NONUNIFORM SELF-SINKHORN STOPPING: SECOND RESEARCH EDITION
18 October 2026. Analytic theorem and proof; research edition, not a manuscript.

Preserved predecessor:
D:/LUMINARY/research/self_transport_nonuniform_extension_20261018/joint_nonuniform_theorem.txt
SHA256 4cd5289f95224cd85f112d6070a7fb8b0853aa1aa12e289515e62bc78e94e02e.
The previous edition and all of its proof/review/numerical evidence are unchanged.

1. Revised theorem and improvements

Let d be 1 or 2. Fix a positive C^6 probability density f on the unit torus
T^d, bounded away from zero. Sample X_i iid from mu=f dx. The density is
fixed as n changes; constants may depend on its norms and lower bound.
Use the exact integral-one periodic Gaussian with Fourier multipliers
  lambda_k=exp(-2 pi^2 epsilon |k|^2).
Assume deterministic epsilon=epsilon_n satisfies
  epsilon->0, n epsilon^(3d/2)->infinity,
  n epsilon^(d/2+3)->0.                                    (H_d)
Thus for epsilon=n^(-kappa), the power windows are
  d=1: 2/7<kappa<2/3; d=2: 1/4<kappa<1/3.
These are sufficient windows, not asserted optimal.

Let a_hat K_n a_hat=1 be the exact empirical self-scale. Starting at b_0=1,
run the actual half-self updates
  b_(j+1)=sqrt(b_j/(K_n b_j)).
Define s_j=2 grad log(K_n b_j), s_hat=2 grad log(K_n a_hat), and the
projective square-root-mass residual
  F_j=sqrt(b_j K_n b_j), S_j=int F_j dmu_n,
  r_j=||F_j/S_j-1||_(L^2(mu_n)).
All queries and masses use the same sample and bandwidth.

Fix ANY relaxation exponent gamma with 0<gamma<d/4, and put
  t_n=n^(-1/2) epsilon^(-gamma),
  A_n=sqrt(n) epsilon^(d/4+1/2), L_n=max(3,ceil(4 log n)).
Let tau_gamma be the FIRST j in {0,...,L_n} with r_j<=t_n; if there is no
hit, return j=L_n. There is NO compulsory burn-in in this stopping rule.

THEOREM. Under (H_d):
(i) The threshold is genuinely hit before the budget with probability
    tending to one, and
      tau_gamma/log(1/epsilon)
            -> (d/4-gamma)/log 2 in probability.             (1)
(ii) For every fixed c<1/2 with c>0,
      P[c t_n<=r_(tau_gamma)<=t_n] ->1.                      (2)
     In particular sqrt(n) r_(tau_gamma)->infinity.
(iii) For each fixed spatial query x,
      A_n |s_(tau_gamma)(x)-s_hat(x)| ->0 in probability,
      A_n[s_(tau_gamma)(x)-grad log f(x)]
             => N_d(0,[4J_d/f(x)]I_d),                       (3)
      J_d=int_Rd (2 pi xi_1)^2/[1+exp(2 pi^2|xi|^2)]^2 dxi.
     The exact empirical score has the same centered Gaussian limit.
(iv) The positive-power range is sharp at its upper endpoint for this
     unit-coefficient threshold, uniformly over the stated density class:
     for gamma>=d/4, uniform sampling f=1 already gives tau_gamma=0 with
     probability tending to one and a DIFFERENT normalized score limit.
     This is a class counterexample, not a necessity assertion for every f.

For two fixed exponents 0<gamma_1<gamma_2<d/4 on the same sample,
  [tau_(gamma_1)-tau_(gamma_2)]/log(1/epsilon)
            ->(gamma_2-gamma_1)/log 2.                       (4)
This quantifies iteration counts, not wall-clock speed or per-iteration cost.

Changes from edition one: d=2 is included, the mandatory-two-update rule
is removed, the single exponent 1/16 becomes every fixed gamma in (0,d/4),
the attained-residual lower constant improves from 1/8 to any c<1/2,
and a leading iteration count and endpoint counterexample are established.

2. Common feature geometry and empirical scaling

We supply the quantitative facts used in the revised argument. Their
one-dimensional versions were proved in the preserved predecessor; all
definitions below now refer to T^d, d=1 or 2.

