FINITE-SAMPLE TRUE-SCORE COVERAGE: HONESTY BARRIER AND SCOPED SALVAGE
18 October 2026. Independent adversarial feasibility audit.

Baseline read, without alteration:
D:/LUMINARY/research/self_transport_nonuniform_revision2_20261018/revised_stopping_theorem.txt
SHA256 7d2ed0072718a9fbb84581ed4c67fe75d4781468592bafa3198e163bbcd12edd.

1. What the existing theorem actually supplies

The baseline fixes one positive C^6 density f on T^d (d=1 or 2), then lets
n tend to infinity and epsilon tend to zero in the stated sufficient window.
Its constants may depend on f, its lower bound, and its derivative norms.
It proves a fixed-query limiting Gaussian law and same-sample Slutsky
studentization after actual adaptive stopping. It states no common C^6 norm
bound and no uniform finite-n coverage error. These are material quantifiers.

Conclusion: nominal pointwise asymptotic confidence regions are justified in
that exact limiting sense; they are not automatically honest finite-sample
regions over the entire stated class.

2. Elementary sharp obstruction for this unbounded smoothness class

Fix an integer sample size n>=1 and a fixed query x0 on T^d. Let F contain
all positive C-infinity probability densities with f>=1/2; this is already a
subclass of the baseline density class. Let theta(f)=partial_1 log f(x0).
Let C_n be any measurable data-based confidence set for theta(f), allowing
auxiliary randomization independent of the data. It can use all sample
information, any kernel bandwidth, every residual and iterate, any stopping
time based on this sample, bootstrap calculations, or sample splitting.

PROPOSITION. If, for a fixed alpha in (0,1),
  inf_(f in F) P_f[theta(f) in C_n] >= 1-alpha,
then, under the uniform density f0=1,
  P_0[q in C_n] >= 1-alpha for every q in R.                 (B1)
Consequently C_n cannot be almost surely bounded under f0. If C_n=[L_n,U_n]
is an interval in the extended real line, then
  P_0[U_n=+infinity] >= 1-alpha,
  P_0[L_n=-infinity] >= 1-alpha,
and for alpha<1/2,
  P_0[C_n=R] >= 1-2alpha.                                  (B2)
Thus any interval with finite endpoints almost surely has infimum coverage
zero over F. The same conclusion excludes almost surely finite-radius
confidence balls for the full score vector.

PROOF. For any fixed q in R and sufficiently large integer m, put
  f_(m,q)(x)=1+[q/(2 pi m)] sin(2 pi m(x_1-x0_1)).
This is a probability density on the unit torus, is C-infinity, and is
uniformly at least 1/2 once m>=|q|/pi. At the fixed query,
  f_(m,q)(x0)=1, partial_1 f_(m,q)(x0)=q,
so theta(f_(m,q))=q exactly. The remaining score coordinates vanish.
The one-observation total variation distance is
  TV(P_(m,q),P_0)
   = (1/2) int |f_(m,q)-1| dx
   = |q|/(2 pi^2 m).
The product-law coupling inequality gives
  TV(P_(m,q)^n,P_0^n) <= n |q|/(2 pi^2 m) -> 0.
Uniform coverage implies P_0[q in C_n]>=1-alpha minus this distance.
Letting m tend to infinity gives (B1). Independent auxiliary randomization
does not change total variation, so the argument covers randomized rules.

If C_n is almost surely bounded, its membership indicators 1{q in C_n}
tend to zero almost surely as q tends to positive infinity. Dominated
convergence contradicts (B1). For intervals, (B1) implies
P_0[U_n>=q]>=1-alpha for every positive q, so continuity from above gives
P_0[U_n=+infinity]>=1-alpha. Apply the same argument to negative q for the
lower endpoint; the union bound gives (B2).

To see that the infimum coverage for an almost surely bounded C_n is zero,
choose q so that P_0[q in C_n] is arbitrarily small, then choose m so that
the above product total variation is arbitrarily small. The corresponding
alternative has score exactly q and coverage arbitrarily close to that
small number. For vector balls apply the same ray q e_1 construction. QED.

The proof works even with a known common lower bound 1/2 and arbitrarily
smooth individual densities. The missing quantitative control is the
smoothness radius, not mere differentiability or positivity. For example,
the sixth derivative norm of f_(m,q) is proportional to |q| m^5.

This is not a contradiction to the baseline's fixed-density asymptotics:
uniform coverage ranges over a different admissible density for each n,
whereas that theorem keeps f fixed. It also does not exclude density-specific
finite-n guarantees with unknown f-dependent constants, honest inference on
a specified bounded smoothness class, smoothed targets, or vacuous unbounded
confidence sets. It makes no impossibility assertion about every meaningful
finite-sample numerical calibration.

