Uniqueness of the KPZ Fixed Point Among Scaling-Covariant Local Fields

Uniqueness of the KPZ Fixed Point Among Scaling-Covariant Local Fields

Definitions and mathematical statement

Use the state space $\mathrm{UC}$ of upper-semicontinuous profiles $h:{\mathbb{R}}\to{\mathbb{R}}\cup\{-\infty\}$, not identically $-\infty$, with $h(x)\le C(1+|x|)$ for some $C$. The KPZ fixed point $\mathfrak h(t,x;h_0)$ is the profile-valued process defined by Matetski--Quastel--Remenik's Definition 3.12 and transition formula $4.12$.

The uniqueness question of Remark 4.12 specifies locality $4.19$ and Theorem 4.5(i), (iii), (iv). Locality requires changes to $h_0$ outside neighborhoods of finitely many observation points to change their joint distribution by $o(t)$ as $t\downarrow0$. Scaling means, for $\alpha>0$, \[ \alpha\mathfrak h(\alpha^{-3}t,\alpha^{-2}x; \alpha^{-1}h_0(\alpha^2\cdot)) \overset{\rm law}=\mathfrak h(t,x;h_0). \] The other two properties are skew time reversibility, interchanging initial and test profiles with sign reversal, and covariance under simultaneous spatial translation. Their full profile-event formulations are retained by the exact theorem reference. The target is uniqueness in law of the nontrivial field satisfying this package, not scaling invariance alone, and not uniqueness of solutions to the KPZ stochastic partial differential equation.

Formulation and concept references: [S1].

Short English statement

Do the specified locality and symmetry requirements characterize the KPZ fixed point uniquely among nontrivial space-time fields?

Research attempt

A normalization issue in the specified axioms.

This attempt was prepared on 27 September 2026. The exact references in [S1] were inspected: Remark 4.12, locality $4.19$, and Theorem 4.5(i), (iii), (iv). The selected package has a time-calibration ambiguity. If a field $H$ satisfies it, then $H_c(t,x;h)=H(ct,x;h)$ satisfies the same package for every fixed $c>0$. For the KPZ fixed point these laws are distinct. A detailed proof is given below, including a nondegeneracy check. Thus literal uniqueness with fixed numerical units cannot follow from only the properties enumerated in the catalogue.

This is a formulation diagnostic, not a claim to settle an intended uniqueness question with an implicit normalization or identification under changes of units. The original statement is preserved. The present attempt remains marked unresolved because it does not establish a characterization after calibration, nor reconstruct all intended hypotheses behind the source's informal conjectural remark. It would be misleading to add a time normalization silently and announce a proof of that changed question.

Recent uniqueness results must also be distinguished by their inputs. Dauvergne--Zhang [E1] characterize a coupling producing the directed landscape when the KPZ fixed-point marginals are already prescribed, together with additional structural conditions. Their primary abstract does not assert that the four symmetries/locality requirements here determine those marginals in the first place. No such theorem is imported into this attempt.

Write the locality and skew events explicitly.

For initial profiles $h,\widetilde h$ agreeing on neighborhoods of the finitely many observed points $x_1,\ldots,x_m$, locality compares their joint distribution functions at fixed thresholds $a_1,\ldots,a_m$: \[ \left|\Pr\{H(t,x_i;h)\le a_i\ \forall i\} -\Pr\{H(t,x_i;\widetilde h)\le a_i\ \forall i\}\right|=o(t). \] The profiles and neighborhoods remain fixed as $t\downarrow0$. This does not specify the distribution when the nearby initial data are the same, nor a coefficient for its evolution in time.

The skew reversal property is the equality of full profile events \begin{equation}\label{wsref-0-61-34-35-32-3a-73-6b-65-77} \Pr\{H(t,\cdot;g)\le-f\} =\Pr\{H(t,\cdot;f)\le-g\}. \end{equation} The inequalities are imposed at every spatial point in the event, as in the source's profile formulation. They are not an assertion that the fields themselves have identical finite-dimensional distributions after arbitrary interchange of profiles. Keeping that distinction prevents an incorrect strengthening of the symmetry.

Lemma 452.1: deterministic changes of time preserve the package.

Let $H$ satisfy the stated scaling, skew reversal, spatial covariance, and locality. For a constant $c>0$ define \[ H_c(t,x;h_0)=H(ct,x;h_0). \] Then $H_c$ satisfies all four properties. If $H$ additionally has the prescribed initial condition, is a time-homogeneous Markov process, or has a Feller semigroup, each of these extra properties is preserved too.

Proof of scaling. The left side of the scaling identity for $H_c$ is \[ \alpha H(c\alpha^{-3}t,\alpha^{-2}x; \alpha^{-1}h_0(\alpha^2\cdot)). \] Apply the identity for $H$ at time $ct$. The result in law is $H(ct,x;h_0)=H_c(t,x;h_0)$. The constants $c$ and $\alpha$ commute; the identity puts no restriction on $c$.

Proof of the other symmetries. Substitute $ct$ for $t$ in (a452:skew). Its two sides become exactly the two required events for $H_c$. Substitution into spatial covariance similarly leaves the same spatial shift and initial-profile shift. No change to the space or height coordinates is required.

