Chromatic splitting: filtrations, attachments and sphere kernels

Source and verification

This article explains family 318 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.

Chromatic homotopy theory studies a spectrum one height at a time. Reassembling those heights requires understanding their overlaps. This family contains five manuscripts: a filtered splitting theorem, an explicit height-three filtration, an obstruction to the corresponding wedge decomposition, a small-prime obstruction to finite assembly, and counterexamples to a canonical retraction. Their conclusions concern different structures and different prime ranges.

🏷️ Preliminaries: localizations and attachments

A spectrum is a stable object representing generalized cohomology; its group ${\pi }_{d}X$ consists of stable maps from the sphere suspended $d$ times. Fix a prime $p$. Morava $K(n)$-theory detects chromatic height $n$; $L_{K(n)}X$ is the universal approximation to $X$ that inverts $K(n)$-homology equivalences. Write $L_r$ for localization with respect to the combined heights $0,\dots ,r$, with height zero meaning rational homology.

Unless stated otherwise, $S$ below is the derived $p$-complete sphere. Put

$$\begin{array}{c}{X}_n=L_{n-1}{L}_{K(n)}S,\\ Q=L_{0}S=H{\mathbb{Q}}_p.\end{array}$$

The overlap $X_n$ is what remains visible at lower heights after height-$n$ localization. The canonical map $i_n:L_{n-1}S\to X_n$ is induced by the localization unit.

A finite filtration is a sequence starting at zero and ending at $X_n$, with prescribed cofibers. A cofiber sequence $A\to B\to C\to \Sigma A$ reconstructs $B$ from $A$ and $C$ together with an attaching map $C\to \Sigma A$. If that attaching map vanishes, the sequence splits, giving $B\simeq A\vee C$. Thus knowing every cofiber does not determine a wedge decomposition.

The ordinary thick subcategory generated by some spectra allows finite sums, suspensions, cofibers and retracts. It does not allow arbitrary tensor products or infinite constructions. Finally, a retraction of $i_n$ is a map back whose composite with $i_n$ is the identity. Such a retraction would make ${\pi }_{\ast }{i}_n$ injective.

🏷️ A filtered splitting at generic primes

For a subset $I\subseteq \{1,\dots ,n\}$, set $d(I)={\sum }_{i\in I}(2i-1)$.

Manuscript's filtered splitting theorem

If $n\ge 1$ and $p>n+1$, the overlap has a finite filtration with $2^n$ cofibers:

$$\begin{array}{rl}{C}_{\varnothing }& =L_{n-1}S,\\ C_I& ={\Sigma }^{-d(I)}{L}_{n-\max I}S,\\ & I\ne \varnothing .\end{array}$$

The nonempty subsets are ordered in blocks by their maximum, and the first-stage map into $X_n$ is the canonical unit. (OpenAI, 2026)

The mechanism is stronger than matching dimensions. The source constructs one compatible family of actual localized-sphere classes $y_i\in {\pi }_{1-2i}{L}_{K(n)}S$. At every lower positive height $t$, the products $y_I$ with $I\subseteq \{1,\dots ,n-t\}$ supply a basis after the appropriate $K(t)$-test. Different independent choices of bases at each height would not give the compatible maps needed for assembly.

There are two substantial inputs. First, the coefficient calculation controls ordinary cooperations by finite covers and a common pole bound; replacing them by a larger analytic ring would change the problem. Second, constant cochains give compatible lifts through the relevant spectral sequence. The connective source supplies the degree bound needed to lift a degree-$b$ cohomology class to filtration $b$; its image is then transported to the periodic target. Connectivity is not asserted for that periodic target.

