Theorem. Block means preserve the hierarchical precision [fdip-009A]
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  • November 3, 2026

Let \(p\ge 2\) and \(k\ge 1\) be integers, and let the sites be the residues modulo \(p^k\). For \(0\le j\le k\), let \(E_j\) replace the value at each site by the mean of the sites congruent to it modulo \(p^{k-j}\). Each class has \(p^j\) sites, so \(E_0\) is the identity and \(E_k\) is the mean of every site. These are the averages of the nested blocks. Set \(P_j=E_j-E_{j+1}\) for \(j<k\) and \(P_k=E_k\). For nonzero real coefficients \(c_0,\ldots ,c_k\), the hierarchical precision \(Q=\sum _{j=0}^{k}c_jP_j\) is invertible.

Let \(A\) send a function \(\xi \) on these sites to the residues modulo \(p^{k-1}\) by the class means \[(A\xi )(r)=\frac {1}{p}\sum _{t=0}^{p-1}\xi (r+tp^{k-1}).\] The matrix \((AQ^{-1}A^{T})^{-1}\) equals the hierarchical precision \(Q'\) of depth \(k-1\) with coefficients \[c'_i=pc_{i+1}\qquad (0\le i\le k-1).\] If every \(c_j\) is positive, \(Q\) is the precision of a centered Gaussian law. In the counting basis, the second-moment matrix of \(A\xi \) is \((Q')^{-1}\).

Fix \(\lambda >0\) and a positive mass \(m\), and take \(c_j=\lambda ^j\) for \(j<k\) together with \(c_k=m\). Replacing \(A\xi \) by \(Z=\sqrt {p\lambda }\,A\xi \) divides the precision matrix \(Q'\) by \(p\lambda \). The resulting matrix is hierarchical of depth \(k-1\), with coefficient \(\lambda ^i\) on each projector below the top level and with top coefficient \(m/\lambda \). That top coefficient grows when \(0<\lambda <1\), stays unchanged when \(\lambda =1\), and shrinks when \(\lambda >1\). In the sense that a mass is relevant when it grows toward coarser scales, relevance holds only for \(0<\lambda <1\). The same coefficient relation repeats at every depth: the second-moment matrix of each block mean is the inverse of the coarser hierarchical precision. These matrix relations do not by themselves produce a Gaussian measure on functions on the \(p\)-adic integers.

Proof. 

  • November 3, 2026

Each \(E_j\) is symmetric, because its matrix is constant on the pairs inside a class and zero across classes, and \(E_j^2=E_j\), because a class mean is unchanged by a second average. If \(i\ge j\), congruence modulo \(p^{k-i}\) is coarser than congruence modulo \(p^{k-j}\), so each coarser class is a union of finer classes. A function constant on a coarser class is constant on each finer class inside it, and therefore \(E_jE_i=E_i\). That identity does not use the sizes of the classes. For the other order, every finer class has \(p^j\) sites, so the mean on a coarser class is the mean of the finer means and \(E_iE_j=E_i\).

In particular \(E_{j+1}E_j=E_jE_{j+1}=E_{j+1}\). Then \(P_jE_{j+1}=E_jE_{j+1}-E_{j+1}^2=E_{j+1}-E_{j+1}=0\). If \(i>j\), then \(i\ge j+1\). The size-free identity gives \(E_{j+1}E_i=E_i\), so \(E_if=E_{j+1}(E_if)\) and the image of \(E_i\) lies in the image of \(E_{j+1}\). When \(i<k\) the same left factor gives \(E_{j+1}E_{i+1}=E_{i+1}\), and both summands of \(P_i\) land in that image. When \(i=k\), \(E_{j+1}E_k=E_k\). A vector in the image of \(P_i\) is therefore \(E_{j+1}\) applied to something, and \(P_jE_{j+1}=0\) gives \(P_jP_i=0\). Both matrices are symmetric, so the opposite product vanishes as well. For \(j<k\) the square expands to \(E_j-E_jE_{j+1}-E_{j+1}E_j+E_{j+1}^2\), which equals \(E_j-E_{j+1}-E_{j+1}+E_{j+1}=P_j\). The case \(P_k=E_k\) is symmetric and idempotent because \(E_k\) is. The sum of the projectors telescopes to \(E_0\). Thus the \(P_j\) are pairwise orthogonal projectors summing to the identity, and \(Q\) multiplies the image of \(P_j\) by \(c_j\). Its inverse multiplies that image by \(c_j^{-1}\). The functions carry the counting inner product. Since \(P_j\) is symmetric and idempotent, \(\xi \cdot P_j\xi =|P_j\xi |^2\), and the quadratic form is \(\xi \cdot Q\xi =\sum _j c_j|P_j\xi |^2\). If \(\xi \neq 0\), some \(P_j\xi \) is nonzero because the projectors sum to the identity. Whenever every coefficient is positive, the form is therefore positive, and it is the precision of a centered Gaussian. By the inverse-precision theorem, that law has covariance operator \(Q^{-1}\). In the counting basis, the second-moment matrix of \(A\xi \) is \(AQ^{-1}A^{T}\).

