INDEPENDENT NONUNIFORM ROUTE REVIEW -- RESEARCH ONLY
Disposition: the d=1 route is plausible with specific corrections below; a full candidate proof has not yet been supplied for final-source review. No proof/manuscript edits. This note audits the parent-provided route and rates, rather than declaring an asymptotic theorem established.

1. D1 MIXED RATES REALLY SUFFICE
Let eta=O_p(n^-1/2 epsilon^-1/2) (the coarse covariance/RKHS evaluation rate, no logarithm needed with an HS expectation argument) and zeta=O_p(n^-1/2 epsilon^-1/4). After the first actual update, a value-norm perturbation delta=O_p(epsilon+eta) and L2_n error O_p(epsilon+zeta) are credible consequences of the potential branch and direct kernel concentration.
The exact relative-log map has ||Psi(e)||infty<=osc(e)/2<=||e||infty, and PSD Markov P gives an L2 contraction 1/2+O(delta). Uniformly over later iterations its linearization error is bounded by C delta*(epsilon+zeta), if the ball and mean/projective handling are specified. Multiplying by sqrt(n) gives the four terms:
 sqrt(n)epsilon^2,
 epsilon^(3/4),
 epsilon^(1/2),
 n^-1/2 epsilon^-3/4.
They vanish when n epsilon^(7/2)->0 and n epsilon^(3/2)->infinity: the first follows because sqrt(n)epsilon^2=(n epsilon^(7/2))^1/2 epsilon^1/4. Thus the coarse mixed bound is sufficient in d=1 even though it was not automatic in d=2. With logarithmic high-probability variants, carry the corresponding log factors explicitly.

2. POPULATION BURN-IN AND REGULARITY
For fixed positive smooth f, e1_population=O(epsilon) in C2 is sufficient to obtain B_n e1_population=O_p(epsilon^2+epsilon eta) in L2_n, hence root-n negligible in the above window. The corrected manuscript proves C1 expansion; obtaining C2 requires explicitly repeating its multiplier/commutator argument one derivative higher, using at least C4 smoothness of sqrt(f). The current 'fixed smooth density' assumption can cover this; it should not be cited as following from the C3/C1 lemma without extension. Later powers of B_n contract in L2_n.

3. STOCHASTIC INITIALIZER FORMULA AND COMMUTATOR
With phi(y)=a_epsilon(y)k_y, C=int phi tensor phi dmu, v=1/a_epsilon, and g_n=(mu_n-mu)(a_epsilon k), one has C v=v, ||v||H=1 and <v,g_n>=0 EXACTLY. The potential derivative is R_C g_n, and the scale-log derivative is -S_epsilon* R_C g_n.
The first-iterate log derivative is -K_delta1/(2 Hf). Replacing the local smooth weights gives -S_epsilon*g_n/2 plus a commutator error, hence relative initializer S_epsilon*(R_C-I/2)g_n=S_epsilon*B_C R_C g_n. The signs and half factors are correct.
The commutator kernel k(x,y)[1-v(x)a(y)] vanishes at x=y, since v(x)a(x)=1. At a sample query Xi this removes the large own-kernel diagonal. Conditional variance has an extra epsilon from |a(x)-a(y)|^2, producing L2_n stochastic size O_p(sqrt(epsilon)*zeta). The conditional missing-self-term bias is O(epsilon/n), provided the population kernel commutator applied to 1 is O(epsilon) uniformly (bounded second derivatives). Root-n times the stochastic bound is epsilon^1/4. A complete proof should condition on Xi and explicitly include its centered population integral, rather than assert evaluation-site independence.

