Decompwlj and famous conjectures

A useful coordinate system is not yet a conjecture-solving machine

The most defensible conclusion is both positive and restrictive. WLJ is a genuine local additive–multiplicative transform: it takes the additive jump to the next term and asks for the first divisor of the shifted value lying beyond that jump. That is mathematically natural for primes and gives exact descriptions of several prime families. But RH, Goldbach, Collatz and BSD are governed by different missing structures—spectral error terms, binary Fourier correlations, directed dynamics, and Euler products with signed local coefficients. The current WLJ coordinates do not supply those structures.

The mathematical core

1.1 Definition and uniqueness

Let a(1)<a(2)<⋯ be a strictly increasing sequence of positive integers. Set

When ℓ(n)≠0, the exact identity is

The minimum makes k(n) unique. Existence is equivalent to

The classification boundary is exactly k=L: a term is level-classified if k>L, and weight-classified otherwise. In logarithmic coordinates the boundary is therefore the undistorted diagonal log k = log L.

1.2 The natural sequence: an exact shifted sieve

For a(n)=n, the jump is 1 and ℓ(n)=n−1. For n≥3,

Consequently, L_ℕ(n)=1 exactly when n−1 is prime. Iterating the same operation on the successive levels recovers the prime factors of n−1 in nondecreasing order. This is a coordinate realization of the Fundamental Theorem of Arithmetic and of the first strike in the sieve of Eratosthenes; it is not a new proof of either theorem.

1.3 The prime sequence: a shifted integer controlled by a true prime gap

For a(n)=p_n, write g_n=p_n+1−p_n. Then

This is the precise additive–multiplicative bridge: the additive datum g_n chooses the shifted integer ℓ_n; the factorization of ℓ_n relative to the moving threshold g_n chooses the weight and level. The attached preprint proves that only 2, 3 and 7 fail the existence condition for the prime sequence.

Four kinds of “connection”

Claims about famous conjectures should be sorted before they are compared. The same formula can be profound in one sense and tautological in another.

The Riemann Hypothesis

The Riemann Hypothesis (RH) says that every nontrivial zero of ζ(s) has real part 1/2. The official Clay description emphasizes its equivalent control of the error in prime counting:

3.1 An exact WLJ reformulation

Define the natural-WLJ prime indicator

Because L_ℕ(m+1)=1 exactly when m is prime,

Therefore RH has the exact WLJ wording

This is mathematically correct and logically equivalent to the classical form. It is also tautological in the precise sense that I_W is exactly the prime indicator in different notation.

3.2 The von Mangoldt function can also be reconstructed

Set Λ_W(1)=0. For m≥2, let k=k_ℕ(m+1) and L=L_ℕ(m+1). Then m is a power of a single prime exactly when L is a nonnegative integral power of k. Hence define

Then Λ_W=Λ, the classical von Mangoldt function, and

This identifies the exact analytic doorway: WLJ would become more than a relabeling only if some property of its weight–level geometry yielded analytic continuation, a functional equation, an explicit formula, or cancellation of the required square-root size.

3.3 Why the prime-WLJ graph does not presently touch the zeros

The prime decomposition uses ℓ_n=2p_n−p_n+1, one consecutive gap away from p_n. Its plots record divisor structure at gap scale. RH controls global oscillation of π(x) around Li(x) and, through explicit formulae, the aggregate contribution of all zeta zeros. A visually sparse lower wing or a limiting class density contains neither the phase nor the signed cancellation of an explicit formula.

• RH does not currently imply Conjecture 9: prime counts do not determine whether the shifted integer ℓ_n has a divisor in a moving interval.
• Conjecture 9 does not imply RH: a density-zero statement has far less information than a square-root error term with oscillation.
• The existence of the exact WLJ Dirichlet series above is not new; its coefficients are exactly Λ.

3.4 GRH and primes in progressions

The same lossless recoding extends to Dirichlet prime counts by restricting the natural-WLJ level-one ray:

Thus GRH for Dirichlet L-functions can be restated as the corresponding square-root error bounds for these residue-restricted level-one counts. Again, this is exact because I_W is the prime indicator. It becomes new only if the weight–level geometry supplies character-sum cancellation rather than merely renaming it.

Goldbach’s conjecture

Binary Goldbach asserts that every even integer N>2 is a sum of two primes. It remains open. The ternary version—every odd integer greater than 5 is a sum of three primes—was proved by Harald Helfgott. The binary conjecture has been computationally verified through 4·10¹⁸, but finite verification cannot settle all even integers.

