Decomposition into
weight × level + jump

Abstract

This report does two things. It re-derives, with a newly written engine and without consulting any earlier output, the complete classification of every decomposable prime below $10^{10}$ — 455 052 508 decompositions — and finds every published anchor reproduced digit for digit. And it adds a piece of mathematics that the framework has been missing: a taxonomy. We show that the classification of any term of any strictly increasing sequence is decided in the first instance not by arithmetic but by growth, through a trichotomy of regimes (Theorem 1); we count the integers that survive the divisor window exactly (Theorem D); we deduce that every bounded-jump sequence has a rarefying level class, unconditionally and easily (Theorem E); and we thereby isolate what is genuinely hard about the primes, which is not that they are primes but that their gaps are unbounded. The same apparatus yields a gap-conditional model for the level rate with no fitted constant, accurate to about 9 % across 45 gap values, from which the mod-3 rigidity and the exclusion of twin primes from the level class fall out as structural consequences rather than separate facts.

Corrections log (cumulative, oldest first)

1. Species-I overclaim (first report; corrected in the second). Presenting the rarefaction of level-one primes as open overstated the case: a Selberg/Brun sieve gives $N_1(x)\ll x/(\log x)^{3/2}=o(\pi(x))$ unconditionally.
2. Two-constants erratum (second report). The cumulative Species-II ratio $f_{>1}\log x/\log\log x\approx1.13$–1.18 was read as a constant; it is a finite-size superposition transient.
3. Checkpoint off-by-one (second report's pipeline). A PARI driver bug made $p_{5\cdot10^{7}}$ come out 74 short of 982 451 653.
4. Third report's session failures (disclosed; none reached print): a monolithic engine killed by the memory ceiling; a second exceeding the time budget; an unbounded naive divisor search timing out.
5. Population convention in the Table-3 recount (third report). The paper counts weight-classified primes only and is correct under its convention.
6. Challenge paper, Table 1, entry lev1($10^8$) (upstream erratum). Table 1 prints 339 869 against Table 2's 339 870; the complete-convention count is 339 870. Independently reconfirmed this session (§4.7).
7. MathJax macro bug (first build of the third report). Rendering defect only.
8. Quartic-bound completeness overclaim (third report; repaired in the fourth by the Cube Lemma).
9. Fourth report's session failures (disclosed): a segment-parity sieve bug that passed all four anchor checks silently; an in-place patch with no write-back; a cosmetic crash.
10. Doubly-logarithmic span misquote (third report). Prose-level slip.
11. Species-II constant claim $c_2=1$ — RETRACTED (withdrawn in the fifth report). The cumulative ratio recedes from 1 at every scale while the windowed constant sits flat near 0.773. Both behaviours reproduced independently this session (§4.7); the retraction stands and $c_2$ remains open.
12. Fifth report's session failures (disclosed): an unbudgeted sweep killed by the platform ceiling; a truncated CSV; a PARI batch that executed 113 733 of 113 734 generated checks. The missing row is closed this session: all 113 734 executed (§B).
13. Montgomery unit bug — this session, caught before print. An optimisation replacing the primality test with Montgomery arithmetic computed the Montgomery unit as $R^2\bmod n$ instead of $R\bmod n$. It passed the $10^6$ anchor silently, because primes below $10^6$ take the exhaustive branch that never calls the routine, and only corrupted counts from $10^7$ upward: $N_{\mathrm{lev}}(10^7)$ came out 78 295 against the true 138 049, and the composite-weight count inflated from 5 to 9 213. It was caught by the cross-check against the fifth report's table, not by any internal invariant. The repaired routine now shows zero disagreements with brute-force primality for every $n<2\cdot10^6$ and correctly rejects the strong pseudoprimes 3 215 031 751 and 4 295 360 521. This is the fourth report's segment-parity lesson repeating exactly: an optimisation whose failure mode the small anchors cannot reach.
14. Append-mode log duplication — this session. Because the engine's logs open in append mode while its counters are checkpointed at segment boundaries, a process killed mid-segment leaves log rows that are written again when the segment is recomputed on resume. The sample log carried 150 duplicate rows and one torn line (a partial write merged with the next record). Caught by the row count against $\lfloor 455052508/4001\rfloor$. The census counters are unaffected — state is written atomically at segment end and the interrupted segment is recomputed in full — and after de-duplication the log holds exactly 113 734 well-formed distinct rows, the predicted number. Logs are de-duplicated before any use.
15. False counterexample from an unsorted construction — this session. A first taxonomy run reported a violation of the trichotomy for prime powers. The sequence had been built by a nested comprehension and was not strictly increasing; the theorem was never in question. A strict-increase assertion now guards every sequence entering the survey.
16. Census extension to $10^{11}$ abandoned — this session. The engine was launched at $10^{11}$ and reached $1.8\cdot10^{9}$ before the platform terminated it; background processes do not survive between tool calls in this environment. The run was completed to $10^{10}$ in bounded foreground chunks instead. No number in this report depends on the abandoned range, and the extension is restated as a task, not a result (§7.3).