Let a be the population scale, v=1/a=K_mu a, w=H_epsilon f. In C^2,
  a sqrt(f)=1+O(epsilon),
  d_1:=log v-(1/2)log w=O(epsilon).                         (5)
Also s_pop=-2 grad log a=grad log f+O(epsilon) in sup norm.
The population proof works in the Banach algebra C^2: the heat-plus
inverse has a uniformly L1-bounded periodized Schwartz multiplier, its
commutator with sqrt(f) is O(sqrt(epsilon)) on C^2, and the forcing
H sqrt(f)-sqrt(f) is O(epsilon). A small quadratic contraction gives (5).
C^6 is sufficient. Uniqueness follows from the symmetric PSD energy identity.

In the Gaussian RKHS H let k_y denote the feature at y and define
  S u=int a k_y u dmu, S* h(y)=a(y)h(y), C=S S*, P=S* S,
  B=(I-C)/2, R=(I+C)^(-1), theta_x=grad_x[a(x)k_x].
P is PSD and Markov, ||S||=||C||=1, ||B||<=1/2, ||R||<=1,
  ||v||_H=1, Cv=v, S*v=1.
Each component of theta_x has norm O(epsilon^(-d/4-1/2)). Feature norms
are O(epsilon^(-d/4)). The population Gram limit is
  epsilon^(d/2+1)<theta_(x,j),R C R theta_(x,l)>
          ->[J_d/f(x)]delta_(jl).                           (6)
Indeed P is unitarily conjugate to M_q H M_q on L^2(dx), q=a sqrt(f),
with operator error O(epsilon) from H. Resolvent and derivative-row
comparison reduce (6) to the Gaussian lattice Riemann sum. The same
conjugation and min-max principle give at least c epsilon^(-d/2)
eigenvalues of C in each fixed positive-width interior spectral interval
in (0,1), counted through a smaller heat-eigenvalue band. Also
  tr C<=C_0 epsilon^(-d/2).                                 (7)

Write delta_mu=mu_n-mu and
  g=delta_mu(a k), g_0=delta_mu(k),
  zeta=n^(-1/2)epsilon^(-d/4), eta=n^(-1/2)epsilon^(-d/2).
Elementary Hilbert second moments give
  ||g||_H+||g_0||_H=O_p(zeta),
  Cov(g)=(C-v tensor v)/n, <v,g>=0 exactly.                 (8)
If C_0n=int a^2 k tensor k dmu_n, its Hilbert--Schmidt error is O_p(eta).
The equation F_n(z)=z-K_n(1/z)=0 near v has derivative I+C_0n. Its inverse
has norm at most one. The reciprocal remainder is bounded in H by
C epsilon^(-d/4)||h||_H^2; this follows from one sup evaluation and the
empirical covariance bound. As epsilon^(-d/4)zeta=eta=o(1), a local
contraction identifies the exact positive empirical scale and gives
  v_hat-v=R g+O_(H,p)(eta zeta),
  ||v_hat-v||_H=O_p(zeta),
  ||a_hat-a||_infinity+sqrt(epsilon)||grad(a_hat-a)||_infinity
                   =O_p(eta).                              (9)
The positive branch agrees at the sample sites with the unique finite
scale; its reciprocal convolution is the same off-sample extension.

Define exact empirical S_n,S_n*,C_hat,P_n by replacing a,mu with a_hat,mu_n.
Then ||S_n||=||C_hat||=1, and
  ||C_hat-C||_op=O_p(eta),
  ||theta_hat_x-theta_x||_H
                 =O_p(eta epsilon^(-d/4-1/2)).               (10)
Evaluation of H-functions in L^2(mu_n) is bounded by a constant times their
H norm on one common event with probability tending to one. None of these
bounds uses a finite-dimensional Bernstein inequality for an infinite RKHS.

3. The essential new two-dimensional burn-in estimate

Let e_j=log(b_j/a_hat). The exact relative-log map is
  Psi_n(e)=[e-log(P_n exp(e))]/2.                            (11)
Write u=e_1-d_1. Expanding the first update b_1=(K_n1)^(-1/2) gives
  u=a Rg-(1/2)g_0/w+O_(L2n,p)(eta zeta),
  ||u||_(L2n)=O_p(zeta), ||u||_infinity=O_p(eta).             (12)

A coarse O(epsilon eta) bound for B_n d_1 would fail to be root-n
negligible in d=2. Here B_n=(I-P_n)/2. Instead use normalized row
differences d_1(y)-d_1(x), whose absolute value is at most
C epsilon dist(x,y). Replacing the row weights a_hat by a changes the
normalized row by at most O_p(eta) times that mean absolute difference.
The empirical row mean of dist(x,y) is O_p(sqrt(epsilon)), uniformly x:
use dist k_epsilon<=C sqrt(epsilon) k_(2epsilon), bounded positive row
denominators, and the same feature second-moment/embedding bound at 2epsilon.
Thus this weight-replacement cost is O_p(epsilon^(3/2)eta).

