Differential cohomology and connective differential K of B-nabla-G

Local trivializations of the geometric bundle identify the vertical Sobolev spaces with fixed Hilbert spaces. A first-order operator whose smooth vertical coefficients vary continuously varies in the bounded $H^1\to L^2$ norm. The same assertion holds with extra compact parameters and after twisting by a vertically smooth Hermitian coefficient bundle. Uniform elliptic constants on each compact parameter set follow from finitely many local elliptic estimates, absorbing sufficiently small coefficient perturbations in the left side. Thus these are continuous families of bounded Fredholm maps, not operators whose domains are left unspecified. Transition maps between Sobolev trivializations and their inverses are bounded and continuous; hence the kernels constructed below glue as vector bundles.

Let V be finite dimensional. A natural q-form operation assigns to each smooth manifold M and each V-valued one-form alpha on M a smooth q-form ${\omega }_M(\alpha )$, commuting with all smooth pullbacks. For $q=0$, restriction to each point shows this operation is constant, independent of alpha. For $q>0$ the value at p depends only on the q-jet of alpha at p. Locality follows by restriction to open neighborhoods. If a change in alpha is $f_0\mathrm{.}\mathrm{.}\mathrm{.}{f}_q$ beta with every $f_i(p)=0$, introduce q+1 auxiliary coordinates $t_0,\mathrm{.}\mathrm{.}\mathrm{.},t_q$ and the universal one-form $\alpha +t_0\mathrm{.}\mathrm{.}\mathrm{.}{t}_q$ beta on M times $R^{q+1}$. At (p,0), any q-tuple of coordinate tangent vectors misses at least one of the q+1 auxiliary directions. Restricting to that coordinate hyperplane makes the added term identically zero, so the two output q-forms agree on the tuple. Multilinearity gives equality of the two q-forms at (p,0). Pull back along the graph of $(f_0,\mathrm{.}\mathrm{.}\mathrm{.},f_q)$. A general one-form difference with vanishing q-jet is a FINITE sum of terms of this kind, by the integral Taylor remainder in local coordinates (monomials of degree q+1 times smooth coefficient functions). Apply the argument successively to each summand. Thus the q-jet assertion holds without an assumed continuity or differential-operator theorem. The resulting map on the finite-dimensional jet space is smooth. To prove this, put the Taylor-polynomial one-form with its coefficients as parameters on $J^q$ times $R^n$, with only coordinate dx-components. The assigned output is an ordinary smooth q-form on this finite-dimensional manifold. Evaluation on the coordinate tangent vectors at $x=0$ gives the jet-value map as a smooth function of the jet parameters. Naturality under each parameter inclusion identifies this smooth map with the one just constructed. The jet component with j derivatives has scaling weight j+1 under coordinate dilations, whereas the output has weight q. A smooth function F satisfying F(lambda J_0,lambda^2 J_1,...,lambda^(q+1) J_q)=lambda^q F(J_0,...,J_q) is a weighted-homogeneous polynomial of weighted degree q. Expand F at the origin through ordinary Taylor degree q. The remainder on the scaled arguments is $O({\lambda }^{q+1})$; divide by ${\lambda }^q$ and let lambda tend to zero. Coefficients of lower weighted powers must vanish, and only weight-q Taylor monomials survive. This proves polynomial dependence from smoothness and scaling, not from an unproved theorem about arbitrary set transformations. Naturality under linear coordinate maps now forces these polynomials to be formed only from alpha and d alpha. Here is the needed invariant tensor argument. An equivariant map from N covector slots to an alternating q-form vanishes unless $N=q$, by scalar dilations. For $N=q$, use independent diagonal changes of basis: a tensor basis vector with a repeated index cannot map to an alternating basis vector, since their diagonal weights differ. For distinct indices, permutation matrices force all coefficients to differ by precisely the alternating sign. Thus the map is a scalar multiple of complete antisymmetrization. Applying this separately to the V-label coefficients proves that every $j\ge 2$ jet term vanishes, since two of its derivative covector slots are symmetric. For $j=1$ it factors through the antisymmetric part d alpha. Zeroth jets give alpha. Conversely all polynomial combinations of alpha and d alpha are natural. Their coefficients are exterior-alternating in degree1 factors and symmetric in degree2 factors. Injectivity is seen by taking sufficiently many coordinate directions, setting independent components of alpha and d alpha, and evaluating on the corresponding distinct coordinate vectors. Consequently the algebra of natural forms is exactly Lambda(V*) tensor Sym(V*), deg alpha=1, deg d alpha=2, d(alpha)=d alpha, d(d alpha)=0. This is the Koszul algebra, with its stated differential, proved here at the level of actual natural operations.

Write $D_b:H_b^{+}\to H_b^{-}$ for such a Fredholm family, temporarily suppressing the additional parameters. At a point choose finitely many vectors projecting onto the cokernel, extend them to local sections, and multiply by a partition of unity. On a compact base finitely many resulting global sections $j_i$ suffice. Surjectivity is open: if $S_0$ is a surjection and $Q_0$ a bounded right inverse, then $S{Q}_0=1+(S-S_0)Q_0$ is invertible when the last term has norm less than one; its Neumann series gives a right inverse. Consequently $S_I=D+j_I:H^{+}\oplus {\mathbb{C}}^I⟶H^{-}$ is surjective everywhere for a suitable finite list. Importantly, ${\mathbb{C}}^I$ is the free vector space on a finite set of formal labels, one for each chosen section. It is not the linear span of their values or the span of the sections. Labels are retained when sections vanish, become dependent, or are pulled back. This distinction prevents dimension loss under pullback. Its kernel $K_I$ is finite-dimensional. Locally the continuous projections $1-Q_I{S}_I$ identify nearby kernels, so $K_I$ is a finite-rank vector bundle. Its index representative is $K_I-{\underset{‾}{\mathbb{C}}}^I$.

Set $V=g$. In the one-form algebra replace d alpha by the curvature variable $F=d$ alpha+(1/2)[alpha,alpha]. This is an invertible triangular change of polynomial generators. For the gauge action alpha mapsto $\mathrm{Ad}(g^{-1})\alpha +g^{-1}dg$, direct differentiation gives F mapsto $\mathrm{Ad}(g^{-1})F$. Pull back along G times M and contract with a tangent vector xi in the G direction at the identity. The resulting contractions satisfy $i_{\xi }$ alpha=xi and $i_{\xi }$ $F=0$. Therefore horizontality for all xi kills precisely the exterior alpha generators: in a basis, successive contractions detect every nonzero positive exterior-degree coefficient. A horizontal element is a polynomial in F alone. Invariance under constant g in ALL components of G says exactly that its coefficient polynomial lies in Sym(g*)^G. Conversely invariant polynomials in F satisfy the full gauge equalizer. The Bianchi identity dF=[F,alpha] makes every such invariant polynomial closed, by infinitesimal invariance. Hence the basic natural-form complex of $B_{\mathrm{\nabla }}$ G is I*(g*)=Sym(g*)^G, with twice polynomial grading and zero differential, for every finite-dimensional Lie group G, without a compactness assumption. The passage to derived maps into the sheaf of closed forms introduces no hidden higher homotopy. For a closed-form sheaf P, regarded as a discrete simplicial sheaf, maps QX times ${\mathrm{\Delta }}^n$ to P factor through ${\pi }_0(QX)$, because ${\mathrm{\Delta }}^n$ is connected. Thus the derived mapping space is discrete, with set of components equal to sheaf maps from the sheafification of ${\pi }_0$ X to P. For an action-groupoid presentation these are exactly the gauge equalizer just computed. Sheafification does not change these maps because P is a sheaf. This proves the actual closed-basic target on the stack, rather than presupposing descent of an underived Hom. For a finite-type graded coefficient algebra $A_C$, apply the calculation degree by degree. On each finite-dimensional test manifold there are only finitely many contributing form degrees, so ordinary and completed coefficient tensor products coincide there. On the universal stack the compatible values form the levelwise inverse limit, giving P_A^q := I_A^q(g*) = product_(k>=0) I^(2k)(g*) tensor (A_C)_(2k-q). The projection towers are surjective degreewise. Their inverse limits therefore have zero ${\lim }^1$: given a sequence $b_j$ in a surjective tower, solve $a_j-f_j(a_{j+1})=b_j$ successively by lifting, which proves surjectivity of 1-shift on the product defining ${\lim }^1$. Thus no additional derived-limit term is discarded in this form computation. The ring structure is the compatible wedge/polynomial multiplication in this completion.

