READER DEFINITION AUDIT — 9 October 2026
Scope: explanatory additions to the existing revised manuscript; no theorem changes.
Files checked: preliminaries-standard.tex, profile.tex, ancient-transfer.tex,
core-cold.tex, inactive.tex, source-history.tex, receiver.tex, selection.tex.
This is a reader audit, not a repeat independent verification of every proof.

1. BASIC NOTATION: preliminaries-standard.tex opening paragraphs.
The existing displayed definitions use '='. To implement the user's request,
change only defining equalities (not identities/PDE conclusions) to ':=':
 X_p := L^1(R^2) cap L^p(R^2),
 ||f||_X_p := ||f||_1+||f||_p,
 ||f||_C_tX := sup_{0<=t<=T} ||f(t)||_X,
 z := x_1+i x_2, x^perp := (-x_2,x_1),
 K(x) := x^perp/(2 pi |x|^2),
 M_m(f) := int |x|^m |f(x)| dx,
 B_m(f) := int bar(z)^m f(x) dx (only when M_m(f)<infinity),
 Gamma(f) := B_0(f) = int f(x) dx,
 P_rad f := (2 pi)^-1 int_0^{2 pi} f(R_angle x) d angle.
The current cut-moment is introduced in prose; a displayed
 B_{m,chi}(f) := int chi(|x|) bar(z)^m f(x) dx
is useful, if a new name is needed elsewhere. Do not add an unused name solely
for its own sake. Explain whole versus cut moments using these formulas.

2. PATTERN ROTATION: preliminaries-standard.tex around lines 62–88.
 (T_theta f)(x) := f(R_{theta/4}x),
 (T_theta V)(x) := R_{-theta/4} V(R_{theta/4}x).
The phase parameter theta is 2pi-periodic on fourfold functions. Physical
rotation of the pattern is -theta/4, not +theta/4. Its protected coordinate
and fourth moment multiply by e^{i theta}. An illustration must not use the
wrong physical rotation sign or treat phase theta as physical angle theta.

3. PROFILE/SEED: profile.tex around lines 83–90 and 333–367.
 rho := sqrt(3/5),
 W_0 := -1_{B_rho}+rho^2 1_{B_1},
 chi_epsilon(x) := epsilon^-2 chi(x/epsilon),
 W_epsilon := chi_epsilon * W_0,
 Omega_epsilon(r) := r^-2 int_0^r s W_epsilon(s) ds.
The unsmoothed inner value is -2/5, the middle value 3/5, and the outer
value 0. Smoothing length epsilon is fixed before q and viscosity.
If illustrating radial graph, distinguish W_0 from W_epsilon and label any
sketch qualitative, not a computed instability profile.
 b_epsilon := 2pi int r^5 f_epsilon(r) dr.
The real neutral seed is precisely
 Phi_theta(r,angle) := e^{i theta}[f_epsilon(r)
      -(b_epsilon/bar b_epsilon)bar f_epsilon(r)] e^{4i angle}
      + complex conjugate.
It obeys B_4(Phi_theta)=0 but has growing coefficient e^{i theta}; neutrality
is cancellation of growing and stable seed moments, not absence of growth.

4. FIXED STACK: Introduction eq:intro-stack and profile.tex cont-stack-data.
 A_j := beta^{j-1}, R_j := q^{j-1}, alpha := q^K,
 L := log(1/q), delta := L^-4,
 omega_{j,0}(x) := A_j[W(x/R_j)+alpha Phi_{vartheta_j}(x/R_j)],
 omega_0 := sum_{j>=1} omega_{j,0}.
One can display groups at viscosity nu_N:
 omega_{<N,0} := sum_{j<N}omega_{j,0},
 omega_{N,0}, omega_{>N,0} := sum_{j>N}omega_{j,0}.
Use these only as bookkeeping groups: every group interacts through the same
velocity; none is an independent Navier–Stokes solution.
Geometry warning: W copies have nested disk support and overlap on constant
cores. Only W' and the seed are in TWO derivative collars per scale. Do not
draw disjoint annular vorticity supports. Collar radii are approximately
rho R_j and R_j; thickness is epsilon R_j.

