No-Have: Negative Existence Theorems

• No-Have is a term for negative existence theorems showing that mathematically natural objects lack an expected property, defining key structural limits.
• The results come in diverse forms—explicit counterexamples, genericity theorems, asymptotic exclusions, and computational barriers—across disciplines such as discrete geometry, dynamics, and quantum information.
• These theorems not only exclude overly rigid formulations but also identify critical variables driving the absence, motivating replacement notions and new modeling approaches.

“No-Have” (Editor's term) denotes a recurrent pattern of nonexistence results in which a mathematically natural object is shown not to possess a sought property. In the supplied literature, the absent property ranges from edge zipper unfoldings and interval-exchange roots to local Nash equilibria, scalar hair, decision-theoretic quantum advantage, and totally geodesic submanifolds. Taken together, these works suggest that negative existence theorems are not merely pathological counterexamples: they identify structural limits of restricted models, sharpen conjectural boundaries, and often motivate replacement notions that remain well-posed (Demaine et al., 2019, Farnia et al., 2020, Bernazzani, 2016, Peng, 2018, Yu, 2024, Murphy et al., 2017).

1. Scope and forms of nonexistence

Domain	Object class	Property shown absent
Discrete geometry	Genus-zero polycubes	Edge zipper unfolding
Dynamical systems	Minimal $m$-IETs of high rank	Nontrivial roots
Generative modeling	Regularized non-realizable GAN games	Local Nash equilibria
Gravitation	Large regular reflecting stars	Static scalar field hair
Quantum information	Constant-depth quantum circuits	Advantage in decision problems
Riemannian geometry	Generic closed $n$-manifolds, $n \ge 4$	Nontrivial immersed totally geodesic submanifolds

The nonexistence statements occur in several logically distinct forms. Some are explicit counterexamples, such as the 14-cube and 44-cube polycubes with no edge zipper unfolding. Some are genericity theorems, such as the statement that most interval exchanges have no roots and that an open dense set of metrics has no nontrivial totally geodesic submanifolds. Some are asymptotic exclusion results, such as the radius bound beyond which a reflecting star cannot support scalar hair. Others are computational or game-theoretic barriers, such as the absence of local Nash equilibria in canonical GAN settings or the absence of quantum advantage for constant-depth decision circuits (Demaine et al., 2019, Bernazzani, 2016, Peng, 2018, Farnia et al., 2020, Yu, 2024, Murphy et al., 2017).

2. Combinatorial obstruction: polycubes with no edge zipper unfolding

For polycubes, the relevant distinction is between an edge unfolding, where the cut set is any spanning tree over the corner vertices of the surface graph $G_P=(V,E)$, and an edge zipper unfolding, where the cut set is restricted to a single simple path spanning all corner vertices. The negative result is specific to the path restriction. Two explicit genus-zero counterexamples are constructed: $P_{44}$, consisting of 44 unit cubes and having no flat vertices, and $P_{14}$, consisting of 14 unit cubes and having three flat vertices. Both admit general edge unfoldings, but neither admits an edge zipper unfolding. The proofs use the bipartiteness of $G_P$ under lattice-parity coloring $(x+y+z)\bmod 2$, together with the lemma that a bipartite graph with parity imbalance greater than $1$ has no Hamiltonian path. For $P_{44}$, all surface vertices are corners and the parity imbalance is $n$0, so any zipper path would have to be Hamiltonian and therefore cannot exist. For $n$1, the argument is subtler because flat vertices may be omitted: the proof analyzes four exhaustive cases depending on which flat vertices $n$2 lie on the zipper path, and each case yields either a parity-imbalance contradiction or the inaccessibility of a corner vertex $n$3 that is reachable only via one of $n$4. The construction of $n$5 also extends to an infinite sequence of larger polycubes with only corner vertices and no zipper unfoldings (Demaine et al., 2019).

