INDEPENDENT ALGEBRA AND FULL MATERIAL OPERATOR AUDIT
Draft checked: tmp/pdfs/euler-nonlinear-proof-draft.tex
Date: 2026-10-07
Scope: exact linear/algebraic identities and exponent bookkeeping. This is
not certification of the nonlinear contour estimates or the proposed theorem.

RESULT
The requested Fourier/moment/detuning/image identities agree with the draft.
The full material operator has a particularly simple bounded shear, so the
suspected derivative-order mismatch does not occur in this linear calculation.
The draft would be substantially clearer if it supplied this exact shear.

1. NORMAL AND TANGENT MATERIAL LINEARIZATION
Write h_i(theta)=r_i exp(i theta)(z_i(theta)+i xi_i(theta)), where z_i and xi_i
are real normal/tangent relative displacements; use positive Fourier m.
The disk coefficients are c_1=-1, c_2=3/5, r_1=rho, r_2=1, rho^2=3/5.
The reference angular velocities are Omega_1=-1/5 and Omega_2=0.

A radial displacement r_j z_j produces the surface vorticity
  c_j r_j z_j(theta) delta(r-r_j).
For its mth harmonic, the streamfunction is
  psi_j,m(r)=-c_j r_j^2/(2m) z_j,m (r/r_j)^m, r<r_j,
  psi_j,m(r)=-c_j r_j^2/(2m) z_j,m (r_j/r)^m, r>r_j.
Use u_r=-r^{-1} partial_theta psi and u_theta=partial_r psi. At the self
interface the averaged tangential shape derivative is zero; combining it
with the averaged reference radial derivative gives the same result obtained
directly from the contour trace. Subtracting Omega_i partial_theta X_i gives

  z_s = L_m z,
  xi_s = T_m xi + C_m z,

  L_m=(i/2)[[2m/5-1, rho^m],[-rho^(m+2), 3/5]],
  T_m=diag(im/5,0),
  C_m=(1/2)[[1,-rho^m],[-rho^(m+2),3/5]].

The exact identity
  H_m L_m - T_m H_m = C_m, H_m=diag(i,-i),
holds entry by entry for every m>0. For negative m use conjugation, hence
H_m=diag(i sign(m),-i sign(m)). This is a Hilbert-transform multiplier,
bounded on C^{1,gamma} and on little c^{1,gamma}. The coordinate
  eta=xi-H z
therefore satisfies eta_s=T eta independently of the normal coordinate.

For P=10 pi, exp(P T_m)=I for integer m. If M_m=exp(P L_m), the full material
period derivative is exactly
  [[M_m,0],[H_m(M_m-I),I]].
Thus B_m=H_m(M_m-I) is bounded, and, on the modes where M-I is invertible,
  B(M-I)^(-1)=H.
No differentiation of z is present in this shear. The draft's shear formula
is correct, but its boundedness is much more transparent through this identity.

The ungauged material DV is also of order zero. Let Omega=diag(-1/5,0).
In (z,xi) coordinates its positive-mode blocks are
  DV=[[N,-Omega],[E,0]],
  N=L_m+im Omega=(i/2)[[-1,rho^m],[-rho^(m+2),3/5]],
  E=C_m+Omega=(1/2)[[3/5,-rho^m],[-rho^(m+2),3/5]].
The sole factors m occur in the label-translation generator, not in DV.
Self symbols are Hilbert transforms; cross symbols are exponentially smoothing.
This verifies boundedness of the linear field derivative on c^{1,gamma}.
It does not independently prove Banach analyticity of the nonlinear field.

For m=0, z_1,z_2 are constant, xi_1,s=0 and
  xi_2,s=(3/5)(z_2-z_1).
This is a finite-dimensional Jordan block of order two, as asserted.

2. SPECTRUM
At m=4,
  sigma=rho^5/2=0.139427400463..., tau=3/10,
  v_+=(1,-i rho), v_-=(1,i rho), ell=(1/2,i/(2rho)).
L_4 v_+=(sigma+i tau)v_+, L_4 v_-=(-sigma+i tau)v_-;
ell L_4=(sigma+i tau)ell, ell v_+=1, ell v_-=0.
The full growing vector has tangent part H v_+; the full growing left
functional is (ell,0), consistent with using the normal moment alone.

For m>=8, put a=(m-4)/5 and d=rho^(2m+2). Then
  delta_m=(a-sqrt(a^2-d))/2=d/[2(a+sqrt(a^2-d))].
The frequencies are m/5-1/2-delta_m and 3/10+delta_m. The discriminant is
positive: a^2>=16/25 while d<= (3/5)^9. In particular
  delta_8=0.00316177603462..., P delta_8=0.09933... .
The period multipliers are -exp(+-i P delta_m), uniformly away from 1.
The largest delta is at m=8, and the center period multiplier distance to 1
there is approximately 1.99753. The growing/stable m=4 period multipliers
are negative real, also away from 1. The exponentially decaying diagonalizer
corrections plus sum min(2,n rho^(2m)/m)=O(1+log(1+n)) give the stated center
power bound; the means give the weaker, sufficient O(1+n) total bound.

