Regular trajectories, pruning and quantum parity

1 Introduction

The class \QAC consists of constant-depth quantum circuits with arbitrary one-qubit gates and generalized Toffoli gates of unbounded arity, using polynomially many total qubits. Ancillary qubits are initialized to zero. The supports of gates in a layer are pairwise disjoint. This last condition makes the availability of fanout a substantive question: one cannot simultaneously use a qubit as the control of arbitrarily many two-qubit gates. Moore asked whether the coherent fanout operation can be implemented in this model, and conjectured a negative answer. Its unitary parity counterpart is equivalent by Hadamard conjugation [8, 16].

We prove Moore’s parity conjecture in the measured-output model and therefore also exclude coherent fanout. The lower bound holds for bounded-error computation with a single measured output, even when the computation changes the input and leaves arbitrary garbage in the other registers.

1.1 The circuit model and the result

For x\in\{0,1\}^{n}, let

An allowed circuit acts on \lvert x\rangle\lvert0\rangle^{\otimes a}. Its gates are arbitrary one-qubit unitaries and generalized Toffoli gates

on distinct control and target qubits. Each gate has unit cost, regardless of its arity. A depth-d circuit is a product of d layers of gates with pairwise disjoint supports. There is no geometric restriction. At the end, one designated qubit is measured in the computational basis. The other registers may be discarded. Intermediate measurements, postselection, and additional fanout gates are not allowed. All gates and their complex entries may depend arbitrarily on n, but not on x.

Here total qubits means the n inputs together with all ancillary qubits. Disjoint nonempty gate supports imply that a depth-d circuit on N qubits contains at most dN gates. Thus the theorem covers the usual fixed-depth model with polynomially many gates and polynomially many ancillas: the ancillary bound gives a polynomial bound on N, and a polynomial bound on N already bounds the number of gates.

In particular, no nonuniform constant-depth family with polynomial resources computes parity with any fixed positive worst-case advantage. The theorem is eventual for each fixed depth, polynomial bound, and advantage; it permits arbitrary finite exceptions at small input lengths.

To make the connection with fanout explicit, let

For the Hadamard gate H, conjugating a controlled-NOT by H\otimes H interchanges its control and target; this follows directly from its action on the two-qubit basis. Writing F_n as the product of its commuting controlled-NOT operations gives

Thus a constant-depth polynomial-resource implementation of F_n, together with two layers of one-qubit gates and a zero target, would contradict Theorem 1.1. This also excludes Moore’s original formulation, in which the ancillary qubits must be restored to zero [16].

Theorem 1.1 also has state-preparation consequences corresponding to those of Gretta, Gupta, and Joshi [9, Corollary 4.10 and Theorem 4.14]. Let \rho be the retained n-qubit state of an allowed unitary preparation from all-zero inputs, with arbitrary final ancillas discarded. Its felinity is 2\sum_y p_y p_{\bar y}, where p_y=\langle y|\rho|y\rangle and \bar y is the bitwise complement of y. At each fixed depth and fixed polynomial total-qubit bound, for every fixed A>0, eventually every such preparation has felinity less than n^{-A}. For every fixed 0<\delta<1, eventually every such preparation and every integer k with n^\delta\le k\le n/2 also satisfy

Here trace distance is one half of the trace norm. The companion states the same bounds [18, Corollary 8 (State-preparation obstructions), Section 7.1]. The argument after the parity proof in Section 7 justifies the pointwise reductions, including exact promotion with arbitrary final garbage and polynomial total-qubit overhead.

1.2 Previous work

The classical class \mathsf{AC}^{0} allows unbounded-fan-in AND and OR gates, NOT gates, and unrestricted reuse of wires. Parity was among the first explicit functions shown to require superpolynomial size at constant depth, by Furst, Saxe, and Sipser and independently Ajtai [1, 7]; Håstad obtained almost optimal exponential bounds through the switching lemma [11]. The quantum question combines the additional expressive power of one-qubit unitaries with the disjoint-support restriction on each layer. Moore’s formulation and the counting classes studied by Green, Homer, Moore, and Pollett made the role of coherent fanout explicit [8, 16]. Constant-depth constructions with fanout further demonstrated the computational importance of that operation [12].

