\subsection{Category \texorpdfstring{$\mathcal O$}{O} and the spectral filtration}\label{fin:kzsection}

Let \(V_\Gamma\) be the original cell lattice of a finite left cell
\(\Gamma\), with its canonical basis. For a root of unity \(\zeta\),
choose a positive real \(\lambda\) with
\(\exp(2\pi\mathrm i\lambda)=\zeta\), and use the complete local ring
\(R=\mathbb C[[u]]\) with \(v=\exp(2\pi\mathrm i(\lambda+u))\).
This identifies \(R\) with the completion at \(v=\zeta\), since the
linear term of \(v-\zeta\) is nonzero. Use the corresponding
Cherednik category \(\mathcal O_R\), with standard module
\(\Delta_R(U)\) for an ordinary group label \(U\). This category has
the finite split highest-weight realization of
\cite[Theorem~5.2]{RouquierSchur}. Its KZ functor is exact,
represented by the projective \(P_{\mathrm{KZ}}\), and sends that
projective to the regular Hecke module. These statements follow
from \cite[Theorems~5.13, 5.15 and 5.16]{GGOR} and
\cite[Proposition~4.36 and \S5.2]{RouquierSchur}, with the KZ
normalization in \cite[\S15]{RouquierErrata}. Set
\[
 F_R(M)=\bigl(\mathrm{KZ}_R(M)\bigr)^\dagger,
\]
where the superscript denotes twisting the Hecke action by \(\dagger\).
For type \(B\) the Dunkl--Griffeth parameters on this line are
\[
 c_0=-2a(\lambda+u),\qquad
 d_0-d_1=-4b(\lambda+u),\qquad d_0+d_1=0.
\]
In the monodromy normalization of \cite[(1.18)]{DunklGriffeth},
the generators \(T_i\) attached to \(s_i\) have roots
\(v^b,-v^{-b}\) for \(i=0\) and \(v^{2a},-1\) for \(i>0\).
Thus \(N_{s_0}=T_0\) and \(N_{s_i}=v^{-a}T_i\) for \(i>0\).

For a finite multiset \(Z\) with decreasing list \(z_1,\ldots,z_m\),
write \(n(Z)=\sum_{i=1}^{m}(i-1)z_i\). For a partition \(U\),
\(n(U)\) applies this definition to its list of row lengths.

\begin{lemma}\label{fin:strict}
In the residue category at each positive logarithmic lift, write
\(\Delta(U)\) for the residue standard and \(L(U)\) for its simple
head. If \(L(U')\) occurs in \(\Delta(U)\) and \(U'\ne U\), then
\(a_{\mathrm{Sch}}(U')<a_{\mathrm{Sch}}(U)\).
The generic fibre of \(\mathcal O_R\) is split semisimple.
The corresponding assertions hold in type \(A\) and for products.
\end{lemma}

\begin{proof}
For a box \((i,j)\) in colour \(k\in\{0,1\}\), its charged content
is \(j-i+b/a\) when \(k=0\) and \(j-i\) when \(k=1\).
The triangular generalized weights in
\cite[Theorem~1.3(a) and (4.3)--(4.4)]{DunklGriffeth}
match the boxes of \(U\) with the boxes of \(U'\), if \(L(U')\)
occurs. For a matched pair at positions \((i,j)\), \((i',j')\)
with colours \(k,k'\) and charged contents \(\eta,\eta'\), the
necessary equations are
\[
 m=d_k-d_{k'}+2c_0\bigl((j-i)-(j'-i')\bigr),\qquad
 m\in\mathbb Z_{\geq0},\qquad m\equiv k-k'\pmod2 .
\]
The comparison uses the lowest joint weight of \(L(U')\)
and the triangular diagonal weights in \(\Delta(U)\).
Substitution of the residue parameters gives
\(m=4a\lambda(\eta'-\eta)\). Thus the contents weakly increase,
and equality preserves the colour.

