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+================================================================
+A COUNTEREXAMPLE TO THE KADISON-KAPLANSKY PROJECTION CONJECTURE
+A plain-language guide for high-school readers
+================================================================
+
+WHAT THIS DOCUMENT IS
+
+In September 2026 a team at OpenAI posted a research paper titled
+"A torsion-free counterexample to the Kadison-Kaplansky projection
+conjecture". The paper claims that a famous 50-year-old open
+problem is false. This guide rewrites that paper for a reader
+who knows some algebra but has not studied operator algebras.
+It is complete in the sense that every logical step of the proof
+appears. Long technical estimates are described rather than
+reproduced symbol by symbol, because those estimates do not carry
+ideas a reader can use; the skeleton of the proof carries all of
+them.
+
+The result, in one sentence: there is a group G built from
+finitely many pieces, with no element of finite order, whose
+"reduced group C*-algebra" C_r*(G) contains an element e that is
+a projection but is not 0 and not 1. If true, this destroys the
+conjecture.
+
+The plan of the document:
+
+ Part 1. The players: groups, C*-algebras, traces.
+ Part 2. The conjecture and why it is hard.
+ Part 3. The theorem: the precise statement.
+ Part 4. The five-step plan of the proof.
+ Part 5. Step A: reduce to a small matrix projection.
+ Part 6. Step B: the expander tower.
+ Part 7. Step C: the deck group and the small trace.
+ Part 8. Step D: choosing voltages, gathering statistics.
+ Part 9. Step E: the geometry and the group G.
+ Part 10. Step F: transferring the band; small trace.
+ Part 11. The end: what just happened; what is new.
+ Part 12. Glossary of every symbol we use.
+
+================================================================
+PART 1. THE PLAYERS
+================================================================
+
+1.1 GROUPS
+
+A group is a set G with an operation "multiply two elements and
+get a third", written gh for g times h, satisfying:
+
+ (i) associativity: (gh)k = g(hk) for all g,h,k in G;
+ (ii) an identity element 1 with 1g = g1 = g for all g;
+ (iii) every g has an inverse g^-1 with gg^-1 = g^-1g = 1.
+
+Examples you have seen before:
+
+ the integers Z under addition (identity 0)
+ the nonzero reals under multiplication (identity 1)
+ invertible n-by-n matrices under usual matrix multiplication
+ rotations of a regular polygon
+
+Two properties are crucial below.
+
+Finitely generated. There is a finite list S = {s_1,...,s_k} of
+elements such that every element of G is a finite product of s_i
+and their inverses. For the integers, S = {1} suffices.
+
+Torsion-free. If g^n = 1 for some n >= 1, then g = 1. No
+element of the group, other than the identity, is "finite order".
+The integers are torsion-free. The group of rotations of a
+regular polygon is not: a rotation by 1/n of a full turn has
+order n.
+
+Our constructed group will be both finitely generated AND
+torsion-free, exactly the kind of group the conjecture is about.
+
+1.2 THE HILBERT SPACE l^2(G)
+
+Fix a group G. Consider all complex-valued functions on G that
+decay fast enough:
+
+ l^2(G) = { a : G -> C | sum over g of |a(g)|^2 < infinity }
+
+This is a complex Hilbert space with inner product
+
+ <a,b> = sum over g of conj(a(g)) b(g).
+
+For each g in G, let delta_g in l^2(G) be the function that is 1
+at g and 0 elsewhere. Then {delta_g : g in G} is an
+orthonormal basis of l^2(G).
+
+Each element h of G acts on l^2(G) by left translation:
+
+ lambda(h) delta_g = delta{hg}.
+
+The map lambda(h) is a unitary operator, and the map h -> lambda(h)
+is a group homomorphism: the group G now sits inside the unitary
+group of the Hilbert space.
+
+1.3 THE REDUCED GROUP C*-ALGEBRA
+
+The group algebra C[G] consists of formal finite sums
+
+ x = sum over g of c(g) g, only finitely many c(g) nonzero.
+
+It acts on l^2(G) by the same left-translation formula extended
+linearly:
+
+ x delta_g = sum over h of c(h) delta{hg}.
+
+The reduced group C*-algebra C_r*(G) is the closure of this space
+of operators in the operator norm:
+
+ C_r*(G) = completion of { x acting on l^2(G) : x in C[G] }.
+
+Concretely: an element of C_r*(G) is a possibly infinite "sum"
+sum c(g) lambda(g) that is a limit, in operator norm, of finite
+sums. It is a C*-algebra: an algebra of operators closed under
+addition, multiplication, taking adjoints, and taking the norm-
+limit.
+
+Why the reduced one? Because l^2(G) gives the "smallest" natural
+completion. Every group admits a universal completion too (the
+full group C*-algebra), but the reduced one is where the trace
+below lives and where the conjecture is usually stated.
+
+1.4 THE TRACE
+
+Every element x of C_r*(G) has a well-defined "coefficient of
+the identity", extracted by pairing against the identity basis
+vector:
+
+ tau(x) = < x delta_1, delta_1 >.
+
+In the group algebra this reads off c(1) from the formula above.
+Properties:
+
+ linear over C
+ tau(xy) = tau(yx) trace property
+ tau(x* x) >= 0 positivity, and =0 iff x = 0
+ tau(1) = 1
+
+In words: tau is a "faithful tracial state". A linear functional
+that behaves like the identity coefficient in the group algebra.
+
+A fact we will use twice:
+
+ If p is a projection in C_r*(G), then tau(p) is a real number
+ in [0,1]. And tau(p) = 0 forces p = 0.
+
+Proof: tau(p) = <p delta_1, delta_1> and p = p* p, so
+tau(p) = <p delta_1, p delta_1> = ||p delta_1||^2 >= 0. Since
+||p|| <= 1, it is at most ||delta_1||^2 = 1. Faithfulness of tau
+gives the second part.
+
+1.5 PROJECTIONS
+
+A projection is an element p of a C*-algebra with
+
+ p* = p and p^2 = p.
+
+Geometrically, p is the orthogonal projection onto its range
+subspace pH. The projections 0 and 1 are the trivial ones.
+
+In the group-algebra setting, projections can be invisible to the
+group elements themselves: they are infinite operator sums, not
+finite combinations. The larger von Neumann closure L(G) is
+more forgiving; the smaller C_r*(G) is more rigid.
+
+The conjecture below asks: is C_r*(G) so rigid that for
+torsion-free G it only contains 0 and 1.
+
+1.6 MATRICES OVER A C*-ALGEBRA
+
+If A is a C*-algebra and N >= 1 is an integer, then M_N(A), the
+algebra of N-by-N matrices with entries in A, is again a
+C*-algebra (there is a natural norm). The trace extends by
+summing the diagonal:
+
+ tau_A,N( (a_ij) ) = sum over j of tau(a_jj),
+
+with NO division by N. This "unnormalized matrix trace" is the
+one we will use. When A = C_r*(G) and a projection p in M_N(A)
+is written out, tau_{G,N}(p) is a nonnegative real number; it can
+exceed 1 because there are N diagonal entries each at most 1.
+
+We sometimes write tr_id for this sort of trace in a large Hilbert
+space, "identity trace". It is the same idea, more carefully
+stated: pick an orthonormal basis {k_a} of the auxiliary space,
+and define Tr(Z) = sum over a of < Z delta_1 (x) k_a, delta_1 (x) k_a >.
+It is the honest (non-normalized) sum of diagonal entries, and it
+stays finite for the operators we construct.
+
+================================================================
+PART 2. THE CONJECTURE AND WHY IT IS HARD
+================================================================
+
+2.1 THE STATEMENT
+
+Irving Kaplansky asked, and Richard Kadison recorded in 1955, the
+following question.
+
+ THE KADISON-KAPLANSKY PROJECTION CONJECTURE.
+ Let G be a torsion-free discrete group. Then the only
+ projections in C_r*(G) are 0 and 1.
+
+Equivalently: if p in C_r*(G) satisfies p = p* = p^2, then p = 0
+or p = 1. Since tau(p) lies in [0,1] and tau(p) = 0 only for
+p = 0, the conjecture is closely tied to a trace quantisation
+statement: for a projection p, tau(p) can only be 0 or 1.
+
+There is also a matrix version, which says: for every N and every
+projection p in M_N(C_r*(G)), the trace tau_{G,N}(p) is an
+integer. (Integer, not merely 0 or 1, because the unnormalized
+matrix trace can be any integer in {0,1,...,N}.
+
+The scalar and matrix versions are equivalent, essentially
+because an N-by-N matrix projection can be "cut down" by a
+full matrix unit to a scalar projection of the same trace in a
+slightly larger algebra. That trick (or a variant) is Step 1.
+
+2.2 WHY TORSION-FREE MATTERS
+
+If G has an element g of finite order k, then
+
+ p = (1/k) ( 1 + g + g^2 + ... + g^(k-1) )
+
+is a projection in C[G] itself (so certainly in C_r*(G)). Its
+trace is 1/k, and it is neither 0 nor 1 whenever k >= 2. So the
+conjecture cannot be true for groups with torsion. Torsion-free
+is a necessary hypothesis.
+
+2.3 WHY IT IS HARD
+
+Here is the difficulty in one sentence: an element of C_r*(G) is
+an infinite operator sum, and it is not obvious, for any
+particular infinite sum, whether it is a projection.
+
+Two routes have been tried:
+
+ROUTE 1: K-THEORY AND ASSEMBLY.
+The algebraic K_0 group K_0(C_r*(G)) is generated by traces of
+projections, modulo relations. If we can show that the trace map
+
+ tau : K_0(C_r*(G)) -> R
+
+takes only integer values, then any projection has an integer
+trace, hence trace 0 or 1, hence is 0 or 1 by faithfulness. So
+conjecture follows from trace-integrality.
+
+Trace-integrality can be attacked through the Baum-Connes
+assembly map
+
+ mu : K_*^{top}(G) -> K_*(C_r^*(G))
+
+which is a homomorphism from a "topological" (geometry of the
+group) K-theory to the operator K-theory. A theorem of Lueck
+(2002), extended by Baum-Guentner-Willett (2016), says: for a
+torsion-free group, the canonical trace tau is integral on the
+IMAGE of the reduced assembly map. So:
+
+ if the assembly map is surjective, the conjecture holds.
+
+Surjectivity (Baum-Connes) is known for sizeable classes:
+
+ - countable amenable groups: Higson-Kasparov (2001)
+ - hyperbolic groups: Mineyev-Yu (2002, without coefficients;
+ Lafforgue 2012, with arbitrary coefficients)
+ - many more: groups acting properly and isometrically on
+ Hilbert space, on trees, etc.
+
+So the conjecture is established for large classes of groups.
+
+ROUTE 2: THE VON NEUMANN ALGEBRA.
+In the larger von Neumann algebra L(G) = weak closure of C_r*(G),
+there is a complete dimension theory: Murray and von Neumann
+showed that projections have traces that can be any real number
+in [0,1]. But C_r*(G) is much smaller than L(G), and its
+projections are rare. Known examples inside L(G), such as
+Austin's irrational-kernel-dimension operators (2013), do not
+immediately give projections inside the smaller C_r*(G).
+
+2.4 WHAT THE NEW PAPER DOES
+
+The paper does NOT try to break the assembly map. It builds an
+explicit projection in a specific group's reduced C*-algebra and
+computes its trace to be a specific non-integer (between 0 and
+1/2). That obviously violates the conjecture, so no assembly
+argument can be true for this group: the group must be outside
+all known covering classes, and its assembly map must not be
+surjective on K_0.
+
+For the same group, a companion paper shows that the complex
+group ring C[G_proj] has no scalar idempotents other than 0 and
+1. So the projection e we build lies in the C*-completion but
+NOT in the algebraic group ring; the gap between the two is where
+the answer hides.
+
+================================================================
+PART 3. THE THEOREM
+================================================================
+
+THEOREM.
+There exist a finitely generated, torsion-free discrete group
+G_proj and a projection e in C_r*(G_proj) such that
+
+ 0 < tau_{G_proj}(e) < 1/2.
+
+In particular e is neither 0 nor 1, and the Kadison-Kaplansky
+projection conjecture is false.
+
+That is the entire result. Every one of the remaining pages is
+about producing that pair (G_proj, e).
+
+================================================================
+PART 4. THE PLAN OF THE PROOF
+================================================================
+
+The proof is broken into six steps. Each step gets a Part below.
+The steps are logically ordered by dependency, not by how the
+paper presents them.
