Rectangular flat-torus string compactification data

Assume the Axiom of Countable Choice. Let $n\ge 1$, let $R_i\in (0,\mathrm{\infty })$ be finite radii with units of length, and let ${\theta }^i\sim {\theta }^i+2\pi$ be dimensionless angular coordinates. Put $S_{R_i}^1:=\mathbb{R}\mathrm{/}(2\pi R_i\mathbb{Z})$, represented by the physical coordinate $y^i=R_i{\theta }^i$, and define the rectangular torus $T_R^n:={\prod }_{i=1}^n{S}_{R_i}^1$ with internal line element $d{s}_n^2={\sum }_{i=1}^n{R}_i^2(d{\theta }^i{)}^2$. The internal metric is constant, diagonal, flat, and Riemannian. The external space is a Lorentzian $d$-manifold $(M_d,g_{\mu \nu })$ with signature $(-,+,\dots ,+)$, and the product background metric is $d{s}_{d+n}^2=g_{\mu \nu }(x)d{x}^{\mu }d{x}^{\nu }+d{s}_n^2$, with no warp factor or moduli-dependent Weyl rescaling. The dilaton is a constant ${\mathrm{\Phi }}_0$, the internal NS two-form $B_2$ vanishes, and background field strengths and localized sources vanish.

For the minimally coupled scalar sector, define $m_0^2$ as the external mass-squared parameter in the separated equation $({\mathrm{\square }}_d+{\mathrm{\Delta }}_{T_R^n}-m_0^2)\mathrm{\Phi }=0$, where ${\mathrm{\square }}_d:=g^{\mu \nu }{\mathrm{\nabla }}_{\mu }{\mathrm{\nabla }}_{\nu }$ and ${\mathrm{\Delta }}_{T_R^n}:=\mathrm{div}\mathrm{grad}$ is the nonpositive-sign Laplace–Beltrami operator. Use natural units $c=\mathrm{\hslash }=1$, so $R_i$ has dimension inverse mass and $m_0$ has dimension mass. The coordinate characters ${\chi }_{\mathbf{n}}(\theta )=e^{i{\sum }_i{n}_i{\theta }^i}$, $\mathbf{n}=(n_1,\dots ,n_n)\in {\mathbb{Z}}^n$, form the Fourier modes of this compact internal space. Their two-derivative Kaluza–Klein masses are $m_{\mathbf{n}}^2=m_0^2+{\sum }_i{n}_i^2\mathrm{/}{R}_i^2$.

For the kinematic free closed-string product sector, use the single-circle data on each factor: for $X^i\simeq X^i+2\pi R_i$, momentum $n_i\in \mathbb{Z}$ and winding $w_i\in \mathbb{Z}$ give $p_{L,i}=n_i\mathrm{/}{R}_i+w_i{R}_i\mathrm{/}{\alpha }^{\mathrm{\prime }}$ and $p_{R,i}=n_i\mathrm{/}{R}_i-w_i{R}_i\mathrm{/}{\alpha }^{\mathrm{\prime }}$, with ${\alpha }^{\mathrm{\prime }}>0$ of dimension length squared. Let ${\mathcal{I}}_i$ be the single-circle orthonormal basis-label set from that data, including $(n_i,w_i)$ and its finite-support left and right oscillator occupations. Define the product Hilbert space to be ${\mathcal{H}}_{\mathbf{R}}:={\mathrm{\ell }}^2({\prod }_{i=1}^n{\mathcal{I}}_i,\mathbb{C})$ with its coordinate orthonormal basis. For a tuple of labels, define $h_i:={\alpha }^{\mathrm{\prime }}{p}_{L,i}^2\mathrm{/}4+N_{L,i}$ and ${\bar{h}}_i:={\alpha }^{\mathrm{\prime }}{p}_{R,i}^2\mathrm{/}4+N_{R,i}$ as in the single-circle data, and define the total product weights by $h:={\sum }_{i=1}^n{h}_i$ and $\bar{h}:={\sum }_{i=1}^n{\bar{h}}_i$. Thus the product convention means a finite product of independent circle basis sectors with additive left and right weight labels. It supplies no further OPE, sewing, modular-invariance, or interacting-string data. This product statement does not extend to off-diagonal metric or $B$-field moduli, the general Narain $O(n,n;\mathbb{Z})$ action, or arbitrary T-duality transformations.

Kaluza–Klein product and warped reduction data

Compact Free String Circle Spectrum

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Flat torus model geometry

The Fourier basis and Parseval's identity on the finite torus

Arvanitakis–Hohm–Hull–Lekeu use dimensionless coordinates of period $2\pi$ and the convention ${\alpha }^{\mathrm{\prime }}=2$ in §4; Ginsparg's compact-boson convention is $p_L=m\mathrm{/}(2r)+nr$, $p_R=m\mathrm{/}(2r)-nr$. These are compatible with the current single-circle formula after the convention change $R=2r$, ${\alpha }^{\mathrm{\prime }}=2$ and the momentum/winding label identification. The cited treatments support the circle data and torus scope, not a locally proved general Narain duality or a consistent truncation theorem.

The scalar Kaluza–Klein argument is complete conditionally on its displayed equation and mathematical suppliers. The free-circle carrier and its batch-59 page placement are current and its Step-1 record is ready. The exact unwarped product clause in the KK-ansatz carrier matches [F3], and no frame-conversion calculation is used. However, the current KK-ansatz Step-1 row remains escalated and is hash-stale: its recorded conventions prerequisite was missing then, while the conventions supplier now has a ready record. This batch does not overwrite that escalation; the item remains escalated until its owner refreshes the stale readiness record.