Bounded Hochschild cohomology: harmonic averaging and rigidity

Source and verification

This article explains family 295 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.

🏷️ Preliminaries: bounded cocycles

A von Neumann algebra $M$ is a unital algebra of bounded operators on a Hilbert space, closed under adjoints and weak operator limits. Its multiplication is associative, but generally not commutative. This article concerns bounded multilinear identities involving that multiplication; no previous course in homological algebra is required.

Let $C_b^k(M,M)$ be the space of bounded complex $k$-linear maps from $M^k$ to $M$, with norm

$$\Vert f\Vert =\underset{\Vert a_1\Vert ,\dots ,\Vert a_k\Vert \le 1}{\sup }\Vert f(a_1,\dots ,a_k)\Vert .$$

For a tuple $\mathbf{a}$, let ${\mathbf{a}}^{[j]}$ denote the shorter tuple obtained by replacing adjacent entries $a_j,a_{j+1}$ with $a_j{a}_{j+1}$. The Hochschild differential forms a $(k+1)$-linear map by multiplying successive inputs and adding two outside multiplication terms:

$$\begin{array}{rl}& df(\mathbf{a})\\ & =a_{1}f(a_2,\dots ,a_{k+1})\\ & +\sum _{j=1}^k(-1{)}^{j}f({\mathbf{a}}^{[j]})\\ & +(-1{)}^{k+1}f(a_1,\dots ,a_k)a_{k+1}.\end{array}$$

Associativity makes the terms in $d(df)$ cancel in pairs, so $d^2=0$. A map satisfying $df=0$ is a cocycle. A map of the form $dg$ is a coboundary, and $g$ is its primitive. The quotient of cocycles by coboundaries is $H_b^k(M,M)$.

The degree-two equation

A bounded bilinear map $f$ is a cocycle when

$$\begin{array}{rl}0=& af(b,c)-f(ab,c)\\ & +f(a,bc)-f(a,b)c.\end{array}$$

Vanishing of its class means that a bounded linear map $g:M\to M$ satisfies

$$\begin{array}{rl}f(a,b)=& ag(b)-g(ab)\\ & +g(a)b.\end{array}$$

Thus the question asks whether every compatible bilinear error can be removed by a bounded linear correction.

Here “bounded” refers to this ordinary operator norm. Complete boundedness, which would require uniform bounds after all matrix amplifications, is a stronger condition and is not an assumption. The quotient uses the actual image of $d$, without taking its closure: a sequence of approximate corrections would not suffice.

🏷️ The manuscript’s vanishing theorem

Higher bounded cohomology

The family manuscript states that for every complex von Neumann algebra $M$ and every integer $k\ge 2$,

$$H_b^k(M,M)=0.$$

Equivalently, every bounded $k$-cocycle has an $M$-valued bounded primitive of degree $k-1$. There is no separability or type restriction. (OpenAI, 2026)

The coefficient space is part of the assertion. If $M$ acts on a Hilbert space $H$, finding a primitive with values in the larger algebra $B(H)$ would not establish this theorem.

For comparison, in degree one the cocycle equation says that a map is a derivation, $f(ab)=af(b)+f(a)b$. The classical inner-derivation theorem handles that degree. The new scope described here starts at degree two.

Two analytic notions will organize the proof. A normal linear map is ultraweakly continuous; a multilinear map is separately normal when this holds in each variable with the others fixed. A faithful normal tracial state $\tau$ on a finite algebra supplies the Hilbert norm

$$\Vert x{\Vert }_2=\tau (x^{\ast }x{)}^{1/2}.$$

A map $T$ is continuous in $2$-norm on bounded balls if $\Vert x_n{\Vert }_2\to 0$ and ${\sup }_n\Vert x_n\Vert <\infty$ imply $\Vert T(x_n){\Vert }_2\to 0$. For linear maps this gives a uniform continuity modulus on each fixed operator norm ball.

The central new argument treats a type ${\mathrm{II}}_1$ algebra with separable predual and faithful normal tracial state. In this usage the algebra has no type-I central summand, but may have a nontrivial center. Separable predual means that its space of normal functionals is separable. For a separately normal cocycle in this setting the construction gives the useful bound $\Vert g\Vert \le \Vert f\Vert$. The later reductions establish the unrestricted theorem; they do not assert this same bound for every original cocycle.