Proof of locality. Denote the original difference of probabilities at time $u$ by $r(u)$, so $r(u)/u\to0$. The difference for $H_c$ is $r(ct)$, and \[ \frac{r(ct)}t=c\frac{r(ct)}{ct}\longrightarrow0. \] For the exponential estimate discussed in [S1], replacing $t$ by $ct$ merely replaces the positive constant in $\exp(-C\delta^3/t^2)$ by $C/c^2$.

Finally, $ct\downarrow0$ with $t\downarrow0$, so an initial-data limit is unchanged. If $P_t$ is the transition semigroup, then $P_t^{(c)}=P_{ct}$ satisfies $P_t^{(c)}P_s^{(c)}=P_{c(t+s)}=P_{t+s}^{(c)}$. The Feller mapping and continuity properties are inherited by the same substitution.$\square$

In particular, adding just Markovness to the listed package does not repair the time-calibration issue. Nor does preservation of a fixed invariant distribution: if $\mu P_t=\mu$ for all $t$, then $\mu P_{ct}=\mu$ for all $t$. The Brownian statement elsewhere in Theorem 4.5 concerns the centered profile $\mathfrak h(t,x;B)-\mathfrak h(t,0;B)$; this centered-profile law is also insensitive to the speed change. The uncentered Brownian law is not invariant, since the evolution adds a random global height.

Lemma 452.2: the KPZ laws in this family are distinct.

Let $Z=\mathfrak h(1,0;0)$ for flat initial data. The one-point law is a nondegenerate, finite real law; [S1], Section 4.4, identifies the flat profile with the Airy$_1$ process, and the corresponding GOE Tracy--Widom distribution is also used in its one-point estimates. Only the fact that $Z$ is not almost surely zero is needed here. Scaling at the flat profile gives \[ \mathfrak h(t,0;0)\overset{\rm law}=t^{1/3}Z, \qquad H_c(1,0;0)\overset{\rm law}=c^{1/3}Z. \]

For any finite real random variable $Z$ not concentrated at zero, and any $a>0$ with $a\ne1$, the laws of $Z$ and $aZ$ differ. Indeed, replace $a$ by its inverse if necessary to assume $a>1$. Equality in law would give, for every $r>0$ and $n\ge1$, \[ \Pr(|Z|>r)=\Pr(|Z|>a^nr). \] The right side tends to zero because $Z$ is finite almost surely. Thus $\Pr(|Z|>r)=0$ for every positive rational $r$, forcing $Z=0$ almost surely, a contradiction. It follows that $H_c$ and $H_1$ already differ at the single observation $(t,x,h_0)=(1,0,0)$ whenever $c\ne1$. $\square$

This argument uses no fitted variance or approximate quantile. It rules out the possibility that the time change is merely a different notation for exactly the same fixed-coordinate law. If the intended uniqueness is instead modulo choices of time units, then these examples belong to one allowed equivalence class and the normalized uniqueness problem is still entirely open within this attempt.

A second diagnostic: the four properties alone do not impose a semigroup.

Let $Z$ take the values $-1$ and $1$ with equal probability, and define on the whole stated profile space \begin{equation}\label{wsref-0-61-34-35-32-3a-76-65-72-74-69-63-61-6c} V(t,x;h_0)=h_0(x)+t^{1/3}Z, \qquad -\infty+z=-\infty. \end{equation} Adding a finite constant preserves upper semicontinuity and the allowed linear upper bound. The profile is not identically $-\infty$ if the initial profile was not. It tends back to its initial profile as $t\downarrow0$.

Scaling is a pathwise identity if the same $Z$ is used on both sides: $\alpha(\alpha^{-3}t)^{1/3}Z=t^{1/3}Z$. Spatial covariance is immediate. For any fixed finite observation set, changing the initial profile away from those points changes none of their values, so locality holds with zero error. Finally the event in (a452:skew) becomes \[ t^{1/3}Z\le\inf_x[-f(x)-g(x)], \] which is symmetric in $f,g$, with the natural extended-real convention when the bound is $+\infty$. Thus this field satisfies the literal four conditions as well.

However its proposed transition kernels do not form a time-homogeneous semigroup. Starting from the zero profile, one step of length two gives height $2^{1/3}Z$ at the origin, whose probability of being at most zero is $1/2$. Two independent kernel steps of length one give $Z_1+Z_2$, whose probability of being at most zero is $3/4$. Therefore $P_2\ne P_1P_1$. This is not proposed as an alternative KPZ Markov process. It shows why an axiom package phrased only as one-time distributional identities does not automatically include temporal consistency. The speed-change family from Lemma 452.1 survives even when that consistency is expressly added.

The identity evolution $V(t,\cdot;h_0)=h_0$ is another degenerate example. Calling the field "nontrivial" presumably excludes frozen evolution, but that phrase alone does not choose among the positive speeds of a genuinely evolving Markov process. The nondegeneracy test above avoids relying on a disputed interpretation of triviality.

A natural noise repair also fails in a simple class.

