The cogood node gives the simple socle and head of the residue induction

We first establish the restriction endpoint locally. Work with left Schur modules through star. In tensor degree $m$, the word of $m_{\alpha }{T}_w$ has $j$th letter the row containing $j$ in $t^{\alpha }w$; at regular type it is $j{w}^{-1}$, with zero-multiplicity labels omitted. Over $K=F(t)$ the evaluations $g^{\otimes m}$ span the tensor Schur algebra: a functional killing them gives a polynomial zero on invertible matrices; multiplication by $\det$ and the infinite-field polynomial identity force that polynomial to be zero. Schur simples are therefore simple polynomial ${\mathrm{G}\mathrm{L}}_N$ modules. The canonical shape/type diagonal is its permutation-summand projection, whose square gives canonical form value one. Semistandard column strictness bounds entries at most $k$ by the first $k$ rows, so all weights are dominated by the highest weight $\xi$ and the highest space is $Kv$. Integer torus characters, rather than Lie eigenvalues modulo $p$, distinguish these weights.

Gaussian factorization $g=u^{-}t{u}^{+}$ on the big cell shows that negative coefficient operators on $v$ span its simple module: a functional killing them kills $gv$ on that open set and hence everywhere by polynomial identity. Similarly, positive coefficient operators detect every nonzero vector $x$ of weight $\xi -\mathrm{\Gamma }$. If every positive highest coefficient vanished, the highest coefficient of $gx$ would vanish on the big cell, because negative operators cannot raise any weight to $\xi$; it would then vanish on the orbit of $x$, which spans the simple and contains $v$. All coefficient operators have integral tensor replacement matrices, and the ranks and Hom kernels of finite matrices preserve extension from $F$ to $K$. These arguments consequently descend to $F$.

Let ${\xi }_N=0$, $l=N-1$, and let $M$ be the full last-coordinate-one subspace of $L_N(\xi )$. A primitive vector means one killed by every nonconstant coefficient of the upper Levi unipotent group, not merely by its Lie generators. Factor the full upper group as $V{U}_l^{+}$, where $V$ has only the last column above the diagonal nontrivial. A positive test of a Levi-primitive vector acts only through $V$, since the right-hand nonconstant Levi coefficients kill it. Raising last coordinate $1$ to highest coordinate $0$ requires a $V$ monomial of degree exactly one. Positive detection therefore gives $E_{rN}x=cv\mathrm{\ne }0$ for some $r$. Hence $\mathrm{wt}(x)=\xi -{\beta }_r$, where ${\beta }_r={\alpha }_r+\cdots +{\alpha }_l$, even when the initial first-level deficit was arbitrary. On that primitive weight space $x\mapsto E_{rN}x$ is injective by the same detection argument, so its dimension is at most one.

Put $a_j={\xi }_j-{\xi }_{j+1}$ and use $B(r,s)=s-r+{\sum }_{j=r}^s{a}_j$ and $C(r,s)=s-r+{\sum }_{j=r}^{s-1}{a}_j$. For nonempty $I$, the canonical $Q(I)$ kills a primitive vector: if $b=\min I$, its ascending-prefix letter $e_{b-1}$ commutes past the remaining ascending letters, whose indices exceed $b$, and becomes a rightmost Levi letter with $b-1<l$. Expand a nonzero primitive vector of weight $\xi -{\beta }_r$ in the essential $T$ basis and select the largest hat among its nonzero coefficients. The triangular pivot isolates a nonzero term. That hat cannot be nonempty, since its $Q$ test would annihilate the vector. Distinct essential hats therefore leave exactly one essential set with empty hat. Empty is linked precisely to a regular set, and the essential-regular matching theorem says such a set exists precisely when $r$ is normal. This proves necessity of normality.

