Parity cancellation and the finite witness

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

This lesson proves that the rank-zero Bott class of the torus \(\langle z,x\rangle\) dies in the reduced group \(C^*\)-algebra of the graphical group, and then in the reduced \(C^*\)-algebra of a finitely generated subgroup. The model contains Bott profiles at all heights. Rotating within the disjoint pairs of heights \((0,1),(2,3),\ldots\) and rescaling moves the even family to the odd family; the pairs \((1,2),(3,4),\ldots\) move the odd family to the even family without its initial term. So the even family equals itself minus its initial term in \(K\)-theory, and the initial term vanishes. This is a swindle in finitely many steps, carried out in a \(C^*\)-algebra generated by bounded operator families. To transfer it to the group, the families are realized only up to errors tending to zero along a sequence of constructions; passing to the quotient of bounded sequences by null sequences makes the realization exact; and a class that vanishes in the quotient vanishes at every late coordinate [OpenAI-BCI, Sections 8.4-8.7 and 9.1, Lemmas 8.5-8.9, Proposition 8.8].

The setting of Carriers, trimming and constant extraction and Finite Cayley formulas for prescribed profiles is in force. We use from the course K-theory of C*-algebras the definition of \(K_0\) of a unital \(C^*\)-algebra through projections in matrix algebras, that homotopic projections are equivalent (Idempotents, projections and their equivalences, Corollary 3.3), and the definition of \(K_0\) of a nonunital algebra as the kernel of \(K_0(B^+)\to K_0(\mathbb C)\) with its normal form (Nonunital algebras: unitization, relative classes and half-exactness, Theorem 1.1).

1. Norm sequence quotients

Take a sequence of constructions indexed by \(\nu\ge1\), with \(b\) fixed throughout. For construction \(\nu\) let \(A_\nu\) be the right regular copy of \(C^*_r(\Gamma_\nu)\) on \(\mathscr L_\nu=\ell^2(\Gamma_\nu)\), and put \[ \begin{gathered} \mathcal Q_r=\frac{\prod_\nu A_\nu}{\bigoplus_\nu A_\nu},\qquad\mathcal Q_{\rm op}=\frac{\prod_\nu B(\mathscr L_\nu)}{\bigoplus_\nu B(\mathscr L_\nu)},\\ \mathcal M=\frac{\prod_\nu B(\mathscr K_\nu)}{\bigoplus_\nu B(\mathscr K_\nu)}, \end{gathered} \] where products mean bounded sequences and direct sums norm-null sequences, and \(\mathscr K_\nu\) is the model space of construction \(\nu\). The faithful coordinate representations make \(\mathcal Q_r\subseteq\mathcal Q_{\rm op}\) an isometric inclusion. Put \(V_\nu=U_\nu D_\nu\mathcal I_\nu:\mathscr K_\nu\to\mathscr L_\nu\). Since \(U^*U=D\), \(D^2=D\), \(\mathcal I^*\mathcal I=1\) and \(\mathcal I=E\mathcal I\), \[ \begin{gathered} \|V_\nu^*V_\nu-1\|\\ =\|\mathcal I^*(D-1)\mathcal I\|\\ \le\|E(1-D)E\|=\delta_\nu^2 \end{gathered}\tag{1.1} \] by Lemma 3.1 of the lesson on carriers. Hence \[ \begin{gathered} \Theta:\mathcal M\to\mathcal Q_{\rm op},\\ [(b_\nu)]\mapsto[(V_\nu b_\nu V_\nu^*)] \end{gathered}\tag{1.2} \] is a \(*\)-homomorphism: it is linear, bounded and respects adjoints, null sequences go to null sequences, and its multiplicative defect \(V_\nu b_\nu(V_\nu^*V_\nu-1)c_\nu V_\nu^*\) has norm at most \(\delta_\nu^2\|b_\nu\|\|c_\nu\|\to0\), once \(\delta_\nu\to0\). Nothing is claimed yet about its image lying in \(\mathcal Q_r\).

