# Schubert geometry, Demazure sections and flag K-theory

Build the scheme and section interfaces for Schubert varieties in general reductive flag varieties: relative incidence and universal bundles, root-coordinate cells and their closures, all-word Bott–Samelson operations, dominant line-bundle cohomology, compatible Frobenius splitting, all-degree standard monomial theory, exact divisor multiplicities and algebraic flag K-rings. The finite-type LS-chain/path engine and its geometric section comparison are part of this construction. Compare the complex realization with holomorphic Lefschetz characters and the existing finite Grassmannian topology. Suggested homes are `TauCeti/AlgebraicGeometry/Schubert/` and `TauCeti/RepresentationTheory/LSPath/`.

Use schemes, sheaves of modules, root data and rational representations at their stated hypotheses. A point-set rank condition is a comparison interface for a scheme, not its definition as a variety. Singular Schubert varieties are not declared smooth.

## Conventions and construction order

Fix a split connected reductive group G over a field k, a pinning when root coordinates are used, T⊂B+ with positive roots Φ+, and the opposite Borel B−. Write X=X*(T), Y=X_*(T), with their actual perfect pairing, W for the Weyl group and I for the simple roots. Keep the central lattice and the isogeny type. Unless separately specified, flags mean right quotients G/P−, with P− containing B−. Minimal representatives belong to W/W_P. The standard Schubert cell is C(w)=B−wP−/P− and X(w) is its schematic closure; its dimension is ℓ(w), e is closed and the longest minimal representative is open. Opposite Schubert loci use B+. Convert left quotients and positive-Borel sources by explicit inversion/opposition maps. An unqualified Schubert union is a finite union for this one fixed Borel, or for one common opposite Borel after the stated transport. Mixed standard/opposite families are named explicitly and retain their own ideal and cohomology conditions.

For λ∈X, the associated line L(λ)=G×^(B−)k_λ has fibre character λ at e. Sections are regular functions f(gb)=λ(b)^−1f(g), with (g·f)(x)=f(g^−1x). Dominant λ have nonnegative pairings with simple coroots; for descent to G/P− require zero pairings on its Levi simple roots. The rank-one restriction has degree ⟨λ,α_i∨⟩. Thus H⁰(G/B−,L(λ)) is ∇(λ), whereas ordinary Δ(λ)* is ∇(−w₀λ). Distinguish this from the pinned contravariant dual D_τΔ(λ)=∇(λ).

Over an arbitrary base S, relative finite-locally-free flags and incidence schemes have their universal functorial meanings. Group flag forms use their actual descent type; individual fundamental line bundles or a chosen pinning need not descend. Integral split calculations use the pinned Chevalley model and its actual character/cocharacter lattices. Characteristic-p splitting is first constructed over perfect fields, including p=2; field extensions and descent are proved before using it for geometric conclusions over other fields. Over a nonreduced S, geometrically reduced fibres do not make the total space reduced.

Construct in this order: S1's existing relative carriers and early root cells; S2's word/incidence API; S3's early line bundles and rank-one derived push-pull; S4.1–5's root-data LS combinatorics; S5's splitting/normalization/coherent comparisons; S7.1–2's divisor valuations; S4.6–7 and S6's geometric path sections and all-degree bases; S7.3's cycle comparison; S8's K-rings; S9's analytic and existing-owner comparisons. S4.1–5 is independent of Schubert splitting, vanishing, SMT and the concrete typeA implementation. S4.6–7 is its geometric application, after S5 and S7.1–2. The generic finite-type engine is the single root-data owner; S9.5 supplies the typeA specialization and operator-level comparison used to refactor the existing concrete path engine without rebuilding its tableau/GT algorithms. The algebraic Steinberg supplier is independent of all these geometric conclusions. The analytic fixed-point argument is a later comparison, not a premise for algebraic vanishing or SMT.

## Exact owners and interfaces

- [ModularCurves §0G](https://github.com/TauCetiProject/TauCetiRoadmap/blob/ba158a94935fcdc3e26804b3ee77ece7fb48d6c1/TauCetiRoadmap/ModularCurves/README.md) owns finite-locally-free Grassmannian representability and its actual universal quotient. Native `Module.Grassmannian R M q` stores a submodule with locally free quotient of rank q; an r-dimensional subspace of k^N corresponds to q=N−r. [AlgebraicModuliForArithmeticGeometry §1.3](https://github.com/TauCetiProject/TauCetiRoadmap/blob/ceba5c798ace9c7aaa13b4f1732828187c904ce5/TauCetiRoadmap/AlgebraicModuliForArithmeticGeometry/README.md) owns coherent quotient Grassmannians, universal Plücker maps and relative flag bundles. Consume those same objects and their base-change/universal properties.
- [ReductiveGroups](https://github.com/TauCetiProject/TauCetiRoadmap/blob/986a551490851718e5394107c45bf446860ba524/TauCetiRoadmap/ReductiveGroups/README.md) owns Hopf/scheme/point and rational-comodule dictionaries, group/root/pinning/central-isogeny structure, and field Bruhat theory. Its integral-distribution/rational-representation extension supplies full-lattice divided-power PBW/Dist(G), Δ/∇/L, induction, pinned extremal vectors and contravariant duality. The algebraic Steinberg input is the simple module Δ((p−1)ρ)=L((p−1)ρ)=∇((p−1)ρ) on the derived simply connected cover, with dimension p^|Φ+| and normalized PBW/extremal pairing. These algebraic constructions do not consume flag splitting or SMT.
- [ShimuraData §3.8, §§3.14–18](https://github.com/TauCetiProject/TauCetiRoadmap/blob/bf2c1dd902eeca2623e32d4d44b81eb942355567/TauCetiRoadmap/ShimuraData/README.md) owns Par_t(G), its G/P comparison, split integral cells, the reduced-word rank-one model and the late integral Schubert/base-change/closure/Richardson comparisons. Consume its early quotient/root-cell/model constructions first. S2 supplies the full-word/crank/subword API on that same model; S5 supplies the geometric cohomology comparisons used in its late section/base-change argument. Only then consume the resulting late comparison for the relative corollaries here. Neither a late vanishing theorem nor completed integral Schubert base change is an early premise for proving itself.
- [GroebnerGeometry G5](https://github.com/TauCetiProject/TauCetiRoadmap/blob/d21bc1c2f895047275495326f5f693e89f0e244c/TauCetiRoadmap/GroebnerGeometry/README.md) owns polynomial minor/shuffle/straightening/Gröbner statements, flag-minor algebra and its GT initial algebra. S6 supplies the scheme, homogeneous coordinate, section and restriction identifications. The original flag algebra is not identified with its toric initial algebra.
- [SchemeAndStackFoundations](https://github.com/TauCetiProject/TauCetiRoadmap/pull/779) supplies generic coherent cohomology, ample Serre, relative derived pushforward/base change and coherent/perfect K-theory at its stated scope; [StableReduction Layer2](https://github.com/TauCetiProject/TauCetiRoadmap/pull/898) supplies relative Proj using AlgebraicVectorBundles L2A's single graded quasicoherent algebra and affine-ring interfaces. AlgebraicVectorBundles supplies tensor, dual, exterior and determinant sheaves and their pullbacks. These generic constructions are not redefined here.
- [TopologicalKTheory T1.5](https://github.com/TauCetiProject/TauCetiRoadmap/blob/f2c908debdefa440e7dfda339a54cb3b74cdc51c/TauCetiRoadmap/TopologicalKTheory/README.md) supplies the single finite subspace Grassmannian and finite Schubert characteristic maps. [TensorProductCombinatorics H6](https://github.com/TauCetiProject/TauCetiRoadmap/blob/a9aea08168bc2ffae16ae87af6af9779f9c8d670/TauCetiRoadmap/TensorProductCombinatorics/README.md) owns its complex Grassmannian integral Schubert cohomology/puzzle comparisons; H8 owns its concrete typeA path/tableau engine. LieGroups supplies compact full flags and their homogeneous complex comparisons. S9 compares these with the algebraic models; it does not rebuild their existing carriers.
- [IndexTheory I8](https://github.com/TauCetiProject/TauCetiRoadmap/blob/592b12da97670f7eaea4fbed01def6b55554af66/TauCetiRoadmap/IndexTheory/README.md) owns the generic holomorphic fixed-map trace construction, in addition to its Dolbeault/Hodge and equivariant index machinery. S9 owns the flag/Bott–Samelson specialization and its section-action convention.
- [ComplexComparisonPartII C−1.1–2, C0.1–5, C1.9 and C2.1–2](https://github.com/TauCetiProject/TauCetiRoadmap/blob/654c956cf2e193efd94e91816658b2fcd1ffb8d8/TauCetiRoadmap/ComplexComparisonPartII/README.md) owns the singular/nonreduced analytic-space carrier, coherent sheaves, analytification and flat coherent pullback, projective all-degree GAGA, relative proper direct-image exchange and its qualified base-change maps. Use its same locally ringed spaces over Spec ℂ, native module sheaves and canonical cohomology/exchange maps. Its C−1/C0/C1 construction consumes the earlier CK/RAG local and Stein theories and is independent of Schubert vanishing and SMT. S9 supplies the application-specific comparisons with the existing smooth flag and word carriers.