🏷️ Harmonic averaging and the primitive

The manuscript constructs a symmetric countably supported probability measure $\mu$ on the unitary group $\mathcal{U}(M)$. Symmetry means that $U$ and $U^{-1}$ have the same law. Its positive-mass atoms include a $2$-norm dense set of unitaries.

The crucial analytic statement is a Liouville theorem: if a bounded linear map $T:M\to M$ is continuous in $2$-norm on bounded balls and satisfies

$$T(x)={\mathbb{E}}_{U\sim \mu }[T(xU)U^{\ast }],$$

then $T(x)=T(1)x$. Positivity and complete boundedness of $T$ are not required. The measure is chosen for $M$, independently of $T$.

First see how this theorem produces a primitive. Write $\mathbf{a}=(a_1,\dots ,a_{k-1})$ and average in the last input:

$$\begin{array}{rl}(P\Phi )(\mathbf{a},x)& =\mathbb{E}[\Phi (\mathbf{a},xU)U^{\ast }],\\ F_n& =\frac{1}{n}\sum _{j=0}^{n-1}{P}^{j}f.\end{array}$$

Unitary multiplication preserves norms, so $P$ is contractive and $\Vert F_n\Vert \le \Vert f\Vert$. Pointwise ultraweak compactness supplies a convergent subnet with limit $F$. Since

$$P{F}_n-F_n=\frac{P^{n}f-f}{n},$$

the limit satisfies $PF=F$. Passing $P$ through the limit is justified by first truncating its countable probability sum and then uniformly bounding the remaining tail.

For fixed $\mathbf{a}$, separate normality of $f$ supplies a $2$-norm continuity modulus for $x\mapsto f(\mathbf{a},x)$. Right multiplication by each walk product preserves both relevant norms, so all the averages share that modulus. Ultraweak lower semicontinuity of $\Vert \cdot {\Vert }_2$ transfers it to $F(\mathbf{a},\cdot )$. The limit need not itself be normal; the retained modulus is sufficient.

Liouville therefore gives

$$\begin{array}{c}F(\mathbf{a},x)=h(\mathbf{a})x,\\ h(\mathbf{a})=F(\mathbf{a},1).\end{array}$$

To identify $dh$, evaluate the original equation $df=0$ at $(a_1,\dots ,a_k,v)$, multiply the output by $v^{\ast }$, and average $v$ over the same Cesàro mixture of walk laws. Put $h_n(\mathbf{a})=F_n(\mathbf{a},1)$. Writing $\mathbf{c}=(a_1,\dots ,a_k)$ and using the adjacent-product notation above, the resulting identity is

$$\begin{array}{rl}0=& a_1{h}_n(a_2,\dots ,a_k)\\ & +\sum _{j=1}^{k-1}(-1{)}^j{h}_n({\mathbf{c}}^{[j]})\\ & +(-1{)}^k{F}_n(\mathbf{c})+(-1{)}^{k+1}f(\mathbf{c}).\end{array}$$

Taking the limit gives $dh+(-1{)}^{k+1}f=0$. Thus $g=(-1{)}^{k}h$ is the required primitive, with $\Vert g\Vert \le \Vert f\Vert$. This argument does not assume that $P$ commutes with the Hochschild differential.

Why normality supplies the continuity modulus

Let $B:M\to M$ be bounded and normal. Suppose a uniformly bounded sequence tends to zero in $2$-norm while its images stay a fixed distance from zero. Spectral truncation replaces each input by an operator supported on a projection of arbitrarily small trace, retaining a definite part of the output norm.

Choose a subsequence whose support traces are summable. The joins of the tail supports decrease to zero. Normality and lower semicontinuity then produce operators $t_j$ on mutually orthogonal two-sided support projections, with $\Vert t_j\Vert \le R$ and $\Vert B(t_j){\Vert }_2\ge \eta >0$. Every signed sum of these inputs has norm at most $R$, whereas independent random signs give

$$\begin{array}{c}\mathbb{E}{\Vert \sum _{j=1}^N{\epsilon }_{j}B(t_j)\Vert }_2^2\\ =\sum _{j=1}^N\Vert B(t_j){\Vert }_2^2\ge N{\eta }^2.\end{array}$$

Boundedness also bounds the left side by $\Vert B{\Vert }^2{R}^2$, a contradiction for large $N$. This establishes the needed continuity for normal slices of the cocycle.