If $I\subset J$ are labeled lists, with $j_J$ restricting to $j_I$, projection to the extra formal coordinates gives an exact sequence $0⟶K_I⟶K_J⟶{\underset{‾}{\mathbb{C}}}^{J\setminus I}⟶0.$ Surjectivity of the last arrow follows by prescribing those coordinates and solving the remaining equation with $S_I$. A continuous right inverse of $S_I$ identifies a preimage explicitly: for $v$ in the extra coordinates take $(-Q_I{j}_{J\setminus I}v,v)$. Thus $K_J\cong K_I\oplus {\underset{‾}{\mathbb{C}}}^{J\setminus I}$. The space of splittings is affine over $\mathrm{Hom}({\underset{‾}{\mathbb{C}}}^{J\setminus I},K_I)$ and therefore contractible. In particular this is an actual stabilization identification of the two virtual bundles, not merely equality of their numerical indices. Two arbitrary lists have the common enlargement formed from their disjoint labeled union, even if all their evaluated sections coincide. Surjectivity persists on an enlargement. The finite good lists inside any fixed universe of labeled sections form a nonempty directed poset: union is an upper bound. Every finite diagram in its nerve is contracted after adjoining an upper bound. This explicitly proves contractibility of that nerve.

For a compact finite-CW space $Z$, let ${\mathcal{V}}_q(Z)$ be the groupoid of finite Hermitian bundles on $Z\times {\mathrm{\Delta }}^q$ and unitary isomorphisms, with continuous singular parameters and pullback along the affine faces and degeneracies. Present bundles as cartesian fields with their local charts: pullback is then precomposition, and successive pullbacks agree literally. The kernels in step 2.1 use the same convention. The total space $X$ has finite CW type: trivialize the pullback family over the characteristic disks of the finite base, use a finite CW model of the compact manifold fiber, replace the attaching maps by cellular maps, and take their finite iterated mapping cylinders. We use compact singular presentations, diagonal singular nerves and thick mapping-cylinder realization throughout. An embedding into $U={\mathrm{colim}}_N{\mathbb{C}}^N$ means a finite-stage embedding on each compact presentation; it does not mean the norm-topologized space of infinite Hilbert projections.

The inclusion of closed basic forms into the full de Rham complex induces q_A:P_A^q -> H^q(BG;A_C). The full de Rham complex is locally quasi-isomorphic to constant coefficients by the explicit radial Poincare homotopy on each Cartesian chart. Derived global sections preserve this local equivalence. Forgetting a connection identifies the shape of $B_{\mathrm{\nabla }}$ G with BG: over a fixed principal bundle the space of connections is a nonempty affine space and hence contractible, and local connections glue by a partition of unity. Equivalently use the bar construction of the gauge action and its contractible affine connection fiber. These identifications give precisely the displayed map $q_A$. They do NOT give a map in the opposite direction for an arbitrary noncompact Lie group. Accordingly the full-scope integral form subgroup is necessarily (P_A^q)_H = {p in P_A^q : q_A(p) belongs to im[c_A:H^q(BG;A)->H^q(BG;A_C)]}. For connective K replace $c_A$ by the actual Chern character c_k:k^q(BG)->H^q(BG;C[u]), (P_k^q)_k = q_k^(-1)(im c_k). When $q_A$ (or $q_k$) is a Chern-Weil isomorphism in an independently established compact case, these agree with the usual image definition after that identification. The inverse-image definition is meaningful and natural for ALL Lie groups, and retains invariant polynomials that lie in the kernel of $q_A$.

For a chain $I_0\subset \cdots \subset I_r$, the corresponding kernel bundles form a flag, with the successive trivial quotients just calculated. The space of splittings of the entire flag is a successive affine bundle and is contractible. Forgetting a sublist forgets a subflag; on quotients it uses the actual short exact sequences of the same flag. These observations give the required higher comparison homotopies, not just pairwise isomorphisms. For a simplex in the nerve of the directed poset, the lists already form ONE chain; all its faces are subchains of this very flag. Choose its splitting comparisons by an inner product on the largest kernel and the induced orthogonal splittings of the subflags. Positive Hermitian metrics form a convex cone, so prescribed metric data on a parameter boundary extend, and applying orthogonal splitting gives the corresponding face homotopies. For a finite diagram with incomparable lists, first adjoin a common good labeled list $L$. Compare EACH vertex to $K_L$ by its actual inclusion $K_I\hookrightarrow K_L$. A chain in the original diagram, followed by $L$, gives a flag in $K_L$; no assertion is made that incomparable vertices belong to one flag. A single metric on $K_L$ supplies all these chain comparisons at once, with their shared subchains. Convex extension of the metric supplies their relative parameter comparisons and comparisons between such choices. Thus coning a finite diagram to $L$ and using its chain simplices gives all the requisite homotopies. There is no demand to extend arbitrary incompatible prescriptions of splittings: the coherence data are built from this common metric system from the outset. Adding one interval parameter gives the comparison of two such systems, and the same convex construction applies to all higher parameters. No assertion that a category with unspecified morphisms is filtered is needed. The directed objects here are formal labeled augmentations; all nonunique splitting data have the stated affine extension property.

For each finite based set $S$, retain bundles $E_U$ for subsets of $S\setminus \{\ast \}$, $E_{\mathrm{\varnothing }}=0$, and isomorphisms $E_U\oplus E_V\cong E_{U\cup V}$ for disjoint subsets, with their unit, interchange and associativity identities. Forgetting to singleton subsets is an equivalence with the corresponding finite product of bundle groupoids; a quasi-inverse takes actual block sums. Preimages of subsets define the finite-based-set functors. Nerve and thick realization give a special $\mathrm{\Gamma }$-space. Its iterated bars have suspension structure maps given by the inclusion of one-simplices in the next bar; the mapping-cylinder telescope of their iterated loop maps gives its connective spectrification $\mathbf{V}(Z)$. The loop-bar assertion follows from the simplicial path object: its initial-vertex degeneracy contracts the total path object, and the special product equivalences identify its fiber with the preceding space when components form a group. Skeletal mapping-cylinder induction preserves this fiber comparison. The first loop-bar group-completes the initial nongrouplike space. Thus this specifies actual spaces, structure maps and their spectrification before any index homomorphism is considered.

We prove the relative embedding extension used in recognizing this spectrum. Let $(Y,Y_0)$ be a finite CW pair and let $E$ be a finite Hermitian bundle on $Y$, with an isometric embedding $e:E{\mathrm{\mid }}_{Y_0}\to {\underset{‾}{\mathbb{C}}}^N$ prescribed. A finite unitary frame cover and subordinate partition ${\rho }_i$ give a global isometric embedding $e_0(v)=(\sqrt{{\rho }_i}{\varphi }_i(v){)}_i$ into some finite trivial bundle. On an NDR neighborhood $N$ of $Y_0$, transport $E$ along the retraction to extend $e$ isometrically to $e_N$, agreeing on $Y_0$. Such transport follows by embedding $E$ globally, subdividing the retraction homotopy until successive projection matrices differ in norm by less than $1\mathrm{/}2$, and polar-normalizing their restricted projections; the subdivision is finite on a compact pair. Choose $h=1$ on $Y_0$ supported in $N$. The map $(h{e}_N,\sqrt{1-h^2}{e}_0)$, extended by zero in the first block outside $N$, is an isometric embedding on $Y$ restricting to $(e,0)$. Hence all disk, sphere, horn and finite face-boundary embedding problems extend after finite ambient stabilization. For several embeddings of one bundle, $(v,t)\mapsto (\sqrt{t_0}{e}_0(v),\dots ,\sqrt{t_r}{e}_r(v))$ gives explicit face comparisons in named blocks. For strict simplicial lifts retain already degenerate simplices as their pullbacks and extend each nondegenerate simplex along its entire prescribed boundary by the relative lemma; the barycentric formula alone is not asserted to preserve degeneracies.

The model must be read in its degree-q shifted form: take the homotopy pullback of ${\mathrm{\Sigma }}^q$ E, H of the full shifted coefficient-form complex, and H of its smart heart (closed degree-q forms in degree0). Take ${\pi }_0$ of derived global sections. For E=HA this is the ordinary Hopkins-Singer group; for $E=k$ it is the source's connective group. A single unshifted smart heart cannot substitute for this degree-dependent truncation in all degrees. Write $E_C^q(BG)=H^q(BG;A_C)$ in the ordinary case and $H^q(BG;C[u])$ in the connective case, and let $E_{C\mathrm{/}Z}$ be the cofiber of $E\to E_C$ (with $A_C\mathrm{/}A$ for ordinary coefficients). The fiber of curvature in the shifted pullback is exactly ${\mathrm{\Sigma }}^{q-1}$ $E_{C\mathrm{/}Z}$. The closed-form corner is zero-truncated, so its global mapping space has no positive homotopy groups. The pullback long exact sequence consequently gives 0 -> E_(C/Z)^(q-1)(BG) --j--> hat E^q(B_nabla G) --R--> (P_E^q)_E -> 0, with I j=beta, where beta is the coefficient connecting homomorphism. Surjectivity of R is explicit: choose a topological cocycle x with c(x)=q(p), and choose a homotopy between its complexified cocycle and the cocycle represented by p. The triple (x,p,homotopy) is precisely a point of the homotopy pullback. Conversely each pullback point supplies this compatibility. Products of triples show that R is a graded-ring map and that the integral targets form a subring. Equivalently the joint (I,R) map fits into the exact sequence 0 -> coker[c:E^(q-1)(BG)->E_C^(q-1)(BG)] -> hat E^q(B_nabla G) -> E^q(BG) x_(E_C^q(BG)) P_E^q -> 0. To see its kernel without any spectral-sequence assertion, two chosen homotopies between the same cocycles differ by an $E_C^{q-1}$ class, and changing the topological cocycle homotopy changes that class by $c(E^{q-1})$. This is exactly the stated cokernel. These two equivalent sequences also imply the precise forgetful restriction on the full curvature kernel: 0 -> coker c_(q-1) -> ker R --I--> ker c_q -> 0, I(ker R)=im beta=ker c_q. There is in general NO injectivity assertion for I on ker R. Injectivity holds when the displayed cokernel vanishes, and only then. The torsion identification also requires the corresponding actual finiteness/complexification hypotheses; it cannot replace this exact sequence for arbitrary completed coefficients.