Conversely, for normal $r$ let $R_r=\{s\in (r,l]:B(r,s)=0\}$ and choose its compensation $f$ with $r<f(s)\le s$ and $B(f(s),s)=0$. Put $D=f(R_r)$. The inverse matching makes $D$ regular, so its hat is empty. The matched norm is ${\mathrm{\Xi }}_{r,l}(D)={\prod }_{s\in [r,l]\setminus R_r}B(r,s)\mathrm{\ne }0$: $B(r,r)=a_r\mathrm{\ne }0$, and $R_r$ removes exactly the other zero factors. Thus $T_{r,l}(D)v$ is nonzero, and the maximal-hat assertion gives $h(I,D)=0$ for every nonempty $I$. For $r\le j<l$, every positive test of $e_{j}T(D)v$ has grade ${\beta }_r-{\alpha }_j$. Expanding its root matrices into simple words and appending the rightmost $e_j$ gives full words with nonempty canonical $Q$ index, since $e_j$ acts before $e_{j+1}$. All highest coefficients vanish, so positive detection gives $e_{j}T(D)v=0$. For $j<r$ the target weight is outside dominance below $\xi$ and is zero. A nonconstant Levi coefficient whose weight is not bounded by ${\beta }_r$ likewise has zero target. At a weight bounded by ${\beta }_r$, root-variable exponents are at most one, so the coefficient is a product of positive root matrices, each an iterated simple-root commutator; its rightmost root kills the vector. This proves full group primitivity. The primitive line at $\xi -{\beta }_r$ is consequently one-dimensional exactly at normal $r$, and we denote any nonzero vector on it by $v_r$.

We separate primitive labels integrally, including $p=2$. In tensor degree $b$ for Levi rank $l$, put $H_a=E_{aa}$ and define $\left(\genfrac{}{}{0px}{}{H_a}{2}\right)$ to act on a weight $\zeta$ by the integer $\left(\genfrac{}{}{0px}{}{{\zeta }_a}{2}\right)$ reduced in $F$. Concretely it is ${\sum }_{x<y}{E}_{aa}^{(x)}{E}_{aa}^{(y)}$, with no division by $2$ in $F$. The tensor action $Z_b$ of the group central transposition sum $z_b={\sum }_{x<y}(x,y)$ satisfies the exact integer identity $Z_b={\sum }_a\left(\genfrac{}{}{0px}{}{H_a}{2}\right)+{\sum }_{a<c}(E_{ac}{E}_{ca}-H_a)$. The equal-position part of $E_{ac}{E}_{ca}$ is $H_a$; after subtracting it, the terms are exactly the two swaps of unequal labels. The diagonal terms count equal-label pairs. Since $z_b$ commutes with the position group and every Schur endomorphism, this operator belongs to the Schur algebra and is central.

On a primitive vector of dominant weight $\zeta$, $E_{ac}$ kills it for $a<c$, so $E_{ac}{E}_{ca}$ acts by $H_a-H_c$. Therefore $Z_b$ has scalar $\kappa (\zeta )={\sum }_a\left(\genfrac{}{}{0px}{}{{\zeta }_a}{2}\right)-{\sum }_c(c-1){\zeta }_c={\sum }_{(c,j)\in [\zeta ]}(j-c)$. The binomial here is the actual diagonal weight operator of step 6.1. Centrality makes this scalar constant on the cyclic module generated by the vector. For first-level ${\zeta }_r=({\xi }_1,\dots ,{\xi }_l)-{ϵ}_r$, it is $\kappa ({\zeta }_r)=\kappa (\xi )-({\xi }_r-r)$, so $\kappa ({\zeta }_t)-\kappa ({\zeta }_r)=-C(t,r)$. Thus the cyclic Levi module ${\mathrm{\Psi }}_r$ generated by $v_r$ cannot contain another primitive $v_t$ when $C(t,r)\mathrm{\ne }0$. Equality of the scalars alone does not prove simplicity; we treat it explicitly next.

Suppose $t<r$ are normal and $C(t,r)=0$. Choose $D$ for the demands of $r$ as in step 5.1. For $w\ge r$, $B(t,w)=B(r,w)$, and $B(t,r)=a_r\mathrm{\ne }0$. The demands of $t$ split into $Y=\{y\in (t,r):B(t,y)=0\}$ and $R_r$. Compensation for $Y$, inherited from normality of $t$, has image $E$ below $r$. Since $B(t,r-1)=C(t,r)-1=-1\mathrm{\ne }0$, all those targets and $E$ lie at most $r-2$. Combining their matching with $D$ gives compensation for every demand of $t$, so choose $v_t=T_{t,l}(E\cup D)v$ and $v_r=T_{r,l}(D)v$. For each $e\in E$ its matched $y$ gives $B(e,y)=B(t,y)=0$, hence $C(t,e)=0$. If $E\mathrm{\ne }\mathrm{\varnothing }$ and $e=\max E$, then $C(e,r)=C(t,r)-C(t,e)=0$.

