History and relation to earlier methods

The Hattori–Stallings trace refines the rank of a projective module by taking values in the quotient of its ring by additive commutators (Hattori 1965; Stallings 1965). For the algebraic group rings \(\mathbb ZG\) and \(\mathbb CG\), this quotient is indexed by conjugacy classes. Bass’s conjecture for the integral group ring asks that this trace be supported at the identity (Bass 1976); the complex-group-ring formulation permits finite-order classes. These support questions connect the algebraic structure of projective modules with the conjugacy structure of a group.

Cyclic homology gives one route to these questions. Burghelea’s computation separates group-ring cyclic homology into contributions from finite- and infinite-order conjugacy classes (Burghelea 1985). Eckmann used this description and the Chern character to prove Bass-type vanishing for several classes of groups (Eckmann 1986). Emmanouil extended this method through nilpotence properties of the cyclic periodicity operator (Emmanouil 1998). Our argument retains cyclic symmetry but works directly with weighted configurations and differential forms.

Berrick, Chatterji, and Mislin formulated the \(\ell^1\)-Bass conjecture in the finite-support form used here. They proved it for countable groups whose Bost assembly map is rationally surjective in degree zero (Berrick et al. 2004, Theorem 1.4). Together with Lafforgue’s Banach-algebra \(K\)-theory (Lafforgue 2002), this includes amenable groups; the passage from countable subgroups to arbitrary groups is explained in (Berrick et al. 2004, Lemma 1.5). Theorem 1 removes the assembly hypothesis while retaining the algebraic finite-support conclusion.

Ji, Ogle, and Ramsey developed related Bass conjectures for rapid-decay completions and higher Chern characters (Ji et al. 2010, 2014). Their work highlights the growth estimates needed when passing from group rings to analytic completions. Here the summability needed in the proof comes from a similarity reduction to an exponentially decaying idempotent. The subsequent estimates use a spatial scale uniform over all classes to be excluded.

The weighted cyclic configurations and Pfaffian detection strategy build on the complex-group-ring argument in (OpenAI 2026, secs. 7–8). In the group-ring setting, nonzero products of coefficients involve group elements from a fixed finite set. For an \(\ell^1\) idempotent these elements may have arbitrarily large word length, even after reduction to exponential decay. The estimates must retain summable length factors without losing control as the number of factors increases. We obtain them from smooth spatial summaries whose derivative bounds are independent of the size of the input support, multiscale time averaging with a positive translation component, and a final ordered-kernel bound uniform in the closing group element. A single spatial radius then excludes all but finitely many finite-order classes. All ingredients used here are proved below; the complex-group-ring theorem is not an input. Standard concentration and transgression arguments are credited at their points of use.

Proof strategy

The argument uses Banach algebra power series, differential forms on finite-dimensional simplices, and elementary estimates for integral operators. We give the needed estimates in full.

First, a finite-support approximation followed by an analytic idempotent correction replaces \(e\) by a similar idempotent \(p\) with an exponential moment in a finitely generated subgroup (Section 2). This step preserves every trace coefficient. It does not assert that \(p\) has finite support.

For a fixed \(g\in G\), consider cyclic lists of group elements whose closing edge is twisted by multiplication by \(g\). Products of the coefficients of \(p\) assign complex weights to these lists. Their total weight is \(\tau_{[g]}(p)\), and idempotency makes the weights compatible with deleting vertices. The ordered times of a list form a simplex. On these simplices, a one-form invariant under simultaneous time translation and equal to one on the translation vector will be called a connection form. A cyclic Stokes argument shows that the weighted average of its curvature Pfaffian equals the trace coefficient, up to sign (Section 3).

The task is then to construct compatible connection forms with small averaged curvature. Interpret a timed list as a step function with values in \(G\), and average its Dirac masses on a sequence of time scales. The resulting probability distributions are summarized by smooth subprobability distributions on auxiliary labels. These summaries record spatial concentration while controlling derivatives caused by moving one boundary of the step function (Section 4). Pairing their differentials with bounded odd time kernels gives one-forms. Their contractions with time translation telescope across the averaging scales. A final equality kernel detects the twist and leaves a strictly positive contraction, which allows normalization to a connection (Section 5).

The curvature estimates use three different features of this construction. At the initial scale, a curvature entry vanishes unless its two times are close; an occupancy estimate controls the resulting sparse Pfaffians on average. At intermediate scales, the short support of the time kernel gives a Hilbert–Schmidt bound. At the last scale, the equality kernel reduces to weighted order kernels, whose singular values satisfy \(s_\nu\le C/\nu\). Expanding the Pfaffian across these stages gives exponential decay in the simplex dimension (Section 6). A single spatial radius works for every infinite-order class and every finite-order class avoiding a fixed finite word ball. This uniformity proves the finite-support conclusion in Section 7.

Tuples and their weights

Fix \(g\in G\) and write \(H=C_G(g)\) for its centralizer. For \(k\ge0\) let \[\mathcal T_k(g)=H\backslash G^{k+1},\] where \(H\) acts by simultaneous left multiplication. For a tuple \(h=(h_0,\ldots,h_k)\) extend its indices to all integers by \(h_{i+k+1}=gh_i\) and define \[ v_{k+1}(h)=\mathop{\mathrm{tr}}\left(\prod_{i=0}^k p(h_i^{-1}h_{i+1})\right), \qquad l_i=1+d(h_{i-1},h_i). \tag{4}\] The product is in the displayed order. Both expressions descend to \(\mathcal T_k(g)\), and \(l_i\) is periodic with period \(k+1\).

