Graduate field guide · 11 September 202601 / Orientation

Eight pages · Number theory · #decompwlj

From prime gaps
to divisor geometry.

A pedagogical introduction to the decomposition into weight × level + jump, for graduate students and researchers meeting it for the first time.

a(n)=k(n)L(n)+d(n)

A canonical rule: the next gap sets the divisor threshold.

Given two consecutive terms of an increasing sequence, decompwlj turns their spacing into a condition on divisors. Its characteristic plots are the geometry of that exact rule. The framework was introduced by Rémi Eismann; this guide develops its elementary foundations, explains the prime patterns, and separates proofs, computations, and conjectures. [1]

All 78,495 decomposable primes up to one million in square log10 weight–level coordinates. Weight class is above the diagonal; level class is below; the twin-prime column is teal. — **One prime, one plotted point.** All decomposable primes $p\le10^6$ ; coincident points overlap. Blue: weight class. Rust: level class. Teal: weight 3, the lesser twin primes.

Reading route

02Build the coordinates
03Recover the natural-number sieve
04Prove the prime structure
05Read columns and level lines
06Measure rarefaction
07Map the open questions
08Reproduce and investigate

Prerequisites. Divisibility, congruences, prime factorization, and elementary asymptotic notation. The advanced sieve result is explicitly attributed; the elementary proofs are given in full.

ExactDefinitions determine the coordinates.

GeometricEqual scales preserve the classification.

AuditableFresh computations through one million.

Framework: Rémi Eismann · Exposition: GPT–Astra01 / 08

Decompwlj · Definitions02 / Construction

Two adjacent terms determine everything

Build the coordinates.

Let $a(1)<a(2)<\cdots$ be a strictly increasing sequence of positive integers. For one pair write $a=a(n)$ , $b=a(n+1)$ . Coordinates depend on the sequence, through the successor.

Definition. First take the jump

d=b-a

. Set

l=\begin{cases}a-d,&a>2d,\\0,&a\le2d.\end{cases}

If

l>0

, define

k=\min\{t\in\mathbb N:t>d,\ t\mid l\}

and

L=l/k

. Otherwise set

k=L=0

. The stored triple always has the order $(k,L,d)$ .

Existence and uniqueness Proved

l>0\quad\Longleftrightarrow\quad a>2d\quad\Longleftrightarrow\quad b<\tfrac32a.

Proof. Substitute $d=b-a$ for the equivalences. If $l>d$ , then $l$ itself is admissible, so the finite divisor set has a unique minimum $k$ . The quotient $L$ is determined, and $a=kL+d$ . Since $0<d<k$ , this is Euclidean division. Uniqueness uses the minimum rule; the equation alone does not select one factor pair. □

Classification. Weight: $k\le L$ (including ties). Level: $k>L$ . Unclassified: $k=L=l=0$ . The zero triple is a sentinel; the decomposition identity is asserted only when $l>0$ .

Modular formulation. For positive integers $t$ ,

a\bmod t=d\iff t>d\ \text{and}\ t\mid(a-d).

Thus $k$ is the smallest modulus that reproduces the next gap as a remainder. The strict threshold is essential.

Worked prime pairs, independently checked against the paper’s Table 2. [1]
Prime p	Next prime	g	l	k	L	Class
5	7	2	3	3	1	Level
7	11	4	0	0	0	Unclassified
11	13	2	9	3	3	Weight · tie
13	17	4	9	9	1	Level
17	19	2	15	3	5	Weight
19	23	4	15	5	3	Level
31	37	6	25	25	1	Level

Read the pair $13\to17$ . Here $l=9$ , but divisor 3 does not exceed the gap 4. Thus $13=9\times1+4$ : the weight is composite, not the smallest prime factor.

Notation: lower-case l is the reflected value; capital L is the level.02 / 08

Decompwlj · Arithmetic foundations03 / Base case

The sieve reappears after a shift

Start with the natural numbers.

For $a(n)=n$ , the jump is 1. For $n\ge3$ , the smallest divisor of $n-1$ exceeding 1 must be prime: a composite minimum would have a smaller nontrivial divisor. Therefore

k(n)=\operatorname{spf}(n-1),\qquad L(n)=\frac{n-1}{\operatorname{spf}(n-1)}.