`Suggested.lean` gives representative native point/rank interfaces and is not an exhaustive checklist. This README is definitive.

## S1. Relative incidence and the root-cell dictionary

1. For a finite locally free E/S and a strictly increasing admissible rank list, consume the universal relative flag with subbundles F_i and locally free quotients. Export empty/single/full flag comparisons, tautological exact sequences, tensor/exterior/determinant pullbacks, forgetful maps, refinement-parameter schemes, annihilator flags in E*, and arbitrary base change. An opposite flag in E needs the displayed splitting/pairing or opposite-Borel data; no canonical opposite/refinement of an arbitrary flag is assumed. Consume Moduli §1.3's closed incidence immersion into the product of quotient-rank Grassmannians and its universal equations F_i→E/F_j for i<j; prove that the source's coordinate model and forgetful/refinement maps identify with those same maps. The product is an ambient space, not the flag itself. On a chosen splitting, compare these sheaf maps with polynomial matrix incidence equations.
2. Consume Par_t(G), actual G/P− and root-coordinate charts. Construct the full source-to-model dictionary: standard and opposite cells, minimal coset representatives, chosen normalizer representatives, T-fixed points, inversion sets, relative dimension and parabolic projection. The left/right quotient map is P−g↦g^−1P− and on Weyl indices W_Pw↦w^−1W_P; prove its root-coordinate and associated-line character formulas before using the source's left-coset statements. The fixed scheme is the finite disjoint union of the indicated sections in the split model; do not identify a general twisted form with that split list. Use actual root subgroup multiplication and quotient universal properties, not merely the cardinality of the Weyl group. Include rank-zero groups, P=G, empty root systems and nontrivial central tori.
3. In typeA, compare row/column conventions, a rank-r subspace with quotient rank N−r, one-based pivots p₁<⋯<p_r and actual zero-based `Fin N` coordinates p_i−1. The increasing basis flag has dim(V∩F_j)=#{i:p_i≤j}; cell dimension is Σ_i(p_i−i). For the rectangle partition λ_i=N−r+i−p_i, this is r(N−r)−|λ|. Rank0/full rank are points and r>N is empty. For N≤1 use the direct point/empty cases, without an A₀/A_(−1) root reference.
4. Construct the determinantal rank-condition schemes relative to an actual complete subbundle flag; compare them with the supplied Schubert closures and locally split integral model. Give their universal ideals, restrictions, saturation/homogeneous presentations, base change and cell decompositions. Two or more arbitrary relative flags give well-defined incidence schemes but not automatic flatness, expected codimension or smooth resolutions. State and prove each such conclusion with its actual versality/transversality/regular-sequence assumptions.
5. Retain the symplectic exercise: for a nondegenerate alternating form, prove dim U≤m for isotropic U⊂k^(2m), and construct the isotropic Grassmannian as the universal zero locus of the restricted form on the tautological bundle. Compare its split group quotient, transitivity, parabolic stabilizer and dimension r(2m−r)−r(r−1)/2, including characteristic2 with an actual alternating form. Rank0 is a point and r>m is empty. Distinguish orthogonal characteristic2/non-smooth stabilizer issues. General root-type flag examples use the actual corresponding reductive model.

## S2. All-word Bott–Samelson and crank interfaces

1. For each simple root use the actual minimal parabolic P_i− and projection G/B−→G/P_i−. For X→G/B− construct the right crank X×_(G/P_i−)G/B−, a P¹-bundle over X; construct the left crank P_i−×^(B−)X, an X-bundle over P¹, with its map to G/B− when X is a B−-space. Distinguish the two projection directions, group actions and universal properties. The right projection is smooth proper of relative dimension1; if X is smooth/proper over k, so is its total space in the corresponding assertion. The left construction is locally an X-bundle over P¹ and inherits smoothness/properness over k when X has that property. Export their base-change and equivariance laws; no absolute properness is asserted for an arbitrary nonproper X.
2. For every finite word i₁…i_r, including empty and nonreduced words, extend the existing iterated-rank-one model as the contracted product P_i₁−×^(B−)…×^(B−)P_ir−/B−. Construct its quotient charts, scheme structure, iterated P¹-bundle maps, universal flag-chain description and multiplication map π to G/B−. Fix factor order and compare the crank order used in the handout. Prove dimension r, smoothness/properness, the B−/T actions, and base-change compatibility. No reducedness of the word is required for these constructions.
3. Construct subword closed embeddings, the 2^r T-fixed sections and the image of each indexed subword. For ε∈{0,1}^r put w_j=s_i₁^(ε₁)⋯s_ij^(ε_j). Prove tangent weights β_j=(−1)^(ε_j)w_(j−1)α_ij from the actual successive rank-one charts; the fibre of π*Lλ has weight w_rλ. For reduced words, identify the open product of root parameters with C(w) and prove π proper birational onto X(w), isomorphic on that open cell. Prove the subword/Bruhat criterion from the existing Coxeter exchange/deletion floor and these actual maps; do not infer an isomorphism from a mere bijection of fixed points. A proper fibre can be nonreduced even when its underlying set is one point.
4. Construct generalized words with parabolic letters and their longest elements, refinement/unrefinement maps and composites. Prove the needed reduced-word comparisons by finite zigzags of refinements and unrefinements from actual Coxeter braid/length data. Ordinary reduced words121 and212 have no further simple-letter refinements, so no single common word refinement is assumed. The resulting diagrams/correspondences are over the flag; a braid move is not asserted to give an isomorphism of the two word schemes. A parabolic longest element reverses the positive subsystem; it need not fix the other simple roots or negate every simple root individually. Give rank-two braid and nonreduced examples.
5. Compare the complex scheme with the compact model using the actual rank-one group T·SU₂α and its finite central quotient, not an unconditional product SU₂×T^(rank−1). Consume the compact rank-one root representatives from LieGroups. Export full groupoid/functorial chart, bundle and projection comparisons, then compare the typeA finite characteristic maps with TopK's existing normalization. Generic word geometry here does not become an extra premise for TopK's finite CW construction.