Compatible forms

For each tuple use the ordered time domain \[\widetilde\Delta_k =\{(t_0,\ldots,t_k):t_0\le t_1\le\cdots\le t_k\le t_0+1\}.\] Extend times by \(t_{i+k+1}=t_i+1\), and set \(x_i=t_{i+1}-t_i\) for \(0\le i\le k\). Thus \(x_i\ge0\) and \(\sum_i x_i=1\). Simultaneous time translation is generated by \(V=\sum_{i=0}^k\partial_{t_i}\). Its quotient is the simplex \[\Delta_k=\{0\le t_1\le\cdots\le t_k\le1\},\] using the section \(t_0=0\), oriented by \(dt_1\wedge\cdots\wedge dt_k\). We regard \([t_i,t_{i+1})\) as carrying the label \(h_i\). The face \(x_i=0\) corresponds to deleting \((h_i,t_i)\): the interval carrying the label \(h_i\) has length zero. At the closing face use the periodic lifts to make this same convention.

Translation invariance and \(\alpha(V)=1\) imply \(\iota_Vd\alpha=0\). Thus \(d\alpha\) is a form on \(\Delta_k\) independent of the choice of translation section. A translation-invariant form whose contraction with \(V\) is zero will be called basic.

For a real skew-symmetric \(2m\times2m\) matrix \(A\), our convention is \[\omega=\sum_{i<j}A_{ij}\,dt_i\wedge dt_j \quad\Longrightarrow\quad \omega^m=m!\mathop{\mathrm{Pf}}(A)\,dt_1\wedge\cdots\wedge dt_{2m}.\] In particular \(\mathop{\mathrm{Pf}}(A)^2=\det A\). This identity follows by reducing a skew form to two-dimensional blocks, with both sides transforming by the square of the determinant of a change of basis.

The proposition reduces trace vanishing to a quantitative construction. We shall choose connections, in arbitrarily large even dimensions, for which the average of the absolute Pfaffian is at most a small exponential factor times \(\prod_{i=1}^n l_i\). Equation (5) then permits summation against the tuple weights. The next section constructs the spatial summaries used to obtain these connections.

Smooth selection of centers

The number of centers that can carry substantial score has a bound independent of \(|\mathop{\mathrm{supp}}P|\).

Choose smooth functions \(\psi_A,\psi_D:\mathbb R\to[0,1]\) such that \[ \begin{array}{c|cc} &\text{identically zero on}&\text{identically one on}\\ \hline \psi_A&(-\infty,0.70]&[0.75,\infty)\\ \psi_D&(-\infty,0.58]&[0.62,\infty). \end{array} \tag{17}\] For completeness, a smooth increasing cutoff from zero to one on \([0,1]\) is \(\eta(t)/(\eta(t)+\eta(1-t))\), where \(\eta(t)=e^{-1/t}\) for \(t>0\) and \(\eta(t)=0\) for \(t\le0\). Every right derivative of \(\eta\) at zero vanishes: away from zero it is \(e^{-1/t}\) times a polynomial in \(1/t\), which tends to zero as \(t\downarrow0\). The denominator in the displayed quotient is everywhere positive, and affine changes of variable give (17). Their first derivatives are bounded by a fixed numerical constant, since they vanish outside fixed compact transition intervals.

Take \(N=M_0r\epsilon^{-2}\) and define \[ \begin{split} w^A_y&=\psi_A(E_r(P,y))^2 e^{N(E_r(P,y)-0.80)},\\ w^D_y&=\psi_D(E_r(P,y))^2 e^{N(E_r(P,y)-0.66)},\\ A(P)_y&=\frac{w^A_y}{1+\sum_z w^A_z},\qquad D(P)_y=\frac{w^D_y}{1+\sum_z w^D_z}. \end{split} \tag{18}\] Here \(M_0\) is fixed in terms of the metric, large enough that for all allowed \(r,\epsilon\), \[ (1+e^{\gamma r})e^{-N\epsilon}\le\epsilon, \qquad e^{\gamma r-0.04N}\le1. \tag{19}\] For example, any \(M_0\ge25\gamma+4\) suffices. Indeed, \(\log(1+e^{\gamma r})+\log(1/\epsilon) \le\gamma r+\log2+1/\epsilon\le(\gamma+2)r/\epsilon\), and \(N\ge25\gamma r\).

The additional \(1\) in each denominator may be regarded as the weight of an unused null label. It allows the channel to have small total mass when no center reaches the exponential threshold. Each displayed sum is finite by Lemma 9.

The final averaging stage will only need \(D\), together with its concentration on a bounded set of labels. We record that consequence separately.

Predictions with an enlarged radius

The early channel can use the average score of the centers chosen by \(D\). All these centers lie near any center selected by \(A\). Enlarging the radius allows us to bound each score contributing at \(y\) by \(E_{6r}(P,y)\). Define \[ \Lambda=G,\qquad L(P)_z=D(P)_zE_r(P,z),\qquad \kappa(y,z)=\mathbf1_{\{d(y,z)\le4r\}}. \tag{27}\] This is a subprobability channel, since \(0\le E_r(P,z)\le1\). When \(d(y,z)\le4r\), every \(h\) with \(\chi_r(d(h,z))>0\) has \(d(h,y)<6r\). On this set \(\chi_{6r}(d(h,y))=1\), and therefore \[ E_r(P,z)\le E_{6r}(P,y). \tag{28}\] After summing against \(D\), this proves \(\operatorname{pred}(P,y)\le E_{6r}(P,y)\).

