The prime-three Kervaire family: detection and order three

Source and verification

This article explains family 309 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.

🏷️ Survival and additive order

The prime-three Kervaire problem asks which members of a distinguished family in the Adams spectral sequence represent actual stable homotopy classes of spheres. Its strong version asks for a representative killed by three. Family 309 claims a complete classification of both questions, with a new order-three representative in stem 322. (OpenAI, 2026)

These are separate conclusions. A spectral sequence records successive quotients of a filtration on homotopy. An element of order three in one quotient can lift to an element of larger order in the underlying group. The manuscript therefore combines a nonexistence argument for the higher family with a geometric construction that retains an actual Moore-spectrum boundary.

The manuscript's classification

In the mod-three Adams spectral sequence, let

$$b_j\in E_2^{2,4\cdot 3^{j+1}},j\ge 0.$$

The manuscript claims that $b_j$ survives nontrivially exactly for $j=0,2,3$. Each of these three detection cosets contains an element of additive order exactly three.

Index	Stable stem	Claimed outcome
$0$	$10$	Classical survivor with an order-three representative
$1$	$34$	Classical nonexistence
$2$	$106$	Classical corrected survivor with an order-three representative
$3$	$322$	The manuscript’s positive construction
$j\ge 4$	$4\cdot 3^{j+1}-2$	Excluded by outgoing detector differentials

The order claim is existential within each detection coset; it does not assign the same order to every representative. This is the odd-primary Adams-family problem. The classical prime-two Kervaire invariant-one problem uses a different family and dimension convention.

🏷️ Preliminaries: stable stems and filtered detection

The stable stem ${\pi }_d^S$ consists of maps between spheres whose dimension difference is $d$, after sufficiently many suspensions. Composition and smash products make these groups into a graded ring. Localization at three makes all integers prime to three invertible. Positive stable stems are finite, so using the three-complete sphere gives the same three-primary groups in the positive degrees relevant here.

The mod-three Adams spectral sequence starts with algebraic extension groups for the Steenrod algebra ${\mathcal{A}}_3$, which encodes mod-three cohomology operations:

$$E_2^{s,t}={\mathrm{Ext}}_{{\mathcal{A}}_3}^{s,t}({\mathbb{F}}_3,{\mathbb{F}}_3).$$

Its differentials have bidegree $(r,r-1)$, and a surviving class in position $(s,t)$ detects an associated-graded piece of ${\pi }_{t-s}^S$. Filtration $s$ measures how deeply a class lies in the resolution. A detection coset consists of actual homotopy classes with a specified nonzero leading symbol, allowing changes in higher filtration.

The Adams–Novikov spectral sequence replaces mod-three homology by Brown–Peterson homology. This makes formal-group information available, but its classes and differentials are distinct from those in the Adams sequence. A comparison map relates them. In particular, killing one Adams–Novikov representative does not automatically kill its Adams image: another representative with the same image may survive.

That distinction is already necessary in stem 106. The classical survivor is a corrected combination involving ${\beta }_{9/9}$ and ${\beta }_7$, rather than survival of ${\beta }_{9/9}$ alone. The manuscript’s higher detection argument retains every relevant two-line preimage.

The other basic object is the Moore spectrum $M=S/3$, defined by the cofiber sequence

$$S\stackrel{3}{\to }S\stackrel{i}{\to }M\stackrel{q}{\to }\Sigma S.$$

If a map $a:S^{d+1}\to M$ satisfies $qa=\Sigma \theta$, exactness gives $3\theta =0$ in actual sphere homotopy. A separate detection argument can then show $\theta \ne 0$, proving exact order three. This mechanism supplies the strong conclusion.