Base-case theorem. If $n-1$ is prime, then $L=1<k$ . If $n-1$ is composite, its smallest prime factor satisfies $k\le\sqrt{n-1}$ , so $k\le L$ . Thus the level class is exactly the primes shifted by one. Assigning each composite to its least prime factor recovers the partition underlying Eratosthenes’ sieve. Repeatedly removing the least prime factor of the current quotient recovers the full factorization. □

Natural numbers from 3 through 30,000 on equal log10 axes. The level class is exactly the horizontal line L equals one; all remaining terms are above or on the diagonal. — **Figure 1 · The base case.** All $3\le n\le30\,000$ . The horizontal line $L=1$ is exactly $n=q+1$ with $q$ prime.

**Figure 1 · The base case.** All $3\le n\le30\,000$ . The horizontal line $L=1$ is exactly $n=q+1$ with $q$ prime.

Factor pairs of 15 on the hyperbola k times L equals 15, with identical linear scales. The prime 17 selects the point 3,5 and prime 19 selects 5,3 because their gaps differ. — **Figure 2 · Same product, different threshold.** The pairs $17\to19$ and $19\to23$ both give $l=15$ . Their jumps choose opposite sides of the diagonal.

**Figure 2 · Same product, different threshold.** The pairs $17\to19$ and $19\to23$ both give $l=15$ . Their jumps choose opposite sides of the diagonal.

The divisor-window criterion Proved

For any decomposable term,

k\le L\iff\exists t\mid l:\quad d<t\le\sqrt l.

Proof. Since $kL=l$ , the inequality $k\le L$ is equivalent to $k\le\sqrt l$ . Such a $k$ lies in the stated window. Conversely, if a divisor $t$ lies there, minimality gives $k\le t\le\sqrt l$ , hence $k\le L$ . □

A useful bound. In the level class, $L<k$ and $L\mid l$ . If $L>d$ , then $L$ would be a smaller admissible divisor than $k$ , a contradiction. Hence $L\le d$ . This bound drives both efficient search and the prime arguments on the next page. □

The natural-number case recovers a sieve partition, not a new factoring complexity bound.03 / 08

Decompwlj · Complete elementary proofs04 / Primes

The threshold meets prime arithmetic

What can be proved directly?

Write $p=p(n)$ , $p'=p(n+1)$ and $g=p'-p$ . For a decomposable prime, $l=p-g=2p-p'$ . The gap is even and $l,k,L$ are odd. Also $\gcd(l,g)=1$ : a common divisor would divide $p$ , whereas $0<g<p$ . Consequently $\gcd(k,g)=\gcd(L,g)=1$ .

The only non-decomposable primes are 2, 3, 7 Proved

Proof. Nagura’s theorem supplies a prime between $x$ and $6x/5$ for every $x\ge25$ . Hence for primes $p\ge29$ , $p'<6p/5<3p/2$ . For the remaining primes $2,3,5,7,11,13,17,19,23$ , direct inspection gives exactly the three failures. In particular $5=3\times1+2$ is decomposable. This proof uses a classical interval theorem; the WLJ step is the criterion on page 2. [3] □

The twin-prime column Proved

p>3:\qquad k=3\quad\Longleftrightarrow\quad g=2.

Proof. If $k=3$ , the positive even gap satisfies $g<3$ , so $g=2$ . Conversely, if $p,p+2$ are prime and $p>3$ , then $p\equiv5\pmod6$ . Thus $l=p-2$ is divisible by 3. Since 3 is the first integer exceeding 2, $k=3$ . □

Rigidity modulo 3 Proved · C7/C8

3\mid l\quad\Longleftrightarrow\quad6\nmid g.

Proof. Both $p$ and $p'$ are nonzero modulo 3. If $3\mid g$ , they have the same residue, and $l=p-g\not\equiv0$ . Otherwise their residues are opposite: $p'\equiv-p$ , so $l=2p-p'\equiv3p\equiv0$ . Because $g$ is even, $3\mid g$ is equivalent to $6\mid g$ . □

Geometric consequence. The level bound

L\le g

sharpens to

L\le g-1

because

L

is odd and

g

even. The missing and populated columns reflect genuine congruence restrictions.