+
+STEP 1. Scalar reduction. Show that if you can produce a
+torsion-free finitely generated group Gamma and a MATRIX
+projection p_0 in M_N(C_r*(Gamma)) with
+
+ 0 < tau_{Gamma,N}(p_0) < 1/2,
+
+then G_proj = Gamma * F_N and a free-product compression formula
+give you a SCALAR projection e in C_r*(G_proj) with the same
+trace. So Step 1 reduces the theorem to producing p_0.
+
+STEP 2. The expander tower. Build an infinite bipartite tower
+of finite layers A_i and B_i (i in Z), of growing sizes, with
+uniformly expanding incidence operators between adjacent layers.
+Two variants Y^0 and Y^# exist, one tuned to expansion, the other
+to unique recovery of long paths from their labels; after
+deleting a proportionally tiny set of vertices, they agree on
+one common graph Y, which has all the good properties at once.
+
+STEP 3. The deck group and the small trace. Build a torsion-
+free nilpotent group F whose arithmetic is rich enough to (a)
+detect short closed walks, and (b) carry small-trace projections
+in its twisted group algebras. The projections come from a
+central Fourier transform: parameter t on the circle, a
+projection p(t) with trace roughly t^n (delta - t)^n, so that
+t -> 0 or t -> delta shrinks the trace.
+
+STEP 4. Realization: choosing voltages and matrices on Y. Put
+F-valued voltages on the edges of Y and simultaneously put finite
+complex matrices W_e on the edges. Two properties must hold at
+once:
+
+ (a) the voltage cover P of Y has no short nonbacktracking
+ closed walks ("girth" hypothesis);
+
+ (b) the resulting regular-representation operator T_P on the
+ cover is within any fixed eta of the model operator
+ Pi O Pi built out of the tower and the projections p(t).
+
+Both properties are proved by finite probabilistic tests.
+
+STEP 5. The geometry and the group G. Given a cover P
+satisfying the girth hypothesis, define
+
+ Gamma = < S | label words of all closed walks in P >,
+
+a finitely generated group presented graphically. Show that
+every component of P embeds in the Cayley graph of Gamma
+(injectivity), that overlaps between different copies are tiny
+tree balls, that the leftover Cayley edges form a forest, and
+that Gamma is torsion-free.
+
+STEP 6. Transfer to the Cayley graph and trace bound. A
+compression + perturbation argument upgrades the approximation
+in Step 4 to a separated band in the actual operator T on the
+Cayley graph of Gamma. The spectral projection p_0 of that band
+sits in M_N(C_r*(Gamma)), is nonzero, and its unnormalized trace
+is bounded by a small weighted sum, which tends to zero.
+
+Then concatenate: Step 2 + Step 3 + Step 4 deliver the cover P;
+Step 5 turns P into Gamma; Step 6 gives p_0; Step 1 turns p_0
+into e over G_proj = Gamma * F_N. Done.
+
+ A note on DEPENDENCY ORDER. In the actual writing, one fixes
+ constants in the order: geometry constants C_* and C_bullet;
+ the nilpotence data m, E_0, A_*, kappa; the operator accuracy
+ eta and finite lists; two finite-moment constants C_1, C_2;
+ and finally ell large enough that all the "for sufficiently
+ large ell" statements hold. Nothing circular happens; each
+ part fixes its constants before the next begins. A parameter
+ ledger in Part 6 spells this out.
+
+================================================================
+PART 5. STEP 1: FREE PRODUCTS AND SMALL MATRIX TRACES
+================================================================
+
+5.1 THE REDUCTION (WHAT WE MUST PRODUCE)
+
+Target (REVISED). Produce a finitely generated, torsion-free
+Gamma and a projection p_0 in M_N(C_r*(Gamma)) for some integer
+N, with 0 < tau_{Gamma,N}(p_0) < 1/2.
+
+If we can do that, this section converts it into the actual
+theorem. If we cannot, we have nothing.
+
+5.2 FREE PRODUCTS
+
+Given groups Lambda and H, the free product Lambda * H is the
+group of finite alternating products l_1 h_1 l_2 h_2 ... with
+each l_i in Lambda and h_i in H, reduced by cancelling adjacent
+pairs (l, l^-1) or (h, h^-1). Intuition: Lambda and H placed
+side by side, sharing only the identity, and any word in them
+gets freely reduced by cancelling adjacent inverses.
+
+The free group F_N on N letters t_1,...,t_N is the free product
+of N copies of Z. Every element of K = Lambda * F_N has a
+UNIQUE reduced form
+
+ gamma v, gamma in Lambda, v a reduced F_N-word.
+
+The set R consists of the identity of K, together with all
+reduced NORMAL WORDS of K whose FIRST factor lies in F_N. Every
+element k of K has a UNIQUE expression k = gamma v with gamma in
+Lambda and v in R. Thus R is a transversal for LEFT Lambda-
+orbits in K. (This is the free-product normal form; the
+first-letter condition is what makes the expression unique.)
+
+5.3 SUBGROUP INCLUSION PRESERVES THE TRACE
+
+LEMMA. If Lambda <= Gamma is a subgroup, then C_r*(Lambda) sits
+inside C_r*(Gamma) as a unital sub-C*-algebra; tau_Gamma extends
+tau_Lambda; and the same holds for finite matrix algebras.
+
+Proof sketch. Each left orbit Lambda g in Gamma is a copy of
+Lambda, and l^2(Lambda g) is a direct summand of l^2(Gamma) that
+the Lambda-action sees as a left regular representation. So
+the left regular rep of Gamma restricts on that orbit to a copy
+of the left regular rep of Lambda. The operator norm of any
+x in C[Lambda] is therefore the same on l^2(Lambda) as on
+l^2(Gamma). The identity coefficient is also preserved.
+
+CONSEQUENCE. If p is a projection in M_N(C_r*(Lambda)), its
+image in M_N(C_r*(K)) is also a projection and has the same
+matrix trace.
+
+5.4 THE COMPRESSION FORMULA
+
+Let p in M_N(C_r*(Lambda)) be a projection with
+
+ a = tau_{Lambda,N}(p), 0 < a < 1/2.
+
+Regard p as a matrix over C_r*(K), K = Lambda * F_N. Let
+
+ U = ( lambda_K(t_1), ..., lambda_K(t_N) )
+
+be a 1-by-N row of the N free generators' left-translation
+unitaries. We look at two products of U with U*.
+
+ U U* = sum over j of lambda_K(t_j) lambda_K(t_j)* = N,
+
+because each t_j on the identity contributes the identity, and
+the N contributions sum to N. So ||U||^2 = N and the operator
+inequality A = p U*U p <= N p holds immediately.
+
+ But A is NOT equal to N p. U* U is an N-by-N matrix over
+C_r*(K) whose (i,j) entry is lambda_K(t_i^{-1} t_j). Its
+diagonal (i = j) is all ones. Its off-diagonal entries (i != j)
+are the DISTINCT non-identity words t_i^{-1} t_j in F_N; no two
+different pairs (i,j) give the same word. So U* U is far from
+a scalar, and it is the free-product structure inside U* U that
+produces the constant 1/2 below.
+
+ Define
+
+ A = p U* U p (an element of p M_N(C_r*(K)) p).
+
+ Then A / N is a positive contraction in the corner, bounded
+ below by c_a^2 / N once section 5.5 is done.
+
+MAIN CLAIM.
+
+ (i) A is invertible in the corner p M_N(C_r*(K)) p.
+ (ii) e = (U p) A^{-1} (p U*) is a projection IN C_r*(K),
+ with tau_K(e) = a.
+
+5.5 WHERE 1/2 COMES FROM: THE LOWER BOUND
+
+The invertibility of A is equivalent to: U is bounded below on
+the subspace p l^2(K)^N. Concretely we will prove
+
+ || U xi || >= c_a ||xi|| for all xi in p l^2(K)^N,
+
+with
+
+ c_a = sqrt(1-a) - sqrt(a),
+
+which is strictly positive exactly when a < 1/2. This is the
+only place 1/2 enters the whole story.
+
+Proof of the lower bound. Use the free-product normal form.
+Write xi in p l^2(K)^N with coordinate functions xi_1,...,xi_N,
+and let
+
+ m_v = || xi restricted to Lambda v ||^2, sum_v m_v = ||xi||^2.
+
+On the orbit Lambda v, the projection p has, by Cauchy-Schwarz
+within that orbit,
+
+ |xi_j(v)|^2 <= a_j m_v, a_j = tau_Lambda(p_jj),
+
+because the "column" p delta_{(v,j)} has squared norm exactly
+a_j = <p delta_{(v,j)}, delta_{(v,j)}>, and Cauchy-Schwarz on
+the orbit gives |xi_j(v)|^2 = |<xi, p delta_{(v,j)}>|^2 <=
+a_j * m_v.
+
+GOOD AND BAD. Call v GOOD in coordinate j if the reduced word
+of v does not begin with t_j^{-1}, BAD otherwise. Write
+xi = xi_g + xi_b.
+
+BAD INPUT. Each bad position determines its bad coordinate j
+from the first letter t_j^{-1}. Summing the Cauchy-Schwarz
+bound over bad (v,j):
+
+ || xi_b ||^2 <= a || xi ||^2.
+
+GOOD OUTPUT. Left-multiplication by t_j maps a good v to a word
+starting with t_j. Outputs for different j are disjoint, and
+multiplication within one coordinate is injective. Therefore
+
+ || U xi_g ||^2 = || xi_g ||^2 >= (1 - a) || xi ||^2.
+
+BAD OUTPUT. Different bad inputs, coming from different j's,
+can collide on the same output word h. Let J(h) be the set of
+coordinates j for which v_{h,j} = t_j^{-1} h is bad. Then
+|(U xi_b)(h)|^2 is bounded by
+
+ (sum_{j in J(h)} a_j) (sum_{j in J(h)} m_{v_{h,j}})
+ <= a sum_{j in J(h)} m_{v_{h,j}},
+
+using Cauchy-Schwarz and sum_j a_j = a. The map (h,j) -> v_{h,j}
+is injective on bad inputs (first letter determines j, and
+h = t_j v_{h,j}), so summing over h gives
+
+ || U xi_b ||^2 <= a || xi ||^2.
+
+ASSEMBLE. Triangle:
+
+ || U xi || = || U xi_g + U xi_b ||
+ >= sqrt(1-a) ||xi|| - sqrt(a) ||xi||
+ = (sqrt(1-a) - sqrt(a)) ||xi||.
+
+That is the claimed c_a bound, and c_a > 0 iff a < 1/2.
+
+5.6 FINISHING STEP 1
+
+From the c_a bound, A = p U*U p >= c_a^2 p in the corner. Also
+U U* = N, so A <= N p. Hence A / N has spectrum in [c_a^2/N, 1]
+on the range of p, and p - A/N has norm <= 1 - c_a^2/N < 1.
+The geometric series
+
+ A^{-1} = (1/N) sum_{k = 0 to infinity} (p - A/N)^k
+
+converges in the corner p M_N(C_r*(K)) p. Continuous functional
+calculus also gives A^{-1/2}.
+
+Set V = U p A^{-1/2} (a 1-by-N row). Then
+
+ V* V = A^{-1/2} p U* U p A^{-1/2} = p,
+
+so e = V V* is a projection in C_r*(K). Writing V = (V_1,...,V_N)
+and using traciality,
+
+ tau_K(e) = sum_j tau_K(V_j V_j*)
+ = sum_j tau_K(V_j* V_j)
+ = tau_{K,N}(p) = a.
+
+So e is a scalar projection of trace a, which lies in (0,1/2).
+If Lambda is torsion-free and finitely generated, so is K
+(adjoining finitely many t_j preserves both).
+
+5.7 SO WHAT STEP 1 BOUGHT US
+
+We traded "small MATRIX trace over Lambda" for "small SCALAR
+trace over Lambda * F_N". Since the scalar form is the actual
+conjecture, Step 1 has reduced everything to:
+
+ BUILD Gamma finitely generated and torsion-free,
+ BUILD p_0 in M_N(C_r*(Gamma)) a projection with
+ 0 < tau_{Gamma,N}(p_0) < 1/2.
+
+That is the rest of the proof. Everything after this is the
+construction of Gamma and p_0.
+
+================================================================
+PART 6. STEP 2: THE EXPANDER TOWER
+================================================================
+
+The next ingredient is not analytic at all: it is an infinite
+bipartite graph, built in layers, whose incidence operators
+between adjacent layers are expanders with a UNIFORM spectral gap.
+We want three things at once:
+
+ (1) expansion, so that the model operator on the tower has
+ a spectral gap;
+ (2) folded labels, so that any nonbacktracking path of
+ length at least L_i is uniquely determined by its word;
+ (3) no short cycles after pruning, so that the eventual
+ voltage cover can have big girth.
