Counterexamples, Nonconverses, and Hypothesis Stress Tests

A counterexample is informative only when it satisfies the hypotheses of the statement it defeats. Removing a hypothesis and losing the conclusion instead tests necessity; finding the conclusion without a hypothesis defeats a converse; and finding that an expression is undefined exposes a domain error rather than a false theorem. This page gives a repeatable method and constructs a Euclidean-invariant Gaussian covariance that is regular away from coincidence but cannot reconstruct a positive Hilbert space.

Required background. Theorem-First Claim Records fixes quantified statements; Existence, Construction, Reconstruction, and Continuum-Limit Claims separates theorem types; Equivalence, Uniqueness, and Comparison Notions fixes the comparison relation; and Positivity, Spectrum, Covariance, and Locality Hypotheses separates structural assumptions. Helpful background. Structural Hypotheses and Their Failure Modes gives physical examples; Basis Dependence, Computational Complexity, and Why There Is No Universal Cure shows why a change of variables is not a universal solution; Rényi Entropies and Analytic Continuation and Analytic-Continuation Failure Modes test continuation claims; Nonlocal Modular Generators and Control tests exceptional geometric cases; Schwinger and Gravitational Production: A Controlled Comparison illustrates limited analogies; and Trans-Planckian Sensitivity and Universality separates robust conclusions from regulator dependence.

The logic of a stress test

Write a proposed theorem as

$$\forall X\in\mathcal C, \qquad H_1(X)\wedge\cdots\wedge H_r(X) \Longrightarrow Q(X).$$

There are four distinct negative results.

1. Counterexample to the theorem: an $X\in\mathcal C$ satisfying every $H_i$ but not $Q$. One such object refutes the statement as written.
2. Necessity witness for $H_k$: an $X$ satisfying the remaining hypotheses and $\neg H_k$, with $\neg Q$. This shows that the proof cannot simply drop $H_k$; it does not prove that every version of the theorem needs exactly that assumption.
3. Counterexample to the converse: an $X$ satisfying $Q$ but not all of the $H_i$. This refutes $Q\Rightarrow\bigwedge_iH_i$ while leaving the forward theorem intact.
4. Failure of definition: an attempted $X$ is outside $\mathcal C$, or $H_i(X)$ and $Q(X)$ are not defined. Plane-wave smearing of a merely compactly supported distribution and an unrenormalized coincident product are examples. They cannot refute a theorem whose objects exclude them.

“No proof is known” is a fifth status, not a negative result. An open converse must remain open until an example or proof settles it. Likewise, a numerical search that finds no counterexample in a finite family establishes only that bounded search result.

A disciplined counterexample construction

A useful sequence is:

Normalize the statement. Record the object class, all quantifiers, domains, hypotheses, and the exact relation in the conclusion. Negate the conclusion literally. The negation of “there exists a unitary intertwiner” is “no such unitary exists,” not merely “the obvious map is not unitary.”

Vary one feature first. Hold the remaining hypotheses fixed while removing or weakening one. This makes the cause of failure identifiable. If several conditions change simultaneously, the example may still refute a universal statement but says less about which condition matters.

Search the smallest adequate class. Finite-dimensional and Gaussian models expose linear-algebra errors; generalized free fields vary the spectrum while retaining Wick factorization; free fields on nontrivial backgrounds test geometry; thermodynamic limits test representation and convergence claims. The model must still belong to the theorem’s declared class.

Check the example independently. Verify every retained hypothesis from the definition, not from a label such as “physical” or “regular.” Compute the failing quantity explicitly whenever possible.

State the survivor. A failed strong conclusion often leaves a useful weaker result—Euclidean invariance but no reflection positivity, local quasiequivalence but no global unitary, or convergence of selected correlators but no reconstructed field theory.

First application: a positive-looking covariance that fails reflection positivity

Let $0<m_1<m_2$ and consider the centered Gaussian Euclidean field with momentum-space covariance

$$C(p_E) = \frac{1}{(p_E^2+m_1^2)(p_E^2+m_2^2)}.$$

This is a tempered, rotationally invariant function of $p_E^2$, smooth for real $p_E$, strictly positive at every real momentum, and ultraviolet-improved relative to a free scalar covariance. These properties make it an effective test of the false implication

$$\text{Euclidean invariance + pointwise positive covariance} \quad\Longrightarrow\quad \text{reflection positivity}.$$

Partial fractions give

$$C(p_E) = \frac{1}{m_2^2-m_1^2} \left( \frac{1}{p_E^2+m_1^2} - \frac{1}{p_E^2+m_2^2} \right).$$

The second free covariance enters with a negative residue. Arici and collaborators prove that a real rational covariance with no pole on the nonnegative real axis is reflection positive exactly when every pole is simple, lies on the negative real axis, and has nonnegative residue Arici et al. 2018, Theorem 3.7, pp. 7–8. Our covariance fails the last condition and therefore cannot yield a positive Osterwalder–Schrader Hilbert space.

