Weight × Level + Jump in Six Pages

§1Definitions

Let be a strictly increasing sequence of positive integers. The decomposition attaches to each term three integers built from the term and its successor alone. Nothing is chosen and nothing is fitted: it is a map, not a model.

Since and , exists and is unique; the identity is exact. Each decomposable term is a lattice point on the hyperbola .

The project brief lists the exceptions as {2, 3, 5}; that is a slip. 5 is decomposable and 7 is not, as the treatise and every engine in this build agree.

Proposition A is the sentence to remember. The classification is a question about divisors in a short interval (Erdős, Tenenbaum, Ford), asked of the integer with the interval's left end tied to the sequence's own jump. The ratio sets the window's width: at it is empty (Proposition B); at it is as wide as it can be.

Already visible, and theorems in §3: ; level one means prime, except 13 and 31 ().

Compute it yourself (PARI/GP)

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])))}
? decomp(1153, 1163)
%1 = [127, 9, 10]

The project kernel decompwlj_fordiv.txt, run as printed; it reproduces Table 1 and the census below 10⁶.

Run the machine on . The jump is , so , and "the least divisor of exceeding 1" is its smallest prime factor. What comes out is not an analogy with the sieve of Eratosthenes; it is the sieve.

The dictionary, entry zero

• The weight column is : exactly the integers (shifted by one) struck at the -th pass of the sieve. Its natural density is .
• Column is non-empty below iff ; the plane has columns, and the stopping rule is the apex of the triangle in Figure 2b.
• The level class is , the numbers the sieve never strikes, and it is one line.

Where the Fundamental Theorem of Arithmetic enters

Proposition 1 itself needs only the elementary fact that the least divisor of an integer is prime. The FTA enters when the construction is iterated on the level. Feed back into the ℕ-decomposition: its weight is , its level , and so on until a level-classified term () stops the chain. The weights met along the way are the prime factors of in non-decreasing order:

So iterated decomposition on ℕ is trial division, and the FTA is exactly the statement that the multiset of weights so produced is an invariant of , independent of how one factors. Two readings follow. The weight is a generalised smallest prime factor, and the level class a generalised set of primes, for an arbitrary increasing sequence. And the level class of ℕ being the shifted primes, the classification of ℕ is a primality test for .

What the base case does not license

Proposition 1 identifies the construction's type; it does not transfer the sieve's power. On another sequence the jump is no longer 1, the window opens up, and the level class stops being one line: on the primes it is a two-dimensional region of lines (Figure 3a). "Level-classified ⇔ prime" survives on the primes only for and only up to a two-element exception set (Lemma 5). The next four pages are about the gap between those two pictures.

Geometry of Figure 2b: with , , , the composites fill the triangle , , . Rendered-raster audit of the 22 Sep plate to 10⁷ confirms the triangle edges to within 3 px (project record).

For with gap , : the decomposition lives on the three-term progression whose upper step is a prime gap. For , is even and odd.

The elementary theory (all proved)

• Lemma 3. On the level class, and . Max below 10⁷: 129, at .
• Lemma 4 (twin column). For : . Below 10⁷: 58,979 each side (A007508 gives 58,980 pairs, counting (3,5)).
• Prop. 2 (mod 3). . Founding Conj. 7, 8; a two-line congruence, never a contribution.
• Prop. 3 (cube bound). Level with composite ⇒ , equality only at 131. The set is up to . Founding Conj. 4.
• Lemma 5 (level one). If : prime. Exceptions: . Below 10⁷, 44,009 level-one primes with prime = 44,011 − 2.

Conjectured truth: (the treatise's §4.4 misprints ; erratum logged 17 Sep). The envelope falls every decade, 0.385 → 0.241 from 10⁵ to 10¹⁰ †.

The deflationary control

A Cramér sequence of matching density gives at 10⁷ against the primes' 0.2077 (0.9 %; treatise: 0.45 % at 10¹⁰). The rate of Theorem 1 is a property of the construction at density , not of the primes. The arithmetic is in the fine structure: odd only, the columns, the lock (§5), , and the level-one ratio to the control, which crosses 1 near †.