If \(A(P)_y>0\), then \(E_r(P,y)>0.70\) and \(q_r(P)>0.70\). For every \(z\) with \(D(P)_z>0\), Lemma 9 gives \(d(y,z)\le4r\). Hence every such \(z\) contributes to the prediction, and (21) gives \[\operatorname{pred}(P,y) =\sum_zD(P)_zE_r(P,z) \ge q_r(P)-2\epsilon \ge E_r(P,y)-2\epsilon.\]

To estimate derivatives, write \(E_z=E_r(P,z)\) again. Where \(D_z>0\), \(E_z>0.58\). Differentiating \(L_z=D_zE_z\) yields \[ \sum_{z:L_z>0}\frac{|\dot L_z|^2}{L_z} \le 2\sum_{z:D_z>0}E_z\frac{|\dot D_z|^2}{D_z} +2\sum_{z:D_z>0}D_z\frac{|\dot E_z|^2}{E_z} \le C\epsilon^{-4}d(h,h')^2. \tag{29}\] In the second sum only \(D_z>0\) is relevant; its denominators are thus bounded below by \(0.58\). We used (23), (22), and \(r\ge1\). At a zero coordinate \(D_z=0\) and \(\dot D_z=0\), so \(\dot L_z=0\). The same finite formulas prove smoothness at faces. This establishes all the early-version claims of Proposition 8.

Accurate finite sample strings

To keep the radius equal to \(r\), an average of center scores is insufficient: we must predict the score separately at each \(y\). A finite string of sample points approximates all scores in a bounded region simultaneously. Random sampling will prove the existence of an accurate string. We will then put smooth positive weights on all strings in a fixed finite alphabet. This distinction is useful at probability faces: the defining weights will not be products of possibly vanishing input probabilities.

Fix a center \(z\in G\). Introduce a null symbol \(\partial\), fixed by the group action, and the alphabet \[\mathcal A_z=B(z,20r)\cup\{\partial\}.\] Define a probability \(Q^z\) on this alphabet by \[ P^z_h=P_h\chi_{10r}(d(h,z)),\qquad Q^z_h=P^z_h\ (h\ne\partial),\qquad Q^z_\partial=1-\sum_hP^z_h. \tag{30}\] The notation \(E_r(P^z,y)\) has the same linear meaning as in (9), although \(P^z\) can have mass less than one. This truncation provides a fixed finite alphabet while preserving every score needed for the lower prediction bound. Indeed, if \(A(P)_yD(P)_z>0\), Lemma 9 gives \(d(y,z)\le4r\). Every \(h\) contributing to \(E_r(P,y)\) then has \(d(h,z)<6r\), where the factor \(\chi_{10r}(d(h,z))\) is one. Consequently \[ E_r(P^z,y)=E_r(P,y) \qquad\text{if }A(P)_yD(P)_z>0. \tag{31}\] For arbitrary \(y\), we still have \(E_r(P^z,y)\le E_r(P,y)\), which is the direction needed for the upper bound. Thus accurate predictions for \(P^z\) will suffice. The Lipschitz taper in (30) will also keep their transfer derivatives proportional to \(d(h,h')/r\).

For a positive integer \(b\) and a string \(w=(w_1,\ldots,w_b)\in\mathcal A_z^b\) let \[ H_y(w)=\frac1b\sum_{i:w_i\ne\partial}\chi_r(d(w_i,y)), \qquad \Delta(P;z,w)=\max_{y\in B(z,22r)} |H_y(w)-E_r(P^z,y)|. \tag{32}\] Both scores vanish when \(d(y,z)>22r\), by the triangle inequality. Thus \(\Delta\) also controls the discrepancy at every \(y\in G\). Always \(0\le\Delta\le1\).

Smooth weights on strings

Fix the integer \(b\) supplied by Lemma 12. To assign smooth weights, replace the maximum of the signed errors by the logarithm of a sum of exponentials. Put \[ S(P;z,w)=\frac1\lambda\log \left(\sum_{y\in B(z,22r)}\sum_{\sigma\in\{-1,1\}} \exp\bigl(\lambda\sigma(H_y(w)-E_r(P^z,y))\bigr)\right). \tag{33}\] Choose \(\lambda>0\), depending only on \(r,\epsilon\), so large that the logarithm of the number of signed tests is at most \(\lambda\epsilon\). The number of tests is independent of \(z\), since left translation identifies the balls. The elementary inequalities \(e^{\lambda\max x_i}\le\sum_i e^{\lambda x_i} \le M e^{\lambda\max x_i}\) give \[ \Delta(P;z,w)\le S(P;z,w)\le\Delta(P;z,w)+\epsilon. \tag{34}\]

The first derivative of this smoothed maximum has a particularly simple bound. The test function \(h\mapsto\chi_{10r}(d(h,z))\chi_r(d(h,y))\) is \(11/(10r)\)-Lipschitz: both factors have absolute value at most one, and their Lipschitz constants add. Thus every signed error in (33) has transfer derivative of absolute value at most \(11d(h,h')/(10r)\). Differentiation of the logarithm gives a convex combination of these derivatives; the factor \(\lambda\) cancels. Consequently \[ |\dot S(P;z,w)|\le\frac{11}{10r}d(h,h'). \tag{35}\] In particular this estimate does not grow with the number of tests or the smoothing parameter.

Now define a probability distribution on the entire fixed set of strings by \[ \pi_z(P)_w =\frac{e^{-N_1S(P;z,w)}}{\sum_{v\in\mathcal A_z^b}e^{-N_1S(P;z,v)}}, \qquad N_1=M_2rb\epsilon^{-2}. \tag{36}\] Every coordinate is strictly positive. The constant \(M_2\) will be fixed in terms of the metric.