🏷️ A random walk with free moments

A unitary is Haar when $\tau (u^j)=0$ for every nonzero integer $j$. A family is free Haar when every nonidentity reduced word in the unitaries and their inverses has trace zero. “Reduced” means that neighboring inverse pairs have been canceled.

The walk is designed so that finitely many independent long products approach the joint moments of free Haar unitaries along selected times. An imported theorem of Popa supplies finite unitary configurations approximating any prescribed finite collection of free-moment tests. The manuscript also treats nonfactor algebras by measurable selection in their factor decomposition.

The probabilistic construction uses rare levels. With

$$\begin{array}{c}{p}_{\ell }=2^{-2^{\ell }},n_{\ell }=p_{\ell }^{-3/2},\\ K_{\ell }=\ell n_{\ell },d_{\ell }=K_{\ell }^4,\end{array}$$

level $\ell$ has probability $p_{\ell }$ and contains $d_{\ell }$ unsigned unitary labels, each sampled with an independent uniform sign. A positive remaining weight is assigned to a symmetric measure on a dense sequence of base unitaries. At stage $\ell$, choose a finite set $F_{\ell }$ capturing base mass greater than $1-K_{\ell }^{-2}$.

Among $\ell$ independent paths of length $n_{\ell }$, the desired event says that each path contains a level-$\ell$ label, none of those labels repeats anywhere regardless of sign, no later level appears, and all base increments lie in $F_{\ell }$. Its failure probability is at most

$$\begin{array}{rl}& K_{\ell }\sum _{j>\ell }{p}_j+K_{\ell }^{-1}\\ & +\frac{\left(\genfrac{}{}{0ex}{}{K_{\ell }}{2}\right)}{d_{\ell }}+\ell (1-p_{\ell }{)}^{n_{\ell }},\end{array}$$

which tends to zero. The four terms control later levels, omitted base atoms, label collisions, and paths missing the current level. For example, $p_{\ell +1}=p_{\ell }^2$ makes the first term $O(\ell \sqrt{p_{\ell }})$, while $n_{\ell }{p}_{\ell }=p_{\ell }^{-1/2}\to \infty$ controls the last.

Each path now has a label unique to it and to the entire path list. Treating all other labels as coefficients, these distinguished letters force every nontrivial reduced word in the path products to have small trace. All requirements at each stage are finite and are imposed simultaneously before constructing the next level.

The conclusion is convergence of scalar joint moments in probability. Operator norms of polynomials in the sampled products are not asserted to converge.

🏷️ From harmonic maps to a rigidity problem

For a harmonic $T$, let $W_n=U_1\cdots U_n$ be the walk and set

$$D(u)=T(u)u^{\ast },Z_n=D(W_n).$$

Harmonicity makes $Z_n$ an $L^2(M,\tau )$-valued martingale, bounded by $\Vert T\Vert$. Orthogonality of its increments gives a limit $Z_{\infty }$ and errors

$$\begin{array}{c}{e}_n=\mathbb{E}\Vert Z_{\infty }-Z_n{\Vert }_2^2⟶0,\\ \mathbb{E}\Vert Z_m-Z_n{\Vert }_2^2\le e_n(m\ge n).\end{array}$$

If the limit is deterministic, conditional expectation forces $D(u)=T(1)$ on every positive-mass atom. Density, the continuity modulus, and linear combinations of unitaries then prove Liouville.

Suppose instead that the limit law has distinct support points $a,b\in M$. For a long product $G$, its forward first half and reversed inverse last half use disjoint increments. Symmetry makes their laws agree, and independence gives a positive probability that $D(G)$ is near $a$ while $D(G^{-1})$ is near $b$. The martingale error bounds justify replacing these halves by the full endpoints. The variables $G$ and $G^{-1}$ themselves are not independent.

Concatenating independent signed blocks with pairwise distinct unsigned indices gives another walk. Thus the value of $D$ on their product is close to its value on the first block. Reusing a block, even with the opposite sign, would invalidate this argument.