Splitting (P) gives $T_{t,r-1}(E)v_r=v_t-T_{t,l}((E\setminus \{e\})\cup \{r\}\cup D)v-C(e,r)T_{t,l}(E\cup \{r\}\cup D)v$. The last term is zero. The middle term has a (Z) witness with endpoint $r-1$, outside its cuts because $E\le r-2$ and $D>r$. Removing matched source $e$ releases its zero-$B$ target $y_e$; assign $t$ to $y_e$ and retain the matching on $E\setminus \{e\}$. All distinct bounded targets are below $r-1$. Hence that term is zero as well. For $E=\mathrm{\varnothing }$, (P) omits the middle term and its last coefficient is $C(t,r)=0$. In both cases $v_t=T_{t,r-1}(E)v_r\in {\mathrm{\Psi }}_r$, without a field denominator.

The Gaussian argument on a cyclic module, without assuming simplicity, shows that ${\mathrm{\Psi }}_r$ is spanned by negative Levi coefficients on $v_r$ and has weights at most ${\zeta }_r$. Any nonzero proper Schur submodule contains a simple Schur submodule and thus a primitive $v_t$ by step 3.1. Dominance gives $t\le r$. For $t=r$, its one-dimensional line is the generating line and would generate all of ${\mathrm{\Psi }}_r$; thus a proper submodule has $t<r$, and the central scalar requires $C(t,r)=0$. Conversely, such an earlier normal $t$ gives a proper nonzero ${\mathrm{\Psi }}_t\subset {\mathrm{\Psi }}_r$ by step 9.1, since its weights are at most ${\zeta }_t<{\zeta }_r$. Consequently ${\mathrm{\Psi }}_r$ is simple precisely when $r$ is normal with no earlier normal $t$ satisfying $C(t,r)=0$. All simple Schur submodules of $M$ are exactly these ${\mathrm{\Psi }}_r$, each once: every simple has a primitive highest vector, and each primitive line has dimension one.

Now assume $\xi$ restricted. The root criterion is its physical signature criterion: $a_r\mathrm{\ne }0$ identifies the removable node in row $r$; scanning $j>r$, a nonzero $a_j$ gives a removable push if $C(r,j)=0$ and an addable pop at row $j+1$ if $B(r,j)=0$. These events have the residue of row $r$; at $a_j=0$ the matching slot supplies and consumes itself and is neutral. For $0<a_j<p$, the two events are disjoint. The compensation-prefix inequalities are exactly survival of the initial push in the signature stack. Also $C(t,r)=({\xi }_t-t)-({\xi }_r-r)$ compares the residues. Thus the simple cyclic modules in step 10.1 are exactly at the good nodes. Their lower labels are restricted: removing row $r$ decreases $a_r$ by one and increases $a_{r-1}$ by one. The only failure, when $r>1$, would be $a_{r-1}=p-1$. Then $C(r-1,r)=0$, and $r-1$ is also normal, since its later demands are those of $r$, $B(r-1,r)=a_r\mathrm{\ne }0$, and the compensation for $r$ is also valid for $r-1$. This contradicts goodness. Hence every Schur socle factor survives the restricted-label corner; no assertion is made about the other composition factors.

To transfer without a held generic classifier, for $\xi ⊢m$ choose $N=m+1$ and the regular composition $\omega =(1^{m-1},0,1)$, so ${\xi }_N=0$ even for columns. Compute the cell corner temporarily in its original right-module convention, then transport through star. Its permutation summand is $M^{\omega }=F{S}_m$, and its actual projection $e$ gives $eSe=F{S}_m$ via ${\rho }_h(x)=hx$. Type-$\omega$ tableaux correspond to standard tableaux by increasing relabeling $1,\dots ,m-1,N$ to $1,\dots ,m$. Their normalized map fibers are singleton, and evaluation at $1$ sends ${\phi }_{st}$ to $m_{st}$. Higher ideals correspond as equality of corner basis spans. The Weyl weight projection keeps exactly these basis vectors; read its cell action using a fixed outer index of the same type and evaluate at $1$ to get the Murphy action. Thus $W^{\zeta }e\cong C_M(\zeta )$ with the actual same-shape normalization.