Predictions at the original radius

We can now finish the later channel. Its label set consists of pairs \((z,w)\), with \(z\in G\) and \(w\in\mathcal A_z^b\). Translation acts on the center and on every nonnull entry of the string. Define \[ L(P)_{z,w}=D(P)_z\pi_z(P)_w, \qquad \kappa(y,(z,w))=H_y(w). \tag{40}\] The total mass of \(L\) is \(\sum_zD(P)_z\le1\), and the kernel takes values in \([0,1]\). All formulas commute with left translation.

By (38), and since \(P^z_h\le P_h\), \[\operatorname{pred}(P,y) \le\sum_zD(P)_z\bigl(E_r(P^z,y)+5\epsilon\bigr) \le E_r(P,y)+5\epsilon.\] Suppose next that \(A(P)_y>0\). For every \(z\) with \(D(P)_z>0\), (31) identifies the predicted truncated score with \(E_r(P,y)\). Moreover \(q_r(P)>0.70\), and (21) gives \(\sum_zD(P)_z\ge1-\epsilon\). It follows that \[\operatorname{pred}(P,y) \ge E_r(P,y)\sum_zD(P)_z-5\epsilon\sum_zD(P)_z \ge E_r(P,y)-6\epsilon.\] These are the required prediction bounds with \(r'=r\).

For the derivative bound it is useful to retain the product structure \(L_{z,w}=D_z\pi_z(w)\). Since \(\sum_w\pi_z(w)=1\) and \(\sum_w\dot\pi_z(w)=0\), expansion of the square gives the exact identity \[ \sum_{z,w:L_{z,w}>0}\frac{|\dot L_{z,w}|^2}{L_{z,w}} =\sum_{z:D_z>0}\frac{|\dot D_z|^2}{D_z} +\sum_zD_z\sum_w\frac{|\dot\pi_z(w)|^2}{\pi_z(w)}. \tag{41}\] All weights and derivatives here are real; the mixed term vanishes by the displayed zero-sum identity. Using (22), (39), and \(b\le Cr\epsilon^{-4}\) yields \[ \mathcal F_L(P;h,h') \le C\bigl(\epsilon^{-2}+N_1/r\bigr)d(h,h') \le Cr\epsilon^{-6}d(h,h'). \tag{42}\]

It remains to check the regularity promised in the channel definition. For a fixed finite input support \(S\), only finitely many centers \(z\) can have \(D_z\ne0\), all in the \(2r\) neighborhood of \(S\). For each such center, the alphabet and string set are finite and independent of \(P\). The functions \(S(P;z,w)\) and \(\pi_z(P)_w\) are smooth functions of the coordinates of \(P\): the scores are linear, and each exponential sum has a strictly positive denominator. Thus (40) is a fixed finite smooth formula on each enlarged support space. If \(L_{z,w}=0\), strict positivity of \(\pi_z(w)\) implies \(D_z=0\); also \(\dot D_z=0\) by Lemma 10. Consequently \(\dot L_{z,w}=0\). This proves smoothness at faces and the required behavior at zero entries.

For use with differential forms, regard the labels of \(A\) and \(L\) as disjoint even when their underlying sets happen to coincide. On their disjoint union define the symmetric kernel \(K\) by \[ \begin{split} K(y,\omega)=K(\omega,y)&=\tfrac12\kappa(y,\omega) \quad(y\in G,\ \omega\in\Lambda),\\ K&=0\quad\text{within either channel}. \end{split} \tag{43}\] For vectors on this union, write \(K(U,V)=\sum_{\xi,\eta}K(\xi,\eta)U_\xi V_\eta\). The combined field \(Y=A\oplus L\) has mass at most two, and \[ K(Y(P),Y(P)) =\sum_yA(P)_y\operatorname{pred}(P,y). \tag{44}\] The kernel is bounded and invariant under simultaneous left translation. The expression in (10) also makes sense for this field of mass at most two, and its square is \(\mathcal F_A(P;h,h')^2+\mathcal F_L(P;h,h')^2\). Thus Proposition 8 controls its derivatives as well. These facts permit both channels to enter a single bilinear pairing in the averaging construction.

Time averages and spatial summaries

Begin with a finite number \(b_*\geq2\) of averaging stages, an even ambient dimension \(n\), and positive time scales \(\delta_1<\cdots<\delta_{b_*}\) satisfying \[ \sum_{i=1}^j\delta_i\leq2\delta_j, \qquad 10\delta_{b_*}<1. \tag{45}\] The construction through positivity uses only these conditions on the time scales. Their precise dependence on \(R\) and \(n\) will be selected afterwards, when preparing the quantitative estimates. The same scales are kept fixed in every dimension \(0\leq k\leq n\). Thus passing to a face never changes the averaging operation.

Choose a nonnegative, even \(C^\infty\) function \(\mu\), supported in \([-1,1]\), with integral one, and put \(\mu_j(s)=\delta_j^{-1}\mu(s/\delta_j)\). On a tuple of dimension \(k\), write its step function and its averages as \[ \begin{split} f(u)&=h_i\quad(t_i\leq u<t_{i+1}),\qquad f(u+1)=gf(u),\\ P_0(u)&=\delta_{f(u)},\qquad P_j(u)=\int_{\mathbb R}\mu_j(s)P_{j-1}(u-s)\,ds. \end{split} \tag{46}\] Zero-length intervals are omitted. These are finitely supported probability distributions at each time, and \(P_j(u+1)=gP_j(u)\). If \(\eta_j=\mu_j*\cdots*\mu_1\), then \[ P_j=\eta_j*P_0,\qquad \mathop{\mathrm{supp}}\eta_j\subset[-2\delta_j,2\delta_j],\qquad \|\eta_j\|_\infty\leq C/\delta_j. \tag{47}\] The support bound follows from (45), and the supremum bound follows by convolving \(\mu_j\) with probability densities.