These constructions also have the relative parameter extension property. Suppose a finite system of augmentations, kernel identifications and face homotopies has been prescribed on the closed subcomplex $A_0$ of a compact parameter complex $A$. Pass to a common disjoint labeled enlargement of the finitely many prescribed lists. The corresponding section maps extend off $A_0$: use a neighborhood deformation retraction for the subcomplex, local Hilbert-bundle trivializations, and a partition of unity to extend continuous sections, agreeing with the prescribed ones on $A_0$. Surjectivity holds on a neighborhood of $A_0$ by openness and compactness. On the remaining compact set choose finitely many extra cokernel sections, multiply them by cutoffs vanishing on a smaller neighborhood of $A_0$, and append their formal labels. This gives a good augmentation on all parameters. Its restriction adds only zero-valued formal coordinates near $A_0$; their identical addition to the kernel and auxiliary terms is the canonical stabilization, so it does not alter the prescribed relative virtual class. Extend splitting/embedding homotopies by step 4.1. This proves the relative assertion, including arbitrary finite higher comparison diagrams; it is stronger than choosing an index separately at each parameter.

Resolve ${\mathcal{V}}_{\bullet }(Z)$ by objects $(E,e)$ with a finite-stage isometric embedding, and arrows $f$ satisfying $e^{\mathrm{\prime }}f=e$. A nerve string of isomorphisms transports an embedding of its first bundle to all its vertices. For a finite parameter disk presentation, its boundary embeddings therefore form precisely the closed-subcomplex prescription in step 4.3. Skeletal extension gives lifts, and applying the same construction with an interval gives relative homotopies and all higher comparisons. This proves that the forgetful map of the diagonal singular realizations is a weak equivalence: a compact disk map has a finite cellular presentation after relative approximation, and induction extends its embeddings over those finitely many characteristic simplices; the same argument proves injectivity on every homotopy group. The embedded groupoid is equivalent to the projector model $P={\coprod }_r{\mathrm{Gr}}_r(U)$ with identity arrows: $(E,e)$ maps to $e{e}^{\ast }$, and the inverse takes the tautological image bundle. Equal projectors give the unique compatible isomorphism $e^{\mathrm{\prime }\ast }e$. Its parameter realization is consequently $M(Z)=\mathrm{Map}(Z,P)$ in the compact finite-stage singular model. For each $\mathrm{\Gamma }$-system choose singleton embeddings and their named orthogonal sums, then transport by its structural identifications. The same proof gives a weak equivalence in every finite based set. Evaluation $(z,e)\mapsto e_z$ is a continuous matrix evaluation before realization and commutes with every restriction and named sum. Thus there is already an evaluation map of the connective spectrum models, not just a correspondence of their components.

A continuous coefficient bundle on $X\times A$ can be used without assuming a preexisting vertical smooth connection. Embed it into a finite trivial bundle over the compact space and write its orthogonal projection as $e$. In finitely many vertical charts smooth its matrix coefficients in the vertical variables, using a partition of unity, to a self-adjoint matrix $a$ with $\mathrm{\parallel }a-e\mathrm{\parallel }<1\mathrm{/}8$ uniformly in all parameters. Its spectrum lies within $1\mathrm{/}8$ of $\{0,1\}$, so the contour formula $e^{\mathrm{\prime }}=\frac{1}{2\pi i}{\int }_{\mathrm{\mid }z-1\mathrm{\mid }=1\mathrm{/}4}(z-a{)}^{-1}dz$ is a vertically smooth projection of the same rank. The map $e^{\mathrm{\prime }}e$ between the ranges is invertible, giving an isomorphism with the original bundle. Two such approximations are joined by their straight-line path, still within $1\mathrm{/}8$ of $e$, followed by the same contour projection. All higher interpolation parameters satisfy the same bound. Relative data are kept using cutoffs and the extension of step 5.1. Hermitian connections form an affine space. Therefore smoothing and connection choices introduce a contractible system of comparisons rather than an extra hypothesis on topological $K$-classes.

The all-cover Čech description has the same relative lifts. Record a finite numerable unitary frame cover, its subordinate partition and the actual transition isomorphisms. It gives the embedding $(\sqrt{{\rho }_i}{\varphi }_i{)}_i$ of step 4.3. Finitely many covers have their simultaneous intersection refinement. On a fixed refinement partitions interpolate in their convex space; boundary partitions extend over a neighborhood and are joined to an interior partition with a cutoff. Changes of frames are their transition-isomorphism nerve arrows, with composition given by the actual coordinate changes, not a claim that frame spaces are convex. Conversely the nonzero coordinate-minor Grassmann charts of an embedded bundle give unitary frames by polar normalization; compactness gives a finite subcover. A common resolution retaining both frame-cover data and an embedding therefore maps to both presentations. Its missing data over a finite boundary diagram extend by step 4.3 and this partition construction. The skeletal disk/homotopy proof of step 5.2 shows both projections are weak equivalences, with all refinement morphisms and higher simplices retained. Evaluation on the common resolution evaluates the same tautological bundle. All constructions commute with restrictions; no global contractibility assertion in a norm topology is used.

Group completion must also be checked with parameters. Form $T_Z=\mathrm{hocolim}(M(Z)\stackrel{\oplus 1}{\to }M(Z)\to \cdots )$ by left trivial-line stabilization. Right direct sum is its coherent monoid action, with named block words making the operations commute. Every bundle $E$ on a finite compact parameter product $Z\times Y$ has an actual complement $E^{\perp }$ with $E\oplus E^{\perp }\cong {\underset{‾}{\mathbb{C}}}^N$, uniformly on that product, by step 4.3. Thus adding any positive bundle to $T_Z$ is a homotopy equivalence: add its complement and undo $N$ telescope shifts. The action bar $B(T_Z,M(Z),\ast )\to BM(Z)$ has homotopy fiber $T_Z$. Over each simplex it is a product with that fiber, and every face transport is the just-proved equivalence; skeletal mapping-cylinder induction proves the fiber assertion. Its total space is contractible, since it is the telescope of $B(M(Z),M(Z),\ast )$, contracted by the extra unit degeneracy. Consequently $T_Z\simeq \mathrm{\Omega }BM(Z)$ is the group-completed zero infinite-loop space. Finitely many connected components of $Z$ cause no problem: a complement trivializes with a single global rank $N$ after adding zero coordinates, and one common telescope denominator works on the entire finite parameter diagram and all its faces. The empty space gives the zero spectrum.

Evaluation gives an actual map $a_Z:T_Z\to \mathrm{Map}(Z,T_{\ast })$, sending the stage-$n$ embedded bundle to its fiberwise stage-$n$ point family. It is a weak equivalence on every homotopy group. Indeed a compact parameter map into the right side is a map $Z\times Y\to T_{\ast }$; its telescope height is bounded, so its image lies in a finite subtelescope, which retracts to its last stage. This supplies a uniform stage and hence a lift from $M(Z)$ at that stage. A homotopy on $Z\times Y\times I$, including its already fixed boundary presentations, has one common later stage by the identical argument. Relative embedding extension supplies a common finite ambient dimension wherever needed. These are surjectivity and injectivity for every disk/sphere test of this specified map, not an inference about higher homotopy from $K^0$. At the point $T_{\ast }$ is the stable virtual-bundle space $\mathbb{Z}\times BU$. The evaluation of step 5.2 respects the entire $\mathrm{\Gamma }$ construction, so gives a spectrum map $\mathbf{V}(Z)\to F(Z_{+},R)$, where $R=\mathbf{V}(\ast )$ is the connective point ring. The displayed comparison identifies its zero infinite-loop map with $a_Z$. The Bott construction below identifies ${\mathrm{\Omega }}^{\mathrm{\infty }}R\to {\mathrm{\Omega }}^{\mathrm{\infty }}KU$ as a weak equivalence, so this yields $\mathbf{V}(Z)\simeq {\tau }_{\ge 0}F(Z_{+},KU)$. Both sides here are connective; the entire function spectrum need not be. The point ring $R$ is $ku$, not periodic $KU$.

