진지하게 수학 피드백이 필요해서 여쭤보고 싶습니다.

현재 제가 작용소대수를 기점으로 미분위상수학의 휘트니 매장정리를 일반화하는 방향으로 연구를 진행중이며, 현재 휘트니 매장정리에 대해서 아래와 같은 작업을 진행 중입니다.
현재까지 단위분할과 유사한 효과를 내는 장치를 작업 중이며, 적어도 작용소대수로 현재까지 아래의 증명과정을 따랐습니다. 그치만 제 눈에도 혹평될 여지가 너무 클 뿐더러, 너무 증명에서 도중에 헐거운 부분이 보였습니다.
그래서 논문을 추천 받을려고 교수님들 잡고 이렇게라도 여쭤봅니다.

\paragraph{Partition of unity on the spectral charts.}
Let \(X\) be a Hausdorff paracompact space equipped with an atlas
\[
\{(U_i,\rho_i)\}_{i=1}^N,
\qquad
\rho_i:U_i\longrightarrow E_i ,
\]
where each \(E_i\) is a Banach space and the coordinate changes
\[
\rho_j\circ\rho_i^{-1}
\]
are \(C^r\) on their domains. In this paper,
\[
\mathcal S_{\mathrm{SpH}}^r(X)
\]
denotes the class of functions \(h:X\to\mathbb R\) such that, for every
chart \((U_i,\rho_i)\),
\[
h\circ\rho_i^{-1}:\rho_i(U_i)\longrightarrow\mathbb R
\]
is \(C^r\). The same convention applies to maps with values in Banach
spaces.

Assume that the cover \(\{U_i\}_{i=1}^N\) is finite and admits smooth
shrinks
\[
V_i\Subset W_i\Subset U_i,
\qquad
X=\bigcup_{i=1}^N V_i .
\]
Choose functions
\[
\eta_i\in\mathcal S_{\mathrm{SpH}}^r(X),
\qquad
0\le \eta_i\le 1,
\]
such that
\[
\eta_i|_{V_i}=1,
\qquad
\operatorname{supp}\eta_i\subset W_i\subset U_i .
\]
Set
\[
S(x)=\sum_{i=1}^N\eta_i(x).
\]
Since the sets \(V_i\) cover \(X\), for every \(x\in X\) there exists
\(i\) with \(x\in V_i\), hence \(\eta_i(x)=1\). Therefore
\[
S(x)>0
\qquad
(x\in X).
\]
Define
\[
\phi_i(x)=\frac{\eta_i(x)}{S(x)} .
\]
Then
\[
\phi_i\in\mathcal S_{\mathrm{SpH}}^r(X),
\qquad
0\le\phi_i\le1,
\qquad
\operatorname{supp}\phi_i\subset U_i,
\]
and
\[
\sum_{i=1}^N\phi_i(x)
=
\frac{\sum_{i=1}^N\eta_i(x)}{S(x)}
=
1 .
\]
Thus
\[
\{\phi_i\}_{i=1}^N
\]
is a \(\mathcal S_{\mathrm{SpH}}^r\)-partition of unity subordinate to
\(\{U_i\}_{i=1}^N\).

For each chart, let \(P_i:E_i\to E_i\) be the Riesz projection associated
to the chosen spectral window,
\[
P_i
=
\frac{1}{2\pi i}
\int_{\partial\Omega_i}
(\lambda-T_i)^{-1}\,d\lambda .
\]
The corresponding local projected coordinate is
\[
P_i\rho_i:U_i\longrightarrow \operatorname{Im}P_i .
\]
Using the partition of unity, define
\[
f:X\longrightarrow
\bigoplus_{i=1}^N
\bigl(R_i\oplus \operatorname{Im}P_i\bigr)
\]
by
\[
f(x)
=
\bigoplus_{i=1}^N
\phi_i(x)
\bigl(R_i\oplus P_i\rho_i(x)\bigr).
\]
Equivalently, the \(i\)-th component is
\[
f_i(x)
=
\phi_i(x)
\bigl(R_i\oplus P_i\rho_i(x)\bigr).
\]
Because \(\operatorname{supp}\phi_i\subset U_i\), this component is zero
outside the chart where \(P_i\rho_i\) is defined. Hence \(f\) is globally
well-defined.

Moreover, on \(V_i\) one has \(\eta_i=1\), so
\[
\phi_i(x)=\frac{1}{S(x)}>0
\qquad
(x\in V_i).
\]
Therefore the local projected coordinate is recoverable from the global map:
\[
R_i\oplus P_i\rho_i(x)
=
\frac{f_i(x)}{\phi_i(x)}
\qquad
(x\in V_i).
\]
Thus the partition of unity only globalizes the local projected coordinates;
it does not add an independent geometric structure.

