diff bfa87cd..2d5618e kadison_kaplansky_projection_counter_v2.txt
diff --git a/kadison_kaplansky_projection_counter_v2.txt b/kadison_kaplansky_projection_counter_v2.txt
index 17f6884..06e8610 100644
--- a/kadison_kaplansky_projection_counter_v2.txt
+++ b/kadison_kaplansky_projection_counter_v2.txt
@@ -1,47 +1,24 @@
-================================================================
-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 projection in a C*-algebra is an element p with p = p* = p^2.
+For a discrete group G, the reduced group C*-algebra C_r*(G) is the
+norm closure of the left regular representation on l^2(G), and its
+canonical trace is tau(x) = <x delta_1, delta_1>. The
+Kadison--Kaplansky projection conjecture asserts that when G is
+torsion-free the only projections in C_r*(G) are 0 and 1. Since
+tau lies in [0,1] on projections and tau(p) = 0 only for p = 0,
+the conjecture says tau(p) is always 0 or 1. It is usually stated
+in the equivalent matrix form: for every N and every projection p
+in M_N(C_r*(G)), the unnormalized trace tau_{G,N}(p) is an integer.
+
+The conjecture is false. A finitely generated torsion-free group
+is constructed below whose reduced group C*-algebra contains a
+projection of trace strictly between 0 and 1/2.
+
+Four tools carry the construction: an expander tower with folded
+labels, a nilpotent deck group whose twisted group algebra holds
+explicit small-trace projections, a van Kampen argument embedding
+the components of a voltage cover into a Cayley graph, and a
+finite-rank comparison in the regular representation that turns a
+model band into a genuine projection. Groups come first.
A group is a set G with an operation "multiply two elements and
get a third", written gh for g times h, satisfying:
@@ -50,7 +27,7 @@ get a third", written gh for g times h, satisfying:
(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:
+Examples:
the integers Z under addition (identity 0)
the nonzero reals under multiplication (identity 1)
@@ -69,11 +46,9 @@ 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
+The constructed group is 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:
@@ -95,8 +70,6 @@ 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.
@@ -122,8 +95,6 @@ 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:
@@ -141,7 +112,7 @@ Properties:
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:
+A fact used 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.
@@ -151,8 +122,6 @@ 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.
@@ -168,8 +137,6 @@ 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
@@ -178,22 +145,16 @@ 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)
+one used throughout. 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
+The notation tr_id denotes this 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
+stays finite for the operators in question.
Irving Kaplansky asked, and Richard Kadison recorded in 1955, the
following question.
@@ -217,8 +178,6 @@ 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) )
@@ -228,8 +187,6 @@ 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.
@@ -238,13 +195,13 @@ 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
+projections, modulo relations. If 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.
+the conjecture follows from trace-integrality.
Trace-integrality can be attacked through the Baum-Connes
assembly map
@@ -253,7 +210,9 @@ assembly map
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
+(Inventiones 149 (2002), 123-152), extended
+by Baum-Guentner-Willett (see the local survey
+bgw-2016-exactness-kadison-kaplansky.pdf), says: for a
torsion-free group, the canonical trace tau is integral on the
IMAGE of the reduced assembly map. So:
@@ -261,7 +220,8 @@ IMAGE of the reduced assembly map. So:
Surjectivity (Baum-Connes) is known for sizeable classes:
- - countable amenable groups: Higson-Kasparov (2001)
+ - countable amenable groups: Higson-Kasparov,
+ Invent. Math. 144 (2001), 23-74
- hyperbolic groups: Mineyev-Yu (2002, without coefficients;
Lafforgue 2012, with arbitrary coefficients)
- many more: groups acting properly and isometrically on
@@ -278,8 +238,6 @@ 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
@@ -288,16 +246,12 @@ 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
+For the same group, a the same methods show 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
+1. So the projection e 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
@@ -310,22 +264,17 @@ 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.
+The steps are logically ordered by dependency, not by their historical order.
-STEP 1. Scalar reduction. Show that if you can produce a
+STEP 1. Scalar reduction. Suppose 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
+give 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
@@ -387,22 +336,14 @@ into e over G_proj = Gamma * F_N. Done.