Cancel common boxes by following the matching through any common
vertex, using the row, column and colour to identify that vertex.
This leaves a matching between the two set differences. On every
remaining path the contents strictly increase. Otherwise every
edge would preserve both colour and content, whereas boxes on one
diagonal of a given partition form an initial interval of that
diagonal. Both set differences cannot contain boxes on the same
coloured diagonal. Sort the two remaining lists of contents in
increasing order. The sorted contents satisfy strict entrywise
inequalities, since the first \(i\) target entries have \(i\)
distinct strictly smaller sources.

Starting with the common diagram, add the unmatched boxes in this
sorted order in both diagrams. Within a row they are added from
left to right. Each matched pair raises two symbol entries,
one in each diagram, by one. The old entry is the charged
content of the added box plus \(N-1\), where \(N\) is the
common padding length in the symbol of Lemma~\ref{fin:orientation}.
The target old entry is therefore larger. The intermediate
row lengths give multisets of entries, which suffice for
the comparison. For a real threshold \(t\), raising
\(x\) to \(x+1\) increases the sum of the positive parts
\(\sum_z\max(z-t,0)\) by
\(\max(x+1-t,0)-\max(x-t,0)\), a nondecreasing function of \(x\).
The increase is strictly larger for some \(t\) when the old
entries differ. The target symbol therefore strictly majorizes
the source symbol when the diagrams differ. Indeed the two
multisets have the same sum, and comparison of the positive-part
sums is the majorization criterion. For a decreasing list
\(z_1,\ldots,z_m\),
\[
 n(Z)=(m-1)\sum_i z_i-\sum_{j=1}^{m-1}\sum_{i=1}^j z_i .
\]
Strict majorization strictly decreases this statistic. The
Schur symbol formula from Lemma~\ref{fin:orientation} gives the
strict inequality.

In the fraction field the same necessary equations use
\(\lambda+u\) in place of \(\lambda\). Since \(m\) is constant,
they force equality of charged contents and then equality
of colours. The two partitions in each colour are determined
by their diagonal lengths, so \(U'=U\).
Every standard is then simple and BGG reciprocity makes every
projective cover simple. This proves generic semisimplicity.
For the symmetric group the weight comparison is the single-colour
case of the same formula, with \(2a\lambda\) instead of
\(4a\lambda\) and ordinary contents. Its Schur statistic is
\(a\,n(U)\) for the ordinary partition \(U\). The same argument
applies. Standards and spectral degrees factor on products.
\end{proof}

By BGG reciprocity and equality of the standard and costandard
classes \cite[Theorem~2.19 and Proposition~3.3]{GGOR},
the projective kernel over a standard has a filtration by
standards with strictly larger Schur valuation. Thus the same
standards give a highest-weight structure ordered by that
valuation, with unequal labels of equal valuation incomparable.
The residue-field criterion
\cite[Theorem~4.15]{RouquierSchur} gives the same order over \(R\).
In particular,
\[
 \operatorname{Ext}^1_{\mathcal O_R}
    (\Delta_R(U),\Delta_R(U'))=0
 \quad\text{if}\quad
 a_{\mathrm{Sch}}(U)\geq a_{\mathrm{Sch}}(U') .
\]
This includes self extensions. Projective standard filtrations
can be ordered with the greater valuations at the bottom, and
sections of equal valuation are direct sums. These are the
standard filtration properties in
\cite[Proposition~4.13]{RouquierSchur}.

We record two compatibility facts needed for the use of KZ.
Its generic labels are the ordinary Tits labels on every such
logarithmic line. To see this, use the holomorphic
standard-module connection on the complex logarithmic line
\cite[\S3.4]{ShanKZ}. On the complement of the discrete nonsemisimplicity set the
primitive central idempotents of the split seminormal representations
vary holomorphically
\cite[(2.4), Corollary~3.12 and Theorem~3.14]{MathasMatrix}.
Each acts on the holomorphic KZ standard by an idempotent of
locally constant rank. At logarithm zero the KZ standard is
the ordinary irreducible \(U\), so its idempotent acts with full
rank and all others with rank zero. The complement is connected.
The same ranks and the seminormal dimension formula therefore
identify the KZ standard with the irreducible of label \(U\)
throughout the complement.
The same comparison over the completion follows by specializing
the holomorphic family and comparing its character germs.