Early quantum lower bounds exposed the importance of the ancillary register. Fang, Fenner, Green, Homer, and Zhang proved a depth bound of order \log(n/(a+1)) for exact parity computation with a ancillas that must be restored to zero [5, Corollary 4.6]. Padé, Fenner, Grier, and Thierauf ruled out exact clean simulation at entangling depth two [19]; Fenner, Grier, Padé, and Thierauf subsequently removed the final-state cleanliness requirement for exact single-output parity at that depth [6, Theorem 6.1]. Here and in the depth-specific comparisons below, entangling depth counts only layers of multiqubit gates; arbitrary one-qubit layers may occur between them. This convention differs from our total layer count by at most a constant factor for fixed depth.

Rosenthal obtained lower bounds for approximate parity and fanout unitaries and related cat-like states, together with constant-depth approximation constructions using superpolynomial resources [20]. Approximation of a parity unitary is a stronger task than computing one measured classical bit: the latter may destroy the input and leave arbitrary garbage. Nadimpalli, Parham, Vasconcelos, and Yuen developed Pauli-spectrum methods with depth-dependent restrictions on the ancillary count [17]. Anshu, Dong, Ou, and Yao obtained lower bounds allowing barely superlinear ancillary counts [2]; Dong, Ou, and Yao developed further low-degree estimates and applications to quantum channels [4]. These resource restrictions do not cover an arbitrary fixed polynomial ancillary bound.

Joshi, Tal, Vasconcelos, and Wright ruled out exact parity at entangling depth three for sufficiently large input lengths, with no bound on the number of gates or ancillas. They also proved approximate parity lower bounds at entangling depth two for one measured output, without requiring input preservation [13]. Kintali’s September 2026 preprint reports exponentially small average parity correlation at entangling depth at most three, independently of the number of gates and ancillas, and an exponential gate lower bound for exact parity at entangling depth at most four [14, Theorems 1–2].¹ These depth-specific statements do not cover every fixed depth. Gretta, Gupta, and Joshi characterize the general parity question through Fourier concentration [9], while Xu and Li extend related fanout reductions to symmetric functions [22]. Our theorem permits every fixed depth and every polynomial total-qubit bound simultaneously. All structural statements needed for its proof are established below.

The companion article Product-projection localization and the QAC0 parity lower bound proves a uniform operator-norm localization theorem and an independent parity lower bound [18, Sections 2 and 6]. The present article instead proves state-dependent propagation, projection insertion, and pruning for regular trajectories. Both arguments use product-projection counts and elementary Chebyshev filters; neither structural theorem is used to prove the other.

1.3 The proof and its main tools

A generalized Toffoli gate is a reflection I-2P, where P is a product of one-qubit rank-one projections. This representation suggests counting how many local factors of a product test fail. A single such gate can couple arbitrarily many qubits, so its conjugation need not preserve small operator support. Moreover, product-projection filters can have large norms away from the count range where they approximate well. Controlling the state on which each approximation is used is therefore essential. Informally, a trajectory sequence is regular when, for every sufficiently small collection of reflections in a layer, the combined count of failures of their local factors has a rapidly decaying tail on the state entering that layer. The precise definition is asymptotic and allows arbitrary gates and states at each circuit size.

Polynomial locality and concentration have been related in work of Kuwahara, Arad, Amico, and Vedral and of Anshu and Metger [3, 15]. In the circuit setting, Pauli concentration and low-degree approximation are central to the works just discussed. An operator has degree at most \ell when it is a sum of operators supported on at most \ell sites each. We retain the elementary link between this degree and the number of product-basis coordinates an operator can change. The additional steps here are two-sided tail transport and stability under a product projection, which permit a trajectory to be made regular by a state-dependent pruning rule.

The first tool, Proposition 4.1, transports operators of low degree along a regular trajectory, with an arbitrarily small fixed loss in the degree exponent. Its conclusion is an approximation on the trajectory vector. The second tool, Proposition 6.1, shows that regularity survives insertion of an arbitrary product projection followed by reverse evolution. The key step is Lemma 5.1, which compares two regular trajectories. One polynomial detects a tail at the starting end; another approximates the product projection at the finishing end. They are applied on different sides of an inner product, so their large global norms are paid for by the appropriate tail estimates.