+
+Points (1) and (2) pull in different directions. The trick is
+to build two variants Y^0 and Y^# on the SAME vertex set with
+the SAME labels, get (1) from Y^0 and (2) from Y^#, and delete a
+proportionally tiny set of vertices so that the induced graphs
+coincide -- the survivor Y has both.
+
+6.1 PARAMETERS
+
+Fix a positive integer ell, eventually large. Let
+
+ d = 2^(4 ell),
+ D = d^2 = 2^(8 ell),
+ s = 10 ell,
+ r = 7 ell,
+ n_0 = 1000,
+ H_0 = (2 n_0 + 1) s + 1 >~ 20010 ell,
+ H_i = H_0 + |i| (i in Z).
+
+The two pieces of d we use: 2^r / 2^s = 2^(7 ell) / 2^(10 ell) =
+2^(-3 ell) = d^(-3/4), and 2*s - r = log_2 d = 4 ell. These
+identities make the degrees below balance.
+
+6.2 BIT STREAMS AND THE VERTEX SETS
+
+Every vertex will be a pair of binary strings. For h >= 0 write
+h = 2 s q + t with 0 <= t < 2 s, and set
+
+ l_A(h) = r + n_0 s + q s + max(t - s, 0),
+ l_B(h) = n_0 s + q s + min(t, s).
+
+Then l_A(h) + l_B(h) = r + 2 n_0 s + h, and l_A(h) - l_B(h)
+-r grows by 1 in one coordinate, then 1 in the other, alternating.
+Note l_A(i) + l_B(i) grows with |i|.
+
+For each i in Z, define
+
+ A_i = F_2^(l_A(i)) x F_2^(r + 1 + l_B(i)),
+ B_i = F_2^(l_B(i)) x F_2^(s - r + 1 + l_A(i)).
+
+A vertex is a pair (u,v) of bit strings of the displayed
+lengths. The string u is called the INPUT stream, and v the
+OUTPUT stream. A quick computation gives the layer sizes
+
+ |A_i| = d * 2^(H_i),
+ |B_i| = 2^(H_i).
+
+So each A-layer is d times as big as the B-layer at the same
+index, and both grow like 2^|i| in the distance from 0.
+
+The A-stream is divided into an initial block of length r (call
+it block 0), followed by blocks of length s (block 1, 2, 3, ...).
+Each B-stream is divided into blocks of length s (block 1, 2, ...).
+Order the blocks from the two sides in an interleaved sequence
+
+ A,0; B,1; A,1; B,2; A,2; B,3; ...
+
+For every |i-k| <= 1, the active portions of u_A at a vertex of
+A_i and u_B at a vertex of B_k form an INITIAL SEGMENT of this
+interleaving: at most one block on each side gets "cut off" part
+way through.
+
+6.3 SLOTS AND EDGES
+
+Fix a pair (i,k) with |i-k| <= 1. On each A_i vertex there are d
+SLOTS, indexed by a = (k - i, j) with 1 <= j <= d. On each B_k
+vertex there are D_{ik} slots, indexed by c = (i - k, j, b) where
+b in F_2^(p - p') for two small integers p, p' defined by
+p = r + 1 + l_B(i) - l_B(k), p' = s - r + 1 + l_A(k) - l_A(i).
+Direct computation gives
+
+ p - p' = 2 s - r + |i| - |k|
+ = log_2 d + |i| - |k|,
+ D_{ik} = d * 2^(p - p')
+ = D * 2^(|i| - |k|), D/2 <= D_{ik} <= 2 D.
+
+At a vertex of A_i, counting all three adjacent pairs (k = i-1,
+i, i+1), we get 3 d slots total. At a vertex of B_k, counting
+the same three pairs, at most 6 D.
+
+An edge over the pair (i,k) is a triple (a, c, x) where x is a
+bit string in F_2^p. The A-endpoint is (vertex a) and the B
+endpoint is (vertex c); the slot pairs are matched by requiring
+the same x on both ends. This defines the bipartite graph
+between A_i and B_k.
+
+6.4 LINEAR EVALUATION AND TRIANGULAR RECOVERY
+
+On each slot we define a linear map K from padded input streams
+to padded output streams. On a full block of length s (or on
+block 0 of length r) K acts by multiplication by an element of
+the finite field F_{2^h}, after choosing an identification of
+F_2^h with F_{2^h}. Additionally, K may send a linear image of
+one block into the FIRST ONE OR TWO bits of the FOLLOWING block
+on its own side. Nothing else is coupled.
+
+The slot EVALUATION is
+
+ Q = v + K u,
+
+a bit string in the output stream, of a specific padded length.
+Both the A-end of an edge and the B-end produce such an
+evaluation, and they must agree on the label x of the edge (with
+an overlap that we have arranged by the lengths above).
+
+RECOVERY. Suppose you are at an A-vertex with slot a. You know
+u_A (input on your side), and you can compute the linear map
+K_a u_A. Subtracting from v_A gives v_A - K_a u_A =: Q_A. The
+edge label x is precisely the first p bits of Q_A; the slot c at
+the other end is determined by the "bucket" data in x. This
+determines the other endpoint's input u_B through the triangular
+system, because the couplings go forward in the block ordering:
+to read block N on one side you only need blocks <= N-1 on the
+other. Once u_B is known, v_B = Q_B + K_c u_B.
+
+The same works from the B-vertex to the A-vertex.
+
+For this to define a genuine bijection between ends, we need the
+diagonal coefficients on distinct slots at a vertex to be
+distinct (so the algebra is invertible), and the "spill" terms
+to be small (so blocks are only coupled to their immediate
+neighbors). Both are arranged by choosing the coefficient maps
+injectively; there are 3d slot names on the A side and at most
+(21/2)D slot names on the B side, both smaller than 2^r and 2^s
+respectively for large ell, so injections into the appropriate
+fields exist.
+
+6.5 UNIQUE PATH RECOVERY
+
+FOLDED. The labeling is folded: at every vertex, the labels on
+the outgoing oriented edges are distinct. This follows from the
+slot structure; different slots differ in one of (offset, serial
+number, bucket, x).
+
+UNIQUE LONG PATHS. If two nonbacktracking paths of length at
+least L_i end or begin at a vertex in layer i, and they read the
+same oriented word in a fixed finite alphabet, they coincide as
+paths. Reason: the triangular equations propagate bit by bit,
+and the block recovery above covers the first half of the path;
+the gated structure past the end lets the remaining bits be read
+off.
+
+6.6 THE TWO GRAPHS Y^0 AND Y^#.
+
+Y^#. One edge label carries, in addition to its slot data, an
+optional "gate" state on the LAST block of the current slot. The
+gate condition is a simple XOR of the four immediately preceding
+full s-blocks of input. When the gate is OPEN, we apply a
+permutation to the last block; when CLOSED, we do nothing. Two
+choices of last-block permutations give the label. In Y^#, this
+gate is used at every edge, so later blocks (in the interleaved
+order) cannot accidentally be read as if they belonged to a
+later layer: the gate isolates the current block from the future
+contents.
+
+Y^0. Same edges and slots, but the last-block permutation is
+applied unconditionally. This gives cleaner analysis of the
+incidences: each block is just a random biregular matching.
+
+Both Y^0 and Y^# have the same vertex set and the same label set.
+
+THE EXPANSION LEMMA. Fix Y^0 with its count-normalized closure.
+For every pair (i,k) with |i-k| <= 1, the incidence between A_i
+and B_k has degrees d and D_{ik}, maps normalized constants to
+normalized constants (and adjoints likewise), and on the
+orthogonal complements acts with norm at most 1 - chi where
+
+ chi = 1/800
+
+is an absolute constant. This is the same constant later used
+for the spectral gap of the eventual Cayley operator.
+
+How is such a gap obtained? Each block is built from independent
+clusters (one per choice of common key), and each cluster is a
+uniform matching between two sets of size 2^t. A matching that
+is well chosen has an incidence matrix whose action on constant-
+orthogonal spaces shrinks by a definite factor. A local-lemma
+argument (the same Erdos-Lovasz style argument as in the usual
+expander constructions) shows that the last-block permutations
+of Y^0 can be fixed so that ALL blocks, at once, obey the gap.
+The gap 1/800 is fixed for concreteness; any positive constant
+below 1 would give a slightly different final trace bound.
+
+6.7 PRUNING: DELETING BAD VERTICES
+
+Y^0 and Y^# do not literally agree edge-by-edge. After deleting
+a proportionally tiny set of vertices, they do.
+
+GATE DELETIONS. Delete every vertex in Y^# where some gate is
+CLOSED at the point where Y^0 would set its slot. By independence
+of the current slot's bits from the preceding inputs, each
+specific pair of gated s-blocks being all-zero has probability
+2^{-4s}; a union bound over at most 6D slots per vertex gives a
+deleted fraction at most
+
+ eps_gate = 6 D * 2^(-4 s) = D / 2^(32 ell).
+
+SEPARATION DELETIONS. Delete every vertex whose own u-bits
+immediately after an EARLIER cutoff (a smaller layer, but the
+same sign and residue of |i| mod K, where K = 32 s) are all zero.
+The probability of this at the smaller layer is
+2^{-min(H_{i'}, mK/2)} with m = (|i| - |i'|)/K, and the union
+bound over all such i' gives a deleted fraction at most
+
+ eps_sep <= 2^(-K/2) / (1 - 2^(-K/2)) + 3 * 2^(-H_0).
+
+This is what prevents the zero padding of a smaller-layer vertex
+from being read as if it were the same layer.
+
+SHORT-CYCLE DELETIONS. Delete every vertex supporting a
+nonempty cyclically nonbacktracking closed path of length at
+most g_i = H_i / (100 log_2 d). A counting argument (each cyclic
+word corresponds to at most one such path, by unique-path
+recovery) bounds the deleted fraction by 2^{-0.9 H_0}.
+
+Total deleted fraction
+
+ eps_ell = eps_gate + eps_sep + 2^(-0.9 H_0) -> 0.
+
+6.8 WHAT SURVIVES
+
+Let Y be the induced subgraph on the vertices surviving all three
+deletions. It is an induced subgraph of BOTH Y^0 and Y^#, so it
+has expansion AND recovery simultaneously. Properties:
+
+ (a) For every |i-k| <= 1, the induced Y-block between A_i
+ and B_k still has degree d and D_{ik} on its sides, has
+ norm <= 1 - chi on the complements of constants, and
+ maps constants to constants.
+
+ (b) The fraction removed from each A_i and B_i is <= eps_ell.
+
+ (c) Labels are folded; positive alphabet size m_ell is
+ O( d 2^r (1+s)^2 ) = O( d^(11/4) (1+s)^2 ), so
+ log_2(2 m_ell) = 11 ell + O(log ell).
+
+ (d) Every nonbacktracking path beginning at layer i of length
+ at least L_i = 100 H_i / log_2 d is uniquely determined
+ by its oriented word.
+
+ (e) No nonempty cyclically nonbacktracking closed path in Y
+ has length at most g_i.
+
+This Y is the combinatorial substrate for the next six parts.
+The A- and B-layers will be placed on a cover, then lifted to
+the Cayley graph of a group, and only the survivor Y matters from
+here on.
+
+================================================================
+PART 7. STEP 3: THE DECK GROUP AND THE SMALL TRACE
+================================================================
+
+7.1 WHAT THE DECK GROUP MUST DO
+
+The deck group F, whose elements will become the voltages on the
+edges of Y, has two jobs.
+
+ (a) DETECT short closed walks. Given a short cyclically
+ nonbacktracking closed walk in the cover, the product of
+ the voltages along its edges must be nontrivial in F.
+ That is what makes the cover have big girth.
+
+ (b) CARRY small-trace projections. Its twisted group
+ algebras must contain projections p_theta whose
+ unnormalized trace is tiny when theta is small.
+
+These pull in opposite directions: (a) wants F rich, (b) wants
+F shrinkable by a twist. A nilpotent group built from upper
+unitriangular matrices delivers both at once.
+
+7.2 THE WORD DETECTOR
+
+Fix an integer m >= 2 (chosen later) and let U_Z be the group of
+upper unitriangular m-by-m integer matrices (upper triangular,
+diagonal entries 1). U_Z is finitely generated, torsion-free,
+nilpotent.
+
+LEMMA (word detector). Let Gamma be a unit-edge graph with
+girth greater than g, and let W be a nonempty, cyclically
+nonbacktracking closed walk of length at most C g. Assign the
+symbol 1 to some edges of the support of W, and generic upper
+unitriangular matrices to the rest (with inverses on reversed
+edges). Then the resulting word in U_Z is NOT the identity
+matrix as long as
+
+ m > L(C), L(C) = ceil(3 C B(C)),
+ B(C) = max(1, ceil(20 C))^2.