Also, relative parabolic induction and restriction are exact,
biadjoint and compatible with KZ over \(R\)
\cite[\S2.8]{LosevDerived}. They preserve filtrations by standards.
Here is the integral argument for the filtration claim.
In the finite \(R\)-algebra realizations, restriction is tensor
with a finite bimodule, projective on the input side by exactness
and on the output side by preservation of projectives. Its
adjoint is Hom from that bimodule, so both functors preserve
\(R\)-freeness and commute with reduction of the defining bimodule.
To identify the reduced restriction, use the completion description
in \cite[\S2.8]{LosevDerived}. Completion at the parabolic point
commutes with reduction modulo \(u\) on the finite polynomial
modules. The centralizer isomorphism specializes by its formula
\cite[Theorem~3.2]{BezrukavnikovEtingof}, as its denominators are units
in the point completion. The restricted object is characterized by
its completion under the equivalence with the completed category.
Its reduction is therefore the field restriction, and biadjunction
identifies the reduced induction. At the residue field induction and
restriction preserve standard filtrations, with ordinary
branching multiplicities \cite[Lemma~2.24]{LosevSurvey}.
To lift a standard filtration of a free \(R\)-module, take a
maximal standard, which is a projective summand \(P\) of the
finite algebra, with \(\operatorname{End}(P)=R\). Its evaluation
map \(P\otimes_R\operatorname{Hom}(P,M)\to M\) reduces to the
inclusion of the maximal-standard trace: the copies of that
projective standard can be placed at the bottom of the residue
filtration. The projective copies move to the bottom because any extension
with the maximal standard as quotient splits. Hom from this
projective to each remaining standard is zero, since the simple
head of the projective does not occur in any remaining standard.
Thus evaluation identifies its source modulo the uniformizer
with the sum of the bottom copies. The source is free, and a
unit minor gives an injective map with free cokernel. This cokernel is a module for the split quasi-hereditary
quotient by the corresponding heredity ideal. Induction on
the labels lifts the whole filtration. Exactness then gives
preservation for all standard-filtered modules.

\begin{lemma}\label{fin:recognition}
Under the proper-rank induction assumptions and the off-wall
actual-cell statement of Lemma~\ref{fin:domino}, every original
left cell of character \(E\) satisfies
\[
 V_\Gamma\otimes R \simeq
 F_R\Delta_R(E\otimes\operatorname{sgn}).
\]
The analogous assertion holds in equal-parameter finite type \(A\).
\end{lemma}

\begin{proof}
First suppose \(E\) is the unique constituent of minimum spectral
degree in the proper induction of Lemma~\ref{fin:orientation}.
By Lemma~\ref{fin:based} its original lattice is a quotient
of the corresponding proper canonical induction by an integral
surjection. The proper lattice is already recognized by induction.
In category \(\mathcal O_R\), the induced standard has exactly
one standard factor labelled \(E\otimes\operatorname{sgn}\) at
the smallest valuation. The Ext vanishing above moves it to a
quotient: an adjacent factor of larger valuation above it
splits off from it, so the two positions can be swapped.
Applying \(F_R\) gives a surjection from the same induced
Hecke lattice to \(F_R\Delta_R(E\otimes\operatorname{sgn})\).
Both quotients are \(R\)-free. Over the fraction field both
kernels are the sum of all isotypic summands except the single
copy of \(E\). Saturated submodules of a free lattice are
intersections with their generic spans. Thus the two kernels
coincide.