Once all basis classes lift compatibly, their outgoing differentials vanish. Since they generate the page, there are no incoming differentials either. Finite convergence then converts this basis into actual homotopy information. Assembly removes the blocks with fixed $\max I$, lowering the remaining chromatic height at each step. The last block is rational: its basis is chosen using the remaining rational dimensions, rather than identified automatically with products involving $y_n$. The construction retains the specified first-stage unit but does not produce a retraction of it. (OpenAI, 2026)

🏷️ Height three: eight layers and a nonzero attachment

For $p\ge 5$, the explicit companion manuscript gives the following ordered cofibers:

$$\begin{array}{c}{L}_{2}S,{\Sigma }^{-1}{L}_{2}S,\\ {\Sigma }^{-3}{L}_{1}S,{\Sigma }^{-4}{L}_{1}S,\\ {\Sigma }^{-5}Q,{\Sigma }^{-6}Q,\\ {\Sigma }^{-8}Q,{\Sigma }^{-9}Q.\end{array}$$

The first two stages form $W=L_{2}S\vee {\Sigma }^{-1}{L}_{2}S$. The map $w:W\to X_3$ uses the unit and a determinant class $\zeta$; it is a $K(2)$-equivalence. Let $Y$ be its cofiber. A $K(1)$-basis involving $1,\zeta ,\delta ,\zeta \delta$, with $|\delta |=-3$, supplies the middle layers. Rational compatibility is necessary: the chosen lift of $\delta$ differs from the rational generator by a nonzero scalar, and the rational map must be rescaled accordingly before the two local pieces can be glued.

A chromatic fracture square reconstructs an object from local pieces and a compatibility map on their overlap. Here a chosen compatibility homotopy gives the middle map into $Y$. Pulling its stages back along $X_3\to Y$ constructs the filtration; the remaining rational quotient supplies its four final stages. The manuscript also provides signed formulas for the attachments relative to these coherent choices. It does not claim every choice is canonical. (OpenAI, 2026b)

The first middle attachment, from ${\Sigma }^{-3}{L}_{1}S$ into $\Sigma W$, is nonzero. This is particularly instructive because the rationalized attaching maps vanish: rationalization alone cannot detect the obstruction to splitting the filtration.

🏷️ Why the strong wedge formula fails

For any spectrum $D$, let ${\tau }_D:L_{0}D\to L_0{L}_{K(2)}D$ be the rationalization of its canonical height-two localization map.

Manuscript's canonical-map obstruction

At every $p\ge 5$, with $D=L_{K(3)}S$, the map ${\tau }_D$ is nonzero on ${\pi }_{-3}$. Consequently the predicted height-three wedge decomposition fails even as an equivalence of underlying $E(2)$-local spectra, without any requirement on summand maps. (OpenAI, 2026a)

To see the implication, form the proposed wedge from the eight displayed layers. Its first two summands have no rational homotopy in degree $-3$, because $L_0{L}_{2}S=Q$. Every remaining summand is $K(2)$-acyclic. Hence its canonical map $\tau$ vanishes in degree $-3$. Naturality makes this property invariant under arbitrary equivalence of spectra. The comparison $D\to L_{2}D$ is both a rational and a $K(2)$-equivalence, so the manuscript’s nonvanishing rules out that wedge.

The nonvanishing proof constructs a detector by taking continuous fixed points of a completed coefficient theory and then evaluating to obtain an ordinary spectrum. The resulting target is $K(2)$-local, so the detector factors through the actual localization unit. A degree-three Lie-cohomology class, represented by a reduced-trace expression $\mathrm{trd}(U[V,W])$, is compared across heights. An integral Witt-vector comparison and a retraction onto constants establish a nonzero scalar relation between the two detected classes; an equality between two possibly zero images would not suffice.