The primes are one sequence. To separate what belongs to the construction from what belongs to arithmetic, run the classifier over a thousand.

1,000 sequences in eleven families, generated from their definitions with fixed seeds, so the corpus is reproducible: polynomials (100), (88), Beatty (70), (58), arithmetic progressions (80), multiplicative predicates (100: squarefree, , , -smooth, -rough, sums of two squares, …), prime families (80), digit families (120: binary weight, palindromes, digit sets, Harshad, digit sums), sieves (20: lucky, Flavius, ludic, Ulam, …), linear recurrences (26) and seeded random models (258). First 20,000 terms each, terms ; duplicates removed. 17,750,424 decompositions, 17,744,571 decomposable; 876 sequences reach full depth.

Engine: complete factorisation of (Miller–Rabin, Pollard–Brent). 181,403 records re-derived by the PARI/GP kernel and 3,000 by SymPy: zero mismatches; prime records identical to §3's sieve engine.

1 · The domain is a cliff

For the decomposable share is 0.934 at , 0.867 at 1.4995, 0.202 at , 0.019 at 1.51 and exactly 0 from 1.52 on (Figure 4d). Twenty sequences are nowhere decomposable (Fibonacci, tribonacci, Jacobsthal, for , …): growth like is outside the construction.

2 · Above θ = ½ the classification is empty

169 sequences have : all 99 polynomials of degree ≥ 2, 53 power sequences , 10 sparse random models, and seven arithmetic ones (, , , primes , , …). Proposition B explains them: for a polynomial of degree and leading coefficient , eventually, so every late term is forced level. The classification carries information only when .

3 · The first moment is universal

Among the 755 non-forced sequences, correlates with at 0.949, and a five-coefficient fit in and gives . ℕ sits at , ; the random family fits to rms 0.031, the arithmetic families to 0.055–0.093. To first order, the level share is a function of how wide the window is.

4 · The rest is local congruence data

Each record's is replaced by a synthetic within ±10 % of , same jump , and classified; controls differ in what inherits.

C₅ is the new control of this build. Since divides both and , the common factor of consecutive terms is arithmetic that the jump cannot see and a residue class mod 210 cannot carry. Inheriting it closes the largest residuals of C₂ outright:

This closes an open end of the 21 Sep build (binary-weight families "unexplained"): consecutive terms share a large power of 2, and at this resolution that accounts for all of it. Grey rows are not closed by C₅: -smooth numbers (level-rich against C₅), odd (squares and twice squares), palindromes, and primes ≡ 1 (mod 4) (not closed by mod-4 or mod-840 controls either; within about 2σ at height 10⁹). Of 738 sequences, 32 lie beyond 5σ of C₅.

Reading. For a generic sequence the level share is predicted to about one point by the window width , the common factor , and modulo small primes. What a sequence adds beyond that is its own arithmetic, now measurable as a residual. The primes add almost nothing to the first moment: §3's deflation, corpus-wide.

Not the 21 Sep corpus; where both measured the same object they agree (the cliff to three digits; binary weight 3: 0.2414 vs 0.2434).

The construction takes an additive datum, the jump to the next term, and uses it to select a multiplicative one, a divisor of the term's deficit. It does so canonically, for every increasing sequence. That is the whole of the "bridge" claim; this page lists where the two sides actually touch, and where they do not.

1 · Divisors in short intervals

By Proposition A, level classification is : no divisor of in . The distribution of integers with a divisor in is Ford's , after Erdős and Tenenbaum; Koukoulopoulos treats shifted primes. What is new here is the normalisation: the integer is , and the left end of the interval is the same . On the primes this couples a divisor condition on to the compositeness of everything in . Theorem 1 works by dropping that coupling; its lower-bound analogue, Problem 2, cannot.

2 · Eratosthenes, and the FTA

At the window is and the least divisor is the smallest prime factor (§2). The multiplicative content is then complete: iterating on the level recovers the factorisation. At the construction sees only divisors above , so it is a truncated sieve, and the truncation point is additive data.