To summarize these averages spatially, choose \(R\geq1\) and an integer \(2\leq J_*\leq b_*\). Put \[ r_j=6^{\min(j,J_*)-1}R,\qquad r=6^{J_*-1}R. \tag{48}\] The radius increases by a factor of six during the first \(J_*-1\) steps and is then fixed at \(r\). Use accuracies \[ \epsilon_j=\epsilon_*(1+j)^{-2}\quad(j\geq1), \qquad 30\sum_{j\geq1}\epsilon_j\leq\frac1{10}, \tag{49}\] where \(0<\epsilon_*\leq10^{-3}\) is fixed sufficiently small. The summable errors will leave a positive quantity after telescoping.

For \(1\leq j<b_*\), apply Proposition 8 to \(P_j(u)\) at parameters \((r_j,\epsilon_j)\). Use its early version when \(j<J_*\) and its later version when \(j\geq J_*\). Denote the resulting channels by \(A_j,L_j\) and their prediction kernel by \(\kappa_j\). On the disjoint union of the two label sets put \[ Y_j(u)=A_j(P_j(u))\oplus L_j(P_j(u)). \tag{50}\] Define a symmetric bilinear kernel \(K_j\) to be zero between labels in the same channel and to have value \(\kappa_j(y,\omega)/2\) between the two channels. Thus \[ K_j(Y_j(u),Y_j(u)) =\sum_y A_j(P_j(u))_y\operatorname{pred}_j(P_j(u),y). \tag{51}\] The total mass of \(Y_j(u)\) is at most two, and \(|K_j|\leq1\).

At the first and last stages use \[ \begin{array}{lll} Y_0=P_0,& K_0(h,h')=\chi_R(d(h,h')),\\[2pt] Y_{b_*}(u)=D(P_{b_*}(u)),& K_{b_*}(h,h')=\mathbf1_{h=h'}, \end{array} \tag{52}\] where \(D\) is the channel of Lemma 11 at parameters \((r,\epsilon_{b_*})\). These two fields have mass at most one. All the fields obey \(Y_j(u+1)=gY_j(u)\), and every kernel is invariant under simultaneous left translation of its labels.

The last channel needs a multiplier to compensate for its possible spread over many labels. Lemma 11 gives \(\sum_zD(P)_z^2\geq\frac14e^{-C_Dr}\) when \(\phi_r(P)>0\), for a fixed constant \(C_D\). Choose a fixed \(C_*\geq C_D\) and set \[ W=\exp(C_*r). \tag{53}\] Since \(0\leq\phi_r\leq1-a_*=1/5\), this gives, for every accuracy parameter in (49), \[ W\sum_zD(P)_z^2\geq\phi_r(P). \tag{54}\] We shall construct the connection under either of the assumptions \[ \begin{array}{ll} \textnormal{(i)}&g\text{ has infinite order},\\[2pt] \textnormal{(ii)}&g\text{ has finite order and } \displaystyle\min_{h\in G}d(h,gh)>4r. \end{array} \tag{55}\] Only the final time-lag kernel and its contraction distinguish these two cases.

Lag pairings and their compatibility

We next turn the fields into one-forms. For \(0\leq j<b_*\) let \(\rho_j=\mu_{j+1}*\mu_{j+1}\) and define the odd function \[ U_j(s)=\operatorname{sign}(s)-2\int_0^s\rho_j(a)\,da. \tag{56}\] It is bounded in absolute value by one and vanishes outside \([-2\delta_{j+1},2\delta_{j+1}]\). In the sense of distributions, \[ U_j'=2(\delta_0-\rho_j). \tag{57}\] Here \(\delta_0\) denotes the Dirac measure at zero, rather than a time scale. At the final stage set \[ \begin{array}{lll} U_{b_*}(s)=\operatorname{sign}(s),&\rho_{b_*}=0, &\text{in case (i)},\\[2pt] U_{b_*}(s)=\operatorname{sign}(s)\mathbf1_{|s|<1},& \rho_{b_*}=\tfrac12(\delta_{-1}+\delta_1), &\text{in case (ii)}. \end{array} \tag{58}\] Formula (57) holds at this stage as well. Values at the jump points have no effect on any integral below.

The differential \(dY_j(u)\) in the following definition acts on the tuple times \(t_0,\ldots,t_k\), with \(u\) fixed. Write \(\int_{\rm per}\) for integration over one period, using a half-open interval when integrating the raw impulse measure, and put \[ \begin{split} B_j&=-\frac12\int_{\rm per}du\int_{\mathbb R}dv\, U_j(v-u)K_j(dY_j(u),Y_j(v)),\\ B&=\sum_{j=0}^{b_*-1}B_j+WB_{b_*}. \end{split} \tag{59}\] For \(j=0\) the differential of the step field is a measure in \(u\). Specifically, set \[ J_i=\delta_{h_{i-1}}-\delta_{h_i},\qquad dP_0(u)=\sum_{i=0}^k\sum_{a\in\mathbb Z} g^aJ_i\,\delta_{t_i+a}(u)\,dt_i. \tag{60}\] The periodic convention gives \(h_{-1}=g^{-1}h_k\). Thus the coefficient of \(dt_i\) in the raw form is \[ (B_0)_i=-\frac12\int_{\mathbb R}U_0(v-t_i)K_0(J_i,P_0(v))\,dv. \tag{61}\] The expression in (60) is a distributional identity in the interior of the ordered time domain. Formula (61) will also specify its smooth extension to the faces.