2. The source algebra and its reduced membership

Let \(A_{\rm ev},A_{\rm od},A_{\rm ev,+},A_0\in M_m(\mathcal M)\) be the classes of the four diagonal pattern families of Section 2 of the lesson on carriers, with \(m=2\); the scalar matrix \(e\) acts on the Bott matrix factor at every height. On a pair of consecutive summands \((i,i+1)\) of \(\mathscr K_\nu\), consider \[ \begin{gathered} \begin{pmatrix}\cos^2t\,a_i&\tfrac12\sin2t\,a_i\\\tfrac12\sin2t\,a_i&\sin^2t\,a_i\end{pmatrix},\\ 0\le t\le\pi/2 . \end{gathered}\tag{2.1} \] Adding \(\operatorname{diag}(e,e)\) gives a projection, by the computation in the proof of Lemma 4.1 below with \(v=1\), using \(ea_i+a_ie+a_i^2=a_i\) and \(\cos^2t+\sin^2t=1\); the off-diagonal function is supported in the intersection of the two carriers. This path moves \(a_i\) from summand \(i\) to summand \(i+1\); then the rescaling path of Section 1 of the lesson on carriers replaces \(a_i\) by \(a_{i+1}\) on summand \(i+1\). Done simultaneously on the disjoint pairs \((0,1),(2,3),\ldots\), this gives a path from the even to the odd family; on \((1,2),(3,4),\ldots\), a path from the odd to the positive-even family. Their norms and moduli of continuity are uniform in \(i\) and \(\nu\): the pairs are orthogonal, and all rescaling ratios are fixed by \(b\).

The source algebra \(\mathcal B\) is the \(C^*\)-subalgebra of \(\mathcal M\) generated by the scalar matrix entries of the four compact differences and of the two paths at rational parameters; by norm continuity it contains them at all parameters. It is an algebra generated by bounded operator families, not a formal infinite sum of \(K\)-classes. Let \(\mathcal B^+=\mathcal B\oplus\mathbb C\) be its unitization, with \((b,\lambda)(c,\mu)=(bc+\lambda c+\mu b,\lambda\mu)\); all the projections above lie in a fixed matrix algebra over \(\mathcal B^+\), with scalar part \(e\) or its block sums.

Lemma 2.1 (reduced membership of the source algebra; [OpenAI-BCI, Lemma 8.5]). The constructions can be chosen so that \(\Theta(\mathcal B)\subseteq\mathcal Q_r\).

Proof. Take \(\eta_\nu\downarrow0\). At stage \(\nu\): choose the extraction length \(k\) from the fixed gap; choose finite meshes of the two model paths with mesh error at most \(\eta_\nu\), using their uniform modulus; collect the finitely many scalar matrix profiles of the mesh points, the four endpoint profiles and the cutoffs; and apply Lemma 3.1 of Finite Cayley formulas for prescribed profiles to this finite family with errors small compared with \(\eta_\nu\). The finite type and weight choices come before \(s_\nu\); choose \(s_\nu\) and then \(H_{0,\nu}\) so that all errors are at most \(\eta_\nu\), including \(\delta_\nu\) and the right side of (3.4) of the lesson on carriers.

Each endpoint family then has a bounded sequence in a fixed matrix algebra over \(A_\nu\) with the same class in \(\mathcal Q_{\rm op}\) as its image under \(\Theta\). For a parameter of a path, interpolate linearly between the physical mesh values; the interpolated sequence differs from the image of the model path by a null sequence, uniformly in the parameter, so every generating path value has image in \(\mathcal Q_r\). Since \(\mathcal Q_r\) is a closed \(C^*\)-subalgebra of \(\mathcal Q_{\rm op}\), it contains \(\Theta(\mathcal B)\). Only a finite profile family was sampled at each stage; the uniform approximation of the compact parameter spaces supplies all path values. \(\square\)

The restricted homomorphism extends to \(\Theta^+:\mathcal B^+\to\mathcal Q_r\), \(\Theta^+(b,\lambda)=\Theta(b)+\lambda1\). This is the only unit extension used; it does not claim that the identity on all model carriers is realized in the reduced algebra.