4. FEATURE COMPARISON
C_hat-C=O_p(eta) via Hilbert-Schmidt plus relative scale weights is a legitimate coarse route and needs no infinite-dimensional matrix Bernstein. Sn* versus S_epsilon* is not a comparison of maps on the same L2 space; formulate all comparisons with the common RKHS covariance and explicit scalar scale factors. Errors of size eta*zeta in feature norm become root-n negligible because sqrt(n)eta*zeta=n^-1/2 epsilon^-3/4=o(1).
For self-adjoint PSD C and C_hat with norm one, B_C=(I-C)/2 and B_C_hat have norms <=1/2. The telescoping power difference bound is j*2^(-(j-1))*||C_hat-C||/2, uniformly O(||C_hat-C||) over j. Thus it does not accumulate an unbounded iteration factor. Resolvent differences are likewise O(eta). These facts support the stated feature proxy comparison but do not alone prove the sample relative-error initializer estimates.
Point-score row factors must be compared too. A weighted-C1 scale error eta gives derivative-feature difference O_p(eta epsilon^-3/4) in d=1, which after multiplying by g_n and A yields O_p(sqrt(n)eta*zeta)=o(1). Thus the pointwise comparison has a compatible rate.

5. MAIN CORRECTION: RESIDUAL MEASURE COMPARISON
Covariance-op control alone gives
 | ||Sn* h||_n - ||S_epsilon* h||_mu |
 <= ||C_hat-C||op^1/2 ||h||H,
up to the separately controlled scale-feature and power errors. With h=B_C^j g_n, the coarse bound is O_p(sqrt(eta)*zeta), or with the spectral decay an additional factor 2^-j.
Its root-n magnitude is O_p(n^-1/4 epsilon^-1/2). Claiming this is o_p(1) would require n epsilon^2->infinity, restricting polynomial kappa<1/2. Therefore a root-n-o(1) residual norm approximation throughout the full kappa<2/3 window is NOT supplied by the proposed covariance argument.
However the first-hit theorem needs only error o_p(t_n), not error o_p(n^-1/2). Choosing gamma=3/16 gives
 sqrt(eta)*zeta / t_n
 =O_p(n^-1/4 epsilon^(gamma-1/2))
 =O_p((n epsilon^(3/2))^-1/4 epsilon^(gamma-1/8)) ->0.
Thus root's adopted gamma=3/16, together with the same n epsilon^(3/2)->infinity condition, repairs this issue while preserving the full d1 inference window. The claimed residual approximation should explicitly be threshold-relative and uniform in j>=2, with root-n-o(1) reserved for the separately controlled initializer/nonlinear errors. A stronger root-n-o(1) statement must not remain in the final theorem.
An alternate repair can exploit 2^-J at the old gamma=1/16, but it is unnecessary with the adopted simpler gamma.

6. NONUNIFORM PROXY COVARIANCE, SPECTRUM, AND QUADRATIC VARIANCE
For j>=1, set h_j=B_C^j g_n and Q_j=||S_epsilon*h_j||mu^2/4. Since Cov(g_n)=(C-v tensor v)/n and B_C v=0,
 E Q_j=tr(C^2 B_C^(2j))/(4n).
At j=0 subtract 1/(4n) from this trace formula: the centered g covariance removes the stationary eigenvector. The intended theorem starts after two mandatory updates, so j>=1 is enough, but a universal j>=0 formula would be wrong without this correction.
The Q_j are nonnegative spectral sums. Their ratio sequence is nondecreasing wherever positive, giving the same log-convex overshoot mechanism as in the uniform proof. Uniform sample Fourier identities cannot be reused verbatim for nonuniform data. For the general Hilbert mean, if A=C B_C^(2j) and Xi=phi(Xi)-v, then
 Var(Q_j)=Var(<Xi,A Xi>)/(16 n^3)
          +(n-1) tr((A(C-v tensor v))^2)/(8 n^3).
The diagonal variance is no longer zero. A pointwise diagonal bound M times its expectation controls it by a term M/(n trace) in relative variance. The proposed M<=C epsilon^-1/2*4^-j is consistent with the feature evaluation bound. This term needs to be carried through the chosen J rather than omitted.
Population P conjugates to M_w H M_w with ||w-1||infty=O(epsilon). Hence operator perturbation O(epsilon), Weyl/min-max, and fixed separated heat eigenvalue shells give nonuniform fixed-shell eigenvalue counts of order epsilon^-1/2. The stationary eigenvalue is harmless because the shell lies away from 1. This is a spectral counting statement, not a claim that the nonuniform eigenvectors are Fourier modes.