Convergence is handled before rationalization. A finite cohomological bound makes each abutment degree depend on a finite Postnikov interval. Central scalar actions annihilate the rational cohomology of nonzero internal degrees. The surviving class in bidegree $(3,0)$ therefore gives actual homotopy in degree $-3$. This argument does not interchange rationalization with an uncontrolled infinite totalization. (OpenAI, 2026a)

The same obstruction explains the nonzero middle attachment above. If that attachment vanished, its middle layer would lift to $X_3$. It is $K(2)$-acyclic, so its composite with height-two localization would vanish. Yet its rational degree-$-3$ class is the generator detected by $\tau$. This contradicts the nonvanishing theorem. (OpenAI, 2026c) (OpenAI, 2026a)

🏷️ At the prime three, finite assembly also fails

The generic-prime theorem requires $p>n+1$. At $p=n=3$, a separate manuscript claims

$$L_2{L}_{K(3)}S\notin \mathrm{Thick}\{L_{0}S,L_{1}S,L_{2}S\}$$

in $E(2)$-local modules over the derived $3$-complete sphere. This excludes all finite attachments and retracts of these generators, including finite Moore-type corrections. It is a statement about ordinary thick generation, not generation as a tensor ideal. (OpenAI, 2026b)

Here the detector remembers a semilinear action. Choose a finite field $k$ containing ${\mathbb{F}}_{3^6}$ and let $G_F={\mathcal{O}}_F^{\times }$, where $F/{\mathbb{Q}}_3$ is the unramified quadratic extension. Apply $K(2)$-localized smash with Morava $E_2$, reduce modulo $3$, and pass from its deformation ring $k[[x]]$ to the Laurent-series field $Q_x=k((x))$. The full unit group $G_F$, acting as height-two automorphisms, acts on $Q_x$ and on the resulting vector spaces. “Semilinear” means $g(av)=g(a)g(v)$. Let $\omega$ denote the one-dimensional periodicity representation.

For the local sphere, all simple constituents of this test are powers ${\omega }^b$. Finite-dimensional semilinear representations with these constituents form a class closed under subobjects, quotients and extensions. The long exact sequence of homotopy groups therefore preserves this condition through finite cofibers, and retracts preserve it as well. After $K(2)$-localization the two lower generators vanish, so this condition is necessary for the claimed finite assembly.

After a finite free constant-field extension $S_k/S$, the diagonal constant-field summand of the overlap test has a degree-$-3$ constituent $Q_x(\delta )$, where $\delta$ is a nontrivial residue norm character. The extended sphere is a finite sum of copies of $S$ as an underlying spectrum, so a finite assembly of the original overlap would also assemble this extension. The critical calculation proves that this line is not any power of $\omega$. Restricting to the finite residue group would lose essential information; the full unit action is used.

The obstruction has an elementary valuation core. The fixed field for the full action is $k$, and an invariant Hasse element trivializes ${\omega }^2$ with a factor of valuation one. If a vector $f{e}^b$ transformed by the order-two character $\delta$, squaring it and dividing by the corresponding power of the Hasse element would give a fixed scalar. Valuations then force $b$ to be even. The trivialization of ${\omega }^2$ would then make $Q_x(\delta )$ trivial. This still needs a semilinear check: an eigenfunction $u$ trivializing that line would satisfy $u^2\in k^{\times }$. The finite field $k$ is relatively algebraically closed in $k((x))$, so $u\in k^{\times }$, contradicting the nontrivial character $\delta$. The fixed-field assertion itself uses the infinite effective action on $k((x))$: a nonconstant invariant of positive valuation would make $k((x))$ a finite extension of its power-series parameter field, which cannot have infinitely many automorphisms over that field.

To reach homotopy rather than just a coefficient calculation, the source controls the action on all finite deformation jets, not only the closed fiber. The generic spectral sequence has rows $0,1$ of periodicity type and rows $3,4$ twisted by $\delta$. Parity leaves only possible $d_3$ maps from untwisted to twisted lines. Any nonzero such map would identify the norm line with a periodicity power, which was excluded. Finite convergence gives the forbidden constituent in actual degree $-3$. The resulting two-step representation need not split; the norm line is a subrepresentation, which is enough. (OpenAI, 2026b)

🏷️ Weak splitting fails through an explicit sphere kernel

The last manuscript concerns the canonical unit itself.

