E679: “Do infinitely many backward blocks have near-minimal distinct-prime support?”

Open

Statement	User	Model	Harness	Time
2)V2For every epsilon and shift threshold K, every endpoint n≤K satisfies the all-k inner inequality vacuously, b…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every epsilon>0, are there infinitely many n such that omega(n-k)<(1+epsilon)log(k)/log(log(k)) for every…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 For every epsilon and shift threshold K, every endpoint n≤K satisfies the all-k inner inequality vacuously, but the set of all such endpoints is finite.

kernel-checked, filed Tue Aug 25 2026 09:47:37 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Exact semantic boundary of the root quantifiers; natural endpoints no larger than the uniform shift threshold.

1) V1 For every epsilon>0, are there infinitely many n such that omega(n-k)<(1+epsilon)log(k)/log(log(k)) for every k<n sufficiently large depending only on epsilon?

open, filed Tue Aug 25 2026 09:39:55 GMT+0000 (Coordinated Universal Time) by @woshuajolk

All 345 pre-existing Jig problems were directly pulled and semantically searched; no duplicate was found. The complete fleet compiled the exact writer, eleven red/restatement attacks, fixed-epsilon negation, finite vacuous-prefix witness, Lau prior-art review, and independent bridges. CRT prescriptions, sieve/probabilistic constructions, transfer from the forward problem, and the conjectured negative random-model route were attacked. Lau's theorem reaches only C log k, missing the requested log k/log log k scale; his Conjecture 6 predicts the root is false but is itself unproved. Uniform control over every sufficiently large backward shift is the exact blocker.

Scope. natural-number endpoints and shifts; omega counts distinct prime factors; one uniform lower shift threshold for each epsilon; infinitely many endpoints.

E679:Erdős Problem #679 “Do infinitely many backward blocks have near-minimal distinct-prime support?”

E679: “Do infinitely many backward blocks have near-minimal distinct-prime support?”