In the Schur cell product choose all four indices $a,s,t,b$ standard of type $\omega$. Evaluation at $1$ turns ${\phi }_{as}{\phi }_{tb}\equiv B_{\zeta }(w_s,w_t){\phi }_{ab}$ into $m_{as}{m}_{tb}\equiv B_{\zeta }(w_s,w_t)m_{ab}$, so the unique Murphy coefficient gives $B_{\zeta }(w_s,w_t)=G_{\zeta }(c_s,c_t)$. The factor is one. Since $e^{\ast }=e$, the Weyl splitting $W=We\oplus W(1-e)$ is orthogonal, giving ${\mathrm{rad}}_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}(We)={\mathrm{rad}}_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}(W)e$, including zero corner forms. Projection is exact by lifting $ye$ through $xe$ and taking kernels fixed by $e$. Hence $L^{\zeta }e\cong C_M(\zeta )\mathrm{/}{\mathrm{rad}}_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}$. The actual $q=1$ form-unit bridge identifies it with $D^{{\zeta }^{\mathrm{\prime }}}\otimes \mathrm{sgn}$, nonzero exactly when ${\zeta }^{\mathrm{\prime }}$ is regular. Transporting these right-cell identities through star gives the corresponding left corners $e{W}^{\zeta }$ and $e{L}^{\zeta }$ used below. Their nonzero images are exactly the distinct absolutely simple restricted-label modules $D(\zeta )$, with the inverse group convention fixed by the nondegenerate quotient form.

Let ${\pi }_1$ be the sum of actual weight projections with last coordinate one, so $M={\pi }_1{L}_N(\xi )$. The lower Schur orbit matrix $E_A$ embeds as $E_{\mathrm{diag}(A,1)}$; on each word with one $N$ it preserves that position and applies $E_A$ to the remaining ordered positions. Products agree and the lower identity is ${\pi }_1$, and transpose/star is respected. The lower regular projection of ${\omega }^{\mathrm{\prime }}=(1^{m-1},0)$ maps to $e$. On the coset-word convention of step 1.1, ${\rho }_h^{\mathrm{\prime }}$ acts by label permutation $h^{-1}$ on the used lower labels, fixing $N$, identically to ${\rho }_h$ for the subgroup fixing the last entry of $S_m$. Thus, in the left convention, $e^{\mathrm{\prime }}M=e{L}_N(\xi ){\downarrow }_{S_{m-1}}$. Write $S_L=S(N-1,m-1)$ for this lower Schur algebra; every socle and corner lift of $M$ below is over $S_L$. The inverse/star equivalence passes between right and left modules and commutes with this embedding, restriction and sign; the ordinary simple labels are fixed by their nondegenerate invariant quotient forms. No self-duality of a degenerate Specht module is assumed.

Socle preservation requires more than exactness. By step 11.1 every Schur socle simple of $M$ survives $e^{\mathrm{\prime }}$. A nonzero Schur submodule killed by $e^{\mathrm{\prime }}$ would contain a simple submodule inside that socle, whose corner is nonzero, a contradiction. Now a group-simple injection $j:D\to e^{\mathrm{\prime }}M$ corresponds to $S_L{e}^{\mathrm{\prime }}{\otimes }_{e^{\mathrm{\prime }}{S}_L{e}^{\mathrm{\prime }}}D\to M$, $s{e}^{\mathrm{\prime }}\otimes x\mapsto sj(x)$, $s\in S_L$. Its image $I$ is generated by its corner $e^{\mathrm{\prime }}I=j(D)$. A proper Schur submodule $J$ of $I$ has either $e^{\mathrm{\prime }}J=0$, excluded by the no-killed-submodule property, or $e^{\mathrm{\prime }}J=e^{\mathrm{\prime }}I$, which generates $I$ and forces $J=I$. Thus $I$ is simple. Every group-simple corner submodule lifts to an actual Schur simple submodule, and conversely every surviving Schur simple has simple corner. Exactness on the semisimple sum, distinct labels and the one-dimensional primitive lines yield ${\mathrm{soc}}_{F{S}_{m-1}}(e^{\mathrm{\prime }}M)=e^{\mathrm{\prime }}{\mathrm{soc}}_{S_L}M$, multiplicity-free with exactly the good-deletion labels. The unused coordinate includes columns and $m=1$, whose lower degree-zero corner is $F$.