The failure can be seen directly. Fourier transform only the Euclidean time coordinate and fix a spatial momentum $\mathbf p$. Write

$$\omega_i(\mathbf p)=\sqrt{\mathbf p^2+m_i^2}.$$

For a test function supported at $t>0$, the reflected quadratic form at this momentum is proportional to

$$Q_{\mathbf p}(f) = \frac{1}{m_2^2-m_1^2} \left[ \frac{\lvert F(\omega_1)\rvert^2}{2\omega_1} - \frac{\lvert F(\omega_2)\rvert^2}{2\omega_2} \right], \qquad F(\omega)=\int_0^\infty e^{-\omega t}f(t)\,dt.$$

Choose $0<b<a$ and first use the distributional probe

$$f_0(t)=\delta(t-a)-e^{-\omega_1(a-b)}\delta(t-b).$$

Then $F_0(\omega_1)=0$, while $F_0(\omega_2)\neq0$ because $\omega_2\neq\omega_1$. Smooth compactly supported bumps around $a$ and $b$ approximate this probe. Localizing the spatial Fourier transform tightly around the chosen $\mathbf p$ keeps the positive term arbitrarily small and the negative term nonzero, so a genuine smooth positive-time test function has $Q(f)<0$. This is the constructive mechanism behind the negative-residue obstruction.

What survives? The covariance remains Euclidean invariant, tempered, pointwise positive as a function, and defines a Gaussian distribution before reflection reconstruction. What fails is positivity of the reflected quadratic form. Without that form, quotient and completion do not produce the desired positive Hilbert space. The relevant reconstruction theorem and its precise hypotheses are developed at Reflection Positivity and Osterwalder–Schrader Reconstruction.

Nonconverses must preserve the direction of proof

Osterwalder–Schrader reflection positivity is sufficient, together with the other Euclidean axioms and the corrected regularity assumptions, for a positive reconstructed theory Osterwalder and Schrader 1975, §§ II–V, pp. 283–297. The example above shows that ordinary pointwise positivity is not a substitute; it does not refute the reconstruction theorem. To challenge that theorem one would need Euclidean data satisfying all of its stated axioms whose reconstruction nevertheless fails.

Similarly, a mass gap often helps prove exponential clustering, but a clustering statement does not automatically imply an isolated particle pole. A generalized free field with purely continuous spectral weight bounded away from zero clusters while having no atomic mass shell. It is a counterexample to that converse, not to the forward decay theorem under its own assumptions.

For equivalence claims, matching a finite family of observables can follow from a true equivalence, but the reverse implication fails without a determining family, continuity, representation data, and an inverse. The distinction between necessary consequences and sufficient hypotheses prevents a check from being promoted into a classification theorem.

Five cases that a free massive example does not test

Verifying a statement only for the standard free massive scalar explores an unusually favorable point: Gaussianity, a unique Poincaré-invariant vacuum, a mass gap, a simple pole, reflection positivity, and no boundary. Before calling the result universal, test at least the following controlled variations.

Massless field

Remove the gap while retaining linearity and locality. Correlations normally decay by a power rather than exponentially, zero modes can obstruct states in low dimension, and infrared limits may depend on the test-function subspace. A proof using $e^{-mL}$ or division by $m$ has exposed its missing hypothesis.

Generalized-free field

Retain Gaussian factorization but replace the single mass pole by a continuous positive spectral measure. Positivity, covariance, locality, and clustering can remain true while the Klein–Gordon equation and isolated-particle conclusion fail. This isolates which steps used a single-shell spectrum rather than Gaussianity.

Thermal state

Replace the vacuum by a KMS state. Spatial translations and a preferred time evolution may remain, but full Lorentz invariance and the vacuum spectrum condition do not. The thermal spectrum is encoded through the KMS analyticity relation, not support only in $\overline V_+$ Haag, Hugenholtz, and Winnink 1967, §§ 2–3, pp. 219–228. A theorem about vacuum reconstruction cannot be applied by renaming a KMS vector “the vacuum.”

Boundary or defect

Keep a free bulk equation but introduce a boundary condition. Normal translations are broken, image singularities appear, and the allowed test functions and causal complements change. Boundary renormalization generally permits local terms absent in the bulk; Symanzik’s construction makes the extra surface interactions explicit Symanzik 1981, §§ 2–4, pp. 5–24. A bulk theorem survives only after its geometry and support assumptions are reformulated.