현재 이런 방식으로 구성 진행중이며, 이것에서 충분히 안 닫혀 보이는 곳이 아래와 같습니다.
\paragraph{Spectral partition of unity.}
Let \(X\) be a Hausdorff paracompact \(S_{\mathrm{SpH}}^r\)-space carrying a
separable pro-Banach spectral atlas. Let
\[
\mathcal U=\{U_\omega\}_{\omega=1}^{N}
\]
be the finite active Riesz-window cover supplied by the bounded spectral
packing hypothesis. On each \(U_\omega\), let
\[
\rho_\omega:U_\omega\longrightarrow E_\omega^{\mathrm{vis}}
\]
be the local visible readout, and let
\[
P_\omega
=
\frac{1}{2\pi i}
\int_{\partial\Omega_\omega}
(\lambda-T_\omega)^{-1}\,d\lambda
\]
be the corresponding Riesz projector. The local window coordinate is
\[
P_\omega\rho_\omega:U_\omega\longrightarrow \operatorname{Im}P_\omega .
\]

Choose an \(S_{\mathrm{SpH}}^r\)-shrink of the cover
\[
V_\omega\Subset W_\omega\Subset U_\omega,
\qquad
X=\bigcup_{\omega=1}^{N}V_\omega ,
\]
and choose \(S_{\mathrm{SpH}}^r\)-bump functions
\[
\eta_\omega:X\longrightarrow[0,1]
\]
such that
\[
\eta_\omega|_{V_\omega}=1,
\qquad
\operatorname{supp}\eta_\omega\subset W_\omega\subset U_\omega .
\]
Since the cover is finite, the sum
\[
S(x)=\sum_{\omega=1}^{N}\eta_\omega(x)
\]
is an \(S_{\mathrm{SpH}}^r\)-function. Moreover \(S(x)>0\) for every
\(x\in X\), because \(x\in V_{\omega_0}\) for some \(\omega_0\), hence
\(\eta_{\omega_0}(x)=1\). Define
\[
\phi_\omega(x)=\frac{\eta_\omega(x)}{S(x)} .
\]
Then
\[
\phi_\omega\in S_{\mathrm{SpH}}^r(X),
\qquad
0\le \phi_\omega\le 1,
\qquad
\operatorname{supp}\phi_\omega\subset U_\omega ,
\]
and
\[
\sum_{\omega=1}^{N}\phi_\omega(x)
=
\frac{\sum_{\omega=1}^{N}\eta_\omega(x)}{S(x)}
=
1 .
\]
Thus
\[
\{\phi_\omega\}_{\omega=1}^{N}
\]
is an \(S_{\mathrm{SpH}}^r\)-partition of unity subordinate to the active
Riesz-window cover.

Using this partition, define the global candidate coordinate map by
\[
f:X\longrightarrow E_{\mathrm{SpW}},
\]
where
\[
E_{\mathrm{SpW}}
=
\bigoplus_{\omega=1}^{N}
\bigl(R_\omega\oplus \operatorname{Im}P_\omega\bigr),
\]
and
\[
f(x)
=
\bigoplus_{\omega=1}^{N}
\phi_\omega(x)
\bigl(R_\omega\oplus P_\omega\rho_\omega(x)\bigr).
\]
Equivalently, the \(\omega\)-component of \(f\) is
\[
f_\omega(x)
=
\phi_\omega(x)
\bigl(R_\omega\oplus P_\omega\rho_\omega(x)\bigr).
\]
The support condition on \(\phi_\omega\) makes this component vanish outside
\(U_\omega\), so \(f\) is globally well-defined.

Finally, if \(x\in V_\omega\), then \(\eta_\omega(x)=1\), hence
\[
\phi_\omega(x)=\frac{1}{S(x)}>0 .
\]
Therefore the local Riesz-window coordinate is recoverable from the
\(\omega\)-component of \(f\):
\[
R_\omega\oplus P_\omega\rho_\omega(x)
=
\frac{f_\omega(x)}{\phi_\omega(x)} .
\]
Thus the partition does not introduce new geometric content. Its only role is
to globalize the local active-window coordinates while preserving their
recoverability on the region where the corresponding window is active.
여기서 \mathbb R \oplus P_{\Omega_i} 의 실수 부분이 붙은 이유는 \paragraph{Partition of unity and weighted projected coordinates.}
Let \(X\) be a Hausdorff paracompact space equipped with a finite atlas
\[
\{(U_i,\rho_i)\}_{i=1}^{N},
\qquad
\rho_i:U_i\longrightarrow E_i ,
\]
where each \(E_i\) is a Banach space. We write
\[
\mathcal S_{\mathrm{SpH}}^r(X)
\]
for the class of functions \(h:X\to\mathbb R\) such that
\[
h\circ\rho_i^{-1}:\rho_i(U_i)\longrightarrow\mathbb R
\]
is \(C^r\) for every chart. The same convention is used for Banach-valued
maps.