We specify a virtual-pair target without invoking a complex-category recognition theorem. Allowing all linear bundle isomorphisms instead of unitary ones gives the same parameter homotopy type: resolve by metrics, whose positive cone is convex, pulls back by isomorphisms and extends relatively. Let $\mathcal{P}(Z)$ be the common-stabilization word bar. Its objects are pairs $(K,A)$; an elementary arrow retains a bundle $Q$ and actual isomorphisms $K\oplus Q\cong K^{\mathrm{\prime }}$ and $A\oplus Q\cong A^{\mathrm{\prime }}$. Isomorphisms between such presentations are retained. A string names every consecutive sum of its $Q$ letters and its associativity maps; faces erase a cut and degeneracies insert zero. Its realization is $B(M(Z)\times M(Z),M(Z),\ast )$ for the diagonal right-sum action. Applying the construction in every finite based set gives a special $\mathrm{\Gamma }$ model. Virtual difference maps this word bar to the group-completed bundle model: the identical positive and negative $Q$ word is canceled using the addition/inverse diagrams of the additive group-completion bar. Those diagrams, with their extra-unit degeneracy, give the actual action homotopies on all faces and substitutions before realization, rather than paths independently assigned to edges.

For clarity, parameter coefficients are the positive bundles in ${\mathcal{V}}_{\bullet }(X)$; their additive inverses, suspensions and Bott shifts will be supplied by the constructions, not by a zero augmentation for an arbitrary complex. Smoothing in step 6.1 and Hermitian vertical connections form a relative contractible system. Let ${\mathcal{V}}_q^{\mathrm{\nabla }}(X)$ have vertically smooth bundles with such connections and connection-preserving unitary isomorphisms. Forgetting this structure is a weak equivalence on parameter models: a continuous family has the relative vertical-smoothing approximation of step 6.1, and affine connection sections extend over every finite closed parameter subcomplex. The extension also applies with the given endpoint isomorphisms; all isomorphisms and their parameter homotopies are retained as actual maps. Apply the construction in each finite direct-sum slot. Thus ${\mathbf{V}}^{\mathrm{\nabla }}(X)\to \mathbf{V}(X)$ is a weak equivalence of the specified connective models. An optional bounded finite-complex coefficient has the alternating sum of its graded bundle coefficients in this model; no equivalence of the whole category of complexes and quasiisomorphisms is required for the public spectrum map.

The virtual-difference map is a weak equivalence. Simultaneous trivial-line stabilization of both entries acts on $\mathcal{P}(Z)$ by a map homotopic to the identity, its homotopy being the elementary common-line arrow and its word-bar simplices. Passing to that telescope changes no weak type. Its coefficient-space telescope is weakly equivalent to $T_Z\times T_Z$: compact disk maps and homotopies have two bounded heights, and raising both to their common maximum gives the required lift. On this grouplike space the coherent shear $(x,y)\mapsto (x-y,y)$, with inverse $(a,y)\mapsto (a+y,y)$, identifies diagonal $M(Z)$ translation with the trivial action on the first coordinate and addition on the second. Retain these actual additive word diagrams in the bar. Therefore $B(T_Z\times T_Z,M(Z),\ast )\simeq T_Z\times B(T_Z,M(Z),\ast )\simeq T_Z$, the second factor being the explicit contractible bar of step 6.3. This identifies the specified virtual-difference map as an equivalence. It works in every finite based set and yields a weak equivalence of connective spectra $\mathbf{P}(Z)\to \mathbf{V}(Z)$. Resolving $K,A,Q$ and their finitely many isomorphisms by step 6.2 gives its restriction-natural Čech evaluation and all disk/sphere/higher-simplex comparisons. No equality only on components has been used.

Let ${\mathcal{A}}_q(p)$ be the augmentation category over ${\mathcal{V}}_q^{\mathrm{\nabla }}(X)$. An object is $(E,A,j)$, where $A$ is a finite Hermitian bundle on $B\times {\mathrm{\Delta }}^q$ and $S=D_E+j:H_E^{+}\oplus A\to H_E^{-}$ is surjective. A morphism is a coefficient isomorphism $u$ and an isometric auxiliary embedding $\alpha :A\hookrightarrow A^{\mathrm{\prime }}$ with $j^{\mathrm{\prime }}\alpha =u^{-}j$; it induces $(u^{+},\alpha )$ on the actual kernels. All maps are continuous in the bounded Sobolev topology of step 1.1, and their compositions, pullbacks and sums are the actual operations. Such objects exist by step 2.1; finite bundle auxiliaries permit tensoring, and complements recover formal trivial lists. For fixed $E$ choose any good augmentation $J$. The functor $A\mapsto A\oplus J$ has natural transformations $\mathrm{i}\mathrm{d}\to (-\oplus J)\leftarrow \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}(J)$ from the two inclusions. These contract its augmentation nerve, including nonunique morphisms. Transport by coefficient isomorphisms identifies the fibers. The two-sided transport nerve shows that projection to the coefficient groupoid is a weak equivalence: over a coefficient string its extra nerve is this contractible nerve; skeletal mapping-cylinder induction and the parameter diagonal preserve the comparison. This works in every sum slot. Together with step 7.3 it gives $\mathbf{A}(p)\to {\mathbf{V}}^{\mathrm{\nabla }}(X)\to \mathbf{V}(X)$ as a specified zigzag of weak equivalences. No global augmentation on an infinite parameter space or filteredness of the morphism category is asserted.

For an augmentation arrow the two exact sequences are $0\to K_A\to K_{A^{\mathrm{\prime }}}\to Q\to 0$ and $0\to A\to A^{\mathrm{\prime }}\to Q\to 0$, with the SAME bundle $Q=A^{\mathrm{\prime }}\mathrm{/}\alpha A$. The first quotient map is the auxiliary-coordinate projection: prescribing that coordinate and solving the remaining equation with a right inverse of $S$ proves its surjectivity and identifies its kernel with $K_A$. A composable $r$-string transports to actual flags $A_0\subset \cdots \subset A_r$ and $K_0\subset \cdots \subset K_r$, with canonical filtered isomorphism $K_r\mathrm{/}{K}_0\cong A_r\mathrm{/}{A}_0$. Split the full $A$ flag, with named successive quotients $Q_i$, and choose a section of $K_r\to K_r\mathrm{/}{K}_0$. The section automatically takes $K_i\mathrm{/}{K}_0$ into $K_i$, its full inverse image. Applied through the filtered quotient isomorphism, this gives simultaneous decompositions $K_i\cong K_0\oplus Q_1\oplus \cdots \oplus Q_i$ and $A_i\cong A_0\oplus Q_1\oplus \cdots \oplus Q_i$, agreeing with every consecutive-interval quotient map. Their choices are iterated affine bundles and are contractible. A subchain retains its decompositions and merges consecutive words. Deleting the middle vertex of $0<1<2$ retains $Q_1\oplus Q_2$ WITH its induced identifications to $A_2\mathrm{/}{A}_0$ and $K_2\mathrm{/}{K}_0$, rather than replacing it by a newly chosen quotient model. Deleting the first vertex transports the remaining decomposition through $K_1$ and $A_1$; a repeated vertex inserts zero. Thus faces and degeneracies have specified compatible maps.

Retain these full-flag decompositions as a simplicial splitting resolution ${\mathcal{A}}_{\bullet }^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}(p)$ of the augmentation nerve, with the face/degeneracy operations just specified and all consecutive-word identifications retained. In each nerve degree its forgetful map is a weak equivalence: over a fixed parameter family the spaces are the affine splitting bundles of step 9.1, and compatible boundary sections of a fixed full flag extend by affine extension and partitions of unity. After singularization, degreewise weak equivalences remain weak equivalences on the diagonal thick realization, by the successive simplex-skeleton mapping-cylinder comparison. This applies in every finite sum slot. Arbitrary unrelated edge splittings need not be restrictions of one strict triangle splitting; no such claim is made. The resolution retains compatible full-flag data, and its mapping-path fibration replacement supplies relative HOMOTOPY lifts for arbitrary disk/sphere diagrams including all their higher boundary data. Literal affine extension applies when a splitting of that fixed full flag is prescribed. This distinction is why the resolution is retained rather than an edgewise choice being declared coherent.

The assignment $(E,A,j)\mapsto (K_A,A)$ is now an actual simplicial map from this resolution to the common-stabilization word bar: a full flag maps to its word $(Q_1,\dots ,Q_r)$ and its two actual decomposition isomorphisms. Erasing a cut or inserting zero is exactly its target face or degeneracy. Kernel continuity follows from the local right-inverse projection formula in step 2.1. The construction consequently gives the spectrum zigzag $\mathbf{V}(X)\stackrel{\simeq }{\leftarrow }\mathbf{A}(p)\stackrel{\simeq }{\leftarrow }{\mathbf{A}}^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}(p)\to \mathbf{P}(B)\stackrel{\simeq }{\to }\mathbf{V}(B)$ before any backwards arrow is inverted. Its component value is $[K_A]-[A]$, since the last map is the specified virtual difference. This verifies the intended finite-augmentation family index; it does not construct the spectrum map from that component calculation. No auxiliary complex-category equivalence is assumed.