## S3. Line bundles, sections and rank-one push-pull

1. Construct L(λ) on the supplied quotient by descent, with its actual linearization, local transition functions and sections. Prove tensor L(λ+μ), dual, pullback, associated-bundle and field/base-change laws. Characterize descent to P− by the Levi pairings and its actual group characters. Prove ampleness for strict positivity on the complementary simple roots and global generation for dominance; a character not in the original lattice is handled only after the specified cover/flag comparison.
2. Construct the regular-function realization ∇λ from the rational representation supplier and the natural section map. Prove its inverse on quotient charts. On the full flag and dominant λ, compare it with the canonical Weyl/induced pair. On X(w), define sections geometrically and identify the dual of restriction with the actual B−-span in Δ(κ), κ=−w₀λ, generated by the extremal vector of weight−wλ. Prove this comparison using the normalized pairing and the actual section-extension theorem of S5. For λ ample on G/P_λ, prove that inclusion of the corresponding geometric Schubert varieties is equivalent to inclusion of these actual extremal-span modules, using the extremal-weight/subword criterion. On walls compare the correct cosets in W/W_λ; a line contracted by the parabolic projection cannot detect distinct full-flag indices. Convert to positive-Borel Demazure modules using pinned contravariant duality/opposition; ordinary dualization alone changes λ.
3. For every integer m, specialize SchemeAndStackFoundations §2.6's projective-space cohomology to H⁰/H¹(P¹,O(m)), and construct its actual rank-one T-weight, functoriality, residue/Serre pairing and zero-case comparisons. Build the relative P¹ pushforward and higher-direct-image comparison on the actual rank-one bundle. Define the integral Demazure Euler operator by finite signed root strings. For dominant string length m=⟨λ,α_i∨⟩≥0 it gives Σ_(j=0)^m e^(λ−jα_i); for m=−1 it is0, and for m≤−2 use the negative H¹ string. Prove its localization identity D_i(eλ)=(eλ−e^(−α_i)e^(s_iλ))/(1−e^(−α_i)).
4. Prove D_i²=D_i, the finite braid relations and word-independence for the required push-pull operators, keeping Euler classes distinct from individual H⁰ outside the vanishing range. For dominant λ, S5 supplies the actual no-higher-cohomology/pullback comparison needed to identify char H⁰(X(w),Lλ)=D_i₁…D_ir(eλ). Prove wall/Levi and longest-word cases, including the full Weyl character formula at characteristic0 and the induced/Weyl formal character in characteristicp. Do not replace a positive-characteristic induced module by a simple module.
5. Construct coordinate maps for the complete linear system and the extremal-line orbit, and prove their scheme-theoretic comparison. Basepoint freeness and injectivity on geometric points do not prove a closed immersion in characteristicp. Own the exact stabilizer/separability and closed-immersion argument for the chosen canonical model; for a general highest-weight representation state the actual stabilizer conditions. Test the SL₂ Frobenius representation, where the orbit map can be purely inseparable.

## S4. Finite-type LS chains, root operators and section indices

1. On a fixed finite root datum build the normal path-crystal category using actual partial maps e_i,f_i:B→Option B, weight, ε_i and φ_i, their partial-inverse/weight/string identities, strict morphisms, components, tensor products with stated tie convention, and transport through actual isomorphisms. The paths below supply the objects; no classification of arbitrary abstract edge-labelled graphs is assumed from rank-one identities alone. For the actual finite root datum construct rational PL paths in X⊗ℝ starting at0 with integral endpoint, their finite rational presentations, equivalence under nondecreasing surjective PL reparametrizations, concatenation, reversal, and equality/normalization APIs. On parametrized representatives define addition of the linear central path t↦tχ for χ annihilating every coroot (equivalently W-fixed); this formula is not an operation on arbitrary reparametrization classes. S4.3 constructs the well-defined central transport on distinguished LS representatives. Remove stationary pieces and merge adjacent positively collinear segments before choosing a rational PL presentation. Prove a normalization/comparison algorithm with its exact rational parameter choices; Euclidean arc lengths need not be rational. The constant path is separate. For an LS path retain its distinguished constant-speed-on-directions presentation with slopes in Wλ and prove uniqueness up to the admitted reparametrizations, including the central/zero-weight cases. Neither all rational paths nor all integral-endpoint paths form a finite crystal.
2. Define running coroot heights H_i(t)=⟨π(t),α_i∨⟩ and the actual raising/lowering operators using record-decreasing/record-increasing segments, equal-height excursions and the final root translation. State the first/last cut-time and tie conventions. Prove presentation/subdivision/reparametrization independence, partial inverses and endpoint changes ±α_i. Before using simplified whole-interval reflections, prove the needed local-minimum integrality on the relevant LS components. Distinguish ambient rational formulas from the normal integral crystal identities ε_i=−min H_i and φ_i=H_i(1)−min H_i.
3. For dominant λ construct the operator component B(λ) of the straight path. Encode it by decreasing cosets τ₁>⋯>τ_s in W/W_λ and rational breaks0=a₀<a₁<⋯<a_s=1. For each internal break a_j require an a_j-chain: a saturated decreasing Bruhat chain of orbit weights ξ₀=τ_jλ,…,ξ_l=τ_(j+1)λ, ξ_h=s_βh ξ_(h−1), β_h positive, ⟨ξ_(h−1),β_h∨⟩<0 and a_j⟨ξ_(h−1),β_h∨⟩ integral. Prove this condition is equivalent to the generated-path definition, with merged repeated directions and the correct coset Bruhat order. Prove finiteness from finite W and bounded break denominators, operator stability, weight/string formulas and a unique highest path. Include λ=0, chamber walls, nontrivial central characters and empty root systems. For a W-fixed character a∈X, let B(μ)^a have weight wt+a and the same root operators and string data; prove it is a normal crystal. For W-fixed χ,σ∈X, construct a strict crystal isomorphism T_{λ,χ}^σ:B(λ)^(σ+χ)→B(λ+χ)^σ. Its underlying path function is independent of σ; σ=0 gives central transport from the weight-twisted B(λ). First prove that an LS class has a unique parametrized representative whose slopes lie in Wλ, after equal adjacent directions are merged; λ=0 uses the constant representative. Apply π(t)↦π(t)+tχ to that representative, so each slope becomes wλ+χ=w(λ+χ). Prove the unchanged simple-coroot heights give the same operator cut times and root actions. Prove the strict-map identity T_{λ,0}^σ=id, the additive composition law T_{λ+χ,ψ}^σ∘T_{λ,χ}^(σ+ψ)=T_{λ,χ+ψ}^σ, and the inverse formula (T_{λ,χ}^σ)^−1=T_{λ+χ,−χ}^(σ+χ), for all central χ,ψ,σ. These offsets make each displayed source, intermediate object and target identical as weighted crystals. Include λ+χ=0, pure central weights and the empty root system. Do not apply the same pointwise formula to an arbitrary representative of the ambient path quotient.
   First export the reflection/order test on the actual quotient W/W_λ. For a positive root β and an orbit weight uλ, negative pairing ⟨uλ,β∨⟩ gives s_βu<u, positive pairing gives s_βu>u, and zero pairing gives the same weight and coset. Compare the minimal quotient representatives, not arbitrary W labels for equal weights. A weak inequality does not give a strict comparison on a chamber wall.