Write \(S\) for the lift that makes a coarse function constant on each class: \((Sg)(r+tp^{k-1})=g(r)\). Averaging a constant returns the constant, so \(AS=I\). The counting pairing gives \((A^{T}g)\cdot \xi =g\cdot A\xi \), and the right-hand side expands to \[\sum _r g(r)\frac {1}{p}\sum _{t=0}^{p-1}\xi (r+tp^{k-1}).\] The coefficient of \(\xi (r+tp^{k-1})\) is \(g(r)/p\), so \(A^{T}=S/p\). The class mean is \((E_1\xi )(r+tp^{k-1})=(A\xi )(r)\). Averaging those \(p\) equal values returns \((A\xi )(r)\), hence \(AE_1=A\) and \(AP_0=0\).

For \(j\ge 1\), the coarse average \(E'_{j-1}\) is formed on the residues modulo \(p^{k-1}\) by classes congruent modulo \(p^{k-j}\). Fix a site \(x=r+tp^{k-1}\). Since \(j\ge 1\), the step \(p^{k-1}\) is divisible by \(p^{k-j}\), so the fine class of \(x\) modulo \(p^{k-j}\) consists of the sites \(r'+t'p^{k-1}\) whose coarse residue satisfies \(r'\equiv r\) modulo \(p^{k-j}\). Each such \(r'\) occurs for all \(p\) choices of \(t'\), and \(Sg\) depends only on \(r'\). The fine mean is therefore the coarse mean of \(g\), which is \((E'_{j-1}g)(r)=(SE'_{j-1}g)(x)\). Thus \(E_jS=SE'_{j-1}\). Applying \(A\) and using \(AS=I\) gives \(AE_jS=E'_{j-1}\), and \(AE_jA^{T}=E'_{j-1}/p\).

For \(1\le j\le k-1\) this difference is \(AP_jA^{T}=(E'_{j-1}-E'_j)/p\). The top term is \(AE_kA^{T}=E'_{k-1}/p\). Because \(AP_0=0\), \[AQ^{-1}A^{T}=\sum _{j=1}^{k}c_j^{-1}AP_jA^{T}.\] Reindex \(i=j-1\). The right-hand side equals \[\sum _{i=0}^{k-1}(pc_{i+1})^{-1}P'_i,\] where \(P'_i=E'_i-E'_{i+1}\) for \(i<k-1\) and \(P'_{k-1}=E'_{k-1}\). The same orthogonal decomposition on the coarser residues says that this sum is the inverse of the hierarchical precision \(Q'\) with coefficients \(c'_i=pc_{i+1}\). The matrix \(AQ^{-1}A^{T}\) is therefore invertible, and its inverse is \(Q'\). If the fine coefficients are positive, the coarse coefficients are positive as well.

Now take \(c_j=\lambda ^j\) for \(j<k\) and \(c_k=m\), with \(\lambda >0\) and \(m>0\). Then \(c'_i=p\lambda ^{i+1}\) for \(i<k-1\), while \(c'_{k-1}=pm\). Dividing \(Q'\) by \(p\lambda \) sends \(c'_i\) to \(\lambda ^i\) below the top level and sends the top coefficient to \(m/\lambda \). If \(Z\) denotes \(\sqrt {p\lambda }\,A\xi \), the second-moment matrix of \(Z\) is \(p\lambda \) times the second-moment matrix of \(A\xi \), so the second-moment matrix of \(Z\) is the inverse of \(Q'/(p\lambda )\). The top coefficient \(m/\lambda \) is larger than \(m\) when \(0<\lambda <1\), equal to \(m\) when \(\lambda =1\), and smaller than \(m\) when \(\lambda >1\).

Repeat the construction at every depth. If positive coefficients \(c^{(k)}_0,\ldots ,c^{(k)}_k\) satisfy \(c^{(k-1)}_i=pc^{(k)}_{i+1}\) for each \(i\), the second-moment matrix of the depth-\(k\) block mean is the inverse of the depth-\((k-1)\) hierarchical precision. The same matrix identity repeats at every depth. These identities do not by themselves produce a Gaussian measure on functions on the \(p\)-adic integers.