The group central $z_m$ acts on $D(\xi )$ by $\kappa (\xi )$, since its tensor action is step 6.1 and its regular-summand action is ${\rho }_{z_m}$; right multiplication equals left multiplication for this central element. On the good lower label it acts by $\kappa (\xi -A)$. Therefore $L_m=z_m-z_{m-1}$ has scalar $\mathrm{res}(A)$ on that simple submodule, with the restricted convention $c-r$. Primary projection selects just that residue, and socles respect the actual direct sum. We obtain $\mathrm{soc}{E}_{i}D(\xi )=D(\xi -A_i)$ when ${\epsilon }_i(\xi )>0$, and zero otherwise. In the latter case $E_{i}D(\xi )=0$, since a nonzero finite-dimensional module contains a simple submodule.

The invariant nondegenerate quotient form makes $D(\xi )$ self-dual. A primary polynomial projector in $L_m$ is self-adjoint, since the inverse involution fixes $L_m$, so its summand has a restricted nondegenerate invariant form. Its head is therefore dual to its simple socle and is the same simple. Thus for every restricted $\eta ,\tau$ of consecutive degrees, ${\dim }_F\mathrm{Hom}(E_{i}D(\eta ),D(\tau ))$ is one if ${\epsilon }_i(\eta )>0$ and $\eta -{\mathrm{good}}_i(\eta )=\tau$, and zero otherwise. The scalar endomorphism ring is $F$ by absolute simplicity. This all-pairs criterion includes the zero case and all initially possible primitive weights; it does not discard middle composition factors.

We justify the needed functor interfaces locally. In $\mathbb{Z}{S}_n$ the Jucys identity is ${\prod }_{r=1}^n(u+L_r)={\sum }_{g\in S_n}{u}^{\mathrm{\#}\mathrm{cycles}(g)}g$. Induct on $n$: choosing $u$ leaves $n$ fixed, and choosing $(a,n)$ inserts it next to $a$ in a cycle, bijectively covering the non-fixed permutations. Hence each elementary symmetric JM coefficient is central over every field. On a simultaneous primary tuple these operators have generalized eigenvalues equal to its elementary symmetric residue coefficients; nilpotence follows by expanding a polynomial in the commuting nilpotent differences. Distinct residue multisets have distinct coefficient tuples, since their monic products of prime-field linear factors retain multiplicities uniquely even in characteristic $p$. For each elementary symmetric operator, construct its univariate primary projectors by the same polynomial CRT construction as in [F5], then multiply these commuting projectors. This gives the content projector equal to the sum of tuple projectors of that content. This proves content centrality without factorial division, position-dependent symmetric-polynomial claims, or content-block primitivity. The projectors are inverse-fixed. For an input in content $\alpha$, induction projection by right $e_{\alpha }{P}_{d+1,i}$ is the same as right $P_{d+1,i}$, because $g{e}_{\alpha }\otimes x=g\otimes e_{\alpha }x$. An absolutely simple $D(\mu )$ lies in exactly one of these central contents, since their scalar idempotents are orthogonal and sum to one; denote it by $\alpha$. Thus these interfaces agree with the actual functor definition.

Put $G=S_{d+1}$, $H=S_d$, and let $E_H:FG\to FH$ truncate group coefficients. For left modules the natural Frobenius map is $\mathrm{\Phi }(g\otimes x)(y)=E_H(yg)x$, with coinduction action $(aq)(y)=q(ya)$. It is balanced and $G$-linear by $E_H(hz)=h{E}_H(z)$ and $E_H(zh)=E_H(z)h$. For a left-coset transversal $T$ of $G\mathrm{/}H$, its inverse is $\mathrm{\Psi }(q)={\sum }_{r\in T}r\otimes q(r^{-1})$. The right cosets $H{r}^{-1}$ partition $G$, so evaluation at $y$ has exactly the term $yr\in H$ and returns $q(y)$. On $g\otimes x$ the unique term with $r^{-1}g\in H$ balances back to $g\otimes x$. These verify both identities and naturality with correct cosets and only finite choices.

