D.12 Conclusion
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The local arithmetic theory now has one common architecture. The finite character coordinate identifies the torsion phase. The source and kernel product reconstructs the Euler coefficient. The kernel valuation records certified precision. Carmichael and Mahler coordinates describe the remaining principal unit motion. Polyphase analysis resolves additional local digits, records carries and shifts exactly, and transports Mahler coefficients through a finite affine kernel.
The odd and dyadic cases agree at the level of convolution and differ at the level of integral geometry. At odd primes the source must be depleted before it becomes a unit mode interpolation. At the dyadic prime the source extends directly to an analytic integral function. The dyadic kernel has exact weight \(4r-1\), giving the optimal uniform modulus \(2^{4M+3}\) after depth \(M\).
Three tasks remain outside the proved region. The first is a complete finite difference estimate after the congruential triangular inverse. The second is an exact higher valuation theory for the affine Mahler kernel. The third is a global existence theorem for local systems that do not already arise from one integral source.
Current status. This dated statement is superseded for the stated divisor-branch family by Theorem 3.7, with the \(j!\) normalization and factorial valuation loss explicitly retained. It makes no claim for unrestricted families.