Contraction and the positive normalizer

The forms are now compatible on all tuple simplices. To normalize them to a connection, we must show that their sum has positive contraction with the translation vector \(V\). The choice of the lag derivative in (57) converts this contraction into a difference between a pairing at one time and its average at the next scale.

Write \[\Phi_j=\int_{\rm per}\phi_{r_j}(P_j(u))\,du \qquad(1\leq j\leq b_*).\] These are the concentration quantities that will telescope.

Differential bounds and the resulting connection

Positivity now allows the normalization. Before applying the trace identity, we also need growth bounds that justify summation over all tuples. More precise bounds on the individual channel differentials will provide the input for the curvature estimates of the next section.

We now specialize the parameters of the preceding construction. At the shortest time scales we use the early channels, whose derivative bound is independent of the spatial radius. The later channels keep that radius fixed; this also fixes the terminal multiplier \(W\) while the time scales continue to increase. Their longer time scales will absorb the polynomial factor \(r^{10}\) in the later derivative bound.

Choose once and for all an integer \(J_*\) with \((3/2)^{J_*-1}>200\). For sufficiently large \(R\) retain the spatial radii, accuracies, and multiplier from (48), (49), and (53), and set \[ \beta_1=R,\qquad \beta_{j+1}=\beta_j^{3/2}. \tag{71}\] Choose \(b_*\) to be the smallest integer such that \[ b_*\geq J_*,\qquad \beta_{b_*}\geq W^2R^{100}. \tag{72}\] For each even integer \(n>10\beta_{b_*}\), use \[ \delta_j=\beta_j/n\qquad(1\leq j\leq b_*). \tag{73}\] The ratio \(\beta_{j+1}/\beta_j\) is at least two when \(R\) is sufficiently large, so these scales satisfy (45). Thus the construction and positivity proved above apply. For each such \(n\), the same actual scales \(\beta_j/n\) are used in every dimension \(k\leq n\); face restriction never changes the denominator to \(k\).

The stage count, spatial radii, accuracies, multiplier, and numbers \(\beta_j\) now depend only on \(R\) and the fixed choices. Since \(\log\beta_j=(3/2)^{j-1}\log R\) and \(\log(W^2R^{100})=2C_*r+100\log R\), with \(r\) a fixed multiple of \(R\), \[ b_*=O(1+\log R). \tag{74}\] The particular exponents in (71) and (72) will enter the curvature bounds. They leave room for the polynomial losses in the channel derivatives.

Choose a sufficiently large fixed constant \(C\) and set \[ a_j=C\epsilon_j^{-10} \begin{cases} 1,&1\leq j<J_*,\\ r^{10},&J_*\leq j\leq b_*. \end{cases} \tag{75}\] As before, \(l_i=1+d(h_{i-1},h_i)\), with periodic indexing, and \(\partial_i=\partial/\partial t_i\).

The construction separates two estimates that the trace argument will use differently. The bounds in Proposition 17 justify the weighted simplex identities, while the scale-dependent bound (76) retains the information needed to make high-dimensional curvature Pfaffians small. We turn to those sharper estimates next.

Smooth stages as weighted operators

Fix a stage \(j>0\) and a point of the simplex. Let \(\Lambda_j\) be its label set and equip \((0,1)\times\Lambda_j\) with the measure \[d\vartheta_j(u,y)=Y_j(u)_y\,du.\] We use the real Hilbert space \(\mathcal H_j=L^2(\vartheta_j)\). Recall that the total mass of \(Y_j(u)\) is at most two, and is at most one at the last stage. Define the operator \(T_j\) by the kernel \[ T_j(u,y;v,z) =\sum_{a\in\mathbb Z}U_j(v+a-u)K_j(y,g^a z), \tag{86}\] where the action of \(g\) on labels is the one used in the channel construction. Oddness of \(U_j\), symmetry of \(K_j\), and simultaneous equivariance of \(K_j\) show that this kernel is skew. Indeed, exchange \((u,y)\) with \((v,z)\) and replace \(a\) by \(-a\) in the sum.

For \(1\le i\le n\), put \[X_i(u,y)=\frac{\partial_iY_j(u)_y}{Y_j(u)_y} \quad\text{when }Y_j(u)_y>0,\] and give \(X_i\) the value zero at the remaining points. The derivatives of the zero weights vanish, and (76) gives \[ \|X_i\|_{\mathcal H_j} \le \frac{a_jl_i}{\sqrt{\delta_j}}. \tag{87}\] The exterior-derivative identity (82) is \[dB_j=\frac12\int_{\rm per}du\int_{\mathbb R}dv\, U_j(v-u)K_j\bigl(dY_j(u)\wedge dY_j(v)\bigr).\] The integral over a period can be taken with fixed endpoints because its integrand is periodic. Reducing \(v\) to \((0,1)\) gives (86); the two antisymmetric terms then agree. Consequently, \[ (dB_j)_{ii'}=\langle X_i,T_jX_{i'}\rangle_{\mathcal H_j}. \tag{88}\]

Here is the finite-dimensional estimate we shall apply to this pairing. Let \(I\) have even size \(k>0\), let \(E\) be the span of the vectors \(X_i\), \(i\in I\), and let \(s_1\ge s_2\ge\cdots\) be the singular values of the compression \(P_ET_j|_E\), completed by zeros. Then \[ |\mathop{\mathrm{Pf}}_I(dB_j)| \le \prod_{i\in I}\|X_i\| \left(\prod_{\nu=1}^k s_\nu\right)^{1/2}. \tag{89}\] To see this, the matrix in (88) has rank less than \(k\) if \(\dim E<k\), so its Pfaffian is zero. Otherwise write the \(X_i\) as columns of a square matrix \(V\) in an orthonormal basis of \(E\). The curvature matrix is \(V^{\mathsf T}T_EV\). Taking determinants and using \(\mathop{\mathrm{Pf}}(A)^2=\det A\) for a skew matrix gives the right side of (89), after Hadamard’s inequality \(|\det V|\le\prod_i\|X_i\|\).

