NONUNIFORM EXTENSION: RATE AUDIT AND FOCUSED TRAJECTORIES
Research only. No manuscript/publication action. Prior proofs and corrected manuscripts were read-only. This is a route/rate audit with finite diagnostics, not a completed nonuniform stopping theorem.

1. WHAT THE EXISTING NONUNIFORM ESTIMATES DO AND DO NOT PROVIDE
The corrected empirical proof gives a weighted-C1 exact-scale error
 eta_old=sqrt(log n/(n epsilon^d)).
It vanishes throughout n epsilon^d/log n->infinity. Comparing the exact scale to the first actual update b1=(K_mu 1)^(-1/2), together with a_epsilon=f^(-1/2)(1+O_C1(epsilon)), gives a projective relative-log burn-in error of order
 O(epsilon)+O_p(eta_old)
in the value supremum norm (direct kernel empirical concentration is no worse). Thus the first actual step lies in a small local relative-log neighborhood under the stated sampling regime. This does NOT itself give a sufficiently small normalized score Taylor remainder. Reusing a Sobolev H^-m->H^m factor epsilon^-m would return the old unhelpful bandwidth condition.

For a true positive Markov P and relative log e, the exact update is
 Psi(e)=[e-log(P exp e)]/2.
The bound ||Psi(e)||_infinity<=osc(e)/2<=||e||_infinity shows that a small uncentered relative-log ball is invariant. In empirical L2, PSD Markov P has ||(I-P)/2||<=1/2; its nonlinear remainder obeys
 ||Psi(e)-(I-P)e/2||_2<=C ||e||_infinity ||e||_2.
Thus after a sufficiently small value-norm burn-in there is a local L2 contraction. Mean-zero log coordinates are not automatically preserved; projective centering or retaining the uncentered chart must be specified. A proof cannot simply assume invariance of centered logs.

A potentially useful finite-dimensional bound: with uniform empirical weights, B=(I-P)/2 has infinity operator norm <=1, since its absolute row sum is 1-Pii, and L2 norm <=1/2. Hence
 ||(I+P)^-1||_infinity
 <=(1/2)sum_k min(1,sqrt(n)*2^-k)=O(log n).
This bound is valid for the finite PSD Markov matrix, without an epsilon-dependent inverse-eigenvalue loss. It does not by itself identify the exact scale via a quantitative continuation argument or prove an empirical functional expansion.

2. DETERMINISTIC POPULATION BURN-IN AND THE UNDERSMOOTHING WINDOW
For fixed positive smooth nonconstant f, the population b1 and exact a_epsilon both agree with f^(-1/2) to O(epsilon) in C1. Consequently their pointwise score error is O(epsilon), provided the relevant bounded C1/positive potential estimates are retained. At A=sqrt(n)epsilon^(d/4+1/2), this deterministic error is negligible if
 sqrt(n)epsilon^(d/4+3/2)->0,
 equivalently epsilon=n^-kappa with kappa>2/(d+6).
In dimension one this is precisely kappa>2/7. Thus a first-step O(epsilon) deterministic bias is compatible with the requested inference-useful undersmoothing condition. Later population errors should remain controlled or improve, but a uniform-in-j population argument is still required; this audit does not infer it merely from the first-step estimate.
If one also wants A->infinity (vanishing unnormalized stochastic score errors), kappa<2/(d+2), which is <1 for d>=1 except its interpretation as a separate requirement. In d=1 this is kappa<2/3. A pointwise triangular-array normalization can still be discussed when A does not diverge, but that would not imply consistency. The requested sampling upper range kappa<1/d alone should not be silently equated with A divergence.

3. RKHS POTENTIAL BRANCH: ROOT'S RATE ROUTE CHECKS
Use the population kernel RKHS and potential v=1/a. The equation
 F_mu(v)=v-K_mu(1/v)=0
has derivative I+C at the exact solution, where
 C=int a(y)^2 k_y tensor k_y dmu(y).
This is self-adjoint PSD. At the population exact scale ||C||=1. The mean-zero empirical forcing has RKHS size
 zeta_H=O_p(n^-1/2 epsilon^-d/4)
(up to logarithms for stronger high-probability events). The value evaluation loss is epsilon^-d/4, so a local potential branch has relative value perturbation about n^-1/2 epsilon^-d/2. Uniform local positivity is therefore compatible with n epsilon^d/log^q n->infinity.
For bounded positive reciprocal weights and ||C_n||=O_p(1), the second derivative satisfies a bound of the form
 ||K_mu(a^3 h1 h2)||_H
 <=C epsilon^-d/4 ||h1||_H ||h2||_H.
One can obtain this via the feature operator, one value evaluation bound, and the L2-to-feature operator norm. The potential remainder is then of size
 epsilon^-d/4 zeta_H^2=O_p(n^-1 epsilon^-3d/4).
A score derivative evaluation costs epsilon^(-d/4-1/2), canceled by the epsilon factor in A. Thus its normalized stochastic quadratic remainder is
 O_p(n^-1/2 epsilon^-3d/4),
so a sufficient polynomial condition is n epsilon^(3d/2)/log^q n->infinity.
This rate calculation is consistent. Positivity alone ensures ||(I+C_n)^-1||<=1; concentration/relative weights are needed to keep ||C_n|| bounded and identify the branch, not to avoid a negative spectrum.
A Hilbert/operator Bernstein covariance bound at scale sqrt(log/(n epsilon^(d/2))) is plausible because ||phi||^2=O(epsilon^-d/2), E phi tensor phi has norm O(1), and the square of a rank-one feature is bounded in form order by its squared norm times that covariance. A complete proof must specify a suitable infinite-dimensional Bernstein/effective-rank argument or supply an elementary substitute; it cannot cite a finite-matrix inequality without accounting for the ambient RKHS.