+
+Sketch of why. The support of W has first Betti number at most
+B(C). Choose a spanning tree of the support and call the
+complement edges PIVOTS. Collapsing the tree leaves a bouquet
+of at most B(C) circles; W crosses each pivot at most
+3 C B(C) = L(C) times. In the free group of the bouquet, W has
+a unique reduced form. Magnus's expansion into noncommutative
+power series gives a nonzero monomial x_{j_1} ... x_{j_k} for
+that reduced form. Substituting x_j = 1 + E_{j, j+1} into U_Z,
+the (1, k+1) entry of the resulting matrix picks out exactly
+that monomial. So the matrix is nonzero as long as k < m.
+
+For us the geometric constant C_* in Part 9 is an absolute
+number. Fix
+
+ m = an integer strictly greater than L(201 C_*).
+
+Then any closed walk in Y with length at most C_* H_i / log_2 d
+is detected by U_Z, uniformly in i.
+
+7.3 THE NILPOTENT GROUP F
+
+Let n be the real vector space of strictly upper triangular
+m-by-m matrices. It has dimension n_f = m(m-1)/2 and is a
+nilpotent Lie algebra. Write U = 1 + n for its exponential
+group (upper unitriangular matrices with real entries and
+diagonal 1).
+
+Build an affine semidirect product E on n. The exact laws are:
+
+ (lambda, b, z) * (mu, b', w)
+ = (lambda + mu . L_{b^-1}, b b',
+ z + w + mu(b^-1 - 1)),
+
+with lambda a linear functional on n (coordinates lambda_ij),
+b in U, and z real. Constants (0, 1, z) are central. Divide
+out the subgroup (0, 1, k) for k in Z to get a quotient bar-E
+with coordinates (lambda, b).
+
+Restrict to INTEGER coordinates: E_Z and bar-E_Z are the
+sublattices where all coordinates of (lambda, b^-1, z) are
+integers. Then define the DECK GROUP
+
+ F = (E_Z x E_Z) / { ((0, 1, k), (0, 1, -k)) : k in Z }.
+
+Equivalently, F has coordinates (gamma_X, gamma_Y, z) with
+gamma_X, gamma_Y in bar-E_Z and z in Z, and multiplication
+
+ (gamma_X, gamma_Y, z) * (eta_X, eta_Y, w)
+ = (gamma_X eta_X, gamma_Y eta_Y,
+ z + w + c(gamma_X, eta_X) + c(gamma_Y, eta_Y)),
+
+where c is the cocycle measuring the failure of bar-E to be a
+group extension. That is our F.
+
+FACTS. F is countable, nilpotent, torsion-free, and finitely
+generated. The integer coordinate z is its CENTRAL coordinate.
+Projection to the upper unitriangular part of EITHER bar-E_Z
+factor is a surjective group homomorphism onto U_Z. That is
+what lets F detect short words (7.2).
+
+7.4 CENTRAL FOURIER DECOMPOSITION
+
+Conjugate by inversion to get right translations, then Fourier-
+transform the central coordinate z: the character z -> e(t z),
+with t in the circle T = R/Z. We get
+
+ l^2(F) = L^2( T ; l^2(bar-E_Z) (x) l^2(bar-E_Z) ).
+
+Under this decomposition, a section lift sigma(gamma) acts on the
+second bar-E_Z factor with a twisted phase e(t c(...)) depending
+on t. The central generator z_0 = (1, 1, 1) acts by the scalar
+e(t). We write u_t for the twisted left action on the first
+factor and v_t for the twisted right action on the second. They
+commute and are twisted-unitary.
+
+7.5 SCHWARTZ IDEMPOTENTS ON THE CONTINUOUS GROUP
+
+Work first with the continuous bar-E (not bar-E_Z).
+
+Twisted convolution of Schwartz functions
+
+ (J *_epsilon K)(h)
+ = integral over bar-E of J(a) K(a^-1 h)
+ e(epsilon c(a, a^-1 h)) da,
+
+with twisted involution
+
+ J^{*_epsilon}(h) = conj(J(h^-1)) e(-epsilon c(h^-1, h)),
+
+makes Schwartz functions bar-E into a *-algebra. Its twisted
+representation
+
+ (pi_epsilon(lambda, b) phi)(x) = e(epsilon lambda(x)) phi(A_{b^-1} x)
+
+on L^2(n) is unitary, and integrating a Schwartz function
+against it gives a bounded operator.
+
+THE KEY CORRESPONDENCE. SCHWARTZ COEFFICIENT FUNCTIONS on
+bar-E are in bijection with SCHWARTZ INTEGRAL KERNELS on n, via
+Fourier transform in the first (lambda) variable. The kernel
+of pi_epsilon(J) is
+
+ K_J(x, y) = (Fourier in lambda of J)(epsilon x, (1+x)(1+y)^-1).
+
+The map J -> K_J is a bijection on Schwartz spaces, an L^2
+isometry (with the correct normalization), and preserves twisted
+convolution.
+
+Therefore a rank-one Schwartz kernel, such as the Gaussian
+projection phig(x) = 2^{n_f/4} exp(-pi |x|^2) of L^2-norm 1,
+has a unique preimage
+
+ J_epsilon in Schwartz(bar-E), with
+
+ J_epsilon * J_epsilon = J_epsilon,
+ J_epsilon^{*_epsilon} = J_epsilon,
+ ||J_epsilon||_{L^2(bar-E)} = 1.
+
+It is a genuine Schwartz idempotent.
+
+7.6 DISCRETIZING: RIEMANN SUMS ON A LATTICE
+
+We need an ACTUAL element of a C*-algebra, not a continuous
+object. Sample J_epsilon on Delta_y bar-E_Z.
+
+Let theta be a small nonzero real, epsilon = sign theta,
+y = |theta|^{1/m}, and
+
+ nu = y^{m n_f} = |theta|^{n_f},
+ K_theta = sum over gamma in bar-E_Z of
+ nu J_epsilon(Delta_y gamma) u_theta(gamma).
+
+Riemann-sum estimates for the WEIGHTED rectangular lattice
+Delta_y bar-E_Z (cell volume nu) give:
+
+ ||K_theta||_{coeff,1} <= C,
+ ||K_theta||_{coeff,2} ~ sqrt(nu),
+ ||K_theta^2 - K_theta||_{coeff,1} <= C y,
+ ||K_theta^2 - K_theta||_{coeff,2} <= C y sqrt(nu).
+
+The coeff-1 norm controls the operator norm, so the spectrum of
+K_theta lies within C y of {0, 1}. Functional calculus supplies
+a genuine projection p_theta with
+
+ || p_theta - K_theta || <= C y,
+ 0 < tr_theta(p_theta) <= C |theta|^{n_f}.
+
+The lower bound 0 < is essential: if p_theta were 0, then
+||K_theta|| <= 1/4, but the identity-coefficient of K_theta is
+O(sqrt(nu)) = O(|theta|^{n_f/2}), strictly positive for
+theta != 0.
+
+7.7 THE MAGNETIC FACTOR: TWO-COORDINATE TWIST
+
+One twist gives small trace as theta -> 0. We want small trace
+at both ends of a window 0 < t < delta. Add a FINITE
+representation R of bar-E_Z on C^N whose central character
+carries a rational shift by delta:
+
+ q = d^{A_*}, delta = q^{-1}, N = q^{n_f}.
+
+R acts on l^2 of the upper unitriangular matrices mod q:
+
+ (R(lambda, b, z) psi)(x)
+ = e( -(z + lambda(x))/q ) psi( b^-1(1+x) - 1 ).
+
+Its central character is R(0, 1, z) = e(-delta z) 1_N.
+
+Let V_t = u_t (x) R be the tensor of the twisted left action
+with R. A diagonal unitary W_t converts V_t into
+u_{t-delta} (x) 1_N:
+
+ W_t^* V_t(gamma) W_t = u_{t-delta}(gamma) (x) 1_N.
+
+This is twisted absorption: the finite factor R absorbs the
+delta shift of the twisted multiplier. Crucially, W_t fixes the
+identity-block, so the identity trace of every projection is
+multiplied by exactly N.
+
+Now, for 0 < t < delta, BOTH t and t - delta fall in the small-
+theta regime of 7.6. Define
+
+ p(t) = p_t (x) p_{t-delta}[V_t],
+
+an equivariant projection. Its identity trace factorizes:
+
+ s_p(t) = N * tr_t(p_t) * tr_{t-delta}(p_{t-delta})
+ <= C N t^{n_f} (delta - t)^{n_f}.
+
+This is small at BOTH ends of the window, because t -> 0 kills
+one factor and t -> delta kills the other.
+
+7.8 THE MOVING WINDOW
+
+The layers on which the model acts are chosen by a smooth bump h.
+Fix a smooth nonnegative h, supported in (-1, 1) and positive on
+[-0.7, 0.7], with
+
+ sum over i of h(v - i)^2 = 1 for every real v.
+
+For 0 < t < delta define
+
+ b(t) = (1/kappa) log_2( t / (delta - t) ),
+ h_i(t) = h(b(t) - i).
+
+The ACTIVE SET J(t) = { i : h_i(t) != 0 } has at most two
+elements, and they are adjacent when there are two. As t runs
+from 0 to delta, b(t) runs monotonically from -infinity to
++infinity, so the active layers SWEEP from -infinity to
++infinity. That is the "moving band".
+
+LAYER WEIGHTS. Let
+
+ w_i = q^{-1} 2^{-kappa |i|}.
+
+On the support of h_i(t):
+
+ t is comparable to q 2^{kappa b} / (1 + 2^{kappa b}),
+ delta-t is comparable to q / (1 + 2^{kappa b}),
+
+so
+
+ t(delta - t) <= 2^kappa q^{-2} 2^{-kappa |i|}.
+
+Multiply by N = q^{n_f} and use the trace bound 7.7:
+
+ w_i s_p(t) <= C 2^{kappa n_f} q^{-n_f} 2^{C_bullet H_i}
+ 2^{-kappa n_f |i|}.
+
+Now substitute H_i = H_0 + |i|, the bound H_0 <= h_* log_2 d =
+4 ell h_*, and q^{n_f} = d^{A_* n_f}. The result is
+
+ w_i s_p(t)
+ <= C 2^{kappa n_f}
+ d^{-(A_* n_f - C_bullet h_*)}
+ 2^{-(kappa n_f - C_bullet) |i|}.
+
+The parameters are chosen in the order
+
+ E_0 > 10 (C_* + C_bullet + 1),
+ A_* in N, A_*/m > 10 E_0 h_*,
+ kappa > 0, kappa/m > 10 (E_0 + A_* + C_bullet + 1).
+
+These ensure A_* n_f > C_bullet h_* (since n_f >= 1 and
+A_*/m > 10 E_0 h_* with E_0 > 10 C_bullet) and kappa n_f >
+C_bullet. So the two exponents on the right are STRICTLY
+POSITIVE, and
+
+ mu_ell = sup over t and i in J(t) of w_i * s_p(t) -> 0.
+
+That is the "small weighted trace" the whole program will
+convert into p_0 with small trace.
+
+7.9 FIXING PARAMETERS IN ORDER
+
+Constants are now set once and never altered again:
+
+ C_*, C_bullet from the geometry (Part 9);
+ m = an integer > L(201 C_*);
+ E_0, A_*, kappa satisfying the displayed inequalities;
+ theta_0 = small radius for the expansion (7.6);
+ ell = the last free parameter, chosen large.
+
+Nothing circular: each choice depends only on choices made
+earlier.
+
+================================================================
+PART 8. STEP 4: REALIZING THE MODEL ON THE COVER
+================================================================
+
+The combinatorial tower Y exists, the deck group F exists, and the
+small-trace projections p(t) exist in the twisted algebras of F.
+The next move is to assemble these into an ACTUAL self-adjoint
+operator on a voltage cover of Y, with two simultaneous
+properties:
+
+ GIRTH: the cover has no short nonbacktracking closed
+ walks;
+ APPROXIMATION: the cover operator is within eta of the model
+ operator on the tower, in the FULL regular norm.
+
+Once that is done, the geometry of Part 9 will turn the cover
+into a finitely generated group.
+
+8.1 THE MODEL OPERATOR AND ITS BAND
+
+Take the linear space
+
+ H^0 = l^2( union of A_i and B_i, i in Z )
+ (x) l^2(F) (x) C^N.