For \(1\le j<b_*\), the support of \(U_j\) is contained in \([-2\delta_{j+1},2\delta_{j+1}]\). At most one integer \(a\) in (86) contributes for a fixed pair \((u,v)\), because \(4\delta_{j+1}<1\). Thus the kernel is bounded by a fixed constant and vanishes unless the circular distance between \(u\) and \(v\) is at most \(2\delta_{j+1}\). The mass bound on the channels implies \[\|T_j\|_{\rm HS}^2 =\int\!\int |T_j(u,y;v,z)|^2\, d\vartheta_j(u,y)d\vartheta_j(v,z) \le C\delta_{j+1}.\] Bessel’s inequality gives the same upper bound for the squared Hilbert–Schmidt norm of every compression. Applying the arithmetic mean inequality to \(s_1^2,\ldots,s_k^2\) gives \[\left(\prod_{\nu=1}^k s_\nu\right)^{1/2} \le C^k\delta_{j+1}^{k/4}k^{-k/4}.\] Combining this with (87) and (89), and using \(\beta_{j+1}=\beta_j^{3/2}\), proves \[ |\mathop{\mathrm{Pf}}_I(dB_j)| \le\bigl(Ca_j\beta_j^{-1/8}\bigr)^k (n/k)^{k/4}\prod_{i\in I}l_i, \qquad 1\le j<b_*. \tag{90}\] In fact, the powers of the time scales simplify as \[\delta_j^{-k/2}\delta_{j+1}^{k/4}k^{-k/4} =\beta_j^{-k/8}(n/k)^{k/4}.\]

The terminal sign kernel

The last lag kernel is not supported at a small time scale. Its useful property is instead that an order comparison is constant between two disjoint ordered intervals. The following elementary approximation works even when the source and target measures differ.

For an even principal set \(I\) of size \(k>0\), insert (91) in (89). Multiplication of the two-form by \(W\) multiplies its size-\(k\) Pfaffian by \(W^{k/2}\). We obtain \[|\mathop{\mathrm{Pf}}_I(WdB_{b_*})| \le W^{k/2}\left(\frac{a_{b_*}}{\sqrt{\delta_{b_*}}}\right)^k \frac{C^{k/2}}{\sqrt{k!}}\prod_{i\in I}l_i.\] For \(k\ge1\), integration of \(\log x\) gives \(\log(k!)\ge\int_1^k\log x\,dx\ge k\log k-k\), and hence \(k!\ge(k/e)^k\). We conclude that \[ |\mathop{\mathrm{Pf}}_I(WdB_{b_*})| \le\bigl(Ca_{b_*}\sqrt W\,\beta_{b_*}^{-1/2}\bigr)^k (n/k)^{k/2}\prod_{i\in I}l_i. \tag{93}\] The estimates (90) and (93) control every smooth stage at every point of the simplex. It remains to control the raw stage in the average required for the full Pfaffian.

The raw stage and nearby coordinates

Recall \(J_i=\delta_{h_{i-1}}-\delta_{h_i}\). The coefficient of \(dt_i\) in \(B_0\) is \[-\frac12\int_{\mathbb R}U_0(v-t_i)K_0(J_i,Y_0(v))\,dv.\] For distinct \(i,i'\) in the interior of the simplex, differentiating at the boundary \(t_{i'}\) and antisymmetrizing gives \[(dB_0)_{ii'} =\sum_{a\in\mathbb Z}U_0(t_{i'}+a-t_i)K_0(J_i,g^aJ_{i'}).\] There is at most one nonzero summand. Since \(K_0(h,h')=\chi_R(d(h,h'))\) and \(\chi_R\) is \(1/R\)-Lipschitz, \[|K_0(J_i,g^aJ_{i'})| \le\frac{2d(h_{i-1},h_i)}R \le\frac{2l_i l_{i'}}R.\] The support of \(U_0\) also shows that this matrix entry is zero if the circular distance between \(t_i\) and \(t_{i'}\) exceeds \(2R/n\).

For \(I\subset\{1,\ldots,n\}\) and \(i\in I\), let \(N_i(I)\) be the number of indices in \(I\), including \(i\) itself, whose times have circular distance at most \(2R/n\) from \(t_i\). If \(k=|I|\) is even, divide row and column \(i\) of the principal curvature matrix by \(l_i\). Each resulting row has Euclidean norm at most \((C/R)\sqrt{N_i(I)}\). Hadamard’s inequality, followed by the Pfaffian–determinant identity, yields \[ |\mathop{\mathrm{Pf}}_I(dB_0)| \le (C/R)^{k/2}\prod_{i\in I}l_i \prod_{i\in I}N_i(I)^{1/4}. \tag{94}\]

We next average the last product. The needed assertion is an average over subsets of a fixed size; it is not a bound asserted separately for each fixed subset of the ordered coordinates.

Combining (94) with Lemma 21 gives a factor of size \((CR^{-1/4})^k\), once the length factors have been extracted and the coordinate subset has been averaged. The remaining task is to combine this subset average with the pointwise estimates for the smooth stages. We first make their scale dependence uniform.