🏷️ The higher-family detector

The negative branch uses height-six Morava $E$-theory $E=E_6$, with an action of a chosen cyclic stabilizer subgroup $G=C_9$ and its subgroup $H=C_3$. Morava $E$-theory is an even-periodic homology theory whose coefficients describe deformations of a height-six formal group. Here the coefficient ring has a complete local degree-zero part and an invertible degree-minus-two period. (OpenAI, 2026)

The homotopy fixed point spectrum $E^{hG}$ records fixed data together with all the homotopies needed to make the group action coherent. Its spectral sequence begins with group cohomology of the coefficient module. The unit of $E$ gives a map from sphere homotopy into its homotopy groups.

The manuscript proves the following detection interface: if $j\ge 2$ and an actual sphere class is detected by $b_j$, its image in

$${\pi }_{4\cdot 3^{j+1}-2}(E^{hG})$$

is nonzero and has finite additive order. The proof compares filtrations, evaluates all possible two-line representatives in an integral coefficient model, and checks the finite layers through which their symbols lift. A nonzero algebraic symbol alone would not establish this assertion about an actual image.

Why a vanishing gap is unavailable

The manuscript also disproves the proposed literal vanishing of ${\pi }_{-2}(E^{hG})$. It constructs infinite-order classes in every even degree by an additive transfer.

The underlying observation is elementary enough to explain. Suppose a finite group acts semilinearly on a rank-one module over a characteristic-zero domain: it acts on the coefficient ring and multiplies the chosen module generator by units. If the kernel of its ring action acts trivially on the module, independence of distinct field automorphisms makes the trace

$$z⟼\sum _{g\in G}gz$$

nonzero. Clearing denominators yields a nonzero integral trace. A homotopy transfer lifts that trace to homotopy fixed points, and its nonzero restriction to the torsion-free coefficient module forces the lift to have infinite order.

This establishes nonvanishing, not torsion-freeness of the entire homotopy group. The manuscript does not assume the proposed 972-periodicity or a torsion-free replacement for the gap; it uses finite-page differentials instead.

🏷️ Outgoing differentials and finite deadlines

The relevant Tate construction is Euler localization: certain representation Euler classes become invertible. It is not rationalization. An actual homotopy-fixed-point class can become an incoming boundary after this localization. Thus the disappearance of a localized page does not, by itself, exclude a sphere class.

An actual class remains a cycle wherever its symbol is present. To contradict its existence, the proof must force a nonzero outgoing differential on that symbol, after excluding an earlier incoming death.

The manuscript first proves actual Euler nilpotence bounds

$$a_{\lambda |H}^{79}=0,a_{\lambda }^{235}=0.$$

These force the localized unit to be hit by differential lengths at most 157 and 469, respectively. The bounds come from genuine equivariant norms and geometric fixed points. A finite-tower exact-couple argument turns nullity of an actual lift into a bounded incoming differential, so this step does not need an assertion about convergence of an infinite Tate tower.

The coefficient calculation then constrains which arrows are possible. It constructs actual cyclically permuted degree-minus-two coordinates $x_0,\dots ,x_8$, with relations

$$x_i+x_{i+3}+x_{i+6}=0(0\le i<3),$$

and proves that the appropriate localization and completion present $E_{\ast }$. The full completed action, fixed formal-module geometry and character constraints matter; a tangent representation or reduced fixed-point set would leave additional possible differentials.

After subgroup calculations and ascent retaining integral restriction kernels, the problem becomes a finite array of one-dimensional groups and possible arrows. Translation classes identify copies of this array, but the proof establishes that those translations and their inverses are cycles on the pages where they are used. No periodicity of the final homotopy groups is inferred from these temporary units.

The forcing mechanism in one row

The final calculation sharpens the deadline: the unit is hit at length 313 and the next page is zero. The already known $b_2$ survivor determines an incoming pairing in a particular row of the array.

Call the relevant entries ${\mathsf{p}}_2$, ${\mathsf{p}}_0$ and $\mathsf{y}$. In that row the detector for $b_2$ occupies ${\mathsf{p}}_2$. Since it comes from an actual sphere class, it cannot support an outgoing differential. Its only available incoming arrow is the length-93 arrow from $\mathsf{y}$, so that arrow must occur before the deadline.

