A Complete Answer to Erdős Problem 690

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certificate_4_2: Separates an imported constant enclosure from a pending finite sum and a proved tail bound.

certificate_4_3: Records the exact large-gap premise and its limited evidentiary status.

certificate_4_4: Records the exact twin-prime premise used for the later ascent.

evidence/: Holds the two independent review records of the symbolic all-k route and the exact snapshots of the reviewed card and problem overlay; no checker and no certificate run are filed here.

lemma_3_1: Derives the exact prime-gap criterion for a strict rise or fall.

lemma_3_2: Bounds the ratio between adjacent CRT densities by reciprocal-prime sums.

lemma_4_1: States the exact analytic inputs used in the finite and uniform ranges.

proposition_5_1: Reconstructs the finite-range reduction with explicit pending numerical certificates.

proposition_6_1: Reconstructs the tail argument above 8600001 with explicit finite-check boundaries.

theorem_1_1: Assembles the finite and uniform ranges at their declared certificate boundaries.

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Shouqiao Wang and Davide Crapis, A Complete Answer to Erdős Problem 690, arXiv:2605.08542v1 [math.NT], submitted 8 May 2026, 18 pages. Versioned record, versioned PDF. arXiv lists the article under CC BY 4.0. The title page carries the subtitle "Discovered by the Multiscalar Fields System" and the affiliation Multiscalar Intelligence; the introduction says the proof was found by that system with limited human interaction, and the acknowledgments (p. 17) name two independent checkers and the AI models used in the system's proof search and verification workflow.

Source artifact and reading coverage

The canonical local artifact is wang_crapis_2026_complete_answer_erdos_problem_690.pdf, the 18-page arXiv v1, 413,851 bytes, downloaded from the versioned PDF link above on 2026-09-07. Printed and PDF page numbers coincide. All eighteen pages were read visually, at compilation, at review and again at filing. A second PDF of the paper in the authors' GitHub repository is not byte-identical to the arXiv v1 and is not the selected artifact; it is not retained. The companion numerical verifier described in Appendix A (p. 18) is not redistributed: the repository that holds it reports no license, and the article's license does not cover it. The arXiv record (https://arxiv.org/abs/2605.08542, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Account

The paper claims that the density $d_k(p)$ of integers whose $k$th smallest distinct prime factor is $p$ is not unimodal for any $k\ge4$ (Theorem 1.1, p. 1). Together with Cambie's theorem for $k=1,2,3$ this gives the exact classification of Corollary 1.2 (p. 2). The proof exhibits, for each $k$, a strict descent and then, at a later prime gap, a strict ascent, rather than estimating the global maximum.

Lemma 3.1 turns a first difference into a comparison between the prime gap plus one and a ratio of adjacent CRT densities. Lemma 3.2 bounds this ratio using elementary symmetric polynomials and reciprocal-prime sums. Proposition 5.1 combines the paper's own exact table for $4\le k\le20$ (Table 1, p. 7; the result page reuses Cambie's accepted range in its place), two medium prime triples and a much larger published gap/twin pair to reach $k=8600001$. Proposition 6.1 constructs a composite block by CRT and finds a later smaller gap by averaging primes in $(4P,8P]$, covering $k\ge8600002$. Theorem 1.1 assembles the adjacent ranges.

Exact dependency and proof limits

The recurrence and the $k\le20$ theorem are reused from Cambie, Claim 6 and Theorem 5, with an explicit zero-based to one-based prime-index translation. Their accepted finite calculation is not repeated; the paper's own Table 1 (p. 7) for $4\le k\le20$ is not transcribed.

Lemma 4.1 states the external analytic estimates with exact source versions and ranges; it does not compile their proofs. Items 2–5 are statements of Dusart 2010; item 1 is Dusart's thesis result as quoted in Axler 2018, whose card also holds the printed digits behind the constant enclosure. Certificate 4.2 separates the imported $B$ enclosure, a pending finite certificate for $C$, and the elementary tail estimates. Certificate 4.3 and Certificate 4.4 expose the two huge prime-record premises. Their primality proofs have not been replayed here.

Every essential source-local symbolic deduction is reconstructed on the result pages. The numerical prime enumeration, sum and logarithm obligations remain pending; this is a conditional proof compilation, not a claim that the source-reported computations have been independently certified. The uniform tail uses neither huge record nor the finite enclosure for $C$.

Certificates and standing

The route rests on forty-two finite certificate obligations: the prime enumeration through 1,999,993 with the two medium consecutive triples and their preceding-prime counts, the exact reciprocal sums and $W$ values of Proposition 5.1, the directed partial sum for $C$ with its exact tail, the large-record index and domain comparisons, and the logarithmic and rational endpoint checks of Proposition 6.1. Each is stated on the page that uses it. Every one of them is pending: no checker has been authored into this folder, imported or executed, and the source's report that its companion verifier succeeds is not a local execution result. Four external premise classes stay uncompiled: the Dusart and Axler analytic estimates, the $B$ enclosure, the large-gap record and the twin-prime record. The accepted Cambie reuse is not reopened.

Certificates 4.3 and 4.4 call the Prime Gap List Project dataset (its commit ef38eb9e428496b3f8eb2a186f512744c0af09a0) and PrimePages entry 136849 "retained"; those captures are not filed in this repository, and the public identifiers on the pages are the reader's route to them.

The symbolic route was reviewed independently on 2026-09-07. The symbolic review records a conditional symbolic pass for the nine result pages and the selected PDF, conditional on the forty-two pending certificates, the four external premise classes and the accepted Cambie reuse, with one locator correction required in the lane's obligation table; the locator delta review records the scoped exact delta pass that closed that hold. The bodies of the nine result pages below their *** separators are the reviewed bodies, except that the end-of-file fixer removed a second trailing newline from theorem_1_1.md (no other byte differs); their "Needs review" standing lines are the reviewed text and are read with the records above. This card was rewritten at filing, and the reviewed card is retained under evidence/assets/. No tier is awarded and no certificate row is moved: a later checker run under the evidence contract is a separate decision, and until it is taken every finite certificate stays pending and the theorem page's conclusion stays conditional.

Read status

Proof partially verified. Every symbolic deduction of Lemmas 3.1 and 3.2, the tail estimate of Certificate 4.2, Propositions 5.1 and 6.1 and Theorem 1.1 was reconstructed and independently reviewed at the records linked above. The forty-two finite inequalities remain unchecked. Lemma 4.1 and Certificates 4.3 and 4.4 are claims checked against the statements of their sources; the proofs of those external statements are not compiled.

Bears on

• Problem 690: the problem stays solved on its cited source, Cambie's Theorem 5, which settles $1\le k\le20$. This paper adds the claimed classification for every $k\ge4$, compiled here as a symbolic route that is conditional on the forty-two pending finite certificates and the four imported premise classes; it changes neither the status nor the compiled Cambie account.