Infinitely many closed geodesic images

Source and verification

This article explains family 345 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

This family reports infinitely many geometrically distinct closed geodesics for every smooth Riemannian metric on a standard sphere of dimension at least two. Its higher-dimensional proof studies loops below an energy threshold, including classes that disappear as the threshold increases. A uniform filling estimate controls these temporary classes; reflection and loop doubling then yield a contradiction to the assumption of finitely many prime images. Finite covering arguments extend the conclusion to spherical space forms and, with an imported theorem, all closed three-manifolds. (OpenAI, 2026)

🏷️ Prime geodesics and geometric distinction

A closed geodesic is a nonconstant periodic curve satisfying the geodesic equation. Parametrize it with constant speed. It is prime if it traverses its periodic trajectory once, rather than repeating a shorter closed geodesic.

The relevant count is the number of images in the manifold. Changing the starting time, reversing orientation, or traversing a geodesic several times does not produce a new image. A prime geodesic may have self-intersections; prime does not mean simple.

Infinitude reported by family 345

For every $n\ge 2$ and every smooth Riemannian metric on the standard smooth sphere $S^n$, the manuscript proves the existence of infinitely many prime closed geodesics with pairwise distinct images. (OpenAI, 2026)

No genericity or nondegeneracy hypothesis is imposed. The two-sphere conclusion is a classical input; the manuscript develops the unrestricted higher-dimensional argument. It also proves the finite-cover consequence and derives the three-manifold corollary discussed below.

The conclusion is qualitative. On a round sphere, infinitely many great circles already have the same prime length. Thus infinitely many images do not imply infinitely many distinct prime lengths or a quantitative counting rate.

🏷️ Energy, loop spaces, and critical points

Let

$$\Lambda =H^1(\mathbb{R}/\mathbb{Z},S^n)$$

be the space of period-one loops with square-integrable velocity, equipped with the strong Sobolev topology. Constant loops form a copy $M$ of $S^n$. Define energy, square-root energy, and length by

$$\begin{array}{rl}E(\gamma )& ={\int }_0^1|\dot{\gamma }{|}^{2}dt,\\ F(\gamma )& =\sqrt{E(\gamma )},\\ \ell (\gamma )& ={\int }_0^1|\dot{\gamma }|dt.\end{array}$$

Cauchy–Schwarz gives $\ell \le F$, with equality for constant-speed loops. Write $X_a=\{F<a\}$ for the open sublevel. A positive threshold is regular when $a^2$ is a regular value of $E$.

The nonconstant critical points of energy are constant-speed closed geodesics. Shifting the starting point gives a whole critical circle. The Morse index counts independent directions in which the second variation is negative; degeneracy means there are additional zero directions beyond the unavoidable parameter-shift direction.

If only finitely many prime images exist, every critical loop is an iterate of one of finitely many oriented prime trajectories. For prime length ${\ell }_i$ and mean index ${\Delta }_i$, iteration theory gives

$$|I_{i,m}-m{\Delta }_i|\le n.$$

Hence local changes in topology lie within fixed-width bands around finitely many degree-versus-length lines of slope ${\beta }_i={\Delta }_i/{\ell }_i$. The manuscript proves uniform bounds for the local ranks, including degenerate iterates and primes with zero mean index.

Ordinary loop-space Betti-number growth does not settle the sphere problem: the degreewise ranks are bounded. Moreover sublevels can contain extra classes that eventually die. A proof must control those classes before comparing sublevel ranks with global ranks.

🏷️ A uniform filling increment

Use coefficients in ${\mathbb{F}}_2$ throughout. A homology class represented below level $a$ is globally null if it bounds a finite chain somewhere in the whole loop space. Its filling need not initially lie near level $a$.

Degree-independent filling estimate

For every smooth metric on $S^n$, $n\ge 3$, the manuscript constructs one constant $D>0$ such that every globally null homology class in $X_a$ already vanishes in $X_{a+D}$, in every degree and for every $a>0$. The same holds for homology relative to constant loops. This estimate does not assume finitely many prime images. (OpenAI, 2026)

For a fixed degree $j$, shortening a family of loops first gives an increment $D_j$. Repeated shortening lowers the starting level by one at each stage. The upper levels of the successive deformation prisms decrease too, so their costs are not added over all stages. Once the family reaches a fixed low level, a finite-dimensional polygonal model supplies finitely many basis fillings with a common bound.