Projection stability then makes regularity available without an initial assumption on the circuit. Starting from a product state, we retain a reflection when its projection has probability at least m^{-6} on the current state, where m is the number of qubits. The retained layer is regular, and the deleted gates change the state by at most 2m^{-2} per layer. Iterating this rule yields a regular trajectory close to the original evolution (Proposition 6.5). The stability and propagation statements concern arbitrary product tests and regular trajectories, rather than parity alone.

Finally, we copy each basis input into a reference register that the subsequent computation leaves untouched. On the uniform coherent input, worst-case parity success becomes a signed acceptance expectation bounded away from zero. The pruned circuit has almost the same expectation. Write W for that circuit, \psi_0 for the initial product state, and P_B for the accepting output projection. The copying gates survive pruning, so conjugating the reference sign back through W sends \psi_0 to a vector with exactly n excitations relative to the initial product state \psi_0. The signed expectation pairs this vector with W^\dagger P_BW\psi_0, whose tail for that same count is bounded by Corollary 6.2. Because the total number m of qubits is polynomially bounded in n, a sufficiently small fixed h>0 can be chosen with m^h=o(n), separating these two scales and giving the contradiction.

Sections 2 and 3 introduce the formalism and the polynomial filters. Section 4 proves low-degree propagation, and Section 5 establishes tail transport. Projection stability and pruning are proved in Section 6; Section 7 completes the parity argument.

2 Product tests and regular trajectories

We work on the Hilbert space (\C^2)^{\otimes m}. Vector norms are Euclidean norms, and operator norms are the induced norms. Unless explicitly stated otherwise, every starting vector has norm at most one. Products of orthogonal projections used in the analysis are applied without renormalizing the resulting vectors. All projection insertions below are analytic devices; the permitted circuit operations remain unitary until the final output measurement.

2.1 The gate model and product tests

The physical circuit has n input qubits, a ancillary qubits initialized to \lvert0\rangle^{\otimes a}, and a designated output qubit measured in the computational basis. Its gates are arbitrary one-qubit unitaries and the generalized Toffoli gates of Section 1. Gates in each layer have pairwise disjoint supports, and each gate counts once regardless of its arity. The gates may depend arbitrarily on the input length, including through their complex entries, but are independent of the input string. No input or ancillary register is required to be restored, and the output qubit may be an original input qubit. All other qubits may be discarded after the output is measured. There are no intermediate measurements, postselection operations, geometric restrictions, or extra fanout gates.

For the analysis, we allow a reflection

where J_g is a nonempty set of sites and each v_{g,j} is a unit vector. The tensor factors are understood to occupy their indicated sites. Each P_g is an orthogonal projection, so R_g=R_g^\dagger=R_g^{-1}. A generalized Toffoli with c\ge1 distinct controls z_1,\ldots,z_c and a distinct target b has this form with

and identities elsewhere. Indeed, it acts trivially unless all controls are one, and on that control subspace its target action is I-2\lvert-\rangle\langle-\rvert=X. This product-reflection viewpoint also appears in Rosenthal [20].

An analytic layer will be either a product of such reflections on pairwise disjoint supports, or a tensor product of one-qubit unitaries. Splitting a physical layer into its two gate types preserves its operation, because its gates have disjoint supports. Thus a physical circuit of depth d gives at most 2d analytic layers. We use t for their number and m for the total number of qubits, including any reference qubits subsequently introduced. Empty layers are permitted. Every reflection layer has at most m gates, since their supports are nonempty and disjoint.

The summands of H_A are commuting orthogonal projections. Choosing the basis (v_{A,j},v_{A,j}^{\perp}) at each tested site diagonalizes them simultaneously. Consequently its spectrum is contained in \{0,1,\ldots,|J_A|\}, and 0\le H_A\le mI. We use spectral inequalities such as H_A\ge x also when x is not an integer.

For a set S of gates in one reflection layer, disjointness makes their specified local factors a single test. We write H_S for its count and P_S=\prod_{g\in S}P_g for its zero projection. If F is the whole reflection layer, then

Indeed, a gate in S commutes with each of its own local factors, while other gates have disjoint supports. Every spectral projection of H_S therefore commutes with F as well. If H_A retains only some sites of this test, then H_A and H_S are diagonal in the same product basis, H_A\le H_S, and

2.2 Asymptotic sequences and uniformity

All asymptotic statements concern an arbitrary sequence indexed by \nu\in\mathbb N, with total qubit counts m_\nu\to\infty and a bounded number of analytic layers. The states, tests, gates, and circuits can all vary with \nu. We suppress \nu and write m for m_\nu; thus “eventually” means at all sufficiently large sequence indices. Every exponent appearing in an estimate is a fixed real number independent of \nu. Padding with empty layers gives a common bound on length. When a fixed layer pattern is convenient, one may restrict to a subsequence with fixed length and layer types: there are only finitely many such patterns. An eventual assertion on every such subsequence yields the assertion on the original sequence, since any infinite set of violations has an infinite subsequence with one fixed pattern.

