Paper IX visual guide · Visual 5 of 23

The final rank-four congruence

For \(n\ge0\), \(k\ge1\), and \(H\ge1\), the actual Euler coefficients satisfy

\[ \sum_{i=0}^k(-1)^{k-i}\binom ki E_{4(n+iH)}^{(4,j)} \equiv0\pmod {2^{\,v_2(k!)+k v_2(H)+A_j(k)}}, \]

where

\[ \begin{aligned} A_0(k)=A_1(k)&=k,\\ A_2(k)&=\max\{k+4,\ 2k+\lfloor k/2\rfloor+1\},\\ A_3(k)&=k+2. \end{aligned} \]

Original Euler congruence exponents at unit step — Original figure from the retained vector PDF.

Reading the figure

The horizontal axis is difference order, one through thirty-two; height is bits of divisibility for the original Euler coefficients. Addresses zero and one share an exact curve. Address three has a proved lower bound. Address two has the larger lower bound, with exactness marked at powers of two. The table uses the recorded theorem-derived arrays.

Recorded theorem-derived unit-step exponents
Order	Addresses 0 and 1 (exact)	Address 2 (lower bound)	Address 3 (lower bound)
1	1	5	3
2	3	7	5
3	4	9	6
4	7	14	9
5	8	16	10
6	10	20	12
7	11	22	13
8	15	28	17
9	16	30	18
10	18	34	20
11	19	36	21
12	22	41	24
13	23	43	25
14	25	47	27
15	26	49	28
16	31	56	33
17	32	58	34
18	34	62	36
19	35	64	37
20	38	69	40
21	39	71	41
22	41	75	43
23	42	77	44
24	46	83	48
25	47	85	49
26	49	89	51
27	50	91	52
28	53	96	55
29	54	98	56
30	56	102	58
31	57	104	59
32	63	112	65

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
4
8 16 24 32
difference order k
0
20
40
60
80
100
120
bits of divisibility for the original Euler coefficient
j = 0, 1: exact
j = 3: proved lower bound
j = 2: proved lower bound
j = 2: exact at powers of two

Original vector figure PDF · Full-size image

Permalink · Source line 191

Original figure caption

The exponent of the final Euler congruence at unit step. The address zero and one curve is exact at every order. Marked address-two orders are universally exact; the other displayed address-two and address-three values are proved lower bounds.

Permalink · Source line 192

At a common step with \(v_2(H)=s\), add \(ks\) to every displayed exponent. Thus a doubling of the step gains one additional bit per difference, rather than one bit for the whole expression.

This statement combines two different proved bounds at address two. A lower bound is not automatically an exact valuation: divisibility may be stronger at unmarked orders or at particular starting rows.

Permalink · Source line 198

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.

Source hashes and conversion checks

Order	Addresses 0 and 1 (exact)	Address 2 (lower bound)	Address 3 (lower bound)
1	1	5	3
2	3	7	5
3	4	9	6
4	7	14	9
5	8	16	10
6	10	20	12
7	11	22	13
8	15	28	17
9	16	30	18
10	18	34	20
11	19	36	21
12	22	41	24
13	23	43	25
14	25	47	27
15	26	49	28
16	31	56	33
17	32	58	34
18	34	62	36
19	35	64	37
20	38	69	40
21	39	71	41
22	41	75	43
23	42	77	44
24	46	83	48
25	47	85	49
26	49	89	51
27	50	91	52
28	53	96	55
29	54	98	56
30	56	102	58
31	57	104	59
32	63	112	65

Order	Addresses 0 and 1 (exact)	Address 2 (lower bound)	Address 3 (lower bound)
1	1	5	3
2	3	7	5
3	4	9	6
4	7	14	9
5	8	16	10
6	10	20	12
7	11	22	13
8	15	28	17
9	16	30	18
10	18	34	20
11	19	36	21
12	22	41	24
13	23	43	25
14	25	47	27
15	26	49	28
16	31	56	33
17	32	58	34
18	34	62	36
19	35	64	37
20	38	69	40
21	39	71	41
22	41	75	43
23	42	77	44
24	46	83	48
25	47	85	49
26	49	89	51
27	50	91	52
28	53	96	55
29	54	98	56
30	56	102	58
31	57	104	59
32	63	112	65

Order	Addresses 0 and 1 (exact)	Address 2 (lower bound)	Address 3 (lower bound)
1	1	5	3
2	3	7	5
3	4	9	6
4	7	14	9
5	8	16	10
6	10	20	12
7	11	22	13
8	15	28	17
9	16	30	18
10	18	34	20
11	19	36	21
12	22	41	24
13	23	43	25
14	25	47	27
15	26	49	28
16	31	56	33
17	32	58	34
18	34	62	36
19	35	64	37
20	38	69	40
21	39	71	41
22	41	75	43
23	42	77	44
24	46	83	48
25	47	85	49
26	49	89	51
27	50	91	52
28	53	96	55
29	54	98	56
30	56	102	58
31	57	104	59
32	63	112	65