Let $e=e_{\alpha }{P}_{d+1,i}$, commuting with $FH$ and inverse-fixed. For any such centralizer element, $E_H(eZ)=E_H(Ze)$: the coefficient at $h$ is the group coefficient trace $\tau (h^{-1}eZ)=\tau (h^{-1}Ze)$, by cyclicity and commuting $e$ with $h^{-1}$. Under $\mathrm{\Phi }$, the value at one of right projection $R_e$ is consequently $q(e)$. If $f:N\to \mathrm{Coind}X$ is $G$-linear, $f(en)(1)=f(n)(e)$. Evaluation therefore identifies projected maps $N\to F_{i}X$ with $H$-maps $E_{i}N\to X$; conversely an $H$-map extends via $q_n(y)=h(eyn)$, with $h:E_{i}N\to X$ evaluated after the projector $e$. For $X=e_{\alpha }X$, every $H$-map already selects the matching content. Thus ${\mathrm{Hom}}_G(N,F_{i}X)\cong {\mathrm{Hom}}_H(E_{i}N,X)$ exactly, with no duality shift and no asserted equality of composition multiplicities.

The labelled signature inverse is local. Adding or removing a corner flips its $i$-sign, and the other boundary statuses that can change have residues $i\pm 1$, not $i$ for $p\ge 2$. In the stack reduction, flipping the last surviving plus gives the first surviving minus and conversely: before that position the minus stack is empty, and its distinguished minus survives exactly when every suffix prefix has at least as many minus pushes as plus pops, equivalent to no later plus surviving after the flip. Good removal preserves restrictedness by step 11.1. Cogood addition also does: its only possible new forbidden difference is $a_r=p-1$, in which case the addable node in row $r+1$ has the same residue, with no $i$-node between the two pluses. If the earlier plus survives, the later one also survives, contradicting that the earlier is cogood. Hence addition remains restricted and the inverse holds in both directions. In particular $B$ is cogood for $\mu$ exactly when it is good for $\nu =\mu \cup B$. No equality between ${\epsilon }_i(\nu )$ and ${\phi }_i(\mu )$ is used.

Apply step 20.1 with $N=D(\eta )$ and $X=D(\mu )$ for every restricted $\eta ⊢d+1$. By step 17.1 the resulting Hom space is one-dimensional precisely when $\eta -{\mathrm{good}}_i(\eta )=\mu$, and zero otherwise. Step 21.1 makes exactly one such label, namely $\nu =\mu \cup B$. The group simple classification covers every simple by these absolutely irreducible labels. The dimensions of all simple-to-module Hom spaces therefore identify the socle of $F_{i}D(\mu )$ as the single $D(\nu )$, with socle multiplicity one, excluding every other label.

Give ${\mathrm{Ind}}_H^{G}D(\mu )$ the invariant nondegenerate coset form $\beta (r\otimes x,s\otimes y)={\delta }_{rs}{\beta }_{\mu }(x,y)$ on a finite left transversal. A group element permutes cosets and acts by $H$ on the fibers, proving invariance. Equivalently its value on general group representatives is ${\beta }_{\mu }(x,E_H(g^{-1}h)y)$. The projection $R_e$ is self-adjoint: moving it to the first argument gives $E_H(e^{\ast }{g}^{-1}h)$, which equals $E_H(g^{-1}h{e}^{\ast })$ by the trace calculation of step 20.1, and $e^{\ast }=e$. Its image is orthogonal to the complement and inherits a nondegenerate invariant form. Thus $F_{i}D(\mu )$ is self-dual, and its head is the dual of its simple socle, namely $D(\nu )$ through the nondegenerate invariant quotient form. This proves both endpoints, including characteristic two.

For $d=0$, the empty partition has one conormal node $(1,1)$ of residue zero and ${\phi }_0=1$. Induction from $S_0$ to $S_1$ is $F$, with $L_1=0$, so $F_{0}D(\mathrm{\varnothing })=D(1)=F$ has these endpoints; other residue functors are zero. Every use of $F(t)$ above descends to the original field: weight projections and coefficient operators have matrices over $F$, finite Hom ranks preserve extension, and the simple Schur and group labels remain absolutely simple. Therefore the endpoint assertions hold over every original field of characteristic $p$.

The total multiplicity is the separate use of [F4]. Apply it to the cogood $B$, which is bottommost conormal. Its conormal nodes weakly above it are exactly all ${\phi }_i(\mu )$ conormal nodes, so $[F_{i}D(\mu ):D(\nu )]={\phi }_i(\mu )$. Combining this count with the independently constructed endpoints proves the full statement. The implication uses no equality of induction and restriction composition counts and no equality ${\epsilon }_i(\nu )={\phi }_i(\mu )$.