For a label for which this induction is obtained only after
the sign twist, use exact longest-element duality on the
original lattices and homological duality on the standards.
More precisely, write \({}^\vee\) for the linear contragredient
with respect to \(N_w\mapsto N_{w^{-1}}\), with the dual basis.
The lattice of \(\Gamma w_0\) is
\((V_\Gamma^\vee)^\dagger\) up to signs, by
\cite[Proposition~11.6 and Corollary~11.7]{Lusztig}.
This formula uses right translation by \(w_0\) and applies
also to type \(A\). It follows from the trace-dual identity
used in Proposition~\ref{fin:close}. If one uses left translation,
the additional diagram automorphism is conjugation by the
integral unit \(N_{w_0}\).
Homological duality takes \(\Delta_R(U)\) to
\(\Delta_R(U^\vee\otimes\operatorname{sgn})\). The target
has the same parameter, since the reflections have order two
\cite[Remark~4.5 and Proposition~4.10]{GGOR}. Over \(R\)
the standard computation uses the polynomial Koszul resolution
over \(R\) and the PBW decomposition, and after localization
the KZ identity is duality of the free flat connection
\cite[\S5.4.2]{GGOR}. For a monodromy generator the
anti-map takes its inverse times the product of its two
eigenvalues. In normalized generators that product is \(-1\),
so the anti-map is \(N_s\mapsto -N_s^{-1}\). The ordinary
labels in these finite Weyl groups
are self-dual. Applying these dualities to the already recognized
paired cell proves the assertion for \(E\).

For type \(A\) induct on the size \(m\) of the symmetric group,
with the two one-dimensional initial cases. For a partition
\(\mu\) of \(m\), let \(W_J\) be the row Young parabolic and
\(w_J\) its longest element. The left ideal generated by
\(C_{w_J}\) is the coordinate span of all \(C_{xw_J}\) for
minimal coset representatives \(x\). In fact the simultaneous
right positive-eigenvalue equations select exactly the
coordinates having all the descents of \(J\), and the
\(N_xC_{w_J}\) are unitriangular in those coordinates.
The left cell of \(w_J\) affords the ordinary Specht character
\(\mu\), by the usual Robinson--Schensted description. Its
coordinate quotient in this ideal is a quotient of induction
from the trivial positive representation of \(W_J\). Type
\(A\) based equivalence supplies the quotient for any cell
of the same label. Young's rule makes \(\mu\) the unique
dominance-minimal constituent, with multiplicity one. In
positive coordinates its spectral degree is
\(a\sum_i\binom{\mu_i}{2}\), strictly increasing under strict
dominance. Hence the preceding direct argument applies when
\(\mu\ne(m)\) and \(W_J\) is proper. The sign label
\((1^m)\) can also be obtained from the empty parabolic as
the unique minimum of the regular character. Longest-element
duality gives \((m)\). This completes the type \(A\)
initialization and the required product statements.
\end{proof}

\begin{lemma}\label{fin:e1}
Off the walls, the recognition of Lemma~\ref{fin:recognition}
and the irreducibility of the actual left cells imply that for
each \(d\) the spectral ideal with generic labels \(A(E)\geq d\)
is spanned by a subset of the original \(C\)-basis.
\end{lemma}

\begin{proof}
Work over the completed ring \(R\). For a simple reflection
\(s\), let \(S_s^+\) be its one-dimensional positive Hecke
module, on which \(N_s\) acts by \(v^{L(s)}\). Put
\[
 T_R=H_R\oplus\bigoplus_s H_R\otimes_{H_{R,s}}S_s^+ .
\]
The rank-one induced summand is \(H_RC_s\), by the
standard coset basis. The functor
\(G_T=\operatorname{Hom}_{H_R}(T_R,-)\) is exact on every
coordinate left-ideal filtration and its subquotients.
Indeed the positive rank-one eigenvectors in a coordinate
subquotient have only left-descent coordinates for \(s\).
The ascent coordinate of \((N_s-v^{L(s)})x\) is its
coefficient in \(x\) times
\(-v^{L(s)}-v^{-L(s)}\), a nonzero element of the domain
\(R\). This follows from the generator formula, whose
off-diagonal outputs are left descents. Each surviving
descent coordinate lifts as a positive eigenvector, proving
the required exactness.