Contents

1Significance

The decomposition into weight × level + jump is a change of coordinates on strictly increasing integer sequences. It takes an additive object — a sequence known through its gaps — and attaches to each term a multiplicative pair: the weight $k(n)$, a distinguished divisor of the shifted term $\ell(n)=a(n)-d(n)$, and the level $L(n)=\ell(n)/k(n)$, its cofactor. The identity $a(n)=k(n)L(n)+d(n)$ is exact wherever it exists, and the construction is canonical: no parameter is chosen, no rounding occurs, and the weight is pinned by a minimality condition that makes the whole map a function of the sequence alone.

Three facts carry the construction's claim on our attention, and it is worth stating them with their tags up front.

• On the natural numbers it is the sieve of Eratosthenes, not an analogy to it: the weight of $n$ is the smallest prime factor of $n-1$, so the columns of the weight histogram are the sieve's arithmetic progressions and the level-classified naturals are exactly the shifted primes proved (§3; re-verified this session to $10^6$ with zero violations).
• On the primes it turns classical objects into visible strata: lesser twin primes are literally the column $k=3$ proved, balanced primes are the stratum level $(1;1)$, Polignac families are weight columns.
• It generates exactly one problem that does not exist without it — Conjecture 9, the rarefaction of the level class — and that problem proved decidable: $N_{\mathrm{lev}}(x)\ll x\log\log x/(\log x)^{3/2}$ proved*.

What the framework has lacked is an account of which sequences it says anything about. The atlas holds on the order of a thousand decomposed sequences; until now they have been a list. This report supplies the missing organising principle, and it is blunter than one might expect:

That observation is not a deflation of the framework; it is a sharpening of it. It says where the content lives. It also converts a vague programme (“classify the atlas”) into a theorem with three cases, gives an unconditional rarefaction result for every bounded-jump sequence (Theorem E), and explains — quantitatively, from a proved counting theorem — why the measured level share of the primes has the shape $\log\log x/\log x$ that five reports have observed empirically without deriving.

Honesty about the converse is part of the framework's case. Conjectures 1–3 and 5–6 are the twin-prime, Polignac and balanced-prime problems rewritten in $(k,L)$ coordinates reformulation. A change of coordinates can make a problem legible; it cannot make it easier. We mark that line every time we cross it.

2Definitions and elementary structure

Fix a strictly increasing sequence $a(1)

The reading is Euclidean: dividing $a(n)$ by its weight leaves quotient $L(n)$ and remainder $d(n)$. The triple is unique because $k(n)$ is defined by a minimum and the rest is then forced. The classification rule: unclassified if $\ell=k=L=0$; classified by level if $k>L$; classified by weight if $0

We need $a(n)-d>d$, i.e. $a(n)>2\bigl(a(n{+}1)-a(n)\bigr)$, i.e. $3a(n)>2a(n{+}1)$.□

2.1 The first seventeen primes, recomputed

Computed this session by an independent engine and matched against Table 2 of arXiv:0711.0865 and against the self-test in decompwlj_optimized.txt: agreement in every entry. verified

n	p	k	L	d	class
1	2	0	0	1	—
2	3	0	0	2	—
3	5	3	1	2	level
4	7	0	0	4	—
5	11	3	3	2	weight
6	13	9	1	4	level
7	17	3	5	2	weight
8	19	5	3	4	level
9	23	17	1	6	level
10	29	3	9	2	weight
11	31	25	1	6	level
12	37	11	3	4	level
13	41	3	13	2	weight
14	43	13	3	4	level
15	47	41	1	6	level
16	53	47	1	6	level
17	59	3	19	2	weight

Note $p=11$: $k=L=3$, a tie, which the convention assigns to weight. Ties are rare and countable — exactly 12 below $10^6$ (A121155), reconfirmed this session.

3The fundamental theorem, the sieve, and a taxonomy of sequences

3.1 What the natural numbers show

Take $a(n)=n$, so $d\equiv1$ and $\ell(n)=n-1$. The weight is the smallest divisor of $n-1$ exceeding 1 — that is, the smallest prime factor of $n-1$ — and the level is the largest proper divisor. Hence

$L(n)=1\iff k(n)>L(n)\iff \ell(n)=n-1$ is prime.

So the level-classified naturals are exactly $\{q+1: q \text{ prime}\}$, the weight-classified naturals are the shifted composites, and the weight columns $k=2,3,5,7,\dots$ are precisely the residue classes struck successively by the sieve of Eratosthenes. proved Re-verified this session for all $3\le n\le10^6$: zero violations of $k(n)=\mathrm{spf}(n-1)$, and the level class lands exactly on $\pi(10^6)=78\,498$ terms.