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)
+ ledger recorded with the tower construction spells this out.
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
+Such a construction converts the target into the theorem.
+Without it the target is exactly the problem.
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
@@ -424,8 +365,6 @@ 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.
@@ -447,8 +386,6 @@ 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.
@@ -458,7 +395,7 @@ 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*.
+unitaries. Two products of U with U* are used.
U U* = sum over j of lambda_K(t_j) lambda_K(t_j)* = N,
@@ -487,10 +424,8 @@ MAIN CLAIM.
(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
+the subspace p l^2(K)^N. Concretely, the claim is
|| U xi || >= c_a ||xi|| for all xi in p l^2(K)^N,
@@ -555,8 +490,6 @@ ASSEMBLE. Triangle:
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.
@@ -582,9 +515,7 @@ 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
+The trade exchanges "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:
@@ -595,14 +526,10 @@ conjecture, Step 1 has reduced everything to:
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:
+Three properties are wanted at once:
(1) expansion, so that the model operator on the tower has
a spectral gap;
@@ -617,8 +544,6 @@ 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),
@@ -629,12 +554,10 @@ Fix a positive integer ell, eventually large. Let
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) =
+The two useful pieces of d: 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
@@ -672,8 +595,6 @@ 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
@@ -687,7 +608,7 @@ Direct computation gives
= 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
+i, i+1), 3 d slots arise in 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
@@ -696,9 +617,7 @@ 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
+Each slot carries 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
@@ -713,31 +632,28 @@ The slot EVALUATION is
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.
+an overlap arranged by the lengths above).
-The same works from the B-vertex to the A-vertex.
+RECOVERY. At an A-vertex with slot a, the input u_A on that side
+and the linear map K_a are known. Subtracting K_a u_A from v_A
+gives Q_A = v_A - K_a u_A. The edge label x is the first p bits
+of Q_A, and the slot c at the other end follows from the "bucket"
+data in x. The triangular system then determines the other
+endpoint's input u_B, because the couplings run forward in the
+block ordering: block N on one side needs only blocks up to N-1 on
+the other. With u_B known, v_B = Q_B + K_c u_B.
-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.
+The same works from the B-vertex to the A-vertex.
-6.5 UNIQUE PATH RECOVERY
+For this to define a genuine bijection between ends, two
+conditions are needed: the diagonal coefficients on distinct
+slots at a vertex must be distinct (so the algebra is invertible),
+and the "spill" terms must 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.
FOLDED. The labeling is folded: at every vertex, the labels on
the outgoing oriented edges are distinct. This follows from the
@@ -747,24 +663,21 @@ 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 in one line: the triangular equations of 6.4
+paths. Reason in one line: the triangular equations
propagate bit by bit along the common word, and the "gate" data
included in each label records the block state sufficiently far
back that a wrong start cannot reproduce the word.
-The full proof of this is deferred to the pruning section of
-6.7 (the "prefix-recovery" lemma), where the gated deletion
-rule and the two-layer comparison are laid out. Nothing
-in 6.5 is used later except the conclusion; the machinery lives
-in 6.7.
-
-6.6 THE TWO GRAPHS Y^0 AND Y^#.
+The proof is the content of the "prefix-recovery" lemma, where
+the gated deletion rule and the two-layer comparison are laid
+out. Nothing earlier in the pruning machinery is used later
+except the conclusion.
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
+full s-blocks of input. When the gate is OPEN, a permutation is
+applied to the last block; when CLOSED, nothing happens. 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
@@ -799,8 +712,6 @@ 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.
@@ -835,8 +746,6 @@ 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:
@@ -864,12 +773,6 @@ 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.
@@ -886,8 +789,6 @@ 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,
@@ -929,7 +830,7 @@ tree of the arc-graph of S. Every complementary arc is LONG,
with length at least g/(3b). From each long arc delete exactly
one of its original edges, calling it the PIVOT of that arc.
The surviving original edges form a spanning tree T of S:
-each long arc of the complement closes one cycle, and we broke
+each long arc of the complement closes one cycle, and that cycle is broken
each one.
