INDEPENDENT SECOND-EDITION PROOF REVIEW -- RESEARCH ONLY
Source: revised_stopping_theorem.txt
SHA256 22d4268df41449978b3eb08cae62460f1d9111048513745a70ef9074a57516ef
Reviewed the full source, including the new d=2 argument, actual unrestricted first hit, every fixed interior gamma, iteration-count asymptotic, improved overshoot, endpoint counterexample and feasible inference pivot.
No root proof, predecessor edition, manuscript, or earlier evidence was edited. This agent developed separate improvement candidates; this report audits the root's integrated proof, especially its independently supplied d=2 burn-in. Literature originality/value review is separate.

DISPOSITION: PASS FOR THE PRECISE STATED D=1,2 FIXED-DENSITY/FIXED-QUERY SCOPE.
No substantive mathematical gap or required correction found. Details and nonclaims below are part of this disposition. This is internal independent AI proof review, not external human peer review or publication approval.

1. WINDOWS AND THEOREM SCOPE
The two conditions n epsilon^(3d/2)->infinity and n epsilon^(d/2+3)->0 yield the displayed nonempty power windows only for d=1,2. They imply eta,zeta->0, A_n->infinity, and n epsilon^(d/2)->infinity. C6 positive fixed density has enough regularity and a fixed positive lower bound for all uniform deterministic constants used here. The same sample and kernel define exact and iterated quantities. The stopping rule genuinely checks j=0 and has a finite fallback budget on every sample.
Gamma is fixed strictly inside (0,d/4); no uniformity in endpoint-varying gamma is asserted. The first-hit iteration-rate and overshoot conclusions are probability statements, not actual residual monotonicity claims. The stochastic score and CLT conclusions are fixed-query; only deterministic/remainder estimates are spatially uniform.

2. POPULATION AND COMMON RKHS FACTS
The C2 contraction proof extends to d=2 without a dimension-specific obstruction. The periodized Schwartz multiplier bounds (I+H)^-1 on C2; C6 supplies more than the C4 heat forcing and C3 differentiated-commutator regularity needed. It yields a sqrt(f)=1+O_C2(epsilon), d1=log v-.5log Hf=O_C2(epsilon), and O(epsilon) population score bias. Positive scaling uniqueness follows from the negative square identity and PSD.
Feature maps S,S*, C,P, norms one, stationary v, and derivative-section scaling are consistent. The population Gram formula has the correct J_d/f(x) factor and vector identity covariance. The derivative-row error under the sqrt(f) conjugation is lower order relative to epsilon^(-d/4-1/2). Min-max compares nonzero population C eigenvalues through P's unitary conjugate to heat eigenvalues. Fixed positive-width bands with interior margin therefore contain order epsilon^-d/2 eigenvalues. Trace C=K0 int a^2 dmu is order at most epsilon^-d/2. No nonuniform Fourier eigenvectors or moving-shell Weyl asymptotic is assumed.

3. EMPIRICAL POTENTIAL BRANCH AND SAME-SAMPLE BOUNDS
Centered feature second moments give zeta, and HS covariance error gives eta. Cov(g)=(C-v tensor v)/n and <v,g>=0 hold exactly, including the zero mean constraint on empirical measures. The reciprocal potential equation near v has PSD derivative I+C0n with inverse norm<=1; one value evaluation and one covariance-controlled L2 evaluation bound its H remainder by C epsilon^-d/4 ||h||H^2. The local contraction radius O_p(zeta) is value-small because epsilon^-d/4 zeta=eta=o(1). Finite-scaling uniqueness identifies it with the exact empirical branch/extension.
The H error eta*zeta, value eta and weighted-C1 eta estimates follow with the displayed exponents. Relative scalar feature weights give C_hat-C=O_p(eta), and derivative sections give the theta comparison. A finite collection of these sample bounds and smallness conditions can be put on one common high-probability event. Subsequent all-j claims are deterministic consequences on that event, not unjustified unions of pointwise probability estimates.

4. NEW D2 NORMALIZED-ROW BURN-IN -- CHECKED IN DETAIL
The refined estimate (13) avoids the older root-n-nonnegligible epsilon*eta loss. Since ||d1||C2=O(epsilon), |d1(y)-d1(x)|<=Cepsilon dist(x,y). Write the empirical Markov row using normalized a_hat weights and compare to normalized population-a weights, so the row sum remains one. Relative weight changes O_p(eta) cost eta times the row mean absolute test difference, rather than eta*||d1||infinity without cancellation.
The heat-kernel bound dist*k_epsilon<=C sqrt(epsilon) k_(2epsilon) follows termwise from Gaussian images. Feature second moments and embedding at 2epsilon give uniformly bounded empirical heat mass; row denominators stay bounded below. Thus the empirical weighted mean distance is O_p(sqrt(epsilon)) uniformly x, proving the weight error O_p(epsilon^(3/2)*eta).
For the remaining deterministic-a empirical numerator, its own diagonal at Xi vanishes exactly. Conditional on Xi=x, its bias is minus the population numerator/n=O(epsilon^2/n), and its variance is <=C epsilon^2/n times int k^2 dist^2=O(epsilon^(1-d/2)), hence <=C epsilon^(3-d/2)/n. Averaging conditional second moments proves L2_n size O_p(epsilon^(3/2)*zeta). The denominator-error times population numerator is O_p(epsilon^2*eta). These are bounded by the retained RHS in (13). The calculation neither assumes independent sample queries nor discards a nonzero diagonal.
Taylor expansion of Psi at deterministic d1 has tilted row Q_d1, with ||Q_d1-P_n||L2op=O(epsilon): it is a product of bounded diagonal multipliers and P_n. Its u-remainder is C||u||infinity||u||2=O_p(eta*zeta). Psi(d1) differs from B_n d1 by O(epsilon^2), so (14) follows. The separate supremum estimate can use the coarser epsilon*eta uniform row error, since it is absorbed by eta in epsilon^2+eta; it is not used as the sharper L2 error. Thus there is no hidden reintroduction of the bad d2 mixed rate.