Manuscript's weak-splitting counterexamples

The map $i_n:L_{n-1}S\to X_n$ has no homotopy retraction when $n=p$, $p\ge 5$, or when $n=p+1$, $p\ge 7$. In both ranges a named nonzero kernel class is the image of

$$\begin{array}{c}{z}_p={\beta }_1^{(p-1{)}^2},\\ |{\beta }_1|=2p(p-1)-2,\end{array}$$

where ${\beta }_1$ is the classical element in the homotopy of the $p$-local sphere. (OpenAI, 2026b)

The proof must establish two facts about this same sphere product. At height $p-1$, a finite-group homotopy fixed point detector receives the sphere by its actual unit. The known calculation identifies the image of ${\beta }_1$ with a nonzero multiple of a filtration-two class $\beta$, and shows ${\beta }^{(p-1{)}^2}\ne 0$ on the limiting page. Multiplicativity of the actual separated homotopy filtration then proves the image of $z_p$ is nonzero. Merely finding a class with the same bidegree would not identify the required sphere element.

At the upper height $a=p$ or $p+1$, the source identifies the local $E_a$-Adams filtration with the relevant descent filtration at finite stages. Its cohomological bound includes a derived-limit allowance and gives

$$F^{a^2+2}{\pi }_{\ast }{L}_{K(a)}S=0.$$

The positive-degree torsion element ${\beta }_1$ maps to zero in torsion-free Morava $E$ coefficients, hence lies in filtration at least one. Even periodicity eliminates its filtration-one associated-graded term, so its filtration is at least two. Multiplicativity puts $z_p$ in filtration at least $2(p-1{)}^2$. The needed inequalities are

$$\begin{array}{rl}& 2(p-1{)}^2-(p^2+1)\\ & =p^2-4p+1>0,\\ & 2(p-1{)}^2-((p+1{)}^2+1)\\ & =p(p-6)>0.\end{array}$$

in the respective prime ranges. Separated convergence makes the constant zero associated-graded tail an actual zero filtration group. Thus the upper unit kills $z_p$. This is an assertion about homotopy elements; it does not by itself prove nullity of an entire tensor-power map.

Completion preserves every positive-height Morava equivalence here. Naturality of completion and localization carries the surviving lower class to ${\pi }_{\ast }{L}_{n-1}S$ and kills it under precisely $i_n$. A retraction would force this map to be injective, which is impossible. At $p=5$ the example is ${\beta }_1^{16}$ in degree $608$; at $p=7$ it is ${\beta }_1^{36}$ in degree $2952$, working at heights seven and eight. (OpenAI, 2026b)

🏷️ A second obstruction: descent exponents

The last manuscript also supplies an independent exponent argument at height $p$ for $p\ge 5$. For a finite group $H$, let $A_H=F(H_{+},S_{(p)})$ be the function spectrum with translation action and pointwise multiplication, and let $J_H$ be the fiber of its unit. The $H$-exponent of an equivariant spectrum $V$ is the least integer $b\ge 1$ (or $\infty$ if none exists) for which the map $J_H^{\wedge b}\wedge V\to V$ is null.

With the pure stabilizer subgroup $H=C_p⋊C_{(p-1{)}^2}$ and $m=p^2+2$, the manuscript proves

$$\begin{array}{c}{\exp }_H(E_{p-1})>m,\\ {\exp }_H(T(L_{K(p)}S))\le m,\\ T(V)=L_{K(p-1)}(E_{p-1}\wedge V).\end{array}$$

For the lower bound, a sufficiently large $p$-divisible power ${\beta }^N$ survives to page $E_{m+1}$ in filtration $2N>m$. Bidegrees exclude incoming differentials and all but one outgoing possibility; the derivation rule kills that remaining differential because $p\mid N$. Finite-page survival suffices for this exponent obstruction: no sphere representative or permanent cycle is needed.

