Jiang–Su stability: comparison, dimension and central splitting

Source and verification

This article explains family 291 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: approximation, rank and central sequences

A $C^{\ast }$-algebra is a norm-closed algebra of bounded operators, closed under adjoints. It is simple when its only closed two-sided ideals are zero and itself, and separable when it has a countable dense subset. A unital algebra is stably finite when every isometry in every matrix algebra over it is unitary. These assumptions describe very different features: ideals, size and the absence of infinite projections.

A completely positive map remains positive after every matrix amplification; c.p.c. means completely positive and contractive. A $C^{\ast }$-algebra $A$ is nuclear if its identity can be approximated on finite sets by c.p.c. maps through finite-dimensional $C^{\ast }$-algebras. Nuclear dimension adds an orthogonality requirement to the return maps. The inequality ${\dim }_{\mathrm{nuc}}A\le d$ means that the finite-dimensional intermediate algebra splits into $d+1$ pieces, and the return map on each piece is c.p.c. and order zero: it sends orthogonal positive elements to orthogonal positive elements. The sum of these return maps need not be contractive.

The Jiang–Su algebra $\mathcal{Z}$ is a particular simple, unital, infinite-dimensional nuclear algebra with a unique tracial state. Its building blocks are prime dimension-drop algebras:

$$\begin{array}{c}{I}_{p,p+1}\subset C([0,1],M_p\otimes M_{p+1}),\\ f(0)\in M_p\otimes 1,\\ f(1)\in 1\otimes M_{p+1}.\end{array}$$

Here the displayed endpoint conditions define the subalgebra. The algebra $A$ is $\mathcal{Z}$-stable if $A\cong A{\otimes }_{\min }\mathcal{Z}$. Absorption means that adding this fixed infinite-dimensional tensor factor does not change the algebra’s isomorphism class.

Comparison concerns positive elements in the stabilization $A\otimes \mathcal{K}$, where $\mathcal{K}$ denotes the compact operators on a separable infinite-dimensional Hilbert space. Write $a\precsim b$ if there are elements $v_j$ with $v_j^{\ast }b{v}_j\to a$ in norm. Identifying mutually comparable elements gives the Cuntz semigroup $\mathrm{Cu}(A)$; addition is orthogonal direct sum. For a lower-semicontinuous trace $\tau$, the associated dimension function is

$$d_{\tau }(a)=\underset{n\to \infty }{\lim }(\tau \otimes \mathrm{Tr})(a^{1/n}).$$

The matrix trace is unnormalized, and the value may be infinite. This measures the trace of the support, rather than the size $\tau (a)$ of the element.

Fix a free ultrafilter $\omega$ on the positive integers. The norm ultrapower $A_{\omega }$ identifies bounded sequences whose norm difference tends to zero along $\omega$. When $A$ is unital with nonempty tracial-state space $T(A)$, its uniform tracial ultrapower $A^{\omega }$ instead kills sequences tending to zero in

$$\Vert x{\Vert }_{2,u}=\underset{\tau \in T(A)}{\sup }\tau (x^{\ast }x{)}^{1/2}.$$

A limit trace on this quotient has the form $\lambda ((x_n))={\lim }_{\omega }{\tau }_n(x_n)$, allowing ${\tau }_n$ to vary. Central elements commute with the constant copy of $A$. The nonunital norm central-sequence algebra is

$$F_{\omega }(A)=\frac{A_{\omega }\cap A^{\prime }}{\mathrm{Ann}(A)},$$

where the denominator consists of elements annihilating $A$ on both sides. This quotient is essential in nonunital absorption arguments.

Uniform property $\Gamma$ asks for central projections $p_1,\dots ,p_m$ in $A^{\omega }$, for every $m$, with

$$\begin{array}{c}{p}_i{p}_j=0(i\ne j),\sum _i{p}_i=1,\\ \lambda (a{p}_i)=\frac{\lambda (a)}{m}(a\in A)\end{array}$$

for every designated limit trace $\lambda$. Equal traces of the projections alone are insufficient: the equal splitting must persist after multiplication by every coefficient $a$.

🏷️ Four regularity results and their hypotheses

This family contains four manuscripts. Their common objective is to turn finite-dimensional approximation or tracial comparison into coherent central structure. Their starting assumptions are different.

