REVISION NOTES
Non-unique vanishing-viscosity selection for planar Euler
9 October 2026

The revised manuscript is a single self-contained LaTeX source. It needs no
local figures, subsidiary TeX files, bibliography database, or earlier periodic
manuscript. Its main non-unique-selection statement is Theorem 1.1 in the
Introduction; the proof is given in the final section.

STRUCTURE AND EDITORIAL CHANGES

1. Reorganized the original nineteen sections into ten logically ordered
   sections: Introduction; Preliminaries; fixed profile and isolated activation;
   canonical orbit and coupling; actual cores and cold packet; inactive prefix;
   actual source history; finite forcing; actual receiver; uniform descent and
   fixed-datum selection.
2. Added the explicit main theorem, background and motivation, proof strategy,
   dependency map, paper outline, and notation table.
3. Collected precise solution classes, kernel estimates, heat and Sobolev tools,
   classical Navier--Stokes theory, finite-Sobolev Euler continuation, scalar
   comparison, passive-colour recovery, compactness, and protected-cap tools.
4. Used verified primary references for the established analytical inputs.
   Strong convergence to an identified Euler limit is distinguished from
   uniqueness of nonlinear selection. The rough fixed-datum compactness result
   is proved directly instead of applying an approximation theorem outside its
   hypotheses.
5. Preserved every original mathematical label, removed duplicate arguments and
   progress commentary, distinguished the adjoint test T_adj from the final
   time T_f, and checked cross-references and compilation.

MATHEMATICAL CORRECTIONS IMPLEMENTED

E1. Corrected physical X_p scaling to A(R^2+R^(2/p)); retained the correct
    velocity and entire-moment scaling.
E2. Specified the modal operator domains and restricted propagator estimates
    to each fixed finite angular sum. Nonlinear remainders use transport
    estimates, without an infinite-angular spectral assertion.
E3. Separated the canonical tail, history replacement, and older-seed errors
    into simultaneous strict noncancellation budgets.
E4. Established full Gaussian energy-space membership with bounded rational
    weights before applying oscillator gaps. Sharp full-weight identities are
    not asserted for arbitrary truncated weights.
E5. Added the cutoff-field time extension required by the normalized comparison
    interval. Proved the zero initial homogeneous H^{-1} trace by low/high
    Fourier splitting and used an integrable coefficient near initial time.
E6. Made cap-relative Gaussian-error conditions explicit. The actual logarithmic
    cap satisfies them; superalgebraic absolute error is not mistaken for an
    estimate relative to an arbitrarily smaller cap.
E7. Added divergence-free approximation hypotheses, strong time representatives,
    endpoint renormalized traces, and a direct norm/finite-propagation proof of
    strong L^1 convergence. Passive observables include the bounded noncompact
    adjoint test actually used in the selection proof.
E8. Derived compact support from the common support family and finite propagation,
    rather than from strong L^1 convergence alone.
E9. Fixed common colours, tests, coefficients and cap levels for both subsequences
    and for the rotation homotopy; retained explicit collar margins.
E10. Wrote cap resets for the full vorticity state and retained the radial heat-base
     defect kappa*t in every downstream comparison.
E11. Controlled actual source history on the entire prospective density-stopped
     interval. Obtained an upper input ball before choosing the stop constant,
     reached a deterministic prospective horizon, and only then used the signed
     coefficient lower bound. Actual histories are truncated and zero extended
     at their controlled endpoint.
E12. Restricted positive slopes to the protected late window, distinguished
     amplitude derivatives from time derivatives, and fixed cap domains and
     endpoint margins.
E13. Restricted inverse-power multipoles to the exterior of the source support and
     provided their absolute convergence and remainder estimate.
E14. Fixed the reference orbit before later finite orders. Proved the cap-to-reference
     advance by one-sided comparison, independent of the expansion endpoint, and
     proved fixed-radius and parent-radius envelope localization separately.
E15. Supplied moment integrability, finite-order quantifiers, logarithmic horizons,
     seed reductions, normalized clocks, and all parameter inequalities required
     by the uniform induction.

Additional repairs: proved every finite Sobolev order and every finite jet of the
same fixed ancient orbit by smaller-amplitude construction, identification and
translation; corrected collar support to the union of the two fixed collars;
covered zero-time Gaussian denominators; proved the generic fixed-small-cap
future-tail estimate using all fixed reference-envelope moments and a moving
initial cutoff; completed the scalar comparison hypothesis for linear-growth
rotating drifts; and made the sparse phase tolerances consistent with the exact
opposite limiting cap coordinates.

VERIFICATION

The revised modules were checked by their respective reviewers. Independent
cross-audits checked the profile/ancient foundations, the core/cold/inactive
estimates, the source-reference handoff, and the global phase closing. A further
adversarial review examined the actual receiving recursion and prospective-stop
ordering. These reviews found the reported repairable issues, which were
integrated. Compilation and page-layout checks supplement the mathematical
checks; they do not constitute a proof of mathematical correctness.

