Common positive threshold minima for reciprocal-sigma sums

a(n) = the smallest number k such that floor(Sum_{d|k} 1/sigma(d)) = n.

Conjecture: a(n) is also the smallest number k such that Sum_{d|k} 1/sigma(d) >= n.

— Jaroslav Krizek, OEIS A265714, revision 23.

This dossier anchors exactly the common-minimum conjecture. For every natural r>0, let g(k) be the exact rational sum of 1/sigma(d) over all positive divisors d of the positive natural k, where sigma(d) is the sum of all positive divisors of d. The sets {k>0 : floor(g(k))=r} and {k>0 : g(k)>=r} must both be nonempty and have the SAME least element. The floor is ordinary floor, r=1 is included, and neither r nor k has a finite upper bound. There is no additional existence or integrality premise. The source's separate integrality question and proposed numerical minima are outside this single assertion.

The two predicates ask respectively for a first floor level and a first threshold crossing of the same exact divisor sum. Their equivalence at the least argument is a first-tier externally named OEIS conjecture. The unchanged whole assertion is preregistered at https://github.com/the-omega-institute/trureturing/issues/23157#issuecomment-6469463476. The frozen theorem D5/S3/Factorization/ReciprocalSigmaThreshold.result addresses precisely this assertion.

Existence of arbitrarily large values alone does not exclude a first crossing that skips the desired floor level. The needed estimate controls the overshoot at the least crossing. The bounded source, repository, pinned Mathlib and selected third-party Lean searches found arithmetic suppliers but no exact prior whole-result supplier in the searched scope. This is not an exhaustive worldwide openness or priority claim.

Prime divisor terms give a lower bound by sums of 1/(p+1). Divergence of the reciprocal-prime sum supplies a positive argument reaching every threshold. Choose the least positive m with g(m)>=r. If m=1, g(1)=1 settles the floor bound.

Otherwise take the largest prime divisor P of m. Multiplicativity, the prime-power bound g(p^a)<=p/(p-1), and the telescoping product over the integers from 2 to P give g(m)<=P. Deleting one copy of P gives a positive q<m with g(m)<=g(q)(1+1/(P+1)). Minimality gives g(q)<r; since r<=P, the estimates imply r<=g(m)<r+1. Hence floor(g(m))=r. Every positive k with floor(g(k))=r also reaches r, so this same m is least for both predicates.

Krizek posed the conjecture. Israel's prior observations include multiplicativity, the prime value, divergence and existence of each level. The prime-power expansion is elementary; the common-minimum proof is a repository derivation, without a claim of literature priority.

A positive natural r for which either set is empty or the two least elements differ would refute the complete assertion. A finite check, a guessed minimum, an integrality assertion or a conditional-existence version cannot settle the unbounded statement.

D5/S3/Factorization/ReciprocalSigmaThreshold.lean defines the exact rational function g, the complete proposition claim, and proves result : claim. The result's statement identity is sha256:c20fbd403ccd97115b5dc48ee1e2b1f3f84739cb14169b914e3d9e796663f622. Its accepted axiom closure is Classical.choice, Quot.sound, and propext.

All seventeen proof declarations are bind-only. The four definitions are necessary and the sixteen private support lemmas have live consumers. The retention basis is open-problem-resolution; no escape witness is certified. The unbounded symbolic proof delivers no bounded enumeration, checker, numerical reduction or certified finite instance, so its utility classification is none.

The faithful compiled Reg mirror remains declared_unresolved, with IE-C050 / source.absent_occurrence and no accepted binding certificate. The missing raw selectors, complete-source reconstruction and current binding are disclosed under CLAUDE section 3.9 at https://github.com/the-omega-institute/trureturing/issues/23192. Compilation of that mirror does not establish declared_validated.

The frozen theorem D5/S3/Factorization/ReciprocalSigmaThreshold.result proves the entire positive-threshold common-minimum statement. The typed Scribe resolution binds this one theorem to this one source conjecture. Semantic fidelity, formal admission, required checks and publication retain their separate roles.

What the settlement shows

The decisive mechanism is the combination of the largest-prime upper bound and the smaller prime-deletion witness at the first crossing. It excludes overshoot of a whole integer interval and identifies both first occurrences for every positive r, including r=1. These conclusions are carried by the theorem above; no numerical minimum is supplied.

No condition beyond positivity is added to the source's thresholds or arguments. The argument concerns this exact reciprocal-sigma function; no extension to arbitrary weighted divisor sums is claimed. The source's integrality question and proposed later numerical minima remain outside the theorem, and no other source result is changed by this resolution.

Source-to-Lean fidelity is a semantic review obligation. Bounded source and library qualification does not prove global absence of another proof. The unresolved Reg binding is not a mathematical refutation or a validated escape registration. No claim of independent discovery, model diversity, worldwide openness or priority follows from the typed resolution.