Comparison and absorption, as stated in the manuscript

A simple, separable, nuclear, non-elementary $C^{\ast }$-algebra with the manuscript’s extended strict comparison is $\mathcal{Z}$-stable. “Non-elementary” excludes algebras isomorphic to compact operators. The statement includes nonunital algebras with unbounded traces. (OpenAI, 2026)

The comparison convention is important. A functional $\phi$ on $\mathrm{Cu}(A)$ takes values in $[0,\infty ]$ and preserves zero, addition, order and increasing suprema. The hypothesis says $x\le y$ whenever

$$\begin{array}{c}\phi (x)\le \phi (y)for\; every\phi ,\\ \phi (x)<\phi (y)if\; 0<\phi (y)<\infty .\end{array}$$

There is no demand for the meaningless strict inequality $\infty <\infty$.

Nuclear dimension without elementary subquotients

For a separable nuclear algebra $A$ with no nonzero elementary ideal subquotient, finite nuclear dimension, $\mathcal{Z}$-stability and ${\dim }_{\mathrm{nuc}}A\le 1$ are equivalent. An ideal subquotient is $I/J$ for closed ideals $J\subseteq I\subseteq A$. Separately, every separable nuclear $A_0$ satisfies ${\dim }_{\mathrm{nuc}}(A_0\otimes \mathcal{Z})\le 1$. (OpenAI, 2026b)

The exclusion is substantive: a matrix algebra has nuclear dimension zero and cannot absorb the infinite-dimensional algebra $\mathcal{Z}$.

Tracial projection routes

For simple, separable, unital, infinite-dimensional, nuclear, stably finite $A$ with traces, real rank zero of the uniform tracial ultrapower of its bounded uniform tracial completion implies uniform $\Gamma$. Real rank zero means that selfadjoint elements admit finite-spectrum approximations in the quotient’s $C^{\ast }$-norm. The manuscript also gives direct comparison routes to uniform $\Gamma$ and $\mathcal{Z}$-absorption, without this real-rank-zero assumption. (OpenAI, 2026c)

The completion here closes bounded subsets of $A$ in $\Vert \cdot {\Vert }_{2,u}$; denote it by $B$. The real-rank-zero assumption is on $B^{\omega }$, for the specified $\omega$, rather than on $A$. The direct unital route uses simple, separable, unital, infinite-dimensional nuclear $A$ with the following strict comparison: for positive $a,b$ in the stabilization, $b\ne 0$ and $d_{\tau }(a)<d_{\tau }(b)$ for every $\tau \in T(A)$ imply $a\precsim b$. It yields absorption and, when traces exist, uniform $\Gamma$. If $T(A)$ is empty, the hypothesis still demands comparison with every nonzero target.

The separate nonunital route assumes that $A$ is nonzero, simple, separable, nuclear and stably projectionless, meaning that $A\otimes \mathcal{K}$ contains no nonzero projections. All densely finite lower-semicontinuous traces must be bounded, and their normalized base must be nonempty and weak-* compact. Its comparison condition says that $a\precsim b$ whenever $b\ne 0$ and $d_{\tau }(a)<1$ for every densely finite trace with $d_{\tau }(b)=1$. These are positive elements in the stabilization; the family of normalized target tests may be empty. These bounded-trace assumptions do not cover the full unbounded-trace theorem of the comparison manuscript.

Equivariant absorption

Suppose $A$ is simple, separable, unital, infinite-dimensional, nuclear, stably finite and already $\mathcal{Z}$-stable. For every action $\alpha$ of a countable discrete amenable group $G$, the manuscript proves that $\alpha$ is cocycle conjugate to $\alpha \otimes {\mathrm{id}}_{\mathcal{Z}}$. No outerness or restriction on trace orbits is imposed. (OpenAI, 2026b)

Amenability supplies finite Følner sets: translating such a set by each element of a prescribed finite subset changes only a small fraction of it. For $\beta =\alpha \otimes {\mathrm{id}}_{\mathcal{Z}}$, cocycle conjugacy means an isomorphism $\Phi :A\to A\otimes \mathcal{Z}$ and unitaries $u_g$ satisfying

$$\begin{array}{c}{u}_e=1,u_{gh}=u_g{\beta }_g(u_h),\\ \Phi {\alpha }_g=\mathrm{Ad}(u_g){\beta }_g\Phi .\end{array}$$

Thus this last theorem upgrades absorption to compatibility with an action; it does not deduce the starting algebra’s $\mathcal{Z}$-stability.