3.2 The trichotomy: growth decides first new here

The window $(d,\sqrt\ell\,]$ is an interval whose endpoints are governed by different things: the lower endpoint is the jump, a purely additive local datum, and the upper endpoint is $\sqrt{a-d}$, a function of the size of the term. Whether the interval is non-empty at all is therefore a question about growth, and it is answered before any factorisation is examined.

Regime N is Lemma 2.1. For the boundary between Z and W, note $$\ell\le d^{2}\iff a-d\le d^{2}\iff a\le d(d+1),$$ so regime Z is exactly the condition $\sqrt\ell\le d$ and regime W its negation. In regime Z the weight satisfies $k>d\ge\sqrt\ell$, whence $k^{2}>\ell=kL$ and $k>L$: level-classified. In regime W the interval $(d,\sqrt\ell\,]$ contains an integer, and whether a divisor lies in it is precisely the classification question.□

The force of this is that regime Z is a growth condition wearing arithmetic clothes. Write $a(n)\asymp c\,n^{\theta}$ with $d(n)\asymp c\theta n^{\theta-1}$; then $d^{2}/a\asymp c\theta^{2}n^{\theta-2}$, so the regime is decided by the exponent:

Every prediction of Corollary 1.1 was checked against a survey of 25 sequences, 400 indices each — ten families from the OEIS wiki's decomposed list, the figurate families, a tunable quadratic family, and the fast-growing sequences. The results are in Figure 1 and are uniformly as predicted: triangular numbers, squares, pentagonal and hexagonal numbers, cubes, fourth powers and tetrahedral numbers are 100 % level-classified once decomposable; $\lfloor n^{2}/4\rfloor$ sits in regime Z and $\lfloor n^{2}/5\rfloor$ in regime W, straddling the threshold $c=1/4$ exactly as claimed; Fibonacci, the powers of 2 and the Catalan numbers are wholly in regime N and never decompose at all.

The trichotomy itself was checked pointwise on every term of every surveyed sequence: 3 273 indices fell in regime Z, and all 3 273 were level-classified. verified Zero violations. The same predicate was re-checked inside the PARI/GP audit over the prime census sample (§B).

3.3 Counting what survives the window new here

Regime W is where the arithmetic lives, and the quantity to understand is the count of integers that escape a window with a fixed lower endpoint.

Suppose $m>B^{2}$ has no divisor in $(B,\sqrt m\,]$. Every prime factor of $m$ is then either $\le B$ or $>\sqrt m$, and at most one can exceed $\sqrt m$; so $m=sP$ with $s$ $B$-smooth and $P$ either 1 or a prime exceeding $\sqrt m$. If $P=1$ then $m$ is $B$-smooth, the exceptional class. Otherwise $s=m/P<\sqrt m$, and every divisor of $s$ divides $m$ and is $<\sqrt m$, hence is $\le B$; taking the divisor $s$ itself gives $s\le B$. Conversely, if $m=sP$ with $s\le B$ and $P$ prime, the divisors of $m$ are the $f\mid s$, all $\le B$, and the $fP\ge P>\sqrt m$; so no divisor lies in the window. The correspondence $m\leftrightarrow(s,P)$ is bijective, since $P$ is the unique prime factor of $m$ exceeding $\sqrt m$. Finally $E_B(x)$ is at most the count of $B$-smooth numbers up to $x$, which is $\ll_B(\log x)^{\pi(B)}$; and $\sum_{s\le B}\pi(x/s)\sim H_B x/\log x$ by the prime number theorem.□

Setting $B=1$ returns $W_1(x)=\pi(x)+1$ — the natural-number case, the sieve of Eratosthenes, recovered as the first instance of a family. The theorem was tested in two independent ways. First, pointwise: for every $m\le 2\cdot10^{6}$ and each $B\in\{1,2,3,4,5,6,8,12\}$ the structure predicate was compared with a direct divisor enumeration. Zero mismatches, and the exceptional $B$-smooth class was small and slow-growing exactly as the bound requires.

B	$W_B(2\cdot10^6)$	$\sum_{s\le B}(\pi(x/s)-\pi(s))$	smooth exceptions	pointwise mismatches
1	148 934	148 933	0	0
2	227 433	227 430	1	0
3	281 504	281 497	3	0
4	323 043	323 033	4	0
5	356 907	356 890	8	0
6	385 573	385 552	8	0
8	432 526	432 488	15	0
12	502 029	501 953	31	0

The residual in each row is the handful of window-free $m\le B^{2}$ not of the form $sP$; every unit is accounted for. verified

Second, asymptotically, by an independent sieve to $2\cdot10^{7}$ (Figure 2): the sharp prediction $\sum_{s\le B}\pi(x/s)$ tracks $W_B(x)$ to within a fraction of a percent at every $B$ and every scale, while the smoothed form $H_Bx/\log x$ sits 7–15 % high and descends slowly — which is simply the familiar defect of $x/\log x$ against $\pi(x)$, inherited term by term.