5. ACTUAL COLOURS/ENVELOPES: preliminaries-standard.tex around 373–380,
inactive.tex around 539–562.
Define the actual colour by its full scalar initial-value problem, explicitly:
 partial_t omega_j^nu + u^nu . grad omega_j^nu = nu Delta omega_j^nu,
 omega_j^nu(0) := omega_{j,0}, u^nu := K*sum_l omega_l^nu.
At a fixed actual velocity, the positive propagator S^nu(t,s) is the solution
operator of that scalar equation. Then
 P_{>=j}^nu(t) := S^nu(t,0)(sum_{l>=j}|omega_{l,0}|),
 omega_{>=j}^nu(t) := sum_{l>=j}omega_l^nu(t),
 |omega_{>=j}^nu| <= sum_{l>=j}|omega_l^nu| <= P_{>=j}^nu.
If avoiding new notation, retain paper's P_j, v_j and use ':=' in existing
display. The sum may not have a finite L-infinity norm; illustrations must
not claim aggregate L-infinity contraction. P_j is a positive envelope,
not the radial projector P_rad.

6. CLOCKS AND ROTATING FRAMES: preliminaries-standard.tex 460–495;
source-history.tex 408–419.
 s_j := A_j t_phys, y_j := x/R_j,
 w_j(s_j,y) := A_j^-1 omega_j^nu(s_j/A_j,R_j y),
 U_j(s_j,y) := (A_j R_j)^-1 u^nu(s_j/A_j,R_j y),
 kappa_j := nu/(A_j R_j^2).
For c=i+1, s_c=beta s_i and kappa_i=beta q^2 kappa_c.
The rigid older-core angular velocity in j units is
 Omega_{<j} := W(0)/2 sum_{l<j}(A_l/A_j).
One can display the rotating-frame colour explicitly:
 tilde w_j(s,y) := w_j(s,R_{Omega_{<j}s}y)
                  = T_{4 Omega_{<j}s}w_j(s,y).
This is the convention that removes drift Omega_{<j} J y. Do not confuse
Omega_{<j} (scalar angular velocity) with Omega_ref (vorticity orbit).

7. SELECTED VISCOSITY AND RELATIVE DIFFUSION: Intro and selection.tex.
 nu_N := q^{3/2} A_N R_N^2 = q^{3/2}(beta q^2)^{N-1},
 kappa_j(nu_N) = q^{3/2}(beta q^2)^{N-j}.
For j<=N this is <=q^{3/2}; for j>N it increases as j grows.
Source selected is N; active transmission runs N -> N-1 -> ... -> 1,
outward in physical scale. The smaller-scale packet is controlled, not zero.
All geometric drawings are schematic since q is very small.

8. PROTECTED COORDINATE/ANCIENT ORBIT: ancient-transfer.tex adjoint;
source-history.tex actual reference; receiver.tex 660–716.
 mathcal A(f) := int T_adj(x) f(x) dx,
 lambda := sigma+i tau (sigma>0).
The normalized reference orbit satisfies A(Omega_ref(0))=delta_ref>0.
It is a FIXED time-and-phase normalization of the orbit from the ancient
construction, not an assertion that the originally denoted Omega_delta_ref
at its original clock origin already has that value exactly.
 c_ref := lim_{u -> -infinity} e^{-lambda u}A(Omega_ref(u)).
Do not set c_ref=delta_ref; the paper gives |c_ref|=delta_ref+O(delta_ref^2).
For q small define its first small cap precisely by
 s_q := inf{u<=0 : |A(Omega_ref(u))|>=delta} < 0.
It is not the fixed reference time u=0. For an actual child cap tau_c,
 t_ref,c := tau_c-s_q (CHILD units),
 t_0 := t_ref,c/beta (RECEIVER units),
 u := s_c-t_ref,c = beta(t_i-clock-t_0).
If describing a child's physical time, t_ref,c/A_c is the corresponding
physical reference time, not t_ref,c itself.
Exact cap phase must be
 theta_c := arg[A(tilde w_c(tau_c))/A(Omega_ref(s_q))] mod 2pi,
so T_{-theta_c}tilde w_c(tau_c) matches the full complex coordinate.
It is NOT just arg A(tilde w_c(tau_c)), since the canonical small-cap
coordinate has its own phase.

