Rigorous Mathematical Referee Report on euler-plane-
revised-1.pdf
This report assesses only mathematical validity. It does not evaluate novelty, significance,
exposition, or style. The conclusions below distinguish between proven errors, unresolved
proof gaps, and steps that were successfully checked. [cite:1]

Overall assessment
The paper cannot presently be certified as mathematically correct on the basis of the checked
material. At least one serious defect appears in a stated lemma, and several later central
arguments rely on estimates or compactness/semigroup steps that are not adequately justified
in the manuscript. [cite:1]

Proven error

Lemma 5.1 (page 54)
Lemma 5.1 claims that for every fourfold-symmetric function                            , one has a
global linear bound                   . The proof expands the Biot–Savart kernel on the far region
          , observes that quarter-turn symmetry kills the low harmonics, and then uses the first
surviving term of order        . [cite:1]
That part is structurally plausible, but the displayed estimate is not justified as written. The
relevant far-field contribution is of the form



and the manuscript bounds it by using           inside a radial integral. This only controls the
near portion of the far region; it does not by itself justify the full tail bound as presented, nor
does it establish the claimed global estimate with the stated hypotheses and proof. [cite:1]
More importantly, the lemma is then used as a uniform origin-growth estimate in later
arguments. The manuscript does not clearly state what the constant may depend on, and the
proof does not provide a dependence strong enough for the subsequent cascade estimates. As
written, the lemma is not established in the form later needed. [cite:1]

Consequences of the error
Lemma 5.1 is used later in Proposition 5.3, Proposition 6.4, Lemma 6.7, and Lemma 7.1 for
origin-growth and foreign-field estimates. Those later arguments therefore are not proved by
the text unless Lemma 5.1 is replaced by a correct statement together with a proof supplying
exactly the required uniformity. [cite:1]
Unresolved proof gaps

Theorem 2.12 (pages 27–30)
Theorem 2.12 proves strong vanishing-viscosity precompactness in                      for
compactly supported zero-circulation data. The argument uses renormalization for the limiting
Euler transport equation with velocity in        , followed by exact conservation of limiting
   and norms and then a Clarkson-type argument for strong convergence. [cite:1]
This strategy is reasonable, but the manuscript does not supply enough detail at several
nontrivial points: the commutator convergence is stated locally; endpoint renormalized
identities are asserted for every prescribed time; spatial cutoffs are used to pass to global
conservation; and the weak initial trace is converted into the precise trace required by
renormalization. These steps are delicate, and the proof given is too compressed to verify
rigorously in its present form. Since the exact norm conservation is then used to deduce strong
convergence, this is a substantive gap in the theorem. [cite:1]

Lemma 3.3 (pages 32–35)
Lemma 3.3 is a central spectral statement. It asserts that after removing the growing Riesz
projections, the remainder propagator obeys an estimate of the form



The proof appeals to a contour argument for a bounded operator after discussing a compact
perturbation of angular multiplication and a finite-dimensional Schur reduction. [cite:1]
What is not fully established is the spectral placement and resolvent control needed to justify
the contour deformation and the explicit semigroup bound on the complement. Excluding
unstable eigenvalues is not, by itself, enough to obtain the stated estimate when the
background multiplication operator contributes continuous spectrum on the imaginary axis.
Because Lemma 3.4 and the subsequent nonlinear activation constructions use this remainder
estimate essentially, the proof gap is central. [cite:1]

Proposition 5.3 (pages 55–57)
Proposition 5.3 is stated for a finite or countable coloured evolution. The proof closes a
bootstrap using continuity of a maximum probe defect over a finite set and then says the
countable case follows because all infinite sums were already controlled. [cite:1]
That is insufficient. Passing from finite families to a countable family requires either a finite-
prefix approximation argument with constants uniform in the prefix, or a direct
continuity/upper-semicontinuity argument for the supremum over all indices. Neither is
supplied, so the countable-colour version is not rigorously established by the proof as written.
[cite:1]
Steps checked successfully

Lemma 2.6
The Biot–Savart bounds in Lemma 2.6 are mathematically consistent with standard arguments.
The     estimate follows from near/far splitting and Hölder, the bound for       follows from
Young’s inequality, and the bound for           is consistent with Calderón–Zygmund theory.
The zero-circulation -velocity conclusion for compactly supported                 is also
plausible from low-frequency cancellation plus high-frequency control. [cite:1]

Lemma 5.2
Lemma 5.2 appears correct. The radial test function            makes the tangential drift
irrelevant, and the stopped Itô computation yields the advertised exponential exit estimate,
provided the standard localization/approximation justifications are inserted. [cite:1]

Lemma 4.10
Lemma 4.10 also appears sound in principle. The stated negative-time asymptotic produces a
nonremovable simple pole in the meromorphic continuation of the bilateral transform, so the
transform cannot vanish identically; analyticity then implies that zeros on a real interval are
discrete. [cite:1]

Final verdict
The mathematics has not been verified. The present draft contains at least one concrete defect
and several major unresolved proof gaps in arguments that appear structurally essential to the
later cascade and nonuniqueness conclusions. [cite:1]
