D.6 Mahler transfer and local analytic branches
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The finite character coordinate resolves the torsion part of a local unit. A sufficiently long arithmetic jump moves the remaining unit into a principal unit neighborhood. The coarse variable then has a convergent Mahler expansion with an explicit valuation slope.
D.6.1 Finite Mahler coordinates
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For a function \(f\) on the nonnegative integers, define
Historical theorem D.48: Finite Mahler reconstruction
Given \(f(0),\ldots,f(N)\), put \(c_r=\Delta^rf(0)\). Then
For every \(s\) with \(0\leq s\leq N\),
Proof
Let
Newton interpolation gives \(P(n)=f(n)\) for \(0\leq n\leq N\). Iterating
gives
When \(0\leq n\leq N-s\), every value used in \(\Delta^sf(n)\) lies among the prescribed data, so \(\Delta^sf(n)=\Delta^sP(n)\).
Historical corollary D.49: Mahler valuation ledger
If \(c_r\in{\mathbb Z_p}\) and \({v_p}(c_r)\geq w(r)\), then
Proof
Apply the nonarchimedean triangle inequality to Historical theorem D.48.
The infinite interpolation theorem states that every continuous function on \({\mathbb Z_p}\) has a unique convergent Mahler expansion [Mahler1958], [Robert2000]. We use that theorem as a standard input when the local variable is allowed to range over all of \({\mathbb Z_p}\).
D.6.2 Carmichael and Mahler transfer for unit modes
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Historical standard fact D.50: Local unit decomposition
For odd \(p\), every unit has a unique factorization
Every dyadic unit has a unique factorization
The first factor gives the finite character address. The second factor admits logarithmic and Mahler coordinates.
For odd \(p\) and \(L\geq1\), put
For \(p=2\) and \(L\geq3\), put
These are exponents of the unit groups modulo the declared prime powers.
Historical theorem D.51: One unit mode
Let \(u\in{\mathbb Z_p}^\times\), let \(H=\Lambda_{p,L}\), and put
For every integer address \(j\) and every \(x\in{\mathbb Z_p}\),
The \(r\)th Mahler coefficient has valuation at least \(Lr\).
Proof
Write \(u^{Hx+j}=u^j(1+q_{u,L})^x\) and use the convergent binomial series on \({\mathbb Z_p}\).
Historical example D.52: Paired principal unit expansions
At the odd prime three,
At the dyadic prime,
The formal expansions are identical. Their first local slopes are one and three because \(4-1=3\) while \(9-1=8\).
Historical corollary D.53: Basic odd and dyadic difference gains
For odd \(p\),
For odd \(u\in\mathbb Z_2^\times\),
Proof
One step difference multiplies a unit mode by \(u^H-1\). Fermat reduction and lifting the exponent give the odd estimate. For odd \(u\),
D.6.3 Polynomial exponential modes
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Historical theorem D.54: Polynomial exponential Mahler transfer
Let
with \(P\in{\mathbb Z_p}[n]\) of degree \(d\). Let \(H=\Lambda_{p,L}\), fix an address \(j\), and write
Then
where
Consequently,
For \(R\geq d\), truncation at depth \(R\) is valid modulo \(p^{L(R+1-d)}\) uniformly on \({\mathbb Z_p}\).
Proof
On the polynomial factor the difference operator acts as \(q+y\Delta\). The two terms commute. Expand \((q+y\Delta)^r\) and evaluate at zero. Since \(\Delta^kQ_j=0\) for \(k>d\), every surviving term contains at least \(q^{r-d}\).
D.6.4 Matrix state systems and finite adelic synchronization
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Let
with \(C\in M_d({\mathbb Z_p})\) invertible modulo \(p\).
Historical theorem D.55: Matrix Carmichael and Mahler transfer
Editorial hypothesis correction. The original displayed statement leaves the vectors untyped. Its integrality conclusion requires \(v_0,w\in{\mathbb Z}_p^d\), \(L\ge1\), and \(H,j,n\in{\mathbb N}\) with \(H>0\); the value at \(Hx+j\) for \(x\in{\mathbb Z}_p\) means the matrix-binomial interpolation. With \(p=3\), \(C=4\), \(L=1\), \(w=1\) and \(v_0=1/3\), one has \(\Delta_1a_0=1\), which is not in \(3{\mathbb Z}_3\). The printed claim without these restrictions is therefore not an as-written certificate. The full original proof follows under the corrected hypotheses. Assume
with \(B\in M_d({\mathbb Z_p})\). For every address \(j\) and every \(x\in{\mathbb Z_p}\),
The series converges on \({\mathbb Z_p}\), and
Proof
Use
and the matrix binomial series. One difference in the coarse variable multiplies the series by \(p^LB\).
Historical theorem D.56: Finite adelic transfer
Editorial hypothesis correction. Read this statement with \(v_0,w\in{\mathbb Z}^d\), \(H>0\) and \(j,n,r\in{\mathbb N}\), and positive integer \(L_p\). These restrictions make the integral congruences and powers well typed. Allowing \(v_0=1/3\) in the one-prime example above loses the advertised digit; invertibility of \(C\) alone does not restore it. The original proof is retained under these hypotheses. Let
for a finite set of primes. Let \(C\in M_d({\mathbb Z})\) be invertible modulo \(Q\), and choose \(H\) such that
with \(B\in M_d({\mathbb Z})\). Then for every nonnegative integer \(x\),
For every \(r\),
Proof
Apply the integer matrix binomial theorem and reduce at every selected prime.