The following elementary observation will also supply uniformity over tests chosen from the current state.

The lemma imposes no cardinality or finiteness requirement on \mathcal D_\nu. In particular, a permitted test may depend on the exact state at its index. The eventual threshold may depend on the fixed tuple \theta; no threshold uniform over all real exponent tuples is asserted.

Commutation with H_S gives

at a reflection layer, so either endpoint can be used in (2.1). It follows that reversing a regular trajectory is regular for the inverse circuit: inverse reflection layers are the same layers, and inverse one-qubit layers impose no condition. Restricting a regular trajectory to any segment or subsequence preserves regularity. Multiplying each trajectory by a scalar of modulus at most one also preserves it. For any fixed use of regularity, finitely many auxiliary exponents and layers may be handled simultaneously by taking the maximum of their eventual thresholds. This does not require a common threshold for all admissible exponents.

2.3 Operator degree and excitation shifts

Degree bounds survive sums, while products have degree at most the sum of the factors’ degrees: expand into supported terms and take the union of each pair of supports. Taking adjoints preserves degree, as does conjugating by a tensor product of one-qubit unitaries. No bound on the number or norms of terms in a support decomposition is part of the definition.

The following support constraint is the elementary locality principle behind the connection between polynomial approximation and Hamming concentration; compare Anshu and Metger [3, Lemmas 2.10–2.11]. We record its exact form for arbitrary product tests.

3 Two polynomial filters

We need two different polynomials in excitation counts. The first approximates the zero projection on an initial part of the integer spectrum. The second detects a tail while remaining nonnegative on the whole real line. Both may grow outside their effective ranges; explicit bounds on that growth will be essential. The zero filter combines normalized Chebyshev approximation with exact roots at the first positive integers, a classical approximation-and-interpolation construction for symmetric predicates; see Sherstov [21, Section 3.1 and Lemma 3.9]. We give the construction and its estimates explicitly because transport requires control of the polynomial over the entire count interval, as well as accuracy on the initial integer range.

3.1 An elementary Chebyshev construction

Define real polynomials C_\ell by

The cosine and hyperbolic cosine addition formulas satisfy this same recurrence and initial values, so

Thus |C_\ell(y)|\le1 for -1\le y\le1, and C_\ell is positive and nondecreasing on [1,\infty). We also have the elementary global bound

Indeed, writing M=2|y|+1\ge1, the initial cases hold, and the recurrence gives 2|y|M^\ell+M^{\ell-1}\le M^{\ell+1} because 2|y|M+1\le M^2.

For 1\le a<R\le m, define

The denominator is a positive real constant. As x increases from zero to a, y(x) decreases from y(0)>1 to one; as x increases from a to R, y(x) decreases from one to minus one. The preceding formulas therefore imply

Whenever a\le R/2, we have R-a\ge1. For every x\in[0,m] it follows that |y(x)|\le1+2m\le3m. Since C_\ell(y(0))\ge1, (3.1) yields the uniform bound

For a useful exact expression for the denominator, put z=\sqrt{a/R}. Defining

direct substitution into the exponential formula for \cosh gives

Moreover, integration of (1-w^2)^{-1} between zero and z gives

Consequently, if \tau_0=2\operatorname{artanh}\sqrt{a/R}, then

as a/R\to0. These facts prove all the approximation estimates below.

3.2 Approximating the zero projection

Since every H_A has integer spectrum, this is a projection approximation in the required sense. More explicitly, with \Pi=\one_{H_A\le m^L}, the spectral theorem gives

For every vector \xi, splitting into the two spectral ranges yields

Its operator degree is at most m^r by Lemma 2.5.