STEP 3 (total pivot crossings). Cyclically start W at a
@@ -991,7 +892,7 @@ with
A closed walk in Y based in layer i of length at most
C_* H_i / log_2 d stays in layers j with
H_j >= H_i - C_* H_i / log_2 d >= H_i / 2 for large ell.
-Property (e) of Part 6.8 says no nonempty cyclically
+Property (e) of the pruning ledger says no nonempty cyclically
nonbacktracking closed path in Y based at layer i has length at
most g_i = H_i / (100 log_2 d). So every embedded circuit in
the support of W has length greater than g_i, in particular
@@ -1000,8 +901,6 @@ length of W is at most C g, so Steps 1 - 6 apply: the
voltage-holonomy word in U_Z is not the identity matrix. This
is uniform 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
@@ -1033,7 +932,7 @@ gamma_X, gamma_Y in bar-E_Z and z in Z, and multiplication
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.
+group extension. That is the deck group F.
FACTS. F is countable, nilpotent, torsion-free, and finitely
generated. The integer coordinate z is its CENTRAL coordinate.
@@ -1041,23 +940,19 @@ 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
+with t in the circle T = R/Z. This gives
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
+e(t). The symbol u_t denotes 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
@@ -1101,9 +996,7 @@ has a unique preimage
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
+An ACTUAL element of a C*-algebra is needed, not a continuous
object. Sample J_epsilon on Delta_y bar-E_Z.
Let theta be a small nonzero real, epsilon = sign theta,
@@ -1133,9 +1026,7 @@ The lower bound 0 < is essential: if p_theta were 0, then
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
+One twist gives small trace as theta -> 0. Small trace is wanted
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:
@@ -1161,7 +1052,7 @@ 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
+theta regime. Define
p(t) = p_t (x) p_{t-delta}[V_t],
@@ -1173,8 +1064,6 @@ an equivariant projection. Its identity trace factorizes:
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
@@ -1234,11 +1123,9 @@ POSITIVE, and
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);
+ C_*, C_bullet from the geometry of the voltage cover;
m = an integer > L(201 C_*);
E_0, A_*, kappa satisfying the displayed inequalities;
theta_0 = small radius for the expansion (7.6);
@@ -1247,10 +1134,6 @@ Constants are now set once and never altered again:
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
@@ -1262,11 +1145,9 @@ properties:
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
+Once that is done, the geometry of the voltage cover 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 )
@@ -1278,8 +1159,8 @@ 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
+sides in Y^0, h_i(t) is the layer window,
+and p(t) is the small-trace projection. 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}
@@ -1305,12 +1186,10 @@ 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.
+beta_ell <= eps_ell -> 0 is the deleted fraction of the tower pruning.
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
+Given any fixed eta > 0 and ell large enough, the construction gives
voltages z_e in F on the positive edges e of Y,
matrices W_e in M_N(C) with ||W_e|| <= C_eta,
@@ -1344,8 +1223,6 @@ 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
@@ -1353,7 +1230,7 @@ The construction has five stages:
(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
+ functions J_epsilon^flat from the Schwartz construction, 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
@@ -1363,8 +1240,6 @@ The construction has five stages:
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
@@ -1380,15 +1255,15 @@ row-and-column sums are bounded by R. Then
where P_x retains the deck indices in B_x.
-Proof. Use the graded coordinate system of Part 7.3: every group
-coordinate carries a positive integer weight w_a, with only
-finitely many coordinates and only finitely many weights, all
-determined by m. Choose one integer T >= x and put
+Proof. Use the graded coordinate system on the nilpotent deck
+group: every group coordinate carries a positive integer weight
+w_a, with only finitely many coordinates and only finitely many
+weights, all determined by m. Choose one integer T >= x and put
B_x = prod over coordinates a of [-T^{w_a}, T^{w_a}] cap Z.
Write W for the sum of all weights and D for the number of
-coordinates; then D and W are fixed. We have
+coordinates; then D and W are fixed. Thus
|B_x| = prod_a (2 T^{w_a} + 1) <= 3^D T^W <= x^c for a fixed c.
Multiplying f in B_x by g right-shifts each coordinate. In a
@@ -1439,8 +1314,6 @@ 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.