4.1 Exact WLJ convolution

Using the natural-WLJ prime indicator I_W from §3, define

Then binary Goldbach is exactly

This is a clean additive–multiplicative expression: each factor is a multiplicative primality condition encoded by a WLJ level, and the sum is an additive convolution. It is arguably the sharpest formal expression of the project’s “bridge” ambition.

4.2 Why the exact formula does not solve Goldbach

Let

Fourier inversion gives

The hard part is therefore phase-sensitive cancellation in S_W, especially on the minor arcs, plus a positive singular-series main term on the major arcs. A scatter plot of (k,L) or the marginal frequency of any WLJ class gives only one-point information; it does not control this two-prime correlation.

• The WLJ algorithm can recognize each prime after factorization, but recognition is not a lower-bound sieve for pairs of primes.
• The classical parity obstruction remains: divisibility-based sieves have difficulty distinguishing the desired “one prime factor” condition from nearby almost-prime conditions strongly enough to prove binary Goldbach.
• Conjecture 9, even if proved, is a density statement about one prime at a time. Goldbach asks whether every even target has at least one correlated pair.

4.3 A nontrivial WLJ research formulation

Partition primes into WLJ strata σ—for example by a fixed weight, fixed level, or classification side—and form

Goldbach representations then split into cross-correlations of these strata. This could reveal which wings carry additive representations, but a proof would require uniform major/minor-arc estimates for the stratum sums and uniform control when the number of strata grows. Since σ(p) depends on the successor prime and on the factorization of 2p−p_next, these sums are at least as delicate as ordinary prime exponential sums.

Twin primes, Polignac-type strata and prime gaps

5.1 The strongest exact classical connection

Thus Conjecture 1—infinitely many primes of weight 3—is exactly the twin-prime conjecture, not merely an analogy. This is a successful classification: an important two-prime pattern is compressed into one WLJ coordinate.

5.2 Fixed weights are not generally fixed gaps

For an arbitrary fixed weight k, the exact constraints are

Except at the especially rigid weight 3, fixing k does not simply fix a Polignac gap. “Infinitely many primes of weight k” is therefore Polignac-like but not, in general, the same as “infinitely many consecutive-prime gaps equal to a prescribed even number.” This distinction matters when interpreting Conjecture 2 and the vertical combs in the figures.

5.3 Hardy–Littlewood tuples in natural-WLJ notation

For a finite admissible set of shifts ℋ, define

This is exactly the number of prime constellations n+ℋ up to X. The Hardy–Littlewood prime-tuples conjecture predicts a singular-series asymptotic for this WLJ expression. The formula shows that the natural-WLJ sieve can encode every finite prime pattern; it also exposes the same limitation as Goldbach: products of level-one indicators require high-order correlation estimates, not marginal class frequencies.

5.4 Level one and balanced primes

L_n=1 means ℓ_n itself is the weight and is prime. If ℓ_n=p_n−1, then

so p_n is balanced. Consequently Conjecture 5 is exactly the conjecture that there are infinitely many balanced primes. The generalized levels (1;i) encode longer backward offsets, but their infinitude remains open.

5.5 Legendre, Andrica and Cramér

If a prime is weight-classified, then k≤L and kL=ℓ, hence k≤√ℓ. Since the definition gives g<k,

This is stronger than the local inequality needed for Andrica at every weight-classified prime. It has two important limitations:

1. It says nothing comparable for a level-classified prime—the very exceptional set under study.
2. Even if Conjecture 9 proves that the exceptional set has density zero, Legendre and Andrica require control at every relevant gap. A density-zero set can still contain infinitely many record gaps.

Cramér’s conjecture predicts the much smaller scale g_n=O(log²p_n). The WLJ inequality above is polynomial-scale and therefore cannot approach Cramér without a new distribution theorem for the jump coordinate.

The Collatz conjecture

For the usual Collatz map, C(n)=n/2 when n is even and 3n+1 when n is odd. The conjecture asserts that every positive starting value eventually reaches 1. The native object is a directed, time-ordered orbit with upward and downward moves. Native WLJ requires a strictly increasing sequence.

