Integral divisibility profiles and a gcd obstruction for minimax levels

Definitions and mathematical statement

Hingston's Question 3.2.1 asks for a metric on $S^n$ and an integral free-loop homology class $X$ with \[ 0<\operatorname{cr}(mX)<\operatorname{cr}(X) \quad\text{for some }m\in{\mathbb{N}}. \] Its simplest stated case is a generator $X\in H_1(\Lambda S^2;{\mathbb{Z}})\cong{\mathbb{Z}}$. Use $\Lambda M=W^{1,2}(S^1,M)$, and fix the length-sublevel convention $\Lambda^aM=\{\gamma:\operatorname{Length}(\gamma)\leq a\}$. Define \[ \operatorname{cr}(X)=\inf\{a:X\in\operatorname{im}[H_k(\Lambda^aM;{\mathbb{Z}})\to H_k(\Lambda M;{\mathbb{Z}})]\}. \] The source does not specify length versus energy normalization; the length choice is needed for the numerical lower bound below. Zero or annihilated torsion classes cannot satisfy the positive inequality.

The source relates this to the global mean frequency $\alpha_{g,A}$, when it exists, defined by the limit of $\deg Z/\operatorname{cr}_A(Z)$ over nonzero homogeneous classes as degree tends to infinity. It cites [Hin-Rad2013] for existence on spheres with field coefficients and asks whether the frequency or critical levels depend on the coefficient field. Its shorthand description of integral loop homology as ${\mathbb{Z}}$ or ${\mathbb{Z}}_2$ is not used here as a complete degreewise homology calculation; exceptional degrees or direct-sum issues require checking. The aid treats a primitive infinite-order class rather than resolving all coefficient behavior.

For any nested integral filtration, put $I_a=\operatorname{im}H_k(\Lambda^aM;{\mathbb{Z}})$. On the infinite cyclic subgroup generated by $X$, there is a unique integer $d(a)\geq0$ with \[ I_a\cap{\mathbb{Z}} X=d(a){\mathbb{Z}} X, \] where $d(a)=0$ means no nonzero multiple appears. The reported exact formulas are \[ \operatorname{cr}(mX)=\inf\{a:d(a)>0,\ d(a)\mid m\}, \] \[ \operatorname{cr}(\gcd(m,n)X) =\max\{\operatorname{cr}(mX),\operatorname{cr}(nX)\}. \] In particular a strict drop is equivalent to finding a level strictly below $\operatorname{cr}(X)$ at which some nonzero multiple of $X$, but not $X$, lies in $I_a$. Equivalently its image in the filtered quotient has finite nonzero order at such a separated level. Nonsaturation at a single endpoint alone need not force a strict difference of infima unless the filtration has an appropriate attainment/persistence property; that qualification is essential for an arbitrary nested filtration.

The aid also reports the rational envelope \[ \operatorname{cr}_{{\mathbb{Q}}}(X\otimes1) =\inf_{m\geq1}\operatorname{cr}_{{\mathbb{Z}}}(mX). \] It is a statement about this class under rational change of coefficients, not a proof that two field-valued global frequencies differ.

The source proposes a hypothetical four-level profile with $a<b<c<d$: critical value $d$ when $m$ is coprime to both $3,7$, $c$ when $3\mid m,7\nmid m$, $b$ when $7\mid m,3\nmid m$, and $a$ when $21\mid m$. The gcd law rules this out for one fixed class: $\operatorname{cr}(X)=\max\{\operatorname{cr}(3X),\operatorname{cr}(7X)\}=c$, contradicting the proposed value $d$. It was an illustrative pattern, not an asserted geometric example.

Finally, for a smooth Riemannian $S^2$ the aid reports \[ \operatorname{cr}(mX)\geq2\operatorname{inj}(S^2,g)>0 \quad(m\geq1) \] for the $H_1$ generator in the length convention. Thus positivity is automatic in that case, but no metric producing the strict drop is constructed. The original existence and field-dependence questions remain unresolved in the supplied account.

Short English statement

Ask whether multiplying an integral loop-homology class can strictly lower its minimax level. The aid gives an exact divisor profile and gcd-maximum law, rules out the source's hypothetical independent 3/7 pattern, and proves positivity for the $S^2$ generator. It supplies no metric realizing a strict drop.

Sources

• Local UnsolvedMath registry, permanent ID $20001878$, code catalogue entry; canonical aim-geometry-notes.json, notes[215]; source tag "question," statement status "reconstructed_unverified."
• Keith Burns and Vladimir S. Matveev, "Open problems and questions about geodesics," Section3.2 "Stability of minimax levels," Question3.2.1 attributed to Hingston, https://aimath.org/pastworkshops/geodesicsproblems.pdf. No PDF inspection is claimed here.
• Source reference label [Hin-Rad2013], for the Resonance Theorem over fields, is retained without inventing its unavailable full bibliographic entry.
• Research attempt 1: "partial_result," problem status at run "open," registry status "partially_solved," rigor label "proved," novelty confidence "low." The research aid is machine generated, not peer reviewed; independently verify before citation.