To specify all tensor coherences, retain the subset-sum systems together with named tensor products of the input slots, every consecutive subword and their universal multilinear identifications. Associativity, unit, symmetry and distributivity and every substitution hold on pure tensors, hence as actual bundle maps. One module slot is the coefficient, augmentation or virtual-pair category; other slots are base bundles. Forgetting named product outputs is an equivalence, its inverse being the actual universal tensor product. Embedded inputs have product embeddings into the named finite ambient tensor word, avoiding an unstated strict identification $U\otimes U=U$. Common intersection frame covers and their tensor frames give the same evaluation maps as step 6.2. In degree $r$ of the two-sided module bar retain the tensor product of every consecutive subword in $M\wedge R^{\wedge r}\wedge R$. A face erases a cut, and a degeneracy inserts the trivial-line unit. Two erased cuts give restriction to the same named diagram; the final-unit extra degeneracy contracts the augmented bar. These specify actual simplicial operations and module structure maps before realization. Splittings in input slots determine their tensor output splittings; independent compatible choices are compared in the same affine spaces as step 10.1. Thus the splitting resolution, projection, virtual-pair map and evaluation are maps of the entire tensor-word resolutions, not a selection of inverses to product equivalences.

For a finite bundle $W$ on the base the augmentation tensor is $(E,A,j)\otimes W=(E\otimes p^{\ast }W,A\otimes W,j\otimes 1)$. In a local frame of $W$ the product vertical connection gives $D_{E\otimes p^{\ast }W}=D_E\otimes 1$, so its stabilized map is surjective and its kernel is canonically $K_A\otimes W$. These identities glue, being identical on pure tensors. They carry both matched quotient sequences and every full flag decomposition of step 9.1 to their tensors by $W$; its virtual-pair word is $(Q_1\otimes W,\dots ,Q_r\otimes W)$. They commute with the actual induced maps on every augmentation arrow, every direct-sum slot and every tensor substitution. Hence step 11.1 is a zigzag of module-spectrum maps over $\mathbf{V}(B)$, and in particular over $R=\mathbf{V}(\ast )$, with $\mathbf{V}(B)$ acting on the source by pullback. The same diagrams with an external compact parameter bundle give external-product compatibility. Resolved evaluation respects these diagrams by step 12.1. This projection formula is proved on actual kernels and all word bars, not inferred from $K^0$.

In the group-completed point model let $\beta :S^2\to {\mathrm{\Omega }}^{\mathrm{\infty }}R$ be the actual stable-bundle map given by the Hopf line minus the trivial line; its chosen basepoint isometry gives its based path. Multiplication in the tensor-word bar gives ${\mathrm{\Sigma }}^{2}M\to M$ for every $R$-module above. Form $M[{\beta }^{-1}]=\mathrm{hocolim}(M\to {\mathrm{\Sigma }}^{-2}M\to {\mathrm{\Sigma }}^{-4}M\to \cdots )$ using mapping cylinders of these multiplication maps and functorial spectrification. This is a construction on whole module maps: step 13.1 commutes with all iterated action words before realization, and therefore maps the entire telescopes. Define the Bott stable-bundle point spectrum to be $KU=R[{\beta }^{-1}]$. This agrees with the stipulated Bott spectrum: step 7.1 at a point identifies its nonnegative source groups with the stable-bundle groups, and [F13] makes multiplication by this actual Hopf-line class the Bott isomorphism. It follows that $R\to KU$ is an isomorphism on every nonnegative homotopy group, hence on zero infinite-loop spaces, as used in step 7.1; no spectrum realization of a group-level supplier has been assumed. The tensor-word bar also identifies $M[{\beta }^{-1}]$ with $M{\otimes }_{R}KU$: both are the same double bar/telescope diagram, and its extra final-unit degeneracy contracts the module bar. Thus localization of the zigzag is a zigzag of $KU$-module maps.

Resolved evaluation and the point Bott map give actual maps $\mathbf{V}(Z)[{\beta }^{-1}]\to F(Z_{+},KU)$ commuting with the whole telescope. For each integer $k$ the source group is ${\mathrm{colim}}_m{\pi }_{k+2m}\mathbf{V}(Z)$. Choose $m$ with $k+2m\ge 0$. The all-degree disk comparison of step 7.1 identifies this term with $K^{-k-2m}(Z)$, and its transition is exterior multiplication by the actual $\beta$, because evaluation already preserves the tensor diagrams. The finite-pair Bott isomorphism [F13] identifies the colimit with $K{U}^{-k}(Z)$, and the displayed map induces that very identification. It is therefore a weak equivalence in every integer degree, with the ordinary point $KU$ action by the same tensor comparison. The virtual-pair comparison of step 8.1 is also a module equivalence and stays so after localization. Applying these specified equivalences to step 11.1 yields ${\tau }_p:F(X_{+},KU)\to F(B_{+},KU)$ as a morphism of $KU$-module spectra. Every map existed before this homotopy-group verification; no inference from a bare $K^0$ homomorphism or classification on components has been used.

For a finite parameter pair $(A,A_0)$ take the homotopy fiber of restriction of this entire zigzag, using a functorial mapping-path model. A boundary trivialization is its actual path to zero with all higher simplices; every map above takes that path and its higher data to the output path and data. The weak equivalences commute with both restrictions and induce weak equivalences on these fibers by the homotopy exact sequence. Step 5.1 displays a single full-pair augmentation when desired, and steps 4.3, 5.2, 6.2 give its actual relative embedding and Čech/refinement comparisons. No extension of incompatible strict splitting prescriptions is asserted: step 10.1 retains their correct relative homotopy lifts. Taking $(D^n,\mathrm{\partial }{D}^n)$ gives suspension, and the based Hopf-line pair gives exactly the action in step 14.1. Thus relative boundary, suspension and Bott compatibility come from the same parameter functors and word diagrams. Positive bundles, virtual inverses, every suspension and every integer Bott shift are all covered; no narrower coefficient hypothesis has been introduced.

Pullback of the whole geometric family pulls back the coefficient, the auxiliary bundle and its section map, the actual kernel, the matched quotient flags and their splitting resolution. With the fiber-field convention of step 3.2 it commutes with composition; in the equivalent ordinary bundle convention the canonical pullback identifications satisfy this identity on every fiber and hence give all its coherent diagrams. The $\mathrm{\Gamma }$ subset systems, all multilinear systems, singular parameter maps, bar construction, Bott telescope and evaluation resolutions are functorial under the same pullback. Therefore the spectrum zigzag and its splitting below form a homotopy-coherent natural transformation on geometric families and closed parameter restrictions. No claim of cofinality of old augmentations among newly chosen ones is used: the augmentation-fiber contraction of step 8.2 supplies the weak equivalence after every pullback.

Under the surjectivity and canonical-line-kernel hypothesis, the pulled-back coefficient $p^{\ast }V$ has the ZERO augmentation. Its finite output pair is literally $(V,0)$. This section is defined on the POSITIVE base-bundle category ${\mathcal{V}}_q(B)$: its objects are finite Hermitian bundles $V$, whose pulled-back operators are $D_{p^{\ast }V}^{+}=D^{+}\otimes 1_V$, surjective with kernel $V$. It preserves direct sums, tensor diagrams, morphisms and parameter restrictions in this category. The natural transformation below is restricted to the augmentation category over these pulled-back positive objects. For any other augmentation $A$ of such a coefficient, the unique map of auxiliary bundles $0\to A$ gives a natural morphism from this zero augmentation to $A$. Its output is the common-stabilization comparison of step 11.1, whose two exact sequences have the same quotient $A$; retaining its full-flag splitting resolution gives its actual cancellation homotopy in $\mathbf{P}(B)$. The nerve of this natural transformation is the splitting homotopy. Its restrictions to every face, degeneracy, direct-sum slot, tensor slot and pullback square are the SAME morphism $0\to A$, so these are its actual higher coherence data. Only AFTER constructing this transformation on positive bundles do the additive $\mathrm{\Gamma }$-group completion and the specified Bott telescope extend its splitting to virtual and suspended classes, carrying it to ${\tau }_p{p}^{\ast }\simeq 1$ on the full function spectra. No zero augmentation is assigned to an arbitrary Fredholm operator with nonzero cokernel, or directly to a finite-complex representative with odd cohomology. For the flag application the assumed harmonic calculation supplies exactly this surjective operator and canonical kernel. No analytic/topological index theorem, topological Thom comparison or differential index theorem has entered this construction.