🏷️ Comparison: rounding trace mass into Cuntz rank

The comparison manuscript isolates two semigroup properties. Almost unperforation says that $(k+1)x\le ky$ implies $x\le y$ for $k\ge 1$. Full almost divisibility requires, for every $x$ and $N\ge 1$, one element $u$ satisfying

$$Nu\le x\le (N+1)u.$$

The same $u$ must work on both sides. Producing unrelated lower and upper approximations would not give this property.

The useful local observation is an integer-rounding lemma. Let $c\ge 0$ and choose an integer $L>\Vert c\Vert$. Form the Cuntz-semigroup element

$$w=\sum _{i=1}^L[(c-i{)}_{+}].$$

For the functional associated with a trace,

$$\phi (w)\le \tau (c)\le \phi (w)+d_{\tau }(c).$$

To see the error, examine a scalar spectral value $t\ge 0$. The number of positive integer thresholds below $t$ differs from $t$ by at most one, and the difference vanishes at zero. Integrating this scalar inequality against the spectral trace measure gives exactly one support-rank error. There is no factor proportional to $L$.

Simplicity and non-elementarity provide arbitrarily small nonzero hereditary corners in the comparison sense. Given a fixed nonzero positive cut $e$ and a large integer $M$, the manuscript obtains $h\ne 0$ with $M[h]\le [e]$. Trace mass can be transferred into the hereditary algebra generated by $h$ before applying the rounding lemma. Operationally, one finds finitely many $z_i$ with

$$\begin{array}{c}\sum _i{z}_i^{\ast }{z}_i=r(a),\\ c=\sum _i{z}_i{z}_i^{\ast }\in \overline{h(A\otimes \mathcal{K})h}.\end{array}$$

Here $r$ is a nonnegative continuous function vanishing near zero. Traciality preserves the mass $\tau (c)=\tau (r(a))$, while the support-rank error is bounded by $d_{\tau }(h)$. Large trace mass and small support rank are compatible because $\Vert c\Vert$ is allowed to grow.

For nonzero $x=[a]$, normalize $\Vert a\Vert =1$ and fix $e=(a-\frac{1}{2}{)}_{+}$. Choose a scalar $s$ strictly between $1/(N+1)$ and $1/N$. Increasing cutoff functions approaching the support of $a$ give positive increments whose total trace mass is $s{d}_{\tau }(a)$. Transfer the increments into successively smaller hereditary corners and round them. Choose their support errors to be summable, with total $E<s-1/(N+1)$.

An orthogonal block sum, rescaled to converge in norm without changing its Cuntz classes, produces a single $u$. For every finite-somewhere nonzero functional the construction gives

$$\begin{array}{c}\phi (u)\le s\phi (x),\\ s\phi (x)\le \phi (u)+E\phi ([e]).\end{array}$$

Such a functional is finite on the fixed positive cut and satisfies $\phi ([e])\le \phi (x)$. When $\phi (x)$ is finite, the choice of $s,E$ gives the strict margins needed to compare $Nu$ with $x$, and $x$ with $(N+1)u$. When $\phi (x)=\infty$, the second inequality and the finite error force $\phi (u)=\infty$. This argument never subtracts infinite quantities. The zero functional and the functional infinite on every nonzero element are treated separately. Extended strict comparison then supplies the full sandwich. (OpenAI, 2026a)

The later absorption construction is substantially longer. Its interface is worth making explicit. Almost unperforation and full almost divisibility feed central order-zero maps and compatible comparison witnesses into $F_{\omega }(A)$. An order-zero map $\theta :M_p\to F_{\omega }(A)$ and an element $v$ obeying

$$\begin{array}{c}{v}^{\ast }v=1-\theta (1),\\ \theta (e_{11})v=v\end{array}$$

encode a unital dimension-drop map. Compatible constructions lead to a unital copy of $\mathcal{Z}$ in the central-sequence algebra, which yields absorption by an imported structural criterion. The nonsimple construction must fix norm-visible ideal multiplicities before choosing tracial errors; trace estimates alone cannot see every ideal obstruction. These middle constructions are not independently verified in this note.