The resulting sufficient window kappa<2/(3d), combined with deterministic undersmoothing kappa>2/(d+6), has nonempty overlap only for d<3. Thus this particular conservative quadratic route plausibly gives inference-useful windows in dimensions 1 and 2, not all dimensions. In d=1, 2/7<kappa<2/3 is nonempty; the focused experiments use .35,.5,.6 inside it.

4. MAIN REMAINING ACTUAL-TRAJECTORY GAP
The potential trajectory is not closed under an update. With relative multiplicative direction u=b/a-1, write S u=K_mu(a u), P=S* S, C=S S*. Then
 u_next=(I-P)u/2+N(u),
 h_next=(I-C)h/2+S N(u), h=S u.
Although ||S N(u)||_H<=C||u||_infinity||u||_2, there is no uniform inequality ||u||_2<=C||S u||_H: arbitrarily small eigenvalues obstruct it. Squaring a high-frequency direction can generate lower-frequency content. Therefore the potential RKHS contraction alone does not establish a uniform nonlinear trajectory expansion. Separate L2/infinity bounds and a structured initialization/noise decomposition are necessary.
Crude cross terms between smooth deterministic burn-in O(epsilon) and stochastic value noise O_p(n^-1/2 epsilon^-d/2) can lose too much if estimated without structure. In particular an unstructured bound can generate normalized factors epsilon^(1-d/2), which is harmless in d=1 but not automatically vanishing in d=2. Multiplication/commutator localization is needed before asserting the cleaner epsilon^(1-d/4) type gain. The proof must state exactly which norm is controlled and where a smooth deterministic factor can be moved through a heat/resolvent operator.
Population conjugation U P U^-1=M_w H M_w with w=a_epsilon sqrt(f)=1+O_C1(epsilon) gives O(epsilon) L2 operator difference. This controls resolvent/power comparisons in L2, but is not alone a pointwise derivative comparison. The derivative-row error and density factors must also be tracked. Growing-j errors can sum to j*epsilon before contraction; with j=O(log(1/epsilon)) this still vanishes, but that estimate needs a proof in the actual norm used.

5. WHAT A FIRST THRESHOLD THEOREM STILL NEEDS
A threshold t=n^-1/2 epsilon^-gamma with gamma>0 does not by itself guarantee a diverging attained sqrt(n)r: a first hit can overshoot. The uniform model's positive spectral Q_j and log-convex ratios are not automatically available for a nonuniform actual finite trajectory.
One needs a quantitative proxy for the actual nonuniform iterate after burn-in, a pointwise score tail bound uniform over all j past a deterministic growing J, and residual lower/concentration bounds up to J. Then an overshoot lower bound is required if the conclusion includes sqrt(n)r_tau->infinity. The initial stochastic residual natural scale is n^-1/2 epsilon^-d/4, so gamma<d/4 is a natural candidate range; it is not a proved universal tolerance. Smooth population bias may dominate the residual at early iterations while already being negligible at score normalization; that can help a counterexample to worst-case necessity but complicates the actual first-hit proof.
For a CLT or inference transfer, the exact empirical estimator must have a matched nonuniform periodic limit law with specified centering/bias. A theorem from another cost/domain cannot be imported automatically. The deterministic population first-step score error and exact-estimator bias are distinct quantities.

6. FOCUSED ACTUAL TRAJECTORY EXPERIMENTS
Source: code/nonuniform_trajectories.py. Authoritative outputs: results/run_01/results.json, iterations.csv, SUMMARY.txt.
Fixed density f=1+.4 cos(2pi x); exact standard start b0=1; actual damped log update. n=512,2048,8192; epsilon=n^-kappa with kappa=.35,.5,.6; seeds 2026101901,2026101902 (18 cases). Query x0=.137 plus 256 grid points. Positive-coefficient finite Fourier kernel, exponent45 cutoff. Empirical solves use low-rank features throughout, not n-by-n matrices. Reference maximum mass residual <1e-14. Separate 4096-point weighted uniform quadrature computes an approximate population scale and its first six actual trajectories. Neither is a rigorous continuum computation.
All 18 runs pass. Maximum empirical reference mass residual 9.33e-15, finite-difference score error 4.95e-10, and small-feature resolvent equation residual 1.22e-15. No packages installed. No old evidence written.
At these modest sample sizes, thresholds with gamma=.0625,.125,.2 often hit at iteration 0 or 1; some kappa=.35 cases hit at iteration 2. They do not yet exhibit the growing-iteration asymptotic regime. For gamma=.125 and first seed, normalized point errors at n=8192 are approximately -.3013,-.3709,-.0641 for kappa .35,.5,.6 respectively; these are reported, not declared asymptotically negligible. The first-step population relative-log error divided by epsilon stays finite in the tested cases but varies substantially, as expected before a genuine small-epsilon regime. Finite floats neither establish nor refute the desired joint theorem.

DISPOSITION
The deterministic first-step observation and the potential-RKHS quadratic rate route are credible and match a useful low-dimensional undersmoothing window. A complete actual-trajectory theorem remains unproved here. The key outstanding item is a structured nonuniform stochastic/trajectory expansion in norms that retain the RKHS rate and control smooth-bias/noise cross terms, followed by a first-hit overshoot argument. Do not advertise the experiments or this rate arithmetic as supplying those missing bounds.
