The zero Harrison parameter follows from continuity on one fixed sphere

Apply [F3] to $C$ to obtain finitely many positively oriented charts ${\varphi }_i:U_i\to V_i\subset {\mathbb{R}}^2$ and nonnegative smooth ${\rho }_i$ with ${\sum }_i{\rho }_i=1$ and $\mathrm{supp}{\rho }_i$ compact in $U_i$. For a continuous two-form $\omega$ along $C$, let $a_i$ be the coefficient of ${\rho }_i{i}^{\ast }\omega$ in the chart, where $i:C\hookrightarrow M$ is the inclusion and $(i^{\ast }\omega {)}_p(v,w)={\omega }_{i(p)}(d{i}_{p}v,d{i}_{p}w)$ defines the continuous pullback pointwise, and extend it by zero to ${\mathbb{R}}^2$. The compact support lies inside $V_i$, so this extension is continuous, including across the chart edge. Choose a closed nondegenerate rectangle $Q_i$ whose interior contains that support and set $I_C(\omega )={\sum }_i{\int }_{Q_i}{a}_i(x)dx$. Each summand exists by [F4]; enlarging a rectangle adds only zero contributions, by coordinate-slice additivity. This defines the claimed finite chart sum for continuous forms.

Nondegeneracy of each $g_q$ gives the continuous inverse through the adjugate divided by its nonzero determinant. The factor $\sqrt{\mathrm{\mid }\det g_q\mathrm{\mid }}$ is continuous as well. The pointwise formula in [F2], interpreted algebraically for continuous fields, therefore defines ${\star }_{g_q}{F}_q$ with jointly continuous coefficients. This algebraic map is coordinate independent because it is characterized pointwise by the metric wedge pairing; it agrees with the smooth Hodge star whenever the metric and form are smooth. Pullback along the fixed smooth inclusion preserves joint continuity, so both pulled-back two-form families have jointly continuous coefficients on $[0,1]\times C$.

For the specified fixed primitive in [F1], the coefficients of $W_q$ are polynomials in $q$ with fixed smooth spacetime coefficients, and hence are jointly continuous. The same holds for $B_q$ and $D_q$. Where $D_q>0$, also $B_q\mathrm{\ne }0$; their reciprocals, $A_q$, $g_q$ and the displayed electromagnetic formula are therefore jointly continuous by their finite sums, products and quotients. The factor $\sqrt{1-q^2}$ is continuous on the closed interval, although no parameter differentiability at one is needed. This proves the asserted sufficient continuity condition for that single family; existence of its primitive and the stated regularity region are supplied hypotheses.

Given a second such localization $({\tau }_j,{\psi }_j)$, insert ${\sum }_j{\tau }_j=1$ into every summand of the first. Each ${\rho }_i{\tau }_j{i}^{\ast }\omega$ has compact support inside $U_i\cap U_j^{\mathrm{\prime }}$. On the overlap the continuous top-form coefficients transform by the determinant of $D({\psi }_j{\varphi }_i^{-1})$. This transition is an orientation-preserving smooth diffeomorphism between open Euclidean sets; the determinant is positive. Apply the compact-support change-of-variables formula in [F4] to these continuous zero-extended coefficients. Its support lies inside the target overlap and its preimage is compact under the inverse transition, so all its hypotheses hold. Thus the two chart integrals for each product agree. Finite linearity identifies both localization sums with the same double sum, proving independence. For smooth $\omega$ this is exactly the chart construction of the pullback integral in [F2], so it agrees with the existing geometric fluxes. Neither construction uses an induced positive-definite metric on $C$.

Use the one fixed localization of step 1.1 for either family in step 1.2 and write its zero-extended coefficients as $a_i(q,x)$. These are jointly continuous on $[0,1]\times {\mathbb{R}}^2$: their common spatial supports lie in the compact chart supports of ${\rho }_i$, away from every artificial chart edge. Choose fixed rectangles $Q_i$ containing those supports. The set $[0,1]\times Q_i$ is a closed bounded box in ${\mathbb{R}}^3$, with its Euclidean metric, hence compact by [F5]. Heine-Cantor applies to $a_i$ on this box and gives ${\sup }_{x\in Q_i}\mathrm{\mid }{a}_i(q,x)-a_i(q_0,x)\mathrm{\mid }\to 0$ as $q\to q_0$, including one-sided limits at the endpoints. By [F4], $\mathrm{\mid }{I}_C({\omega }_q)-I_C({\omega }_{q_0})\mathrm{\mid }\le {\sum }_i\mathrm{vol}(Q_i){\sup }_{x\in Q_i}\mathrm{\mid }{a}_i(q,x)-a_i(q_0,x)\mathrm{\mid }\to 0$. The finite sum proves continuity of both integrals and, after division by $4\pi$, both fluxes.

Apply the continuity just proved to the identities assumed on $0<q<1$. Letting $q\downarrow 0$ gives $Q_E(C,0)=1$ and $Q_M(C,0)=0$; letting $q\uparrow 1$ gives $Q_E(C,1)=Q_M(C,1)=0$. Independently, the regular electromagnetic formula in [F1] has $B_1=2$ and finite $A_1$, and its factor $\sqrt{1-q^2}$ vanishes at one, so $F_1=0$. Every integral and parameter limit has been taken on the same compact sphere; no limit at spatial infinity or exchange with such a limit occurs.