The manuscript further states that, for separable nuclear $A$, an abstract isomorphism $\mathrm{Cu}(A)\cong \mathrm{Cu}(A\otimes \mathcal{Z})$ already characterizes $\mathcal{Z}$-stability. The semigroup isomorphism need not initially come from an algebra homomorphism.

🏷️ Nuclear dimension: two order-zero pieces

The nuclear-dimension manuscript approaches absorption from finite-dimensional approximation. Its forward approximation construction starts with $A_0\otimes \mathcal{Z}$ and uses extra tensor copies of $\mathcal{Z}$ for separate tasks: a matrix model, comparison of an orthogonal copy, and complementary positive weights $k$ and $1-k$. Here $k$ is a positive contraction with full spectrum $[0,1]$.

Apply the construction to $A=A_0\otimes \mathcal{Z}$. At the final stage, two unitary conjugations match the model to the first-corner inclusion of $A$. Schematically, the outgoing pieces have the form

$${\varphi }_i(b)=U_i\left(\begin{array}{cc}0& 0\\ 0& k_i{\beta }^{\prime }(b)\end{array}\right)U_i^{\ast },$$

where $k_0=k$, $k_1=1-k$, and each weight commutes with the model c.p.c. order-zero map ${\beta }^{\prime }$. Multiplying an order-zero map by a commuting positive contraction preserves order zero: if $b,c\ge 0$ and $bc=0$, then the product of the weighted images is $k_i^2{\beta }^{\prime }(b){\beta }^{\prime }(c)=0$. Each piece is contractive, while their sum approximates the desired corner inclusion. The incoming map duplicates the finite-dimensional approximation into a direct sum, whose norm is the maximum of the two coordinate norms.

Strong self-absorption of $\mathcal{Z}$ removes the auxiliary tensor factors. Stabilization and passage to a hereditary corner preserve the two-piece bound; they do not introduce new approximation pieces. Unitization followed by passage to an ideal handles nonunital $A_0$. The detailed matrix models and relative-commutant comparison supply the inputs to this final argument. (OpenAI, 2026c)

The reverse implication uses a different obstruction. A character is a nonzero homomorphism to $\mathbb{C}$. Finite nuclear dimension and the absence of elementary ideal subquotients give central finite-dimensional order-zero coverings by matrix blocks of size at least two. These coverings prevent characters: a nonzero order-zero map from $M_r$ to $\mathbb{C}$, $r\ge 2$, cannot preserve the required orthogonality.

The manuscript proves an auxiliary theorem for any nonzero unital characterless algebra $D$: some finite maximal tensor power of $D$ contains two orthogonal full positive elements. “Full” means that the generated closed ideal is the entire algebra.

The mechanism builds a selfadjoint tensor polynomial $h$ whose positive and negative signs survive every irreducible representation. The absence of characters gives a finite commutator witness, hence uniformly separated scalar evaluations. A Gaussian tensor polynomial finds both signs simultaneously on the resulting families of choices; a large separated set makes the probability of a one-sign failure smaller than the cost of a finite approximation net.

Representations of a maximal tensor product need not be spatial tensor-product representations. The proof therefore uses pure-state excision in the commuting factor images. Positive contractions of norm one chosen in these factors still have a product of norm one: nonzero spectral projections have nonzero product because restriction to a nonzero commuting projection is faithful on a factor. Excision transfers the scalar positive and negative evaluations to compressions of the represented $h$. Consequently neither sign can disappear. The positive and negative parts $h_{+},h_{-}$ are orthogonal and nonzero in every irreducible representation, which is the required fullness.

Reindexing central sequences supplies commuting copies of a separable characterless algebra inside $F_{\omega }(A)$. The resulting homomorphism from the maximal tensor power is unital; injectivity is unnecessary for preserving fullness. An imported absorption criterion then combines these two full orthogonal elements with finite nuclear dimension to obtain $\mathcal{Z}$-stability.

