Fully localized controlled completion of the massive IIA DGKT flux family

For the quantized flux data of the smeared/averaged classical constant-mode massive Type IIA DGKT family on the $T^6\mathrm{/}{\mathbb{Z}}_3$ orientifold with its further freely acting ${\mathbb{Z}}_3$ quotient, including its three bulk and nine blow-up Kähler moduli and source tadpole, can one construct a globally compact, source-localized ten-dimensional completion, including an explicit equation prescription, microscopic O6-core description, observables and correction estimates, and establish that it solves those equations and all required Bianchi identities? The construction must recover smooth closed $H_3$ with no NS5 magnetic source and an $F_2$ current allowed to be singular at the D6/O6 loci in its classical exterior description. The equation prescription, microscopic description and estimates are themselves requested outputs, not already supplied objects or assumptions that an exact background exists. Here the bulk flux lattice is $e_{i,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\kappa }^{1\mathrm{/}3}(2\pi {\mathrm{\ell }}_s{)}^3{f}_4^i\mathrm{/}\sqrt{2}$ with $f_4^i\in \mathbb{Z}$ in the specified source cycles and R-R basis. In string units put $c_4={\kappa }^{1\mathrm{/}3}(2\pi {)}^3\mathrm{/}\sqrt{2}$ and $e_i=e_{i,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{\mathrm{\ell }}_s^3=c_4{f}_4^i$. An explicit allowed large-flux ray is $f_4^i=-N$ for all three bulk indices with $N\in \mathbb{N}$, fixed $c_4$ and allowed exceptional fluxes, and $\bar{e}=c_{4}N\to \mathrm{\infty }$; the coefficient $\bar{e}$ is not assumed integral. The sought large-$F_4$ limit (large $\bar{e}$ along such quantized source families) must retain the source family's four-dimensional moduli stabilization, weak coupling, large internal volume, and parametric AdS/KK scale separation while all string and loop corrections remain controlled. The question concerns the exact compact localized completion; first-order approximate localization that does not supply a controlled compact O6 core, first-order localization of a distinct massless-IIA T-dual family, and singular candidate brane geometries are not solutions of this statement. It does not assume a holographic dual.

The requested construction has the following tasks and types. They specify what must be constructed and checked; they do not assume that a complete exact-string equation system is currently known.

Index the family by positive integers $N\ge N_0$, for some integer $N_0\ge 1$; no vacuum at $N=0$ is required. First specify a fixed connected compact smooth six-manifold $X$ without boundary, resolving the stated orientifold covering topology, a smooth involution $\sigma :X\to X$ with ${\sigma }^2=\mathrm{i}\mathrm{d}$, an orientation, and finitely many compact oriented embedded three-manifolds $S_a\subset X$ without boundary representing the D6/O6 source loci, with the prescribed intersections and orientifold fixed loci. On $X\setminus S$, where $S={\bigcup }_a{S}_a$, construct smooth real fields $(g_6,A,\varphi ,H_3,F_0,F_2,F_4,F_6)$: a positive Riemannian metric $g_6$, dimensionless warp and dilaton functions $A,\varphi :X\setminus S\to \mathbb{R}$, a three-form, a constant and even-degree forms, respectively; require $H_3$ to extend to a smooth closed form on $X$. The classical exterior ten-dimensional string metric has the form $e^{2A}{g}_{{\mathrm{A}\mathrm{d}\mathrm{S}}_4}+g_6$, where $g_{{\mathrm{A}\mathrm{d}\mathrm{S}}_4}$ is the Lorentzian metric of signature $(-,+,+,+)$ on the universal cover of four-dimensional anti-de Sitter space with constant sectional curvature $-{\mu }^2$, hence $\mathrm{Ric}(g_{{\mathrm{A}\mathrm{d}\mathrm{S}}_4})=-3{\mu }^2{g}_{{\mathrm{A}\mathrm{d}\mathrm{S}}_4}$, and $\mu >0$ has units of inverse length. Work in natural units $\mathrm{\hslash }=c=1$ with length-valued local coordinates: the metric coefficients, $A$ and $\varphi$ are dimensionless; smooth NS/RR field components have units of inverse length in the adopted normalization. The exterior parameter $\mu$ is not identified with the separately normalized four-dimensional Einstein curvature below. External RR components, orientations, source stress tensors, charge units and Hodge-duality signs must be given in one convention. Use ${\mathrm{\ell }}_s=\sqrt{{\alpha }^{\mathrm{\prime }}}$; any source with ${\mathrm{\ell }}_s=2\pi \sqrt{{\alpha }^{\mathrm{\prime }}}$ must be converted explicitly. No Ricci-flat metric or strict $SU(3)$ structure is imposed on the localized completion.