9. THREE-PIECE CONTROLLED HISTORY: receiver.tex 740–759.
 E_q := C_reset delta^{M+1}+q^chi L^D,
 B_q(u) := B_4(T_{-theta_c}tilde w_c(t_ref,c+u))
            for -t_ref,c <=u<=U_q, and 0 otherwise.
The three pieces are [-t_ref,c,s_q], [s_q,0], [0,U_q]. Actual cap occurs at
u=s_q, fixed reference at u=0. Past envelope is C_past delta
exp(sigma(s_c-tau_c)); after change of origin it contains exp(sigma(u-s_q)).
For a history picture mark the zero extension for u<-t_ref,c and u>U_q.
Do not label B_q on all R as the true actual future.
 mu_beta := [sigma+i(tau-4/5)]/beta,
 F_ref(mu) := int_R e^{-mu u} B_4(Omega_ref(u)) du,
 C_* := gamma_epsilon F_ref(mu_beta).
IMPORTANT FACTOR CHECK: There is no extra beta in C_*. The forcing is
beta q^6 gamma_epsilon ...; u=beta(t-t_0) cancels beta in the transform.
 gamma_epsilon := A(F_epsilon e^{4i angle}),
 F_epsilon(r) := -i W'_epsilon(r)/(4pi r^5).
No unrestricted actual infinite-time-history claim is established.

10. RECEIVER GAUGE: receiver.tex 1002–1008.
The child's own older-rotation-removed reference phase is theta_c. In the
receiver's frame at t_0 define
 phi_c := theta_c+4t_0/5 mod 2pi.
Because the receiver core rotation is -1/5, the retained source fourth
forcing in receiver units is
 beta q^6 gamma_epsilon e^{4i(t-t_0)/5} B_c(beta(t-t_0)).
Here B_c can be made unambiguous by defining it from the phase-normalized
child history: B_c(u) := e^{i phi_c} B_q(u) on its controlled interval.
It is therefore a receiver-reference-gauged child moment, not the raw moment
of the child still carrying the parent rotation. Multiplying by BOTH
exp(4it/5) and phi_c would count the reference rotation twice.

11. DENSITY STOP VERSUS COORDINATE CAP: receiver.tex 1042–1062.
 epsilon := q^6,
 a(t) := epsilon exp(sigma(t-t_0)),
 W_{kappa_i}(t) := exp(kappa_i t Delta)W,
 T_s := inf{t>=0: ||w_i(t)-W_{kappa_i}(t)||_X=C_s delta},
 S_q^* := sigma^-1 log[delta/(q^6 |C_*|)],
 H_+ := t_0+S_q^*+L_0,
 Hhat := min(H_+,T_s),
 U_s := beta(Hhat-t_0), U_+ := beta(H_+-t_0).
Coordinate cap is t_i := inf{t>=0: |A(w_i(t))|>=delta}.
T_s and t_i measure different quantities. The proof obtains upper history
bounds on [0,Hhat], computes C_a, selects
 C_s > max{1,2 C_a exp(sigma L_0)/|C_*|},
and proves T_s>H_+. ONLY THEN define full C_q from the history to H_+ and
use its nonzero lower bound to obtain t_i. Draw this ordering as boxes or
an arrow flow, not as a signed coefficient feedback bootstrap.