Now consider ${\mathsf{p}}_0$ in the same row. One proposed incoming arrow belongs to another residue row and is unavailable; its only other incoming arrow would use the $\mathsf{y}$ already consumed at length 93. Therefore ${\mathsf{p}}_0$ must disappear by an outgoing arrow. The exhaustive list allows length 85 or length 157; the argument need not decide which.

All detectors for $j\ge 4$ are translates of this ${\mathsf{p}}_0$ entry through the established unit cycles, with no earlier incoming death. Their forced outgoing differential contradicts actual sphere detection. The $j=3$ detector lies in a different residue class, so this argument leaves it undecided and a separate construction is necessary.

The coefficient geometry, full arrow enumeration and translation proofs are substantial manuscript inputs here. The row argument explains what they must deliver and why the direction of the arrows matters.

🏷️ The positive branch: a corrected extended cube

Begin with the known order-three class $x\in {\pi }_{106}^S$. The strong Kervaire criterion realizes it through a Moore map $a_0:S^{107}\to M$ whose cone has a nonzero reduced power ${\mathcal{P}}^{27}$. At three, ${\mathcal{P}}^m$ raises cohomological degree by $4m$.

A truncated extended cube takes three smash factors and their permutation symmetry over a finite lens-space construction. The resulting map lands in a four-cell spectrum $T$ fitting into

$${\Sigma }^{3}M⟶M\stackrel{j_T}{\to }T\stackrel{{\pi }_T}{\to }{\Sigma }^{4}M.$$

It produces a map $F:{\Sigma }^{322}M\to T$ with quotient $x^3{1}_M$ and a nonzero operation ${\mathcal{P}}^{81}$ from degree zero to degree 324 in its cone.

A useful cellular detail is that the bottom Moore attachment remains degree three. Passing to permutation coinvariants identifies the three one-upper-cell generators; it does not sum their boundaries into a degree-nine attachment. The two-upper-cell layer disappears because the transposition acts by the sign on two odd sphere factors and two is invertible at three.

The map into $T$ is not yet the desired Moore map. The manuscript constructs a correction ${\overline{F}}^{\prime }$ of Adams–Novikov filtration at least two with the same quotient under ${\pi }_T$. Then

$$F-{\overline{F}}^{\prime }=j_{T}Z$$

for some $Z:{\Sigma }^{322}M\to M$. The primary Adams invariant is additive and vanishes on maps of filtration at least two. Subtraction therefore preserves the nonzero ${\mathcal{P}}^{81}$ coefficient.

🏷️ The finite calculation needed for the correction

The correction has two obstruction components, and they are removed at different stages. First, the manuscript proves

$$i{x}^3\in F^{10}{\pi }_{318}M.$$

One obstruction component vanishes by an injectivity statement for the first two layers of the restricted Adams–Novikov tower. The other is induced by a map acting as a scalar multiple of $v_1$ on $B{P}_{\ast }M$. Cyclicity of that homology module identifies the induced map on the spectral sequence; no multiplicativity of the underlying map is assumed.

The finite calculation supplies precisely

$$\begin{array}{c}{v}_1:E_6^{10,328}(M)⟶E_6^{10,332}(M),\\ v_1=0\text{ on this map},\\ E_{10}^{f,322+f}(M)=0,\\ f\ge 14,f\equiv 2(\mathrm{mod}4).\end{array}$$

The first assertion is a zero map, not vanishing of the target group. It removes the filtration-ten contribution before higher-filtration vanishing is used. This order prevents a false comparison: an incoming full $d_9$ at filtration ten could start at filtration one, which the restricted tower omits. Increasing connectivity makes the remaining filtration finite, so vanishing of all its associated-graded pieces really kills the obstruction.

Universal bounds instead of choosing a differential

The source computes finite cobar complexes and encodes necessary constraints on the unknown fifth differential. Leibniz identities give affine equations; linear consequences of its square-zero identity refine them. The final affine family may still contain maps that are not square-zero. What is established is that it contains the actual differential.

