Definition. Compatible compact-open reference subgroups [fcap-004C]
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    Let \(V\) be a finite-dimensional rational vector space and let \(Q\) be nondegenerate. For each rational prime \(p\), put \(V_p=\mathbb Q_p\otimes _{\mathbb Q}V\) and let \(Q_p\) be the extended form. Write \[\begin {gathered} O_p=O(Q_p),\\ SO_p=SO(Q_p),\\ \operatorname {Spin}_p=\operatorname {Spin}(Q_p),\\ \operatorname {Spin}_p\xrightarrow {\rho _p}SO_p\xrightarrow {i_p}O_p. \end {gathered}\] Here \(i_p\) is inclusion and \(\rho _p\) is the vector action. Spin is the actual vector-generated even norm-one carrier of Remark [fcap-0028], with the canonical local topology of Definition [fcap-0044]. The actions are continuous by Lemma [fcap-0045]. For a rational orthogonal or Spin point \(x\), write \(x_p\) for its scalar extension; the action compatibility is Theorem [fcap-003N].

    A compatible compact-open reference tuple for \(Q\) consists of subgroups \[U_p^O\leq O_p,\qquad U_p^{\operatorname {Spin}}\leq \operatorname {Spin}_p\] at every prime, subject to the following requirements.

    1. Each subgroup is compact and open in its ambient local group.
    2. The reference groups respect the vector action: \[i_p\rho _p(U_p^{\operatorname {Spin}})\subseteq U_p^O.\]
    3. For each \(g\in O(Q)\), there is a finite set \(S_O(g)\) of primes such that \(g_p\in U_p^O\) whenever \(p\notin S_O(g)\).
    4. For each \(s\in \operatorname {Spin}(Q)\), there is a finite set \(S_{\operatorname {Spin}}(s)\) such that \(s_p\in U_p^{\operatorname {Spin}}\) whenever \(p\notin S_{\operatorname {Spin}}(s)\).

    The exceptional sets may depend on the rational point. The special orthogonal reference subgroup is determined by these data: \[U_p^{SO}=i_p^{-1}(U_p^O).\] Thus an element of \(SO_p\) belongs to \(U_p^{SO}\) exactly when its underlying orthogonal transformation belongs to \(U_p^O\), and \(\rho _p\) restricts to \(U_p^{\operatorname {Spin}}\to U_p^{SO}\).

    The subgroup \(U_p^{SO}\) is open by continuity of \(i_p\). It is compact because \(i_p\) is the closed embedding cut out by \(\det =1\), as in Lemma [fcap-0045]: its image inside \(U_p^O\) is closed in that compact set. If \(g\in SO(Q)\), determinant commutes with scalar extension, so the rational orthogonal membership condition gives \(g_p\in U_p^{SO}\) outside \(S_O(i(g))\), where \(i:SO(Q)\to O(Q)\) is inclusion. No further special orthogonal family or exceptional-set axiom is needed.

    For an affine finite-type scheme over a discrete valuation ring, its integral points are open and closed in its fraction-field points, as in [conrad2011adelic, Example 2.3, p. 3]. The paired orthogonal and Spin data above additionally impose compatibility and rational membership. TauCeti packages those exact fields in TauCeti.QuadraticMap.OrthogonalCompactOpens and defines the derived subgroup in TauCeti.QuadraticMap.OrthogonalCompactOpens.specialOrthogonal. Its compactness is TauCeti.QuadraticMap.OrthogonalCompactOpens.isCompact_specialOrthogonal. The formal structure allows arbitrary rational quadratic forms; the nondegenerate setting here supplies the existence construction in Proposition [fcap-004D].