Could the vertical example be repaired by taking a process $Z_t$ with independent stationary increments and the same $1/3$ self-similarity? Assume finite second moments and $Z_0=0$. Such increments imply $\operatorname{Var}(Z_n)=n\operatorname{Var}(Z_1)$ at positive integers. Self-similarity would give $\operatorname{Var}(Z_n)=n^{2/3}\operatorname{Var}(Z_1)$. At $n=2$ these equalities force the variance to be zero. Likewise $\mathbb E Z_n=n\mathbb E Z_1=n^{1/3}\mathbb E Z_1$ forces the mean to be zero. At every fixed $t$, self-similarity then makes $Z_t=0$ almost surely. With a continuous or cadlag version the entire process is zero, by first intersecting the probability-one events at rational times and then using path regularity.

This proves failure of this particular finite-variance additive-noise repair. It is not a nonexistence theorem for KPZ fluctuations: their profile dynamics do not consist of adding one independent scalar noise increment uniformly over all space at every step.

Possible calibrations are additional assumptions.

One can exclude the family $H_c$ by prescribing the flat one-point law at time one, because Lemma 452.2 then forces $c=1$. Alternatively the full affine-covariance formula in Theorem 4.5(vi), which is not among the three selected symmetry items, contains numerical time coefficients. For a slope $v$, the standard formula has spatial displacement $vt/2$ and vertical correction $v^2t/4$. For $H_c$ these become $cvt/2$ and $cv^2t/4$, respectively. Fixing those coefficients addresses this speed freedom in a way the exponent-only scaling law does not.

These are examples of extra data, not axioms silently inserted into the catalogue. Prescribing one marginal or an affine coefficient would still not prove that every other initial profile has the correct transition law. Conversely prescribing the entire transition formula $4.12$ would define the KPZ process directly and would no longer establish a characterization from the smaller locality/symmetry package.

An explicit sufficient comparison lemma for a normalized route.

Suppose two Markov semigroups $P_t,Q_t$ act as contractions on a Banach space of bounded observables, and satisfy the much stronger estimate \begin{equation}\label{wsref-0-61-34-35-32-3a-6f-70-65-72-61-74-6f-72} \|P_\delta-Q_\delta\|_{\mathrm{op}}=o(\delta) \quad(\delta\downarrow0). \end{equation} Then $P_t=Q_t$ for every $t\ge0$.

Proof. For $\delta=t/m$, the telescoping identity gives \[ P_\delta^m-Q_\delta^m =\sum_{j=0}^{m-1}P_\delta^{m-1-j} (P_\delta-Q_\delta)Q_\delta^j. \] Contraction bounds its operator norm by $m\|P_\delta-Q_\delta\|=t\,o(\delta)/\delta\to0$. The semigroup identities identify the left side with $P_t-Q_t$, independently of $m$. Therefore it is zero.$\square$

The estimate in the catalogue is not (a452:operator). It compares two initial profiles for the same evolution near fixed observations; it does not compare two candidate evolutions uniformly over profiles and observables. Confusing these statements would erase the main uniqueness problem. A more flexible generator or martingale-problem route would still need a common, sufficiently determining domain and a uniqueness theorem for its evolution. Neither is obtained from the four properties here. In particular time rescaling multiplies any existing generator by $c$ while preserving those properties.

Finite diagnostics and outcome.

The companion checks verify exact finite-profile skew events for the vertical toy, its scaling under rational cube time changes, and the $1/2$ versus $3/4$ two-step distribution discrepancy. They are tests of the counterexample mechanisms and conventions, not simulations of the KPZ fixed point. Distinctness of the genuine time-rescaled KPZ laws rests on the proved dilation lemma and the attributed nondegenerate one-point law, not on a discretized numerical distribution.

Outcome: UNRESOLVED as an intended normalized characterization; the literal listed package has a proved calibration defect. The four enumerated properties cannot distinguish positive speeds of the same nontrivial KPZ Markov evolution. A normalized interpretation requires an explicit additional convention or quotient by changes of units. After that issue is addressed, this attempt supplies no proof that locality and the remaining symmetries identify all transition laws.

Sources

• S1 K. Matetski, J. Quastel, and D. Remenik, The KPZ fixed point, Remark 4.12, Acta Mathematica 2021. https://arxiv.org/html/1701.00018v3. Formulation source. Consulted 24 September 2026: Remark 4.12, locality $4.19$, and Theorem 4.5(i), (iii), (iv).
• E1 Duncan Dauvergne and Lingfu Zhang, Characterization of the directed landscape from the KPZ fixed point, arXiv:2412.13032 $2024$. https://arxiv.org/abs/2412.13032. Primary abstract and authors' publication link inspected 27 September 2026. Its prescribed KPZ marginals and coupling hypotheses are distinguished from the package in [S1].
• E2 K. Matetski, J. Quastel, and D. Remenik, same primary text as [S1], Theorem 4.5, Remarks 4.11--4.12, and Section 4.4, especially $4.24$. https://arxiv.org/html/1701.00018v3. Full symmetry/locality statements and flat-data marginal identification inspected 27 September 2026. The time-calibration diagnostic is proved explicitly above.