NONUNIFORM STOPPING THEOREM -- SECOND RESEARCH PROOF CHECKPOINT
18 October 2026. Completed analysis/improvement and internal verification.

Authoritative source: revised_stopping_theorem.txt
SHA256 7d2ed0072718a9fbb84581ed4c67fe75d4781468592bafa3198e163bbcd12edd

Established scope: fixed positive C6 density on T^d, d=1 or 2; iid sample;
exact normalized periodic Gaussian; fixed-query score inference. Conditions:
epsilon->0, n epsilon^(3d/2)->infinity, n epsilon^(d/2+3)->0.
Power bandwidths epsilon=n^-kappa: d1 (2/7,2/3), d2 (1/4,1/3).

The genuine first residual hit from b0=1 works for every fixed 0<gamma<d/4
at t=n^-1/2 epsilon^-gamma. It preserves the fully converged estimator's
true-score Gaussian law. Its attained residual is between c*t and t with
high probability for every fixed 0<c<1/2. Its leading update count is
[(d/4-gamma)/log2] log(1/epsilon). The unit threshold fails to preserve that
limit uniformly over this class at gamma>=d/4, already at f=1, because it
stops at zero and returns a different covariance. KDE studentization gives
direct pointwise confidence regions and stationarity tests.

The main improvement is a d2-compatible normalized-row cancellation plus
common iteration-dependent stochastic error control. The result is a
successor in the same theorem family, not several separate new discoveries.

Verification: earlier full-source AI proof review preserved; fresh full
final-source cross-review passed. A high-band concentration justification
was sharpened and separately rechecked. Source-bound code/diff and final
repair reviews passed. Review files record their actual source versions.
This is internal AI verification, not external human refereeing or formal
proof-assistant certification. No blocking proof gap was reported after repair.

Both supplementary probes reproduced exactly as parsed JSON in an isolated
output tree. Independent radial/lattice calculations checked d1,d2 constants.
These are floating diagnostics, not certified numerical premises of the proof.
Earlier finite cases, including failures of an asymptotic lower ratio at the
tested n, are retained without reinterpretation as successful validation.

Originality/value: no checked primary predecessor covers the combined actual
nonlinear first-hit result. Closest sources and overlap are in
FINAL_ORIGINALITY_AND_VALUE.txt and LITERATURE_AND_APPLICATIONS.txt. Kokot's
fixed-epsilon RKHS derivative/CLT and two-update population matching, existing
score asymptotics and statistically adequate approximate normalization are
credited. Priority remains provisional. This is a worthwhile focused specialist
contribution and a stronger theorem-family result, not a broad breakthrough.

Applications and frontier: APPLICATIONS_AND_RESEARCH_FRONTIER.txt. The direct
benefit is inference-preserving stopping for periodic density scores. Finite-n
calibration, runtime gains, selected queries, confidence bands, diffusion paths,
boundaries, vanishing densities and d>=3 remain unproved. At d=3 the present
bandwidth conditions are incompatible. See CLAIM_LEDGER.json for exact limits.

Prior editions and evidence are untouched. PRESERVATION.json records the final
read-only manifest checks; MANIFEST.json inventories this separate research
edition. REVISION_RECORD.json binds changed sources, reviews and dispositions.

Recommended next decision: retain this as the strongest research edition and
prepare one coherent specialist manuscript if requested. The most direct next
research gain would be finite-sample tolerance and inference calibration. No
new manuscript, publication, submission or cross-chat handoff was performed.