5. NONLINEAR CONTROL STARTS AT E2, NOT E1
The exact log map is projectively invariant and does not preserve mean-zero logs. Its uncentered sup ball is invariant by Markov extrema. From e2, its infinity radius is O_p(epsilon^2+eta), and its L2 size is O_p(epsilon^2+zeta). PSD bounds the linear half-step by 1/2; the nonlinear L2 remainder is radius times L2 norm. This yields a common q=3/4 contraction for every later j and the uniform error (15).
Every root-n term vanishes under H_d. In particular sqrt(n)epsilon^(3/2)eta=epsilon^(3/2-d/2)->0 for both dimensions; sqrt(n)eta*zeta=(n epsilon^(3d/2))^-1/2->0; and sqrt(n)epsilon^2->0 is guaranteed for d2 exactly by n epsilon^4->0 and is stronger for d1. The later product (epsilon^2+eta)(epsilon^2+zeta) has only smaller mixed terms. The intended actual d2 extension is therefore not an unproved extrapolation of the d1 coarse product bound.

6. INITIALIZER COMMUTATOR AND ITERATION-DEPENDENT ERROR
The kernel k(x,y)[1-v(x)a(y)] vanishes at y=x. Its conditional variance is Cepsilon^(1-d/2)/n and conditional bias O(epsilon/n), giving sqrt(epsilon)*zeta in sample L2. Replacing v/w by a and a by a_hat gives the stated initializer feature direction S_n* B Rg plus eta*zeta and commutator errors.
The commutator is propagated by actual B_n powers of norm at most 1/2. Therefore its root-n size is epsilon^(1/2-d/4)*2^-j: negligible already at fixed j in d1 but only O_p(2^-j) in d2. Formula (17) explicitly interprets its bound with common random constants for every j, avoiding the invalid notation of a supremum with an unresolved free index. Intertwining and half-step telescoping add only a uniform eta*zeta error, with no accumulating j factor.
The source correctly retains the d2 commutator through the point-score and residual arguments. It does not call it uniformly root-n negligible from j=2.

7. POINT-SCORE AND PROJECTIVE RESIDUAL PROXIES
The exact score error is 2 grad log(P_n exp(e_j)). Derivative-row L2 norms are controlled by theta_hat and ||S_n||=1. Nonlinear gradient terms use e2's small radius and L2 size, so their normalized error tends to zero in d1,d2. Operator/section replacements cost sqrt(n)eta*zeta. The proxy is V_j=2<theta,C B^j Rg>, with correct signs, extra heat/covariance factor and score factor 2. Its covariance gives E[A^2|V_j(x)|^2]<=C4^-j. Summing over later iterations works at each fixed x and permits substitution of a sample-dependent stopping index.
The square-root mass linear field is (I+P_n)e_j/2. Feature intertwining cancels R with I+C; empirical C_hat-C costs eta*zeta. Its sample mean is exactly <v_hat-v,B^j g>, since Bv=0. The size zeta^2*2^-j is root-n negligible because sqrt(n)zeta^2=eta->0. Centering and self-normalizing therefore fit the error bound, including any retained d2 commutator term.
The empirical/population norm comparison uses sqrt(||C_hat-C||op)||B^j g||H, giving n^-1/4 epsilon^-d/2*2^-j after root-n scaling. This is D_n epsilon^-d/8*2^-j as written. The geometric factor and threshold-relative use are essential; no unjustified uniform root-n norm approximation remains.