The highest spectral degree standard of rank one has
\(F_R\)-image \(S_s^+\) and is projective by the order in
Lemma~\ref{fin:strict}. Denote it by \(\Delta_s^+\). The
projective
\[
 Q_R=P_{\mathrm{KZ}}\oplus
       \bigoplus_s\operatorname{Ind}_s\Delta_s^+
\]
maps to \(T_R\) under \(F_R\), with the same endomorphism
algebra by full faithfulness on projectives.
Its additive closure is generated by the projective covers
of residue simples whose support has codimension at most one,
lifted to \(R\).
For rank one \(P_{\mathrm{KZ},s}\) together with
\(\Delta_s^+\) generates the projectives. If the two Hecke
roots are different then KZ already detects both simple
heads. If they coincide, exactly the high standard has
the finite-support simple head and the other head survives
KZ. This follows directly from the two rank-one standard
modules, or the rank-one calculation in
\cite[proof of Lemma~4.4]{LosevDerived}.
In higher rank, restriction to the rank-one stabilizer
of a generic point on a reflection hyperplane detects
the simple heads with support of codimension one, by
parabolic support detection in
\cite[\S2.8 and proof of Lemma~4.4]{LosevDerived}.
Induction and adjunction give the corresponding
projective summands. Inducing \(P_{\mathrm{KZ},s}\)
gives the regular-image projective already represented by
\(P_{\mathrm{KZ}}\). This verifies the assertion over the
residue field, and projective summands lift over \(R\).

Let \(\widetilde F_R=\operatorname{Hom}_{\mathcal O_R}(Q_R,-)\). The enlarged functor is still fully faithful on projectives.
A homomorphism between its images of two projectives restricts,
via the idempotent for \(P_{\mathrm{KZ}}\), to a homomorphism
between their KZ images. Lift it by KZ full faithfulness.
After subtracting that lift, the homomorphism vanishes after
precomposition with every map \(P_{\mathrm{KZ}}\to Q_R\).
KZ faithfulness on projectives then makes it zero.
Its residue quotient is fully faithful on standard-filtered
objects by \cite[Lemma~4.3]{LosevDerived}, which uses precisely
the projective covers for support of codimension at most one.
Indeed a standard-filtered object is free over the polynomial
ring, since standards are free over the polynomial ring and extensions
split over that ring. Objects in the kernel have support of
codimension at least two. They admit no map to an object free over the polynomial ring. An extension of one by an object free over the polynomial ring splits:
off the support the extension is identified with that free
object, and taking sections there extends it back by Hartogs
across codimension at least two. The restriction and extension
map respects the Cherednik action, which sheafifies over
the quotient of the reflection space by the finite group.
These Hom and Ext vanishings identify the morphisms between
standard-filtered objects with their morphisms in the Serre
quotient.
Since \(R\) is a discrete valuation ring and the generic
category is split semisimple, \cite[Propositions~4.41--4.42]{RouquierSchur}
then identify \(\widetilde F_R\) as an exact equivalence
from standard-filtered objects onto modules filtered by its
standard images.

For a standard-filtered object \(M\) the natural map
\(\widetilde F_R M\to G_T(F_R M)\) is an isomorphism.
On the \(P_{\mathrm{KZ}}\)-summand this is the defining
comparison. On the rank-one summands use relative adjunction
and compatibility with restriction. In rank one the
comparison
\(\operatorname{Hom}(\Delta_s^+,M)\to
 \operatorname{Hom}(S_s^+,F_R M)\)
is an isomorphism on both standards, since their generic
labels are different, the Hom modules are torsion-free,
and the identity maps to the identity. Induct on the
standard filtration. A last factor \(\Delta_s^+\) splits
by projectivity. For the other last factor both Hom spaces
into that factor vanish, so both functors have the same
value as on the preceding subobject. This proves the
natural comparison, including compatibility with the
endomorphism actions.

