Adopted compact-boson flat-gauging and rational self-duality model

Fix the idealized compact-boson full-local-CFT package $C_R$ at dimensionless radius $R>0$, with local Ginsparg radius $r=R\mathrm{/}\sqrt{2}$ and genuine primary labels $V_{n,w}$, $n,w\in \mathbb{Z}$. Adopt the source's noncompact SymTFT and two-step flat-gauging construction as restricted physical model data. First gauge $U(1{)}_w$ with flat connections, producing the noncompact intermediate scalar theory. Then gauge the period-$2\pi \mathrm{/}R$ subgroup $\mathbb{Z}\subset \mathbb{R}$ of its momentum-shift symmetry. This supplies a topological radius-changing interface $I_R:C_R\to C_{1\mathrm{/}R}$. Compose it with the selected invertible radius-duality interface $U_{1\mathrm{/}R\to R}:C_{1\mathrm{/}R}\to C_R$ to form $N_R=U_{1\mathrm{/}R\to R}\circ I_R:C_R\to C_R$.

Adopt the source-reported charge-label action, up to a nonzero overall normalization, $N_R:V_{n,w}⟼\{\begin{array}{ll}{V}_{R^{2}w,n\mathrm{/}{R}^2},& R^{2}w\in \mathbb{Z}\text{ and }n\mathrm{/}{R}^2\in \mathbb{Z},\\ \text{a non-genuine defect endpoint},& \text{otherwise}\mathrm{.}\end{array}$ A non-genuine endpoint belongs to an endpoint sector of the interface data and is not a genuine bulk local-field label. This assumption supplies the construction and its label action; it does not supply a continuum functional-measure foundation or an absolute normalization at irrational $R^2$.

For reduced rational $R^2=p\mathrm{/}q$, with positive coprime integers $p,q$, adopt the source's anomaly-free finite subgroup gauging $G_R\cong {\mathbb{Z}}_p\times {\mathbb{Z}}_q$ and select an invertible self-duality interface $U_R:C_R\mathrm{/}{G}_R\to C_R$. Under this identification, the specified gauging symmetry lines $L_{a,b}$ become the lines $g_{a,b}$ acting by $g_{a,b}{V}_{n,w}=e^{2\pi i(an\mathrm{/}p+bw\mathrm{/}q)}{V}_{n,w},0\le a<p,0\le b<q\mathrm{.}$ Use the untwisted endpoint-framed finite gauging wall $W_{G_R}:C_R\to C_R\mathrm{/}{G}_R$ with the finite-groupoid-cardinality convention: hold both endpoint trivializations fixed, weight a framed flat-bundle sector by $1\mathrm{/}\mathrm{\mid }\mathrm{Aut}\mathrm{\mid }$, and include no extra division by $\mathrm{\mid }{G}_R\mathrm{\mid }$ at the endpoints. Define the rationally normalized defect to be exactly $N_R=U_R\circ W_{G_R}$, with no additional scalar.

The two-step construction and the finite rational construction are identified here only in their source-reported charge-label action up to normalization. The exact finite-wall normalization is a selected rational model input. It is not a claim that the source proves an exact scalar match between these constructions. No continuous-gauging normalization is inferred.

Compact free boson as a full local CFT

Compact-boson momentum-winding lattice and cocycle

Topological defects, interfaces, orientation, and fusion conventions

Finite-group gauging wall conventions

Argurio et al., sections 3–3.1 and 4.2

This is an explicitly adopted source-described model construction, rather than a proof that every compact-boson algebra has the required continuum gauging realization. It records no empirical measurements or tests. The source obtains the construction using functional gauge-field integrals; this item does not provide a rigorously constructed continuum measure. Its irrational absolute normalization and twisted-Hilbert-space interpretation are not asserted.

Finite semisimplicity of the rational group-line sector and pairwise non-isomorphic simplicity of its lines remain separate hypotheses of the consuming fusion theorem and example. Neither their fusion direct sum nor noninvertibility is postulated here: those are deductions requiring the finite-wall counting result, the declared line-sector hypotheses, and the genuine-field argument. No finite fusion-category structure is assumed for an irrational-radius continuous construction.