   For comparable u≥v in this quotient define the positive cover bond b_λ(x,y)=−⟨xλ,β∨⟩=⟨yλ,β∨⟩ when x>y is a cover and yλ=s_βxλ, β positive. Prove existence and uniqueness of its positive root label and integrality/strict positivity of the bond. For a saturated chain C from u to v, let d(C) be the gcd of its bonds. Construct the chain-independent interval gcd d_λ(u,v), with the empty-chain convention d_λ(u,u)=0. Give its combinatorial proof before S5: construct maximal-chain connectivity by rank-two intermediate replacements in every parabolic Bruhat interval using a reflection/deletion-order shelling or an explicit chain-sorting induction from RootSystems' exchange/deletion API; then prove equality of the two local bond gcds for each actual rank-two interval with two intermediates. Carry the actual conjugated root/coroot labels and λ through the A₁×A₁, A₂, B₂/C₂ and G₂ calculations, including singular quotient cases. Some quotient rank-two intervals have only one intermediate; no alternative chain is assumed there. These connectivity and arithmetic lemmas are construction targets here, not assumed consequences of Matsumoto or of abstract diamond shape. Gcd associativity then propagates local equality along the chain moves. Extend the same edge-integrality definition of a saturated a-chain to every rational a; actual internal LS breaks still require 0<a<1. For a=p/q in lowest terms with q>0, prove that one saturated a-chain exists iff q divides d_λ(u,v) iff every saturated chain from u to v is an a-chain. Use the native finite-gcd divisibility API and coprime denominator cancellation. Incomparable endpoints have no chain and are outside this comparable-interval gcd API. Keep actual coroots in nonsimply-laced types. Central translation leaves stabilizers and bonds unchanged; zero/pure-central weights and empty root systems give a singleton quotient. This proof uses no Schubert divisor, vanishing or SMT premise. S7 subsequently compares the already-defined bonds with its geometric multiplicities.

   Prove the convexity API for the distinguished LS representative: π(t)∈t·Conv(Wλ), or equivalently π(t)/t∈Conv(Wλ) for 0<t≤1; at t=0 prove π(0)=0 directly. Write π(t) as the finite sum of its elapsed nonnegative segment durations times slopes in Wλ, with total duration t, and apply the native `Convex.sum_mem`, `subset_convexHull` and `convex_convexHull` APIs to the normalized durations. If 0∈Conv(Wλ), deduce π(t)∈Conv(Wλ); in semisimple finite type establish this premise by averaging the orbit. A nonzero pure central λ has orbit {λ} and π(t)=tλ, so the unscaled containment is not asserted for an arbitrary reductive weight. This parametrized formula concerns the distinguished representative; the unparametrized path image has the corresponding representative-independent containment when its premise holds.

4. Construct the generic finite-type character/component theorem, not just its typeA specialization. Prove the root-operator word-distance bound, linked-path subdivision/gluing lemmas and positive-integer stretching required to compare arbitrary dominant rational PL paths of endpoint λ with the straight component. Prove the tensor-component theorem and translated-chamber multiplicity rule, including the actual tensor-factor convention. Derive the formal Weyl/Demazure characters from these finite sets and the rational representation supplier. In characteristic0 compare with simple rational modules; in characteristicp compare the formal Weyl/induced characters, which need not be the simple L(λ) characters; a crystal graph is not a basis on which every actual Lie lowering operator sends a vector to a multiple of one other vector. The adjoint SL₃ two-dimensional zero-weight space is a regression against that false inference.
5. Construct fixed-chain simplices and their face identifications after equal adjacent directions are merged. The finite LS set is the subset satisfying the break-chain integrality conditions; it is not the whole continuum of a simplex. Give the six-vertex S₃ Bruhat-chain complex and its four maximal tetrahedra, distinguishing this realization from the fixed-λ finite path set. The comparison with TPC H8 is S9.5, after the generic root-data construction. No concrete tableau theorem is a premise of this generic component theorem.
6. After S5 and the divisor calculation S7.1–2, construct geometric path sections in all tensor degrees and their defining-chain API. Use the Schubert poset, extremal sections and the actual codimension-one valuations with Chevalley bonds to build the chain-indexed LS lattice/filtration. Prove the normalized leaves are one-dimensional and identify their indices and character count with S4.1–5. Construct sections lifting those leaves and prove triangular spanning/independence, reduced-word compatibility, weight and restriction support. This is the geometric finite-flag filtration application, not an assumed generic quantum Frobenius/canonical-basis theorem. If a distribution/contraction model is also used for a comparison, prove its exact integral/denominator and specialization maps separately; it is not needed as an unstated premise of this geometric construction.
7. Print the opposition/dual dictionary. Set κ=−w₀λ and θ(w)=w₀ww₀. Conjugating G/B− to G/B+ by the chosen longest representative sends X(w) to the positive-Borel X(θ(w)) and pulls the positive-Borel line with fibre−κ back to L(λ), with the actual linearization comparison. In the positive-Borel convention, LS path sections indexed by π∈B(κ) have weight−π(1) and survive on X(θ(w)) exactly when their initial direction is≤θ(w). Transporting gives weight−w₀π(1) on the chosen X(w). For partial flags use the actual minimal-coset projection W/W_P→W/W_λ before making this comparison; do not compare indices in different stabilizer quotients as though they were equal. Prove the terminal-direction formulation in the notebooks by explicit opposition/reversal, rather than equating their untransported symbols w. The point and longest-word tests must recover the single fibre λ and the full ∇λ.

