F4 Singular Wrapped M2 Physical Transfer Obligation

The classical action and selected matter surface presently carried into this pair do not construct the actual singular wrapped-M2 protected state sector, the light branch and vertex states, a closed transfer/domain, a regulator limit, or the combined fermion/C-field twist that identifies an index with ${\int }_{{\mathcal{S}}_R}{g}_4$. The required inputs are listed in def-f4-singular-wrapped-m2-sector-realization-data. A finite matrix or a point-matching prescription chosen to give a desired Euler characteristic is not a derivation of those inputs. This is a scoped local authoring hold, not a new physical postulate and not a claim that the standard relation is a substantiated open research problem or that no literature construction exists. The symbol ${\int }_{{\mathcal{S}}_R}{g}_4$ is the intended comparison target only when a smooth compact resolved model supplies the oriented surface cycle and flux class in the scope of the zero-mode postulate below; this remark does not assert that the integral is defined or computed for every singular or noncompact matter surface in the retained scope.

The named smooth comparison target is the adopted postulate post-f4-m2-matter-curve-zero-mode-dictionary, with its own regime: a smooth compact resolved flat sectioned elliptic Calabi--Yau fourfold, a selected smooth compact matter surface $S_R^{\mathrm{s}\mathrm{m}}$ embedded by $i:S_R^{\mathrm{s}\mathrm{m}}\hookrightarrow {\widehat{Y}}_4$ and a proper smooth submersion $q:S_R^{\mathrm{s}\mathrm{m}}\to C_R$ with closed oriented real two-dimensional fibres equal to the chosen rational curve, over a smooth compact matter curve $C_R$, algebraic gauge-background flux, the oriented fibre-integrated real class $\alpha =q_{\ast }(i^{\ast }[g_4])\in H_{\mathrm{d}\mathrm{R}}^2(C_R;\mathbb{R})$, a chosen integral lift $c\in H^2(C_R;\mathbb{Z})$ of $\alpha$, a supplied holomorphic line bundle ${\mathcal{L}}_R$ with $c_1({\mathcal{L}}_R)=c$, and a spin line ${\mathcal{S}}_R^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}$ with $({\mathcal{S}}_R^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}{)}^{\otimes 2}\cong K_{C_R}$. The postulate itself denotes the smooth matter surface and spin line by $S_R$ and ${\mathcal{S}}_R$, respectively. Under those assumptions it adopts $\mathrm{deg}{\mathcal{L}}_R={\int }_{S_R^{\mathrm{s}\mathrm{m}}}{g}_4$ and ${\chi }_R=h^0(C_R,{\mathcal{L}}_R\otimes {\mathcal{S}}_R^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}})-h^1(C_R,{\mathcal{L}}_R\otimes {\mathcal{S}}_R^{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}})$. Here $H^q(C_R,\mathcal{E})$ is cohomology of the sheaf of holomorphic sections and $h^q$ its complex dimension, finite for the supplied smooth-curve line by the postulate's declared general-line Dolbeault/sheaf comparison. The coefficient map is its degree-two integral-to-de-Rham map, and degree and $K_{C_R}$ have its explicit characteristic-class and holomorphic-cotangent meanings. The selected singular-model surface ${\mathcal{S}}_R$ in the target ${\int }_{{\mathcal{S}}_R}{g}_4$ above is a different object. The smooth zero-mode postulate records net chirality only and excludes singular curves, general non-algebraic flux and microscopic M2 quantization.

The separate postulate post-f4-mtheory-g4-shifted-quantization gives only a conditional free-period convention: for a smooth compact connected resolved Calabi--Yau fourfold ${\widehat{Y}}_4$ with holomorphically trivial canonical bundle, an oriented Hausdorff second-countable smooth eleven-manifold $Y_{11}$ with a smooth mostly-plus Lorentzian metric $g$, a time orientation $\tau$, and a principal ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(10,1)$-bundle covering its oriented time-oriented orthonormal frame bundle as specified by that postulate, and internal inclusion $i:{\widehat{Y}}_4\hookrightarrow Y_{11}$ with $i^{\ast }T{Y}_{11}\cong T{\widehat{Y}}_4\oplus {\epsilon }^3$, the adopted tension $T_{M2}=(2\pi {)}^{-2}{\mathrm{\ell }}_p^{-3}$ and unit-curvature identification give $g_4=[G_{4,\mathrm{C}\mathrm{J}\mathrm{S}}]\mathrm{/}((2\pi {)}^3{\mathrm{\ell }}_p^3)$ and $g_4+\frac{1}{2}{\rho }_{{\widehat{Y}}_4}(c_2(T{\widehat{Y}}_4))\in \mathrm{im}{\rho }_{{\widehat{Y}}_4},{\rho }_{{\widehat{Y}}_4}:H^4({\widehat{Y}}_4;\mathbb{Z})\to H_{\mathrm{d}\mathrm{R}}^4({\widehat{Y}}_4;\mathbb{R})\mathrm{.}$ It does not choose the integral lift, holomorphic line bundle or spin line and retains no torsion or differential C-field data; it has no singular-space conclusion. These postulates identify smooth comparison conventions only; neither supplies singular states, a transfer map, an index proof or the twist.

F4 Singular Wrapped M2 Sector Realization Data

M2 matter-curve zero modes for smooth algebraic flux backgrounds

Shifted M-theory G4 quantization on a resolved Calabi–Yau fourfold

The regulated protected-sector carrier and combined mathematical data carrier specify conditional input contracts. Their current source and readiness must be assessed using the native records after any interface correction. The full-AC hypothesis inherited from the spinor/characteristic prerequisites is retained explicitly in the required-data definition. Neither a current ready input contract nor the correction of an obsolete carrier-absence statement constructs the physical singular sector or its transfer comparison. This remark may be locally ready as a nonlogical statement of that obligation while the physical construction itself remains unproved. No Pfaffian/electric family theorem or physical realization is assumed to clear a status-record defect.