12. COMPLETE GROWING COEFFICIENT: receiver.tex eq:ar-full-Cq.
 C_q := (alpha/epsilon) exp(lambda t_0) A(Phi)
    -(1/epsilon) int_0^{H_+} exp(-lambda(t-t_0))
         [int T_adj(x) b_J(t,x).grad W(x) dx] dt.
The seed coefficient here is the actual own-seed amplitude (possibly <=the
global alpha along the amplitude homotopy); it may equivalently be absorbed
into Phi. The field b_J contains all retained actual harmonic sectors, not
only the child fourth moment. It is extended by zero after H_+ for the
finite-coefficient construction. C_q=e^{i phi_c}C_*+o(1).
The generic finite expansion's settled cap center is
 S_q(C_q) := sigma^-1 log[delta/(epsilon |C_q|)].
The prospective S_q^* uses |C_*|, and the two differ by o(1). Avoid assigning
the same exact name/formula to both centers in a picture.

13. EXACT CAP AND ITS PROTECTION: caps.tex opening; receiver.tex late window.
 tau(a) := inf{t in [0,T_f] : |a(t)|>=delta},
 mathcal P(a) := a(tau(a))/delta.
The early strict margin and integrated positive slope are distinct controls.
Slope d|A|/dt>=sigma delta/2 is only in the protected late window and level
band, not a global-in-time estimate. The final limiting cap times may
DIFFER for the two solutions. Their coordinates are +delta and -delta.
A final picture must not imply equal first-cap times or monotonic modulus
for all later times.

14. FINITE RECURSION OBJECTS: receiver.tex beginning, eq:ffr-reserve etc.
If adding a pedagogical boxed dictionary, define
 mathcal L f := -(K*W).grad f -(K*f).grad W,
 Q(f,g) := -(K*f).grad g,
 C := {x: rho-epsilon<=|x|<=rho+epsilon}
      union {x:1-epsilon<=|x|<=1+epsilon}
for this fixed mollified profile, or retain a larger fixed collar chosen in
paper if intended. Do not identify actual derivative support with every
point of the closed collar unless justified; it is contained there.
 g_{j ell} defined by its displayed recursive IVP (eq:ffr-recursion),
 F(t) := sum_{j=1}^M sum_{ell=0}^N epsilon^j kappa^ell g_{j ell}(t),
 w_app(t) := W_kappa(t)+F(t),
 r_{j ell} := k_*+2(M-j)+2(N-ell),
 R := k_*+2M+2N+3.
These are finite sums/orders. A plot must not suggest an infinite convergent
power series; the proof uses arbitrary fixed finite orders plus an actual
remainder estimate. Whole-sector pattern rotation differs from multiplying
an order-j coefficient by one scalar complex phase.

15. PHASE HOMOTOPY: selection.tex prop:next-phase-degree.
With fixed N and all phases except theta_N, define
 r_j^{N,r} := 1 if j=N, and r if j!=N,
 theta_j^{N,theta} := theta if j=N, and the previously fixed theta_j otherwise,
 omega_0^{N,r,theta}(x) := sum_j A_j[W(x/R_j)
                      +alpha r_j^{N,r} Phi_{theta_j^{N,theta}}(x/R_j)].
 a_{N,r,theta}(t) := int T_adj(x) w_1^{nu_N;N,r,theta}(t,x) dx,
 F_N(r,theta) := mathcal P(a_{N,r,theta}) in S^1.
At r=0, F_N(0,theta)=e^{i theta}F_N(0,0). At r=1 only degree(F_N(1,.))=1
is guaranteed, not exact linear phase response. The r-homotopy changes an
auxiliary initial datum and its outer colour (unless N=1); it is not a
change of the fixed final datum with viscosity. The final selection occurs
at r=1 with the fixed outer phase theta_1=0, N>=2.

