FINITE-SAMPLE SCORE CALIBRATION: VERIFIED LIMITATION AND STOP DECISION
18 October 2026. Separate research record; not a new publication theorem.

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

1. Scope and conclusion

The baseline proves pointwise-in-a-fixed-density asymptotic Gaussian coverage
after actual stopping. It imposes no common derivative-norm radius on its C6
density class. This is insufficient for bounded confidence sets with a positive
uniform finite-sample coverage guarantee for the true unsmoothed point score.
The exact limitation is proved below. Quantitative smoothness-class calibration
is possible by standard methods, but no meaningful new self-transport-specific
finite-sample theorem was established in this investigation. The branch stops
at the user's mathematical-value condition. The baseline theorem is retained.

2. Definitions and quantifiers

Fix d=1 or 2, n>=1, and a deterministic spatial query x0 in the unit torus.
Let F* be all C-infinity probability densities satisfying 1/2<=f<=3/2.
This is a subclass of the baseline positive C6 class. Write P_f for the
one-observation distribution and theta(f)=partial_1 log f(x0).
A confidence procedure C_n is a measurable random set in R, based on the n
iid observations, optionally with independent auxiliary randomization. Events
{q in C_n} must be measurable. The procedure may use a data-dependent solver
stopping index, sample splitting, a bootstrap or a data-dependent bandwidth.
It remains one procedure and cannot use the unknown f as an oracle input.

PROPOSITION A. If C_n is almost surely bounded under the uniform density f0=1,
then
  inf_(f in F*) P_f^n{theta(f) in C_n}=0.                    (A)
The same assertion holds for a score-vector confidence set in R^d that is almost
surely bounded under f0. A fixed n is arbitrary; no limit in n is being taken.

PROPOSITION B. Suppose instead that for some 0<alpha<1 the procedure is honest:
  inf_(f in F*) P_f^n{theta(f) in C_n}>=1-alpha.
Then
  P_0^n{q in C_n}>=1-alpha for EVERY fixed q in R.            (B1)
If C_n=[L_n,U_n] with measurable extended-real endpoints, it follows that
  P_0^n{U_n=+infinity}>=1-alpha,
  P_0^n{L_n=-infinity}>=1-alpha.
In particular, when alpha<1/2,
  P_0^n{C_n=R}>=1-2alpha.                                  (B2)
The interval assumption is essential for the whole-line conclusion. For
general sets, (B1) does not say the entire line is contained simultaneously.

3. Direct proof by smooth periodic alternatives

For a fixed q in R and any sufficiently large integer m, define
  f_(m,q)(x)=1+[q/(2pi m)]sin(2pi m(x_1-x0_1)).
These are smooth probability densities in F* once m>=max(1,|q|/pi).
At the query f_(m,q)(x0)=1 and partial_1 f_(m,q)(x0)=q, hence
  theta(f_(m,q))=q.
All other score coordinates are zero. With TV denoting half the L1 distance,
  TV(P_(m,q),P_0)=|q|/(2pi^2 m),
  TV(P_(m,q)^n,P_0^n)<=n|q|/(2pi^2 m)->0.                  (C)
The first identity integrates |sin| on an integer number of torus periods.
The product bound follows by independent coupling or telescoping product
measures. An independent random seed has the same law under both models and
does not increase this bound for any output event.

For (A), boundedness under f0 implies 1{q in C_n}->0 almost surely as positive
q tends to infinity. Dominated convergence gives P_0^n{q in C_n}->0. For any
eta>0 choose q with this probability <eta/2, then m so that (C)<eta/2. The
alternative f_(m,q) has target q and coverage below eta. Thus the infimum is
zero. The vector proof uses the ray q e_1 and is identical.

For (B1), honesty and (C) imply
  P_0^n{q in C_n}>=1-alpha-n|q|/(2pi^2 m).
Let m tend to infinity with q fixed. For intervals, {q in C_n} implies U_n>=q
for q>0. Let q tend to infinity through integers and use continuity from above
to get P_0(U_n=infinity)>=1-alpha. Apply the negative ray to L_n. The union
bound gives P_0(L_n=-infinity,U_n=infinity)>=1-2alpha, proving (B2). QED.

An equivalent useful divergent-target alternative is
  f_m(x)=1+m^(-1/2)sin(2pi m(x_1-x0_1)), m>=4.
It belongs to F*, has theta(f_m)=2pi sqrt(m), and
  TV(P_m^n,P_0^n)<=n/(pi sqrt(m))->0.
Its score escapes every bounded confidence set while the data law approaches
the uniform law. Its sixth-derivative norm grows without bound. Mere C6 or
even C-infinity membership supplies no common control of this behavior.