@@ -1462,7 +1335,7 @@ Each X_j - EX_1 is bounded in norm by 2 B_0, and its coefficient
support lies inside the coordinate box of edge q^a. So the
coefficient row and column sums of Y are bounded by 2 B_0.
-Step 1 (finite test). Apply the box lemma of 8.4 with x = q,
+Step 1 (finite test). Apply the box lemma with x = q,
exponent B, and operator A = Y. This gives a coordinate box B_q
of size q^{c_0} with
@@ -1534,15 +1407,13 @@ ess-sup_t ||E_emp(t) - (h_i h_k p)(t)|| <= eps_emp
UNIFORMLY over the finitely many distributions. This proves the
lemma.
-8.6 COMPACT COEFFICIENT TEMPLATES
-
-We want a FINITE list of coefficient templates, uniform in the
+A FINITE list of coefficient templates is needed, uniform in the
layer index. Two replacements make this possible.
-COMPACT SUPPORT. the compactness result of Part 7 (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.
+COMPACT SUPPORT. The compactness result for Schwartz coefficient
+functions 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
@@ -1578,13 +1449,11 @@ 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,
+So for i < I, FINITELY MANY ordinary empirical M-lists are used,
one per template.
Blocks with i >= I. Introduce the subgroup
@@ -1596,7 +1465,7 @@ 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
+v in V_q. The finite magnetic representation R satisfies
R(sigma(y)) = 1 for y in Lambda_q.
Build the coset average
@@ -1623,9 +1492,7 @@ 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
+For each block (i,k), ONE distribution is specified 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.
@@ -1686,8 +1553,6 @@ PROPERTIES OF THE DISTRIBUTIONS.
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.
@@ -1730,14 +1595,12 @@ 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
+be the CENTERED edge sum. The aim is ||X_{ik}|| small, uniformly
in the layers.
MOMENTS. Expand
@@ -1751,7 +1614,6 @@ 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.
@@ -1767,8 +1629,6 @@ 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,
@@ -1787,20 +1647,14 @@ Therefore
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
+The voltage cover P just sampled is taken (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
+Notation: "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
+matrix size m of the unitriangular model. The two are unrelated.
Let z_e in F be the voltage of the positively oriented edge e of
Y. The voltage cover P has vertex set
@@ -1815,14 +1669,12 @@ 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
+GIRTH HYPOTHESIS (what the sampling 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
+Let S be the finite positive alphabet of Y. Define
Gamma = < S | label words of all closed walks in P >,
@@ -1840,10 +1692,8 @@ 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.
+It remains to show: every phi is injective on vertices and edges. If
+not, a contradiction to minimality follows.
Suppose phi identifies two distinct source vertices. Reading a
source path between them yields a closed image word in X, hence
@@ -1925,12 +1775,10 @@ chosen injectivity failure (they were assumed 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
+injective, the components of P embed genuinely 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,
+With a component C, a map phi, and a base vertex fixed,
"indexed copies" of X are defined up to a deck alignment:
(C, phi) ~ (delta_a C, phi delta_a^-1), a in F.
@@ -1973,8 +1821,6 @@ 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)).
@@ -1991,8 +1837,6 @@ 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)
@@ -2012,13 +1856,11 @@ is canonically identified with
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.
+correspondence h <-> a is an INJECTION. F is torsion-free by
+construction, so 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
@@ -2031,7 +1873,7 @@ 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
+of each copy permits replacement of 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
@@ -2060,16 +1902,13 @@ 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
+Combining the embedding, the disjoint domains with residual
+forest, the incidence bijection, and torsion-freeness proves the
+geometric theorem: the voltage cover P produces a finitely
+generated torsion-free group Gamma with indexed copies whose
+geometry is under full control.
-All constants are those produced by 9.3 -- 9.4:
+The constants are as follows:
D_0 = 10^6,
L* = 4 D_0 + 8 (about 4 * 10^6),
@@ -2077,21 +1916,16 @@ All constants are those produced by 9.3 -- 9.4:
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
-================================================================
+what makes the dependency ledger of parameter choices correct.
-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.
+A cover P now exists whose operator matches the model band.
