Results on separated spectral bands, the tower, voltage covers and their Cayley graphs, the nilpotent voltage group and the random realization, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• A torsion-free counterexample to the Kadison-Kaplansky conjecture (OpenAI) — Theorem 1.1. There are a finitely generated torsion-free group G_proj and a projection e in C_r*(G_proj) with 0 < tau(e) < 1/2. Proved in lessons 2 to 18. No hypothesis.
• Reduced subgroup inclusion — Lemma 2.1. For a subgroup Lambda of a countable group K, lambda_Lambda(x) -> lambda_K(x) extends to an isometric unital *-homomorphism C_r*(Lambda) -> C_r*(K) preserving the canonical traces. No hypothesis.
• Torsion in free products — Lemma 3.2. A free product of torsion-free groups is torsion-free, and finitely many finitely generated factors give a finitely generated group. No hypothesis.
• Scalar compression (OpenAI) — Proposition 4.1. A projection p in M_N(C_r*(Lambda)) with 0 < tau_(Lambda,N)(p) < 1/2 gives a projection e in C_r*(Lambda * F_N) of the same trace. No hypothesis.
• The matrix construction (OpenAI) — Theorem 5.1. There are a finitely generated torsion-free group Gamma and a self-adjoint T in M_N(C Gamma) with a separated spectral projection p_0 in M_N(C_r*(Gamma)) of unnormalized trace in (0, 1/2). Proved in the last lesson. No hypothesis.
• Separated spectral projections — Lemma 1.1. The projection onto a separated part of the spectrum of T = T* lies in every unital C*-algebra containing T, commutes with the commutant of T, and is a contour integral of the resolvent. No hypothesis.
• The identity trace of equivariant operators — Lemma 2.1. Tr(A*A) = Tr(AA*) for equivariant A; the trace is additive, monotone and basis independent; equivariant projections of trace zero vanish; injective equivariant maps and equivalences compare traces. No hypothesis.
• Compression of a separated upper space — Lemma 3.1. Cutting A = A* with a separated upper space E by a projection J with ||(1-J)E|| = eps <= 1/2 gives an operator within a + 12 eps of JAJ with an upper space equivalent to E and the same gap. No hypothesis.
• A contour estimate — Lemma 4.2. If ||A - B|| <= u < r_0 <= b/2 and B has the eigenvalue 1 on F and spectrum in [-2, 1-b] elsewhere, the contour projection F_A satisfies ||F_A - F|| <= u/(r_0 - u) and is equivalent to F when u < r_0/2. No hypothesis.
• Cardinalities of the layers — Lemma 2.1. |A_i| = d 2^(H_i), |B_i| = 2^(H_i), and all stream lengths at layer i are at most H_i. No hypothesis.
• Degrees — Lemma 3.1. A vertex of B_k has D_ik = D 2^(|i|-|k|) slots for the pair (i,k), between D/2 and 2D, and d|A_i| = D_ik |B_k|. No hypothesis.
• Coefficients and spills (OpenAI) — Lemma 6.1. The injections and spills can be chosen so that, for every family, side and nonzero functional zeta, every value of zeta M_a restricted to an initial segment of the inputs occurs for at most 0.55 n_sl slots, and every value of the full functional for at most 10^-3 n_sl slots. For large l.
• One edge per slot — Lemma 3.1. The edge equations have exactly one solution for every vertex and slot, from either side, so every slot carries exactly one edge. No hypothesis.
• Folding and the alphabet — Lemma 4.1. Both towers are folded, and the number m_l of positive labels satisfies log2(2 m_l) = 11 l + O(log l). No hypothesis.
• Independent matching clusters (OpenAI) — Lemma 5.1. With uniform permutations at the marked block, the edges of a pair form independent uniform matchings within clusters. No hypothesis.
• Uniform gap of the mean incidence (OpenAI) — Lemma 4.1. The mean incidence of every adjacent pair contracts functions of mean zero by a factor less than 0.8. Uses Lemma 6.1 of the lesson on streams and slots.