4. Exact implication for the baseline Gaussian confidence balls

For any deterministic finite n and epsilon>0, the baseline's Gaussian ball
has zero coverage at some f in F*, regardless of the solver stopping rule,
if the Gaussian critical value is used as a universal finite-n guarantee.
This does not question its stated fixed-f asymptotic coverage.

Here is a direct proof for the exact periodic kernel. Compactness and strict
positivity give finite deterministic constants
  k_min=min_z k_epsilon(z)>0,
  G=max_z ||grad k_epsilon(z)||<infinity.
For ANY positive weights b_i and any sample, the score
  s_b(x0)=2[sum b_i grad k_epsilon(x0-X_i)]/
                                         [sum b_i k_epsilon(x0-X_i)]
satisfies ||s_b(x0)||<=2G/k_min. Every finite positive half-self iterate,
any data-based selection of those iterates, and the exact empirical scale
obey this bound. Also f_tilde(x0)=n^(-1)sum k_epsilon(x0-X_i)>=k_min.

With A=sqrt(n)epsilon^(d/4+1/2), the nominal Gaussian confidence ball from
the baseline has radius
  R_n=2sqrt(J_d chi-square_(d,1-alpha))/(A sqrt(f_tilde(x0)))
     <=R_max=2sqrt(J_d chi-square_(d,1-alpha))/(A sqrt(k_min)).
It is therefore contained, for EVERY possible sample, in the deterministic
ball centered at zero with radius D=2G/k_min+R_max. Choose an integer m>=4
with 2pi sqrt(m)>D and use f_m from Section 3. Its true score lies outside
the reported confidence ball for every sample: its coverage is exactly zero.
No numerical evaluation of k_min or G is needed for this finite-n statement.
The selected density may have very large derivative norms and depends on n
and epsilon; the baseline theorem fixes f before taking its asymptotic limit.

5. What quantitative assumptions can salvage

With declared known m0>0, M0, M2, M3,j such that
  f>=m0, ||f||_infinity<=M0,
  ||Delta f||_infinity<=M2, ||partial_j Delta f||_infinity<=M3,j,
ordinary bias-aware kernel concentration gives valid finite-n score sets.
The complete construction and proof are in BARRIERS_AND_SALVAGE.txt Section3.
In brief, heat-semigroup bias bounds epsilon M2/2 and epsilon M3,j/2,
bounded-variable Bernstein bounds for a periodic KDE f_tilde and derivative
g_tilde_j, and a union bound yield simultaneous errors c0,cj. The ratio
z_j=g_tilde_j/f_tilde then satisfies, with probability at least 1-alpha,
  |z_j-partial_j log f(x0)|<=[cj+|z_j|c0]/m0.
This is a standard explicit finite-sample reference region on that known class.
For a stopped self-transport center s_tau, enlarging its component radius by
|s_tau,j-z_j| preserves coverage on the same event, even for the same-sample
adaptive tau. This follows from the triangle inequality; it does not prove
that such enlargement is small or that the self-transport stopping rule is
more efficient. No new optimality or Gaussian coverage claim is made.

More generally, if an exact-reference confidence region has justified coverage
and a numerical certificate bounds ||s_tau-s_hat|| by a computed radius, set
inclusion transfers coverage to the inflated region, with the appropriate
union-bound loss if the certificate is probabilistic. Neither this transfer
nor a generic Newton/adjoint corrector establishes the missing reference law.
The independent construction audit retains a Newton route as exploratory
standard methodology, not as an accepted new finite-sample statistical theorem.

6. Originality, value and disposition

The smooth alternatives above instantiate established discontinuous-functional
and nonidentification impossibility arguments, not a new general statistical
principle. Dufour's Proposition3.1 and Theorem3.3 give the relevant general
coverage/unboundedness mechanism. Pfanzagl's work explicitly addresses
discontinuous functionals; access limits and exact comparisons are recorded
in LITERATURE_AND_VALUE.txt. Low's confidence-interval theory and Armstrong--
Kolesar's bias-aware inference supply additional context, with different
precise models. Ordinary kernel concentration, confidence-set inflation,
and Newton/adjoint numerical error estimates are existing tools.

No meaningful new self-transport-specific finite-sample tolerance/coverage
theorem was established beyond the preserved research edition. This is a
useful scope check and negative research outcome, not a publication increment.
Stop this branch under the user's condition. Preserve the prior theorem and
all attempted derivations, including limitations and failed routes.

The finding excludes uniformly honest bounded true-score intervals over the
unbounded-smoothness class. It does NOT exclude fixed-density asymptotics,
specified bounded smoothness classes, density-specific bounds with declared
constants, finite-model Monte Carlo calibration, exact-empirical numerical
error certificates, confidence sets for smoothed targets, or unbounded sets.
No manuscript, submission, publication, or external-chat handoff is performed.