For a matrix family $A(u)=A_0+{\sum }_{i=1}^r{u}_i{A}_i$, define universal bounds

$$\begin{array}{rl}{K}^{-}& =\bigcap _{i=0}^r\ker A_i,\\ K^{+}& =A_0^{-1}\left(\sum _{i=1}^r\mathrm{im}{A}_i\right),\\ I^{-}& =A_0\left(\bigcap _{i=1}^r\ker A_i\right),\\ I^{+}& =\sum _{i=0}^r\mathrm{im}{A}_i.\end{array}$$

For every allowed parameter value,

$$\begin{array}{c}{K}^{-}\subseteq \ker A(u)\subseteq K^{+},\\ I^{-}\subseteq \mathrm{im}A(u)\subseteq I^{+}.\end{array}$$

For example, $A(u)z=0$ implies $A_{0}z=-{\sum }_{i>0}{u}_i{A}_{i}z$, which proves the upper kernel bound. On the common kernel of the parameter matrices, $A(u)=A_0$, proving the lower image bound. Parameter correlations do not affect these implications.

If multiplication by $v_1$ sends the universal upper cycle space into the universal lower boundary space, its map on actual cohomology is zero. This proves the needed conclusion for every differential consistent with the source constraints, without selecting one of the remaining parameter values. A separate weight estimate handles all higher filtrations beyond the finite tables.

The tables and their recurrence are source-reported exact finite algebra. This note has checked the use of universal bounds and the correction interfaces; it has not independently regenerated the matrices or replayed their rank calculations.

🏷️ Recovering order three and the unstable consequence

A final endomorphism splitting removes a filtration-at-least-two section component of $Z$, again preserving the primary operation. What remains factors through the upper Moore quotient and yields

$$a_1:S^{323}\to M,q{a}_1=\Sigma \theta .$$

The cone of $a_1$ still has nonzero ${\mathcal{P}}^{81}$. The imported strong Kervaire criterion identifies $\theta$ as a nonzero class detected by $b_3$. Exactness of the actual Moore cofiber sequence gives $3\theta =0$, so its order is exactly three. The order statement comes from this boundary, not from spectral-sequence survival alone.

After localization at three, using previously established unstable equivalences, the manuscript translates the strong classification into product decompositions of looped degree-three sphere fibers. For $n>1$, the claimed decomposable indices are exactly $n=3,27,81$. At the new index, writing $S^{163}\{3\}$ for the fiber of the degree-three map and $T^d(3)$ for the relevant Anick H-space, it obtains

$$\Omega S^{163}\{3\}\simeq T^{163}(3)\times \Omega T^{487}(3).$$

This consequence uses the index change $n=3^{j+1}$ and the order-three version of the classification. The external unstable theorems, full coefficient-action and fixed-geometry proofs, complete Tate arrow tables, extended-power cone calculation and large finite rank computations remain outside this selected review.

🏷️ Key steps and proof outlines

A finite tower bounds incoming death without Tate convergence

Let $F^{s+1}\to F^s$ be a decreasing tower and let an actual class $\alpha \in {\pi }_V{F}^s$ become zero in ${\pi }_V{F}^0$. Its symbol in the layer $F^s/F^{s+1}$ is zero by page $E_{s+1}$. If it starts nonzero, it disappears by an incoming differential of length at most $s$.

The exact-couple proof identifies the cumulative boundaries in filtration $s$ as

$$B_r^s=j_s\ker ({\pi }_V{F}^s\to {\pi }_V{F}^{s-r+1}),1\le r\le s+1.$$

At $r=1$ this group is zero. Deriving the exact couple adds classes killed by one further tower map, giving the formula inductively. Since $\alpha$ is an actual class at level $s$, its layer symbol has zero connecting image and cannot support an outgoing differential while it remains nonzero. Its image at level zero is zero, so it lies in $B_{s+1}^s$.