Fix a finite number of blocks and an error tolerance first. The endpoint event has a positive probability, possibly very small. Choose the walk time only afterwards, so the combined failure probability of the free-moment and concatenation tests is smaller. A deterministic successful tuple then exists.

A tracial ultrapower identifies bounded sequences whose $2$-norm difference tends to zero along a free ultrafilter. This turns those increasingly accurate finite tuples into exact free Haar unitaries $g_i$ in $N=M^{{\omega }_1}$. Bounded-ball continuity makes the coordinatewise map $R=T^{{\omega }_1}$ well defined. For every nonempty signed word $w$ with distinct unsigned indices,

$$R(w)=\{\begin{array}{ll}aw,& \text{first sign }+,\\ bw,& \text{first sign }-.\end{array}$$

The remaining rigidity theorem says that a bounded map with this continuity and these identities in a finite tracial algebra must have $a=b$. Freeness from $a$ or $b$ is not needed.

🏷️ The rigidity mechanism

The first-letter identities cannot immediately be applied to powers of one generator, because such powers repeat its index. The manuscript instead averages the exact free family:

$$\begin{array}{c}{s}_r=r^{-1/2}\sum _{j=1}^r{g}_j,\\ \Vert s_r{\Vert }_2=1,\Vert s_r\Vert \le 2.\end{array}$$

The norm bound follows from the free-word length estimate

$$\begin{array}{c}\Vert \sum _{|w|=q}{c}_{w}w\Vert \\ \le (q+1){\left(\sum _{|w|=q}|c_w{|}^2\right)}^{1/2}.\end{array}$$

It is applied only after exact freeness has been obtained. A second ultrapower supplies $s=(s_r{)}^{{\omega }_2}$ and the induced map $S=R^{{\omega }_2}$.

Expand ordered products of blocks $s_r(s_r^{\ast }{s}_r{)}^d$. Their reduced words begin and end positively and have no scalar term. Among words of a fixed reduced length $q$, those with a repeated surviving index number $O(r^{q-1})$, while each coefficient is $O(r^{-q/2})$. Their total $2$-norm is therefore $O(r^{-1/2})$; the length estimate upgrades this to operator norm for each fixed polynomial. Discarding them leaves words covered by the first-letter identity.

Consequently $S$ acts by $a$ on these ordered block products in $s$, and by $b$ on their adjoints. A moment calculation identifies the distribution of $A=s^{\ast }s$:

$$\begin{array}{c}\tau (A^d)=\frac{1}{d+1}\left(\genfrac{}{}{0ex}{}{2d}{d}\right),\\ d\nu (x)=\frac{1}{2\pi }\sqrt{\frac{4-x}{x}}{1}_{(0,4)}(x)dx.\end{array}$$

The moments count noncrossing cancellation pairings. The displayed probability measure has those moments and no atom at zero, so faithfulness implies that $s$ has zero kernel. Its polar part $V$ is unitary because the algebra is finite.

Polynomial approximation first transfers the block identities to $s(A+\epsilon {)}^{-1/2}$. These are contractions converging to $V$ in $2$-norm; bounded-ball continuity then gives

$$\begin{array}{c}S(V^j)=a{V}^j,\\ S(V^{-j})=b{V}^{-j}(j\ge 1).\end{array}$$

To force $a=b$, use the analytic polynomials

$$H_m(z)=\sum _{j=1}^m\frac{z^j}{j},h_m=\sum _{j=1}^m\frac{1}{j}.$$

Their imaginary parts are uniformly bounded on the circle, whereas their real parts are at least $\cos (1)h_m$ within angular distance $1/m$ of $1$. Boundedness of $S$ and the two power identities therefore bound $(a-b)\mathrm{Re}{H}_m(\lambda V)$ uniformly for every rotation $\lambda$.

On each short spectral arc, rotate it near $1$ and invert the real part within that spectral corner. For its projection $e$, this yields $\Vert (a-b)e\Vert \le C/h_m$, with $C$ independent of the arc and $m$. Partitioning the circle into such arcs gives

$$\begin{array}{c}\Vert a-b{\Vert }_2^2\\ =\sum _e\tau (e(a-b{)}^{\ast }(a-b)e)\\ \le \frac{C^2}{h_m^2}\sum _e\tau (e)=\frac{C^2}{h_m^2}⟶0.\end{array}$$

The trace weights sum to one, so the number of arcs introduces no loss. This proves rigidity, rules out a nonconstant martingale limit, and establishes Liouville. The finite trace is essential to this final step.

