handedness of the asteroid: the mirror image matches all 32 components exactly and lies at orbit distance at least 5 m. The odd bispectra do determine it. The even pool does not determine the keep-out radius either. The requirement Tout ≤ 277 m holds for c and fails for d+ . The requirement Tout ≤ 273 m holds for d− and fails for c. For every admissible set K containing c, d+ and d− , the task interval of Tout at the value Φ6 (c) with componentwise relative tolerance 10−14 has width at least 279.588 − 272.172 = 7.416 m. By Prop. 8.2, every estimate of the keep-out radius from that value has worst-case error at least 3.708 m on these three tuples. The modeled volumes of d+ and d− differ from that of c by at most 2 · 10−16 m3 . The continuation that produced d± imposes no centroid constraint. The example therefore audits the representation with the center-of-mass origin of the radar model held fixed, as in Sec. 10.1, and not a pipeline that recenters every candidate shape. These statements concern the degree-six model of a radar shape model with a vertex spacing of about 25 m; see [14]. They do not describe the surface of Bennu at finer scales.

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12. Conclusion The fiber radius measures geometric ambiguity, but does not by itself determine task utility or attribute confidentiality. Continuous ambiguity, discrete mirror ambiguity, finite-precision instability, and task ambiguity are distinct. The spectral pool gives a concrete example: its fibers can leave the shape undetermined while fixing enclosed volume exactly. The explicit clearance pair shows, without a numerical search, that a second geometric decision can remain impossible from the same descriptor. In an audit seeded by the radar shape model of Bennu, these phenomena appear together: the even pool fixes the modeled volume, misses the handedness, and, at componentwise relative descriptor tolerance 10−14 , leaves the keep-out radius uncertain by more than 7 m. Together with task intervals, observer-centered reconstruction bounds, and the analysis of combined releases, this gives a precise basis for evaluating what a retained geometric representation can certify and what sharing it discloses. References [1] A. S. Bandeira, B. Blum-Smith, J. Kileel, A. Perry, J. Niles-Weed, and A. S. Wein, Estimation under group actions: recovering orbits from invariants, Appl. Comput. Harmon. Anal. 66 (2023), 236–319. ↑6 [2] T. Bendory, D. Edidin, J. Katz, and S. Kreymer, Orbit recovery for spherical functions, 2025. arXiv:2508.02674. ↑14 [3] L. Beshaj, Reduction theory of binary forms, Advances on superelliptic curves and their applications, 2015, pp. 84–116. arXiv:1502.06289. ↑3, 4 [4] L. Beshaj, Integral binary forms with minimal height, ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–Oakland University. MR3579531 ↑3, 4 [5] J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36, Springer, Berlin, 1998. ↑6, 9, 23 [6] D. Edidin and M. Satriano, Orbit recovery for band-limited functions, SIAM J. Appl. Algebra Geom. 8 (2024), 733–755. ↑14 [7] Z. Fan, R. R. Lederman, Y. Sun, T. Wang, and S. Xu, Maximum likelihood for high-noise group orbit estimation and single-particle cryo-EM, Ann. Statist. 52 (2024), no. 1, 52–77. arXiv:2107.01305. ↑2, 14 [8] A. J. Hoffman and H. W. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. J. 20 (1953), 37–39. ↑2, 10 [9] R. Kakarala, The bispectrum as a source of phase-sensitive invariants for Fourier descriptors: a grouptheoretic approach, J. Math. Imaging Vision 44 (2012), no. 3, 341–353. ↑2, 10, 13 [10] M. Kazhdan, T. Funkhouser, and S. Rusinkiewicz, Rotation invariant spherical harmonic representation of 3D shape descriptors, Symposium on Geometry Processing (SGP 2003), 2003, pp. 156–164. ↑2, 9, 13 [11] J. M. Lee, Introduction to smooth manifolds, Second, Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. ↑8 [12] L. C. Maximon, 3j, 6j, 9j symbols. §34.3, https://dlmf.nist.gov/34.3. ↑21, 22 [13] M. C. Nolan, C. Magri, E. S. Howell, L. A. M. Benner, J. D. Giorgini, C. W. Hergenrother, R. S. Hudson, D. S. Lauretta, J.-L. Margot, S. J. Ostro, and D. J. Scheeres, Shape model and surface properties of the OSIRIS-REx target asteroid (101955) Bennu from radar and lightcurve observations, Icarus 226 (2013), no. 1, 629–640. ↑25, 27 [14] , Asteroid (101955) Bennu radar shape model V1.1, 2021. ↑25, 27 [15] T. Shaska, Exact Euclidean-invariant surface descriptors for data-efficient learning from segmented anatomy, 2026. Preprint. ↑2, 13, 14 [16] T. Shaska and M.-R. Siadat, Invariant shape analysis of surfaces with spherical topology, 2026. arXiv:2609.39567. ↑2, 3, 4, 5, 6, 7, 12, 13, 15 Department of Computer Science and Engineering, Oakland University, Rochester, MI 48309 Department of Mathematical Sciences, United States Military Academy, West Point, NY 10996

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