## S5. Splitting, normality, restriction and cohomology

1. In characteristicp>0 construct the actual Frobenius module F_*O and O-linear Cartier trace on smooth root-coordinate charts. Identify Hom_O(F_*O,O) with F_*(ω^(1−p)) using the specified duality/pullback convention. On affine coordinates the normalized trace selects exponents congruent to p−1 modulo p; it is p^−1-linear over a perfect field and sends the chosen normalized anticanonical section to1. Prove independence under changes of root charts and line trivializations. Construct the section from the algebraic Steinberg module and its normalized extremal pairing before using geometric vanishing or SMT.
2. Prove the resulting splitting is compatible with every standard and opposite Schubert ideal by actual B-canonical/rank-one root-subgroup calculations and word induction. For an ideal I the required statement is σ(F_*I)⊂I, not just reducedness of its zero set. Prove closure under finite sums/intersections of compatible ideals, then conclude scheme-theoretic intersections and reduced unions in this family are reduced in positive characteristic. Item6 constructs their characteristic-zero mixed-family comparison. The rank-one, point, empty and whole-flag ideals are explicit tests. Do not infer arbitrary Richardson transversality from this reducedness statement.
3. Carry out the construction on the derived simply connected cover, where ρ and the Steinberg weight exist in the actual character lattice. Prove the induced map of flags is an fpqc quotient isomorphism and transport the splitting/ideal identities, even when the central kernel is nonreduced. No étale-cover or differential-isomorphism assumption is substituted for that quotient argument. Keep actual line linearizations when passing between isogeny types.
4. Prove normality of each Schubert variety. Construct geometrically connected fibres of the reduced-word map by the actual rank-one gallery/partial-flag induction, with the closed fibre pieces and their intersections; do not infer connectedness merely from a fixed-point count. Apply proper Stein factorization and finite normalization to prove geometric unibranchness. In characteristicp prove the needed weak-normality consequence of Frobenius purity: principal ideals are Frobenius closed, and a finite universal-homeomorphic normalization in the total fraction ring must equal the original ring. Combine this with the constructed unibranchness; splitting alone is not a normality theorem. Item6 constructs the corresponding characteristic-zero/integral comparison from the actual identified integral fibres, before using the characteristic-zero conclusions of this paragraph. Once X(w) is normal, prove π_*O_Z=O_X(w) by the actual proper birational/Stein comparison. Reduced unions need not be normal; retain their actual ideal exact sequences.
5. In positive characteristic, for dominant λ first prove H^q(Z,π*Lλ)=0 for q>0 by the actual rank-one/word splitting calculation: construct a stable splitting along a boundary-supported ample divisor, prove the required iterated splitting and tensor/pushforward maps, and combine it with nefness and Serre vanishing. Construct the ample combination with positive coefficients on every word-boundary component from the iterated projective bundles; compatibility with the reduced boundary alone is not sufficient. Deduce R^qπ_*O_Z=0 for q>0 by twisting with all sufficiently high powers of an ample line on X(w), using projection formula, coherent Leray and Serre to detect the vanishing direct-image sheaves. This deduction does not use the same direct-image theorem as its premise. Then prove H^q(X(w),Lλ)=0, restriction surjectivity from the full flag, and the same statements for finite reduced unions of Schubert varieties for one fixed Borel (or for their common opposite-Borel transforms). Compare H⁰ on X(w) and Z using the already proved π_*O statement. For ample λ use Frobenius iteration and Serre vanishing with the actual ideal sequence. For merely nef/dominant λ, including0, prove the needed parabolic projection/fibre O-cohomology and pullback comparison; ampleness is not silently assumed. Include empty unions, H⁰(point,L)=its fibre and H⁰(O_Y)=k for a nonempty common-Borel Schubert union Y. Mixed standard/opposite compatible unions retain S5.2's ideal and reducedness conclusions, but their section comparison uses the actual ideal/cohomology exact sequence and proved vanishing hypotheses. It does not assert dominant restriction surjectivity or a common-Borel path basis for every mixed union: in P¹, Y={0,∞} and λ=0 give H⁰(O_Y)=k² and the diagonal restriction k→k², with H¹(I_Y)=k.
6. Supply the following integral inputs to ShimuraData §3.15's single integral-Schubert comparison, before consuming its completed base-change theorem. Start with its early smooth projective integral full flag F/ℤ and each smooth iterated-P¹ word model Z/ℤ, their multiplication map π:Z→F and the open root-cell isomorphism; the schematic Schubert closure is not yet assumed to commute with base change. Choose λ in the actual character lattice strictly positive off the chosen parabolic, so Lλ is ample on that flag, and set A_m=H⁰(Z,π*Lλ^(⊗m)), m≥0. First prove the characteristic-p vanishing of these actual nef pullback bundles for every m≥0 by item5's stable ample-divisor splitting construction, not by splitting alone. Do the same on F. Construct derived cohomology/base-change on these smooth proper integral sources using their flat bundles, actual coherent complexes and the finite affine Čech comparison. Prove the complexes are perfect over ℤ; the positive-characteristic fibre vanishing and constant Euler ranks then give finite-free H⁰, no higher cohomology, and the required arbitrary-ring base-change maps. This step uses neither flatness nor fibre identification of a putative Schubert closure.

   Construct every restriction map H⁰(F,Lλ^(⊗m))→A_m, including m=0. Their reductions at each prime are surjective by the independently proved characteristic-p Schubert restriction and word direct-image comparisons. The cokernel is a finite ℤ-module; its reduction at every prime is zero, so it is zero. Prove this argument with the actual H⁰ base-change identifications. The resulting graded algebra map is surjective in every degree. Prove finite generation of A via its finite coherent module over the ample section algebra on F, construct Proj A and its closed immersion into F, and compare the morphism from Z given by the pulled-back ample line. Identify its scheme-theoretic image with the schematic root-cell closure using the open-cell isomorphism and the actual homogeneous kernel, not just an equality of geometric point sets. Prove A is normal from the normal integral Z and its invertible sheaf, using the intersection-of-local-valuation or total-line-space argument; hence its Proj is normal. This establishes the characteristic-zero normality needed in item4, as well as the integral fibre/section comparison. Treat partial flags with an actual ample character of their lattice and the specified projection, not an assumed fundamental weight outside that lattice.

   For the mixed standard/opposite comparison, construct its additional homological inputs on those same supplied loci. The word boundary has affine-space complement; prove the stable ample-boundary splitting plus strong F-regularity of that regular affine complement gives global F-regularity. Push the actual Frobenius retractions through the proper word map with π_*O=O to obtain global F-regularity of field Schubert images, and prove their Cohen–Macaulay consequence at the stated F-finite/perfect-field hypotheses. Use the identified flat integral models and the precise openness/spreading comparison to obtain the characteristic-zero Cohen–Macaulay statement. Prove proper standard/opposite intersection and Tor independence in the smooth flag by the actual root-cell dimension and Cohen–Macaulay intersection calculation; construct the corresponding flat integral Richardson model and its fibre ideals before spreading reducedness to characteristic0. Finite compatible mixed unions/intersections require their actual ideal-lattice and exact-sequence/base-change induction. Neither flatness of an intersection nor base change of an arbitrary ideal intersection is assumed for free. These constructions give reducedness/ideal comparisons, not the false all-dominant mixed-union section-surjectivity theorem.