The upper bound needs an actual null map. A general nilpotence theorem first gives some finite exponent, before testing against finite spectra. The coefficient bound then annihilates the specified map on all finite tests. Brown–Comenetz duality for the local sphere rules out a residual phantom map, proving nullity with the explicit exponent $m$. Its finite layers are transported by $T$; a coherent mixed-height trace gives each transported layer exponent at most one. A hypothetical retraction of $i_p$ would make $E_{p-1}$ an equivariant retract of the upper object, contradicting these two bounds. (OpenAI, 2026b)

🏷️ How the five manuscripts fit together

The generic-prime filtration is compatible with nontrivial attachments, including the height-three attachment detected rationally after further localization. The prime-three finite-assembly obstruction lies outside the generic-prime range. The weak-splitting counterexamples occur at heights $p$ and $p+1$, also outside that range; they obstruct a particular canonical map by an actual kernel element. None of these distinctions can be replaced by a single assertion that an overlap “splits.”

This post explains the stated results and selected proof mechanisms. The completed coefficient comparisons, full deformation-jet calculation, descent equivalences and external fixed-point computations remain substantial inputs; this exposition is not an independent certification of all five proofs.

🏷️ Key steps and proof outlines

The family’s conclusions use three different kinds of evidence: compatible maps build a filtration, a natural localization map obstructs a wedge, and a bound on an actual tensor-power map controls a descent exponent. The following details explain why these tests cannot be interchanged.

Building the filtration without discarding its attachments

For $n>1$, suppose the compatible classes $z_I$ in the filtered-splitting criterion have been constructed. Start with $Q_0=X_n$ and remove the first block

$$B_1=L_{n-1}S\vee {\Sigma }^{-1}{L}_{n-1}S⟶Q_0⟶Q_1.$$

The map is a $K(n-1)$-equivalence, so its cofiber has height at most $n-2$. At step $j$, the current quotient $Q_{j-1}$ has height at most $n-j$. Compose the original maps $z_I$ with the quotient map, for all $I$ with $\max I=j$. Localization of their sources then gives

$$B_j=\underset{\max I=j}{\bigvee }{\Sigma }^{-d(I)}{L}_{n-j}S⟶Q_{j-1}⟶Q_j.$$

Why is this a $K(n-j)$-equivalence? At that height the original compatible family gives one direct-sum basis. Earlier steps removed exactly its components with smaller maximum. Because the maps at every step are composites of those same original components, the remaining components are precisely $B_j$. This identifies the quotient, rather than merely comparing its dimension. Chromatic fracture lowers the height once more.

The last quotient is rational. There is a small injectivity issue before subtracting rational dimensions: a cofiber could gain a shifted kernel. The source excludes this by mapping the $z_I$ to the height-one test. Their images are nonzero rationalized units in distinct shifted summands, even when several subsets have the same degree $d(I)$. They are therefore rationally independent. Each rationalized block map is injective onto its designated vectors, so no shifted kernel appears. The remaining dimensions determine a rational basis for the final block.

To recover a filtration of $X_n$, put $G_j=\mathrm{fib}(X_n\to Q_j)$. The pullback construction gives

$$G_{j-1}⟶G_j⟶B_j.$$

Pulling back the partial sums of each $B_j$ refines these into the individual layers. Put the unit component first in $B_1$ to retain the specified canonical map. Crucially, these pullbacks retain the boundary maps into $\Sigma G_{j-1}$; the construction never supplies nullhomotopies for them. (OpenAI, 2026d)

The same rational class tests the first middle attachment

At height three let $C={\Sigma }^{-3}{L}_{1}S$ be the first middle layer. If its attachment into $\Sigma W$ vanished, the map from $C$ into the quotient by $W$ would lift to $X_3$. Its composite with height-two localization would be zero because $C$ is $K(2)$-acyclic. On rational degree $-3$, however, the coherent choice of this middle map identifies its generator with the class on which the canonical map $\tau$ is nonzero. Naturality forces the two computations to agree, giving the contradiction.