The Cube Lemma Proved · sharp

If a prime is level-classified and its weight is composite, then

l\le(g-1)^3.

Proof. Write $k=uv$ with $1<u,v<k$ . Both $u$ and $v$ divide $l$ . Minimality forces $u,v\le g$ ; parity strengthens this to $u,v\le g-1$ . The preceding level bound gives $L\le g-1$ . Thus $l=uvL\le(g-1)^3$ . □

Sharpness. The consecutive pair $131\to137$ gives $g=6$ , $l=125$ , $k=25$ , $L=5$ and $125=5^3$ . This proves sharpness of the inequality. It does not prove that a finite list exhausts composite-weight level primes: $g$ is unbounded.

Proof labels in this guide describe the exact statement displayed.04 / 08

Decompwlj · Reading the graphs05 / Geometry

Learn to read the two wings

Columns are arithmetic events.

Set $u=\log_{10}k$ and $v=\log_{10}L$ . The diagonal $u=v$ separates the two classes. With equal scales, the signed perpendicular distance from that diagonal is $\log_{10}(L/k)/\sqrt2$ . It directly measures the weight–level ratio.

Prime weight–level plane through one million. Vertical bands encode fixed weights, horizontal bands fixed levels. The two classes lie on opposite sides of the dashed diagonal. — **Figure 3 · Prime fingerprint.** Every decomposable prime $p\le10^6$ is plotted. The three unclassified primes are omitted because $\log0$ is undefined.

Four reading rules

Vertical columns: a fixed weight $k$ ; the teal $k=3$ column is exactly the lesser twin primes.

Horizontal lines: a fixed level $L$ . The lowest line is $L=1$ .

Upper envelope: $kL=l<p\le x$ , so $u+v<\log_{10}x$ . The triangular boundary partly comes from the cutoff.

Equal products: $kL=l$ is a hyperbola on linear axes and a line $u+v=\log_{10}l$ on logarithmic axes. Different terms generally have different $l$ .

Colour identifies class. Overplotting obscures density; use the census for counts.

Two exact column identities Proved

k=5\iff g=4\ \text{and}\ 5\mid l.

Proof. If $k=5$ , then $g=2$ or $4$ . Gap 2 forces weight 3, leaving gap 4. Conversely, if $g=4$ and $5\mid l$ , the first integer above the threshold is already a divisor. □

k=7\iff\begin{cases}g=6,\ 7\mid l,\ \text{or}\\g=4,\ 7\mid l,\ 5\nmid l.\end{cases}

Proof. Again $g=2$ is excluded. At gap 6, 7 is the first candidate. At gap 4, the only smaller odd candidate is 5; even divisors cannot divide $l$ . □

Counts of primes up to one million in odd weight columns 3 through 25. Weight 3 has 8168 primes, weight 5 has 2702, and weight 7 has 3677. Composite weights are rust. — **Figure 4 · Column occupancy is uneven.** Full counts for the displayed weights, $p\le10^6$ . Rust bars mark composite weights; teal marks weight 3. Weights 5 and 7 draw from different gap sources.

The bridge is exact: $r\mid l\iff p\equiv g\pmod r$ . Divisibility becomes a residue condition on a gap event. A probability such as $1/(r-2)$ needs additional hypotheses. [2, §6]

All weight–level figures use square canvases and equal horizontal and vertical scales.05 / 08

Decompwlj · Data and asymptotics06 / Rarefaction

A measured trend and a separate proof question

How rare is the level class?

Let $N_{\mathrm{lev}}(x)$ count level-classified primes $p\le x$ , and let $N_1(x)$ count those with $L=1$ . For $x\ge7$ , the decomposable population is $D(x)=\pi(x)-3$ . The observed share is $f(x)=N_{\mathrm{lev}}(x)/D(x)$ .