3. Parity identities

Put \(b_I=[e+A_I]-[e]\in K_0(\mathcal B)\) for \(I\in\{\mathrm{ev},\mathrm{od},(\mathrm{ev},+),0\}\); the common scalar rank puts each difference in the kernel of \(K_0(\mathcal B^+)\to K_0(\mathbb C)\), which is \(K_0(\mathcal B)\). The two explicit paths of projections and Corollary 3.3 of the lesson on idempotents give \[ b_{\rm ev}=b_{\rm od}=b_{\rm ev,+} .\tag{3.1} \]

Lemma 3.1 (finite additivity). Let \(a',a''\) be self-adjoint with \(a'a''=a''a'=0\), such that \(e+a'\) and \(e+a''\) are projections. Then \[ P_t=\begin{pmatrix}e+a'+\cos^2t\,a''&\cos t\sin t\,a''\\\cos t\sin t\,a''&e+\sin^2t\,a''\end{pmatrix} \] is a path of projections from \(\operatorname{diag}(e+a'+a'',e)\) to \(\operatorname{diag}(e+a',e+a'')\). Consequently \([e+a'+a'']-[e]\) equals \(([e+a']-[e])+([e+a'']-[e])\).

Proof. With \(W=(\cos t,\sin t)^T\) and \(E=\operatorname{diag}(e+a',e)\), \(P_t=E+Wa''W^*\), \(W^*W=1\), and \(EW a''=We a''\), \(a''W^*E=a''eW^*\) because \(a'a''=a''a'=0\). So \(P_t^2=E+W(ea''+a''e+a''^2)W^*=P_t\), since \((e+a'')^2=e+a''\). \(\square\)

Applied to \(a'=A_0\) and \(a''=A_{\rm ev,+}\), whose height supports are disjoint, Lemma 3.1 gives \(b_{\rm ev}=b_0+b_{\rm ev,+}\). With (3.1), \[ b_0=0\quad\text{in }K_0(\mathcal B).\tag{3.2} \] Only finitely many additions and subtractions of \(K\)-classes occur, and the all-height objects are elements of \(\mathcal B\), whose images are reduced by Lemma 2.1.

4. The physical root and supported Bott transport

Let \(J_{0,\nu}:e_0^{\rm car}\mathscr L_\nu\to\mathscr K_\nu\) be the inclusion as the zeroth summand, \(n_0^B\) the number of surviving \(\mathscr B_0\)-vertices, and \[ \begin{aligned} S_\nu&=(n_0^B)^{-1/2}\\ &\qquad\times\sum_{v\in\mathscr B_0^{\rm surv}}R(t_v^{-1})T_0,\\ v_\nu&=S_\nu\chi_0\in A_\nu . \end{aligned}\tag{4.1} \] The sums are finite; the norm of \(S_\nu\) on all of \(\mathscr L_\nu\) need not be bounded uniformly. For a coset \(h\mathsf H\), \(R(t_v^{-1})\delta_{hf}=\delta_{hft_v}\), so (4.1) is the normalized constant transport into the sheet coordinates, after spreading: on the localized source space \(v_\nu=\mathcal J\mathcal I_\nu J_{0,\nu}\chi_0\). The single-height case of Lemma 3.2 of the lesson on carriers gives \[ \begin{aligned} &\|v_\nu-V_\nu J_{0,\nu}\chi_0\|\\ &\quad\le C_\nu2^{-344.5H_{0,\nu}}\to0 ; \end{aligned}\tag{4.2} \] in particular the \(v_\nu\) are uniformly bounded, since \(\|V_\nu\|\le1\) and \(\|\chi_0\|\le1\). The factor \(n_0^B\) is accounted for by the multiplicity estimate, not by assuming that summation over sheets is isometric.

Let \(a^{\rm phys}_\nu=a_0(R(z),R(x))\) and \(p^{\rm phys}_\nu=e+a^{\rm phys}_\nu\), exact projections in \(M_m(A_\nu)\), and write \(a^{\rm phys}\), \(v\) for their classes in \(\mathcal Q_r\), with \(v\) tensored with \(I_m\) on the Bott matrices. The cutoff \(\chi_0\) is real, supported in the carrier, and equal to one on \(\operatorname{supp}a_0\). By (1.1) and (4.2), \(v^*v\) and \(\chi_0^2\) agree up to a null sequence after multiplication by \(a^{\rm phys}\), and so \[ \begin{gathered} v^*v\,a^{\rm phys}=a^{\rm phys}\\ =a^{\rm phys}v^*v,\\ \Theta(A_0)=v\,a^{\rm phys}v^* , \end{gathered}\tag{4.3} \] the second by multiplying (4.2) on both sides of the compact difference.