• The local lemma, asymmetric form — Lemma 3.1. If each event is independent of the family of its non-neighbours and P(E_z) <= x_z prod over neighbours (1 - x_z') with 0 <= x_z < 1, all events are avoided with probability at least prod (1 - x_z) > 0. No hypothesis.
• Expansion of the realized incidence (OpenAI) — Proposition 5.1. The tables can be chosen so that the normalized incidence of every adjacent pair maps normalized constants to normalized constants and has norm at most 1 - 1/800 off the constants. For large l.
• Recovery of prefixes (OpenAI) — Lemma 1.1. Two nonbacktracking paths with the same word have equal stream prefixes at corresponding positions, of length growing with the radius. No hypothesis.
• The tower (OpenAI) — Proposition 4.1. After deleting a fraction at most eps_l -> 0 of every layer, the tower Y has expanding incidences, folded labels from at most m_l letters, recovery of long paths from their words, and no short cyclically nonbacktracking closed paths. For large l.
• The geometric conclusion (OpenAI) — Theorem 3.1. Under the girth condition, every component of the voltage cover embeds in the Cayley graph of Gamma, the indexed copies have nested overlap balls, the remaining edges form a forest, and Gamma is torsion-free. For large l.
• Embeddings of the components (OpenAI) — Proposition 7.1. Every label map from a component of the cover to the Cayley graph is injective on vertices and formal edges. No hypothesis.
• Short chain fillings (OpenAI) — Lemma 1.1. The chain of every closed walk of length n in the coned Cayley graph is a sum of at most 12n chains of closed walks of length at most four. No hypothesis.
• Thin triangles from linear chain fillings — Lemma 2.1. In a connected graph in which the chain of every closed walk of length n is a sum of at most 12n chains of closed walks of length at most four, geodesic triangles are 10^6-thin. No hypothesis.
• One ball for all earlier overlaps (OpenAI) — Proposition 4.1. In a well-order of the indexed copies, each copy has one ball containing all its vertices shared with earlier copies, on which its edges form a forest, of size at most 2^(C_bullet H_i). No hypothesis.
• Assigned domains and the residual forest (OpenAI) — Proposition 5.2. The new vertices D_Q of the copies are disjoint, and the Cayley edges outside the copies' own domains form a forest. No hypothesis.
• The incidence count — Proposition 6.1. Pairs of an indexed copy and one of its vertices correspond bijectively and equivariantly to pairs of a group element and a vertex of the tower. No hypothesis.
• Torsion-freeness (OpenAI) — Proposition 4.1. The group Gamma defined by the closed walks of the cover is finitely generated and torsion-free. No hypothesis.
• The rank bound — Lemma 1.1. A finite connected metric graph of total length at most Cg and girth greater than g has first Betti number at most max(1, ceil(20C))^2. No hypothesis.
• The matrix detector — Lemma 3.1. A nontrivial reduced word of length less than m is not an identity on upper unitriangular integer m x m matrices; its entries are polynomials of degree at most m-1, also after fixed factors. No hypothesis.
• Grading, Haar measure and integer points — Lemma 2.1. The affine group has integer polynomial laws, dilation automorphisms scaling the cocycle by y^m, Lebesgue Haar measure, and torsion-free nilpotent integer points. No hypothesis.
• The fiber representations — Lemma 5.1. Fourier transform in the central coordinate decomposes the regular representation of F into twisted representations u_t of the two factors, commuting with twisted right operators. No hypothesis.
• Schwartz idempotents (OpenAI) — Lemma 1.1. For each sign there is a Schwartz function J on the quotient group with J *_eps J = J, J^(*eps) = J and ||J||_2 = 1. Uses the Hermite expansion of the index theory course.
• Nonzero discrete twisted projections (OpenAI) — Proposition 3.1. For 0 < |theta| < theta_0, the lattice samples of a rescaled Schwartz idempotent give a projection p_theta in C_r*(L, theta c) with 0 < tr(p_theta) <= C|theta|^(n_f). No hypothesis.