3 · Three-term progressions

On the primes, . Lemma 5 turns "level one" into " is prime": a Hardy–Littlewood triple , singular series , which vanishes unless and averages over to (Lemma 1). A multiplicative stratum () acquires an additive constant (twice the twin-prime constant) through an identity between local factors.

4 · The congruence lock

A divisibility (multiplicative) is a residue condition (additive). Among gap- primes, avoids and mod , so a flat law predicts for — larger than , which is why primes are level-poorer than a size-only control (first 20,000 primes: 0.260 against 0.279, −4.2σ; at height 10⁹ a control matching closes it, −0.1σ). Measured below 10⁷, pooled over :

Exact at (gaps 2, 4, where the law is provably exact), then a 2–3 % deficit. The treatise's interior-credit law explains its shape with no free parameter (consecutiveness makes residues with at interior cheaper; the lock class earns no such credit) and needs one global factor for its size. The law is heuristic; the fit is verified at 10¹⁰ over 391 cells †. The weight columns are the same lock read at small : (19,495 each below 10⁷) and (26,974 = 19,115 + 7,859), both proved.

5 · The common factor

For any sequence, divides both and . The corpus of §4 shows this is the dominant arithmetic input to the level share outside the primes: inheriting halves the residual rms left by (0.027 → 0.011) and removes every case. For the primes and the lock (item 4) takes its place.

What the framework does not carry

Difficulty conservation. Recasting a hard problem in these coordinates does not make it easier. "Infinitely many primes of weight 3" is the twin-prime conjecture (Lemma 4); fixed columns and lines are Polignac-type (founding Conj. 2, 3, 5, 6); "the level class is infinite" contains the balanced-prime problem. Goldbach does not appear anywhere in the framework. Nothing here bears on the Riemann Hypothesis.

The native yield, on which the framework should be judged, is: Proposition 1 (the sieve, exactly), Lemmas 2–5 and Propositions 2–3 (small, sharp, permanent), Theorem 1 (a genuine theorem answering a question only these coordinates pose, pending refereeing), the column identities and the lock, Lemma 1's constant, and — from §4 — the observation that the first moment of the classification is universal and its second-order content is local congruence data.

Table of correspondences

Labelled by kind. Program: a concrete direction with an identifiable technique. Conjecture: a precise statement with stated evidence. Open: a question with no route yet. Nothing on this page is a result.

Inherited from the treatise

Program 2 — function fields. Over the conditioning "no irreducible in between" is a computation, and Hardy–Littlewood analogues are theorems (Sawin–Shusterman). Order monic irreducibles by degree, then lexicographically; ask for the analogue of Proposition 1, Theorem 1 with the right exponent, and Theorem 2 with its constant.

Program 3 — consecutiveness conditioning. State the interior-credit law independently of weight and level, derive the second-order factor instead of fitting it, and compare with Lemke Oliver–Soundararajan, whose consecutive-prime biases use the same bookkeeping.

Program 4 — Species II. The level-above-one stratum is a divisors-of-shifted-primes problem with a moving window; its constant (, treatise Problem 3) is uncomputed. Buchstab-type methods; Ford, Koukoulopoulos.

Native to the corpus (new with this build)

Program 5 — the universality function. For random sequences with local density , derive as a function of and from the probability that a random integer of size has no divisor in (a Ford–Tenenbaum quantity at ). Test: the random family's residual (rms 0.031) should vanish.

Program 6 — a predictive control. Prove, for a class of "generic" sequences, that equals its C₅ prediction up to ; then treat the residual against C₅ as the definition of a sequence's arithmetic signal.

Open — the C₅ residuals. Why are -smooth numbers level-rich against C₅ (+11σ at ), and -odd numbers level-poor (−12σ)? Candidate mechanism for smooth numbers: is itself smooth, so its divisors crowd below differently from a random cofactor; untested. The primes ≡ 1 (mod 4) residual (−4σ at small height) is unexplained and may be finite-size.