+There is Gamma, finitely generated and 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
+The sampled cover operator T_P satisfies
|| T_P - Pi O Pi || <= eta,
@@ -2106,13 +1940,11 @@ 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,
+The choices also satisfy theta <= 1/2 and 2^{C_bullet} theta^2 <= 1/2,
which just amounts to eta small enough (a fixed choice).
-10.1A TWO HILBERT-SPACE LEMMAS
-
Both are used repeatedly. Both are stated for a general complex
-Hilbert space; the operator content of Part 10 is the same in
+Hilbert space; the operator content of the transfer to the Cayley graph is the same in
each use.
COMPRESSION LEMMA. Let A = A* be an operator on H with ||A||
@@ -2239,7 +2071,7 @@ circle has length 2 pi r_0, so || F_A - F || <= u/(r_0 - u), which
is the first display of (b). If a real t satisfies
dist(t, spec B) > u, the same Neumann trick shows t is away from
spec A. Since spec B on F H is {1} and spec B on (1 - F) H lies
-in [-2, 1 - b], we get the two interval statements of (b).
+in [-2, 1 - b], the two interval statements of (b) follow.
For (c): if u < r_0/2 then || F_A - F || <= u/(r_0 - u) < 1. For
y in F H,
@@ -2258,8 +2090,6 @@ is a partial isometry from F H to F_A H, built from F and F_A
alone, preserving common equivariance. The identity-trace
equality is again the adjoint-square identity.
-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
@@ -2267,12 +2097,12 @@ the resolvent over the circle |z - 1| = r_0 = chi/3. Then
|| (T_P - I) E_P || <= alpha,
T_P on (1 - E_P) H <= (1 - chi + alpha) I.
-Proof. Set a = 0 and b = chi in the compression lemma of 10.1A,
+Proof. Set a = 0 and b = chi in the compression lemma,
with A = O, E = E_*, and J = Pi. Its three hypotheses hold:
|| (O - I) E_* || = 0 because O E_* = E_* (Section 8.1); O on
(1 - E_*)H is at most (1 - chi)I by the same model band; and
eps := || (1 - Pi) E_* || <= sqrt(beta_ell) by the compression
-estimate in 8.1, with sqrt(beta_ell) <= 1/2 for large ell.
+estimate, with sqrt(beta_ell) <= 1/2 for large ell.
The compression lemma gives a projection E_*^Pi (the range of
Pi E_*) on Pi H and an operator B_Pi on Pi H with
@@ -2282,11 +2112,11 @@ Pi E_*) on Pi H and an operator B_Pi on Pi H with
Since ||O|| <= 1 (Section 8.1) and B_Pi is the direct sum of 1
on E_*^Pi H and (Pi - E_*^Pi)O(Pi - E_*^Pi) on the complementary
-corner of Pi H, we have ||B_Pi|| <= 1.
+corner of Pi H, ||B_Pi|| <= 1.
Extend B_Pi by zero on (1 - Pi)H, obtaining an operator B on H.
The projection E_*^Pi reduces B, and B on (1 - E_*^Pi)H lies
-between -1 and 1 - chi. The input from 10.1 closes the
+between -1 and 1 - chi. The cover estimate closes the
comparison:
|| T_P - B ||
@@ -2309,8 +2139,6 @@ perturbation lemma gives
The three displays of the statement follow immediately.
-10.3 EXACT FIBER TRACE
-
For almost every t in the circle T = R / Z,
tr_id E_P(t)
@@ -2321,7 +2149,7 @@ Proof. Work fiberwise in the central Fourier decomposition.
The direct integral respects resolvents and contour integrals,
so E_P(t) is the contour projection of T_P(t) for almost every t.
-OUTSIDE THE WINDOW. By Part 8 (G3), ||T_P(t)|| <= eta < r_0
+OUTSIDE THE WINDOW. By the off-window bound, ||T_P(t)|| <= eta < r_0
almost everywhere. So the spectrum of T_P(t) lies in
[-eta, eta], which is at distance r_0 - eta > 0 from C_{r_0},
and the contour integral vanishes.
@@ -2329,24 +2157,24 @@ and the contour integral vanishes.