Only the finite quotient involving these $s$ layers is used. This is the mechanism converting actual Euler nilpotence into deadlines in the localized detector, without assuming convergence of an infinite Tate filtration. The later finite array then distinguishes classes that can disappear by this incoming mechanism from detector symbols forced to support outgoing differentials. (OpenAI, 2026)

Removing the correction obstruction requires two different page arguments

For the positive construction, write $F_{a}Z$ for stage $a$ of the Adams–Novikov tower of a spectrum $Z$. The cube calculation gives a lift of $i{x}^3$ through $F_{10}M$, and hence through $F_{2}M$. A section into the Moore endomorphism spectrum converts it into a filtered version of $x^3{1}_M$.

The triangle defining $T$ supplies an obstruction to lifting that filtered endomorphism. The Moore endomorphism splitting divides it into two components. The first becomes zero after forgetting filtration because the original cube map $F$ is already an unfiltered lift. To conclude it was zero before forgetting, the source proves injectivity of

$${\pi }_{321}{F}_{2}M⟶{\pi }_{321}M.$$

Indeed, the quotient $M/F_{2}M$ has only its first two tower layers. Their homotopy is supported in degrees zero and three modulo four, respectively, whereas $322$ is two modulo four. Thus ${\pi }_{322}(M/F_{2}M)=0$, and the exact sequence gives the injection.

The second component is the image of the lifted cube under a fixed degree-four map $h:{\Sigma }^{4}M\to M$. Since $B{P}_{\ast }M$ is cyclic and its degree-four part is spanned by $v_1$, the induced homology map is a scalar multiple of multiplication by $v_1$. Functoriality, rather than multiplicativity of $h$, transports this description to the relevant spectral-sequence pages.

The finite calculation says that the map

$$v_1:E_6^{10,328}(M)⟶E_6^{10,332}(M)$$

is zero. The comparison with the restricted tower is valid at these slots, so the obstruction moves beyond filtration ten. Sparseness moves the next possible filtration to fourteen. Only then does the proof use the $E_{10}$ vanishing in all relevant filtrations at least fourteen. Increasing connectivity makes the filtration finite, so vanishing of all remaining successive quotients kills the obstruction itself.

The order is essential. An incoming $d_9$ at filtration ten could originate in filtration one, which the restricted tower omits. The proof does not compare those $E_{10}$ slots. It first removes filtration ten using the zero map on $E_6$, and then applies higher-filtration vanishing where comparison is valid.

The last subtraction preserves detection and produces actual order three

The corrected map ${\overline{F}}^{\prime }$ has the same quotient as $F$ and filtration at least two. Their difference therefore lifts to an endomorphism $Z:{\Sigma }^{322}M\to M$. The primary invariant is additive on the maps in question and vanishes on the correction, so $Z$ retains the nonzero cone operation ${\mathcal{P}}^{81}$.

There is one more correction. Split $Z$ into its section component and its component factoring through the upper quotient of the source Moore spectrum. The section component comes from ${\pi }_{322}M$ and also has filtration at least two, by the same low-layer degree supports. Subtracting it retains the primary invariant and leaves

$${\Sigma }^{322}M\stackrel{{\Sigma }^{322}q}{\to }{S}^{323}\stackrel{a_1}{\to }M.$$

The cone comparison preserves the bottom class and the tested degree-324 class, so $a_1$ satisfies the strong detection criterion. Finally the actual composite $q{a}_1=\Sigma \theta$ lies in the image of the Moore boundary. Exactness of the cofiber sequence gives $3\theta =0$ in sphere homotopy, while the nonzero primary detector gives $\theta \ne 0$. Thus the chosen representative has exact order three. Its order is established by a genuine map and exact sequence, beyond the order of its spectral-sequence symbol.

🏷️ Manuscripts and verification scope

• The Kervaire invariant problem at the prime three — pinned manuscript and companion source.

No family scope page was present at lean/docs/309.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. The Kervaire invariant problem at the prime three.