slug: visomirski-griffin-2026-frozen-wave-conjecture
bibkey: visomirskigriffin2026spatial
doi: 10.1016/j.chaos.2026.118260
url: https://arxiv.org/abs/2601.06314v1
triage: theorem
motivation_gids:
  - D5/S3/FluidDynamics/ReactionDiffusion/VisomirskiGriffinFrozenWaveRefutation.result

Visomirski–Griffin frozen-wave conjecture

Problem

Matthew Visomirski and Christopher Griffin, Coherent Structures and Travelling Waves in Spatial Replicators from a Biased Volterra Lattice, arXiv:2601.06314v1, Section 4.2, Conjecture 2 (page 19), asserts:

Consider the one-dimensional spatial replicator, i.e., Eq. (6) with reaction term generated by a $2n$-cycle with $n\ge 2$. Assuming periodic boundary conditions and initial conditions of the form,

$$u_i(x,0)=\{\begin{array}{ll}(1+\sin x)/(2n),& i\text{ odd},\\ (1-\sin x)/(2n),& i\text{ even},\end{array}$$

the dynamics will evolve to a frozen wave stationary solution. That is, ecological niches will be maintained.

The simulated parameters are $a=-1/2$, $D=1/100$, on $[-\pi ,\pi ]$ with periodic boundary conditions. Preregistration: #14205.

Motivation

The settlement declaration is D5/S3/FluidDynamics/ReactionDiffusion/VisomirskiGriffinFrozenWaveRefutation.result. It refutes the fixed-parameter assertion for every even-cycle size, using $n=25$. The proof also establishes the family homogenization statement VG-R on its live path. The supporting result D5/S3/FluidDynamics/Fourier/PeriodicAllenCahnDecay.odd_allen_cahn_decay is the generic scalar decay theorem.

Gap

The source reports frozen-wave simulations for small even cycles. Those simulations do not establish the universal size quantifier. The implemented claim asks that, for every $n\ge 2$, there exist a classical solution with the stated data and a continuous, $2\pi$-periodic, nonhomogeneous limit approached in squared $L^2$ distance. The stationary PDE is omitted from the limit predicate; the source’s stationary-wave assertion implies this weaker claim. Refuting it therefore excludes the source’s frozen-wave reading at the simulated parameters.

The classical-solution record retains continuity at time zero, spatial periodicity, the time and first and second space derivatives and their joint continuity at positive times, the literal spatial equation, and the initial data. It supplies no solution-existence theorem or positivity assumption.

Route

Uniqueness follows from finite-strip boundedness, local Lipschitz continuity of the finite-dimensional reaction, and a vector energy estimate. The shift by two species and the reflection combined with a shift by one preserve the equation and data, so uniqueness enforces two parity profiles $p,q$ and $p(t,-x)=q(t,x)$. A zero-data linear energy argument gives $p+q=1/n$.

Consequently $w=n(p-q)$ is odd and solves

$$w_t=D{w}_{xx}+\mu (w-w^3),\mu =-a/(2n).$$

For $Y(t)={\int }_{-\pi }^{\pi }w(t,x{)}^{2}dx$, periodic integration by parts and the mean-zero Wirtinger inequality yield

$$\frac{1}{2}{Y}^{\prime }=-D\int w_x^2+\mu Y-\mu \int w^4\le -(D-\mu )Y-\mu \int w^4.$$

When $D>\mu \ge 0$, the exponential comparison gives $Y\to 0$. When $D=\mu >0$, the bound $Y^2\le 2\pi \int w^4$ gives the Riccati estimate $Y(t)\le Y(0)/(1+(\mu /\pi )Y(0)t)$, again tending to zero. The species deviations are $\pm w/(2n)$, so every squared distance to $1/(2n)$ tends to zero.

Falsifier

At $n=25$, $a=-1/2$, $D=1/100$, one has $\mu =D$. Every classical solution with the specified data homogenizes. A continuous periodic profile differing from the homogeneous equilibrium has strictly positive squared distance from it; the squared-distance comparison excludes convergence to such a profile. This contradicts claim without assuming that a classical solution exists.

