Independent upstream audit, 9 October 2026

Scope. I read profile.tex in its entirety and ancient-transfer.tex in its
entirety, with particular attention to the spectral, finite-expansion,
canonical-ancient-orbit, adjoint, and nonzero-transform ingredients. This
is a local mathematical audit. It does not certify the coupled colour
estimates in other modules or the main fixed-datum selection theorem.
No manuscript edits were made.

1. Fixed smooth profile and modal spectrum

The signs and masses of W_epsilon' are consistent. Convolution of a disk
indicator by a radial decreasing positive mollifier is radial decreasing.
For epsilon < rho, its center value is exactly one; hence each positive
derivative-layer measure has mass one. The two collars are disjoint.
The core and outer interface angular velocities are -1/5 and zero.

I independently derived the interface matrix from f(r)r dr = sum_i
c_i r_i^2 q_i delta_{r_i}, where (c_1,c_2)=(-1,rho^2). The equation is
lambda q_i = -im Omega_i q_i + i/(2r_i^2) sum_j h_m(r_i,r_j)c_j r_j^2 q_j.
It gives exactly M_m = (i/2)[[2m/5-1,rho^m],[-rho^(m+2),3/5]].
For m=4 the diagonal entries coincide, and the off-diagonal product is
rho^10/4. Therefore lambda_+ = rho^5/2+3i/10 and its vector is (1,-i rho).
For m >= 8 in 4N, the real matrix discriminant is
[2(m-4)/5]^2 - 4rho^(2m+2), strictly positive at m=8 and increasing
thereafter. Thus these limiting eigenvalues are purely imaginary.

The high-mode estimate is valid in the weighted Hilbert space
L^2(|W_epsilon'|dr): angular multiplication and the two same-sign collar
blocks are skew-adjoint. The two cross blocks have Hilbert-Schmidt norm
bounded by C a_*^m with a_*<1, uniformly in epsilon. The real part of an
eigenvalue is consequently at most C a_*^m. This is only an eigenvalue
estimate; the later compact-Fredholm argument supplies the reduction of
positive spectrum to eigenvalues on the H^k(I) spaces actually used.

For the remaining finitely many modes, the Schur complement based on
evaluation at collar centers is valid: receiving-radius oscillation of
the Green integral is O(epsilon) in sup norm on a fixed right half-plane.
The complementary block is invertible by a Neumann series. Uniform
determinant convergence, together with a large-|lambda| resolvent bound,
excludes extra roots in Re lambda >= eta. The growing simple root has
algebraic multiplicity one via the displayed Woodbury resolvent formula.
Local elliptic bootstrapping transfers that statement to every fixed
H^k(I) without requiring a globally Sobolev zero extension at the
endpoints of I. The stated finite direct-sum semigroup estimate then
follows by bounded-operator contour calculus. None of this establishes
an estimate for an infinite angular direct sum, and the manuscript
explicitly does not claim one.

The sentence 'Its spectrum is imaginary' has an ambiguous antecedent:
it must describe the angular multiplication operator, not the complete
linearized generator. This is a minor wording correction already
identified by the other audit.

2. Seed and finite-order nonlinear closure

The growing eigenfunction measure converges to
-rho^2 delta_rho - i rho^3 delta_1. Thus its fourth moment tends to
2pi(-rho^6-i rho^3), which is nonzero. Complex conjugation within the
positive fourth sector gives eigenvalue -bar(lambda), so the displayed
combination cancels its initial fourth moment while leaving growing
projection e^(i theta). Circulation and fourfold symmetry follow.

The two-parameter seed/viscosity recursion is triangular in total degree.
Its derivative loss is finite: for a target coefficient norm, assign
sufficiently high derivative orders to the finitely many predecessors.
All noncompact terms come from the exact radial heat base's Taylor
remainder; every omitted term retains a seed factor. The residual orders
a^(M+1) and kappa^(N+1)(1+t)^D a are therefore consistent. Comparison
of actual and approximate vorticity in L1 intersect Lp uses only the
approximation gradient, and the choice C1 before M,N avoids circular
energy exponents. The required inequalities on M,N give the stated
error bounds on logarithmic windows.

The radial exponential Kato tail argument is valid. The radial heat
base velocity does not contribute to the radial weight; the integral
of the nonradial velocity contributes only O(delta). Optimizing the
exponential weight yields the stated Gaussian tails and fourth-moment
tightness. This supplies whole-moment estimates instead of incorrectly
deducing them from L1 convergence alone.

