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1 \section{The fixed seven period target}\label{sec:fixedseven}
2 At seven, principal units are not automatically seventh powers. The residue class of $8=1+7$ is a simple warning. The away from seven period simplification cannot be copied without an error bound. The inherited integral marked family supplies an error in $1+7^{7c}\Oloc_F$, where $c=v_7(C)\ge1$ and $F=\Q_7(\sqrt5)$. That specific error is sufficiently deep for the stated period window.
3 
4 Let $X=[\phi]$ and $H=[3]$ in the mod seven local Kummer group. Exact logarithmic first digits give
5 \[
6 L(\phi)=5\overline{\sqrt5},\qquad L(3)=6,\qquad L(\sqrt5)=3,
7 \quad L(u)=\frac{u^{48}-1}{7}\pmod7.
8 \]
9 The two unit coordinates $X,H$ are independent, and $[\sqrt5]=4H$. For a rational seven adic unit $A$, put $q_7(A)=(A^6-1)/7$ modulo seven. The local marked period comparison has
10 \[
11 P=ZI+2XR,\qquad Z=(3q_7(A)+5)H,
12 \]
13 with $R$ from \eqref{eq:periodmatrix}.
14 
15 The actual coefficient action must intertwine the toric submodule and quotient period matrices. Their valuation matrix is exactly $7cI$, so the two action matrices agree after cancellation over $\Z_7$. Their reduction commutes with $R$ and has characteristic polynomial $T^2-5$. The retained centralizer argument then gives one orientation $\epsilon\in\{1,-1\}$ and local scalar class
16 \begin{equation}\label{eq:kappa7}
17 \kappa_7=2\epsilon wX+(3q_7(A)+5)H.
18 \end{equation}
19 Merely knowing $R^2=5I$ would not identify this as the actual coefficient action. The period intertwiner is the extra geometric input.
20 
21 The globally normalized natural class \eqref{eq:rigidalpha} restricts to $vwX+(r+4)H$. Comparing it with \eqref{eq:kappa7} allows one common nonzero graded scalar $g=v/(2\epsilon)$. The remaining coordinate gives
22 \[
23 2r+1=\epsilon v(3q_7(A)+5),
24 \]
25 and hence
26 \begin{equation}\label{eq:fermatquotient}
27 (2r+1)^2=v^2(3q_7(A)+5)^2\quad\text{in }\F_7.
28 \end{equation}
29 The scalar is not independently adjustable in the two unit coordinates.
30 
31 For the final bounded patterns $r=6$. Patterns A and B, with $u=\pm3w$, require $q_7(A)\in\{0,6\}$ and $A^6\equiv1$ or $43$ modulo $49$. Patterns C and D, with $u=\pm w$, require $q_7(A)\in\{1,5\}$ and $A^6\equiv8$ or $36$. Each tuple still permits twelve rational unit residues $A$ modulo $49$. None is eliminated.
32 
33 \subsection{Higher precision retains a valuation coordinate}
34 For $1\le j\le7c-1$, the deep period errors vanish in $F^\times/F^{\times7^j}$. This follows because seventh powering takes $1+7^s\Oloc_F$ bijectively onto $1+7^{s+1}\Oloc_F$ for $s\ge1$. The guaranteed window does not assert that actual errors become nontrivial at the next precision.
35 
36 Use the three coordinate basis
37 \[
38 X_j=[\phi],\qquad H_j=[3],\qquad T_j=[7].
39 \]
40 The third coordinate is valuation. The actual local scalar class in the window is
41 \begin{equation}\label{eq:higherkappa}
42 \kappa_j=2\epsilon wX_j+\Lambda_jH_j+7cT_j,
43 \end{equation}
44 where
45 \[
46 \Lambda_j=7\ell_j(C_0)-3\ell_j(A)-3-5d_j,
47 \quad \ell_j(a)=\frac{\log(a^6)}{\log(3^6)}\pmod{7^j},
48 \quad d_j=\frac{\ell_j(5)}2.
49 \]
50 At level $49$, with $Q_A=(A^6-1)/7$ modulo $49$ and $q_0=(C_0^6-1)/7$ modulo seven,
51 \begin{equation}\label{eq:level49}
52 \Lambda_2=47+38Q_A+14Q_A^2-7q_0\pmod{49}.
53 \end{equation}
54 Both $C_0$ and the coordinate $7c[7]$ disappeared modulo seven. Neither may be omitted modulo forty nine.
55 
56 At higher actual lattice depth, put $m=\lfloor n/2\rfloor+1$. The inherited diagonal theorem supplies an actual global upper extension $\alpha_j$ for $1\le j\le m$. In the common range $1\le j\le\min(m,7c-1)$, its local coordinates satisfy $U_j=2\epsilon wg_j$, $V_j=\Lambda_jg_j$, $W_j=7cg_j$ for one unit $g_j$. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision $1\le j\le m$, even beyond the proved unit period window. Consequently
57 \[
58 v_7^{\mathrm{trunc}}(W_j)=\min(j,1+v_7(c)).
59 \]
60 At depth one, only the residual global triangular precision is supplied. Replacing the actual mod forty nine representation by an arbitrary triangular Kummer lift would erase its lower $7\beta$ entry. The local higher class exists, but its global comparison must retain the nontriangular deformation.
61 
62 The archive contains exact seven adic fifth root positive controls for all admitted local residues. Their $B$ values solve $B^5=C^7-A^3$ in $\Q_7$. They are not asserted rational or ordinary integers. The higher local period criterion therefore provides real conditions while still leaving every bounded tuple locally possible.

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