For the remaining fixed-scale empirical numerator
  delta_mu_y k(x,y)a(y)[d_1(y)-d_1(x)],
the diagonal at x=X_i vanishes. Conditional on X_i=x its bias is
O(epsilon^2/n), and its variance is at most
  (C/n)epsilon^2 int k(x,y)^2 dist(x,y)^2 dmu(y)
            <=C epsilon^(3-d/2)/n.
Its L^2(mu_n) fluctuation is therefore O_p(epsilon^(3/2)zeta).
Denominator fluctuations cost O_p(epsilon^2 eta), since the population
row numerator is O(epsilon^2). The population Taylor estimate itself is
||(I-P)d_1||_infinity=O(epsilon^2). Combining these proves
  ||B_n d_1||_(L2n)=O_p(epsilon^2+epsilon^(3/2)eta).          (13)

Taylor-expand (11) at the small deterministic function d_1. Its derivative
is (I-Q_(n,d1))/2, where Q_(n,d1) is P_n tilted by exp(d_1) and normalized
by its row sum. On L^2(mu_n), Q_(n,d1)-P_n has norm O(epsilon).
The Taylor remainder in u has norm at most C||u||_infinity||u||_2.
Using (12)--(13),
  e_2=B_n u+E_2,
  ||E_2||_2=O_p(epsilon^2+epsilon^(3/2)eta+epsilon zeta+eta zeta),
  ||e_2||_2=O_p(epsilon^2+zeta),
  ||e_2||_infinity=O_p(epsilon^2+eta).                       (14)
For the last bound, Psi_n is sup-norm Lipschitz at most one and its
value at d_1 is O_p(epsilon^2+epsilon eta), by a coarser uniform row bound.

The sup norm in (11) is nonincreasing. PSD gives ||B_n||_2<=1/2, while
the nonlinear deviation is bounded by C||e||_infinity||e||_2. Starting
from e_2, the entire later trajectory is L^2-contractive with factor 3/4
on one common high-probability event. Hence
  sup_(j>=2)||e_j-B_n^(j-2)e_2||_2
       =O_p[(epsilon^2+eta)(epsilon^2+zeta)].                 (15)
All root-n-scaled errors in (14)--(15) tend to zero for d=1,2 under (H_d).
In particular sqrt(n)epsilon^(3/2)eta=epsilon^(3/2-d/2)->0,
sqrt(n)eta zeta=(n epsilon^(3d/2))^(-1/2)->0,
and sqrt(n)epsilon^2->0 follows from the bias condition for d<=2.
This is the refinement that the previous one-dimensional proof lacked.

4. A common residual proxy and the correct iteration-dependent errors

The first-update commutator is
  q_n=g_0-v g=delta_mu_y k(x,y)[1-v(x)a(y)].
Its diagonal is zero at x=X_i. Conditional variance and bias give
  ||q_n||_(L2n)=O_p(sqrt(epsilon)zeta),                      (16)
since int k^2 dist^2=O(epsilon^(1-d/2)); the conditional bias is O(epsilon/n).
As v/w-a=O(epsilon) and R-(1/2)I=B R, (12) becomes
  u=S_n* B Rg+O_(L2n,p)[(eta+sqrt(epsilon))zeta].
The exact intertwining B_n S_n*=S_n*(I-C_hat)/2, (10), and contraction
of both feature half-filters imply for all j>=2
  e_j=S_n* B^j Rg+E_(j,n),
  sqrt(n)||E_(j,n)||_2
       <=E_n+K_n epsilon^(1/2-d/4)2^(-j), all j>=2,          (17)
where E_n=o_p(1), K_n=O_p(1) are common to all j. Thus (17) is a
simultaneous per-j bound, retaining its iteration-dependent decay.
For d=2 the commutator term is NOT root-n negligible at a fixed j.