   Export the resulting finite-free section/restriction modules, saturated homogeneous ideals and arbitrary-base-change maps at the existing integral owner. For common-Borel unions use their actual ideal/section exact sequences and the field restriction/vanishing theorem to construct and compare the integral union; do not normalize a reducible union or infer fibrewise reducedness from saturation alone. Establish the characteristic-zero and arbitrary-field versions of items4–5 on the identified schemes; descend geometric conclusions from perfect algebraic closures by their actual faithfully flat comparison. The final relative split Schubert bundle and specified twisted descent applications consume this completed Shimura comparison. No late §3.15 result is a premise for the smooth-source H⁰ argument that supplies its proof.

## S6. Coordinate rings and all-degree standard monomials

1. Construct the natural graded map from the homogeneous coordinate algebra of a chosen Plücker/highest-line embedding to ⊕_(m≥0)H⁰(X,L^(⊗m)), with actual multiplication and restriction maps. Keep degree0, reduced unions and all m≥0. Establish its kernel, saturation, surjectivity in every degree and the section/Proj comparison; a projective embedding alone does not identify these two graded rings. For non-ample dominant λ compare with the contracted partial flag image instead of falsely asserting an embedding of the full flag.
2. For typeA Grassmannians and flags consume GG's polynomial maximal-minor/flag-minor straightening bases, certified orders and initial algebra. Identify every coordinate minor with the actual determinant universal section and its dual T-weight. Prove the complete homogeneous ideal and geometric Schubert restriction ideal, standard-on-X(w) basis and Schubert-union support criterion. The scheme ideal, section restriction and multigrading are part of this comparison; algebraic pointwise vanishing over a field is not the full relative scheme proof.
3. Give the source's minuscule induction in full on G/P_λ with its actual ample minuscule polarization; full-flag pullbacks use the already proved parabolic cohomology comparison before any Serre argument. The extremal section divisor is a reduced lower-dimensional Schubert union; multiplication gives 0→O_X(m−1)→O_X(m)→O_D(m)→0. Prove the union intersection ideals are reduced by S5 before using Mayer–Vietoris. Prove both the higher-cohomology descent from actual Serre vanishing and the H¹/restriction/dimension step by the exact maps and standard sections. An unexplained dimension count cannot assume the restriction surjectivity being proved. The m=0 boundary case and all nonnegative twists stay included.
4. For arbitrary finite root type and dominant λ use S4's geometric path sections and defining chains. For an ordered tuple of dominant shapes, a standard product is one whose successive path directions admit a common decreasing lift in W with the specified stabilizer quotients; build its canonical minimal/maximal defining-chain tests and equivalence with the path concatenation criterion. Prove spanning by a well-founded triangular straightening and independence by the normalized filtration. Pairwise weak standardness is not substituted for a common defining chain. Export every multidegree, the single-shape Veronese rings, Demazure character and restriction formulas, and a basis on a finite reduced common-Borel Schubert union by sections standard on at least one component.
5. Give the compatible ideal and multiplication API: for the fixed-Borel Schubert family, basis vectors failing its support test span the vanishing ideal, ideal intersections/sums match its reduced unions/intersections, and restriction maps are coordinate projections in that compatible basis. Mixed standard/opposite intersections keep their actual S5.2 ideal identities; any section basis or surjectivity comparison there requires its own proved restriction/vanishing hypothesis, not this fixed-Borel assertion. For each normal Schubert variety and the full flag, prove projective normality and normal generation in the ample/actual embedding lane using these all-degree maps and integrality, not by defining the section ring to be the coordinate ring. Reduced unions retain their all-degree coordinate/section comparison and normal-generation theorem, but are not asserted normal without an actual normality hypothesis; a reducible union can fail normality along its intersections. For the source's toric degeneration consume GG's Rees/SAGBI comparison and identify the initial algebra; an associated graded toric algebra does not make the original flag algebra toric. General path-based degenerations may be unions of toric components and must keep their actual fan/valuation hypotheses.

## S7. Chevalley divisors and incidence multiplicities

1. Choose the normalized extremal section nonzero on C(w). For v=w s_β with β>0 and ℓ(v)=ℓ(w)−1, prove that its divisor along X(v) has coefficient ⟨λ,β∨⟩. Show support, valuation at the codimension-one local ring, actual rank-one transverse parameter and compatibility with the chosen left/right quotient and weight. Equivalently express it using the transported extremal weight at v; prove the equality of these formulas rather than guessing the source's β side.
2. Distinguish the effective Cartier divisor of the full section from individual Schubert boundary Weil divisors, which need not be Cartier. In the minuscule lane all nonzero coefficients are1 and recover the reduced hyperplane/Pieri boundary used in S6. For general λ retain multiplicities in the cycle/class formula and use an actual coherent boundary filtration; do not tensor a fictitious invertible O(−X(v)) for every component. Include zero coefficients, λ=0 and the empty divisor.
3. Construct the flag-specific Chow-to-integral-cohomology comparison from the generic algebraic cycles/localization/refined-intersection API of SchemeAndStackFoundations §5 and the existing LieGroups flag CW/cohomology carrier. Prove the affine-cell filtration spans the Chow group by Schubert closures, and prove independence using opposite-cycle intersection and degree. Send a Schubert generator to the ambient homology class obtained by pushing the complex-oriented fundamental class of its reduced-word resolution; identify its coefficient1 on the open cell and the corresponding CW generator. Prove the pairing, divisor/Chern class and parabolic projection laws, yielding the flag comparison and its ring compatibility. This is an application-specific construction, not a claim that AVB's proposed general Betti cycle-class programme already exists. Compare the source Chevalley–Pieri degree calculation, including positivity and the actual multiplicities. General singular intersections retain their refined-intersection hypotheses; the transverse complex Grassmannian point-count comparison of TPC remains at its existing owner.

## S8. Algebraic flag K-rings and characteristic maps

1. Consume the generic coherent/exact Grothendieck and perfect-complex owners. On smooth projective G/P verify the hypotheses of the supplied regular/quasi-projective finite-resolution comparison and identify its vector-bundle K₀≃coherent G₀ maps. Do not assert this equality on singular Schubert varieties. Their structure sheaves define ambient classes by proper coherent pushforward and the regular ambient resolution. Export the supplied products, duals, tensor twists, derived proper pushforwards and Tor/flat/perfect-range pullbacks on these flag objects, and construct their actual equivariant character/Schubert comparisons.
2. Construct c:ℤ[X]=R(T)→K₀(G/B−), eλ↦[Lλ], its augmentation, restriction and Weyl comparisons. If the derived group of G is simply connected, prove the integral presentation R(T)⊗_(R(G))ℤ≅K₀(G/B−), with ℤ carrying dimension augmentation. After identifying the character image with R(T)^W, the ideal is generated by f−ε(f), f invariant; it is not the ideal generated by the whole invariant ring, which contains1. Prove the integral basis/rank and actual induction/restriction/projective-bundle or Schubert-filtration comparisons establishing surjectivity and kernel.
3. In the same simply-connected-derived range construct K₀^T(G/B−)≅R(T)⊗_(R(G))R(T), specify which factor is the base character and which gives associated lines, and compare its augmentation to item2. For a partial flag use R(P−)=R(L) and the corresponding character presentation, with the actual Levi Weyl subgroup and bundle comparison. Prove the finite free/basis statements needed for these tensors; do not assume rational invariants imply their integral version.
4. For a general isogeny type retain c and its actual integral image. Use the derived simply connected cover's flag isomorphism to present ordinary K₀ with its larger available line lattice, proving compatibility and independence of the displayed choice. Rational comparison from the original lattice is separate. For PGL₂, its flag is P¹ but the root-character line is O(2); its image misses a primitive integral degree-one class. This is a required counterexample to an unqualified integral original-lattice presentation. Keep central torus characters and augmentation relations explicit.
5. Construct push-pull Demazure operators on these K-groups and compare them with S3's integral character operators, including word/Levi laws. Compute Schubert structure-sheaf and ideal-sheaf bases and pairings using the actual coherent duality and Euler pushforward. Distinguish those bases from line-bundle characters, ordinary cohomology classes and quantum/K-theoretic puzzle rules; none of those extra rules follows from the ordinary TPC puzzle theorem.