Choose shrinks
\[
V_i\Subset W_i\Subset U_i,
\qquad
X=\bigcup_{i=1}^{N}V_i ,
\]
and functions
\[
\eta_i\in\mathcal S_{\mathrm{SpH}}^r(X),
\qquad
0\le \eta_i\le 1,
\]
with
\[
\eta_i|_{V_i}=1,
\qquad
\operatorname{supp}\eta_i\subset W_i\subset U_i .
\]
Set
\[
S(x)=\sum_{j=1}^{N}\eta_j(x).
\]
Since the \(V_i\) cover \(X\),
\[
S(x)>0
\qquad (x\in X).
\]
Define
\[
\phi_i(x)=\frac{\eta_i(x)}{S(x)} .
\]
Then
\[
\phi_i\in\mathcal S_{\mathrm{SpH}}^r(X),
\qquad
0\le \phi_i\le 1,
\qquad
\operatorname{supp}\phi_i\subset U_i,
\]
and
\[
\sum_{i=1}^{N}\phi_i(x)=1 .
\]

For each \(i\), let
\[
P_{\Omega_i}
=
\frac{1}{2\pi i}
\int_{\partial\Omega_i}
(\lambda-T_i)^{-1}\,d\lambda
\]
be the Riesz projection associated to the chosen spectral window
\(\Omega_i\). The projected local coordinate is
\[
P_{\Omega_i}\rho_i:U_i\longrightarrow \operatorname{Im}P_{\Omega_i}.
\]

Define the local weighted projected coordinate
\[
g_i:X\longrightarrow \mathbb R\oplus\operatorname{Im}P_{\Omega_i}
\]
by
\[
g_i(x)
=
\begin{cases}
\bigl(\phi_i(x),\phi_i(x)P_{\Omega_i}\rho_i(x)\bigr),
&x\in U_i,\\[1mm]
(0,0),
&x\notin U_i .
\end{cases}
\]
This is well-defined because
\[
\operatorname{supp}\phi_i\subset U_i .
\]

The global coordinate map is
\[
f:X\longrightarrow
\bigoplus_{i=1}^{N}
\left(\mathbb R\oplus\operatorname{Im}P_{\Omega_i}\right),
\]
\[
f(x)
=
\bigoplus_{i=1}^{N}g_i(x)
=
\bigoplus_{i=1}^{N}
\bigl(\phi_i(x),\phi_i(x)P_{\Omega_i}\rho_i(x)\bigr).
\]

The scalar component records the cutoff weight. Hence, whenever
\[
\phi_i(x)>0,
\]
the projected local coordinate is recovered by
\[
P_{\Omega_i}\rho_i(x)
=
\frac{\phi_i(x)P_{\Omega_i}\rho_i(x)}{\phi_i(x)} .
\]
In particular, since \(\eta_i=1\) on \(V_i\),
\[
x\in V_i
\Longrightarrow
\phi_i(x)=\frac{1}{S(x)}>0,
\]
and therefore
\[
P_{\Omega_i}\rho_i(x)
=
\frac{\operatorname{pr}_2(g_i(x))}
{\operatorname{pr}_1(g_i(x))}
\qquad
(x\in V_i).
\]

Thus the summand
\[
\mathbb R\oplus\operatorname{Im}P_{\Omega_i}
\]
is used to store both the cutoff weight and the weighted projected coordinate:
\[
g_i(x)
=
\bigl(\phi_i(x),\phi_i(x)P_{\Omega_i}\rho_i(x)\bigr).
\]

이렇게 나왔습니다. 여기서 드는 의구심은 X^{[2]}=(X\times X)\setminus \Delta_X

여기서

\Delta_X=\{(p,p):p\in X\}

일때,

서로 다른 두 점 (p,q)에 대해 collision section 을 둔다면, (s_0(p,q)=f(p)-f(q)) 이것을 번들로 두고 따라서 bad set은

\operatorname{Bad}_\Delta
=
\{
A:\exists p\ne q,\; A K_0(p,q)=0
\}

좋은 class는

A_\Delta^\circ
=
A_{\mathrm{SpW}}\setminus \operatorname{Bad}_\Delta 를 둘 때, A\in\operatorname{Bad}_\Delta라는 걸 어떤 p\ne q에 대해

0\in \sigma_{\mathrm{SpW}}(A K_0(p,q))

로 읽을 수 있습니다. 그리고 A\in A_\Delta^\circ이면

Af(p)\ne Af(q)

라서 injective 라는 것을 끌어내는 관점입니다. 하지만 이 가정, 즉 A_\Delta^\circ\neq\varnothing 이 너무 무거워서 고민입니다. 린드스키와 Nest-AP 바나흐 공간, 그리고 위의 글루잉 조건으로 충분한지가 너무 의구심이 들고요, 기초 가정을 너무 좁히지 않고도 이 간극을 메울 수 있는 좋은 정리들을 알고 싶습니다. 부탁드립니다.