3.3 A nonnegative tail detector

Evaluating T at a count operator gives a positive semidefinite operator of degree at most m^k and norm at most e^{m^k}. In particular, its global nonnegativity prevents contributions from outside [m^h,m^u] from cancelling the detected tail in a quadratic form.

4 Propagation of low-degree perturbations

Pauli expansions and low-degree approximations have been used to study \QAC [2, 4, 17]. Here regularity lets us replace the large product projections arising from conjugation by polynomials of small degree. The resulting approximation controls a perturbed vector along an exact reference trajectory.

5 Transporting excitation tails

The next lemma compares two regular trajectories joined at their final endpoint by a product projection. Its proof uses the two filters of Section 3 in different directions. A tail filter T is propagated forward along the projected trajectory, producing an effective range on which the final projection can be approximated by a polynomial D. That polynomial is then propagated backwards along the original trajectory. The resulting scalar product is small because T suppresses low counts while the original vector has small high-count tails. Positivity of T converts this estimate into a tail bound for the projected vector.

Figure 1 records the two uses of regularity. The forward propagation concerns T\chi, whereas the reverse propagation concerns DV\psi. Neither filtered vector is assumed to have a regular trajectory. In the diagram, K is the degree exponent produced by forward propagation. The zero filter D has degree at most m^r. Reverse propagation approximates V^\dagger DV\psi by G\psi, where G has degree at most m^{r'}; the proof below constructs these objects and proves the displayed bounds.

The proof descends from the automatic zero tail above m to the requested threshold. At one step, a known tail bound above m^u controls the tail filter’s global growth while the filter detects the new band m^h\le H_A\le m^u. The following lemma supplies compatible exponent gaps for this step. Here u,h are the exponents for the old and new count thresholds, with tail precision exponents u-\alpha and h-\beta. The tail filter has suppression exponent s and degree and logarithmic norm bounds with exponent k, enlarged to K under forward propagation; the zero filter has precision exponent q and degree and logarithmic norm bounds with exponent r, enlarged to r' under reverse propagation. Finally, v_0<v are initial count cutoff exponents, and \theta is the precision exponent used for the assumed input tail above m^{v_0}.

6 Projection insertion and pruning

The tail-transport lemma assumes regularity of two trajectories. We first use it on successively longer suffixes to show that inserting a product projection preserves regularity. Two consequences for product starting states will then allow us to construct regular trajectories by deleting reflections with small projection probability. Gate deletion by a telescoping estimate also appears in the Pauli-spectrum approach of Nadimpalli et al. [17, Lemma 23 and Claim 24], where width controls a Frobenius-norm error. Here deletion depends on projection probability in the actual pruned-prefix state; the retained layer must then satisfy the regularity condition for every fixed exponent triple.

7 The parity contradiction

We now prove Theorem 1.1. Reference qubits retain the classical input labels while the computation may freely change all its own registers. Their signed acceptance expectation will be bounded away from zero by correctness and forced to zero by Corollary 6.2.

Majority and symmetric-function consequences. Xu and Li’s Theorem 1 and Corollaries 2–3 [22], combined with Theorem 1.1, also exclude strict majority and broader symmetric families in our circuit model. More precisely, let f_n:\{0,1\}^n\to\{0,1\} be symmetric, let f_{n,i} be its value at Hamming weight i, and set \rho(f_n)=\max_{0\le i<n:\,f_{n,i}\ne f_{n,i+1}}\min\{i+1,n-i\}, with zero for constant functions. Its pointwise-error-1/3 approximate degree is the least total degree of a real polynomial p satisfying \lvert p(x)-f_n(x)\rvert\le1/3 for every x\in\{0,1\}^n. Assume either \rho(f_n)\ge n^\delta for all sufficiently large n for some fixed \delta>0, or this approximate degree is \Omega(n^{1/2+\eta}) for some fixed \eta>0. Then no fixed-depth, polynomial-total-qubit family achieves success at least 1/2+1/(\log n)^\gamma on every input at every sufficiently large length, for any fixed \gamma>0. Strict majority \operatorname{MAJ}_n(x)=\mathbf1_{\{|x|>n/2\}} has \rho=\lceil n/2\rceil.