Choose a left-cell filtration of the regular \(H_R\)
given by a linear extension of its left-cell order.
Apply \(G_T\), which is exact on that filtration. Its
factors are \(\widetilde F_R\)-images of standards by
Lemma~\ref{fin:recognition}. The preceding equivalence
lifts the entire flag to a standard-filtered object
whose total image is \(\widetilde F_R(P_{\mathrm{KZ}})\).
Full faithfulness identifies the total object with
\(P_{\mathrm{KZ}}\), and the idempotent selecting its
summand in \(Q_R\) recovers the original Hecke flag.
The same argument lifts each two-cell subquotient.

Suppose two adjacent cells of different spectral values
are ordered with the smaller value below the larger.
Their lifted extension splits by the displayed Ext
vanishing. The generic projector onto the lower cell
is unique because the generic labels are distinct.
Thus this projector is regular at every root of unity,
by completing at each of them as above. The factors of the
Schur elements vanish at finite nonzero values only at
roots of unity. At any other such value the Hecke algebra
is split semisimple and its generic block projectors are regular
by \cite[(2.4) and Corollary~3.12]{MathasMatrix}.
The projector matrix is therefore Laurent polynomial.
It is fixed by bar, by uniqueness of the projection
onto the coordinate submodule. For positive real \(v\)
the original trace gives the Euclidean form in the
standard basis. The canonical Gram matrix tends to the
identity at infinity, as does the Schur complement
on the two-cell subquotient. The generic projector is
orthogonal for this form and hence bounded at infinity.
A bar-fixed Laurent polynomial bounded there is constant.
The limiting Gram identity makes this constant projector
the ordinary coordinate projection. Its coordinate
complement can now be placed below the other cell.
Repeating these adjacent swaps sorts the original
coordinate flag by decreasing spectral value. Its
initial segments are precisely the spectral threshold
ideals.
\end{proof}

\begin{lemma}\label{fin:e2}
If every spectral threshold of \(B_n\) is a \(C\)-coordinate
ideal, no generator edge at a fixed spectral degree joins
distinct actual two-sided cells.
\end{lemma}

\begin{proof}
The threshold property identifies the spectral value of
each cell and is Geck's E1
\cite[Lemma~3.2 and Proposition~3.6]{GeckFinite}.
In the generic regular representation the threshold
spans exactly the simple blocks of the indicated degrees.
On \(P_{\mathrm{KZ}}\), order a standard filtration by
Schur valuation using Lemma~\ref{fin:strict}. The section
at each valuation is a direct sum of standards by the
Ext vanishing. Its KZ image is therefore a filtration
of \(H_R\) with each exact section a direct sum of KZ
standards. Twisting by \(\dagger\) converts the labels
to the positive convention. Both this filtration and
the original coordinate spectral filtration are
saturated, since their successive quotients are
\(R\)-free. Their generic spans agree, so the filtrations
coincide.

Inside an exact degree layer, filter by actual two-sided
cells in a linear extension of their preorder. The
generic constituents of distinct two-sided cells are
disjoint by \cite[Remark~3.3]{GeckFinite}.
Each initial segment is saturated and its generic
span consists of complete distinct simple matrix blocks.
In the direct sum of KZ standards just obtained, its
intersection is exactly the sum of the standard lattices
belonging to those labels. Hence every such initial
segment splits integrally at every root of unity.
The unique spectral projection onto it is Laurent,
bar fixed and bounded at infinity, by the same
semisimplicity and trace argument as in
Lemma~\ref{fin:e1}. It is therefore the coordinate
projection. Differences of successive projections give
the individual two-sided-cell projections, which commute
with both Hecke actions. This proves the claim.
\end{proof}

\begin{lemma}\label{fin:wall}
Suppose the off-wall spectral thresholds of rank \(n\) are
\(C\)-coordinate ideals. The same holds on each positive
integral ratio \(b/a\) in that rank.
\end{lemma}