The universal flag tower and its finite-stage bases are constructed as follows. Let $G$ be compact connected. Peter–Weyl [F14] gives continuous finite-dimensional unitary representations separating points. Each such representation $\rho :G\to U(m)$ is smooth: choose a left-invariant Riemannian volume $d\mu$ on the compact Lie group and a smooth function $\varphi$ of integral $1$ supported near the identity, so $A_{\rho }={\int }_G\rho (h)\varphi (h)d\mu (h)$ is invertible by continuity of $\rho$. Then $\rho (g)A_{\rho }={\int }_G\rho (k)\varphi (g^{-1}k)d\mu (k)$, which is smooth in $g$ by differentiation under the compact integral; hence $\rho$ is smooth. Finite-dimensionality of $\mathfrak{g}$ lets us choose finitely many such representations with differentials of zero common kernel. Their direct sum has a discrete compact kernel, hence finite; adding one separating representation for each nonidentity kernel element gives a faithful smooth unitary embedding $G\hookrightarrow U(r)$. Choose a maximal abelian subalgebra $\mathfrak{t}\subseteq \mathfrak{g}$, which exists by maximizing its finite dimension. By [F19], $\exp (\mathfrak{t})$ is a connected abelian subgroup. Its closure $T$ is compact connected abelian and is an embedded Lie subgroup by [F19]. Its Lie algebra is abelian and contains $\mathfrak{t}$, so maximality makes it exactly $\mathfrak{t}$. The local exponential theorem makes $\exp (\mathfrak{t})$ open in $T$; connectedness then makes it all of $T$. Thus $T$ is a maximal torus: any larger connected compact abelian subgroup has an abelian Lie algebra containing $\mathfrak{t}$ and the same dimension, hence equals $T$. Explicitly $T=\mathfrak{t}\mathrm{/}\mathrm{\Lambda }$, with $\mathrm{\Lambda }=\ker (\exp {\mathrm{\mid }}_{\mathfrak{t}})$ discrete and cocompact; a lattice basis identifies this with $(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^{\dim T}$. No conjugacy theorem is needed for this choice. Put $E_n=V_r({\mathbb{C}}^{r+n})$, the space of orthonormal $r$-frames, with the free right action through this embedding; freeness follows because a frame is injective. Put $B_n=E_n\mathrm{/}G$ and $X_n=E_n\mathrm{/}T$. Here is the required quotient construction, without invoking Lee’s theorem. For a compact Lie group $K$ acting freely on a smooth manifold $E$, the differential of its orbit map at a point $x$ is injective: an infinitesimal stabilizer integrates to a one-parameter subgroup fixing $x$, which freeness makes trivial. Choose a small transverse submanifold $S$ at $x$ complementary to this orbit tangent. By [F17], $(k,s)\mapsto sk$ is a diffeomorphism near $(e,x)$ and is a local diffeomorphism on $K\times S$ after shrinking $S$. It becomes globally injective after a further shrinking. Otherwise there are $s_j,s_j^{\mathrm{\prime }}\to x$ and $k_j\mathrm{\ne }e$ with $s_j{k}_j=s_j^{\mathrm{\prime }}$; compactness gives a subsequence $k_j\to k$, freeness forces $k=e$, and the local product chart then forces $k_j=e$ and $s_j=s_j^{\mathrm{\prime }}$, a contradiction. Consequently $K\times S$ maps diffeomorphically onto a saturated open neighborhood, and $S$ is a quotient chart. The product-chart inverse gives smooth chart transitions and makes the quotient projection a submersion. The orbit relation is closed: for converging orbit pairs, a convergent subsequence of the relating elements of $K$ relates the limits. Since the quotient map is open, this makes the quotient Hausdorff; images of a countable basis give second countability. These charts apply to the free actions of both $G$ and $T$ on $E_n$. Thus $B_n$ and $X_n$ are compact smooth quotients with quotient submersions; the invariant Stiefel metric descends using the orthogonal complements to the orbit tangents. The map $X_n\to B_n$ is the associated $G\mathrm{/}T$ bundle. Coordinate inclusions define the two towers, and [F18] gives finite CW type for every $B_n$ and $X_n$. No dimension-dependent elliptic estimates are required uniformly in $n$: each fixed compact stage supplies its own family map, and pullback gives all transition comparisons.

Use the FUNCTORIAL parameter models before taking homotopy inverse limits. Pullback along $B_n\hookrightarrow B_{n+1}$ gives strict towers of the coefficient, connection, augmentation, split-flag and virtual-pair models from steps 3.2, 4.2, 4.3, 5.2, 6.2, 6.3, 7.1, 7.2, 8.1, 7.3, 8.2, 9.1, 10.1, 11.1, and commutes with their $\mathrm{\Gamma }$ systems, all tensor systems and Bott telescopes. Thus the arrows $\mathbf{V}(X_n)\leftarrow \mathbf{A}(p_n)\leftarrow {\mathbf{A}}^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}(p_n)\to \mathbf{P}(B_n)\to \mathbf{V}(B_n)$ give a map of tower zigzags before any inverse of a weak equivalence is selected. Apply functorial fibrant replacement and take their homotopy limits; a levelwise weak equivalence of these towers remains a weak equivalence of their homotopy limits. This last assertion follows directly from the sequential path-object construction as the homotopy fiber of $1-\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}$ on the product: products of weak equivalences of fibrant spectra are weak equivalences, and the long homotopy exact sequence gives the same for their homotopy fibers. The evaluation maps in step 15.1 identify the localized towers with $C_n=F((X_n{)}_{+},KU)$ and $A_n=F((B_n{)}_{+},KU)$. Hence this gives the actual map between their complete homotopy limits. Equivalently, on a path model it takes $c_n$ and its path to $r_n{c}_{n+1}$ to its transferred value and the transferred path, with the comparison path supplied by this tower zigzag. The splitting is defined by the zero-augmentation section and the natural morphism $0\to A$ of step 17.1; these already commute with the tower restriction functors. Taking the same bar, Bott and path constructions therefore carries that natural transformation to a splitting of the complete homotopy-limit maps. No uniform-in-$n$ finite augmentation or equality only on ${\pi }_0$ has been used.

For completeness the inverse-limit assertion follows directly from this model. The homotopy limit of a spectrum tower is the homotopy fiber of $1-\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}:\prod C_n\to \prod C_n$. In degree $q$ its homotopy exact sequence is $0\to \mathrm{coker}(1-\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}:\prod K{U}^{q-1}(X_n)\to \prod K{U}^{q-1}(X_n))\to K{U}^q(\mathrm{hocolim}{X}_n)\to \ker (1-\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}:\prod K{U}^q(X_n)\to \prod K{U}^q(X_n))\to 0.$ The last identification with the function spectrum of the homotopy colimit can be seen without an extra convergence theorem: the mapping telescope is made from $X_n\times [n,n+1]$ with the endpoint identifications. A map out of it is precisely a sequence of maps and the connecting paths above. The kernel is $\lim$ and the cokernel is the sequential ${\lim }^1$ by these explicit formulas. Step 18.1 maps this homotopy-fiber construction, including its connecting paths, so induces the corresponding maps on the kernel and cokernel. Its splitting therefore retains phantom classes and all ${\lim }^1$ terms. No injectivity assertion is deduced by discarding a nonzero ${\lim }^1$.

For the ordinary-cohomology assertion one can use a finite-dimensional collapse, independently of the family analytic index. Embed the complex flag $F$ equivariantly into a real orthogonal $G$-representation: its projective embedding sends a line to its orthogonal rank-one projection in the real vector space of Hermitian endomorphisms. This is a smooth embedding, and compactness gives a $G$-invariant tubular neighborhood with a positive constant radius after choosing invariant metrics. For $P\mathrm{/}T\to B$, associate this embedding to $P$; it embeds the flag bundle in the vector bundle $W=P{\times }_{G}V$. Its normal bundle $\nu$ has the orientation specified by $TF\oplus \nu =W{\mathrm{\mid }}_F$. Collapsing the complement of the tubular neighborhood gives $\mathrm{Th}(W)\to \mathrm{Th}(\nu )$. Compose its cohomology pullback with the two oriented Thom isomorphisms from [F15]. This defines $p_{!}:H^q(P\mathrm{/}T;\mathbb{C})\to H^{q-\dim F}(B;\mathbb{C})$. The Thom module identity gives $p_{!}(p^{\ast }ab)=a{p}_{!}(b)$ on the actual cochain-derived maps. The ambient representation, its metric, the tubular radius and the fiber embedding are the SAME on every base; associating them to a pullback principal bundle pulls back the entire collapse. Thus the construction gives coherent maps of the cochain spectra on the compact flag tower, and the path model of step 18.1 extends them to its homotopy limit. This does not assume a topological-to-analytic index theorem.

Forgetting the last frame vector is a sphere fibration; induction on $r$ shows that $V_r({\mathbb{C}}^{r+n})$ is at least $2n$-connected. Thus the mapping telescope $E_{\mathrm{\infty }}$ is weakly contractible: a sphere or homotopy disk, being compact, maps into a finite subtelescope, and becomes null in a later sufficiently connected stage. It has CW type and is therefore contractible (extend a contraction cell by cell, using the vanishing relative homotopy obstructions). The telescope of the actual principal bundles $E_n\to B_n$ is a numerable principal $G$-bundle over the base telescope; local trivializations on adjacent mapping cylinders give its numerable cover. Choose its one classifying map to the contractible universal model of [F14], and restrict that map to the stages. Thus the classifying maps and all their tower homotopies are restrictions of one map, rather than independently chosen stage maps. The pullback square of universal bundles identifies the homotopy fiber of $B_n\to BG$ with $E_n$: the long exact sequences for the two principal $G$ fibrations give this identification on every homotopy group, since the universal total space is contractible. Its connectivity tends to infinity. It follows that the compatible maps from the telescope of the $B_n$ to $BG$ induce isomorphisms on all homotopy groups, and hence a homotopy equivalence between their CW replacements. Thus the argument uses the genuine classifying property of [F14], not an unproved assertion that any free contractible total space automatically has that property. The same construction for $T$ identifies the telescope of $X_n$ with $BT$, and the projection is the actual map $BT\to BG$. It is not a fictitious universal compact manifold. With the fixed $G$-invariant flag geometry, each stage has the same untwisted kernel and cokernel hypothesis. Steps 18.1, 19.1 now give the splitting on the complete function spectra of $BG$ and $BT$. Taking any homotopy degree proves split injectivity in every integer degree, completing the $KU$ assertion.