The research step removes the dependence on $j$. A loop product joins loops at a common point and lowers the sum of their homological degrees by $n$. A complementary cut operation, denoted $D_T$, uses a cohomology class $T$ to discard a sufficiently long loop segment and lower degree.

The manuscript chooses a product-power cycle $S$ carried below length $l$ and a cocycle $T$ that vanishes below length $L$. A resonance theorem relates critical length to degree and allows these choices with $l-L$ bounded independently of the power. Thus cutting and subsequently joining restore the degree with only a bounded net length cost.

Two identities make this useful. For suitable degrees and with a uniform additive filtration allowance,

$$\mu (D_{T}x,S)\simeq \mu (D_{T}S,x),$$

where $\mu$ denotes the chain-level join and $\simeq$ means homologous in the specified length sublevel. The second identity is

$$[D_{T}S]=[M]\text{in }{H}_n(\Lambda ).$$

The constant-loop fundamental class $[M]$ is the unit for the loop product. The fact that this is an absolute unit statement is essential: a relative class alone would not provide the same filling correction.

For a high-degree null cycle $x$, choose the power so that $D_{T}x$ lies in a fixed finite range of degrees. Fill that cut cycle using one of the finitely many $D_j$. Also fill the difference between $D_{T}S$ and a constant-loop fundamental cycle. Join the two fillings, use the interchange homology, and then use the unit homotopy. Their total support lies below the original level plus one fixed increment.

The construction first uses length sublevels. Strongly continuous arclength normalization gives $F=\ell$ without raising the original energy sublevel, transferring the conclusion to $X_a$. To pass to relative homology, evaluation at time zero retracts the constant loops. The chain map

$$P=\mathrm{id}-i_{\#}({\mathrm{ev}}_0{)}_{\#}$$

annihilates constant-loop chains and preserves every positive sublevel. Applying it to a relative cycle and a global relative filling reduces the assertion to the absolute one. The same constant $D$ works at critical thresholds as well as regular ones.

🏷️ Reflection, doubling, and transient classes

Loop reflection and half-period translation act by

$$\begin{array}{rl}(r\gamma )(t)& =\gamma (-t),\\ (\tau \gamma )(t)& =\gamma (t+\frac{1}{2}).\end{array}$$

They commute. The fixed loops of $\tau$ are precisely double loops. The map $\delta \gamma (t)=\gamma (2t)$ identifies the whole loop space with that fixed locus and satisfies

$$F(\delta \gamma )=2F(\gamma ).$$

Thus the fixed part of $X_{2a}$ is a copy of $X_a$.

The proof uses equivariant cohomology to retain reflection symmetry. For an invariant pair $(Y,Z)$, it is the ordinary cohomology of $(ER{\times }_{R}Y,ER{\times }_{R}Z)$, where $ER$ is contractible with a free reflection action. This construction records the action while avoiding the singularities of a direct quotient.

Put

$$\begin{array}{c}{B}_a^j=H_R^j(X_a,M),\\ B_{\infty }^j=H_R^j(\Lambda ,M).\end{array}$$

Restriction maps global classes into sublevel classes. Define the extra, or transient, quotient and its total dimension by

$$\begin{array}{c}{Q}_a^j=\mathrm{coker}(B_{\infty }^j\to B_a^j),\\ T(a)=\sum _j\dim Q_a^j.\end{array}$$

Under the finite-prime assumption, the sum is finite at each regular threshold. The central intermediate conclusion is stronger: $T(a)$ is bounded by one constant independent of $a$.

🏷️ Two degree filtrations and a linear bound

Restriction to double loops has two degree labels: the actual equivariant degree in the ambient sublevel and the degree of the half-length root loop. Inverting the class associated with half-translation eventually makes restriction an isomorphism. That eventual statement alone loses the finite-degree information needed here.

The manuscript instead keeps the actual restriction image $E_J$ in each ambient degree $J$. Let $P_j$ project its root coefficients to root degrees at most $j$. It measures four nonnegative defects: missing low-degree root classes in degrees $2j$ and $2j-1$, actual-image vectors supported above root degree $j$, and ambient classes invisible on the fixed locus.