3. Canonical ancient Euler orbit

The high-Sobolev error estimate uses the order-zero Fourier multiplier
bound for grad(K*g) and standard finite Leibniz/commutator estimates.
For k>=3, grad(zeta) is bounded, and no derivative above k of the actual
error is needed. Grad(omega_app) at the next finite derivative order is
available from the smooth finite expansion. The energy coefficient can
be fixed before M,N by making only the approximate perturbation small.

The residual's pure-seed constant can be fixed independently of N;
all other seed-order residual terms contain a positive power of kappa.
This permits delta_* to be fixed independently of the initial polynomial
exponent K_*, while N and the q threshold depend on K_*.

The cap-shifted compactness argument is valid: uniform H^k plus the
equation give time equicontinuity in lower Sobolev norms; the weighted
tail estimate gives global, not merely local, compactness. Zero
circulation and a common initial support give uniform finite kinetic
energy, so the velocity has the claimed H^(k+1) control. The limiting
orbit has a common compact support for negative times and the finite
polynomial error O((delta exp(sigma s))^(M+1)).

Uniqueness of ancient subsequential limits uses X=L1 intersect Lp,
not a derivative-losing H^k difference estimate. The comparison orbit's
gradient is uniformly in X. Starting at -T gives the factor
exp(-[(M+1)sigma-E_k]T), which tends to zero by the strict stated choice
of M. This yields full convergence and independence of K_*.

The cap-translation identity is consistent with the rotation convention.
The all-finite-order corollary does not intersect infinitely many
amplitude thresholds: for each target order, construct at a smaller
amplitude and identify its X limit with the original orbit, then use
the time/rotation translation and smooth Euler continuation. Its
constants and onset times are explicitly allowed to depend on the
finite target order.

The forward compact-window Navier-Stokes convergence is a valid
X stability argument against the smooth Euler orbit. Its separate
forward Gaussian propagation estimate correctly removes the whole
fourth-moment cutoff. It asserts only fixed windows at this stage.

4. Adjoint, transfer constants, and noncancellation

The adjoint integral equation has the correct sign for a bilinear
pairing against f(r)e^(4i theta). Its denominator lambda+4i Omega never
vanishes. Extension gives r^4 e^(-4i theta)=bar(z)^4 near zero and
r^(-4)e^(-4i theta)=z^(-4) outside the profile. Thus the global test is
smooth and bounded with harmonic inner and outer pieces. It annihilates
the stable eigenfunction because their eigenvalues are distinct.

The exterior moment expansion and the inverse-quintic forcing sign
are correct. In interface variables the forcing is
h=(i/(4pi rho^6),i/(4pi)). Pairing against
ell(q)=(q_1+i q_2/rho)/2 gives
gamma_0=(i rho^(-6)-rho^(-1))/(8pi), as stated. Similarly the older
interior cubic field gives (rho^2+i/rho)/4, which is also nonzero.
The resolvent/Schur convergence on a fixed contour legitimately gives
these smooth-profile projection limits.

The child-time substitution cancels the prefactor beta. The resulting
discount is mu_beta=(sigma+i(tau-4/5))/beta, so the transfer constant
is gamma times the bilateral moment transform, without an extra beta.
The rotation factor e^(4is/5) agrees with the parent core velocity -1/5.

The bilateral transform is absolutely analytic on 0<Re z<sigma. After
subtracting the leading negative-time exponential, it extends as
c b/(lambda-z)+G(z), with G analytic on 0<Re z<2sigma. Since c b !=0,
the pole is nonremovable and the transform is not identically zero.
The curve beta -> mu_beta is nonconstant and lies inside the analytic
strip for beta>1; its zeros cannot have an interior accumulation. Hence
arbitrarily large nonexceptional beta can be chosen. No lower bound
uniform in beta or assertion at the boundary z=0 is claimed or needed.

The prescribed-input receiver theorem is explicitly auxiliary, and its
statement does not pretend that the prescribed source and receiver
form an unforced interacting system. Its coefficient recursion and
comparison follow the same valid finite-order estimates. This theorem
alone would be insufficient to prove the actual cascade.

Conclusion. I found no theorem-invalidating calculation or logical
defect in these scoped upstream ingredients. Their essential restrictions
(finite angular and derivative sets, fixed future windows before the
later interacting estimates, and constants depending on the selected
profile) are mathematically material and are stated. This conclusion
does not convert this local audit into a certification of the entire
119-page manuscript, nor does it independently verify all external
classical references or every estimate in the other modules.
