APPLICATIONS, VALUE AND NEXT DECISIVE QUESTIONS
18 October 2026. Research assessment of the second nonuniform stopping edition.
Read the exact theorem and final source-bound reviews for mathematical authority.

What the theorem makes possible

For a fixed positive smooth density on the circle or two-dimensional torus,
one may check the actual residual from the initial iterate and use a relaxed
tolerance n^(-1/2) epsilon^(-gamma), with any fixed 0<gamma<d/4 in the proved
bandwidth window. The returned score has the same pointwise Gaussian law,
centered at the true log-density gradient, as the fully converged estimator.
The attained residual is larger than root-n by a diverging factor. Thus an
arbitrary-vector worst-case error certificate need not prescribe the accuracy
needed along this random nonlinear solver trajectory. This is a precise result
for this estimator and model, not a new general principle of statistical stopping.

The leading update count is
  tau_gamma ~ [(d/4-gamma)/log 2] log(1/epsilon)
in probability. Two fixed exponents therefore yield a quantified leading count
difference. This is not a wall-clock benchmark or optimal complexity theorem.
At gamma>=d/4 the unit-coefficient rule already stops at j=0 for uniform f,
giving the wrong limiting variance. Increasing the tolerance without restriction
can therefore invalidate an inference law even when the returned score remains
consistent. The endpoint result is for this rule over the stated class, not
for every density or every constant multiplying the endpoint threshold.

Direct statistical applications

1. Pointwise confidence regions for the density score grad log f(x). A positive
   consistent periodic KDE estimates the variance factor f(x), even on the same
   sample as the stopped estimator. This gives the displayed Gaussian pivot,
   directional intervals and chi-square balls.
2. Testing grad log f(x)=0 at a prespecified location. This is a local gradient
   test; it neither locates an unknown critical point nor classifies a mode.
3. Local log-density slope inference for circular/phase data, or paired phases,
   when iid sampling, positivity and periodic smoothness are appropriate.
4. A fixed-point value of a drift proportional to grad log f can inherit the
   same uncertainty law by deterministic rescaling. Whole diffusion trajectories
   and stationary sampling distributions require additional results.

Useful constants, independently checked as supplementary calculations

  d=1: J_1 approximately 0.06392985462807312,
       4J_1 approximately 0.25571941851229246.
  d=2: J_2=(pi^2/12-log 2)/(2pi),
       4J_2 approximately 0.08232757529299570.

For d=2, polar coordinates and u=2pi^2 r^2 reduce J_2 to
  (2pi)^(-1) integral_0^infinity u/(1+exp u)^2 du.
Integration by parts, or the convergent alternating exponential expansion,
gives integral=pi^2/12-log 2. This elementary evaluation is standard, with
no separate originality claim. Floating radial/lattice agreement is a sanity
check; the analytic theorem does not depend on an unenclosed numerical integral.
At uniform density the initial-score variance is 2(4pi)^(-d/2), namely
approximately 0.5641895835477563 in d1 and 0.15915494309189535 in d2, both
strictly greater than the corresponding exact-score variance 4J_d.

Contribution judgment

The defensible candidate originality is the combined actual nonlinear iid
first-hit theorem: genuine initial checking, a strict family of relaxed
tolerances, a matching class-level endpoint obstruction, leading update counts,
and true-score inference in a nonempty d2 bandwidth range. Its new technical
step is the normalized-row cancellation that closes the d2 mixed-error loss,
combined with iteration-dependent stochastic norm comparison and spectral lower
control. Generic RKHS differentiation, population matching after two updates,
spectral contraction, CLTs, log-rate inversion and studentization are credited
prior or standard mathematics. No checked primary theorem covers the full joint
statement; that focused comparison does not certify worldwide priority.

This is a worthwhile focused contribution to numerical statistics and entropic
self-transport. It substantially improves this theorem family's completeness,
but does not establish a broad advance in optimal transport or machine learning.
It merits development as one coherent specialist theorem, with the predecessor
relationship explicit. Extra editions or corollaries should not be counted as
separate independent contributions.

What is still required for publication or practical use

The research proof and reviews are not yet a submission-ready manuscript. A
professional treatment should expand the shared population/Gram lemmas into
numbered, inspectable arguments, give a normalization-matched related-work
comparison and bibliography, and state the stopping rule and endpoint limits
without editorial ratings. External human review remains valuable. There is
no known blocking proof objection after the completed internal reviews; these
presentation requirements do not substitute for their exact source dispositions.

Finite experiments remain diagnostics. Earlier n values often hit at the first
permitted iterate and did not display the asymptotic residual lower ratio. The
new FFT calculation probes population orders, not empirical stopped-score
coverage, and does not repair or conceal those earlier finite observations.
Finite-n confidence coverage, calibrated constants and measured runtime gains
have not been established.

Next research, ranked for its relation to the current theorem

First: derive explicit finite-sample remainder/probability bounds and test a
calibrated tolerance/coverage rule. This most directly turns the asymptotic
theorem into a usable procedure; it requires real new bounds, not more favorable
plots alone. It should precede practical speed or coverage claims.

Second: obtain spatial or integrated score-error control for the stopped rule.
That would support selected queries, functional inference or a separately proved
diffusion-drift transfer. Fixed-query Slutsky alone cannot supply these results.

Higher dimensions are a deeper change. At d=3 the present conditions become
n epsilon^(9/2)->infinity and n epsilon^(9/2)->0, an empty window. A credible
extension needs improved stochastic control and/or bias correction; merely
replacing d by 3 is invalid, and bias correction alone does not automatically
repair every mixed-row error. Boundaries and vanishing densities also require
new analysis. No such extension is asserted or begun in this checkpoint.
