Exact first differences, least periods, and dyadic images

Proof of Lemma 7.1

At a natural argument the exact Newton expansion is finite. Vandermonde gives

For \(1\leq i\leq H\), \(v_2(\binom Hi)\geq v_2(H)-v_2(i) \geq v_2(H)-\lfloor\log_2k\rfloor\); terms with \(i>H\) vanish. Thus each Newton summand with \(k\geq2\) has valuation at least \(b+v_2(H)+1\). The \(k=1\) summand is \(Ha_1\), of exact valuation \(b+v_2(H)\), proving (7.2).

The coefficient hypothesis implies \(a_k\to0\) dyadically, so the Newton series converges uniformly on \(\mathbb Z_2\), where \(\binom xk\) is integral-valued. For distinct \(x,y\in\mathbb Z_2\), approximate the pair by natural integer pairs with their difference valuation eventually equal to \(v_2(x-y)\). The corresponding function differences have the fixed exact valuation \(b+v_2(x-y)\). A limit of elements with that fixed valuation has the same valuation: the normalized residues are units, and \(\mathbb Z_2^\times\) is closed.

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Proof of Theorem 7.2

The actual reciprocal recurrence is

For \(N=4\), the four kernels are

The threshold-free theorem gives \(v_2(a_k)\geq k\) in every address. The following exact values cover all orders where that bound does not already imply (7.1):

They are finite rational evaluations of (7.6), followed by \(a_k=\sum_{h=0}^k(-1)^{k-h}\binom kh C_j(h)\). For addresses zero and one, \(k\geq3\) is covered by the generic bound; for address two, \(k\geq7\) is covered; for address three, \(k\geq5\) is covered. In each range the elementary inequality is \(k-\lfloor\log_2k\rfloor\geq b_j+1\). Lemma 7.1 proves the normalized identity. Multiplication by \(j!\) proves the original identity.

A row displacement \(H\) preserves the original coefficient modulo \(2^r\) at even one row exactly when \(v_2(H)+B_j\geq r\). The least positive such integer is \(2^{\max(0,r-B_j)}\), and the same displacement works at every row. This proves both optimality and the absence of a starting threshold.

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Proof of Corollary 7.3

The exact difference identity gives \(v_2(T_j(x)-T_j(y))=v_2(x-y)\) and \(T_j(0)=0\). For every \(s\geq1\), the map induces an injection of \(\mathbb Z_2/2^s\mathbb Z_2\) into itself. It is a bijection because both sets have \(2^s\) elements. These quotient bijections are compatible with reduction. Inverting them gives a compatible preimage system for every dyadic integer, proving surjectivity. Multiplication by \(j!\) changes the image ideal by exactly \(v_2(j!)\). The finite quotient bijection is also precisely the residue enumeration in a least row period.

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Proof of Theorem 7.4

Every slope \(N/i\) in the source has valuation at least two, since \(i\leq j<N/2\). Hence \(g(1)\equiv1\pmod4\) and \(g(2)\equiv1\pmod8\). The first two Newton coefficients from the reciprocal recurrence are

The first has valuation exactly one. Also \(\binom{2N}{N}\equiv6\pmod8\) for every \(N=2^a\geq2\). To see this, write

The even-\(h\) factors give \(B_{a-1}\); the odd-\(h\) factors are one modulo four for \(a\geq2\). Since \(B_{a-1}\) has valuation one, \(B_a\equiv B_{a-1}\pmod8\), and \(B_1=6\). Therefore \(a_2\equiv6-1+2+1=0\pmod8\). The global Mahler bound covers all \(k\geq3\), completing the criterion with \(b=1\). The period and image deductions are the same as above. The proof also covers \(N=2,j=0\), whose kernel is identically one.

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Proof of Proposition 7.5

Rank four was proved above. At ranks two and eight the finite criterion is verified by the actual rational recurrence (7.6). The following valuation lists record every initial Newton coefficient needed in addition to the generic tail \(v_2(a_k)\geq k\):

The first entries can also be read directly from \(a_1=-1-\binom{N+j}{j}^{-1}\). At rank eight they are respectively

All later entries in the table are finite rational arithmetic, with full numerator and denominator records retained in the accompanying exact certificate all_rank_extensions_verified.json. Every entry with \(k\geq2\) meets \(b_{N,j}+1+\lfloor\log_2k\rfloor\). Beyond the displayed cutoff, the inequality \(k\geq b_{N,j}+1+\lfloor\log_2k\rfloor\) holds at every order. Thus the finite table and the proved infinite tail establish the full hypothesis of Lemma 7.1. This is a finite coefficient proof followed by a universal tail argument, rather than extrapolation from sampled rows or steps.

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