7. ADOPTED J AND THRESHOLD POWERS CHECK
For J=floor[log(1/epsilon)/(128 log2)], a fixed shell lambda in [1/4,1/2] gives
 T_J>=c epsilon^-1/2*16^-J>=c epsilon^-15/32.
Thus sqrt(n) proxy residual is at least order epsilon^-15/64, which dominates the threshold root-n scale epsilon^-3/16. These powers are correct, with room for floor rounding. Off-diagonal and diagonal relative variance terms vanish for this very small J throughout the stated window, provided shell constants are chosen with margin under the O(epsilon) spectral perturbation.
One must show no threshold hit among all allowed j from the mandatory two updates through J. Monotonic Q_j plus the single uniform o_p(t) norm comparison suffices. A first-hit definition restricted to j>=2 is honest but should be stated explicitly; it is not identical to an unrestricted first hit from j=0.
For overshoot, high-shell domination and spectral log-convexity can propagate a fixed lower ratio for Q_j to all later j. Norm comparison at scale o_p(t), rather than root-n scale, still yields r_tau>=c t with high probability from r_(tau-1)>t. The empirical r_j itself need not be monotone. A budget must be defined on every sample and proved to hit on the high-probability local event.

8. WHAT IS NOT YET CERTIFIED
This route audit does not certify a complete nonuniform theorem. The full candidate must supply: (a) quantitative RKHS branch with empirical and population covariance bounds; (b) actual first-iterate L2 initializer and sample-dependent commutator calculation; (c) all-j common-RKHS proxy comparison, including derivative row factors and centering; (d) threshold-relative residual measure comparison; (e) general Hilbert quadratic variance including its nonconstant diagonal; (f) fixed-shell spectral counting with perturbation margins; (g) a simultaneous-in-iteration pointwise score tail and first-hit budget/overshoot proof; and (h) a matched nonuniform periodic exact-score law and explicit centering if inference is claimed.
The d1 interval 2/7<kappa<2/3 is arithmetically compatible with these rates. No result for d2, arbitrary kappa<1, a spatial supremum CLT, or general solver trajectories follows from this audit.

PARAMETER RECONCILIATION AFTER PARENT UPDATE
The parent now proposes retaining the original J=floor[log(1/epsilon)/(16 log2)] and gamma=1/16 instead of the simpler gamma=3/16 option discussed above. This is also rate-compatible, provided the 2^-j factor in the measure-norm comparison is retained.
For j>=J,
 sqrt(n)*measure_norm_error <=O_p(2^-J n^-1/4 epsilon^-1/2)
                           =O_p(n^-1/4 epsilon^-7/16).
Dividing by the threshold root-n scale epsilon^-1/16 gives O_p((n epsilon^(3/2))^-1/4)=o_p(1). For 2<=j<=J, the rough uniform measure error O_p(n^-1/4 epsilon^-1/2), divided by the proxy lower residual scale epsilon^-1/8 at J, is again O_p((n epsilon^(3/2))^-1/4)=o_p(1). A fixed-shell T_J>=c epsilon^-1/4 provides that lower scale. Thus both the no-early-hit and later-threshold comparisons can work without claiming a uniform root-n-o(1) residual norm approximation.
Empirical mean-centering of the population feature residual costs <g,B^j g> of size O_p(zeta^2*2^-j), because <v,B^j g>=0. Its root-n size n^-1/2 epsilon^-1/2 is negligible. Empirical scale-feature substitutions add eta*zeta errors, also root-n negligible. The full candidate should spell out these comparisons, rather than treating population and empirical means as identical.