\begin{proof}
Fix \(r=b/a\in\mathbb Z_{>0}\). Form the two-variable Hecke
algebra over \(\mathbb Z[v^{\pm1},t^{\pm1}]\) with
lexicographic exponent order, primary variable \(v\),
and weights \((a,0)\), \((ra,1)\). Specializing \(v=z^N\)
and \(t=z\) for sufficiently large integral \(N\)
preserves all required strict signs in the finite canonical
basis and the orders of the Schur degrees. For a fixed
primary spectral threshold the selected simple labels thus
form a spectral threshold at \((Na,Nra+1)\), which is
off the wall. The corresponding coordinate span lifts to
a two-variable coordinate ideal. Indeed one chooses \(N\)
so that the specialization is injective on each finite
exponent support of the relevant structure coefficients
and the matrix coefficients in integral generic Specht
models. Vanishing into excluded coordinates and vanishing
in excluded irreducibles lift. Its dimension is the sum
of the squares of dimensions of the selected labels, as
at the one-variable specialization, so the lifted ideal
has precisely their generic support.

At \(t=1\) the specialized basis changes to the wall
canonical basis by a constant integral unitriangular
matrix. Every lexicographically negative term has
nonpositive primary exponent, so the transition and its
inverse have coefficients in \(\mathbb Z[v^{-1}]\).
The transition is bar invariant and hence constant.
The primary Schur degrees are the wall degrees. In the
cancellation-free Schur product
\cite[Theorem~3.2]{ChlouverakiJacon}, use the Ariki--Koike
parameters \(q=v^{2a}\), \(Q_0=v^{2ra}t^2\), \(Q_1=-1\).
Each cross factor
with zero primary exponent is \(1+t^2\) or \(1+t^{-2}\),
which becomes \(2\), and the highest primary coefficient
of every other factor survives. Over \(\mathbb Q(v)\)
the wall Hecke algebra is split semisimple. Its generic
primitive idempotents are regular at \(t=1\), since the
finite Schur elements are units in that local ring and
the trace-dual basis is integral. The lifted coordinate
ideal is saturated, so it is the image of the sum of
the selected idempotents over that local ring: this
equality holds generically and both submodules are
saturated. It follows that its specialized constant
rational coordinate subspace is exactly the required
spectral ideal.

It remains to show that this subspace is coordinate
in the wall basis. The Schur maximum degree is constant
on Rouquier blocks of the two-colour integral-charge
cyclotomic algebra by \cite[Proposition~3.18]{BroueKim}.
Apply this to the Ariki--Koike generators \(v^bN_{s_0}\)
and \(v^aN_{s_i}\) for \(i>0\), with parameters
\(x,x^r,-1\), where \(x=v^{2a}\). The change from
normalized generators uses only Laurent units.
Substituting \(x=v^{2a}\)
sends each allowed denominator \(x^m-1\) to
\(v^{2am}-1\), so the aggregate block idempotent
specializes in the same integral localization.
Thus the spectral projector
belongs to the Hecke algebra over
\(\mathbb Z[v^{\pm1},(v^m-1)^{-1}:m\geq1]\).
This ring expands in integer Laurent series in
\(v^{-1}\), since every inverted denominator is monic
at infinity. The projector is orthogonal for the
positive original trace form, whose wall canonical
Gram matrix tends to the identity. Its range is the
constant subspace just obtained. Therefore its limit
is the Euclidean orthogonal projector onto that
subspace. Each diagonal entry of the limit is the
limit of the trace of \(C_wC_w^*\) multiplied by the
central idempotent. This trace is in the displayed
integral ring and lies between zero and
\(\tau(C_wC_w^*)\to1\). Its limit is an integer,
hence zero or one. An orthogonal projector with such
diagonal entries is coordinate, since the squared
norm of its image of a unit coordinate vector equals
the corresponding diagonal entry. The wall spectral
ideal is therefore a \(C\)-coordinate span.
\end{proof}