Supply the full localized massive-IIA classical equation system recovered in the controlled exterior limit, including Einstein, dilaton, NS, RR and supersymmetry conditions. Its degree-three current equation must agree with the adopted source/tadpole convention. In a source-free patch the Bianchi components in that convention are $d{F}_0=0$, $d{H}_3=0$, $d{F}_2-F_0{H}_3=0$, $d{F}_4-H_3\wedge F_2=0$ and $d{F}_6-H_3\wedge F_4=0$; any induced higher source charges require their own specified currents. For a degree-$k$ current on $X$, use a continuous real-linear functional on smooth compactly supported $(6-k)$-forms, with $⟨dT,\eta ⟩=(-1{)}^{k+1}⟨T,d\eta ⟩$ and $T_{\alpha }(\eta )={\int }_X\alpha \wedge \eta$. A candidate must define the extensions of its singular fields and products; no product of arbitrary currents is presumed meaningful. Any regularization used to define the weak equations must have a demonstrated prescription-independent physical limit. An integrated charge balance alone is not the requested equation system.

Construct neighborhoods $C_{N,a}$ of every source locus and their intersections, a microscopic description on their union $C_N$, and overlap/matching maps with the exterior on collar regions. Specify its state/field spaces, equation or amplitude maps, source charge and stress-tensor matching, orientifold identifications, flux quantization and conservation laws. Establish well-defined finite four-dimensional kinetic coefficients and vacuum energy in that prescription and compatibility of the microscopic and exterior equations. Excision with arbitrary boundary values, or an unexplained singular supergravity expression, does not supply this construction. A strongly coupled core is allowed if its microscopic dynamics and its contribution to the observables are controlled; weak exterior coupling alone supplies no such control. Identifying a suitable microscopic massive-O6 prescription is part of the question.

Construct the physical spectrum in a fixed four-dimensional Einstein normalization, including a positive completed warped kinetic inner product, the removal of infinitesimal gauge directions, and self-adjoint realizations of the physical mass-squared fluctuation operators on explicitly specified dense domains with core matching and AdS boundary conditions. Identify the bulk Kaluza–Klein tower (the modes arising from internal excitation), its retained-modulus complement and a discrete positive mass gap; these spectral properties must be established, not assumed from the smeared geometry. Let $m_{\mathrm{K}\mathrm{K},N}>0$ be the bottom positive mass in the bulk Kaluza–Klein tower in this normalization, excluding gauge directions and the retained moduli. Define $L_{\mathrm{A}\mathrm{d}\mathrm{S},N}$ in the same Einstein normalization by $\mathrm{Ric}(g_{4,E,N})=-3L_{\mathrm{A}\mathrm{d}\mathrm{S},N}^{-2}{g}_{4,E,N}$. Supply the prescription-defined finite positive string-frame volume measure ${\nu }_N$ on $X$, agreeing with $d{\mathrm{v}\mathrm{o}\mathrm{l}}_{g_{6,N}}$ in the classical exterior and with its microscopically computed contribution in the core; writing the completed volume as an integral over $X$ refers to this measure. Establish $V_N:={\mathrm{\ell }}_s^{-6}{\nu }_N(X)\to \mathrm{\infty }$, ${\sup }_{X\setminus C_N}{e}^{{\varphi }_N}\to 0$, and $m_{\mathrm{K}\mathrm{K},N}{L}_{\mathrm{A}\mathrm{d}\mathrm{S},N}\to \mathrm{\infty }$. Retain the source family's stabilized moduli: their vacuum is isolated modulo the declared gauge/periodic identifications, and its canonically normalized physical scalar spectrum is nonsingular on the stabilized directions and obeys $m^2{L}_{\mathrm{A}\mathrm{d}\mathrm{S},N}^2\ge -9\mathrm{/}4$ with the prescribed AdS boundary conditions. Any unstabilized directions must be disclosed. Computing these quantities in the completed warped geometry is required; the smeared scaling exponents alone do not establish them.

For correction control, construct a string-theoretic observable/amplitude prescription (or an equivalent microscopic description), with explicitly specified spaces, domains, regularization and physical interpretation; recover the required classical bulk/source limits rather than assigning arbitrary equation maps. It must include bulk higher-derivative, string-genus, localized source, intersecting-source and nonperturbative contributions. In the canonically normalized low-energy sector below a declared fixed fraction of $m_{\mathrm{K}\mathrm{K},N}$, specify norms on the equations and the first two variations needed to determine the vacuum and spectrum. Here an external leg is an argument of a multilinear interaction coefficient in the expansion in canonically normalized low-energy fluctuations; derivative order counts spacetime derivatives in that coefficient. For each fixed derivative order and finite number of legs, let ${\mathrm{\Gamma }}_N$ denote the corresponding exact prescription on a stated open neighborhood $U_N$ of the vacuum in the physical fluctuation space. Give a common twice continuously Fréchet differentiable map realization for ${\mathrm{\Gamma }}_N$ and its approximants on $U_N$, specifying domain and target normed spaces. Construct finite approximants ${\mathrm{\Gamma }}_N^{(q)}$, indexed by integers $q\ge 1$, and finite nonnegative dimensionless bounds $R_q(N)$ for the prescription and its first two variations on that neighborhood in the specified normalized operator norms: require ${\sup }_{u\in U_N}\mathrm{\parallel }{D}^j({\mathrm{\Gamma }}_N-{\mathrm{\Gamma }}_N^{(q)})(u){\mathrm{\parallel }}_{N,j}\le R_q(N)$ for $j=0,1,2$, with $D^0$ denoting the map itself and $D^j$ its $j$-th Fréchet derivative, and ${\inf }_{q\ge 1}{R}_q(N)\to 0$ as $N\to \mathrm{\infty }$. Norms must be tied to the physical kinetic normalization with equivalence constants independent of $N$ and $q$, so a vanishing bound cannot be produced merely by rescaling the norm. The approximation index may depend on $N$; no exchange of a flux limit and an infinite formal series is assumed. Establish relative errors tending to zero for the nonzero Einstein coefficient, AdS curvature and KK scale, and errors small enough to preserve isolation and stability on the stabilized directions. These bounds must include integrated core and intersection effects on the corrected solution. Smallness of each fixed-order term without any bound on the total remainder does not suffice. No convergence of a formal genus/derivative series is presumed; an alternative nonperturbative prescription with equally explicit accuracy bounds is admissible.