🏷️ Uniform Gamma: weighted projection packing

The tracial-projection manuscript gives two mechanisms for equal splitting. In the real-rank-zero route, nuclear averaging first makes selected matrix compressions nearly scalar. A subprojection can then be cut from a matrix corner and transported back. That subprojection need not already be central.

The estimates retain its trace mass. For a unitary $x$ and projection $f$, the exact identity

$$\Vert [x,f]{\Vert }_{2,\tau }^2=2\tau (f)-2\tau (|fxf{|}^2)$$

turns approximate scalar compression into a commutator estimate. Packing many orthogonal pieces must sum these weighted errors, rather than multiply a worst-case bound by an uncontrolled number of pieces.

One terminal estimate illustrates the order of choices. Let $H>0$ be a total packing parameter, use $N$ rounds, and put $\eta =H/N$. The accumulated error has the form

$$\begin{array}{c}{E}_N=\eta ((1+1000\eta {)}^N-1),\\ E_N\le \eta e^{1000H}.\end{array}$$

The remaining trace mass is bounded by $e^{-H}+3N{\eta }^2+3E_N$. First choose $H$ large enough to suppress $e^{-H}$, then choose $N$ sufficiently large with that $H$ fixed. Taking both parameters large without this order would not control the exponential factor.

Finite-set halving projections in $B^{\omega }$ are lifted to positive contractions and selected coordinate by coordinate. The resulting errors tend to zero along the original ultrafilter. Centrality extends from the separable dense algebra $A$ to its possibly nonseparable completion $B$ by uniform $2$-density. The construction does not replace $\omega$ by an iterated ultrafilter. Coefficient trace identities are retained through the selection. (OpenAI, 2026d)

The direct strict-comparison route constructs support and moment data without the real-rank-zero assumption. It produces commuting rows of selfadjoint variables $d_{N,j}$ whose first two moments are approximately zero and one against the whole earlier coefficient algebra, and whose fourth moments are at most three. For

$$s_N=\frac{1}{\sqrt{N}}\sum _{j=1}^N{d}_{N,j},$$

a Taylor expansion of the exponential gives Gaussian characteristic functions uniformly over the relevant traces. The cubic remainder is controlled by Hölder’s inequality, $\tau (|d{|}^3)\le \tau (d^4{)}^{3/4}$. Applying the conditional moment estimates to the coefficient containing all earlier exponential factors makes the errors telescope. Independence and identical distributions are not assumed.

The norms of $s_N$ may grow. Consequently the proof does not use $(s_N)$ as a bounded ultrapower representative. Instead it chooses continuous functions approximating indicators of consecutive intervals of Gaussian mass $1/m$. The bounded contractions $f_i(s_N)$ have small projection defects and the desired coefficient traces. Those contractions admit the coordinate selection that yields the projections of uniform $\Gamma$.

In the nonunital absorption endpoint, compactness of the normalized trace base controls the normalization factors of generalized limit traces that are finite on a fixed nonzero positive cut. This is what extends tracial largeness from normalized limit traces to the trace tests required by the nonunital absorption criterion.

🏷️ Equivariance: Gaussian fluctuations with a fixed variance

The equivariant manuscript must make these central splittings invariant under the action. Existing $\mathcal{Z}$-absorption provides central tail copies of $\mathcal{Z}$. Reindexing makes their entire $G$-orbits commute across different rows. The generated algebra $C$ is separable and invariant. At an extreme trace of $C$, the commuting row algebras become independent in the associated finite factor. Independence is across rows, not among different translates within a row, and is not asserted for every trace. (OpenAI, 2026b)

Normalized sums of centered row variables have Gaussian limits. Along a sequence of possibly changing extreme traces, their finite covariance matrices have subsequential limits $R$. Write such a scalar covariance matrix as a Gram matrix of unit vectors $v_g$. There is no need for $R$ to be translation invariant.