6.1 Why sorting an orbit is not faithful

Consider the orbit

Sorting its distinct values gives 1,2,3,4,5,6,8,10,16, which can be decomposed by WLJ. But its new jumps are gaps between order statistics, not Collatz transitions. The operation has forgotten direction, revisit counts, stopping time, and the location of the odd steps. Those are exactly the data the conjecture concerns.

6.2 Monotone derived sequences: useful but not equivalent

6.3 What a faithful extension would need

A directed WLJ theory would have to keep the transition label, allow signed increments, and retain the 2-adic valuation in the accelerated odd map

At minimum it needs a reconstruction theorem: from the proposed coordinates one must be able to recover the directed orbit, not just its set of values. It would also need a Lyapunov function, invariant measure, contraction estimate, or control of all exceptional orbits. The present minimum-divisor-above-gap rule supplies none of these.

Tao’s major result shows that almost all Collatz orbits attain almost bounded values in logarithmic density, using probabilistic transport and 3-adic harmonic analysis. This illustrates the scale of missing structure: a density statement about factorization coordinates is not a substitute for control of an iterated dynamical system.

Birch and Swinnerton-Dyer

For an elliptic curve E/ℚ and a good prime p, let

The local data form the Euler product

The BSD conjecture states, in its rank form,

7.1 The data-type obstruction

The ordinary prime WLJ coordinates (k_p,L_p,g_p) depend only on the ordered primes; they are identical for every elliptic curve. BSD varies with E. The curve-dependent coefficients a_p(E) are signed, are not increasing, and must remain attached to their prime labels because the Euler product uses the pair (p,a_p).

• Sorting the values a_p or #E(𝔽_p) destroys the prime labels and hence destroys the Euler product.
• Applying WLJ only to the good primes gives essentially the ordinary prime sequence with finitely many deletions; it cannot distinguish ranks of different curves.
• Applying WLJ to conductors or ranks across a database may be exploratory data analysis, but it does not encode the analytic continuation or the zero at s=1.

7.2 A meaningful “decorated WLJ” experiment

One may retain the prime geometry and attach the elliptic coefficient as a fourth label:

Then test weighted sums such as

This is a legitimate research observable. It could test whether local Frobenius traces correlate with prime-gap/divisor strata. But to bear on BSD one would need to prove that the family of weighted sums preserves enough of the logarithmic derivative of L(E,s) to recover its order of vanishing. No such theorem is presently known.

7.3 A nearer algebraic-number-theory neighborhood

For a fixed curve and modulus m, one can form increasing prime subsequences such as

WLJ of these subsequences could study gaps inside Chebotarev-type sets. That is closer to Galois representations, Sato–Tate phenomena and effective Chebotarev estimates than to BSD itself. It is valuable if framed as a new distribution experiment, not as a rank formula.

Audit of the supplied graphs

Every two-dimensional graph is interpreted with weight on the horizontal axis and level on the vertical axis. Original rectangular images are placed inside square frames without stretching. The three regenerated plots are exactly 1800×1800 pixels.

8.1 The natural-number sieve

8.2 The two prime wings

8.3 The three-dimensional fans

8.4 What the pictures can and cannot support

Audit of the conjecture ledger

9.1 Published and current-source status

The attached arXiv v4 preprint (2010) states C9 as a conjecture based on computation. The current OEIS project page also calls it “Conjecture 9.” The supplied fifth-edition internal report presents a “Theorem B” proof architecture and labels the result PROVED*, explicitly pending outside refereeing.

The supplied report does not include the referenced full submission manuscript. Its summary invokes a uniform Selberg upper-bound sieve over growing parameters and a summed singular-series estimate, but does not provide the detailed uniform lemmas needed to audit those steps. Accordingly this report uses the following status:

9.2 Status table

9.3 A concrete route that deserves a complete write-up

A level-classified prime satisfies L≤g−1 and can be parameterized by two prime linear forms

For an upper bound, one may drop the requirement that the second prime be the immediate successor. Split at g≤M; large gaps contribute at most O(X/M) by summing gap lengths, while the small-gap part invites a uniform two-linear-forms upper-bound sieve summed over L≤g≤M. This is a credible route to density zero. To become a theorem it needs, in one public proof, the exact injection, degenerate local cases, uniformity in L and g, a summable bound for the singular series, and an optimized choice of M.