Expansion of the normalized curvature

First record the dependence of the preceding estimates on \(R\). Since \(r\) is a fixed multiple of \(R\), the stopping condition compares \[\log\beta_j=(3/2)^{j-1}\log R, \qquad \log(W^2R^{100})=O(R+\log R).\] Taking \(j\) of order \(1+\log R\) makes the first expression at least the second. As in (74), minimality therefore gives \[ b_*=O(1+\log R). \tag{96}\] For \(j<J_*\), the constants \(a_j\) are bounded independently of \(R\), because this is a fixed finite set of stages. Their bases in (90) are at most \(CR^{-1/8}\). At the later stages, \[a_j\le CR^{10}(1+b_*)^{20}, \qquad \beta_j\ge R^{200},\] so the intermediate bases are at most \(CR^{-15}(1+\log R)^{20}\). At the last stage, the stopping condition \(\beta_{b_*}\ge W^2R^{100}\) gives \[Ca_{b_*}\sqrt W\,\beta_{b_*}^{-1/2} \le CR^{-40}(1+\log R)^{20}W^{-1/2}.\] Together with the raw factor just obtained, all these bases are at most \[ \varepsilon_R=C_1R^{-1/10}\le1 \tag{97}\] for sufficiently large \(R\), after choosing a fixed \(C_1\) large enough.

The normalization of the connection gives \[ d\alpha=C_{\rm nor}^{-1}\sum_{j<b_*}dB_j +C_{\rm nor}^{-1}W\,dB_{b_*}+Q, \qquad Q=-C_{\rm nor}^{-2}dC_{\rm nor}\wedge B. \tag{98}\] The lower bound \(C_{\rm nor}\ge c_0>0\) from Proposition 17 changes the bases in (97) only by a fixed factor; enlarge \(C_1\) accordingly. The last term in (98) is decomposable, so \(Q\wedge Q=0\). Moreover, (77) implies \[ |Q_{ii'}|\le C_Rl_i l_{i'}. \tag{99}\] In particular, this stage can contribute to a Pfaffian partition only on zero or two indices. The constant \(C_R\) in (99) is independent of \(n\).

There are \(S=b_*+2\) stages in (98): the raw stage, the \(b_*\) smooth stages, and \(Q\). Denote their matrices by \(M_0,\ldots,M_{S-1}\), with \(M_{S-1}=Q\). The Pfaffian expansion is \[ \mathop{\mathrm{Pf}}\left(\sum_{j=0}^{S-1}M_j\right) =\sum_{I_0\sqcup\cdots\sqcup I_{S-1}=\{1,\ldots,n\}} \sigma(I_0,\ldots,I_{S-1}) \prod_{j=0}^{S-1}\mathop{\mathrm{Pf}}_{I_j}(M_j), \tag{100}\] where all \(|I_j|\) are even and each sign \(\sigma\) is \(1\) or \(-1\). There is no multiplicity factor for a fixed ordered partition. To check this normalization, expand the divided exterior power \((\sum_jM_j)^m/m!\) as \(\sum_{\sum m_j=m}\prod_j M_j^{m_j}/m_j!\), and then extract the coordinate volume coefficient using (84) on each coordinate subset.

Take absolute values in (100). In each term the length factors multiply to \(\prod_{i=1}^nl_i\), independently of the partition. Fix now the even sizes \(k_j=|I_j|\). For any given raw subset \(I_0\) of size \(k_0\), the number of completions to a partition of these sizes is \[\frac{(n-k_0)!}{\prod_{j=1}^{S-1}k_j!},\] which is independent of \(I_0\). Every factor other than the raw neighbor counts has already been bounded independently of the simplex point and of the actual subset. Thus, upon summing over these partitions and integrating over the simplex, the count product in (94) is averaged uniformly over all raw subsets of size \(k_0\). Lemma 21 applies without any assertion about an individual fixed subset.

The smooth estimates also contain factors \((n/k_j)^{k_j/4}\) or \((n/k_j)^{k_j/2}\). Their product is at most \[ \prod_{j:k_j>0}(n/k_j)^{k_j/2}\le S^{n/2}. \tag{101}\] For completeness, put \(p_j=k_j/n\) on the positive sizes. Concavity of the logarithm gives \[\sum_jp_j\log(1/p_j) \le\log\left(\sum_jp_j/p_j\right)\le\log S;\] exponentiating \(n/2\) times this inequality proves (101). Factors for the raw and decomposable stages may be inserted here because they are at least one.

Let \(q=k_{S-1}\). A nonzero term has \(q=0\) or \(q=2\). The small bases in the other stages give \(\varepsilon_R^{n-q}\), while the decomposable stage costs at most \(C_R\). Since \(0<\varepsilon_R\le1\), both possibilities are bounded by \(C_R\varepsilon_R^n\) after absorbing the single factor \(\varepsilon_R^{-2}\) into \(C_R\). This absorption introduces no dependence on \(n\).

For fixed sizes, the number of ordered partitions is \(n!/\prod_j k_j!\). Summed over all admissible sizes this is at most \(S^n\), the number of all assignments of the \(n\) indices to \(S\) stages. Combining this count, (101), and the preceding uniform subset average proves \[\mathbb E_{\Delta_n}|\mathop{\mathrm{Pf}}(d\alpha)| \le C_R\varepsilon_R^n S^{3n/2}\prod_{i=1}^n l_i.\] Substitution of (97) proves (85), and (96) supplies the assertion about \(S\). This completes the proof of Theorem 18.