Their sum, denoted ${\Phi }_j(a)$, obeys

$$\sum _{j\ge 1}{\Phi }_j(a)\le C(1+a).$$

The reason for linear growth is geometric: finitely many prime trajectories have only $O(a)$ iterates below a length threshold. A common invariant descending flow defines the full and fixed crossing pairs, so their restriction maps and exact sequences agree. Away from the index window of a crossing, the combined defect cannot increase. Each crossing increases it by only a bounded amount. The preceding-degree term is needed to accommodate the degree shift in the connecting map.

Let $w$ be the degree-one reflection class. Away from fixed-width bands around the finitely many index slopes and the global resonance slope, ordinary filling and the relevant exact sequences make multiplication by $w$ an isomorphism between neighboring transient quotients.

Consequently a nonzero transient in a long degree gap would extend to a string of length comparable to $a$. Many independent root classes from one part of that string can be lifted into actual ambient restriction images. Multiplying by $w$ moves their root degrees upward; multiplying by the half-translation class changes ambient degree without changing those root coefficients. This creates order-$a$ independent high-root vectors at order-$a$ different cutoffs, forcing an order-$a^2$ contribution to the defect sum. The linear bound excludes it.

Only a bounded number of degree bands remain. Positive mean-index primes contribute uniformly bounded rank in each such degree. Zero mean-index primes require a separate low-degree argument: the uniform filling increment restricts the relevant deaths to a length window of bounded width, containing only a bounded number of critical events. Together these estimates prove the uniform bound for $T(a)$.

🏷️ Equality under doubling forces the contradiction

The global reflection cohomology has a basis arranged in pairs. Write $d=n-1$. The lower classes, called bottoms in the manuscript, have degrees

$$(2m-1)d+k,0\le k\le d,$$

and each has a partner obtained by cup product with

$$e={\mathrm{ev}}_0^{\ast }[S^n{]}^{\ast }.$$

Here $e$ has degree $n$ and is unrelated to the energy notation $E$. The label $m$ comes from the round-metric filtration used to compute the global classes; it does not label a prime geodesic of the arbitrary metric.

The full rotation-and-reflection action gives

$$T(2a)\ge T(a)$$

whenever $2a$ is regular. Its equality statement is decisive: every nonzero restricted bottom from an odd round-filtration stage must have a partner whose restriction is independent modulo all bottom restrictions in that degree. Mere nonvanishing of the partner would be weaker than the condition used in the proof.

Because $T$ is a bounded nonnegative integer, it attains a maximum on an open interval without critical values. Doubling that interval preserves maximality at regular points. This uses monotonicity under doubling only; $T$ is not asserted to be monotone between arbitrary thresholds.

In a sufficiently large doubled interval, the degree-gap comparison forces an odd-stage bottom class to change from invisible to visible. Choose its first critical crossing. The exact sequence says its newly visible restriction comes from the local crossing group.

Cup product by $e$ vanishes on that local group. To see the geometry, choose a point of $S^n$ outside the finite union of prime images. The critical neighborhoods can be chosen so that evaluation at zero lands in the punctured sphere, whose top cohomology vanishes. A common descending flow transfers this vanishing to the actual crossing, even if several prime iterates cross at once.

Therefore the newly visible bottom has zero partner immediately after its first appearance. Equality under doubling required an independent partner there. This contradiction excludes the finite-prime-image assumption. The local rank argument and the common-flow construction retain arbitrary degeneracy throughout. (OpenAI, 2026)

🏷️ Finite covers and closed three-manifolds

If $\tilde{M}\to M$ is a finite smooth cover of degree $r$, equip the cover with the pulled-back metric. It is a local isometry, so closed geodesics upstairs project to closed geodesics downstairs.

For one oriented prime trajectory downstairs, lifting a period permutes the $r$ points over its starting point. Each lift returns after at most $r$ periods, including its velocity. Thus at most $r$ upstairs prime images can project to a fixed downstairs image. The argument permits self-intersections and does not require a regular covering.

