Paper IX visual guide · Visual 6 of 23

Why address two has a different slope

The normalized address-two sequence obeys both

\[ \begin{aligned} v_2(\Delta_{H_1}\cdots\Delta_{H_k}C_2(n)) &\ge k+v_2(k!)+3+\sum_i v_2(H_i),\\ v_2(\Delta_{H_1}\cdots\Delta_{H_k}C_2(n)) &\ge2k+v_2(k!)+\lfloor k/2\rfloor+\sum_i v_2(H_i). \end{aligned} \]

Their maximum supplies the useful envelope.

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The two normalized address-two bounds and their envelope — Original figure from the retained vector PDF.

Reading the figure

The dashed curve is the constant-surplus bound. The dotted curve adds the paired-filtration gain. The solid curve takes their maximum. Powers of two mark the universally exact orders. These are normalized coefficients; the original Euler normalization adds one to the displayed bounds.

Mathematical labels

\(k+v_2(k!)+3\)

\(2k+v_2(k!)+\lfloor k/2\rfloor\)

Selectable labels from the original PDF

Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.

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normalized valuation lower bound
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Original vector figure PDF · Full-size image

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Original figure caption

Normalized address-two precision. The paired filtration controls the large-order slope. At order one the constant-surplus bound is stronger.

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For every base row and every positive step list, equality is proved when \(k\) is a power of two:

\[ v_2(\Delta_{H_1}\cdots\Delta_{H_k}C_2(n)) =V_2(k)+\sum_i v_2(H_i), \]

with \(V_2(1)=4\), \(V_2(2)=6\), and \(V_2(k)=2k+v_2(k!)+k/2\) for \(k=2^r\ge4\). The infinite marked family establishes sharpness at unbounded orders.

For original Euler coefficients add one, since \(v_2(2!)=1\). The extra \(\lfloor k/2\rfloor\) comes from a paired divided-power filtration in the Bernoulli transformation. It does not arise by fitting the plotted points.

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Source and conversion record

The full edition has 23 sections and 20 figures, including eight worked examples. The older visual README describes 15 sections and 12 figures. Original wording, equations and finite-check scope are retained. Descriptions and data tables are reading aids. This conversion renders existing figure PDFs without rebuilding TeX, running the mathematical generators, auditing proofs or compiling Lean.

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