For a finite Følner set $F$, put

$$P_F=\frac{1}{|F|}\sum _{g\in F}{v}_g{v}_g^{\ast }.$$

This positive operator has trace one and rank at most $|F|$. For a standard real Gaussian vector $W$ and a symmetric matrix $S$, the centered quadratic form

$$q_S=⟨W,SW⟩-\mathrm{Tr}(S)$$

has variance $2\Vert S{\Vert }_{\mathrm{HS}}^2$. Using $S=P_F$ can make the variance shrink with $|F|$. Using $S=\sqrt{P_F}$ instead fixes the variance at two. Meanwhile,

$$\Vert \sqrt{P}-\sqrt{Q}{\Vert }_{\mathrm{HS}}^2\le \Vert P-Q{\Vert }_1$$

converts a small Følner boundary into a small change in the quadratic form.

The square root is applied only to this scalar Gram operator. The empirical operator-valued covariance array need not be positive, so taking its matrix square root would be unjustified. The proof chooses a polynomial $p$ with $p(0)=0$ approximating $\sqrt{t}$ on $[0,1]$ to within $\eta /\sqrt{|F|}$. The rank bound then gives a Hilbert–Schmidt error at most $\eta$, even for singular covariance matrices. Expanding $q_{p(P_F)}$ as a fixed polynomial allows substitution of the original row sums and empirical covariances.

The order of parameters is again essential: choose the desired Fourier cancellation and invariance tolerances, then the Følner set and polynomial, and only afterwards the number of rows. A moving-trace compactness argument makes the estimates uniform over extreme traces; convexity extends them to all traces of $C$.

Coefficient traces enter through the positive functionals $c\mapsto \tau (ac)$ for $a\in A_{+}$. Since $C$ commutes with $A$, normalizing such a functional gives a trace on $C$ whenever $\tau (a)>0$. Thus scalar cancellation becomes the mixed cancellation needed for uniform $\Gamma$.

A bounded coordinate selection produces an invariant central unitary with Haar-distributed moments against every coefficient. Continuous approximations to equal arcs of the circle, followed by a second bounded selection, produce invariant equal-trace projections. This avoids applying discontinuous indicator functions directly inside a $C^{\ast }$-algebra. An imported equivariant absorption theorem then yields the stated cocycle conjugacy.

🏷️ Key steps and proof outlines

Transfer mass before rounding, so the error is a small support

The comparison route uses a concrete factorization. If a nonnegative continuous function $r$ vanishes on $[0,\delta ]$, fullness of a nonzero hereditary support $h$ supplies a column $v=(v_i)$ with

$$v^{\ast }v=(a-\delta /2{)}_{+},v{v}^{\ast }\in M_m(\overline{hBh}).$$

Set $g(t)=\sqrt{r(t)/(t-\delta /2{)}_{+}}$, extended by zero where the denominator vanishes. The cutoff makes this continuous. For $z_i=v_{i}g(a)$, one has ${\sum }_i{z}_i^{\ast }{z}_i=r(a)$, while $c={\sum }_i{z}_i{z}_i^{\ast }$ stays in the small hereditary support. Traciality preserves $\tau (c)=\tau (r(a))$ simultaneously for every extended trace. (OpenAI, 2026a)

For $q(t)=\min (t,1)$ and integer $L>\Vert c\Vert$, the exact staircase identity is

$$c=\sum _{i=1}^{L}q((c-(i-1){)}_{+}).$$

The trace of each term lies between the ranks of the cuts at $i$ and $i-1$. Summing telescopes, leaving only $d_{\tau }(c)$ as error. Thus an arbitrarily tall staircase still costs only one support rank. Choosing successively smaller supports makes these errors summable while retaining one final Cuntz-semigroup element for both inequalities of almost divisibility.

Stabilization does not add approximation colors

The nuclear-dimension route ends with two order-zero return maps approximating $A\to M_2(A)$. To approximate a finite set in $A\otimes M_m$, request accuracy on all its entries and amplify each map by ${\mathrm{id}}_{M_m}$. An entrywise error at most $\delta$ gives matrix error at most $m\delta$, so choosing the original accuracy after fixing $m$ controls it. (OpenAI, 2026c)

An order-zero map has the form $h\pi (\cdot )$ with commuting positive contraction $h$ and a supporting homomorphism $\pi$. Its amplification is $(h\otimes 1)(\pi \otimes \mathrm{id})(\cdot )$ and remains order zero. Embed the two target blocks into the stabilization, retaining the prescribed first $m$-corner. General finite sets are first approximated by finite-corner compressions. These operations change accuracy and matrix size but preserve exactly two colors. Hereditary-subalgebra permanence then returns the result to the original corner.