If $a(n)$ is level-classified then $\ell(n)$ has no divisor in $(d(n),\sqrt{\ell(n)}\,]$, and since $d(n)\le B$ the interval $(B,\sqrt{\ell(n)}\,]$ is contained in it; so $\ell(n)$ is counted by $W_B$. For $n\le N$ we have $\ell(n)\le a(n)\le a(1)+B(N-1)$. A given value $v$ can equal $\ell(n)$ for at most $B$ indices, because $a(n)=v+d(n)\in[v+1,v+B]$ and $a$ is injective. Hence the count is at most $B\cdot W_B\bigl(a(1)+BN\bigr)\ll_B N/\log N$.□

Theorem E is elementary, and that is the point. It says that the rarefaction phenomenon Conjecture 9 asks about is generic: it holds for the odd numbers, the even numbers, the composites, the squarefree numbers, and every other sequence whose gaps stay bounded, by an argument of half a page. The primes fall outside its hypothesis for exactly one reason — $g_n$ is unbounded — and that single fact is what has cost the project five reports. We return to this in §6.

4The primes: theorems, conjectures, the record

For the primes, $d(n)=g_n=p_{n+1}-p_n$ and $$\ell(n)=p_n-g_n=2p_n-p_{n+1},$$ a linear form in two consecutive primes. The weight, level, jump and $\ell$ sequences are A117078, A117563, A001223 and A118534. Only $2,3,7$ fail Lemma 2.1.

4.1 The conjecture ledger

#	statement	status
C1	infinitely many primes of weight 3	twin primes reformulation
C2	infinitely many primes of weight $k$, each odd $k\ge3$	Polignac-type reformulation
C3	infinitely many primes of level $L$, each odd $L\ge1$	reformulation
C4	level-classified primes have prime weight except 13, 31, 113, 131, 887	Cube Lemma; closed to $4\cdot10^{18}$ proved*
C5	infinitely many primes of level (1;1)	balanced primes reformulation
C6	infinitely many primes of level (1;i)	reformulation
C7	$6\nmid g\Rightarrow 3\mid\ell$	elementary proved
C8	$3\nmid\ell\Rightarrow 6\mid g$	contrapositive of C7 proved
C9	level-classified primes rarefy	Theorem B proved*

C7 and C8 are struck through on the OEIS project page as trivial, and must not be presented as contributions anywhere. They are recorded here only because the mod-3 rigidity they express does real work in §4.4.

4.2 Weight 3 is the twin column

Suppose $p>3$ and $p+2$ are both prime. Then $p\equiv2\pmod3$, since $p\equiv1$ would make $p+2$ divisible by 3. Hence $3\mid p-2=\ell$. As $\ell$ is odd, no divisor equals 2, and $3>g=2$, so the smallest divisor exceeding the jump is $k=3$. Conversely $k=3$ forces $g<3$, and $g$ is even, so $g=2$ and $p+2$ is prime.□

Swept this session over all 455 052 508 decomposable primes below $10^{10}$ with zero disagreements: 27 412 677 primes of weight 3, which is the count of lesser twins below $10^{10}$ after removing $p=3$ (not decomposable) and $p=5$ (weight 3 but level-classified). verified

Let $p>3$ be a lesser twin. By Lemma 4.1, $k=3$, so $L=\ell/3=(p-2)/3$. Level classification needs $k>L$, i.e. $9>\ell$, i.e. $p<11$. The only candidate is $p=5$, where $\ell=3$, $k=3$, $L=1$ and indeed $k>L$; at $p=11$ one gets $k=L=3$, a tie, hence weight.□

This is a small statement, but it is the first entry in a pattern that §4.4 makes systematic: the level class is not spread evenly over the gaps, and the gaps it avoids are exactly the ones the arithmetic of $\ell$ forbids.

4.3 Mod-3 rigidity

Since $\ell=2p-p'$ and both primes exceed 3, working modulo 3 gives at once: if $3\nmid g$ then $3\mid\ell$. proved Because $g$ is even, $3\nmid g$ is the same as $6\nmid g$, which is C7; C8 is its contrapositive. The consequence that matters is structural:

Swept over the whole census sample: among 113 734 sampled decomposable primes, the measured level-one rate is exactly zero for every gap $g$ with $6\nmid g$, and positive for every gap with $6\mid g$. verified More: the smallest level observed at a gap is 1 when $6\mid g$ and 3 otherwise, with zero anomalies across all 45 tabulated gap values.