3. Rank obstruction: interval exchanges with no roots

For interval exchange transformations, the absent property is being a nontrivial power. If $n$6 is a minimal $n$7-interval exchange transformation and its rank $n$8 exceeds $n$9, then $n \ge 4$0 cannot be written as $n \ge 4$1 for any IET $n \ge 4$2 and integer $n \ge 4$3. Here rank means the $n \ge 4$4-dimension of the span of the exchanged interval lengths. The argument combines a count on maximal chains of discontinuities with first-return maps and tower decompositions: if $n \ge 4$5, then the number of maximal $n \ge 4$6-chains is at most $n \ge 4$7, which yields $n \ge 4$8. The estimate is sharp. Explicit even and odd families of minimal $n \ge 4$9-IETs are constructed with $G_P=(V,E)$0 that are nevertheless nontrivial powers, using periodic permutations of equal-length base intervals together with interval-wise rotations and a linear system modulo $G_P=(V,E)$1 that produces a root. The $G_P=(V,E)$2-IET case is stronger: a minimal $G_P=(V,E)$3-IET that is not of rotation type has a nontrivial root if and only if it fails Keane’s infinite distinct orbit condition. That equivalence rests on a classification of minimal IETs whose discontinuities all lie in one orbit: up to conjugacy, they are towers over a minimal rotation, and such towers always have roots. For fixed separating irreducible permutation, the set of length vectors producing no root is residual and of full measure, which justifies the paper’s title formulation that most interval exchanges have no roots (Bernazzani, 2016).

4. Equilibrium failure in GANs and the proximal replacement

In GAN theory, the absent property is a local Nash equilibrium of the simultaneous-move zero-sum game. A positive realizable case remains: if $G_P=(V,E)$4 reproduces the data distribution $G_P=(V,E)$5, then there exists a constant discriminator $G_P=(V,E)$6 such that $G_P=(V,E)$7 is a Nash equilibrium for vanilla GAN, general $G_P=(V,E)$8-GANs, and Wasserstein GANs. The negative result concerns regularized non-realizable regimes. When learning $G_P=(V,E)$9 with $P_{44}$0 using a linear generator $P_{44}$1 constrained by $P_{44}$2, the paper proves: for $P_{44}$3-GANs with convex $P_{44}$4 and nondecreasing $P_{44}$5, the minimax has no Nash equilibrium; for W2GAN with $P_{44}$6-concave discriminators, the minimax has no Nash equilibrium and even no local Nash equilibria for quadratic discriminators; and for one-dimensional WGAN with a $P_{44}$7-Lipschitz discriminator, the minimax has no Nash equilibrium. In each case, the obstruction arises because, after fixing an optimal discriminator, the generator objective fails to have a local minimum in the translation parameter $P_{44}$8. Empirically, after 200,000 iterations of WGAN-WC, WGAN-GP, and SN-GAN on MNIST and CelebA, fixing the final discriminator and further optimizing the generator causes the objective $P_{44}$9 to decrease rapidly and sample quality to collapse, behavior inconsistent with local Nash optimality.

The proposed replacement is the proximal equilibrium, defined through the discriminator-localized objective

$P_{14}$0

This notion captures the sequential character in which the generator moves first and the discriminator responds locally. The paper proves that, under stated convexity and transport conditions, the Wasserstein-distance minimizing generator yields a proximal equilibrium for both W2GAN and WGAN when the discriminator space is equipped with the Sobolev semi-norm $P_{14}$1. It then introduces proximal training, where the discriminator performs proximal maximization and the generator updates against the proximal best response. On CIFAR-10, proximal training improved inception scores from $P_{14}$2 to $P_{14}$3 for WGAN-WC (DIM$P_{14}$4), from $P_{14}$5 to $P_{14}$6 for WGAN-WC (DIM$P_{14}$7), from $P_{14}$8 to $P_{14}$9 for SN-GAN (DIM$G_P$0), and from $G_P$1 to $G_P$2 for SN-GAN (DIM$G_P$3) (Farnia et al., 2020).

5. Large reflecting stars and the absence of scalar hair

For asymptotically flat regular reflecting stars, the absent property is a nontrivial static scalar condensate outside the star. The system consists of a static, spherically symmetric, horizonless compact object with reflecting surface radius $G_P$4, a Maxwell field $G_P$5, and a minimally coupled, static, massive, charged scalar field with full gravitational backreaction. The reflecting condition is Dirichlet: $G_P$6, and asymptotic flatness requires $G_P$7, $G_P$8, $G_P$9, and $(x+y+z)\bmod 2$0.

The proof introduces $(x+y+z)\bmod 2$1. Because $(x+y+z)\bmod 2$2 and $(x+y+z)\bmod 2$3, a nontrivial solution forces $(x+y+z)\bmod 2$4 to have an extremum at some $(x+y+z)\bmod 2$5. Evaluating the transformed scalar equation at that extremum yields a key inequality relating $(x+y+z)\bmod 2$6 to gauge and geometric terms. Asymptotic flatness, the decay of the scalar, and the fact that the energy density at large radius is dominated by the Maxwell sector imply control of $(x+y+z)\bmod 2$7, $(x+y+z)\bmod 2$8, and $(x+y+z)\bmod 2$9 for sufficiently large $1$0. The resulting bound is

$1$1

If $1$2 exceeds this bound, the only static solution compatible with the reflecting boundary condition and asymptotic flatness is the trivial one, so large reflecting stars cannot have scalar field hair. The argument explicitly incorporates backreaction through $1$3, $1$4, and the induced control of $1$5. The result excludes hair only for sufficiently large stars; it does not preclude scalar configurations for small $1$6 (Peng, 2018).