Weighted packing balances trace and commutator errors together

In the projection method, write $P,S$ for the sums of orthogonal inserted pieces with the two signs and $Z_N=P+S$. If $\eta =H/N$, the source’s accumulated energy is

$$E_N=\eta ((1+1000\eta {)}^N-1)\le \eta e^{1000H}.$$

Its final coefficient estimate is more precise than a trace-only split:

$$\left|\tau (Px)-\frac{1}{2}\tau (x)\right|\le \frac{1}{2}{e}^{-H}+5H\eta +\frac{3}{2}{E}_N+\sqrt{15H{E}_N}.$$

The square-root term comes from summing movement errors by Cauchy–Schwarz with the projection masses as weights. The signed target contributions cancel before errors are estimated. This avoids charging a constant error for every small piece. First fix large $H$, then let $N\to \infty$; both this coefficient error and $\Vert [x,P]{\Vert }_2\le \sqrt{E_N}$ vanish uniformly over the prescribed tests. (OpenAI, 2026d)

Square-root normalization prevents equivariant fluctuations from vanishing

For the equivariant route, let $P$ be a positive covariance-derived matrix of trace one. A centered Gaussian quadratic $q_B=⟨W,BW⟩-\mathrm{Tr}(B)$ has variance $2\Vert B{\Vert }_{\mathrm{HS}}^2$. Using $B=P$ could give variance $2/n$ when $P=I_n/n$. Using $B=P^{1/2}$ instead gives exactly two for every rank.

This normalization also respects Følner perturbations because

$$\Vert S^{1/2}-T^{1/2}{\Vert }_{\mathrm{HS}}^2\le \Vert S-T{\Vert }_1.$$

To prove it, put $a=S^{1/2}$, $b=T^{1/2}$, and $d=a-b$. Trace against $\mathrm{sign}(d)$ and use $a^2-b^2=ad+db$. In a basis diagonalizing $d$, positivity gives $a_{jj}+b_{jj}\ge |d_j|$, so the resulting trace is at least $\mathrm{Tr}(d^2)$; trace-norm duality supplies the upper bound. (OpenAI, 2026b)

Finally approximate $\sqrt{s}$ on $[0,1]$ by a polynomial vanishing at zero, with error $\eta /\sqrt{|F|}$ for the already chosen Følner set $F$. The rank bound converts this into Hilbert–Schmidt error at most $\eta$. Choose the oscillation parameter, tolerance, Følner set, and polynomial before the row count tends to infinity. This fixed-variance construction produces almost invariant unitaries with small nonzero moments, which feed the invariant central splitting and absorption argument.

🏷️ Connections and verification boundaries

The similarity family also turns local operator estimates into uniform structure, but its derivation bounds serve a different objective. The bicentralizer family likewise needs bounded representatives before passing to a limit; here the bounded objects are spectral cutoffs or unitaries, rather than the potentially unbounded Gaussian sums.

The arguments above explain selected mechanisms in all four manuscripts. The extended comparison and dimension-drop assembly, finite-dimensional model comparison, and full projection-packing constructions remain research inputs whose complete proofs have not been independently certified here.

🏷️ Manuscripts and verification scope

• Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case — pinned manuscript and companion source.
• Cuntz comparison and Jiang–Su absorption — pinned manuscript and companion source.
• Nuclear dimension and Jiang–Su stability without elementary subquotients — pinned manuscript and companion source.
• Tracial projection methods and uniform property Γ — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The documented selection proves uniform Gamma from real rank zero of the specified uniform tracial ultrapower of the uniform tracial completion, with the stated simple, separable, unital, nuclear, stably finite hypotheses and traces. It also lists linked direct-comparison results in the unital case and the stably projectionless case with bounded densely finite traces, a nonempty compact normalized trace base and finite-target-rank comparison. It does not list formal coverage of all four manuscripts’ headline theorems. 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. Cuntz comparison and Jiang–Su absorption.

🐻  OpenAI 2026b. Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case.

🐻  OpenAI 2026c. Nuclear dimension and Jiang–Su stability without elementary subquotients.

🐻  OpenAI 2026d. Tracial projection methods and uniform property Γ.