**Table 1 · Fresh exact census.** All starting primes $p\le x$ , with the next prime retained even beyond the cutoff.
x	D(x)	N_lev	N₁	Ties	Share
1,000	165	75	24	3	45.45%
10,000	1,226	390	135	6	31.81%
100,000	9,589	2,658	880	8	27.72%
1,000,000	78,495	18,353	5,953	12	23.38%

Measured level-class shares at cutoffs 1000, 10000, 100000 and one million. Prime shares fall from 45.45 percent to 23.38 percent. Two explicitly simulated random controls also fall, but differ from each other. — **Figure 5 · Model choice matters.** One realization per control. Retain integers independently with probability $\min(1,1/\log n)$ , or only odd integers with probability $\min(1,2/\log n)$ ; apply the same classifier to consecutive retained terms.

At $10^6$ , the three shares are 23.38%, 24.08%, and 28.85%, respectively. Similar decay in the controls is not evidence of universality or prime-specific structure. Residues and column statistics retain finer arithmetic information.

Project Theorem 1: rarefaction.Internal proof*

N_{\mathrm{lev}}(x)\ll\frac{x\log\log x}{(\log x)^{3/2}}.

The project marks this PROVED*: internally audited, awaiting external refereeing. The later treatise agrees; public founding sources still state a conjecture. [2, §3 and §8] [6, §6.2]

What the proof architecture establishes

Take $M=\lceil(\log x)^{3/2}\rceil$ . Telescoping gives $O(x/M)$ gaps $g>2M$ . For $g\le2M$ , composite-weight level terms have $p=O(M^3)$ by the Cube Lemma. The others have prime weight $P=k$ , yielding

P,\qquad LP+2m,\qquad LP+4m,\qquad g=2m,\ L<2m.

A dimension-three Selberg sieve then sums over coefficients. Consecutiveness may be dropped for an upper bound. This outline omits the required uniform sieve and averaged singular-series estimates.

The implication is elementary. By $\pi(x)\sim x/\log x$ , the stated estimate gives $f(x)\ll\log\log x/\sqrt{\log x}\to0$ . Zero relative density does not decide whether the level class is infinite. □

All logarithms in asymptotic formulas are natural logarithms.06 / 08

Decompwlj · The theorem–conjecture boundary07 / Open questions

A dictionary preserves the difficulty of a problem

What remains to be understood?

If $l$ is prime, then $k=l$ and $L=1$ . The converse fails: $13\to17$ and $31\to37$ have composite reflected values 9 and 25. When $l(n)=p(n-i)$ , the notation level $(1;i)$ records which earlier prime is reflected.

l(n)=p(n-1)\iff p(n)-p(n-1)=p(n+1)-p(n).

Thus level $(1;1)$ is exactly the balanced-prime condition. Infinitely many balanced primes would imply infinitely many level-classified primes. The reverse implication does not follow.

The conditional level-one law.Conditional

N_1(x)\sim2C_2\frac{x}{(\log x)^2},\qquad C_2=\prod_{\substack{q>2\\q\ \mathrm{prime}}}\left(1-\frac1{(q-1)^2}\right).

The project derives this under uniform Hardy–Littlewood hypotheses for prime triples and a Gallagher-type hypothesis for consecutive gaps. The local Euler factors yield the exact constant $2C_2\approx1.32032$ ; a correct constant identity does not prove the asymptotic. [2, §4]

One exact algebraic step. For $q\ge5$ , cancellation gives

\frac{1-3/q}{(1-1/q)^3}\,\frac{(q-1)(q-2)}{q(q-3)}=\frac{q(q-2)}{(q-1)^2}=1-\frac1{(q-1)^2}.

This is the Euler factor of the twin-prime constant. The special factors at 2 and 3 and the averaging normalization must also be included in the full calculation.