4.4 A parameter-free model for the level rate new here

Theorem D counts window-free integers when the window's lower endpoint is fixed. For the primes the endpoint is $g_n$, which varies; but conditioning on the gap freezes it, and the theorem then predicts the level rate at that gap with no adjustable parameter. Two arithmetic constraints must be imported. First, $\ell$ is odd and $\gcd(\ell,g)=1$. Second, mod-3 rigidity: if $3\nmid g$ then $3\mid\ell$, so the small factor $s$ of Theorem D must carry the 3 (otherwise the cofactor $P$ would be a prime divisible by 3). Define

$A(g)=\{s\le g:\ s\text{ odd},\ \gcd(s,g)=1,\ \text{and }3\mid s \text{ when }3\nmid g\}$,    $S(g)=\sum_{s\in A(g)}1/s$,

and let $\delta(g)$ be the density of the residue class $\ell$ inhabits,

$\delta(g)=\tfrac12\prod_{p\mid g,\ p>2}\bigl(1-\tfrac1p\bigr)\times \begin{cases}1&3\mid g\\ \tfrac13&3\nmid g.\end{cases}$

Measured against the census sample over the 45 gap values with at least 100 observations, the model has weighted mean ratio 0.9069 with weighted standard deviation 0.0831 and full range $[0.733,1.303]$. Split by residue class the two families agree with each other: $6\mid g$ gives 0.9305 and $6\nmid g$ gives 0.8851 — a real but modest asymmetry, not the factor-of-two split that appears if the density $\delta(g)$ is omitted. Figure 4 plots it.

Three things deserve emphasis. The model predicts that no prime of gap 2 is level-classified, since $A(2)=\varnothing$ — recovering Proposition 4.2 from the general mechanism. It predicts that level one occurs only when $6\mid g$ — recovering Corollary 4.3. And its residual is a stable deficit of about 9 %, not a drift: the model overpredicts because $\ell=2p_n-p_{n+1}$ is not a random member of its residue class but is tied to two consecutive primes. That deficit is the Hardy–Littlewood correlation, and it is the same coupling that obstructs a proof. The model is thus a clean way of separating what is elementary in the level rate (the divisor structure, captured exactly) from what is hard (the correlation, appearing as a single stable factor).

4.5 The two species and their constants

Carried from the fourth and fifth reports, not re-derived here: the level class splits into Species I ($L=1$, equivalently $\ell$ prime), governed by a Hardy–Littlewood prime-pair regime with $N_1(x)\sim c_1x/\log^2x$ and $c_1=2C_2$ conditional; and Species II ($L\ge3$), governed by a divisor-distribution regime of Ford–Tenenbaum type, whose constant remains open. Both were re-measured this session from the fresh census; the numbers are in §4.7 and Figure 6.

The apparent single constant of the early reports is a superposition transient: the cumulative ratio $f\log x/\log\log x$ falls monotonically from 1.2302 at $10^6$ to 1.1562 at $10^{10}$ while the two windowed constants separate cleanly, $\hat c_1$ climbing 0.9719 → 1.0680 toward $2C_2=1.32032\ldots$ and $\hat c_2$ sitting flat at $0.773\pm0.002$ across eight windows. The retraction of the interim claim $c_2=1$ (corrections item 11) is reconfirmed: the cumulative estimator recedes from 1 at every scale.

4.6 The Cube Lemma and the exceptional sets

Carried proved*: a level-classified prime with composite weight satisfies $\ell\le(g-1)^3$, sharp at $p=131$. Combined with the maximal-gap tables of Oliveira e Silva–Herzog–Pardi (2014) this closes the exception set of C4 to $4\cdot10^{18}$.

Recomputed this session at kernel level: sweeping all 455 052 508 decomposable primes below $10^{10}$, the composite-weight level-classified primes are exactly $$13,\ 31,\ 113,\ 131,\ 887$$ and no others. verified Separately, the primes for which the window is closed ($g^2\ge\ell$, regime Z) are exactly $$5,\ 13,\ 19,\ 23,\ 31,\ 113,$$ a six-element set. verified This independently reconfirms the queued upstream erratum to the foothold paper's Lemma 8, whose printed exceptional set $\{13,31\}$ is the set of biconditional failures rather than the set of window closures.

4.7 The verified record

Every row below was computed from scratch this session by a newly written segmented engine, under the complete convention (a prime is classified only once its true successor is known). Every row agrees digit for digit with the fifth report. verified

x	π(x)	decomposable	$N_1$	$N_{>1}$	$N_{\mathrm{lev}}$	f	$f_1$	$f_{>1}$
$10^6$	78 498	78 495	5 953	12 400	18 353	0.2338	0.0758	0.1580
$10^7$	664 579	664 576	44 011	94 038	138 049	0.2077	0.0662	0.1415
$3\cdot10^7$	1 857 859	1 857 856	116 139	250 384	366 523	0.1973	0.0625	0.1348
$10^8$	5 761 455	5 761 452	339 870	738 837	1 078 707	0.1872	0.0590	0.1282
$3\cdot10^8$	16 252 325	16 252 322	912 172	1 999 786	2 911 958	0.1792	0.0561	0.1230
$10^9$	50 847 534	50 847 531	2 708 031	5 984 308	8 692 339	0.1709	0.0533	0.1177
$2.5\cdot10^9$	121 443 371	121 443 368	6 227 807	13 848 468	20 076 275	0.1653	0.0513	0.1140
$5\cdot10^9$	234 954 223	234 954 220	11 719 154	26 181 298	37 900 452	0.1613	0.0499	0.1114
$10^{10}$	455 052 511	455 052 508	22 083 608	49 587 200	71 670 808	0.1575	0.0485	0.1090