A research program ordered by leverage

1. Publish and independently audit the complete C9 proof. This is the highest-value native target because it is sharply stated, supported by data, and plausibly approachable by classical upper-bound sieve methods.
2. Build a WLJ explicit-formula program. For test functions Φ, study D_Φ(s)=Σ_pΦ(k_p,L_p,g_p)p^−s. First prove convergence domains and mean values; only then ask about continuation or zeros.
3. Measure additive energy of WLJ strata. For Goldbach, compute residue-class biases and exponential sums of fixed-weight and fixed-level prime subsets. Pre-register the statistics and compare against matched random models to avoid visual overfitting.
4. Separate value cutoffs from index cutoffs everywhere. A statement for p≤X and one for n≤N have different normalizations. Every CSV, plot and caption should record the convention.
5. Develop a directed extension before touching Collatz data. State axioms: signed transition, time label, valuation label, and exact orbit reconstruction. Reject any transform that only sorts orbit values.
6. Use elliptic curves only with decorated prime labels. Test a_p(E) against WLJ strata across rank-controlled families, with correction for conductor and local reduction type. Treat all observed rank signals as exploratory until an Euler-product identity is proved.
7. Establish null models for every graph. Compare primes with Cramér-style random primes, shuffled gaps, and randomized factorizations that preserve marginals. A visual feature is arithmetically interesting only if it survives the appropriate null comparison.

Milestones that would change the assessment

Conclusion

WLJ’s strongest contribution is organizational: it places additive gaps and multiplicative divisors in the same exact coordinate system and makes several prime families visible as strata. That is already worthwhile. The correct next step is not to attach every famous conjecture to the pictures, but to identify the missing invariant in each case and ask whether WLJ can generate it.

Reproducibility, PARI/GP kernel and sources

A.1 Optimized PARI/GP decomposition

The following uses fordiv, which enumerates divisors in increasing order and stops at the first divisor exceeding the jump. It returns [weight, level, jump].

decompwlj(a, b) =
{
  my(d, ell);
  if (a >= b, error("the sequence must be strictly increasing"));
  d = b - a;
  if (a <= 2*d, return([0, 0, d]));
  ell = a - d;
  fordiv(ell, k,
    if (k > d, return([k, ell/k, d]))
  );
}

primewlj(p) = decompwlj(p, nextprime(p + 1));

The core agrees with the supplied decompwlj_fordiv.txt. Factoring ℓ dominates the cost. The “newSieve” scan in the OEIS material can be faster for individual level-classified inputs, while fordiv is simpler and exact for audited batches.

A.2 Independent chart census

• Prime-value cutoff: p≤1,000,000.
• Total primes: 78,498; nondecomposable: {2,3,7}; decomposable: 78,495.
• Level-classified (k>L): 18,353; share: 23.381107%.
• Weight 3: 8,168; level 1: 5,953; boundary k=L: 12.
• Every row was checked against p=kL+g, and the minimum divisor was independently verified from the full divisor list.

A.3 Source discipline

The attached 2010 preprint is the authority for the framework’s definitions and original conjecture status. The current OEIS project page is used for the sequence map and for the explicit note that C7 and C8 are trivial. The supplied fifth-edition HTML is treated as an internal research report whose starred proof claims still require a public, independently checkable manuscript.

A.4 References

1. Rémi Eismann, “Decomposition into weight × level + jump and application to a new classification of primes”, arXiv:0711.0865v4 (2010).
2. OEIS Wiki, “Decomposition into weight × level + jump”, definitions, algorithms, sequences and conjectures.
3. Rémi Eismann, decompwlj.com, atlas of decomposed sequences and visualizations.
4. Enrico Bombieri, official Clay problem description: “The Riemann Hypothesis”.
5. Harald Andrés Helfgott, “The ternary Goldbach problem”, including the distinction between binary and ternary Goldbach and the circle-method architecture.
6. Tomás Oliveira e Silva, Siegfried Herzog and Silvio Pardi, “Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4·10¹⁸”, Mathematics of Computation 83 (2014), 2033–2060.
7. Terence Tao, “Almost all orbits of the Collatz map attain almost bounded values”, Forum of Mathematics, Pi 10 (2022), e12.
8. Andrew Wiles, official Clay problem description: “The Birch and Swinnerton-Dyer Conjecture”.

Local materials audited: 0711.0865v4.pdf, algos.txt, decompwlj_fordiv.txt, Fable5_decompwlj_deep_analysis_5th_edition.html, and seven supplied figures. All images are embedded in this file; no local paths or network requests are required to view the report.