+
+Let Pi denote orthogonal projection onto the surviving vertices
+Y. For 0 < t < delta and each pair (i,k) with |i-k| <= 1, put
+
+ O(t)_{A_i B_k} = U^0_{ik} (x) h_i(t) h_k(t) p(t),
+
+where U^0_{ik} is the normalized incidence between the layer
+sides in Y^0 (from Part 6), h_i(t) is the layer window (Part 7),
+and p(t) is the small-trace projection (Part 7). Set the
+opposite block to the adjoint and all other blocks to 0.
+
+BAND. Let a_i = |A_i|^{-1/2} 1_{A_i} and b_k = |B_k|^{-1/2} 1_{B_k}
+be the normalized constant vectors. Define an isometry
+
+ V_*(t) zeta
+ = 2^{-1/2} * ( + ( h_i(t) a_i (x) zeta )_i
+ + ( h_i(t) b_i (x) zeta )_i ),
+
+for zeta in the range of p(t). Set E_*(t) = V_*(t) V_*(t)^*.
+Then
+
+ O(t) E_*(t) = E_*(t), (eigenvalue 1)
+ ||O(t)|| <= 1,
+ O(t) on (1 - E_*(t)) H^0_t <= (1 - chi) I,
+ tr_id E_*(t) = tr_id p(t).
+
+Reason: on the diagonal side the U^0_{ik} act as the rank-one
+operator |h_i><h_k|; the symmetric two-sided subspace receives
+U^0_{ik} and its adjoint, giving eigenvalue 1; the antisymmetric
+subspace sits at -1 and does not affect the UPPER trace; and on
+the complement of the constants, the U^0_{ik} shrink by 1 - chi.
+
+COMPRESSION TO Y. Cutting out the deleted vertices in Y
+costs at most sqrt(beta_ell) in norm on E_*(t), where
+beta_ell <= eps_ell -> 0 is the deleted fraction from Part 6.
+So Pi O(t) Pi still has a band near 1.
+
+8.2 WHAT MUST BE REALIZED
+
+Given any fixed eta > 0 and ell large enough, we construct
+
+ voltages z_e in F on the positive edges e of Y,
+ matrices W_e in M_N(C) with ||W_e|| <= C_eta,
+
+so that:
+
+ (G1) Augmenting each original label (of Y) by its matrix
+ weight gives a finite positive alphabet of size o(dD).
+ The label's ORIGINAL part still determines the degree
+ normalizer D_{ik}.
+
+ (G2) The voltage cover P of Y has no nonempty cyclically
+ nonbacktracking closed walk based in layer i of length
+ at most C_* H_i / log_2 d.
+
+ (G3) Let T_P be the cover's incidence operator, with entries
+ from B_k to A_i given by
+
+ (T_P)_{ab} = (d D_{ik})^{-1/2}
+ * sum over edges e : a -> b of
+ W_e * rho(z_e).
+
+ Then ||T_P - Pi O Pi|| <= eta in the full regular
+ norm. Both operators have LAYER WIDTH ONE (only
+ |i-k| <= 1 is nonzero), so their central Fourier
+ fields satisfy the same inequality almost everywhere and
+ ||T_P(t)|| <= eta outside 0 < t < delta.
+
+The point of (G3) is that T_P is a FINITE sum of group
+translations with matrix coefficients; it is manifestly an
+element of M_N(C_r*(Gamma_proj)) once Gamma_proj is defined
+from the cover. So T_P is a real object, not an approximation.
+
+8.3 THE STRATEGY
+
+The construction has five stages:
+
+ (1) FINITE TESTS. Reduce statements about the FULL regular
+ norm to statements about finitely many block-matrices.
+ (2) EMPIRICAL LISTS. Replace expectations by M-term empirical
+ averages that are close in that norm.
+ (3) COMPACT TEMPLATES. Work with compactly supported smooth
+ functions J_epsilon^flat as in Part 7, and use finite
+ log-grid templates of the p(t) window.
+ (4) TWO-BLOCK CASES. For the small index i < I, use finitely
+ many templates; for the large index i >= I, use subgroup
+ factorization modulo q to just one fixed finite list.
+ (5) ONE DRAW PER EDGE. Fix independent distributions on the
+ blocks, draw once, and use compactness (or counting) to
+ find a single sample that satisfies all conditions at
+ once.
+
+8.4 FINITE TESTS
+
+Lemma. Let x >= 2 and let B be a fixed positive exponent. There
+is a finite coordinate box B_x in F, of size x^c, such that for
+any fixed finite set of group elements g with all coordinates
+bounded by x^a, the symmetric difference ratio
+
+ |B_x sym-diff B_x g| / |B_x| <= C x^(-B).
+
+Let A be a self-adjoint matrix of convolution sums on
+l^2(J) (x) l^2(F) (x) C^n, with J a finite set, whose coefficient
+row-and-column sums are bounded by R. Then
+
+ ||P_x A P_x|| <= ||A|| <= ||P_x A P_x|| + C x^(-B) R,
+
+where P_x retains the deck indices in B_x.
+
+Reason. The box B_x is a product of coordinate intervals whose
+sizes grow with graded weights. Multiplying by a single group
+element shifts each coordinate by a bounded amount, so only a
+thin boundary strip of B_x moves outside; the strip fraction is
+C x^(-B). Averaging the translate P_s A P_s over s in F is
+controlled by this same strip count and reduces the norm to a
+finite test that does not depend on the amplification n.
+
+The key consequence: the full regular norm of a block operator
+is determined up to C x^(-B) * (row sum) by ONE finite matrix
+test.
+
+8.5 EMPIRICAL LISTS
+
+Fix M = d^{1/32} = q^{1/(32 A_*)}, a number that grows with ell
+but stays much smaller than d.
+
+Lemma. Suppose a finite set of operator-valued distributions on
+l^2(F) (x) C^N is given, with all constituent operators
+uniformly norm-bounded by B_0, all group coordinates bounded by
+q^a, and matrix size N <= q^a. For each fixed epsilon_emp > 0,
+and ell large enough, each expectation has an M-term empirical
+average within epsilon_emp of itself in the full regular norm,
+UNIFORMLY over the finitely many distributions.
+
+Proof sketch. Do the finite test of 8.4 at x = q. Expand the
+2k-th moment trace of the empirical average at the test matrix
+and condition on all but one sample; a term whose index occurs
+exactly once vanishes. Bound the surviving terms by walk
+counting: for k = ceil(log q) and M = q^{1/(32 A_*)}, all
+quantities shrink to 0 super-polynomially fast in q. Union over
+the finitely many distributions: done.
+
+8.6 COMPACT COEFFICIENT TEMPLATES
+
+We want a FINITE list of coefficient templates, uniform in the
+layer index. Two replacements make this possible.
+
+COMPACT SUPPORT. Lemma nil:templates (Part 7) says: J_epsilon
+can be replaced by compactly supported smooth J_epsilon^flat at
+arbitrarily small L^1 coefficient cost, uniformly in the
+lattice mesh, for small enough ell.
+
+LOG-GRID PARTITION OF UNITY. Let psi_nu(t) be a smooth partition
+of unity on 0 < t < delta, pulled back from the natural
+log_2 coordinate. For each cell nu on a fixed-grid partition of
+each of the two logarithmic charts, its associated magnitude
+templates a_nu, b_omega satisfy
+
+ |log(a_nu / t)| <= u, |log(b_omega / (delta-t))| <= u,
+
+for a small fixed u = u(epsilon).
+
+For each block (i,k), let T_{ik} be the set of pairs (nu, omega)
+whose supports meet h_i h_k. This set has size at most
+K_epsilon, independent of i, k, ell. So the block-averaged
+operator
+
+ sum over (nu,omega) in T_{ik} of
+ f_{ik,nu,omega}(t)
+ * L_+(a_nu ; u_t)
+ * L_-(b_omega ; V_t)
+
+approximates h_i(t) h_k(t) p(t) to error epsilon, uniformly,
+where f_{ik,nu,omega} = h_i h_k psi_nu theta_omega.
+
+MASK DECAY. Each f_{ik,nu,omega} is smooth with compact support
+and derivatives of size at most C_j w_i^{-j} in the L^1 norm.
+Consequently its Fourier coefficients decay:
+
+ |hat{f}(n)| <= C_j w_i (1 + |n| w_i)^(-j) for every j >= 1,
+
+so summing over n gives sum |hat{f}(n)| <= C, and truncating at
+K_i = ceil(L / w_i) for a small fixed L adds error C_j (1+L)^(1-j).
+Replacing the mask by its truncated Fourier series costs a
+DETERMINED amount, uniformly in layer.
+
+8.7 TWO CASES ON BLOCK INDEX
+
+Blocks with i < I. Here the layer weights h_i force
+b_omega^(-1) <= q * 2^{kappa (I+1) + C}. There are only
+O(1 + log q) such templates over all such blocks, and their
+coordinate supports are bounded by a fixed power of q.
+So for i < I we use FINITELY MANY ordinary empirical M-lists,
+one per template.
+
+Blocks with i >= I. Introduce the subgroup
+
+ Lambda_q = { (lambda, b) in bar-E_Z :
+ lambda_ij = (b-1)_ij = 0 mod q }.
+
+Reduction modulo q is a surjective group homomorphism, so
+Lambda_q has index Q_q = q^(2 n_f). Choose one representative
+v per coset into a set V_q with coordinates in {0,...,q-1}.
+Then every gamma is uniquely gamma = yv with y in Lambda_q and
+v in V_q. The finite representation R from Part 7 satisfies
+R(sigma(y)) = 1 for y in Lambda_q.
+
+Build the coset average
+
+ C_q = Q_q^{-1} sum over v in V_q of
+ R(sigma(v)) * rho(sigma_Y(v)).
+
+Then for any i >= I, the block template can be replaced by
+
+ [ Q_q sum over y in Lambda_q of
+ b_omega^{n_f} J_-^flat(Delta_{b_omega^{1/m}} y)
+ rho(sigma_Y(y)) ] * C_q,
+
+i.e. by the SAME empirical list C_q for every large-i block. The
+individual sum over y is bounded in coefficient L^1 by a
+constant depending only on ell (and on the fact that its mesh in
+Lambda_q is FINE, since b_omega^(-1/m) / q >= 2^{2 E_0 H_i} for
+large i).
+
+The comparison with the ordinary template uses three facts
+jointly: the mesh is much finer than q (subgroup side), the
+cocycle phase is nearly constant on cosets (same yv as y), and
+the coefficient function is smooth. The upshot is that the
+difference in operator norm is o(1) uniformly in the block and
+in t.
+
+8.8 ONE DRAW PER EDGE
+
+For each block (i,k) we specify ONE distribution for a single
+pair (z, W). Sample the distribution on each undirected edge
+independently, keeping the positive-orientation pair and taking
+the inverse/adjoint on the reversed edge.
+
+The distribution on one block:
+
+ (a) Pick (nu, omega) uniformly in T_{ik}, and pick an integer
+ n uniformly in {-K_i, ..., K_i}.
+
+ (b) Pick x uniformly from a fixed rectangle covering
+ J_+^flat(Delta_{a_nu^{1/m}} x). Attach sigma_X(x) with
+ scalar multiplier
+
+ n_nu^+ * a_nu^{n_f} * J_+^flat(Delta_{a_nu^{1/m}} x).
+
+ (c) If i < I: pick v' from the fixed M-empirical list for
+ b_omega, and attach sigma(v'), R(sigma(v')), and its
+ scalar multiplier.
+
+ If i >= I: pick y uniformly in the Lambda_q rectangle and
+ v uniformly in V_q, and attach sigma_Y(y) sigma_Y(v),
+ R(sigma(v)), and multiplier
+
+ n_omega^Lambda * Q_q * b_omega^{n_f}
+ * J_-^flat(Delta_{b_omega^{1/m}} y).
+
+ (d) Multiply the group elements in the indicated order: this
+ is z. Multiply the scalar multipliers: this is c_e. Keep
+ the single matrix R(v).
+
+ (e) Round c_e to a fixed finite complex grid in the disk of
+ radius C_epsilon + 1. Set W = c_e^flat * R(v).
+
+The matrix R(v) is unitary, so rounding changes the norm of the
+edge weight by at most a prescribed constant. Voltages retain
+their full value even if the weight becomes zero.
+
+PROPERTIES OF THE DISTRIBUTIONS.
+
+ - Each distribution is FINITE, supported on a finite set.
+ - The scalar multipliers are uniformly bounded.
+ - The matrix products are uniformly bounded in norm by
+ C_eta = C_epsilon + 1.
+ - The augmented alphabet (original label, matrix weight) has
+ size O_epsilon(d^{89/32} (1 + log d)^3) = o(d D).
+ - Term-by-term linearity, plus empirical list cancellation,
+ plus scalar rounding, gives
+
+ sup over t of || E[W rho(z)](t) - h_i(t) h_k(t) p(t) ||
+ <= epsilon,
+
+ uniformly in (i, k).