UNRESOLVED OR UNPROVED EXTENSIONS

No unresolved central step has been identified in the retained fixed-datum
non-unique-selection theorem after the integrated corrections and stated scope
restrictions. The following broader assertions from earlier formulations are
not included as established results:

* An unrestricted uniform propagator estimate on an infinite angular direct sum.
  Only modal and fixed finite-sector estimates are proved and used.
* A cap-relative error bound for arbitrarily tiny cap scales without the explicit
  relative Gaussian-error condition. The actual cap is logarithmic.
* The signed cold-tail suppression estimate on arbitrary horizons or arbitrary
  restarts. Its proved power budget uses the stated logarithmic horizon and
  neutral original compact data.
* Control or convergence of the actual infinite future after the parent
  density/core stop. Only the controlled truncated history enters the proof.

These limitations are explicit in the revised statements. No assertion about
the full set of viscosity accumulation points, or selection of every Euler
solution, is made.


READER DEFINITIONS AND FIGURES — 9 OCTOBER 2026

* Added Introduction Section 1.4, a construction guide with displayed :=
  definitions for the profile and neutral seeds, geometric stack, actual
  colours and positive envelopes, scale groups, physical/local clocks,
  older-core rotation, growing coordinate, canonical cap, reference clock,
  phase gauge, three source-history intervals, transform and transfer,
  receiver density stop and coordinate cap, and outer phase selection.
* Added six self-contained vector TikZ figures: the radial profile; nested
  collars and outward transmission; three-part discounted history; receiver
  stop ordering; phase shooting and sparse subsequences; finite recursion
  and its derivative reserve. Captions mark schematic geometry and explicitly
  distinguish overlapping supports, clocks, finite controlled histories,
  stop types, phase degree and possibly different limiting cap times.
* Made the finite recursion spaces, norms, operators, coefficients, initial
  conditions and truncations explicit. Defined whole-plane extensions of
  radial coefficient functions.
* Added the exact geometric phase-tail bound, recursive finite/final phase
  assignments and a positive half-supremum definition of each fixed-viscosity
  continuity radius. Specified N_1 >= 2 to preserve the fixed outer phase.
* Distinguished Gaussian weight beta_G=3/10 from geometric ratio beta>1,
  fixed smoothing epsilon from receiver forcing varepsilon=q^6, and finite
  viscosity truncation order N from selected cascade depth N. Expanded the
  notation table with clocks, stops, envelopes and transfer coefficients.
* Stated explicitly that q, alpha and delta are held fixed during the final
  vanishing-viscosity sequences.

Verification of this update: independently checked the new definitions and
formulas against the retained proof, including frame signs, exact cap-phase
ratio, transfer normalization, tail sum and continuity-radius quantifiers.
The main result statements and all 64 formal result environments are retained;
no theorem is strengthened by these explanatory changes. The source contains
all six figures and requires no external graphics, bibliography file or local
input file. Final PDF compilation is clean, and the new figures and affected
page layouts were visually checked. This update does not constitute a new
independent verification of every unchanged proof. The scope restrictions
listed above remain unchanged.


SUPPLIED FRIEND REPORT COMPARISON — 9 OCTOBER 2026

All M1–M6 and B1–C4 comments in the supplied report.tex were compared
with the current revision. The report itself was not edited.

Already addressed: B1 compact support from the common admissible support
ball, C1 physical L1/Lp scaling, C2 actual-packet wording, C3 epsilon
notation; substantive M2 weighted absolute three-part history, M3 upper
receiver stop, M4 individual/aggregate two-scale localization, and M5
protected caps and the whole sparse variable tail.

Changes made in this pass:
* Added explicit M_input(H), E_adv, E_in and five positive receiving
  exponents in Proposition 10.1, replacing its unspecified inherited-reset
  o(1); added the finite parameter-choice table with depth/phase-independent
  thresholds and factor-two invariant-class slack.
* Extracted Corollary 8.3 on upper continuation to the prospective endpoint.
  Its full colour/initial-seed/regularity and weighted input hypotheses
  are explicit; no signed transmission lower bound is assumed.
* Displayed uniform fourth-moment tightness from the sixth moment for both
  envelopes, using an independent cutoff radius. The separate parent-radius
  Gaussian remains an independent estimate.
* Specified common normalized e,h,c and their physical clock factors,
  and justified the scale-uniform coordinate Lipschitz bound.
* Credited the CFMS two-interface prototype through the exact identity
  W_0=-rho^2 w_bar, and added primary-source comparisons with Mengual,
  Brue–Colombo–Kumar and Albritton–Colombo–Mescolini.
* Clarified that the angular multiplication operator has imaginary spectrum;
  the full linearized generator has its stated unstable pair.

Theorem 1.1 is byte-for-byte unchanged from the start of this pass. The
standalone source retains six embedded TikZ figures and all previous labels;
there are now 65 formal result environments and eight bibliography entries.
Compilation is clean, with no undefined references/citations or box warnings.
Affected manuscript pages, including the spectral clarification, and all
five comparison-report pages were
rendered and visually checked.

Scoped independent audits examined each disputed dependency and upstream
spectral/ancient, core/cold/inactive, compactness and passive-colour inputs.
No new major obstruction corresponding to the supplied objections was found.
This is a targeted report comparison, not a fresh exhaustive independent
certification of every statement and classical external theorem in the full
manuscript. The previously listed unproved extensions remain outside scope.
See report-comparison.pdf (and its TeX source) for the point-by-point
disposition and quantitative checks.
