Paper IX visual guide · Visual 19 of 23

Newton coordinates turn precision into a finite computation

Let \(b_k=\Delta_1^kC_{4,0}(0)\). The first coefficients obtained from the difference triangle are

\[ (b_0,b_1,b_2,b_3,b_4)=(1,-2,72,-33872,60511872). \]

The exact Newton identity recovers each nonnegative integer row:

\[ C_{4,0}(n)=\sum_{k=0}^{n}b_k\binom nk. \]

Permalink · Source line 53 · Included exact Newton coefficients

Newton coefficient valuations and the eight-bit cutoff — Original figure from the retained vector PDF.

Reading the figure

Points show the recorded Newton valuations for orders one through sixteen at rank four, address zero. The dashed line is eight bits. Orders five and above lie at or above it. The source uses the coefficient theorem for the uniform truncation and separately records finite reconstruction checks through row thirty-two.

Recorded Newton valuations
Order	Valuation
1	1
2	3
3	4
4	7
5	8
6	10
7	11
8	15
9	16
10	18
11	19
12	22
13	23
14	25
15	26
16	31

Selectable labels from the original PDF

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

1
2
3
4
5
8 12 16
Newton order k
05
10
15
20
25
30
v2 of the Newton coefficient
precision cutoff: 8 bits

Original vector figure PDF · Full-size image

Permalink · Source line 59

For this address the proved law is \(v_2(b_k)=2k-s_2(k)\) for \(k\ge1\). Every \(k\ge5\) therefore has \(v_2(b_k)\ge8\). Since \(\binom nk\) is integral, those terms vanish modulo \(256\). Thus, for every \(n\ge0\),

\[ C_{4,0}(n)\equiv1+254\binom n1+72\binom n2+ 176\binom n3+128\binom n4\pmod{256}. \]

The uniform truncation follows from the coefficient theorem. The script separately checks the exact Newton reconstruction and this reduction for rows \(0\) through \(32\). Those finite checks validate the worked arithmetic without replacing the theorem.

Permalink · Source line 60

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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