3. MOMENT NORMALIZATION AND SOURCE
For a circle r_i and normal displacement r_i z_i(theta),
  d/dh integral_D bar(z)^4 dA
    =r_i^6 integral_0^{2pi} z_i(theta) exp(-4i theta) dtheta
    =2pi r_i^6 z_i,4.
Therefore Q_i=z_i,4+O(||h||^2) with the draft's 1/(2pi r_i^6) normalization.
For the signed colour,
  B=2pi[-rho^6 Q_1+rho^2 Q_2],
and on the growing vector its leading coefficient is
  2pi(-rho^6-i rho^3)a, which is nonzero.

An exterior zero-circulation source with B_phys=integral bar(z)^4 omega dA
has streamfunction fourth coefficient -B_phys/(16pi r^4). Thus
  (u_r/r)_4=i B_phys/(4pi r^6).
Applying ell at the receiving radii gives
  D_4=(i rho^(-6)-rho^(-1))/(8pi), |D_4|=0.191235017419... .
The factor 4pi is correct for complex Fourier coefficients, rather than
cosine amplitudes.

4. FRAME DETUNING AND GENERIC RATIO
Under physical rotation phi, the fourth conjugate moment transforms by
exp(-4i phi). Since Omega_k^ext-Omega_j^ext=-A_j/5,
the receiver exponent in the incoming frame is
  A_j[sigma+i(3/10)-4i/5]=A_j(sigma-i/2).
After s=A_k(t-T_k), its exponent is z_beta=(sigma-i/2)/beta and its source is
D_4 q^6 B_*(s). In receiver time theta=s/beta, the source is beta D_4 B_*(beta
theta); the beta cancels in its discounted integral.

If the previously asserted canonical moment bounds and nonzero backward
asymptotic hold, F is holomorphic on 0<Re z<sigma and is not identically zero
by Fourier uniqueness on any vertical line. The image of beta in [4,5] is
a nonconstant compact segment strictly inside this strip. The bad ratios
are therefore finite, not merely non-full-measure. At least one permissible
beta exists. Translating the orbit gives F_delta(z)=exp(z t_delta)F(z), and
delta asymptotic exp(sigma t_delta) yields |F_delta(z_beta)| asymptotic
delta^(1/beta). This part is logically conditional on the canonical orbit
and its established global moment bounds, not a numerical computation of F.

5. TORUS IMAGE CANCELLATION
For a complete constrained colour, circulation and holomorphic moments
M_1,M_2,M_3 vanish. The quadratic regular Green term gives no velocity.
The degree-four term has only M_4, multiplied by a constant target power,
so its velocity also vanishes. The next degree-eight term has
  (z-y)^8 -> binom(8,4) z^4 M_4 = 70 z^4 M_4,
whose derivative is 280 z^3 M_4. Thus the first relative image velocity has
size A_l R_l^6 R_i^2 x_l. Compared with the direct inner-source field it
gains R_i^8; compared with an outer-source field it gains R_l^8. The self
gain is R_i^8. All are at most q^8. This is a complete-colour cancellation,
not a cancellation for either single disk separately.

6. WEIGHTS AND EXPONENTS
With epsilon=q^6 and w_i=(Dq^6)^(j-i), an inner source l=i+d contributes
  A_i (beta/D)^d,
and an outer source l=i-d contributes
  A_i (Dq^8/beta)^d.
The feedback into the recipient is O(A_j Dq^8 I_j), and the quadratic
inactive self-term divided by w_i is O(A_i Dq^6 I_j^2). These powers agree.

The deadline 9L/sigma gives backward history loss q^(-h),
  h=9/(beta-1) in [9/4,3].
Consequently: recipient prehistory q^(12-h) is at least q^9=o(q^6);
reverse-prefix/image prehistory q^(14-h) is at least q^11=o(q^6);
older normalized moment q^(6-h) is at least q^3=o(1);
higher harmonics q^(10-h) are at least q^7=o(q^6).
All fixed powers of log(1/q) preserve these comparisons.

The q^3 threshold is reached after <=3L/sigma+O(1) if the asserted signed
seed lower bound holds. Growth at sigma_0=3sigma/4 then reaches a fixed cap
in <=4L/sigma+O(1), totaling 7L/sigma+O(1)<9L/sigma for small q.
For eta<sigma/16, reset exponent 6-8eta/sigma>11/2>0.
For a_i<sigma/2, the inactive residual exponent 6-9a_i/sigma>3/2>0.
The negative tail integrability condition sigma_0>sigma/beta holds uniformly.

COMPUTATIONAL ARTIFACTS
check_annulus_algebra.py uses exact rational arithmetic in Q(i,rho), checks
all allowed modes through m=256, and writes annulus-algebra-results.json.
All assertions passed. The script uses ordinary floating arithmetic only
for the displayed stable-frequency samples and numerical constants.