🏷️ Removing separability and type restrictions

For a possibly nonseparable type ${\mathrm{II}}_1$ algebra with faithful normal tracial state, work first with a finite set $E$ of inputs. Place $E$, all values $f(E^k)$, and unital matrix systems of every finite size inside a countably generated von Neumann subalgebra $N_E$. The matrix systems rule out finite type-I central components of $N_E$.

The trace-preserving conditional expectation $E_E:M\to N_E$ is normal and respects multiplication on either side by elements of $N_E$. Hence $E_{E}f{|}_{N_E^k}$ is a separately normal cocycle, and the separable argument gives a primitive of norm at most $\Vert f\Vert$. Precomposing that primitive with $E_E$ in each input yields a global cochain whose differential agrees with $f$ exactly on $E^k$.

These cochains have a common norm bound. Direct the finite sets by inclusion and take a pointwise ultraweak subnet: every fixed input tuple eventually lies in $E^k$, so the limit is a primitive everywhere. The subalgebras and expectations need not be nested.

For general $M$, an imported normal-reduction theorem first writes an arbitrary cocycle as a separately normal one plus a coboundary. Classical results handle the central summand with no type-${\mathrm{II}}_1$ part. The remaining center splits into orthogonal summands, each admitting a faithful normal tracial state.

There is one further compatibility issue: solving a cocycle on a central corner does not automatically solve it on inputs mixing that corner and its complement. For a central projection $z$, define

$$\begin{array}{c}({\mathrm{cut}}_z\psi )(a_1,\dots ,a_k)\\ =\psi (z{a}_1,\dots ,z{a}_k),\end{array}$$

where $\psi$ has values in $zM$. The manuscript constructs an insertion homotopy $J_z$ satisfying

$$\begin{array}{c}d{J}_z+J_{z}d=I-{\mathrm{cut}}_z,\\ \Vert J_z\psi \Vert \le (k-1)\Vert \psi \Vert .\end{array}$$

Operationally, $J_z$ inserts $1-z$ just before the first complementary input in the expansion into $z$ and $1-z$ components. Adjacent differential terms cancel; merging the inserted projection with that input recovers the omitted term.

Adding $J_z(zf)$ to the pulled-back corner primitive corrects the mixed inputs. On each tracial type-${\mathrm{II}}_1$ piece its norm is at most $k\Vert f\Vert$. On the complementary piece, with classical primitive $b_0$, the bound is $\Vert b_0\Vert +(k-1)\Vert f\Vert$. The maximum of these two bounds permits assembly across all central pieces, including an uncountable family, using the bounded central product and the net of finite partial sums. Finally add back the primitive from normal reduction. This completes the manuscript’s route to an actual bounded primitive for every original cocycle. (OpenAI, 2026)

🏷️ Key steps and proof outlines

A normal map cannot retain large outputs on infinitely many tiny supports

The Liouville input requires $2$-norm continuity on bounded balls. For a bounded normal map $B$, suppose $\Vert x_j\Vert \le R$, $\Vert x_j{\Vert }_2\to 0$, but $\Vert B(x_j){\Vert }_2\ge \eta$. Choose a fixed spectral cutoff $\alpha$ with $\Vert B\Vert \alpha <\eta /4$. The part $y_j$ of $x_j$ above that cutoff retains output norm at least $3\eta /4$ and has a two-sided support projection $e_j$ satisfying

$$\tau (e_j)\le 2{\alpha }^{-2}\Vert x_j{\Vert }_2^2.$$

Pass to a subsequence with summable support traces and put $p_j={\bigvee }_{h\ge j}{e}_h$. Then $p_j\downarrow 0$. Normality lets one compress $y_j$ into sufficiently long disjoint bands $p_j-p_h$ while retaining output norm at least $\eta /2$. This gives elements $t_s$ with orthogonal two-sided supports and $\Vert t_s\Vert \le R$. (OpenAI, 2026)

Every signed sum ${\sum }_{s\le N}{ϵ}_s{t}_s$ still has operator norm at most $R$, while independent signs give

$$\mathbb{E}{\Vert \sum _{s\le N}{ϵ}_{s}B(t_s)\Vert }_2^2=\sum _{s\le N}\Vert B(t_s){\Vert }_2^2\ge N{\eta }^2/4.$$

Boundedness gives the incompatible upper bound $\Vert B{\Vert }^2{R}^2$. The common continuity modulus this proves survives every unitary average, and then survives the ultraweak limit by lower semicontinuity. Normality of the limit itself is not needed.