It follows that every smooth metric on a manifold finitely covered by a standard sphere has infinitely many prime images. In dimension three, the manuscript combines this with two imported results. For infinite fundamental group, the theorem of Rademacher–Taimanov gives infinitude. For finite fundamental group, geometrization identifies the manifold as a spherical space form, so the finite-cover argument applies. Nonorientable manifolds are included; for a disconnected closed manifold it suffices to work on one component.

🏷️ Key steps and proof outlines

Degree reduction has an explicit bounded remainder

The uniform filling theorem begins with a high-degree globally null cycle $x$ of degree $j$. Put $d=n-1$ and choose

$$m=\lfloor \frac{j-2}{2d}\rfloor ,q=2md,r=j-q.$$

Then $2\le r\le 2n-1$. The cut operation determined by the chosen cocycle lowers the cycle to this fixed finite range. Its product-power partner has degree $n+q$, so joining restores the original degree. The resonance estimates make the difference between the partner’s support length $l$ and the cut threshold $L$ one constant $C_{\mathrm{gap}}$ independent of $j$. (OpenAI, 2026)

Let $D_{\ast }$ be the maximum of the fixed-degree filling constants in a finite range containing all these remainders. The cut cycle fills by a chain $W$ below length $a-L+1+D_{\ast }$. If that threshold is nonpositive, the cut has no surviving terms and $W=0$ suffices. The absolute unit statement also supplies a filling $V$ of $D_{T}S-c$, where $c$ is a constant-loop fundamental cycle, below $C_{\mathrm{gap}}+1+D_{\ast }$.

Two joined fillings and one interchange fill the original cycle

The chain $\mu (W,S)$ has boundary $\mu (D_{T}x,S)$. The chain $\mu (V,x)$ has boundary $\mu (D_{T}S,x)-\mu (c,x)$. The filtered interchange homology joins the first boundary term to $\mu (D_{T}S,x)$, and the unit homotopy joins $\mu (c,x)$ to $x$.

Every chain lies below $a+C_{\mathrm{gap}}+D_{\ast }+6$, including the join and subdivision allowances. Their sum fills $x$ with one degree-independent increment. The key point is cancellation of the large lengths $l$ and $L$; separately bounding them would give a cost increasing with degree.

Arclength normalization converts the resulting length filling into a square-root-energy filling, and its deformation prism stays inside the original energy sublevel. For relative chains, $P=I-i_{\#}({\mathrm{ev}}_0{)}_{\#}$ is a chain map killing constants. If $\partial z$ consists of constants, then $Pz$ is an absolute cycle; applying $P$ to a relative global filling makes it absolutely null. Thus exactly the same increment works relative to constants, including at critical thresholds.

A linear crossing budget excludes a quadratic transient cost

Under the hypothetical finite-prime assumption, there are only $O(a)$ critical iterates below level $a$, and each crossing changes the finite-cut defect by a uniformly bounded amount. The total defect budget is therefore linear in $a$. A transient class in a long gap, however, propagates through many neighboring degrees by the reflection class. Lifting a long segment of this string into actual restriction images and shifting ambient degree by the half-translation class produces order-$a$ independent defects at order-$a$ cutoffs.

This quadratic demand contradicts the linear budget. Remaining bounded-width degree bands are treated by local ranks and the uniform filling increment, including zero mean-index primes. The final doubling argument uses the maximum of the resulting bounded integer-valued transient count. Equality at that maximum requires an independent partner for each newly visible bottom class; local evaluation in a punctured sphere kills that partner at its first crossing. The filling theorem, finite-cut bookkeeping, and equality case of doubling therefore have distinct roles in the contradiction.

🏷️ Connections

The metric Blaschke family studies rigidity under a common minimizing distance and prime period. The present result allows arbitrary metrics and counts distinct images, even when many share a length.

Simple closed geodesics on hyperbolic surfaces develops quantitative counting for simple curves on negatively curved surfaces. Here prime curves may self-intersect, the topology is spherical, and the conclusion is qualitative infinitude.

🏷️ Manuscripts and verification scope

• Infinitely many closed geodesic images on every Riemannian sphere — pinned manuscript and companion source.

No family scope page was present at lean/docs/345.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Infinitely many closed geodesic images on every Riemannian sphere.