The question is therefore to construct these completion, core, observable and remainder-control data and establish their stated properties along the quantized family, or identify a genuine obstruction. It is not an existence assertion inside an already supplied exact equation theory, and it is not restricted to a four-dimensional EFT. The mathematical typing and accuracy requirements above are an explicit precision target for this research question, not results attributed to the cited partial constructions. None is asserted to have been achieved.

Use the free-period convention of Massive IIA Romans-mass and NS three-form flux quantization: $F_0$ is constant, $H_3$ is smooth and closed with no NS5 magnetic current, $(2\pi {\mathrm{\ell }}_s)F_0\in \mathbb{Z}$, and $(2\pi {\mathrm{\ell }}_s{)}^{-2}{\int }_{{\mathrm{\Sigma }}_3}{H}_3\in \mathbb{Z}$ on every free integral three-cycle. The bulk four-form lattice and its fixed coefficient spacing $c_4$ are part of the same source-model input: the equal-bulk large-flux sequence means the integer quanta $f_4^i=-N$ with $N\to \mathrm{\infty }$, not arbitrary integer-valued coefficients $e_i$. The smeared source tadpole, flux conventions, and large-$\bar{e}$ scaling data are inputs of DGKT classical smeared four-dimensional effective solution, not equations for a localized source distribution.

For any candidate whose compact internal geometry and localized sources satisfy the hypotheses of Massive IIA D6/O6 tadpole on a compact Calabi–Yau threefold, that result imposes the necessary condition $N_{D6}({\mathrm{\Sigma }}_3)-2N_{O{6}^{-}}({\mathrm{\Sigma }}_3)+\frac{F_0}{2\pi {\mathrm{\ell }}_s}{\int }_{{\mathrm{\Sigma }}_3}{H}_3=0$ on each compact oriented embedded three-manifold ${\mathrm{\Sigma }}_3\subseteq X$ without boundary that is transverse to the source cycles, with $N_{D6}-2N_{O{6}^{-}}$ the signed source-first intersection combination defined there. This conditional integrated charge identity does not solve the localized field equations or provide an O6-core completion.

The cited source record, bounded by the focused readings through 2026-10-31, leaves this construction question open. This is a source-qualified status report, not a claim that no unreviewed result exists. In particular, the absence of a complete microscopic equation/core prescription is one of the requested construction tasks; it is not used as a premise proving either existence or nonexistence. DGKT constructs the classical constant-mode/smeared four-dimensional family and leaves the fully varying warp/source profiles unresolved. Marchesano, Palti, Quirant, and Tomasiello construct an approximate localized supersymmetric DGKT-type solution with $SU(3)\times SU(3)$ structure at first order in the weak-coupling expansion and at two-derivative supergravity order (arXiv:2003.13578v1, §§6.2–6.3 and 7). They state that intersecting-source interactions first enter at the next order and leave those terms unresolved; their approximation also breaks down in a strongly coupled, substringy O6-core region. This is a substantive partial localization result, not an exact compact solution or a controlled core completion. Cribiori et al. study first-order localization for a distinct massless-IIA family related by formal T-dualities and identify uncontrolled source-core regions. Apers–Montero–Valenzuela's flux-backtracking construction gives a strongly coupled singular DGKT brane geometry and describes a worldvolume CFT as conjectural if it exists. Apers's 2026 brane-dual paper and the later Apers–Sánchez González absorption analysis provide decoupling evidence within their candidate brane geometry; the latter assumes a one-modulus truncation and finds ${\mathcal{P}}_{\mathrm{a}\mathrm{b}\mathrm{s}}\sim {\omega }^{27\mathrm{/}7}$ for DGKT, where $\omega \to 0$ is the scalar-wave frequency in that calculation and ${\mathcal{P}}_{\mathrm{a}\mathrm{b}\mathrm{s}}$ is its dimensionless absorption probability. This is a model calculation, not an empirical measurement. None of these reviewed results constructs a globally compact localized O6 solution with a controlled core and complete compact equations, or settles the stated existence question.