REPORT MAPPING: DEFINITE CORRECTIONS AND SPARSE PHASE CLOSURE
Reviewed entire upload/report.tex, all 285 lines, including C1--C4 and the
final recommendation. Current source reviewed: output/euler-publication/
euler-plane-revised.tex (9111 lines before this turn's edits).

B1 -- ALREADY ADDRESSED.
Current conditional sparse-phase proof, label ph:sparse, lines1899--1903:
"Circulation passes by strong $L^1$ convergence. Compact support follows
from membership of $g_\infty$ in the assumed compact-data phase
family; a common support ball, as in the scaled stack construction,
also gives it directly."
Module caps.tex182--185. For the concrete initial stack, cont-stack-data
proof/current2157/module profile.tex23 explicitly says "All supports are
in a common ball." All W and phase seeds have one fixed support radius;
R_j=q^(j-1)<=1, so every physical summand has that same support bound.
The limit vanishes outside it by L1 convergence. Integral preservation
follows from continuity of f -> integral f on L1. No false general
support-passing statement remains in the checked passages.

B2/M6 -- NOT ADDRESSED in the current revision.
The current bibliography has GW01, DL89, MB02 and CCS21, and the
Introduction does not cite CFMS, Mengual, BCK or ACM. The revised text
does explicitly distinguish identified-limit convergence from uniqueness:
Introduction lines117--124, CCS21 comparison. This addresses that last
part of M6, but neither the closest-work comparison nor the precise
two-ring-profile attribution. These are genuine scholarly omissions;
they do not themselves demonstrate a mathematical failure. Primary
papers must be consulted before adding precise descriptions.

C1 -- ALREADY ADDRESSED.
Isolated activation scaling remark, current2817--2822,
module profile.tex685ff:
"Thus the separate density bounds are $CAR^2\delta_q$ in $L^1$ and
$CAR^{2/p}\delta_q$ in $L^p$, and their sum is" followed by
||w(t)-G(t)||_{L1 cap Lp} <= CA(R^2+R^(2/p)) delta_q.
This is correct: under f(x)=A g(x/R), ||f||_r=A R^(2/r)||g||_r.

C2 -- ALREADY ADDRESSED.
Inactive-density subsection, label sec:inactive-density, current5254--5255,
module inactive.tex150--151:
"The first two closure corollaries assume an explicit packet bound;
Section~\ref{sec:next-packet-force} derives a bound from the actual packet."
The potentially misleading "stronger actual packet estimate" is gone.

C3 -- ALREADY ADDRESSED.
Literal epsilon(t) occurrences in the current active source and the
current source-history.tex module all have the backslash \epsilon(t).
Current6385,6963,7019,7026 checked. A stale source-history-original.tex
still contains a bare epsilon(t), but it is not assembled or delivered.

C4 -- STILL PRESENT; MINOR FIX REQUIRED.
Spectral proof, label smooth-active-spectrum, current2424--2428,
module profile.tex280--284:
"Thus the coupling ... is compact on $H^k(I)$ and the angular multiplier
is bounded there. Its spectrum is imaginary: off the multiplier's value
range the inverse is multiplication by ...".
Replace "Its spectrum is imaginary" with "The angular multiplication
operator has purely imaginary spectrum". The argument thereafter does
refer to the full operator via the compact-operator Fredholm alternative.
This is an antecedent ambiguity, not a false claim in context. It can be
fixed autonomously without changing any mathematical claim.

INDEPENDENT CHECK: HALF-SUPREMUM CONTINUITY RADIUS.
selection.tex328ff/current9016ff defines r_k as half the supremum over
0<r<=1 of radii giving the desired <2^-k phase difference for every
admissible stack with outer phase fixed0 and X-distance<r. At fixed
viscosity the initial-total-data solution map is continuous in CtX;
the fixed smooth outer-colour map is continuous in CtL1; the protected
first-cap map is continuous in the uniform observable topology.
Hence the set of acceptable radii is nonempty and downward closed.
Writing s=sup S>0, r_k=s/2<s gives a radius r in S with r>r_k; the
property at r implies the property at r_k. No attainment of the
supremum is required. Thus the half-supremum definition is sound.

INDEPENDENT CHECK: ENTIRE VARIABLE TAIL.
The reader-guide eq:guide-phase-tail/current758 defines
Delta_tail(m)=2alpha[||Phi_0||_1 (beta q^2)^(m-1)/(1-beta q^2)
 +||Phi_0||_p (beta q^(2/p))^(m-1)/(1-beta q^(2/p))].
Rotation preserves both norms, and ||Phi_theta-Phi_eta||_r<=2||Phi_0||_r,
so this uniformly bounds ANY phase changes at all j>=m. Both geometric
ratios<1 under the fixed p>2 condition beta q^(2/p)<1. Once N_(k+1)
is chosen with Delta_tail(N_(k+1))<min_(i<=k) r_i, all later assigned
indices are >=N_(k+1), so the entire final-minus-kth datum lies inside
r_k. No sum of individually allocated tolerances is needed. The proof
keeps theta1=0, giving the same outer initial colour at every step.
Each assigned coordinate is changed once; uniform absolute convergence
identifies the X limit with the final phase sequence. Outputs differ
from the k-th shot by <2^-k. Along each parity these tolerances vanish;
protected-cap closure gives limiting coordinates +delta and -delta.
Cap times may differ, as the manuscript correctly allows. I found no
remaining error in this sparse closure or common-support argument.

BOUNDARY OF THIS AUDIT.
The validity of the uniform first-cap hypothesis is not established by
fixed-positive-viscosity continuity. It rests on the interacting-cascade
estimates and the invariant-stage proposition. This memo verifies the
conditional sparse/continuity/tail deduction only; it is not independent
certification of those nonlinear cascade estimates.