3. An elementary quantitative salvage with stated smoothness bounds

The following is supplied to show feasibility, not proposed as an original
self-transport theorem. Suppose constants m0>0, M0, M2, M3,j are known and
  f>=m0, ||f||_infinity<=M0,
  ||Delta f||_infinity<=M2,
  ||partial_j Delta f||_infinity<=M3,j, j=1,...,d.
C^3 is enough for these assertions; a known C^6 radius can imply them.
Let H_epsilon be the exact periodic Gaussian heat convolution with Fourier
multiplier exp(-2 pi^2 epsilon |k|^2), so H_epsilon=exp(epsilon Delta/2).
For any deterministic epsilon>0 and fixed x0, set
  f_tilde=n^(-1) sum k_epsilon(x0-X_i)>0,
  g_tilde_j=n^(-1) sum partial_j k_epsilon(x0-X_i),
  z_j=g_tilde_j/f_tilde.
Use actual periodic-kernel bounds (or rigorous upper bounds)
  K0=||k_epsilon||_infinity,
  Kj=||partial_j k_epsilon||_infinity,
  V0=M0||k_epsilon||_(L2(dx))^2,
  Vj=M0||partial_j k_epsilon||_(L2(dx))^2.
For alpha in (0,1), let u=log(2(d+1)/alpha), and define conservative errors
  c0=epsilon M2/2+sqrt(2V0 u/n)+2K0 u/(3n),
  cj=epsilon M3,j/2+sqrt(2Vj u/n)+4Kj u/(3n).
Then, with probability at least 1-alpha, simultaneously
  |f_tilde-f(x0)|<=c0,
  |g_tilde_j-partial_j f(x0)|<=cj, all j.

For completeness, the heat-semigroup integral identity and sup-norm
contraction give the displayed epsilon-times-Laplacian bias bounds.
Bernstein applies to each bounded iid kernel evaluation: variance is at
most V0 or Vj, centered envelope at most K0 for the nonnegative density
kernel and at most 2Kj for the signed derivative kernel. The constants
2/3 and 4/3 above are conservative inversions of Bernstein's tail bound.
Two-sided tails and a union bound over d+1 quantities give the event.

On this event, the exact ratio identity gives
  |z_j-partial_j log f(x0)| <= [cj+|z_j|c0]/m0.              (S1)
For example, g_tilde_j-z_j f_tilde=0, and
  z_j-partial_j f/f
    = (g_tilde_j-partial_j f)/f-z_j(f_tilde-f)/f.
This proves (S1), using f>=m0. The product rectangle, or a Euclidean ball
with radius the L2 norm of these component bounds, therefore has honest
finite-n coverage on this quantitative class.

If one insists on centering at an adaptively stopped self-transport score
s_tau, the componentwise radii can be enlarged by the computable numbers
  |s_tau,j(x0)-z_j|.
The triangle inequality then yields valid finite-n regions on the same
event for any same-sample measurable choice of tau. There is no optional
stopping penalty because the statistical event was simultaneous for the
fixed KDE quantities before tau was chosen. But the enlargement may be
large; this construction alone does not establish efficient self-transport
confidence regions or a valuable new stopping theorem. It should not be
sold as such. Bandwidth selection would require a predeclared finite-grid
union bound or another valid simultaneous construction; the displayed
calculation only states one deterministic bandwidth.

4. Implications for the research decision

(a) Distinguish numerical tolerance from unknown statistical bias. A residual
certificate bounds errors relative to the exact sample optimizer. It cannot
by itself certify that an unsmoothed derivative of f is covered: the
high-frequency alternatives above remain statistically indistinguishable.

(b) Monte Carlo calibration on a finite family is useful evidence about
those stated models and parameter ranges. It cannot establish finite-n
honesty over the entire baseline class. Coverage should be recorded with
the exact target, confidence level, replication count, Monte Carlo error,
bandwidth, tolerance, sample size, and density model.

(c) A mathematically sound extension must either impose quantitative class
bounds and prove statistical plus algorithmic error control, target a
specified smoothed functional, prove a scoped transfer of coverage from an
already justified confidence set, or explicitly remain a simulation study.
An exact transfer statement may be useful but its elementary triangle-
inequality component is standard. Meaningful originality would have to lie
in a substantially sharper, computable self-transport-specific certificate
or an otherwise unproved quantitative distributional approximation.

(d) The barrier proposition and the elementary KDE salvage are standard
statistical mechanisms in this specialized notation. They are valuable as
honesty checks and research design constraints; they are not a substantial
standalone novel mathematical advance. Do not manufacture a new-paper claim
from them if no stronger self-transport-specific result survives review.

5. Primary literature consulted for context (not needed for the proof)

Mark G. Low (1997), On nonparametric confidence intervals, Annals of
Statistics 25(6), 2547-2554, DOI 10.1214/aos/1030741084.
https://projecteuclid.org/journals/annals-of-statistics/volume-25/issue-6/On-nonparametric-confidence-intervals/10.1214/aos/1030741084.full
The accessible publisher abstract describes lower bounds on expected
interval length and the obstruction to adaptation over derivative
smoothness classes while maintaining uniform coverage. The above simple
unbounded-radius score proposition is proved directly and is not attributed
to an identical statement in that paper.

Timothy B. Armstrong and Michal Kolesar (2019 arXiv v6; published 2020),
Simple and Honest Confidence Intervals in Nonparametric Regression.
https://arxiv.org/pdf/1606.01200
Read introduction pages 1-5 and Section 2's bias-aware construction.
It explains the distinction between pointwise-in-f asymptotics and honest
coverage, the need for explicit smoothness-radius information without
additional class restrictions, and bias-aware critical values. These are
well-developed ideas. Its main setting is not the exact torus self-Sinkhorn
stopping problem, so it is contextual precedent rather than a claimed
covering predecessor for the root's eventual specialized theorem.

6. Review scope

This audit independently read the baseline source and derived the above
counterexample and elementary bounded-class alternative. It is not an
external human review, a review of a future root candidate, or an endorsement
of any claimed finite-n Berry--Esseen/self-transport bias bound. No original
edition, frozen evidence, manuscript or publication was changed.
