Indecomposable tilting objects exist and are classified in parabolic category O

Construction from actual exact sequences. Let $\bar{P}(A)$ be the labelled opposite projective cover. Its highest-weight sequence is $0\to \bar{K}(A)\to \bar{P}(A)\to \bar{\mathrm{\Delta }}(A)\to 0$, with $\bar{K}(A)$ standard-filtered by labels strictly above $A$. Applying $\mathrm{\Psi }$ gives $0\to M(A)\to X(A)\to \mathrm{\Psi }\bar{K}(A)\to 0$, where $X(A)=\mathrm{\Psi }\bar{P}(A)$ and the quotient has a finite standard flag with labels strictly below $A$. Its endomorphism algebra is the opposite of the local endomorphism algebra of $\bar{P}(A)$, so $X(A)$ is indecomposable. For an arbitrary extension $0\to X(A)\to Z\to M(B)\to 0$, extension closure makes $Z$ standard-filtered. Inverting $\mathrm{\Psi }$ turns this into an extension with source projective $\bar{P}(A)$ as quotient, which splits; exact equivalence transfers the splitting back. Hence ${\mathrm{Ext}}^1(M(B),X(A))=0$ for every $B$. The proved homological criterion of [F1] gives a costandard flag of $X(A)$, so it is the required actual tilting object with normalized bottom-standard inclusion.

Uniqueness. Let $T,T^{\mathrm{\prime }}$ be two indecomposable tilting objects with $M(A)$ once at the bottom and all other labels $\prec A$. Extending the common normalized bottom inclusion both ways uses ${\mathrm{Ext}}^1$ vanishing against standard flags, and the resulting composites restrict to the identity on the bottom standard, hence are not nilpotent; finite-dimensional local $\mathrm{End}$ and Fitting's lemma force them to be isomorphisms. So $T\cong T^{\mathrm{\prime }}$, and [F1] identifies its unique maximal support label as $A$.

Every tilting is a sum of these. Let $T\mathrm{\ne }0$ be an arbitrary tilting object and let $\mathrm{\Gamma }$ be the lower ideal generated by its entire composition support. Choose one maximal support label $A$, which is maximal in $\mathrm{\Gamma }$, and put $m=[T:L(A)]\ge 1$. All standard and costandard sections of $T$ have labels in $\mathrm{\Gamma }$, so it is tilting in that truncated highest-weight category. The proved maximal-standard trace in [F1] injects $\mathrm{\Delta }(A{)}^{\oplus m}$ into $T$ with standard-filtered quotient. Choose one copy $i:\mathrm{\Delta }(A)\hookrightarrow T$; its quotient is still standard-filtered, since the other $m-1$ copies and the trace quotient have such flags. Let $j:\mathrm{\Delta }(A)\hookrightarrow X(A)$ be the actual normalized inclusion of step 1.1. Since $X(A)\mathrm{/}\mathrm{\Delta }(A)$ is standard-filtered and $T$ is tilting, ${\mathrm{Ext}}^1(X(A)\mathrm{/}\mathrm{\Delta }(A),T)=0$ extends $i$ to $f:X(A)\to T$ with $fj=i$. Similarly ${\mathrm{Ext}}^1(T\mathrm{/}\mathrm{\Delta }(A),X(A))=0$ extends $j$ to $g:T\to X(A)$ with $gi=j$. The composite $gf$ fixes the nonzero bottom standard of the indecomposable $X(A)$, so it is invertible by the finite local-End/Fitting argument. Hence $f$ splits with retraction $(gf{)}^{-1}g$, and $T\cong X(A)\oplus T^{\mathrm{\prime }}$. The complement is tilting by summand closure and has strictly smaller length; induction proves the stated finite decomposition. Incomparable maximal support labels and multiplicities $m>1$ are both allowed: no normalized flag of the entire $T$ was assumed.