Expanding the exact score error 2 grad log(P_n exp(e_j)) and using (10)
gives the same form of simultaneous normalized error, uniformly in x:
  V_j(x)=2<theta_x,C B^j Rg>,
  A_n|s_j(x)-s_hat(x)-V_j(x)|
       <=o_p(1)+O_p(epsilon^(1/2-d/4)2^(-j)), j>=2.          (18)
The nonlinear terms are controlled starting at e_2, not e_1.
From (8), at each fixed x,
  E[A_n^2 |V_j(x)|^2]<=C4^(-j).
Thus for every deterministic J->infinity,
  sup_(j>=J) A_n|s_j(x)-s_hat(x)|->0 in probability.           (19)
The stochastic limit in (19) is fixed-query, not a spatial supremum law.

For the residual set
  Q_j=(1/4)<g,C B^(2j)g>, q_j=sqrt(Q_j).
Linearizing its centered square-root-mass field yields (simultaneously j>=2)
  sqrt(n)|r_j-q_j| <= E_n+K_n[
       epsilon^(1/2-d/4)2^(-j)+n^(-1/4)epsilon^(-d/2)2^(-j)], (20)
with E_n=o_p(1), K_n=O_p(1). The exact feature mean identity
  <1,S_n*B^j g>_n=<v_hat-v,B^j g>=O_p(zeta^2 2^(-j))
handles normalization without an independence assumption. The last term
in (20) is the square-root covariance comparison
  | ||S_n*h||_n-||S*h||_mu |<=||C_hat-C||_op^(1/2)||h||_H.
It cannot be omitted or called uniformly root-n negligible. For convenience
write D_n=(n epsilon^(3d/2))^(-1/4)->0, so that this term is
  D_n epsilon^(-d/8)2^(-j).

5. Quadratic concentration and the leading iteration count

For j>=1 let T_j=tr(C^2 B^(2j)). Then E Q_j=T_j/(4n). If a spectral
band of C is retained, the same assertion holds with its projection.
For centered iid Hilbert vectors Z, direct expansion gives
  Var||n^(-1)sum Z_i||^2
    =[E||Z||^4-(tr Sigma)^2]/n^3
                    +2(n-1)tr(Sigma^2)/n^3.
Apply this to Z=(1/2)C^(1/2)B^j[a(X)k_X-v]. Its covariance is
(1/4)C^2 B^(2j); its squared norm is at most Cepsilon^(-d/2)4^(-j).
Consequently
  Var Q_j/(E Q_j)^2
    <=C max_l(lambda_l^2 beta_l^(2j))/T_j
                    +Cepsilon^(-d/2)4^(-j)/(n T_j),          (21)
where beta_l=(1-lambda_l)/2. This retains the nonconstant diagonal term.

Put u_*=d/4-gamma>0. Fix a small delta>0 with delta<min(u_*,d/8)/2,
and let
  j_-=floor[(u_*-delta)log(1/epsilon)/log 2],
  j_+=ceil[(u_*+delta)log(1/epsilon)/log 2].
Choose a fixed beta_*<1/2 sufficiently close to 1/2 that, with
sigma=|log beta_*|/log 2>1 and u_-=u_*-delta,
  u_- sigma<u_*, u_-(sigma-1)<d/8.                          (22)
Such a fixed choice always exists. A fixed spectral band with beta>=beta_*
and lambda bounded away from zero contains c epsilon^(-d/2) eigenvalues.
For every 2<=j<=j_-,
  T_j>=c epsilon^(-d/2) beta_*^(2j).
By (21), its relative variance is bounded by
  Cepsilon^(d/2)(1/(4beta_*^2))^j+C(1/(4beta_*^2))^j/n.
Summing these geometric bounds through j_- gives o(1): (22) makes the
largest epsilon exponent positive, and n epsilon^(3d/2)->infinity controls
the diagonal bound. Hence with probability tending to one, simultaneously
  sqrt(n)q_j>=c epsilon^(-d/4) beta_*^j, 2<=j<=j_-.           (23)
At j_- this lower bound dominates epsilon^(-gamma), by (22).

The ratio of the measurement term in (20) to the bound (23) is at most
  C D_n epsilon^(d/8)(1/(2beta_*))^(j_-)
       <=C D_n epsilon^(d/8-u_-(sigma-1))->0.
Its other terms are negligible because d<=2 and the smallest lower bound
in (23) diverges. Thus r_j>t_n for every 2<=j<=j_- with probability
tending to one. In particular r_2/t_n->infinity.