**Founding conjectures, with their statements kept distinct.** C-labels refer to the paper and OEIS page. [1] [4]
Item	Statement	Status and interpretation
C1	Infinitely many primes have weight 3.	Open Exactly the twin-prime conjecture.
C2	Every odd weight $k\ge3$ occurs infinitely often.	Open Contains C1; requires gap and divisor-minimality conditions.
C3	Every odd level $L\ge1$ occurs infinitely often.	Open Fixed-level infinitude.
C4	Only $13,31,113,131,887$ are level-classified with composite weight.	Separate closure needed The Cube Lemma is proved. This exact list is checked here through $10^6$ ; that does not establish global closure.
C5 / C6	Infinitely many level $(1;1)$ primes / infinitely many level $(1;i)$ primes for each fixed $i\ge1$ .	Open C5 is balanced-prime infinitude; C6 is a stronger family.
C7 / C8	$6\nmid g\Rightarrow3\mid l$ and its contrapositive.	Proved Elementary modulo-3 argument, page 4.
C9	$N_{\mathrm{lev}}(x)/\pi(x)\to0$ .	Project theorem* Internally audited sieve argument; page 6.

A native question. Is the level class infinite? It is weaker than proving infinitude of the balanced subfamily and is not settled by a zero-density upper bound. Identifying the asymptotic contribution from levels $L>1$ is another distinct problem.

No implication for the Riemann hypothesis or Goldbach is established by these results.07 / 08

Decompwlj · Reproducibility and references08 / Research practice

A small, reproducible starting point

Compute carefully. Ask precisely.

The following is the supplied PARI/GP kernel, reformatted without changing its rule. Its input contract is a pair of positive integers $a<b$ ; it returns $[k,L,d]$ . The official GP documentation states that fordiv visits integer divisors in increasing order, so the first divisor exceeding the gap is the weight. [5]

decomp(a,b) = {
  my(d = b-a, l);
  if(a <= 2*d, return([0,0,d]));
  l = a-d;
  fordiv(l,k, if(k > d, return([k,l/k,d])))
}
\\ Examples: decomp(13,17) = [9,1,4]
\\           decomp(131,137) = [25,5,6]

Source: supplied decompwlj_fordiv.txt. Factoring $l$ has a cost; this is not a factoring complexity claim. GP was unavailable here, so the audit used an independent integer implementation of the definition.

Verification record

All graph coordinates and counts were recomputed. The first 17 prime triples and four census checkpoints match the references. All decomposable rows through $10^6$ pass reconstruction, threshold, parity, divisor-window, twin-column, modulo-3, and Cube-Lemma checks.

The last pair $999983\to1000003$ gives $(k,L,g)=(137,7299,20)$ . Retain the successor beyond the cutoff.

What the controls mean

The controls use NumPy’s PCG64 generator, seeds 20260911 and 20260912. These are individual realizations. Exact plotted shares are embedded in the HTML’s audit record.

The constant was checked by a prime product through $10^6$ , with a logarithmic tail bound of order $10^{-6}$ , sufficient for the printed digits.

Three productive next questions. Separate cutoff geometry from congruences; count columns at fixed gaps, retaining minimality; compare level-one and higher-level strata with specified controls. Name the population, normalization, and hypotheses of each conjecture.

Source and status note

The numbered ninth edition mentioned in the project brief was not among the supplied files and was not located. The consulted ledger is the supplied eighth edition (7 August 2026), checked against the later pedagogical treatise (20 August 2026). The rarefaction asterisk is retained; the Cube Lemma is distinguished from C4’s global exception assertion. Supplied historical graphs were inspected; all figures here are newly computed.

Rémi Eismann. Decomposition into weight × level + jump and application to a new classification of primes, arXiv:0711.0865v4 (2010). Definitions, original conjectures, Table 2.
Supplied project exposition. Fable5_decompwlj_deep_analysis_8th_edition_final_2026_08_07.html, §§3–8 and cumulative corrections log. Internal theorem statuses and analytic outline.
Jitsuro Nagura. On the interval containing at least one prime number, Proc. Japan Acad. 28 (1952), 177–181.
OEIS. Construction and conjectures; A117078 (weight), A117563 (level). Atlas: decompwlj.com. Public sources checked 11 September 2026.
PARI/GP. Official control-statement reference: fordiv. Supplied reference kernel: decompwlj_fordiv.txt.
Later project treatise. decompwlj_pedagogical_treatise_census_1e8_2026-08-20.html, §§6.1–6.3. The ledger preserves the distinction between elementary proof, finite verification, and internal analytic claim.

Self-contained HTML · Embedded figures and mathematics · Eight A4 print pages08 / 08

From prime gapsto divisor geometry.