Lemma 4.1 (supported Bott transport; [OpenAI-BCI, Lemma 8.6]). Let \(B\) be a \(C^*\)-algebra, \(e\in M_m(\mathbb C)\) a projection, \(a=a^*\in M_m(B)\) with \(e+a\) a projection in \(M_m(B^+)\), and \(v\in B\), acting as \(v\otimes I_m\), with \(v^*va=a\). Then \[ \begin{gathered} [e+a]-[e]=[e+vav^*]-[e]\\ \text{in }K_0(B). \end{gathered} \] No contraction assumption on \(v\) is needed.

Proof. Also \(av^*v=a\), by taking adjoints. Put \(c=\cos t\), \(s=\sin t\), \(W_t=(cI_m,sv)^T\), \(E_{\rm sc}=\operatorname{diag}(e,e)\), and \[ \begin{aligned} P_t&=E_{\rm sc}+W_taW_t^*\\ &=\begin{pmatrix}e+c^2a&cs\,av^*\\ cs\,va&e+s^2vav^*\end{pmatrix}. \end{aligned} \] It is self-adjoint. Since \(e\) is scalar on the matrix factor and \(v\) acts as \(v\otimes I_m\), \(E_{\rm sc}W_t=W_te\). Moreover \(aW_t^*W_ta=c^2a^2+s^2av^*va=a^2\). So \(P_t^2=E_{\rm sc}+W_t(ea+ae+a^2)W_t^*=P_t\). The path is norm continuous, with endpoints \(\operatorname{diag}(e+a,e)\) and \(\operatorname{diag}(e,e+vav^*)\), and all nonscalar entries lie in \(M_{2m}(B)\). Its modulus of continuity depends only on \(\|a\|\) and \(\|v\|\). \(\square\)

Apply Lemma 4.1 in \(\mathcal Q_r\) to (4.3); by (3.2) and the homomorphism \(\Theta^+\), \[ \begin{gathered} [(p^{\rm phys}_\nu)]-[e]=0\\ \text{in }K_0(\mathcal Q_r) . \end{gathered}\tag{4.4} \] The matrix size in Lemma 4.1 is \(2m\), independent of the number of transports, layers, sheets or constructions.

5. From the quotient to one finitely generated group

Lemma 5.1 (eventual coordinate equality; [OpenAI-BCI, Lemma 8.7]). Let \(B_\nu\) be unital \(C^*\)-algebras and \(P_\nu,Q_\nu\in M_m(B_\nu)\) projections of a fixed size. If their classes agree in \(K_0\bigl(\prod B_\nu/\bigoplus B_\nu\bigr)\), then \([P_\nu]=[Q_\nu]\) in \(K_0(B_\nu)\) for all large \(\nu\).

Proof. By the definition of \(K_0\) there are a projection \(R\) in a matrix algebra over the quotient and, after padding with zeros, a partial isometry \(w\) with \(w^*w=P\oplus R=:E\) and \(ww^*=Q\oplus R=:F\). Lift \(R\) to a bounded self-adjoint sequence \(r_\nu\); its defect \(\|r_\nu^2-r_\nu\|\) tends to zero, so for late \(\nu\) the spectrum of \(r_\nu\) avoids \(\frac12\), and the continuous functional calculus with a function equal to \(0\) below \(\frac12\) and \(1\) above gives projections \(R_\nu\) with \(\|R_\nu-r_\nu\|\to0\). Form \(E_\nu,F_\nu\), lift \(w\) to a bounded sequence \(b_\nu\), and put \(z_\nu=F_\nu b_\nu E_\nu\). Then \[ \begin{gathered} \|z_\nu^*z_\nu-E_\nu\|\to0,\\ \|z_\nu z_\nu^*-F_\nu\|\to0 . \end{gathered} \] For large \(\nu\), \(z_\nu^*z_\nu\) is invertible in the corner \(E_\nu M(B_\nu)E_\nu\) and \(z_\nu z_\nu^*\) in the corner of \(F_\nu\). The element \(z_\nu(z_\nu^*z_\nu)^{-1/2}\), with the inverse square root taken in the corner, is a partial isometry with initial projection \(E_\nu\), and its final projection is \(F_\nu\) by the invertibility in the \(F_\nu\)-corner. So \([P_\nu]+[R_\nu]=[Q_\nu]+[R_\nu]\), and cancellation in the group \(K_0(B_\nu)\) gives the claim. All stabilizations are finite and fixed once the witness in the quotient is chosen. \(\square\)