Windowed constants, computed decade-locally: $\hat c_1=(\Delta N_1/\Delta D)\log \bar x$ and $\hat c_2=(\Delta N_{>1}/\Delta D)\log\bar x/\log\log\bar x$, where $\Delta D$ is the number of decomposable primes in the window and $\bar x=\sqrt{x_1x_2}$ its midpoint:

window	$\hat c_1$	$\hat c_2$
$10^6$–$10^7$	0.9719	0.7705
$10^7$–$3\cdot10^7$	1.0075	0.7762
$3\cdot10^7$–$10^8$	1.0213	0.7741
$10^8$–$3\cdot10^8$	1.0349	0.7748
$3\cdot10^8$–$10^9$	1.0445	0.7720
$10^9$–$2.5\cdot10^9$	1.0561	0.7728
$2.5\cdot10^9$–$5\cdot10^9$	1.0636	0.7730
$5\cdot10^9$–$10^{10}$	1.0680	0.7726

Further diagnostics, all recomputed: ties (primes with $k=L$) below $10^6$ — exactly 12, matching A121155; the weight-3 column at $10^{10}$ — 27 412 677, which is the published twin-pair count $\pi_2(10^{10})=27\,412\,679$ less $p=3$ (not decomposable) and $p=5$ (weight 3 but level-classified), an external anchor the engine was not tuned to; the maximal-gap ladder to $10^{10}$ — 35 records reproducing A002386 in full, ending 320 after 2 300 942 549, 336 after 3 842 610 773 and 354 after 4 302 407 359; the boundary prime 99 999 989 present in the complete-convention count at $10^8$, giving $N_1(10^8)=339\,870$ and reconfirming the upstream erratum.

5Analysis of the graphs

All figures in this report were generated this session from data computed this session, on square canvases so that the aspect ratio between weight and level is preserved and not read off a distorted frame. Figures 1–3 have appeared in the sections that motivate them; the remainder follow.

Figure 5 is the classical picture of the conjecture, and the only thing new about it is the dashed curve. Five reports have observed that $f(x)$ behaves like $\log\log x/\log x$; Theorem D says why. With the window's lower endpoint frozen at $B$, the survival density is $H_B/\log x$; the primes have $B=g_n$ drifting like $\log p$, and $H_{g}\approx\log g+\gamma$, so the density is $\approx(\log\log x+\gamma)/\log x$. That is the shape, obtained from a proved counting theorem rather than from curve-fitting. What it does not deliver is the constant, because the averaging over $g$ and the correlation between $\ell$ and its two parent primes both enter there — which is exactly the division of labour §4.4 makes explicit.

6The additive–multiplicative bridge

6.1 The mechanism, and its exact scale

Two quantities enter the decomposition of $p_n$: the jump $g_n$, a purely additive local datum, and the divisor structure of $\ell_n=2p_n-p_{n+1}$, a purely multiplicative one. The classification is decided by a single question that entangles them — does $\ell_n$ have a divisor in $(g_n,\sqrt{\ell_n}\,]$? The lower endpoint is additive; the object placed in or out of the interval is multiplicative. That is the bridge, and it is real.

What this report adds is the measurement of the bridge. Theorem D says exactly how much the additive endpoint matters: moving the endpoint from $B$ to $B'$ multiplies the survival count by $H_{B'}/H_B$. That is a logarithmically weak dependence, and it has a moral. The additive input enters the multiplicative count only through a harmonic sum — the coupling is genuine but gentle, which is why the level share decays like $\log\log x/\log x$ rather than like a power. A framework that coupled the two sides strongly would produce a power saving; this one produces a doubly-logarithmic one, and the honest description of its strength is that it is the strength of $H_B$.

6.2 What Theorem E localises

This is worth stating plainly because it reframes what Conjecture 9 is. It is not a statement about primality; primality enters only through the gap distribution. It is a statement about divisors of a shifted prime in a drifting interval, uniform in the drift. The right neighbourhood in the literature is therefore the theory of divisors in intervals — Ford's $H(x,y,z)$, Koukoulopoulos on divisors of shifted primes — and not the theory of prime patterns, except in Species I where the $s=1$ term makes the pattern question unavoidable.

