B. General finite logarithmic recovery (Block 20)

Let \(A\) be a rational algebra, without any commutativity assumption on \(A\). For a unipotent element \(U\) and a positive integer \(d\) satisfying \((U-1)^d=0\), put

The value is independent of the chosen annihilating exponent. The finite logarithm and the nilpotent exponential satisfy

Both maps preserve every annihilating exponent, and hence preserve the exact nilpotency class of \(N\) and \(\exp(N)-1\), or of \(U-1\) and \(L(U)\). The Lean finite sum includes its zero-index term with coefficient zero; the displayed mathematical sum starts at one. The formal statements also handle the zero algebra and the exponent-zero case.

The proof evaluates Mathlib's formal power-series inverse identities at constant nilpotent series. Such evaluation is exactly a finite polynomial. For an arbitrary noncommutative ambient algebra, perform the calculation inside the commutative subalgebra generated by the element and then apply its inclusion. No analytic convergence or topology is assumed. The finite logarithm commutes with ring homomorphisms between rational algebras.

Let \(K\) be a characteristic-zero field and \(N\) an endomorphism with \(N^d=0\). For \(t\ne0\) the actual recovery formula is

All coefficients lie in \(K\). The vector spaces need not be finite dimensional, but nilpotence still means that some finite power of the whole operator is zero. A linear map intertwines two nilpotent operators exactly when it intertwines their exponential samples at a common nonzero parameter. Also

For supplied covariance data and the twisted cocycle of Block 13, let \(\rho_D(g)=\exp(t(g)N_D)r_D(g)\). If a common element \(u\) satisfies \(r_D(u)=r_E(u)=1\) and \(t(u)\ne0\), every intertwiner of \(\rho_D\) and \(\rho_E\) recovers both the original \(r\)-intertwiner and the \(N\)-intertwiner. There is no cubic nilpotence restriction. This recovers equivariant projectors for both original components. Reconstructed-action isomorphisms induce actual isomorphisms of the inertia-fixed monodromy kernels. Their local characteristic polynomials agree under the existing finite-dimensional and normal-subgroup hypotheses.

For actual finite original inertia images the common kernel sample is derived. Suppose \(u\in H\) satisfies \(\nu(u)=1\) and \(t(u)\ne0\) and both \(r_D(H)\) and \(r_E(H)\) are finite. Their sampled actions have positive finite orders \(a,b\). At \(u^{ab}\) both original actions are trivial, and

by the cocycle identity and characteristic zero. Thus the same full intertwiner equivalence holds using the actual finite-image hypotheses, without a separately supplied common trivial-action sample. The initial nonzero cocycle sample and the arithmetic interpretation of the data remain requirements.

Actual affine representations with monodromy chains of every finite length invoke these theorems. In particular a length-six example has \(N^5\ne0\) and \(N^6=0\) and is recovered from its actual reconstructed action. The length-four example still defeats the old quadratic logarithm, while the full finite logarithm recovers it.

The sample hypotheses retain substantive content. A zero parameter loses \(N\). For unrestricted supplied algebraic data, one can also absorb \(\exp(tN)\) into \(r\) and set the new monodromy to zero; the reconstructed representation is unchanged. The absorbed original action fails the required trivial-action sample condition. This is a counterexample in the algebraic datum class, not a failure of uniqueness of arithmetic Weil–Deligne attachment with finite original inertia action.

Block 20 removes the general logarithmic-extraction gap from the local algebraic comparison. It does not construct the actual arithmetic attachment, identify every intended packet, prove the coefficient and Hodge–Tate/period comparisons, or prove arithmetic Frobenius coverage. At the Block 20 checkpoint, the ledger recorded fifteen checked claims and five partial claims; see the Block 23 scope correction above.

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