Historical example D.57: A simultaneous Fibonacci jump
Let
The matrix \(C\) is invertible modulo twenty four, and
For every scalar coordinate of the state sequence \(C^nv_0\),
One integral jump therefore gives simultaneous congruence ladders at the primes two and three.
D.6.5 Amice pullback
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Historical definition D.58: Amice pullback
For a formal exponential generating series
define
Historical theorem D.59: Derivative conjugacy
Proof
Differentiate the composition \(F(\log(1+T))\).
Historical proposition D.60: Weighted convergence
If
and
then the series converges whenever \({v_p}(T)\geq\alpha\).
Proof
The valuation of the \(r\)th term is \({v_p}(c_r)+r{v_p}(T)\), which tends to positive infinity.
D.6.6 Odd prime depletion
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For an odd prime \(p\), the source \(S_n(p)\) contains one nonunit base. Put
and
Historical theorem D.61: Depleted odd prime reconstruction
For every even \(n\),
Proof
The removed source mode contributes \(\sigma_p\cosh(pt)\). Multiplication by \(\operatorname{sech}^2(pt)\) gives \(\sigma_p\operatorname{sech}(pt)\). Subtract it from the full source and kernel product.
Fix an even address \(j\in\{0,2,\ldots,p-3\}\). For \(x\in{\mathbb Z_p}\), define
Historical theorem D.62: Odd prime analytic Euler branch
The series defining \(\mathscr E_{p,j}^\times\) converges in the Tate algebra on \({\mathbb Z_p}\). For every nonnegative integer \(x\),
Proof
Every retained base is a unit, and its \((p-1)\)st power lies in \(1+p{\mathbb Z_p}\). The polynomial
has Gauss norm at most
This tends to zero because \(p\) is odd. The remaining factors have bounded Gauss norm. Thus the series converges in the Tate algebra. At a nonnegative integer \(x\), the binomial factor vanishes for large \(r\), and Historical theorem D.61 gives the value.
Historical remark D.63
The analytic function is defined by its unit source series. The expression \(p^{(p-1)x+j}\) in the interpolation identity is evaluated only at nonnegative integers. No analytic function \(x\mapsto p^x\) on all of \({\mathbb Z_p}\) is used.
D.6.7 Direct dyadic interpolation
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For \(x\in\mathbb Z_2\), define
The apparent singularity at \(x=-1\) is removable, with value \(-\log(9)/8\).
Historical lemma D.64: Analytic dyadic source
The function \(\mathscr R\) belongs to the dyadic Tate algebra and takes values in \(\mathbb Z_2\). For every nonnegative integer \(n\),
Proof
Put \(y=x+1\) and let \(\ell=\log 9\). Since \({v_2}(\ell)=3\),
The coefficient of \(y^m\) has valuation
which is nonnegative and tends to positive infinity. The quotient therefore belongs to \(\mathbb Z_2\langle y\rangle\). The factor \(2x+1=1+2x\) is a unit in \(\mathbb Z_2\langle x\rangle\), with inverse \(\sum_{m\geq0}(-2x)^m\). Hence \(\mathscr R\) is integral and analytic. The integer values follow from the definition of \(R_{2n}\).
Define
Historical theorem D.65: Direct dyadic Euler interpolation
The series defining \(\mathscr E_2\) converges in the dyadic Tate algebra. For every nonnegative integer \(n\),
Truncation after depth \(M\) is uniformly valid modulo \(2^{4M+3}\) on \(\mathbb Z_2\).
Proof
The monomial Gauss norm of \(\binom{2x}{2r}\) is at most \(2^{{v_2}((2r)!)}\). Since
the product tends to zero in Gauss norm. Translation by the integer \(r\) preserves the unit disc, so \(\mathscr R(x-r)\) remains an integral analytic function with uniformly bounded supremum norm. The series therefore converges in the Tate algebra.
At a nonnegative integer \(n\), every term with \(r>n\) vanishes and Historical theorem D.39 remains. For arbitrary \(x\in\mathbb Z_2\), the integer valued polynomial \(\binom{2x}{2r}\) and the source value \(\mathscr R(x-r)\) lie in \(\mathbb Z_2\). Every omitted term after depth \(M\) is therefore divisible by \(2^{4M+3}\). This proves the uniform truncation statement.
Historical theorem D.66: Unified local interpolation statement
The odd branch \(\mathscr E_{p,j}^\times\) and the dyadic branch \(\mathscr E_2\) are analytic local realizations of the source and kernel law in Historical theorem D.40. At integer arguments they recover, respectively,
on each even odd prime character branch and
on the dyadic even branch. The two realizations have distinct source corrections and distinct kernel valuation slopes.
Proof
Combine Historical theorem D.62 and Historical theorem D.65 with the source and kernel definitions in Historical theorem D.40.