Evidence

Proved (kernel-checked declaration): result : ¬ claim in the settling module. Its live proof path contains private vg_family, solution_pair, classical_unique, homogenizes_of_parity_scalar, and homogenizes_excludes_frozen, and consumes the supporting public odd_allen_cahn_decay. All declarations use only the permitted axiom closure.

Literature reading: the source expressions and conjecture are recorded in the Library note, with locators for Sections 2, 3 and 4.2. The Chafee–Infante stationary-profile reading below uses Moreira–Valero, arXiv:2107.04334v1, equations (4)–(5), Theorem 2.1 (page 4), with diffusion coefficient identically one and $f(u)=u-u^3$.

Triage

What the settlement shows

• Proved — mechanism; kernel-checked declarations. Private solution_pair forces the two parity profiles, $p+q=1/n$ and their reflection relation. Private vg_family derives the odd Allen–Cahn equation with $\mu =-a/(2n)$ and applies odd_allen_cahn_decay. For $-1<a\le 0$, $D>0$, $n\ge 2$ and $2nD+a\ge 0$, every classical solution with the source data homogenizes. The critical equality is included; loss of niches is not restricted to a strictly diffusive inequality.
• Proved — authors’ threshold; kernel-checked specialization and paper argument. At the simulated parameters $\mu =1/(4n)$. For $n\ge 2$, multiplying $1/(4n)\le 1/100$ by the positive number $400n$ gives $100\le 4n$, equivalently $n\ge 25$. Thus every cycle with at least 50 species homogenizes by VG-R. The boundary $n=25$ is the numerical specialization used in result.
• Proved — nonzero stationary profiles when $D<\mu$; literature reading and paper argument. With $D>0$, rescale time by $D$ and put $\lambda =\mu /D>1$. The Chafee–Infante Dirichlet problem on $(0,\pi )$ is $w_t=w_{xx}+\lambda (w-w^3)$. Moreira–Valero, arXiv:2107.04334v1, Theorem 2.1 (page 4), applied to constant diffusion one, gives a nonzero pair after the first bifurcation at $\lambda =1$; equations (4)–(5), pages 2–3, state the classical problem and its stationary equation. Odd reflection across the zero endpoints produces odd $2\pi$-periodic stationary profiles: the ODE gives $w_{xx}=0$ at an endpoint, so value, first derivative and second derivative match in the extension. This correspondence is a paper argument, not a Lean existence theorem. At the authors’ parameters this covers $2\le n\le 24$, including six- and eight-cycles ($n=3,4$).
• Open — convergence of the conjecture’s data when $D<\mu$; literature boundary. Whether the alternating sine data converge to one of these nonzero profiles is not settled in this delivery. Stationary existence alone does not prove that convergence.
• Open — positive bias $a>0$; energy-sign boundary. Then $\mu <0$ and the cubic energy contribution $-\mu \int w^4$ is nonnegative. The delivered theorem assumes $\mu \ge 0$ and does not settle this regime. A maximum-principle bound such as $|w|\le 1$ is a separate obligation.
• Open — per-size parameter choice; quantifier boundary. The reading “for each $n$ there are $a,D$” is not refuted by a fixed-parameter universal claim’s failure. The delivered result does not settle that reading.
• Open — other source results; scope boundary. The refutation does not invalidate the source’s four-cycle analysis, the reported six- and eight-cycle simulations, or its separate travelling-wave results. A theorem or interpretation requiring the conjecture uniformly in cycle size must retain the diffusion threshold or replace that premise. No dependency analysis of the paper’s other results is proved here.
• Open — other domain lengths. Rescaling suggests a threshold involving the first Laplacian eigenvalue; the formal delivery fixes period $2\pi$. No general-period theorem is supplied.

There are no computed Triage claims or numerical simulation scripts in this dossier. The threshold equivalence is the exact paper argument above, with the boundary specialization checked in the Lean refutation.

ASSUMED-UNVERIFIED

The source and literature checks establishing that the named conjecture was unsettled are the preregistration’s recorded literature evidence. They are not Lean statements. Existence of a classical replicator solution, convergence in the complementary diffusion regime, positive bias, and the per-size parameter reading remain outside the formal result. The stationary-profile citation and its odd-reflection correspondence are literature and paper evidence, not kernel-checked existence or convergence results.