Indefinite-metric field

Keep manifest covariance and local field equations but permit an indefinite inner product, as in an auxiliary covariant gauge description. Then $\langle\Psi,\Psi\rangle$ may be negative before a physical-state quotient. A Hilbert-positivity theorem cannot use the auxiliary space as its target; one must identify the constraint, null subspace, quotient, and physical observables.

These tests are not a claim that every theorem must include all five cases. They determine its correct domain. A theorem explicitly restricted to massive scalar vacuum models is not weakened by excluding thermal or boundary systems; only an unwarranted universal label is removed.

Independent checks

Residue check. Compute every pole and residue of a rational covariance symbolically and numerically. A sign error in the partial fraction decomposition reverses the reflection-positivity diagnosis.

Smooth-test check. Replace delta probes by compactly supported bumps strictly inside $t>0$. Negativity persisting under sufficiently narrow approximation confirms that the counterexample lies in the actual test-function class.

Quantifier check. Negate the exact theorem in formal order. The negation of $\forall X\,\exists Y\,R(X,Y)$ is $\exists X\,\forall Y\,\neg R(X,Y)$; failing to find one preferred $Y$ is insufficient.

Limit check. Follow the example as $m_2\to m_1$. The covariance tends, after rescaling, toward a double pole, which also violates the rational reflection-positivity criterion. Thus the obstruction does not disappear at degeneracy; its form changes from a negative residue pair to a multiple pole.

Neighboring-theorem check. Confirm that the example does not accidentally violate a hypothesis that was meant to remain fixed. If it does, state only the weaker role it actually establishes.

Common pitfalls

Choosing an object outside the theorem. A non-tempered kernel cannot refute a theorem about tempered distributions. It can show that temperedness is not automatic in a broader class.

Varying every assumption at once. Such an example may refute universality but rarely identifies the operative hypothesis. A one-feature deformation is usually more explanatory.

Calling a failed search a no-go theorem. Finite scans and numerical optimization have bounded domains and tolerances. They do not establish a universal negative statement without a completeness argument.

Exercises

1. Classify three negative results. Let $P\Rightarrow Q$ be a theorem. Classify objects satisfying (a) $P\wedge\neg Q$, (b) $\neg P\wedge\neg Q$, and (c) $\neg P\wedge Q$.

Solution

(a) is a counterexample to the theorem. (b) is consistent with the theorem and may show why $P$ is useful, but it does not prove necessity in every formulation. (c) is a counterexample to the converse $Q\Rightarrow P$. The truth table prevents a nonexample from being reported as a refutation.

2. Produce the negative reflection form. Verify that the two-delta probe above cancels the $m_1$ contribution but not the $m_2$ contribution.

Solution

Its Laplace transform is

$$F_0(\omega)=e^{-\omega a} -e^{-\omega_1(a-b)}e^{-\omega b}.$$

At $\omega=\omega_1$ the two terms agree, so $F_0(\omega_1)=0$. If $F_0(\omega_2)=0$, then $e^{-(\omega_2-\omega_1)(a-b)}=1$, impossible because $\omega_2>\omega_1$ and $a>b$. Hence only the negative-residue term remains in $Q_{\mathbf p}$ and the form is negative. Smooth bumps approximate the calculation by continuity.

3. Test a universal clustering claim. The massive free scalar clusters exponentially. Explain why this one example cannot prove that every local QFT clusters exponentially.

Solution

The proof uses an isolated positive mass gap. A massless field has long-range power-law correlations, a generalized free field can have a continuum reaching zero, and a thermal or symmetry-broken phase may have different large-distance behavior. The strongest conclusion from the example is exponential clustering for that massive model under its vacuum assumptions. A universal theorem would need hypotheses controlling the infrared spectrum and vacuum sector.

4. Separate a boundary failure from a theorem failure. A Poincaré-covariant theorem is applied on a half-space with reflecting boundary and fails under normal translations. What follows?

Solution

The half-space is not invariant under the full Poincaré group, so it lies outside the original background class. The example does not refute the theorem. It shows that a boundary version must replace the symmetry group, specify boundary conditions and boundary-local counterterms, and reprove the conclusion under those altered hypotheses.

References

• Arici, Francesca, Daniel Becker, Chris J. Fewster, Rainer Verch, and Miles Visser. “Reflection Positivity in Higher Derivative Scalar Theories.” Journal of Mathematical Physics 59 (2018): 082301. DOI. Open PDF.
• Haag, Rudolf, Nicolaas M. Hugenholtz, and Marius Winnink. “On the Equilibrium States in Quantum Statistical Mechanics.” Communications in Mathematical Physics 5 (1967): 215–236. DOI.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II.” Communications in Mathematical Physics 42 (1975): 281–305. DOI. Open PDF.
• Symanzik, Kurt. “Schrödinger Representation and Casimir Effect in Renormalizable Quantum Field Theory.” Nuclear Physics B 190 (1981): 1–44. DOI.