+
+ - Each coordinate of every possible z in a block beginning in
+ layer i, and of its inverse, is bounded by 2^{C H_i}.
+
+ - Every first-factor coordinate interval contains at least
+ 2^{E_0 H_i} points; likewise (for large i) for the
+ second-factor subgroup intervals.
+
+8.9 GIRTH VIA SCHWARTZ-ZIPPEL
+
+Fix a nonempty cyclically nonbacktracking base walk w of length
+<= C_* H_i / log_2 d, based in layer i. Consider a formal upper
+unitriangular matrix on each edge in the support of w.
+
+If SOME block along w has first index j_0 < I, all such first
+indices are at most 2 I, so use the first-factor quotient of F
+onto U_Z. Otherwise every first index is >= I, and use the
+second upper unitriangular quotient.
+
+Either way, some entry of the word (minus identity) is a
+nonzero polynomial in the entries of the edge matrices, of
+total degree at most m - 1 (because every contributing monomial
+has superdiagonal weight <= m - 1).
+
+Assign independent uniform coordinate choices at each edge:
+in the first-factor case, from integer intervals of size at
+least 2^{E_0 H_j}; in the second-factor case, from a linear
+image of the integer Lambda_q points scaled by q. Each
+coordinate set has at least 2^{E_0 H_j} points.
+
+SCHWARTZ-ZIPPEL: a nonzero polynomial of total degree r
+evaluated on independent uniforms from sets of size at least
+L vanishes with probability at most r/L. So a specific walk
+has trivial holonomy with probability at most
+
+ (m - 1) * 2^{-E_0 min_j H_j}.
+
+The number of walks to test from layer i is at most
+
+ (d+1) * 2^{H_i} * sum over 1 <= a <= C_* H_i / log_2 d
+ of (6D)^a <= 2^{(3 C_* + 3) H_i}.
+
+Since min_j H_j >= (1 - o(1)) H_i and E_0 > 10 (C_* + C_bullet + 1),
+the union bound over all such walks is at most C 2^{-2 H_i} for
+each layer i, and summing over i gives a bound tending to 0 as
+ell -> infinity.
+
+CLOSED LIFTS = CLOSED BASE WALKS. A closed lift projects to a
+closed base walk, and nonbacktracking survives; trivial holonomy
+depends only on the base walk, not on the lift's starting vertex.
+So the tests above exclude every forbidden closed walk in P.
+
+8.10 CENTERED MOMENT BOUNDS
+
+Fix a block (i,k). Let
+
+ X_{ik} = sum over positive edges e : a -> b of
+ ( W_e rho(z_e) - E[W_e rho(z_e)] )
+
+be the CENTERED edge sum. We want ||X_{ik}|| small, uniformly
+in the layers.
+
+MOMENTS. Expand
+
+ E[ Tr( hat{X}_{ik}^{2 l} ) ]
+
+where l = ceil(C_2 H_i / log_2 d) is a FIXED even power.
+Summands whose walk uses a single undirected edge exactly once
+vanish by centering. For the remaining summands, the support has
+bounded cycle rank (varying length, bounded b), so the walk-
+counting lemma gives at most C_b^l Delta^l walks, with
+Delta <= 6D. Because each factor is bounded by 2 C_epsilon and
+each compressed test matrix has at most 2^{C_1 H_i} dimensions,
+we get
+
+ E[Tr(hat{X}_{ik}^{2 l})] <= 2^{C_1 H_i} (C_4 D)^l.
+
+MARKOV AT THRESHOLD epsilon sqrt(d D_{ik}). The finite test in
+8.4 contributes a fixed absolute error, which is smaller than
+epsilon sqrt(d D_{ik}) / 2 for large ell. Since D/2 <= D_{ik}
+<= 2 D, the probability that ||X_{ik}|| exceeds that threshold is
+at most
+
+ 2^{C_1 H_i} (C_5 / d)^l.
+
+Choosing C_2 > 2 (C_1 + 4) gives 2^{(C_1 - C_2/2) H_i} <=
+2^{-4 H_i}. Summing over at most three blocks per first index
+gives total failure probability tending to 0.
+
+8.11 SIMULTANEOUS CHOICE AND GLOBAL NORM
+
+Sum the girth failure bound (8.9) and the block norm failure
+bounds (8.10). For large ell the sum is < 1. There is
+positive probability of satisfying ALL conditions at once (or,
+equivalently, a compactness argument over the countable product
+of finite edge distributions stabilises every finite constraint
+simultaneously). Fix such an assignment.
+
+Combine the block estimates. Each block's expectation matches
+its model block to within epsilon, and each centered deviation
+is bounded by epsilon. A width-one block matrix with entries
+bounded by epsilon has norm at most 3 epsilon; doubling the
+off-diagonal makes it self-adjoint without changing the norm.
+Therefore
+
+ || T_P - Pi O Pi || <= 6 epsilon.
+
+Taking epsilon < eta/6 gives (G2) and (G3).
+
+================================================================
+PART 9. STEP 5: THE GEOMETRY AND THE GROUP Gamma
+================================================================
+
+Now we take the voltage cover P from Part 8 (it exists, it has
+big girth, and its operator matches the model), and turn it into
+a finitely generated torsion-free group Gamma. The Cayley graph
+of Gamma will be where the operator T from the theorem lives.
+
+A short note on notation: in this Part, "M_mark" always means
+the number of marked vertices in a plane diagram. It is not the
+matrix size m from Part 7. The two are unrelated.
+
+9.1 THE VOLTAGE COVER
+
+Let z_e in F be the voltage of the positively oriented edge e of
+Y. The voltage cover P has vertex set
+
+ V(P) = V(Y) x F,
+
+and for every positively oriented edge e : v -> w of Y with
+voltage z_e, an edge in P from (v, f) to (w, f z_e). Reversing
+gives (w, f) -> (v, f z_e^-1).
+
+Left deck action: delta_a(v, f) = (v, a f). This action is
+FREE on V(P), and it is transitive on the lifts of any fixed
+base vertex or fixed base path.
+
+GIRTH HYPOTHESIS (what Part 8 supplies). There is an absolute
+constant C_* such that for every i in Z, no nonempty cyclically
+nonbacktracking closed walk in P based in layer i has length at
+most C_* H_i / log_2 d.
+
+9.2 THE GROUP
+
+Let S be the finite positive alphabet of Y (Part 6). Define
+
+ Gamma = < S | label words of all closed walks in P >,
+
+i.e. each closed walk in P contributes its label word as a
+relator. This group is finitely generated (S is finite) and is
+presented graphically.
+
+The Cayley graph X = Cay(Gamma, S) has the formal vertices g in
+Gamma and formal edges (g, sigma) for g in Gamma, sigma in S.
+
+For each connected component C of P, each h in Gamma, and each
+fixed base vertex v_0 of C, there is a UNIQUE label-preserving
+graph map phi : C -> X with phi(v_0) = h. The path-independence
+uses the presentation: any two paths in C from v_0 to a vertex
+of C read words that differ by a closed walk, hence by a
+relator. The maps phi are not assumed injective at this stage.
+
+9.3 INJECTIVITY: VAN KAMPEN DIAGRAMS
+
+We must show: every phi is injective on vertices and edges. If
+not, we get a contradiction by minimality.
+
+Suppose phi identifies two distinct source vertices. Reading a
+source path between them yields a closed image word in X, hence
+by the van Kampen Lemma bounds a finite PLANE DIAGRAM with
+bounded faces reading relators. Choose among all injectivity
+failures one that minimizes, lexicographically, the pair
+
+ (J, number of edges),
+
+where J = number of bounded faces, and the diagram's exterior
+word is the source path. All source incidences are retained as
+part of the diagram.
+
+A REDUCTION LEMMA eliminates obvious slack:
+
+ (R1) No bounded face has a freely trivial word (deleting it
+ would reduce J).
+ (R2) No edge between two distinct bounded faces has aligned
+ source incidences (compare before/after deck action and
+ delete the edge).
+ (R3) No unmarked vertex of degree one (would be a backtrack,
+ contradicting edge minimality); no bridge with both
+ endpoints on a bounded face.
+
+Compress every unmarked degree-two vertex. The result is a
+plane graph with ARCS (compressed edges) and marked vertices.
+Each arc remembers its whole original edge path.
+
+ARC LENGTH. If an arc on a bounded face has length >= L_i (for
+any layer i of an endpoint on the face), then its freely reduced
+word determines the projected nonbacktracking path in Y. Both
+source lifts of that path then differ by ONE deck transformation,
+so the two incidences are aligned -- forbidden by (R2). So
+every arc on a bounded face has length < L_i.
+
+FACE CYCLE. If a bounded face had fewer than 20 arcs, its
+perimeter would be at most 19 * L_i < 20 L_i = 2000 H_i / log_2 d.
+Cancel backtracks, cancel cyclically. Foldedness keeps the
+cancellations actual, and the remaining nonempty cyclic path
+would contradict the girth hypothesis (2^C_* = 10^4 > 5000).
+So each bounded face has at least 20 arcs.
+
+EULER COUNT. Let b be the number of arcs and M_mark be the
+number of marked vertices. A plane Euler formula b = v + J - 1
+with each unmarked vertex of degree >= 3 gives
+
+ 20 J <= 2 b, b <= 3 J + 3 M_mark - 3,
+ b <= 6 M_mark.
+
+INJECTIVITY. Take an injectivity failure with M_mark = 1.
+The two inequalities 20J <= 2b and b <= 6 give
+
+ 20 J <= 6 J, so J = 0.
+
+The diagram is a tree, its exterior word is freely trivial. A
+path with freely-trivial word in a folded graph returns to its
+source. That contradicts the chosen injectivity failure (the
+endpoints were supposed distinct).
+
+Consequently every component map phi is injective, and the same
+argument applied to edges handles the edge case. Since phi is
+injective, we get genuine EMBEDDINGS of the components of P into
+the Cayley graph X.
+
+9.4 INDEXED COPIES AND BALL OVERLAP
+
+Because we fixed a component C, a map phi, and a base vertex,
+"indexed copies" of X are defined up to a deck alignment:
+
+ (C, phi) ~ (delta_a C, phi delta_a^-1), a in F.
+
+An INDEXED COPY Q therefore remembers its source component and
+its source vertices up to this alignment, not just its image.
+
+Adjoin for each indexed copy Q one CONE CENTER c_Q and a spoke
+from c_Q to each vertex of Q. Let X-hat be the resulting cone
+graph.
+
+THIN TRIANGLES. Lemma (short chain fillings): every edge loop
+in X-hat of length n is a sum of at most 12n oriented loops of
+length at most 4. From such a filling, every geodesic triangle
+in X-hat is D_0-thin, with D_0 = 10^6.
+
+WELL-ORDER. Fix one base point o in X. Order the indexed
+copies first by grid distance d_Xhat(c_Q, o), and then by an
+enumeration tie-break. For each Q choose one geodesic from
+c_Q to o and let u in Q be its first vertex past c_Q. This
+"root vertex" u will anchor the trimming ball.
+
+AVOIDANCE. Lemma: if a path in X-hat of length at most L joins
+u, x in Q avoiding c_Q, then the INTRINSIC distance in Q
+satisfies d_Q(u, x) <= 24 (L + 1) L_i.
+
+BALL CONTAINMENT. For any vertex x in Q that ALSO belongs to
+some earlier copy Q', a short path in X-hat from u to x avoids
+c_Q, of length <= L* = 4 D_0 + 8. By avoidance,
+
+ d_Q(u, x) <= 24 (L* + 1) L_i.
+
+So define
+
+ B(Q) = { v in Q : d_Q(u, v) <= R_Q },
+ R_Q = ceil( 25 (L* + 1) L_i ).
+
+Every earlier-overlap vertex of Q lies in B(Q). The ball B(Q)
+is a TREE (else a short cycle in Q, contradicting girth after
+proper scaling), and no outside vertex has two incidences into
+B(Q).
+
+9.5 DISJOINT DOMAINS AND THE RESIDUAL FOREST
+
+Define the ASSIGNED DOMAIN of Q by
+
+ D_Q = V(Q) minus (union over earlier copies R < Q of V(R)).
+
+These are pairwise disjoint, and V(Q) \ D_Q is contained in
+B(Q).
+
+Delete from X every edge of Q whose two endpoints both lie in
+D_Q (for every Q). The remaining graph is a FOREST.
+
+Reason: if a circuit existed, its image in the Cayley graph
+would give a nonzero cycle. Directness of the embedding (below)
+means the LATEST copy Q with a nonzero component would have a
+cycle supported on the tree B(Q), which forces that component to
+be zero -- a contradiction.