Check the primitive formula in the first nontrivial degree

For a two-cocycle $f$, the original equation is

$$af(b,v)-f(ab,v)+f(a,bv)-f(a,b)v=0.$$

Multiply by $v^{\ast }$ on the right and average over the walk’s Cesàro law. With $h_n(a)$ denoting the average of $f(a,v)v^{\ast }$, this becomes

$$a{h}_n(b)-h_n(ab)+F_n(a,b)-f(a,b)=0.$$

The harmonic limit has $F(a,b)=h(a)b$ by Liouville, so $ah(b)-h(ab)+h(a)b=f(a,b)$: precisely $dh=f$. In degree $k$, the same calculation contributes the sign $g=(-1{)}^{k}h$. This small example explains why the proof averages the original cocycle equation instead of claiming that averaging commutes with its differential.

Obtain exact equations on finite tests before taking a global limit

To remove separability in the tracial type-II case, fix a finite set $E\subset M$. Choose a countably generated subalgebra $N_E$ containing both $E$ and the finite set $f(E^k)$. Also include unital matrix systems of every size, ensuring $N_E$ has no finite type-I central part. Its trace representation has separable predual.

Let ${\mathcal{E}}_E:M\to N_E$ be the trace-preserving expectation and define $f_E={\mathcal{E}}_{E}f{|}_{N_E^k}$. Bimodularity makes $d{f}_E=0$. The separable theorem supplies $g_E$ with $d{g}_E=f_E$ and $\Vert g_E\Vert \le \Vert f\Vert$. Extend it to all of $M$ by applying ${\mathcal{E}}_E$ in every input slot.

For a tuple from $E$, its adjacent products lie in $N_E$, and its cocycle value was explicitly included there. Hence this extension satisfies the original equation exactly on $E^k$. Pointwise ultraweak compactness gives a global limit with the same norm bound. Directed finite subsets eventually contain every fixed tuple, so its equation passes through finitely many terms to the limit. The algebras $N_E$ need not form an increasing chain or have mutually compatible expectations.

Correct mixed central inputs before assembling summands

A primitive inside a central corner $zM$ accounts only for cocycle evaluations whose inputs have all been cut by $z$. Put $q=1-z$. The manuscript’s homotopy inserts $q$ immediately before the first input from $qM$, with the corresponding alternating sign.

Expanding each input into its $zM$ and $qM$ parts makes the homotopy identity transparent. Merging the inserted $q$ with the next $qM$ input reproduces that input with positive total sign. Every other adjacent merger cancels with its counterpart in the differential; a merger crossing $zM$ and $qM$ vanishes. Thus $d{J}_z+J_{z}d=\mathrm{id}-{\mathrm{cut}}_z$, with norm bound $k-1$ in degree $k$.

This bounded correction supplies the mixed-input terms that a cornerwise argument would otherwise omit. Uniform primitive bounds then permit central assembly. The cohomology section proves these analytic reductions; the free-walk Liouville and harmonic rigidity theorems remain the principal prior inputs.

🏷️ Connections

Central sequences and regularity also uses tracial approximation, but its comparison and stability conclusions concern different properties. Here the decisive approximation produces harmonic rigidity and a cochain primitive. The two ultrapowers in this proof have distinct roles: the first makes moment identities exact; the second averages that exact free family with a uniform operator norm bound.

🏷️ Manuscripts and verification scope

• Vanishing of higher bounded Hochschild cohomology — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The documented selection covers ordinary bounded Hochschild cocycles with values in the complex von Neumann algebra itself, in every degree at least two, without separability or type restrictions. It requires an actual bounded primitive. The degree-one inner-derivation theorem is outside that selection. 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. Vanishing of higher bounded Hochschild cohomology.