6. Low-depth quantum circuits: no advantage and one-copy learnability

In the quantum-information setting, the absent property is computational advantage in decision problems for constant-depth circuits. A depth-$1$7 circuit is built from $1$8 layers of disjoint two-qubit gates, with $1$9. The decisive structural fact is the shallow light cone: under Heisenberg evolution, a local observable spreads to at most $P_{44}$0 qubits. If such a circuit computes a Boolean function $P_{44}$1 with bounded error by reading the first output qubit, then

$P_{44}$2

for some subset $P_{44}$3 of size $P_{44}$4 and an operator $P_{44}$5 acting only on those qubits. Hence the acceptance probability depends on only $P_{44}$6 input bits and can be computed classically in $P_{44}$7 time. Under the paper’s model, low-depth quantum circuits therefore have no advantage in decision problems.

The same locality drives a one-copy estimation result. Fix a global Pauli product $P_{44}$8, and consider observables

$P_{44}$9

with constant length $n$00, constant degree $n$01, and small norm $n$02. A single global measurement in the $n$03-basis yields an estimator $n$04 satisfying

$n$05

where $n$06 depends only on depth and locality. For $n$07, one has $n$08, so the failure probability is $n$09. The analysis extends to noisy shallow circuits modeled by layers of local two-qubit CPTP maps via Stinespring dilation, and it has a classical analogue for low-complexity distributions generated by constant-depth local Markov transitions. The same paper emphasizes NISQ verification: a classical prediction derived from the constant-size light cone can be compared against single-copy experimental measurements with confidence improving linearly in $n$10 for fixed $n$11 (Yu, 2024).

7. Generic metrics, structural consequences, and open directions

In Riemannian geometry, the absent property is the existence of nontrivial immersed totally geodesic submanifolds. For every compact smooth manifold $n$12 of dimension at least $n$13 and every finite $n$14, the set of $n$15 Riemannian metrics with no nontrivial immersed totally geodesic submanifolds contains a set that is open and dense in the $n$16-topology. The stronger statement is formulated in terms of partially geodesic $n$17-planes: an $n$18-plane $n$19 is partially geodesic if for every $n$20, the Jacobi operator $n$21 preserves $n$22 in the sense that $n$23. Since the tangent plane of a totally geodesic submanifold must be partially geodesic, it suffices to eliminate partially geodesic $n$24-planes for $n$25. The proof does so by local curvature deformations on the Grassmann bundle $n$26: in an adapted orthonormal $n$27-frame $n$28, two compactly supported functions $n$29 are inserted into mixed metric components so that $n$30 and $n$31, forcing a normal component in $n$32. The use of two independent normal directions explains the dimensional restriction $n$33. An immediate corollary constrains isometry groups: if a smooth effective action has subgroup fixed point sets of dimensions incompatible with the theorem’s representation-theoretic conditions, then generic metrics are not $n$34-invariant (Murphy et al., 2017).

These negative results preserve substantial open terrain. For polycubes, open problems include whether every polycube tree admits an edge zipper unfolding, whether every polycube admits some edge unfolding, whether $n$35 is minimal, and whether zipper existence can be efficiently recognized (Demaine et al., 2019). For GANs, unresolved directions include extending proximal-equilibrium existence beyond W1/W2 to general $n$36-GANs, sharpening the connection to Stackelberg equilibria, and understanding the tradeoff between $n$37 and inner maximization accuracy (Farnia et al., 2020). For shallow quantum models, the paper explicitly leaves open what happens at logarithmic depth $n$38, as well as extensions to broader gate sets, geometries, and measurement models (Yu, 2024). The manifold result itself notes that a modification may work in dimension $n$39, and Bryant is cited as having outlined a local proof.

This suggests that “No-Have” theorems are most informative when they do two things simultaneously: they exclude an overly rigid formulation and they identify the structural variable that caused the exclusion. In the supplied papers, those variables are path-restricted cuts rather than cut trees, high rank rather than tower structure, simultaneous Nash equilibrium rather than a proximal sequential response, large radius rather than small compact objects, constant-depth light cones rather than deeper causal spread, and generic curvature perturbations rather than highly symmetric metrics.