The full formulation and output-model adjustment are recorded in the companion article [18, Section 7.3, Corollary 10 and its proof]. Xu and Li require the measured output to be an initially zero ancilla. Appending one fresh zero qubit and a CNOT from our designated output preserves its measurement distribution even with arbitrary garbage, at a cost of one qubit and one layer. A family with the stated success would, after this adjustment, yield exact clean fanout through the cited reduction. The Hadamard-to-parity argument in Section 1 excludes such fanout using our own Theorem 1.1. Under the same circuit conventions, adjoining unit-cost primitive reversible unbounded threshold gates therefore strictly enlarges the class computed with worst-case success at least 2/3: the reversible gate \lvert x,b\rangle\mapsto\lvert x,b\mathbin\oplus\operatorname{MAJ}_n(x)\rangle computes majority exactly from a zero target.

State-preparation consequences. We justify the bounds stated in Section 1. Keep every register of a preparation when using its unitary and inverse. For felinity \eta>0 on h retained qubits of a depth-d preparation on M total qubits, the constructions in the proofs of Gretta et al. [9, Lemmas 4.9 and 4.1] give target probabilities a\ge1/2 on 0^h and b\ge\eta^2/2 on 1^h, in depth O(d+1) on O(M+h) qubits. The square occurs because the top Fourier coefficient has magnitude at least \eta, so its weight is at least \eta^2.

The exact step is the one-round case of Grier et al. [10, Theorem 7]. For unit vectors g,e, if a unitary V prepares V|0\rangle=(|g,0\rangle+\sqrt3|e,1\rangle)/2, where |0\rangle denotes its entire zero input and the last qubit is a flag, then

Indeed, \langle0|V^\dagger Z_{\rm flag}V|0\rangle=-1/2. A flag initialized to one can be given coefficient c=1/(2\sqrt{a+b})\le1 on zero separately under the two extreme target tests: use a one-qubit reflection sending |1\rangle to c|0\rangle+\sqrt{1-c^2}|1\rangle. Each conditional operation is a product reflection, so the two tests use two sequential Toffoli gates up to local conjugations. The flag-zero component is one half of the normalized union of the extreme branches. The displayed identity prepares that union exactly, with one-branch probability q=b/(a+b)\in[\eta^2/2,1/2], since a+b\le1 and b\le1-a\le a.

Prepare r=\lceil(\log 2)/(-\log(1-q))\rceil=O(\eta^{-2}) independent copies on fresh zeros. Disjoint columnwise reversible ORs into h new targets give probabilities u=(1-q)^r\in[1/4,1/2] on 0^h and v=1-u\in[1/2,3/4] on 1^h. Here (1-q)/2\le(1-q)^r\le1/2 and r\le(\log 2)/q+1. Giving these two branches flag-zero coefficients 1/(2\sqrt{2u}) and 1/(2\sqrt{2v}), respectively, and applying the same exact round yields

with unit branch states \alpha_0,\alpha_1 containing arbitrary garbage. The all-zero reflection is likewise a local conjugate of a Toffoli, and the ORs use Toffoli gates with local X gates. The source-dependent flag coefficients are permitted by nonuniformity. Rosenthal’s exact case [20, Theorem 3.1] converts this balanced state, using its complete preparation and inverse, to the exact parity unitary on arbitrary states of the h+1 data qubits, with every auxiliary qubit restored to zero, and hence to clean F_h by Hadamard conjugation. All copies run in parallel, and only a constant number of preparation and reflection stages are sequential. The resulting depth is O(d+1) and the total-qubit bound is O((M+h)\eta^{-2}+h).

For the Dicke bound, use the block calculation in the proof of the companion’s state-preparation corollary [18, Section 7.1, Corollary 8 and its proof]. Uniformly for n\ge2k and all sufficiently large k, it shows that an input at trace distance at most 1/(80k) from |D_k^n\rangle produces, with \ell=\lfloor k/\log k\rfloor fresh targets and three additional allowed layers, a retained state of felinity greater than 1/(40k). Applying the preceding promotion at arity \ell gives exact parity in fixed depth on O((M+n)k^2) total qubits. For fixed 0<\delta<1 and k\ge n^\delta, eventually \ell\ge\sqrt k, so n\le k^{1/\delta}\le\ell^{2/\delta} and k^2\le\ell^4 make this one fixed polynomial bound in \ell. A felinity violation \eta\ge n^{-A} similarly preserves one fixed polynomial bound at arity n. Both parity arities grow along any unbounded sequence of violations, contradicting the eventual pointwise Theorem 1.1.