+
+9.6 THE INCIDENCE BIJECTION
+
+There is a canonical bijection
+
+ (Q, x) : Q is an indexed copy, x in V(Q)
+ <--> ( x, v ) : x in Gamma, v in V(Y),
+
+where v is the base vertex under the unique source vertex of the
+embedding. In particular, for a fixed x and v there is EXACTLY
+ONE incidence at x of type v.
+
+Consequence: the equivariant vertex space
+
+ oplus over Q of l^2(V(Q))
+
+is canonically identified with
+
+ l^2(Gamma) (x) l^2(V(Y)),
+
+and with a further C^N factor if matrix indices are used.
+
+9.7 TORSION-FREENESS OF Gamma
+
+STABILIZERS. If h in Gamma stabilizes an indexed copy, its
+action corresponds to a deck transformation delta_a in F
+preserving the representative source component, and the
+correspondence h <-> a is an INJECTION. Since F is torsion-free
+(Part 7), the stabilizer has no finite-order element.
+
+CONE COMPLEX. Let Z be the two-dimensional cellular complex
+obtained by gluing each cone on Q into X. Its augmented chain
+complex
+
+ 0 -> C_2(Z;k) -> C_1(Z;k) -> C_0(Z;k) -> k -> 0
+
+is EXACT for every field k.
+
+Why: X is connected and each cone attached to it, so exactness
+at C_0 and k is immediate. For exactness at C_1, take a cycle
+of X-hat, its spokes sum to zero per center, and connectedness
+of each copy lets us replace that spoke chain by a finite
+combination of edges on Q; then the residual X-cycle is a finite
+sum of copy cycles by directness, each of which bounds the two-
+cell faces on its own copy. For injectivity at C_2, a finite
+sum of triangles determines one edge chain c_Q per copy, and
+vanishing of the boundary says boundary(c_Q) = 0 in each copy;
+directness forces every c_Q to be zero.
+
+TORSION CONTRADICTION. Suppose Gamma has an element of finite
+order. Some power generates a subgroup C_p of prime order p.
+Working over k = F_p and using freeness of the cell action
+(9.6), every C_j(Z; F_p) is a FREE MODULE over A = F_p[C_p].
+The exact chain complex above then gives a FREE RESOLUTION of
+the trivial module F_p of length 2 over A.
+
+But F_p[C_p] has a PERIODIC free resolution of INFINITE length:
+
+ ... -> A --z--> A --z^(p-1)--> A --z--> A --z^(p-1)--> ... -> F_p.
+
+Tensoring the length-2 resolution with F_p over A gives one copy
+of F_p in degrees 0, 1, 2, then zero after that. Tensoring the
+periodic resolution gives one copy of F_p in EVERY degree. But
+any two free resolutions of F_p over A are chain-homotopy
+equivalent, so their homology after tensoring with F_p must
+coincide. Contradiction.
+
+Therefore Gamma has no nonidentity finite-order element. Gamma
+is torsion-free.
+
+Combining 9.3 (embeddings), 9.5 (disjoint domains + residual
+forest), 9.6 (incidence bijection), and 9.7 (torsion-free)
+proves the geometric theorem that part 9 was aiming at: the
+voltage cover P of Part 8 produces a finitely generated
+torsion-free group Gamma with indexed copies whose geometry is
+under full control.
+
+9.8 CONSTANTS FIXED BEFORE ell
+
+All constants are those produced by 9.3 -- 9.4:
+
+ D_0 = 10^6,
+ L* = 4 D_0 + 8 (about 4 * 10^6),
+ C_* = 10^7 (L* + 1),
+ C_bullet = 2 * 10^4 (L* + 1).
+
+They are absolute and are fixed BEFORE ell is chosen. This is
+what makes the dependency ledger in Part 7.9 correct.
+
+================================================================
+PART 10. STEP 6: TRANSFER TO THE CAYLEY GRAPH
+================================================================
+
+We now have a cover P whose operator matches the model band.
+We have Gamma, f.g. torsion-free, with copies of P covering its
+Cayley graph. The remaining job: turn the band on P into a
+spectral band for the actual FINITE group-algebra operator T on
+Gamma, and bound the unnormalized trace of its upper projection.
+
+10.1 THE INPUTS
+
+From Part 8, the cover operator T_P satisfies
+
+ || T_P - Pi O Pi || <= eta,
+
+where Pi is the projection onto surviving vertices and O is the
+model operator, whose band E_*(t) has eigenvalue 1 on the
+symmetric constants and eigen-at-most 1 - chi on the orthogonal
+complement.
+
+Take ell large enough that beta_ell (deleted fraction) also
+satisfies 20 sqrt(beta_ell) <= alpha / 2, where alpha = chi / 1000.
+Fix eta <= alpha / 2. Let
+
+ theta = C_chi eta, C_chi = 4 / chi, r_0 = chi / 3.
+
+We also require theta <= 1/2 and 2^{C_bullet} theta^2 <= 1/2,
+which just amounts to eta small enough (a fixed choice).
+
+10.2 THE COVER PROJECTION E_P
+
+Let E_P be the spectral projection of T_P obtained by integrating
+the resolvent over the circle |z - 1| = r_0 = chi/3. Then
+
+ ||T_P|| <= 2,
+ || (T_P - I) E_P || <= alpha,
+ T_P on (1 - E_P) H <= (1 - chi + alpha) I.
+
+Reason. The compression lemma (below, sketched) says: if
+J = Pi compresses a separated upper space E_* with small loss,
+then JE_* has closed range and the compressed comparison operator
+stays an upper space on J H with the gap preserved up to a
+quantity proportional to the loss. The perturbation lemma (also
+below) says: if ||A - B|| <= u < r_0 and B has a spectral band
+with contour at radius r_0, then A has a nearby projection and
+the band survives with widened margins. Both apply here, with
+J = Pi and r_0 = chi/3.
+
+10.3 EXACT FIBER TRACE
+
+For almost every t in the circle T = R / Z,
+
+ tr_id E_P(t)
+ = tr_id p(t) if 0 < t < delta,
+ = 0 otherwise.
+
+Reason. Inside the window, compression and perturbation both
+act by equivariant partial isometries (see the lemmas), which
+preserve identity trace. Outside the window, ||T_P(t)|| <= eta
+almost everywhere by Part 8 (G3), so its contour projection is
+exactly zero there.
+
+In particular E_P is a nonzero projection (since p(t) has
+positive trace over the positive-measure window), and its
+diagonal at a lifted vertex depends only on the base vertex.
+
+10.4 WEIGHTED SIZE S
+
+For a surviving base vertex v, define
+
+ e_v = Tr_{C^N}( E_P( (v, 1_F) ) ) >= 0,
+
+the matrix-identity diagonal of E_P at v. Let w_i = 2^{C_bullet H_i}
+be the layer weight. Define the weighted size
+
+ S = sum over surviving base vertices v of w_{i(v)} * e_v.
+
+LOCALIZATION. The model O(t) is supported on the active layers
+J(t). The difference Delta(t) = T_P(t) - Pi O(t) Pi has layer
+width one. A resolvent series for T_P(t) in terms of O(t) gives
+
+ || P_i E_P(t) || <= 4 theta^{ dist(i, J(t)) },
+
+which decays in the layer distance from the active band. That is
+why layer weights and layer distances combine to give a small
+number.
+
+WEIGHTED TRACE BOUND. A short union-bound over closest layers
+in J(t), plus Equation (10.3), gives
+
+ S <= 96 mu_ell,
+
+where mu_ell is the small weighted trace of Part 7. Since
+mu_ell -> 0 as ell -> infinity, S -> 0 too.
+
+10.5 THE CAYLEY OPERATOR T
+
+Use the finite alphabet S from Part 8 (original label plus matrix
+weight tag). For sigma in S with matrix weight W_sigma and
+degree normalizer D_sigma = D_{ik}, define
+
+ w_sigma = (d D_sigma)^{-1/2} W_sigma.
+
+The actual Cayley operator is
+
+ T = sum over sigma in S of
+ ( w_sigma * lambda(sigma)
+ + w_sigma^* * lambda(sigma^{-1}) ) .
+
+T is a finite sum of matrix coefficients times left-translations.
+It is self-adjoint and belongs to M_N( C_r*(Gamma) ).
+
+10.6 LOCAL PIECES ON EACH COPY
+
+Each indexed copy Q inherits a copy operator T_Q (using only its
+own edges) and an upper projection E_Q. The diagonal of E_Q at
+a vertex u in Q is e_v where v is u's base vertex (from 10.3).
+
+BALL TRIMMING. For the assigned domain D_Q and trimming ball
+B(Q) from Part 9,
+
+ || 1_{B(Q)} E_Q ||_{HS}^2 <= S.
+
+Reason: the Hilbert-Schmidt norm squared on ID (identity
+diagonal) is bounded by the vertex-sum of e_v over the ball, and
+part 9 bounds how many lifts of a base vertex appear in B(Q) by
+the layer weight w_i, giving S.
+
+CONSEQUENCE. If S <= 1/4, then the compression of T_Q to
+l^2(D_Q) (x) C^N is within alpha + 20 sqrt(S) of an operator
+C_Q that satisfies
+
+ C_Q = 1 on a projection F_Q equivalent to E_Q,
+ ||C_Q|| <= 2,
+ C_Q <= (1 - chi + alpha) I on (1 - F_Q) H_Q.
+
+In particular F_Q != 0 whenever E_Q != 0.
+
+10.7 GLOBAL COMPARISON
+
+Form the orthogonal direct sums
+
+ C = oplus_Q C_Q + 0 on unassigned vertices,
+ T_D = oplus_Q T_Q|_{D_Q} + 0 on unassigned vertices,
+ F_D = oplus_Q F_Q + 0 on unassigned vertices.
+
+The assigned domains D_Q are disjoint, so both C and T_D are
+bounded operators on l^2(Gamma) (x) C^N. From 10.6,
+
+ || T_D - C || <= 2 alpha.
+
+RESIDUAL EDGES. The difference T - T_D is exactly the matrix
+adjacency on the residual Cayley edges (edges NOT internal to
+any assigned D_Q). By Part 9.5 those edges form a FOREST.
+
+FOREST NORM LEMMA. A self-adjoint matrix adjacency on a
+countable forest of maximum degree m_1 and matrix edge norms at
+most w_1 has norm at most 2 w_1 sqrt(m_1).
+
+Applied to T - T_D, with m_1 <= 2 |S| and w_1 <= sqrt(2) C_eta /
+sqrt(dD):
+
+ || T - T_D || <= 4 C_eta sqrt( |S| / (dD) ) =: f_ell.
+
+Since |S| = o(dD), f_ell -> 0 as ell -> infinity. Increase ell
+so f_ell <= alpha.
+
+Then ||T - C|| <= 3 alpha.
+
+10.8 SPECTRAL SPLIT
+
+Recall C is self-adjoint, equals 1 on F_D, and is at most
+1 - chi + alpha on the orthogonal complement. Also C_D and T
+differ by at most 3 alpha. Therefore
+
+ spectrum(T) contained in
+ (-infinity, 1 - chi + 4 alpha] UNION [1 - 3 alpha, 1 + 3 alpha].
+
+The upper interval is nonempty (F_D != 0). Therefore the number
+1 - chi/4 lies strictly between the two intervals, and the
+spectral indicator
+
+ p_0 = 1_{[1 - chi/4, infinity)}(T)
+
+is a nonzero projection in M_N(C_r*(Gamma)). It is the spectral
+projection of T for the upper band.
+
+10.9 TRACE BOUND FOR p_0
+
+This is the subtle part: how to bound the unnormalized trace of
+p_0 from above.
+
+Use ALL the indexed copies at once. The equivariant vertex
+space is
+
+ L = oplus_Q l^2(V(Q)) (x) C^N
+ = l^2(Gamma) (x) l^2(V(Y)) (x) C^N = H (x) l^2(V(Y))
+ (identification from Part 9.6).
+
+Let E = oplus_Q E_Q. Its identity trace is
+
+ Tr_L(E) = S_0 = sum over surviving base vertices v of e_v
+ <= S < infinity.
+
+So the total upper space of all copies has small trace.
+
+RESTRICTION MAP. Define A : H -> L by
+
+ A xi = ( E_Q( xi|_{V(Q)} ) )_Q.
+
+Because the D_Q are disjoint and the ball-loss bound 10.6 holds,
+A is bounded with
+
+ ||A|| <= 1 + sqrt(S).
