GEOMETRY SECTION DERIVATION NOTES
18 October 2026. Supporting derivation for the publication revision.

Scope: manuscript/geometry.tex only. Source baseline:
D:/LUMINARY/research/self_transport_nonuniform_revision2_20261018/revised_stopping_theorem.txt
and preserved predecessor joint_nonuniform_theorem.txt. No baseline was edited.

Population approximation is written out as a Banach C2 contraction. The
uniform inverse uses the periodized Schwartz multiplier F(xi)=1/(1+exp
(2pi^2|xi|^2)); the commutator is O(sqrt(epsilon)) on C2, and the forcing
is O(epsilon). C6 is more than needed for this local calculation but is
retained from the theorem. The PSD uniqueness identity is explicit.

The first derivation used the valid common-space Q=M_q H M_q comparison
in L2(dx). The section now uses the shorter independently checked direct
RKHS comparison suggested by the adversarial review. Define
C_H=int k_y tensor k_y dy on the SAME H. Since C=int q(y)^2 k_y tensor
k_y dy, its operator distance from C_H is O(epsilon) by quadratic forms.
C_H has eigenvalues lambda_k and norm one. Hence R C R differs by
O(epsilon) from R_H C_H R_H. The section is a(x)partial k_x plus
(partial a(x))k_x; the second term is sqrt(epsilon) smaller in H norm.
Normalized Gram errors therefore vanish, and Fourier orthogonality gives
the lambda_k^2/(1+lambda_k)^2 lattice sum with factor a(x)^2 ->1/f(x).
This directly verifies Jd/f with zero cross terms.

Fixed spectral band counts now use the same-H C versus C_H min-max
comparison, followed by a smaller interior heat annulus. No moving-band
or changing-frequency Weyl assertion is used.

Empirical branch: the exact reciprocal nonlinear remainder has an H norm
bounded by C epsilon^(-d/4)||h||_H^2 via one sup factor and the empirical
covariance operator; this is stronger than using two sup evaluations.
Its contraction size is eta=epsilon^(-d/4)*zeta. Resolvent replacement
costs eta*zeta. The branch is identified with the unique positive finite
scaling and its reciprocal-convolution extension, without signed-measure
implicit differentiation. Empirical covariance uses a Hilbert--Schmidt
second moment, not an infinite-dimensional Bernstein bound.

No new claim was added, and no mathematical objection to these supporting
lemmas was found during the derivation. This is a section-level derivation,
not an independent verification of the assembled manuscript.