Its normalization can also be checked locally. Choose $h\in \mathfrak{t}$ whose one-parameter subgroup is dense in $T$ and whose value on every root is nonzero. Such choices avoid finitely many root hyperplanes and countably many rational-character hyperplanes. The induced vector field on $F=G\mathrm{/}T$ vanishes exactly at the $T$-fixed flags, namely $N_G(T)\mathrm{/}T=W_G$: if a flag $gT$ is fixed by $T$, then $g^{-1}Tg\subset T$, so equality holds by dimension, and conversely every normalizer element gives a fixed flag. At each zero the derivative on each real root two-plane is a nonzero rotation, whose determinant is positive. Therefore the zero is transverse with oriented local index $+1$. The Euler class is the zero-section pullback of its Thom class. Pulling that class back by this transverse section, excising disjoint small zero neighborhoods and using the orientation on each normal disk proves directly that its integral is the sum of these local indices, namely $\mathrm{\mid }{W}_G\mathrm{\mid }$. This is a calculation with Thom classes, rather than an appeal to an unproved Euler-characteristic formula. Consequently $p_{!}e(Tp)=\mathrm{\mid }{W}_G\mathrm{\mid }1$ in degree zero on every base component. The map $\mathrm{\mid }{W}_G{\mathrm{\mid }}^{-1}{p}_{!}(e(Tp)-)$ therefore splits $p^{\ast }$ over $\mathbb{C}$, already on the coherent tower maps. Passing to their actual homotopy limits as in step 18.1 preserves the splitting and its inverse-limit and ${\lim }^1$ components.

Apply the internal family-index and flag-splitting claim proved in steps 1.1, 2.1, 3.1, 3.2, 4.1, 4.2, 4.3, 5.1, 5.2, 6.1, 6.2, 6.3, 7.1, 7.2, 7.3, 8.1, 8.2, 9.1, 10.1, 11.1, 12.1, 13.1, 14.1, 15.1, 16.1, 16.2, 17.1, 17.2, 18.1, 19.1, 19.2, 20.1 and 20.2 to the actual $H^1\to L^2$ families of [F7]. Its coefficient, augmentation and matched-quotient splitting resolutions, virtual-pair word bar, all-arity tensor maps, actual Hopf-line Bott telescope and resolved evaluation are constructed in steps 3.2, 4.1, 4.2, 4.3, 5.1, 5.2, 6.1, 6.2, 6.3, 7.1, 7.2, 7.3, 8.1, 8.2, 9.1, 10.1, 11.1, 12.1, 13.1, 14.1, 15.1 and 16.1. Thus the resulting map is the $KU$-module spectrum morphism of step 15.1, retaining the relative boundary paths of step 16.1. The positive zero-augmentation splitting is step 17.1. The compact Stiefel flag tower and complete homotopy-limit construction are steps 17.2, 18.1, 19.1, 19.2 and 20.1, with ordinary Thom/Euler normalization in steps 19.2 and 20.2. These are locally proved internal steps, not a load-bearing reference to a separately stated unproved helper or an inference from its $K^0$ classes. No finite augmentation on an infinite universal base is chosen.

Let G be compact connected and T a maximal torus. The compact Lie algebra is the orthogonal sum of its center and its derived semisimple algebra: for an invariant positive inner product, B(x,x)=-||ad $x\mathrm{\mid }{\mathrm{\mid }}_{HS}^2$ on the center complement, so Cartan's criterion applies there. Complexify the semisimple part. Choose positive roots and the decomposable line $\mathrm{\ell }={\mathrm{\Lambda }}^N(n_{+})$ in ${\mathrm{\Lambda }}^N(g_C)$, N the number of positive roots. This is an ACTUAL G-representation under the exterior power of the adjoint action, so no Lie-algebra representation is silently assumed to descend to G. The line has regular highest weight2rho. The existing projective flag construction identifies the orbit of this line under the complex semisimple group with its smooth projective flag; its line stabilizer is the Borel, equivalently the normalizer of $n_{+}$. The stabilizer inside G is precisely T. Indeed a compact-real-form element preserving $n_{+}$ also preserves its conjugate $n_{-}$, so belongs to the intersection of their Borel normalizers, the complex Cartan. Its compact intersection is T. At the Lie-algebra level each positive complex root contributes its compact real two-plane, so the G-orbit map has full real tangent dimension in the complex flag. Its image is open, and compactness makes it closed; connectedness of the complex flag makes it all of that flag. Thus G/T is identified with this smooth complex flag. Give ${\mathrm{\Lambda }}^N(g_C)$ its G-invariant Hermitian metric by compact averaging. The induced Fubini-Study form restricts to a G-invariant Kähler form on the projective flag. Central torus factors act trivially and do not alter the argument. This constructs the exact analytic flag geometry used below without an algebraic-to-analytic cohomology comparison. Equivalently, an explicit invariant Kähler form can be obtained by choosing a regular element h in the positive Cartan chamber. The coadjoint-orbit form at h is ${\omega }_h(X^{\mathrm{\#}},Y^{\mathrm{\#}})=⟨h,[X,Y]⟩$, with an invariant positive inner product on the compact Lie algebra. Its exterior differential vanishes by Jacobi. On each positive-root real two-plane, choose the complex structure for which ${\omega }_h(X,IX)>0$. This is the complex flag structure just identified, and gives an invariant Kähler metric. Its (0,1)-cotangent isotropy weights all have one sign in the positive-root cone. Every sum of a positive number of these weights is nonzero, since evaluation on h is strictly of that sign. We can now prove the required analytic Dolbeault vanishing directly, avoiding a hidden GAGA or analytic Borel–Weil–Bott premise. The existing compact Kähler Hodge and commutator suppliers say ${\mathrm{\Delta }}_d=2{\mathrm{\Delta }}_{dbar}$ and that every de Rham class has a unique harmonic representative. Every g in connected G is isotopic to the identity, so it induces the identity on de Rham cohomology (Cartan's homotopy formula along the isotopy). Since the metric is G-invariant, each harmonic form is consequently G-invariant. A harmonic (0,j)-form is also de Rham harmonic; its value at the base point must be T-invariant. For $j>0$, the isotropy weight computation shows that the exterior (0,j)-space has no invariant vector. Thus every such harmonic form is zero. In degree0 the harmonic functions are precisely constants on the connected compact flag. Therefore the vertical Dolbeault Dirac operator has its even kernel canonically C with trivial G-action and its odd kernel zero. The same reasoning is valid when the flag is a point. These are the only harmonic spaces used below. Associate this complex to any principal G-bundle P->B. The family pi:P/T->B is locally the fixed flag family with transition operators given by the unitary G-action. The kernel of the vertical Dolbeault family is canonically the constant-function line; the opposite kernel is zero. Consequently ind_family(1)=1 in KU^0(B). For coefficients pulled back from B the same kernel is exactly the pulled-back coefficient. For this untwisted family the positive chiral map is surjective with kernel the canonical constant line. For a coefficient pulled back from any compact parameter base, the same map is surjective with kernel exactly that coefficient bundle; the zero auxiliary space therefore gives the actual output pair $(V,0)$. For any other augmentation the canonical morphism $0\to A$ gives the common-stabilization comparison with matched quotient $A$, exactly as in step 17.1. Its nerve is the natural splitting homotopy in step 17.1, and the same morphism defines every restriction and tensor coherence; this is not merely equality of fiber indices. Construct the universal tower from a faithful unitary embedding $G\hookrightarrow U(r)$: put $E_n=V_r({\mathbb{C}}^{r+n})$, $B_n=E_n\mathrm{/}G$ and $X_n=E_n\mathrm{/}T$. Each is compact, $X_n\to B_n$ is this flag family, and the closed inclusions pull back its invariant geometric data. The connectivity of $E_n$ tends to infinity, so their homotopy colimits give $BG$ and $BT$. The relative-parameter naturality proved in steps 16.1, 16.2, 17.1, 17.2 and 18.1 supplies maps of these function-spectrum towers and a coherent identity ${\tau }_n{p}_n^{\ast }\simeq 1$. Its path-fiber construction gives $\tau p^{\ast }\simeq 1$ on the FULL homotopy-limit function spectra. Thus tau_pi pi*=id: KU^*(BG)->KU^*(BG) and restriction to $BT$ is split injective in every degree, including any Milnor classes. This conclusion uses only existence of the elementary family Fredholm index and the explicit harmonic flag calculation. It does NOT equate an analytic family index to a topological Thom pushforward, invoke a local heat-index formula, or use differential K-theory. It therefore does not create a circular prerequisite for the differential index theorem.