+
+Its kernel K is a closed invariant subspace, and eq. (10.6) says
+that for every xi in K,
+
+ E_Q( xi|_{D_Q} ) = - E_Q( xi|_{I_Q} ),
+
+so the local loss on D_Q is paid by the trimming ball, giving
+
+ sum over Q of || E_Q( xi|_{D_Q} ) ||^2 <= S ||xi||^2.
+
+SANDWICH. For xi in K, using the local estimate from 10.2 on
+each copy, and the forest error:
+
+ <T xi, xi> <= (1 - chi + 3 alpha) ||xi||^2.
+
+On the OTHER HAND, on the range of p_0:
+
+ <T xi, xi> >= (1 - 3 alpha) ||xi||^2.
+
+These two inequalities are incompatible unless xi = 0. So
+p_0 H intersects K only at 0. The map P_{K^perp} : p_0 H -> K^perp
+is injective and equivariant.
+
+TRACE CONSEQUENCE. A polar decomposition identifies p_0 H with a
+closed invariant subspace of K^perp. Therefore
+
+ tau_{Gamma,N}(p_0)
+ = Tr_H( p_0 ) <= Tr_L( P_{K^perp} ) <= Tr_L( E )
+ = S_0 <= S.
+
+Since S < 1/2 (we chose ell well past S <= 1/4), we conclude
+
+ 0 < tau_{Gamma,N}(p_0) < 1/2.
+
+The lower bound 0 < is because p_0 != 0 (from 10.8) and the
+trace is faithful.
+
+10.10 WHAT PART 10 PRODUCES
+
+We have produced:
+
+ a finitely generated torsion-free group Gamma;
+ a projection p_0 in M_N( C_r*(Gamma) );
+ with 0 < tau_{Gamma,N}(p_0) < 1/2.
+
+That is EXACTLY the matrix target that Step 1 (Part 5) needed.
+
+================================================================
+PART 11. THE CONCLUSION
+================================================================
+
+11.1 PUTTING THE PIECES TOGETHER
+
+We now have, from Parts 6 -- 10:
+
+ - a finitely generated torsion-free group Gamma;
+ - an integer N = q^{n_f};
+ - a matrix projection p_0 in M_N(C_r*(Gamma));
+ - with 0 < tau_{Gamma,N}(p_0) < 1/2.
+
+That is exactly the "matrix target" that Step 1 (Part 5) needs.
+
+Apply Step 1 with Lambda = Gamma and p = the matrix projection
+p_0. Let
+
+ G_proj = Gamma * F_N
+
+be the free product with the free group on N letters. Since
+Gamma is f.g. and torsion-free, so is G_proj. Let t_1,...,t_N be
+the free generators, U the row (lambda(t_1),...,lambda(t_N)),
+A = p_0 U* U p_0 in the corner, V = U p_0 A^{-1/2}, and
+
+ e = V V* in C_r*( G_proj ).
+
+This e is a projection, and by the trace computation of 5.6,
+
+ tau_{G_proj}(e) = tau_{Gamma,N}(p_0) in (0, 1/2).
+
+Therefore e is neither 0 nor 1. Since G_proj is finitely
+generated and torsion-free, e is a genuine counterexample to
+the Kadison-Kaplansky projection conjecture.
+
+The theorem is proved.
+
+11.2 WHAT ELSE THE CONSTRUCTION GIVES
+
+(a) TRACE IS NOT INTEGER ON K_0. The K_0 class [p_0] in
+K_0(C_r*(Gamma)) has trace tau_{Gamma,N}(p_0) in (0, 1/2), which
+is not an integer. Lueck (2002) and Baum-Guentner-Willett (2016)
+prove that the reduced assembly map with trivial coefficients
+lands in the integral part of K_0-trace for torsion-free groups.
+Therefore [p_0] lies OUTSIDE the image of the reduced assembly
+map for Gamma. This was not known before at this level of
+explicitness.
+
+(b) C[G_proj] HAS NO SCALAR IDEMPOTENTS OTHER THAN 0 AND 1.
+A companion paper (OpenAI, "The Bass trace conjecture and the
+characteristic-zero Kaplansky idempotent conjecture", September
+2026) proves that the complex group ring C[G_proj] has no
+scalar idempotents other than 0 and 1. Therefore the projection
+e lies in the C*-completion C_r*(G_proj) but NOT in the
+algebraic group ring C[G_proj]. This is the sharpest possible
+illustration that the norm-closure is much larger than the
+algebraic group ring.
+
+(c) STABLE VERSION. The conjecture also has a "stable" form
+about projections in M_N(C_r*(G)) for all N. The companion
+paper's results make the stable form fail for G_proj via the
+matrix projection p_0 itself; the scalar projection e handles
+the original (scalar) form.
+
+11.3 WHAT IS NOT CLAIMED
+
+The argument is an EXISTENCE proof. It does not claim:
+
+ - an explicit list of generators of G_proj (only that a finite
+ set exists, of size swamped by all the layer parameters);
+ - a useful NUMERICAL BOUND on N = q^{n_f};
+ - a "short" or "understandable" presentation of G_proj;
+ - any minimality or optimality of the construction.
+
+Every parameter is chosen in the ordering of Part 7.9, and
+"ell large enough" is existence only.
+
+This is the end of the proof.
+
+================================================================
+PART 12. GLOSSARY
+================================================================
+
+Here is every symbol used above, alphabetically. One-line
+definitions; cross-references give the Part where the object was
+properly introduced.
+
+ A_i A-side layer at index i. |A_i| = d 2^(H_i). [6.2]
+ B_i B-side layer at index i. |B_i| = 2^(H_i). [6.2]
+ bar-E Affine semidirect product on n; quotient by the
+ central subgroup (0, 1, k), k in Z. [7.3]
+ bar-E_Z Integer-coordinate lattice inside bar-E. [7.3]
+ B(Q) Trimming ball of an indexed copy Q: intrinsic
+ radius R_Q around the root u. [9.4]
+ C[G] Group algebra of G (finite formal sums). [1.3]
+ C_r*(G) Reduced group C*-algebra of G. [1.3]
+ C_bullet Absolute constant 2*10^4 (L* + 1). [9.8]
+ C_* Absolute constant 10^7 (L* + 1), the girth scale
+ in the cover hypothesis. [9.8]
+ chi Expansion gap constant 1/800. [6.6]
+ D d^2 = 2^(8 ell). [6.1]
+ D_{ik} B-side degree of the pair (i, k): D 2^(|i|-|k|).
+ [6.3]
+ d A-side degree of a single (i, k) pair; d = 2^(4 ell).
+ [6.1]
+ delta q^(-1), the width of the t window. [7.7]
+ D_Q Assigned domain of copy Q: V(Q) minus all earlier
+ copies. [9.5]
+ ell Integer seed; all "for ell large enough" bounds
+ hold simultaneously. [6.1]
+ eta Fixed small accuracy of T_P around Pi O Pi. [10.1]
+ e Final SCALAR projection in C_r*(G_proj). [11.1]
+ F Deck group, torsion-free nilpotent integer group
+ built from two copies of bar-E_Z. [7.3]
+ F_N Free group on N letters t_1,...,t_N. [5.2]
+ G_proj Final group Gamma * F_N. [11.1]
+ Gamma Group presented graphically by the cover P. [9.2]
+ h_i(t) Smooth bump in the t window indexed by layer.
+ [7.8]
+ H_i H_0 + |i|, the layer "height". [6.1]
+ H_0 (2 n_0 + 1) s + 1, base height. [6.1]
+ J(t) Active layers at parameter t. At most two,
+ adjacent. [7.8]
+ K 32 s, the period separating repeated tower families.
+ [6.1]
+ M_mark Number of marked vertices in a plane diagram. [9.3]
+ m Rank parameter for upper unitriangular matrices,
+ fixed above L(201 C_*). [7.2]
+ mu_ell sup over t and i in J(t) of w_i * s_p(t), the
+ "small weighted trace". Tends to 0. [7.8]
+ n The nilpotent Lie algebra of strictly upper
+ triangular m-by-m matrices. [7.3]
+ n_f m(m-1)/2, dimension of n. [7.3]
+ N q^(n_f), the matrix size of the magnetic factor. [7.7]
+ O(t) Model operator (tower incidence tensor p(t)). [8.1]
+ p(t) Small-trace projection p_t (x) p_{t-delta}[V_t].
+ [7.7]
+ p_0 Final matrix projection in M_N(C_r*(Gamma)). [10.8]
+ p_theta Individual twisted projection in Proposition from
+ 7.6.
+ P Voltage cover of Y. [9.1]
+ Pi Orthogonal projection onto surviving vertices Y.
+ [8.1]
+ q d^(A_*), the finite magnetic factor modulus. [7.7]
+ r 7 ell, length of block 0 of the A-stream. [6.1]
+ r_0 chi/3, contour radius for the E_P projection. [10.1]
+ R Not used for complex numbers in this document
+ (Part 1 only).
+ R_Q Trimming ball radius: ceil(25 (L*+1) L_i). [9.4]
+ S sum over surviving base vertices v of w_{i(v)} e_v;
+ the "weighted size" of E_P. [10.4]
+ s 10 ell, the block size in both streams. [6.1]
+ s_p(t) Identity trace of p(t): N tr_t(p_t)
+ tr_{t-delta}(p_{t-delta}). [7.7]
+ tau The faithful tracial state on C_r*(G). [1.4]
+ tau_{G,N} Unnormalized matrix trace on M_N(C_r*(G)). [1.6]
+ T Actual Cayley operator in M_N(C_r*(Gamma)). [10.5]
+ T_P Voltage-cover operator on P. [8.2]
+ theta C_chi * eta, a small real. [10.1]
+ u, v The two bit-string coordinates of a vertex.
+ [6.2]
+ U Row of free-generator unitaries in Step 1. [5.4]
+ U_Z Upper unitriangular m-by-m integer matrices. [7.2]
+ V(Q) Vertex set of indexed copy Q. [9.4]
+ w_i Layer weight 2^(C_bullet H_i). [7.8]
+ w_sigma (d D_sigma)^(-1/2) W_sigma, the normalized Cayley
+ generator weight. [10.5]
+ W_e Matrix edge weight on e (part of the augmented
+ label). [8.2]
+ X Cayley graph of Gamma. [9.2]
+ Y^0 Auxiliary tower with unconditional last-block
+ permutations (expansion). [6.6]
+ Y^# Auxiliary tower with gated rules (path recovery).
+ [6.6]
+ Y Pruned survivor of Y^0/Y^#; induced subgraph with
+ both properties. [6.8]
+ z_e Voltage on edge e, an element of F. [8.2]
+
+Small aliases used in the text:
+
+ delta_g Basis vector at g in l^2(G). [1.2]
+ lambda(g) Left translation unitary on l^2(G). [1.2]
+ rho(a) Right translation unitary on l^2(G). [part 7,
+ part 8 uses right convention].
+ [_]_flat Compactly supported smooth approximation of the
+ Schwartz coefficient function. [7.5, 8.6]
+ sym-diff Symmetric difference of two sets.
+
+================================================================
+FURTHER READING
+================================================================
+
+ - OpenAI, "A torsion-free counterexample to the Kadison-
+ Kaplansky projection conjecture", September 2026. The
+ primary source; contains all technical estimates.
+
+ - OpenAI, "The Bass trace conjecture and the characteristic-
+ zero Kaplansky idempotent conjecture", September 2026. The
+ companion: C[G] has no scalar idempotents for the same G.
+
+ - P. Baum, E. Guentner, R. Willett, "Exactness and the
+ Kadison-Kaplansky Conjecture", Contemp. Math. 671 (2016).
+ Integrality of the trace on the reduced assembly image for
+ torsion-free groups.
+
+ - W. Lueck, "The relation between the Baum-Connes conjecture
+ and the trace conjecture", Invent. Math. 149 (2002). The
+ earlier version of the same integrality theorem.
+
+ - N. Higson, G. Kasparov, "E-theory and KK-theory for groups
+ which act properly and isometrically on Hilbert space",
+ Invent. Math. 144 (2001). Baum-Connes with coefficients
+ for amenable groups.
+
+ - I. Mineyev, G. Yu, "The Baum-Connes conjecture for
+ hyperbolic groups", Invent. Math. 149 (2002). The
+ hyperbolic case of Baum-Connes.
+
+ - V. Lafforgue, "La conjecture de Baum-Connes a coefficients
+ pour les groupes hyperboliques", J. Noncommut. Geom. 6
+ (2012). The hyperbolic case with arbitrary coefficients.
+
+ - M. Pimsner, D. Voiculescu, "K-groups of reduced
+ crossed products by free groups", J. Operator Theory 8
+ (1982). The free-group case of the projection conjecture.
+
+================================================================
+END OF DOCUMENT
+================================================================