• Absorption of the finite factor (OpenAI) — Lemma 3.1. Tensoring u_t with the finite magnetic representation R gives N copies of u_(t - delta), preserving norms and multiplying identity traces by N. No hypothesis.
• Uniform weighted trace bound (OpenAI) — Proposition 6.1. w_i N t^(n_f) (delta - t)^(n_f) tends to zero uniformly in the active layers i, so the weighted fiber trace mu_l tends to zero. For large l.
• The model band and its trace (OpenAI) — Lemma 2.1. The model operator has eigenvalue 1 on layerwise constant vectors of the active layers tensored with p(t), the rest of its spectrum below 1 - chi, identity trace equal to that of p(t), and deletion loss at most sqrt(eps_l). No hypothesis.
• Realization of the model (OpenAI) — Theorem 3.1. For every eta > 0 there are voltages and finitely many matrix weights on the edges of Y such that the cover has the girth of Theorem 3.1 of the voltage cover lesson and its weighted adjacency is within eta of the compressed model. For large l.
• Finite norm tests (OpenAI) — Lemma 4.1. Coordinate boxes in F with small boundary under bounded right translations test the regular norm of convolution matrices up to C x^(-B) R. No hypothesis.
• Empirical averages (OpenAI) — Lemma 5.1. Expectations of at most C(1 + log q) finite operator distributions have M-term empirical averages within any fixed epsilon in the regular norm. For large l.
• The expectation of one draw (OpenAI) — Proposition 4.1. For every epsilon there is a finite distribution of one weight and one voltage per block whose expectation is within epsilon of h_i h_k p(t) in every fiber. For large l.
• The alphabet and the voltage coordinates — Lemma 5.1. The matrix weights number O(d^(1/32) log d), so refined labels number o(dD); voltages have coordinates at most 2^(C H_i) and wide independent unitriangular coordinates. For large l.
• The Schwartz-Zippel bound — Lemma 1.1. A nonzero polynomial of degree D vanishes at independent uniform points of sets of size at least s with probability at most D/s. No hypothesis.
• Walks without singleton edges (OpenAI) — Lemma 3.1. Closed walks of length 2l without singleton edges and with supports of first Betti number at most b number at most (72(2b+2)^2 Delta)^l. No hypothesis.
• A simultaneous choice (OpenAI) — Proposition 5.1. One assignment of weights and voltages gives nontrivial holonomy to all short cyclically nonbacktracking closed walks and keeps every block within epsilon sqrt(dD_ik) of its mean. For large l.
• The cover projection and its exact fiber trace (OpenAI) — Proposition 2.1. The weighted adjacency of the cover has a separated upper projection whose fiber traces equal those of p(t). For large l.
• The weighted size (OpenAI) — Proposition 3.1. The layer-weighted identity trace of the cover projection is at most 96 mu_l. For large l.
• A nonzero upper band (OpenAI) — Proposition 4.3. The weighted Cayley adjacency T has spectrum in (-inf, 1 - chi + 4 alpha] and [1 - 3 alpha, 1 + 3 alpha], with a nonzero upper projection in M_N(C_r*(Gamma)). For large l.
• The matrix projection (OpenAI) — Theorem 6.1. The construction gives Theorem 5.1 of the first lesson: 0 < tau_(Gamma,N)(p_0) < 1/2. No hypothesis.

Reading order

Eighteen lessons. The first reduces the theorem to a matrix with a separated spectral band and the second proves the spectral and trace tools; lessons 3 to 7 build and prune the tower; lessons 8 to 10 prove the geometry of the voltage cover, its Cayley graph and torsion-freeness; lessons 11 to 14 build the nilpotent voltage group, its Schwartz idempotents and the magnetic factor; lessons 15 to 17 realize the model operator on the cover with random voltages; the last lesson transfers the band and bounds its trace. Van Kampen diagrams and plane maps are used as named statements with free locators.