For a torus of rank r, $BT=(C{P}^{\mathrm{\infty }}{)}^r$. Approximate it by $(C{P}^N{)}^r$. These spaces have finite CW decompositions with cells only in even dimensions. The cofibration long exact sequence in KU and Bott periodicity show by induction over cells that $K{U}^1$ is zero and $K{U}^0$ is a finite-rank free abelian group. Restriction along increasing N is surjective in $K{U}^0$, because the relative spaces likewise have only even cells and their relative $K{U}^1$ vanishes. For this countable compact tower the mapping telescope and its function spectrum have the explicit model $\mathrm{hofib}(1-\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}:\prod F((({\mathbb{C}\mathbb{P}}^N{)}^r{)}_{+},KU)\to \prod F((({\mathbb{C}\mathbb{P}}^N{)}^r{)}_{+},KU))$. Its homotopy exact sequence has kernel $\lim$ and cokernel ${\lim }^1$, with no other derived-limit term. The flag transfer of steps 18.1, 19.1, 19.2 and 20.1 is a map on the complete function spectra obtained from its compact Stiefel flag tower. The projective-space tower displayed here gives a different presentation of its $BT$ source: the actual mapping-telescope evaluation equivalence identifies its homotopy limit with $F(B{T}_{+},KU)$. Compose with that equivalence to express the same complete transfer on this presentation. No levelwise flag-transfer map on the individual $({\mathbb{C}\mathbb{P}}^N{)}^r$ stages is asserted or needed; in particular no Milnor class is lost by equating two unrelated towers. The Milnor sequence therefore gives $K{U}^{odd}(BT)=0$ and $K{U}^{even}(BT)={\lim }_N$ $K{U}^0((C{P}^N{)}^r)$, a torsion-free group. The ${\lim }^1$ term of the even-group tower is zero by surjectivity, and the odd-group tower is identically zero. The complexified Chern character on BT is injective. On each finite approximation the cellular Atiyah–Hirzebruch filtration has free even associated-graded groups and the Chern character induces the usual rational isomorphism, so its kernel is torsion and hence zero. If a global class has zero complexified character, each finite restriction is zero; the displayed Milnor calculation then makes the global class zero. Split restriction and naturality now imply KU^odd(BG)=0, KU^even(BG) is torsion-free, c:KU^even(BG)->KU_C^even(BG) is injective. The same holds in the connective degrees $q\le 0$. Indeed the cofiber of k->KU has no homotopy in nonnegative degrees. For $q\le 0$ its q-fold suspension is still strictly coconnective, whereas the suspension spectrum of BG is connective. The cofiber long exact sequence therefore makes $k^q(BG)\to K{U}^q(BG)$ an isomorphism, with the same conclusion one degree below q. The complex cohomology of BG vanishes in odd degrees as well. The independent ordinary-cohomology splitting is the explicit transfer of steps 19.2, 20.1 and 20.2. Embed the flag into its real orthogonal representation of Hermitian projection matrices; associate the embedding, a fixed invariant tubular neighborhood and its oriented normal bundle to each $E_n\to B_n$. Thom collapse and the two oriented Thom isomorphisms give $p_{!}$ with its actual pullback and module identities. A dense regular torus vector field has precisely the Weyl-fixed flags as zeros, with a nonzero rotation on every root two-plane; all local zero indices are therefore $+1$. The defining Thom-class calculation of the Euler class consequently gives $p_{!}e(Tp)=\mathrm{\mid }W\mathrm{\mid }$. Thus $\mathrm{\mid }W{\mathrm{\mid }}^{-1}{p}_{!}(e(Tp)-)$ splits ordinary complex restriction at each compact stage and on the coherent homotopy-limit cochain spectra. In particular $H^{\ast }(BG;\mathbb{C})$ injects into $H^{\ast }(BT;\mathbb{C})$ and is even. This argument does not assume a finite global stabilization on an infinite skeleton or discard a Milnor term. It introduces no differential index premise.

For the integral torsion specialization, assume throughout this step that $G$ is compact and connected, as in the compact-connected clauses of the Statement. Only finite generation of $H^q(BG;\mathbb{Z})$ in each fixed degree is needed. The Peter–Weyl supplier applies because a compact Lie group is compact Hausdorff; its uniform density conclusion implies the finite-dimensional unitary representations separate points. Their differentials have common kernel zero: otherwise exp(tx) would be in every representation kernel for all t, contradicting point separation for x nonzero. Finite-dimensionality of the Lie algebra lets one choose finitely many of these differentials with zero common kernel. Their direct-sum representation has a compact zero-dimensional kernel, hence a finite kernel. Add one representation separating each of its finitely many nonidentity elements. This constructs a faithful finite-dimensional unitary embedding G into U(r). The Stiefel manifold $V_r(C^N)$ of orthonormal r-frames then carries a free right G-action. It is increasingly connected as N grows: forgetting the last frame vector is the sphere bundle with sphere $S^{2(N-r)+1}$, and induction starting with the unit sphere proves the required connectivity by its homotopy exact sequence. Its quotient by G is a compact smooth manifold; the map to BG has that increasingly connected Stiefel fiber. Thus each fixed $H^q(BG;Z)$ agrees with the integral cohomology of a sufficiently large compact finite-dimensional approximation, and is finitely generated. The coefficient-extension kernel is therefore exactly its torsion subgroup. This supplies the finite-type assertion used in the ordinary even-degree clause and introduces no completion theorem.

For $T\cong U(1{)}^r$, $BT\cong ({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}{)}^r$. For a character $\lambda :T\to U(1)$, the normalized linear invariant polynomial is ${\mathrm{\ell }}_{\lambda }=-(2\pi i{)}^{-1}d{\lambda }_e\in {\mathfrak{t}}^{\vee }$; its Chern–Weil form is the normalized curvature of the associated universal line. This identity can be checked on the finite stage ${\mathbb{C}\mathbb{P}}^1$: [F9] identifies that curvature class with the first Chern class, and [F20] identifies its restriction as the degree-two generator. By products and the finite even-cell calculation, this gives the completed graded Chern–Weil isomorphism for $T$ with any finite-type graded complex coefficients. Choose $h\in \mathfrak{t}$ off the finitely many nonzero root hyperplanes, so its Lie centralizer is $\mathfrak{t}$. For any $x\in \mathfrak{g}$, maximize $⟨\mathrm{Ad}(g)x,h⟩$ on compact $G$. At a maximum, differentiation in every $Y$ gives $⟨Y,[\mathrm{Ad}(g)x,h]⟩=0$, hence $\mathrm{Ad}(g)x\in \mathfrak{t}$. This proves directly that every element of the compact Lie algebra is conjugate into $\mathfrak{t}$. Restriction of invariant polynomials is therefore injective: a polynomial vanishing on $\mathfrak{t}$ vanishes everywhere. Its image is the Weyl-invariant subring by the actual Chevalley restriction/extension supplier, applied to the complexified semisimple part and tensoring with the central polynomial algebra. On cohomology, restriction $H^{\ast }(BG;\mathbb{C})\to H^{\ast }(BT;\mathbb{C})$ is injective by the ordinary transfer above, and its image is Weyl-invariant because normalizer conjugations of $T$ induce maps of $BT$ over $BG$. If $x$ is a complex cohomology class of $BG$, choose by Chevalley an invariant polynomial $p$ with restriction equal to $\mathrm{res}(x)$ under the torus comparison. Naturality gives $\mathrm{res}(q_G(p)-x)=0$, and injectivity makes $q_G(p)=x$. The same restriction argument proves injectivity of $q_G$. Completion with graded finite-type coefficients is levelwise, with the surjective towers and zero ${\lim }^1$ proved above. Hence the natural map from invariant forms to complex cohomology is a Chern–Weil isomorphism for compact connected $G$; it has not been inverted outside that scope.

For compact connected G, step 23.1 makes the odd complex group in degree q-1 zero and the even integral Chern character in degree q injective for even $q\le 0$. The coefficient exact sequence therefore makes $k_{C\mathrm{/}Z}^{q-1}(BG)=0$, and step 4.4 proves the asserted curvature isomorphism. In ordinary cohomology, step 24.1 identifies $q_A$ with Chern–Weil; if $H^{q-1}(BG;A_C)=0$, the cokernel in the exact sequence of step 4.4 vanishes, giving precisely the stated forgetful identification. For $A=Z$ and even q, compact connectedness gives odd complex cohomology zero by step 23.1; finite-type integral cohomology in each fixed degree is proved by the explicit compact Stiefel approximation in step 23.2, so the kernel of complexification is exactly its finite torsion subgroup. This proves the stated ordinary short exact sequence in that scope. All general-Lie flat information is retained in the exact sequences, and no curvature-kernel injectivity outside its stated vanishing hypothesis is inferred. Naturality and products were constructed at the pullback level in step 4.4, completing every claim.