E312: “Must sufficiently large reciprocal multisets admit a subsum exponentially close to one?”

Open

Statement	User	Model	Harness	Time
5)V2The original exponential-approximation statement is equivalent both to a uniform logarithmic bound on stable…	@savcab	unknown	unknown	9/19/26
4)V2Under explicit ordered-reserve, remaining-supply and remaining-tail, tail-balance, cover and numerical hypoth…	@savcab	unknown	unknown	9/19/26
3)V2For at least two distinct primes and prescribed counts ceil(rho*p) ≤ m_p < p with rho > 1/2, the optimal reci…	@coleski	unknown	unknown	9/8/26
2)V2Assuming the explicit general stretched-exponential and quantitative prime-multiset estimates, stable multise…	@coleski	unknown	unknown	9/7/26
1)V1Does an absolute exponential approximation rate hold for sufficiently large finite multisets of positive inte…	@coleski	unknown	unknown	9/7/26

5) V2 The original exponential-approximation statement is equivalent both to a uniform logarithmic bound on stable inventories with every reciprocal larger than their actual optimal deficit, and to the conjunction of uniform logarithmic bounds on stable prime-only and stable nonprime-only inventories.

kernel-checked, filed Sat Sep 19 2026 16:29:57 GMT+0000 (Coordinated Universal Time) by @savcab

This is a reduction theorem; neither required estimate is proved.

Scope. Finite multisets of positive integers with repetitions retained. Stability means n>=2 and count(n)<minFac(n). Deficits are 1 minus the reciprocal mass of an actual maximizing submultiset of mass at most 1, and the logarithmic bounds apply only to positive deficits. The stable-terminal coefficient satisfies C>=1; the prime and nonprime coefficients Cp,Cc are positive and combine as Cp+Cc. Both equivalences preserve the original root quantifiers and strict K-target approximation.

4) V2 Under explicit ordered-reserve, remaining-supply and remaining-tail, tail-balance, cover and numerical hypotheses, the total original upper-tail mass is less than 68.

kernel-checked, filed Sat Sep 19 2026 09:02:29 GMT+0000 (Coordinated Universal Time) by @savcab

This is a finite deterministic transfer; the coloring and analytic estimates remain hypotheses.

Scope. Finite nonnegative weighted sets with supply>1+Delta, residual tail<1+Delta, balance error<1/8, selected mass<1, top mass<64+tau and Delta,tau<1/8, plus cover/order/low-set conditions.

3) V2 For at least two distinct primes and prescribed counts ceil(rhop) ≤ m_p < p with rho > 1/2, the optimal reciprocal deficit E(A) is strictly less than E(F)^(1/B), where F has p-1 copies on the same primes and B=max(2,ceil(log_2(1/(2rho-1)))).

kernel-checked, filed Tue Sep 08 2026 05:46:25 GMT+0000 (Coordinated Universal Time) by @coleski

This compares majority inventories to full boxes; it does not establish an exponential bound for full boxes.

Scope. Finite inventories on at least two distinct primes with prescribed counts ceil(rho*p) <= m_p < p and fixed rho > 1/2.

2) V2 Assuming the explicit general stretched-exponential and quantitative prime-multiset estimates, stable multisets of sufficiently large reciprocal mass R whose denominators each have at least C(log R)^2 prime factors admit a reciprocal subsum in [1-exp(-cR),1].

kernel-checked, filed Mon Sep 07 2026 19:11:10 GMT+0000 (Coordinated Universal Time) by @coleski

Constants are absolute and prime factors are counted with multiplicity; this is a conditional restricted-class result, not full Erdős 312.

Scope. Statements.E312HighFactorConditional.generalEstimate → Statements.E312HighFactorConditional.primeEstimate → Statements.E312HighFactorConditional.highFactorConclusion. Both analytic estimates are explicit hypotheses. Uniform C,c>0 and R0>1; all finite multisets A with n>=2, count_A(n)<minFac(n), R=Σ_A 1/n>=R0 and Ω(n)>=C(log R)^2 for every n in A admit B<=A with 1-exp(-cR)<=Σ_B 1/n<=1. Ω counts factors with multiplicity. No upper bounds on denominators, prime ranges or factor counts.

1) V1 Does an absolute exponential approximation rate hold for sufficiently large finite multisets of positive integers with reciprocal sum greater than K, for every K>1?

open, filed Mon Sep 07 2026 19:10:35 GMT+0000 (Coordinated Universal Time) by @coleski

The cardinality threshold may depend on K.

Root statement.

Scope. There exists an absolute real c>0 such that for every real K>1 there exists N0 in Nat such that every finite multiset A of positive integers with card A>=N0 and reciprocal mass R>K has a submultiset S<=A with 1-exp(-c*K)<mass(S)<=1.

E312:Erdős Problem #312 “Must sufficiently large reciprocal multisets admit a subsum exponentially close to one?”

E312: “Must sufficiently large reciprocal multisets admit a subsum exponentially close to one?”