Independent upstream compactness / passive-colour audit
=======================================================

Scope: current modules compactness.tex, passive.tex, kernel.tex, the
viscous inputs in preliminaries-standard.tex, and the abstract cap/
sparse-phase implications in caps.tex. This is not a verification of
the actual cascade estimates in source-history.tex, receiver.tex, or
selection.tex.

Conclusion within this scope: no mathematical blocker was found.
Theorem cp:compact and Proposition ph:colour have rigorous proofs under
their stated hypotheses. Their use does not presuppose Euler uniqueness.
The unconditional non-unique-selection theorem still depends on the
separately disputed uniform cap and phase-shooting cascade estimates.

1. Kernel and finite energy.
   For p>2, the near Biot--Savart kernel lies in L^{p'}, its far part is
   bounded, and the far part lies in L^p. The stated L-infinity and L^p
   bounds follow by Holder / Young. The velocity Holder difference bound
   uses near disks of radius comparable to d and the far derivative
   |DK| <= C r^{-2}; both terms scale as d^{1-2/p}. The derivative
   singular-integral argument supplies the required global W^{1,p}
   bound. Compact support and zero circulation give
   |omega_hat(xi)| <= |xi| integral |x| |omega(x)| dx at low frequencies;
   interpolation gives omega in L^2 for the high frequencies. Thus the
   initial finite-energy assumption used by cp:compact is satisfied.

2. Velocity compactness.
   The kinetic energy bound and bounded velocity give
   ||u_n tensor u_n||_2 <= V E. Leray projection is bounded on H^s, so
   the velocity equation yields a uniform H^{-2} time derivative bound.
   The scaled mollifier has H^2 norm at most C h^{-3} in dimension two.
   Combining the convolution pairing with spatial C^alpha bounds gives
   ||u_n(t)-u_n(s)||_infinity <= C h^alpha + C h^{-3}|t-s|.
   The chosen h proves a common time modulus. Local uniform convergence
   on compact cylinders is therefore valid; no vorticity compactness
   theorem or Euler uniqueness is being smuggled into this step.

3. Weak vorticity paths.
   Compact smooth pairings have a uniform Lipschitz bound from scalar
   contraction and bounded drift. A countable dense test set plus the
   uniform L^p bound gives weak convergence uniformly in time for every
   L^{p'} test. Slice-wise lower semicontinuity gives the global L^1
   bound, and the limiting trace at zero is exactly the fixed datum.
   Local uniform velocity convergence is enough to pass the transport
   product in compactly supported tests. The viscosity term vanishes.

4. Total-vorticity renormalization and endpoint traces.
   The commutator formula with the subtracted omega(x) is legitimate:
   the scalar multiple integral that is subtracted vanishes by
   divergence freedom. Minkowski, Sobolev difference quotients, and
   Holder give its L^{p/2} bound by ||Du||_p times the L^p translation
   modulus of omega. Dominated convergence gives L^1_t L^{p/2}_x
   convergence. Since p/2>1, this suffices for local L^1 errors.
   For fixed mollifier scale the scalar path is absolutely continuous
   locally in spatial norms, so integration between any two prescribed
   times is justified. Spatial approximation at those fixed endpoints
   uses the already constructed weak L^p representative; it does not
   require strong time continuity in advance. This proves the exact
   renormalized endpoint traces without a circular argument.

5. Conserved norms and finite propagation.
   Bounded-derivative renormalizations with beta(0)=0 have integrable
   flux, and expanding cutoffs remove it using bounded velocity and
   the global L^1 bound. The convex tangent truncations of |s|^p are
   increasing to |s|^p; their displayed derivative in the truncation
   parameter is correct. Absolute-value approximations give conservation
   of L^1 and beta(s)=s gives circulation. The exterior test has material
   derivative psi' (u dot x/|x| - V) <= 0. Cutoffs and the L^1 bound
   justify the noncompact exterior test. Thus compact support of the
   Euler limit is obtained from finite propagation, not from bare
   strong L^1 convergence.

6. Strong convergence uniformly in time.
   The displayed Clarkson inequality has the correct direction for
   p>=2. Weak convergence plus the limiting equality of norms and
   approximating contraction gives strong convergence even at variable
   bad times t_n. Weak time continuity plus constant norm gives strong
   time continuity of the limit. This proves uniform strong L^p
   convergence. For a ball containing the Euler support, the L^1 tail
   inequality follows from contraction of the approximants and equality
   of the limiting L^1 norm. It is valid for signed vorticities because
   it uses absolute values. Hence uniform strong L^1 follows. Kernel
   boundedness then identifies the locally obtained velocity limit with
   K*omega, removing the deliberately deferred identification step.

7. Fixed passive colours.
   A bounded colour belongs locally to L^{p'}, including when 1<p<2.
   The local commutator bound with Du in L^p is therefore sufficient.
   Spatially convolving the divergence-free drift and then using colour
   translation continuity proves commutator convergence; its error is
   controlled in L^1_t W^{1,p}_loc. Extension to negative times by the
   initial colour and zero drift introduces no jump because the given
   weak initial trace is exact. Integration of the mollified equation
   between prescribed times verifies all endpoint traces. Square
   renormalization on the bounded colour range proves both L^2
   conservation and uniqueness, applied to the difference. This is
   uniqueness for a prescribed Sobolev velocity, not uniqueness for
   the nonlinear Euler equation.

8. Passive limits and observables.
   Uniform drift convergence and fixed colour datum identify every weak
   passive limit with that unique transported colour. Conservation and
   uniform convexity provide C_t L^2 convergence, and the same tail
   argument provides C_t L^1 convergence. The bounded C^2 observable
   tests are justified by expanding cutoffs; the derivative and mixed
   diffusion cutoff errors vanish. Thus equal total Euler paths yield
   equal fixed colour paths and equal cap observables. Actual fourth
   moments can only be recovered using the claimed common support ball
   in the Euler limit, or appropriate separate moment estimates in the
   viscous construction; the passive proposition does not supply
   viscous moment tightness on its own.

9. Report B1 and sparse implication.
   In ph:sparse the circulation conclusion now comes from strong L^1,
   while compact support comes from membership in the assumed compact
   phase family / a common stack support ball. This repairs report B1.
   The abstract sparse argument is valid provided its cap protection,
   shooting, and relative compactness hypotheses actually hold. The
   half-radius / summable-tail mechanism does not rely on any uniform
   positive-viscosity continuity constant as viscosity tends to zero.

Unresolved within the full paper, but outside this memo's scope:
The actual interacting cascade must establish all hypotheses of
ph:degree and ph:sparse uniformly in depth and phase homotopies.
Compactness and passive-colour identification cannot replace any of
those transmission, localization, source-history, or protected-cap
estimates. No assertion that the main theorem is correct follows from
this upstream audit alone.