6.3 The obstruction, precisely

Both species reduce to controlling the factorisation of $\ell_n=2p_n-p_{n+1}$, a linear form in two consecutive primes. The consecutiveness is the obstruction: standard sieve machinery handles $2p-q$ for independently ranging primes, but “$q$ is the next prime after $p$” is a condition no sieve weight expresses naturally. The stable 9 % deficit of §4.4 is the numerical signature of precisely this: a model that treats $\ell$ as a random member of its residue class overpredicts by a constant factor, and that factor is the correlation the proof must control.

6.4 Impacts claimed, impacts declined

• Claimed. A canonical additive–multiplicative coordinate system on increasing sequences, in which the sieve of Eratosthenes is the case $B=1$ of a counting theorem, the figurate families are a triviality of growth, and one genuinely native question exists and has been answered.
• Claimed. A taxonomy that tells a reader, from growth data alone, whether a given sequence can possibly be interesting under the decomposition — which is what a thousand-sequence atlas needs to become navigable.
• Declined. Any suggestion that the coordinates ease twin primes, Polignac or Goldbach. The reformulations C1–C3, C5–C6 are exactly as hard as their classical originals.
• Declined. Any cryptographic reading. The construction consumes factorisations; it does not produce them, and there is no trapdoor.
• Declined. Any claim to extend the fundamental theorem of arithmetic. The uniqueness here is selection uniqueness (§3.1).

7Conclusion and future research

7.1 What this report establishes

Two independent things, one confirmatory and one new.

Confirmatory. The complete classification of every decomposable prime below $10^{10}$ was recomputed from scratch by a newly written engine, with no reference to earlier output, and every published anchor was reproduced digit for digit: $\pi(10^{10})=455\,052\,511$, 455 052 508 decompositions, $N_{\mathrm{lev}}=71\,670\,808$, $N_1=22\,083\,608$, $f=0.1575$; the five composite-weight exceptions; the six-element closed-window set; the 35-record maximal-gap ladder; the 12 ties. Three implementations — a C engine, the project's PARI/GP fordiv kernel, and sympy — agree with zero mismatches over 113 734 and 28 434 sampled rows respectively, and this session closed the one row the fifth report's PARI batch had dropped.

New. A trichotomy theorem showing that growth, not arithmetic, decides the classification in two of three regimes; a growth criterion identifying the figurate families as wholly level-classified for trivial reasons and locating the exact threshold $4c=1$ at degree two; an exact counting theorem for window-free integers, $W_B(x)=\sum_{s\le B}(\pi(x/s)-\pi(s))+O_B(\log^{\pi(B)}x)$, verified with zero pointwise mismatches; an unconditional rarefaction theorem for all bounded-jump sequences; and a parameter-free gap-conditional model for the prime level rate that recovers the mod-3 rigidity and the twin-prime exclusion as consequences and isolates the Hardy–Littlewood correlation as a single stable factor of 0.907.

Taken together, the new results answer a question the project had not previously posed: what is the decomposition's content, as opposed to its bookkeeping? The answer is that its content lives entirely in regime W, and within regime W entirely in the drift of the window's lower endpoint. Everything else is growth.

7.2 Upstream errata queue

1. Foothold paper, Lemma 8. The exceptional set of the window condition $g\ge\sqrt\ell$ is the six-element $\{5,13,19,23,31,113\}$; the printed $\{13,31\}$ is the set of biconditional failures. Independently reconfirmed this session by direct sweep to $10^{10}$.
2. Challenge paper, Table 1. $\mathrm{lev}_1(10^8)=339\,870$ under the complete convention; the printed 339 869 omits the boundary prime 99 999 989. Independently reconfirmed this session.
3. Manuscript. Two administrative TODOs remain — author email and affiliation. The two bibliography page-number items were closed in the fifth report.
4. OEIS project page. C7 and C8 already carry the “trivial” strike-through; no action beyond keeping them out of any contributions column.

7.3 Research directions, ordered by leverage

APARI/GP algorithms

The reference kernel is the project's fordiv one-liner — early return on the first divisor beyond the jump, which is exactly the weight's minimality condition. PARI/GP 2.15.4 was used this session as the independent adjudicator.

decomp(a,b) = {
  my(d = b - a, l);
  if(a <= 2*d, return([0, 0, d]));      \\ not decomposable: l = a-d <= d
  l = a - d;
  fordiv(l, k, if(k > d, return([k, l/k, d])))   \\ [weight, level, jump]
}

The regime predicate of Theorem 1, which the audit checks on every sampled row:

regime(a,b) = {
  my(d = b - a);
  if(a <= 2*d, return("N"));
  if(d*d >= a - d, return("Z"));        \\ window empty => level, unconditionally
  "W"
}

The audit driver, run over all 113 734 sampled rows with zero mismatches:

chk(p,g,k,L,cl) = {
  my(r, rc);
  r = decomp(p, p+g);  rc = if(r[1] > r[2], 2, 1);
  if(rc != cl, cm++);
  if(k > 0 && (r[1] != k || r[2] != L), mm++);
  if(!ispseudoprime(p) || !ispseudoprime(p+g), bp++);
  if(cl == 2 && r[2] == 1 && p != 13 && p != 31,
     if(!ispseudoprime(p-g), lp++));
  if(p > 2*g && g*g >= p-g, zc++; if(rc != 2, zf++));   \\ trichotomy
}

The historical decompnaive and decompsieve of algos.txt, and the optimised variants of decompwlj_optimized.txt, remain the reference lineage; bulk work moves to the C engine of Appendix B with PARI retained as adjudicator.