8. VARIABLE-DIAGONAL HILBERT VARIANCE AND RATE BRACKETS
For j>=1, B annihilates stationary v and Q_j is a centered Hilbert sample-mean squared norm with covariance C^2B^(2j)/4. The variance formula retains E||Z||^4-(tr Sigma)^2 over n^3 and the equal/reversed off-diagonal term. Diagonal/off-diagonal cross covariances vanish by conditioning on the remaining independent centered variable, not by assuming zero third moments. Deterministic spectral projections preserve the formula. The fourth-moment envelope yields exactly the second term in (21).
For each fixed bracket slack delta, a fixed beta band sufficiently close to 1/2 satisfies (22), with lambda bounded below by a positive band-dependent constant. Chebyshev summed geometrically over all j=2..j_- gives a simultaneous lower-proxy event; a single pointwise lower bound would not have been enough for early measurement errors. The exponent margins in (22) make both diagonal and off-diagonal union bounds vanish. Comparing the common norm error to the band lower scale proves all early actual residuals exceed t_n.
At j_+, Markov using the trace upper bound gives q_j/t_n=o_p(1), and the actual norm comparison is smaller still. The brackets imply the asserted logarithmic iteration-rate limit. Their upper index is well below the budget under H_d. Subtracting two finite-pair convergence statements on the same data gives the iteration-count difference limit; it does not establish wall-clock savings.

9. GENUINE FIRST HIT FROM J=0
For projectively centered small logs, the linear residual field (I+P_n)e/2 has norm between ||e||2/2 and ||e||2, and Taylor remainders are relative O(||e||infinity). Centering Psi adds no unsupported mean-preservation assumption. Thus r(Psi(e))<=C r(e) with a uniform finite C in the local ball. Since r2/t_n diverges, the established small e1 excludes j=1. For fixed nonconstant f, r0 tends to the positive coefficient of variation of sqrt(f) under f dx; heat bias and uniform empirical convolution error justify this limit. t_n->0, so j=0 is excluded. For f=1, e0 is already small by the branch estimate and the same local residual comparison excludes j=0. These arguments do not assume global residual monotonicity or a uniform density-class lower variance bound.

10. ONE-HALF NORM OVERSHOOT
Q_j is a positive spectral sum, so q_j=sqrt(Q_j) has nondecreasing successive ratios bounded above by 1/2. A fixed low beta band and a narrowly separated higher band make the low contribution negligible at J=j_-. The chosen width makes both off-diagonal and diagonal concentration exponents positive. For every beta_L<1/2, q_(J+1)/q_J>=beta_L-o_p(1); monotonicity propagates this simultaneously to every later j. This proves the sup norm-ratio limit in (24). The squared Q ratio approaches 1/4, not 1/2; the source uses q correctly.
For all j>=J, the actual norm error is o_p(t_n), since the measurement exponent is d/8-delta>0. At the first hit the prior residual exceeded t_n, yielding the lower bound for every fixed c<1/2. The source appropriately avoids asserting P(r_tau>=t_n/2)->1 or a unique limit of r_tau/t_n; crossing phase can remain.

11. TRUE-SCORE LAW AND FEASIBLE INFERENCE
Exact-score expansion from the potential branch gives 2<theta,Rg> plus normalized negligible errors. Covariance subtracts the stationary rank-one term, which vanishes because <theta,v>=grad(a v)=0. The population Gram limit yields the stated scalar identity matrix. Influence summands are bounded after normalization by C/(sqrt(n)epsilon^(d/4))->0. Multivariate triangular-array Lindeberg applies. Bias to grad log f is O(epsilon), killed by n epsilon^(d/2+3)->0. The stopped score shares that same fixed-query law via the simultaneous later-iteration score tail.
The periodic KDE K_n1(x) is pointwise consistent under H_d, positive with high probability (indeed positive as an exact heat sum), and estimates f(x). Same-sample Slutsky gives the studentized N_d(0,I) pivot and chi-square norm statistic. These support pointwise balls, predetermined directional intervals, and the stated fixed-query score-zero test. They do not locate an unknown mode or justify selected-query/global confidence claims; the source states these limits correctly.

12. ENDPOINT SHARPNESS
For uniform f and j=0, the residual field is H delta/2. Its continuum quadratic concentration yields r0/[n^-1/2 epsilon^-d/4]->c_d=.5(4pi)^(-d/4)<1. The nonlinear residual error is smaller by eta; the empirical-norm change is smaller by sqrt(eta), so the actual initial residual has the same limit under H_d. Thus the UNIT-coefficient threshold hits at j=0 with high probability for every gamma>=d/4, including equality. No random-index endpoint argument is needed.
The actual initial direct-kernel score has covariance W_d=4 int qq^T Lambda^2, whereas the exact self-score has V_d=4 int qq^T[Lambda/(1+Lambda)]^2. Their difference is positive definite. Actual triangular-array expansions, not fixed-epsilon proxy CLTs, establish these different limits under H_d. Therefore the upper endpoint class counterexample is valid and stronger than merely showing a nonzero numerical-error covariance. The source correctly does not claim failure for every nonconstant f or arbitrary endpoint multipliers.

FINAL ASSESSMENT
No necessary mathematical correction identified in this source. The revision closes the d2 mixed-rate issue with a normalized-row estimate and a Taylor expansion at the deterministic first-step function, and integrates the threshold/iteration improvements without losing common-event or normalization control. The claimed mathematical scope is considerably more precise than generic solver convergence or arbitrary-density inference. Validity of this proof does not establish originality, optimality, empirical runtime benefits, or external acceptance; those remain separate.