Proposition 5.2 (reduced Bott vanishing; [OpenAI-BCI, Proposition 8.8]). For all large \(\nu\) in the compatible sequence, the class \([p^{\rm phys}_\nu]-[e]\), which is the image of the fixed rank-zero Bott class \(\beta\) of (1.2) of the lesson on carriers under \(C(\mathbb T^2)\cong C^*_r(\langle z,x\rangle)\to C^*_r(\Gamma_\nu)\), is zero in \(K_0(C^*_r(\Gamma_\nu))\).

Proof. Lemma 5.1 applied to (4.4) gives vanishing in \(K_0(A_\nu)\) for large \(\nu\). In every coordinate, (1.2) of the lesson on carriers identifies the difference with the image of \(\beta\). The unitary \(I\) of Section 1 of the lesson on carriers carries the right regular copy to the usual reduced algebra, without changing the group elements or their functional calculus. \(\square\)

Lemma 5.3 (a finite witness lies in a finitely generated subgroup; [OpenAI-BCI, Lemma 8.9]). Let \(H\le\Gamma\) be finitely generated and \(P,Q\) projections in a matrix algebra over \(C^*_r(H)\). If their images have equal classes in \(K_0(C^*_r(\Gamma))\), then some finitely generated subgroup \(G\) with \(H\le G\le\Gamma\) already has \([P]=[Q]\) in \(K_0(C^*_r(G))\).

Proof. For a subgroup \(G\le\Gamma\), the restriction of the regular representation of \(\Gamma\) to \(G\) is a direct sum of copies of the regular representation of \(G\), over the cosets; so \(C^*_r(G)\to C^*_r(\Gamma)\) is isometric. Choose one stabilization projection \(R\) and a partial isometry \(w\) witnessing \([P]=[Q]\) over \(C^*_r(\Gamma)\), and approximate the finitely many entries of \(R\) and \(w\) by elements of the group algebra, self-adjoint for \(R\). Let \(G\) be generated by \(H\) and the finitely many group elements occurring. The defects of the approximations, as an almost projection and an almost partial isometry, are the same in \(C^*_r(G)\) by isometry. Spectral cutting gives a projection in a matrix algebra over \(C^*_r(G)\), and compressing the approximate partial isometry to the two corners and normalizing it as in Lemma 5.1 gives an exact witness there; the functional calculus stays inside \(C^*_r(G)\). The projections \(P,Q\) themselves are kept exactly. \(\square\)

6. The order of parameters

There are two indices: the height \(i\ge0\) in one tower, and the construction index \(\nu\ge1\). Degrees do not depend on \(i\); no bound uniform in \(\nu\) is needed.

First fix the integer \(b\) with \(\sqrt{5\cdot2^{1-b}}<3/5\), which gives the gap \(\rho=1-0.05^2/2\) of Expansion and the sequential law. Keep the compact Bott projection, its class and its matrix size fixed, and fix enlarged supports for all neighboring scales; their ratios depend only on \(b\), so the rescaling and rotation paths have one modulus of continuity. For each \(\nu\), take a target error \(\epsilon_\nu\downarrow0\), and choose in this order:

  1. the extraction length \(k_\nu\), so that \(\rho^{2k_\nu}\) is small in (4.3) of the lesson on carriers; finite meshes of the two paths; the resulting finite collection of same-height and adjacent-height types; and per-factor tolerances small enough for the telescoping bound;
  2. the finite Fourier truncations and weight set of the product sampler, the exponent of the finite norm test, the moment constant, and a deletion fraction with \(k_\nu\sqrt{\varepsilon_{\rm del}}\) small;
  3. \(s_\nu\), a large multiple of \(20000\) and \(5b\), with, besides the coefficient, probability and moment requirements, \[ \begin{gathered} \frac{s_\nu}b\ge10^{12},\quad\frac b{0.50005s_\nu}\le10^{-6},\\ 2^{-0.000025s_\nu+O(\log s_\nu)+O_\nu(1)}\ll\epsilon_\nu , \end{gathered} \] the last for the refined alphabet after the type and weight choices; and the gate budget and the height term of the separation budget small;
  4. \(n_{0,\nu}\), hence \(H_{0,\nu}=(2n_{0,\nu}+1)s_\nu\), with \(H_{0,\nu}/s_\nu\ge10^{12}\) and all initial-height requirements of the preceding lessons: the remaining separation budget, the sum of the expected pruning fractions, the sum of all voltage and norm failure probabilities below one, and the trimming and synthesis errors, of the forms \(C_\nu2^{-345H_{0,\nu}}\) and \(C_\nu2^{-344.5H_{0,\nu}}/(1-2^{-344.5b})\), below their allocations; \(C_\nu\) does not depend on this final increase.

The random data come after these parameters. The coefficient pools are chosen once (Lemma 3.1 of Streams, coefficient pools and universal corrections) and reused on all full blocks. The sequential law gives expanding auxiliary incidences for every realization in its support, the deletion estimates of the labelled tower hold for every realization, and the pruning selects one realization with all layer deletions small. Only after this graph is fixed are the voltages and weights drawn, by one independent law that controls both kinds of failures before their simultaneous avoidance; no change of measure is hidden in the centered moment estimate. The alphabet does not grow with the height; it is refined only by the finite weight set, and the private completion adds a countable alphabet whose coefficients in every operator polynomial are zero. Finite generation is obtained only at the end, from the finite witness. At stage \(\nu\) only finitely many profiles are required, and their meshes become finer with \(\nu\); their uniform modulus supplies the complete paths in the quotient, so no separate approximation is chosen for each height or path parameter.

7. Exercises

Exercise 7.1. In Lemma 3.1, why must \(a'a''=0\) be assumed? Find the defect \(P_t^2-P_t\) without it.

Exercise 7.2. Show that the swindle cannot be performed in \(K_0\) of a single construction by summing the classes \([e+a_i]-[e]\) over all \(i\): what is missing?

Exercise 7.3. In Lemma 5.1, why is it necessary to cut \(r_\nu\) spectrally before forming \(E_\nu\) and \(F_\nu\)?

Exercise 7.4. Show that \(v^*va=a\) cannot be weakened to \(\|v\|\le1\) in Lemma 4.1: give \(v\) and \(a\) for which \(vav^*\) is not of the form required.

8. Solutions

7.1. In general \(EWa''=We a''+(c\,a'a'',0)^T\), and \(P_t^2-P_t\) contains the terms \(c^2(a'a''+a''a')\) and \(cs\,a'a''\); they vanish exactly when the products vanish.

7.2. \(K_0\) is defined through finitely many matrix entries, and an infinite sum of classes is not defined. The construction replaces the infinite sum by one bounded operator family, an element of the source algebra, whose class is a single element of \(K_0\); the homotopies are paths of such families.

7.3. \(E_\nu=P_\nu\oplus r_\nu\) must be a projection for the corner \(E_\nu M(B_\nu)E_\nu\) and the partial isometry with initial projection \(E_\nu\) to make sense; \(r_\nu\) is only approximately idempotent.

7.4. With \(B=\mathbb C\), \(m=1\), \(e=0\) and \(a=1\), and \(v=\frac12\), \(vav^*=\frac14\) is not a projection, so \(e+vav^*\) is not a projection. The condition \(v^*va=a\) says that \(v\) is isometric on the support of \(a\).

References