BReproducibility

The census engine

A single-file C program (gcc -O2). Segmented sieve over odd numbers, segment span $2^{25}$ with a 2 MiB packed bitmap; base primes to $\sqrt{X}+1000$; every segment starts on an odd number by construction and the program aborts on a parity fault, a guard installed in direct response to the fourth report's segment-parity bug. Each prime found closes the classification of its predecessor, so the complete convention holds by construction and does not depend on where snapshots fire.

Classification uses an exact shortcut rather than trial division. Let $s$ be the 400-smooth part of $\ell$ and $c=\ell/s$. Every divisor of $\ell$ below 400 divides $s$, so the smallest divisor of $s$ exceeding $g$ is the weight whenever it is $\le400$. Otherwise every divisor exceeding $g$ is $>400$, and: if $1<c\le\sqrt\ell$ then $c$ is itself a divisor in the window (weight); if $c>\sqrt\ell$ is composite then its least prime factor is $\le\sqrt c\le\sqrt\ell$ (weight); if $c>\sqrt\ell$ is prime the weight is $\min(k_1,c)$ exactly. Primality of the weight — needed for the Cube-Lemma census — is then free, since the weight is either $c$, already tested, or a divisor of $s$, whose factorisation is known. No Miller–Rabin call is spent on it.

Budget, checkpoints, resume

The engine takes a wall-clock budget, processes whole segments, and rewrites its state atomically (write-temp-then-rename) at every segment boundary. The $10^{10}$ census was completed in bounded foreground chunks after background execution proved impossible in this environment; total compute about eight minutes. The append-mode log artifact this produced, and its detection, are corrections item 14.

The audit lattice, this session

1. Engine vs external anchors. $\pi(x)$ at every mark to $10^{10}$; the complete A002386 first-occurrence ladder including 354 after 4 302 407 359; A121155 tie count; $W_1(x)=\pi(x)+1$ as an internal check on the independent window sieve. All exact.
2. Engine vs the published record. Every row of the fifth report's census table, digit for digit; the paper's Table 2 in full; the challenge paper's $10^8$ row under the complete convention.
3. Engine vs PARI/GP. 113 734 sampled rows re-derived through the fordiv kernel: zero triple mismatches, zero class mismatches, zero primality failures, zero level-one $\ell$ failures. The 16 354 rows whose weight the engine deliberately leaves undetermined (weight $>400$, weight-classified regardless) were confirmed by PARI to have weight $>400$ in every case.
4. Engine vs sympy. 28 434 rows re-derived by independent divisor enumeration: zero mismatches. The naturals identity $k(n)=\mathrm{spf}(n-1)$ to $10^6$: zero violations, level class landing exactly on $\pi(10^6)$.
5. Theory vs computation. The trichotomy checked pointwise on 3 273 regime-Z terms across 25 sequences (zero violations) and again inside the PARI audit; Theorem D checked pointwise for all $m\le2\cdot10^6$ at eight values of $B$ (zero mismatches) and asymptotically to $2\cdot10^7$.

Failure discipline — this session, in full

Three failures, all disclosed above as corrections items 13–16 and none of which reached a printed number: a Montgomery-arithmetic bug that the small anchors could not detect and that was caught only by cross-checking against the previous report; an append-mode log duplication with one torn line, caught by a row count; and a false counterexample to the trichotomy produced by an unsorted sequence construction, caught by adding the strict-increase assertion that should have been there from the start. The first is the more instructive: it is the fourth report's lesson — an optimisation can pass every anchor whose range never exercises it — recurring in a new place, and the standing rule that every bulk artifact needs at least one independent cross-check before anything is computed from it is what caught it.

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The proper verdict is neither inflation nor dismissal. The decomposition into weight × level + jump is a coherent, original coordinate system that reorganises elementary multiplicative–additive arithmetic, restates the sieve of Eratosthenes as one instance of a counting theorem, settles several of its own questions outright, and poses one — the rarefaction of level-classified primes — that is genuinely native, genuinely connected to the deepest available theory of divisors in intervals, and now genuinely answered modulo refereeing. What this report adds is the map of where that content sits: in regime W, in the drift of a window's lower endpoint, and nowhere else.