## S9. Complex realization, fixed-map traces and existing comparisons

1. Over ℂ compare the actual algebraic flag, Schubert and word schemes with their analytic/topological carriers, actions, tautological/associated bundles and maps. Use analytification and projective coherent GAGA at the actual objects. Identify the compact homogeneous flag model and its rank-one factors through LieGroups, and compare typeA subspace/quotient/Plücker models with MC0G, TopK and TPC. Ordinary relative forms and field changes retain their actual algebraic hypotheses; the complex comparison is not their substitute.

   Construct the downstream smooth-carrier comparison with [FormalConnectionsOpers F2.4](https://github.com/TauCetiProject/TauCetiRoadmap/blob/c9f4ac867e9f22bbf6f71d6f5c4a361ab9f0e622/TauCetiRoadmap/FormalConnectionsOpers/README.md): for every separated smooth finite-type ℂ-scheme, work on its clopen strata of constant differential rank and identify its independently constructed manifold with the smooth carrier of ComplexComparisonPartII's same analytification. Prove the identity on ℂ-points is biholomorphic and compares the affine Euclidean topology, standard-étale/implicit charts, maximal atlases, holomorphic structure sheaves, tangent/Kähler-differential maps and finite-locally-free bundles, with restriction and morphism naturality. For varying differential dimensions use the clopen dimension strata. Specialize these comparisons to the actual flag and word objects and their actions. This comparison follows both independent constructions and is not a premise of the generic analytic-space, coherent or GAGA theory.
2. Consume the following map-level holomorphic Lefschetz interface from IndexTheory's fixed-map trace construction. If f:M→M is holomorphic on a compact Hausdorff second-countable finite-dimensional boundaryless complex manifold, all fixed points are isolated with det_ℂ(1−df_x)≠0, E is a holomorphic vector bundle and φ:f*E→E is a specified holomorphic bundle map, then the alternating trace of φ∘f* on coherent/Dolbeault cohomology is Σ_x tr(φ_x)/det_ℂ(1−df_x). The generic construction owns positive-time heat smoothing/trace class, cohomological supertrace and the local graph/diagonal calculation; arbitrary pullback is not assumed L²-bounded and f need not be an isometry.
3. For section action (g·s)(x)=g_Es(g^−1x), apply that formula with f=g^−1 and φ=g_E on pulled-back fibres. The differential pullback has the eigenvalues of df, not automatically their inverses. At e of G/B−, tangent weight+α for t gives df for t^−1 with weight−α; the denominator is1−e^(−α) and the fibre contributes eλ. For the word model use S2's actual subword weights to prove χ_T(Z,π*Lλ)=Σ_ε e^(w_rλ)/∏_j(1−e^(−β_j)) in the stated localization. Prove for generic t that its fixed points are exactly those sections and every denominator is nonzero. A finite whole-torus fixed locus does not make the expression defined for every t.
4. Sum the rank-one contributions to obtain S3's exact Demazure recurrence and compare with the independently constructed algebraic Euler/vanishing argument. Prove polynomial cancellation of the localization formula and its integral character meaning. The full longest-word result recovers the Weyl character formula in the stated characteristic-zero analytic lane. It does not prove positive-characteristic geometric vanishing by itself.


5. Specialize the generic S4.1–5 engine to the full typeA root datum, including the GLₙ central lattice and actual SLₙ restriction, and construct the comparison with TPC H8's `RationalPath`, straight-path component, column paths and row/semistandard-tableau paths. Map finite presentations and their reparametrization equivalences, endpoints, concatenations and each raising/lowering operator; prove partial inverses, string lengths, highest elements and tensor-component multiplicities are transported. Rational break/scaling and opposite/dual conventions are part of the maps, not an equality inferred from the same character count. Use these isomorphisms to give TPC's typeA path statements as specializations of the single generic engine while retaining its concrete bounded SSYT, GT, row/column, hive and algorithmic recurrences and their independent constructions. The generic engine does not consume these specializations; this comparison adds no return edge from TPC to S4.1–5. The finite complex Grassmannian CW/cohomology/puzzle targets remain at their existing TopK/TPC owners.


6. Compare the resulting integral full-flag CW classes with the Schubert Chow/degree comparison of S7.3; no new Grassmannian characteristic-map owner is introduced. As the compact-normalizer application, for compact connected K and maximal torus T use the actual free finite covering K/T→K/N(T) with deck W, AlgebraicTopology Stage5.3's singular-cohomology transfer over ℚ and the flag's even-cell Euler characteristic |W|. Prove H⁰(K/N(T);ℚ)=ℚ and all positive-degree groups vanish. Over ℤ averaging is unavailable: for SU₂ identify K/N(T)=ℝP² and compute H⁰=ℤ, H²=ℤ/2 and other positive cohomology zero. The generic compact BG/BT invariant/Chern–Weil comparison remains at CharacteristicClasses C5; the source's invariant-degree/Coxeter-exponent and exterior-cohomology formulas require that owner's actual invariant-theory/transgression interfaces and are not inferred from this rational-normalizer calculation.

7. Construct the compact moment-Morse comparison with [HamiltonianReduction R2a](https://github.com/TauCetiProject/TauCetiRoadmap/pull/1049) on its SAME homogeneous coadjoint orbit. For compact connected K, a real dominant μ∈t and H=K_μ, use LieGroups Layer8 and S9.1 to identify K/H with the analytic partial flag G/P− whose Levi roots vanish on μ. No integrality or prequantization of μ is required for this diffeomorphism; an associated line still requires the actual character lattice. Compare the inclusion derivative, root-plane coordinates, T action and the actual KKS form. At p∈O_μ∩t construct the invariant complex structure by J_flag|E_α=sign(α(p))J_α on the tangent planes α(p)≠0 and prove that it is the integrable structure transported from this flag. Then ω(v,J_flag w)=B(v,w)/|α(p)| on those planes. Extend the equality equivariantly and construct the positive K-invariant Kähler metric, rather than assume the arbitrary metric used for the Morse chart has this property.