Duality, ideals and Grothendieck basis. Restricted duality makes $DT(A)$ an indecomposable tilting object with the same composition support $\le A$ and $[DT(A):L(A)]=1$. In the lower highest-weight ideal of labels $\le A$, the maximal-standard trace proved in [F1] gives an inclusion $M(A)\hookrightarrow DT(A)$ with standard-filtered quotient of strictly smaller labels, since its multiplicity space has dimension one. This supplies an actual normalized standard flag, so uniqueness identifies $DT(A)\cong T(A)$. If $\mathrm{\Gamma }$ is a lower ideal containing $A$, every support label of $T(A)$ lies in $\mathrm{\Gamma }$, so $T(A)$ computed in the ambient block lies in the Serre subcategory of $\mathrm{\Gamma }$ and coincides with the partial tilting there; the classes $[T(A)]$ are unitriangular in the standard basis by the flag multiplicities, hence form a Grothendieck basis.

The projection kernel and standard embeddings. The projection $\pi$ of [F4] has kernel spanned by the $L$-vectors off ${\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$ and is $U$-stable; the residue functors have nonnegative composition coefficients, so iterating them on an off-standard simple never produces a standard composition factor. For this embedding argument, enlarge a finite upper alphabet if necessary while keeping the lower endpoint and the Levi column heights fixed. The blocks of the original labels are unchanged, because their entry multisets are unchanged; the established Fock projection and its off-standard kernel apply on the enlarged alphabet, where the standard predicate for those labels is the same. Every generalized standard $M(B)$ then embeds into an iterated $e$-word applied to an irreducible standard $M(C)$: add successively huge offsets to the later columns so that all later entries lie above the earlier ones, making $C$ the unique minimum in its fixed-multiset tableau order and hence $M(C)=L(C)$ simple; then lower the later-column entries back, column by column and bottom to top, with the explicit residue word. A later column stays above the active residues, so a wrong move can only occur in an earlier column and strictly increases an earlier prefix weight; hence $B$ is the unique maximal final standard section and occurs once, and the canonical maximal trace embeds $M(B)$ at the bottom. If $L(A)$ embedded into $M(B)$ with $A$ off standard, adjunction would give the standard $L(C)$ as a quotient of an $f$-word applied to $L(A)$, impossible by kernel stability.

Converse and prinjectivity. Conversely every standard label is reached from the ground simple projective by the crystal $f$-word, and the simple-head clause of [F4] makes $P(A)$ a summand of that translated projective-injective, hence injective; projectivity of $P(A)$ is by definition. A projective-injective indecomposable is self-dual: $DP(A)=P(B)$ with $[P(A)]=[P(B)]$ because simple classes are duality-fixed and the Cartan matrix is $D{D}^t$ with unitriangular invertible $D$, so $A=B$. A self-dual $P(A)$ has its head in its socle, and its standard flag embeds that simple into a standard section; conversely if $A$ is standard then $P(A)$ is projective-injective by the previous sentence. BGG reciprocity and Parabolic Kazhdan-Lusztig polynomials give the dual-canonical transition matrix identify the standard coefficients of $[P(A)]$ with the transpose composition matrix defining the quasi-canonical vector $P_A$ at $s=1$. Thus the normalization $P_A=s^a{T}_{A^{\uparrow }}$ of [F4] identifies that actual projective class with the source canonical vector at $s=1$, whose unitriangular monomial expansion has unique maximal label $A^{\uparrow }$. Since $P(A)$ is an indecomposable tilting object its unique maximal standard label identifies it as $T(A^{\uparrow })$.

Conclusion. Every label has a unique normalized indecomposable tilting object with the stated maximal-support flag, every tilting is a finite sum of these, restricted duality fixes them, lower ideals stabilize the same objects, and $P(A)$ is projective-injective exactly for standard $A$, in which case $P(A)=T(A^{\uparrow })$; no Irving criterion or future Schur-theoretic input is used.