Balanced auxiliary-prime density

../

evidence/: Exact reviewed subject and scoped independent assessment and contract-grade records for the balanced-prime loss refinement.

loss_refinement: The full balanced-schedule proof, with its exact reviewed subject and qualified independent review and distinct contract-grade records.

---

This lead's notes that cite no source are author-recorded research, not claims.

Exact result

Put $\gamma=\sqrt6-2$. There exist one infinite set $A\subseteq\mathbb Z_{>0}$, a real constant $C>0$, and a real $x_0>e$ such that, writing

$$A(x)=|A\cap[1,x]|,\qquad r_A(n)=|\{(a,b)\in A^2:a\le b,\ a+b=n\}|,$$

one has $r_A(n)\le2$ for every integer $n$ and

$$A(x)\ge x^\gamma \exp\{-C\sqrt{\ln x\,\ln\ln x}\} \qquad\text{for every real }x\ge x_0.$$

The representations are unordered and a diagonal is counted once. The same set consequently has

$$\liminf_{x\to\infty}\frac{\log A(x)}{\log x}\ge\gamma,$$

and, for each real $\alpha<\gamma$, it satisfies $A(x)\ge x^\alpha$ for every sufficiently large real $x$. Only that latter threshold may depend on $\alpha$.

The full proof chooses independent auxiliary primes with dyadic exponent $w_j=\lceil j\log_2(j+1)\rceil$. It supplies the arithmetic, conditional expectation, simultaneous selection and all-cutoff deductions, rather than applying a theorem outside its parameter range.

Current scoped standing

The exact original author proof is retained as a byte-exact review subject. It received a fresh-context, independent whole-argument refutation-failed assessment. A distinct grader's PASS accepted the original report contract and reviewer independence.

That verdict relies on exact standard external theorem interfaces and does not recursively certify their proofs. The review did not authenticate the Cilleruelo method attribution, L2 standing, historical author reading, or novelty. Those exclusions remain explicit. The reviewer, author and grader are distinct roles and contexts; the grade is not a second blind review.

This native filing was prepared by the exposed author. The original mathematical assessment and contract grade are retained without extending them to an unchecked transfer, later repair, numerical tier, formal proof, publication, or public record. No mathematical computation or formal reproduction is claimed. The source artifact identities, exact locators, reading depth and raw-transfer accounting limitations accompany the review.

Ownership and relationships

This is a separate research lead for the balanced-schedule loss refinement, not a new L-claim and not an amendment to L4. L4's existing statement has loss $\exp\{-C\sqrt{\ln x}\,\ln\ln x\}$ and records author standing at tier 0. The retained refinement improves the order of that loss; it does not improve the leading exponent $\gamma$. No L4 statement, status, tier, proof or historical evidence is altered by this record.

The original author attributes the shifted discrete-logarithm method to Cilleruelo's selected manuscript, arXiv:1209.0326v2, dated 15 May 2013. Its density theorem is background, not an established premise of this proof. The canonical source index supplies source identity and its own separate reading qualifications.

The result bears on Erdős Problem 158, whose intended question concerns zero square-root lower density for every infinite B2[2] set. Since $\gamma<1/2$, this result does not settle that question. There is no claim of positive square-root lower density, a positive constant multiple of $x^\gamma$, sharpness, priority, or a current public record. The problem page's dated status assessment is not renewed by retaining this argument.