   For regular X in the negative chamber, prove grad(f_X)=J_flag X_O and the complete negative-gradient flow is the complexified action exp(−itX). Construct its coordinate action in the actual root-subgroup charts and prove its unstable manifolds are the standard B− cells C(w), of complex dimension ℓ(w), with backward limit wμ; its stable manifolds are the opposite cells. The compact flow model is being compared to those root-coordinate cells, not used to define the algebraic Schubert schemes. For positive X, retain index(f_X)=dim(O_μ)−2ℓ(w) and use −f_X for the same index-2ℓ(w) model. Zero stabilizer roots are omitted on partial flags, and central components change only the height constant. Prove the convergence/limit and cell-coordinate statements and the parabolic projection when μ lies on a wall, with X still regular in the chosen chamber, not just agreement of cell counts or characters. If instead X lies on a wall, use R2a's centralizer-orbit Morse–Bott strata and their negative normal bundles; do not assert the same isolated-critical-point/Bruhat-cell statement for that height. An arbitrary invariant metric need not have these Bruhat unstable manifolds.

   Compare the MorseTheory M2 negative-handle characteristic maps and the existing LieGroups/TopK flag characteristic maps by actual relative-pair maps, root-coordinate orientations and compatible finite filtrations. Distinct-value height choices and simultaneous critical levels must give the same labelled comparison under the constructed handle-continuation maps. Use the complex orientation of each negative root-plane sum; check independence from torus choices of normalizer representatives, whose root-plane rotations have positive real determinant. Compare the resulting integral cellular/singular generators with S7.3's pushed-forward resolution classes and their coefficient1 on the open cells, including the boundary/attaching-map comparison. A singular Schubert closure need not be a ball and is not declared to be the image of an embedded Morse disc. Export the finite even-cell Poincaré polynomial Σ_(w∈W/W_μ)q^(2ℓ(w)), torsion-free integral groups and |W/W_μ| Euler characteristic with these actual comparison maps. Multiplicative Schubert/cohomology formulas retain their existing S7/TopK/TPC construction; they do not follow merely from a vanishing cellular differential.

   This is a later application of HR R2a, the independent compact LieGroups foundations, the existing root-cell/complex-realization maps and MorseTheory's generic attachments. It is not a prerequisite for LieGroups Layer6 conjugacy, S4's generic LS engine, ReflectionInvariantTheory §§1–3 polynomiality, or CharacteristicClasses C5/C5a Borel/compact cohomology. Retain S9.6's normalizer/Weyl transfer and SU₂/ℝP² test at that single owner.

## Acceptance calculations and source-specific interfaces

- TypeA: the six minors of an actual two-frame in k⁴, their signs/change of frame, the Gr(2,4) Plücker quadric, coordinate-chart inverse, Schubert ideals, dimensions and all-degree bases. Include signed determinant-one SL representatives, the full block-monomial normalizer/stabilizer dictionary, the symplectic antidiagonal-form block parabolic and the source root-action/contragredient wedge signs. Include zero subspace/full subspace and the determinant line; claims that a tautological subbundle has no global sections need the correct nontrivial-rank range.
- For 1≤r≤dimV and nonzero θ∈ΛʳV, prove decomposability iff (ι_ηθ)∧θ=0 for every η∈Λ^(r−1)V*, by constructing the vector-valued contraction-image subspace and its basis. Include θ=0 as the affine-cone origin separately. For r=0 use Λ⁰V=k and its unique projective point, without that vector-valued contraction criterion; for r>dimV the exterior power is zero and its projectivization is empty. Compare this basis-free criterion with the already owned polynomial Plücker ideal and the actual universal image, retaining the exterior-algebra signs in every characteristic.
- For the rank-d tautological subbundle on the full flag over an algebraically closed field, prove H⁰(S_d)=0 when 0<d<N, and explicitly handle d=0 and d=N. Identify H⁰(S_d*) and its determinant sections through the actual induced-module/evaluation map. Multiplication of fundamental section spaces gives the Cartan-product quotient, not an unqualified isomorphism with the entire tensor product.
- For nonempty integral finite-type projective X and nonempty open U, compare the homogeneous degree-zero fraction field with the actual function field k(X)=k(U), and dimX=trdeg_k k(X) over a field. The local regular-section ring O(U) is not itself that fraction field; give the homogeneous affine-chart/homogenization maps and apply them to the Schubert cell dimension calculations.
- The complete flag→incidence product→Plücker→positive Veronese→Segre composite has its actual closed immersion and line pullback. Zero exponents contract flag directions and are not automatically embeddings; strictly positive relevant coefficients give the intended ample partial/full-flag model.
- For the finite chain/convexity API include a single cover, an equal endpoint with gcd0, incomparable endpoints, the arithmetic a-chain criterion with denominator1 (outside the internal LS break range), a chamber-wall stabilizer, nonsimply-laced rank-two bonds, and a pure central straight path. The wall case must give equality of cosets when the coroot pairing vanishes; the pure central case must retain the scaled convexity statement.
- For central transport in GL₂, take π(t)=(t,0), χ=(1,1) and the increasing rational PL reparametrization φ(t)=t/2 for t≤1/2 and φ(t)=3t/2−1/2 for t≥1/2. The paths π and π∘φ represent the same class. The shifted π lies on x=2y, whereas (π∘φ)(1/2)+(1/2)χ=(3/4,1/2) does not. Verify this regression against an ambient quotient-level central action, and recover B(λ)→B(λ+χ) through the canonical LS representative instead.
- For SL₂ use P¹, O(m), the m+1 character string and degree m. In characteristicp compare induced, Weyl and simple modules, including the pth symmetric-power/Frobenius stabilizer example and the n=4,p=3 binomial cyclicity counterexample. A unique highest line need not imply a module is generated by that line.
- For SL₃ use word121: construct its map onto the full flag, the exceptional P¹ locus and its actual P¹-bundle, proving triviality by a transition calculation rather than from the mere existence of a section. Give all eight fixed subwords and the four-tetrahedron Bruhat-chain model. Compare the two zero-weight adjoint path indices without inventing unread diagram labels.
- For general root type retain parabolic projection, minuscule versus nonminuscule divisor multiplicities, all-degree Demazure/path bases, scheme-theoretic reduced union/intersection and the degree/Chevalley formula. Kac–Moody and general quantum-group/canonical-basis statements are separate source programmes; the finite-type engine here is not a proof of them. Integral divided-power and rational-module source assertions are consumed from the exact RG owner, not silently reproved by geometric characteristic-zero arguments.

Proof routes include the original Knutson Bott–Samelson/Demazure handouts and algebraic-varieties notes for these applications; Littelmann's *Paths and root operators in representation theory* for the finite path/linking construction; Lakshmibai–Littelmann–Magyar's *Standard monomial theory for Bott–Samelson varieties* for the geometric path-section and all-degree comparison; and the root-coordinate/splitting/coherent constructions stated in S5. For the finite saturated-chain bond theorem use Chirivì–Fang–Littelmann, [*LS algebras, valuations and Schubert varieties*, Definition2.1 and §7.1](https://arxiv.org/html/2203.12992), in its dominant-weight/minimal-parabolic-coset setting; construct its chain-independent gcd through the combinatorial targets of S4.3. Each cited generic foundation retains its single owner.