For the upper bound use T_j<=4^(-j)tr(C^2)<=Cepsilon^(-d/2)4^(-j).
At j_+, E[n Q_(j_+)]/epsilon^(-2gamma)<=Cepsilon^(2delta)->0.
Markov gives q_(j_+)/t_n->0. The measurement term (20) divided by the
threshold is O_p[D_n epsilon^(u_*+delta+gamma-d/8)]
=O_p[D_n epsilon^(d/8+delta)]->0. Its other terms also vanish relative
to t_n. Therefore r_(j_+)<t_n with probability tending to one.
These two estimates bracket the first hit among j>=2 between j_- and j_+.
Also j_+<L_n eventually under (H_d).

6. Exclusion of early hits: the actual first hit starts at j=0

For a small relative-log oscillation, projectively center e to have mean
zero. Taylor expansion gives
  c_0||e||_2<=r(e)<=C_0||e||_2
with constants independent of n,epsilon: the linear field is (I+P_n)e/2,
whose norm is between ||e||_2/2 and ||e||_2. Markov centering and the local
L2 contraction of Psi_n show r(Psi_n(e))<=C_1 r(e), uniformly in this
small neighborhood. Adding a constant to e does not change r and is killed
by Psi_n, so centering introduces no assumption about mean preservation.
Equation (12) and (5) give ||e_1||_infinity=o_p(1). Thus r_2<=C_1 r_1
with probability tending to one. As r_2/t_n->infinity, j=1 cannot be a hit.

For fixed nonconstant f, r_0 converges in probability to
  sd_(f dx)(sqrt(f))/E_(f dx)sqrt(f)>0.
Indeed K_n1-Hf has sup error O_p(eta), Hf->f, and bounded continuous
population moments obey the law of large numbers. Since t_n->0, j=0 is
excluded. For the remaining constant density f=1, e_0=o_p(1) in sup norm
by (9), so the same local comparison r_1<=C_1 r_0 excludes j=0 as well.
This proves (1) for the unrestricted first hit. Equation (19) at tau_gamma
already proves the numerical-equivalence part of (3).

7. Sharper attained residual: any lower constant below one half

For every realized g, Q_j is a positive spectral sum. It is decreasing
and log-convex, so q_(j+1)/q_j is nondecreasing and at most 1/2 when q_j>0.
Let J=j_- for a small delta as above. To show this ratio tends to 1/2,
fix beta_L<1/2 and then beta_H,beta_H2 with
  beta_L<beta_H<beta_H2<1/2.
Low-beta trace is at most Cepsilon^(-d/2)beta_L^(2J). A high band with
beta in [beta_H,beta_H2] has trace at least
cepsilon^(-d/2)beta_H^(2J). For this projected band, the Hilbert vector
in the variance calculation has the sharper squared-norm envelope
  ||Z_H||^2<=C epsilon^(-d/2)beta_H2^(2J),
rather than the global 4^(-J) envelope. Both relative variance terms are
therefore bounded by
  C[epsilon^(d/2)+1/n](beta_H2/beta_H)^(2J).
Choose the band's positive width small enough that
2u_- log(beta_H2/beta_H)/log 2<d/2. Both terms then vanish under (H_d).
The low/high expectation ratio tends to zero. Markov and concentration
therefore show low modes contribute
o_p(Q_J). It follows that q_(J+1)/q_J>=beta_L-o_p(1).
As beta_L can approach 1/2 and the ratios are nondecreasing,
  sup_(j>=J)|q_(j+1)/q_j-1/2|->0 in probability.              (24)

For every j>=J, (20) divided by t_n is o_p(1): at j_-, the exponent of
its measurement term is u_-+gamma-d/8=d/8-delta>0. Thus at the actual
first hit, r_(tau-1)>t_n and (24) imply
  r_tau>=(1/2-o_p(1))t_n-o_p(t_n).
The definition gives r_tau<=t_n. This proves (2). It does not assert
that r_tau/t_n has a single limit; geometric iteration allows a rounding phase.
Equation (4) follows by subtracting two instances of (1) on the same sample.