This also explains the need to rescale the rational comparison when gluing the middle map: it must identify the same generator detected by the canonical map. A nonzero class in an unrelated rational vector space would not prove that this particular attachment is nonzero. Conversely, a split rationalized attachment does not imply a split attachment before rationalization. (OpenAI, 2026e) (OpenAI, 2026a)

Why a finite-page class can obstruct an exponent

For the lower exponent bound, write $t=p-1$ and $m=p^2+2$. Choose $N$ divisible by $p$ with $2N>m$. The class ${\beta }^N$ lies in filtration $2N$. The source’s bidegree calculation leaves only a possible outgoing differential of length $2t+1$ through page $m$; all incoming sources are excluded. On that remaining page the derivation rule gives

$$d_{2t+1}({\beta }^N)=N{\beta }^{N-1}{d}_{2t+1}(\beta )=0,$$

because the target is $p$-torsion. Thus the class survives to $E_{m+1}$ in filtration greater than $m$. An exponent at most $m$ would force the corresponding finite-page vanishing line, contradicting this class. For example, at $p=5$ one can take $m=27$, $N=15$: filtration $30$ survives to $E_{28}$. This argument requires neither a permanent cycle nor a sphere representative. The separate sphere-kernel argument does require identification of the actual sphere product and its limiting homotopy class. (OpenAI, 2026b)

From all finite tests to one null map

For the upper bound, vanishing of associated-graded groups alone would not establish nullity of

$$e_m:I_h^{{\otimes }_{h}m}⟶B_h,B_h=L_{K(h)}S.$$

The source first imports a nilpotence theorem giving some finite exponent, independent of every test object. With that in hand, the cohomological bound in the finite partial totalizations proves that $e_m$ vanishes after precomposition by every map from an ordinary finite spectrum.

The remaining issue is a possible phantom map. The local sphere has a special duality description. If $M_{h}S$ is the monochromatic sphere and $I_{\mathrm{BC}}$ the Brown–Comenetz dualizing spectrum, the imported duality theorem makes $D_h=F(M_{h}S,I_{\mathrm{BC}})$ invertible in the local category. Consequently, for the ordinary spectrum $C=D_h\wedge M_{h}S$,

$$[U,B_h]\cong {\mathrm{Hom}}_{\mathbb{Z}}({\pi }_0(U\wedge C),\mathbb{Q}/{\mathbb{Z}}_{(p)}).$$

Express $U$ as a filtered homotopy colimit of ordinary finite spectra $U_i$. Ordinary smash and homotopy groups commute with that colimit. A functional vanishing on every ${\pi }_0(U_i\wedge C)$ therefore vanishes on the whole group. This proves that the finitely tested map is actually null, without commuting localizations through that colimit.

Its nullhomotopy makes $B_h$ a retract of $\mathrm{cofib}(e_m)$. The cofibers of the successive tensor-power maps give exactly $m$ layers, each of the form $E_h{\otimes }_h{I}_h^{{\otimes }_{h}i}$. These are the finite layers transported to the lower height in the exponent comparison. Thus the numerical exponent is backed by an actual null map and an actual finite construction. (OpenAI, 2026b)

🏷️ Manuscripts and verification scope

• Filtered chromatic splitting at generic primes — pinned manuscript and companion source.
• The height-three chromatic overlap: an explicit filtration and its attachments — pinned manuscript and companion source.
• A rational obstruction to strong chromatic splitting at height three — pinned manuscript and companion source.
• Failure of finite assembly for a chromatic overlap at the prime three — pinned manuscript and companion source.
• Counterexamples to weak chromatic splitting: sphere kernels and descent exponents — pinned manuscript and companion source.

No family scope page was present at lean/docs/318.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 2026a. A rational obstruction to strong chromatic splitting at height three.

🐻  OpenAI 2026b. Counterexamples to weak chromatic splitting: sphere kernels and descent exponents.

🐻  OpenAI 2026c. Failure of finite assembly for a chromatic overlap at the prime three.

🐻  OpenAI 2026d. Filtered chromatic splitting at generic primes.

🐻  OpenAI 2026e. The height-three chromatic overlap: an explicit filtration and its attachments.