16. SPARSE TAIL/FINAL FIXED DATUM: selection.tex main proof.
A useful explicit tail bound, avoiding vague 'small tail', is
 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})].
For phase changes only at j>=m, this bounds ||omega_0^theta-omega_0^theta'||_X.
It tends to zero and supports the required choices
 Delta_tail(N_{k+1}) < min_{i<=k}r_i,
where r_i is the fixed-viscosity continuity radius selected for tolerance
2^-i at the i-th finite assigned datum. Each sparse phase coordinate changes
once. Define
 zeta_k := (-1)^k,
 nu_k := nu_{N_k},
 vartheta_j^infinity := lim_k vartheta_j^{(k)},
 omega_0^infinity := sum_j A_j[W(x/R_j)+alpha Phi_{vartheta_j^infinity}(x/R_j)].
Even/odd strong limits are extracted subsequentially; do not claim that
EVERY even/odd selected solution converges without extracting subsequences.
The final datum is the same along both parity subsequences.

17. FIGURE RECOMMENDATIONS / CONSISTENCY CHECKLIST.
A. Concentric nested disk/collar scale diagram: N source, j<N older stack,
   j>N actual cold packet; arrows N->N-1->...->1 outward; not to scale.
B. Clock ladder showing s_c=beta s_i and child cap/reference at tau_c,
   t_ref,c; receiver reference t_0=t_ref,c/beta.
C. Three-piece reference history with u=s_q and u=0, actual endpoints,
   zero extension and exponentially discounted future.
D. Receiver density closure order upper input -> C_a -> C_s -> T_s>H_+
   -> completed C_q/nonzero -> protected first coordinate cap/reset.
E. Phase shooting: r=0 exact degree-one rotation, r=1 continuous degree-one
   map, sparse selected viscosities targets +/-; true subsequential limits.
F. Finite recursion triangle j/ell and derivative reserve, if there is room.
All diagrams qualitative unless explicitly constructed numerical values
are supplied. Their captions should state the precise object/definition
references rather than introduce additional unnamed quantities.

18. CHECK OF NEW reader-guide.tex (root version reviewed).
All main displayed formulas are consistent with the current proof: seed
neutrality, geometric scales, actual scalar colours, clock normalization,
older-frame sign, exact phase ratio, three history intervals, transform
factor, distinct density/coordinate stops, completed coefficient and parity
selection. Cross-reference section labels all exist in the current final tex.
Suggested completions/clarifications, beyond the s_q infimum change already
planned by root:
(a) Put lambda:=sigma+i tau, sigma:=Re lambda>0, tau:=Im lambda before mu_beta.
(b) Add displayed scalar and vector T_theta actions before use in history.
(c) Define phi_c:=phi+4t_0/5 mod 2pi explicitly, not only by external prose.
    If needed, B_c(u):=e^{i phi_c} mathcal B_q(u); the instantaneous forcing
    then includes exp(4i(t-t_0)/5) as in the proof.
(d) Define the homotopy initial datum via its piecewise seed amplitudes/phases,
    as in item 15; new omega_{1;N,r,theta} needs this explicit datum definition.
(e) Define Delta_tail and the continuity radius used in sparse shooting;
    item 16 gives the exact uniform geometric sum formula.
(f) After q is chosen, delta is fixed for every viscosity. Add this sentence
    to prevent 'shrinking cap' from suggesting viscosity-dependent delta.
(g) The transform display may state its domain 0<Re mu<sigma directly.
(h) Clarify the equal inner/outer signed-mass statement refers to W_0's disk
    and annulus; the smooth profile has transition collars.
(i) F_epsilon(r) vanishes near r=0, so extend its displayed quotient by zero
    there if it is defined as a global smooth function.
(j) If an explicit displayed formula for Omega_ref itself is desired:
    s_ref:=inf{s in R : |A(Omega_{delta_ref}(s))|>=delta_ref},
    theta_ref:=-arg A(Omega_{delta_ref}(s_ref)) mod 2pi,
    Omega_ref(u):=T_{theta_ref}Omega_{delta_ref}(u+s_ref).
    This fixed protected cap exists on the small ancient branch for a
    sufficiently small fixed delta_ref, by its negative asymptotics and
    positive slope. This is optional: do not add new symbols if the current
    fixed time-and-phase normalization is deemed clear enough.
No mathematical theorem change is needed for these reader additions.