8. True-score limit and directly justified inference

Equation (9) gives the exact empirical score expansion
  s_hat(x)-s_pop(x)=2<theta_x,Rg>+o_p(A_n^(-1)).
The normalized covariance tends by (6) to [4J_d/f(x)]I_d. The Perron
rank-one subtraction vanishes because Rv=v/2 and <theta_x,v>=0.
Each normalized iid influence summand is bounded by
C/[sqrt(n)epsilon^(d/4)]->0, by the feature and derivative-section norms.
The multivariate triangular-array Lindeberg CLT applies. The O(epsilon)
population bias vanishes at scale A_n by n epsilon^(d/2+3)->0. Combining
with (19) at the actual tau proves (3).

The same-sample periodic KDE f_tilde(x)=K_n1(x) converges in probability to
f(x)>0: its bias tends to zero and variance is O(1/(n epsilon^(d/2))).
Therefore the feasible pivot
  A_n sqrt(f_tilde(x))/(2 sqrt(J_d))
                  [s_tau(x)-grad log f(x)] =>N_d(0,I_d).     (25)
No independence from the stopped score is required by Slutsky. In particular
  A_n^2 f_tilde(x)||s_tau(x)-grad log f(x)||^2/(4J_d)
                    =>chi-square_d.                        (26)
This supplies pointwise confidence balls, directional intervals and a
fixed-query test of grad log f(x)=0 with asymptotic level alpha. It does not
locate a stationary point, classify a mode, justify a data-selected query,
or give simultaneous confidence bands. Studentization is standard inference,
an application of the theorem, not a separate originality claim.

9. Endpoint counterexample and precise meaning of sharpness

Take f=1, which belongs to the stated density class. Write
Lambda(xi)=exp(-2 pi^2|xi|^2). The initial residual has linear field
(1/2)H delta_mu. Fourier orthogonality and quadratic concentration give
  r_0/[n^(-1/2)epsilon^(-d/4)]
       ->c_d=(1/2)(4pi)^(-d/4)<1 in probability.              (27)
Its actual nonlinear/empirical-norm remainder is o_p of this scale by
the same eta embedding and covariance bounds; no fixed-epsilon limit is
used with a shrinking epsilon. Indeed its continuum proxy mean squared is
(4n)^(-1)sum_(k!=0)lambda_k^2, its relative variance is O(epsilon^(d/2)),
and its empirical and nonlinear norm errors are smaller by factors
O_p(sqrt(eta)) and O_p(eta), respectively.
For gamma>=d/4, (27) makes the unit threshold hit at j=0 with probability
tending to one. At that iteration s_0=2grad log(H mu_n) has limit
  A_n s_0(x)=>N_d(0,W_d),
  W_d=4 int_Rd (2pi xi)(2pi xi)^T Lambda(xi)^2 dxi.
The exact self-score covariance is
  V_d=4 int_Rd (2pi xi)(2pi xi)^T
                         [Lambda(xi)/(1+Lambda(xi))]^2 dxi.
Their difference is positive definite. Both claims follow from the actual
kernel triangular-array expansions; their normalized nonlinear remainders
are O_p(sqrt(n)eta zeta)=o_p(1) under (H_d). Thus the stopped score has a
different limiting distribution, proving (iv). This addresses the unit
threshold and the model class. Arbitrary endpoint multipliers, necessity
for each nonconstant f, or nonpower tolerance frontiers are not asserted.

10. Attribution, value and limitations

Kokot's publicly deposited UW dissertation (16 September 2026), Chapter3
Theorem3.6/Corollary3.3/Lemma3.29, precedes the fixed-epsilon RKHS scaling
derivative and Hilbert CLT; Section4.4.1/Lemma4.4 precedes the two-update
population approximation. These are credited ingredients. Mordant supplies
decreasing-regularization score asymptotics in a different Euclidean model;
Wormell--Reich and Cheng--Landa already analyze nonuniform normalization
and statistically adequate approximate scaling. General spectral statistical
stopping, covariance calculus, CLTs and Slutsky are established mathematics.

The candidate increment is the actual nonlinear sampled-trajectory theorem
with no delayed stopping restriction, a family of relaxed tolerances and its
endpoint obstruction, leading iteration counts, and preservation of true-score
inference in d=1,2. Originality must be assessed against primary literature;
the absence of a found predecessor cannot certify worldwide priority.

The geometry remains periodic and the density fixed and bounded below.
There is no claim for d>=3, boundaries, selected queries, global stochastic
processes, pathwise Langevin error, invariant measures, diffusion-map spectra,
runtime savings or finite-n tolerance calibration. Equation (4) concerns
iteration counts only. All prior experimental evidence remains preserved,
including finite cases not satisfying an asymptotic residual lower inequality.