INSIDE THE WINDOW. Throughout this paragraph fix a t in
(0, delta) at which the above fiberwise identities hold. The
same comparison as 10.2 gives ||T_P(t) - B(t)|| <= alpha, with
-B(t) extended by zero off Pi as in 10.2, and the operator
-hypotheses of 10.1A are satisfied with a = 0, b = chi, and
+B(t) extended by zero off Pi, and the operator
+hypotheses are satisfied with a = 0, b = chi, and
u = alpha < r_0/2 = chi/6.
-Apply the compression lemma of 10.1A (e) to A = O(t),
+Apply the compression lemma, part (e), to A = O(t),
E = E_*(t), J = Pi. It gives
tr_id E_*(t) = tr_id E_*^Pi(t).
-Apply the perturbation lemma of 10.1A (c) to A = T_P(t) and
+Apply the perturbation lemma, part (c), to A = T_P(t) and
B = B(t). It gives an equivariant partial isometry V(t) with
V(t)* V(t) = E_*^Pi(t) and V(t) V(t)* = E_P(t), so by the
adjoint-square identity of the identity trace,
tr_id E_*^Pi(t) = tr_id E_P(t).
-Combining with the model-band trace from 8.1
-(tr_id E_*(t) = tr_id p(t)), we obtain
+Combining with the model-band trace
+(tr_id E_*(t) = tr_id p(t)), this gives
tr_id E_P(t) = tr_id p(t) for 0 < t < delta.
@@ -2355,11 +2183,11 @@ theoretic density: both sides are measurable and finite, and
the equalities proved above hold on the full-measure set where
the fiberwise operators are defined.
-NONVANISHING. Part 7.7 shows tr_id p(t) is at least a positive
+NONVANISHING. The magnetic shift shows tr_id p(t) is at least a positive
constant times t^{n_f} (delta - t)^{n_f} on 0 < t < delta, so
strictly positive there. The window (0, delta) has positive
measure. Since tr_id E_P(t) = tr_id p(t) on that window, and
-tr_id E_P(t) = 0 outside, we get tr_id E_P > 0. A projection
+tr_id E_P(t) = 0 outside, so tr_id E_P > 0. A projection
with positive identity trace is nonzero, so E_P != 0.
DIAGONAL INVARIANCE. The operator T_P is F-equivariant: its
@@ -2374,8 +2202,6 @@ base vertex v of Y,
Therefore the C^N-matrix trace of the diagonal of E_P at (v, f)
depends only on the base vertex v.
-10.4 WEIGHTED SIZE S
-
For a surviving base vertex v, define
e_v = Tr_{C^N}( E_P( (v, 1_F) ) ) >= 0,
@@ -2386,9 +2212,10 @@ be the layer weight. Define the weighted size
S = sum over surviving base vertices v of w_{i(v)} * e_v.
LOCALIZATION. Write A_0(t) = Pi O(t) Pi and
-Delta(t) = T_P(t) - A_0(t). From 8.1, ||Delta(t)|| <= eta. From
-the compression lemma of 10.1A with a = 0, b = chi, applied as in
-10.2, we have ||A_0(t) - B(t)|| <= 20 sqrt(beta_ell) <= alpha/2,
+Delta(t) = T_P(t) - A_0(t). The cover estimate gives
+||Delta(t)|| <= eta. The compression lemma with a = 0 and b = chi,
+applied to the model band, gives
+||A_0(t) - B(t)|| <= 20 sqrt(beta_ell) <= alpha/2,
where B(t) is the compression-lemma operator on Pi H. B(t) has
spectrum in [-1, 1-chi] together with eigenvalue 1 at the
reducing projection E_*^Pi(t).
@@ -2439,9 +2266,9 @@ k = dist(i, J(t)). For k = 0 there is no vanishing segment to
remove; the same bound is valid.
WEIGHTED TRACE BOUND. Fix t in (0, delta). Write tr_id for the
-identity trace of 10.3 (sum over base and matrix indices, or over
+identity trace (sum over base and matrix indices, or over
its central Fourier fiber; the countable amplification is handled
-by the adjoint-square identity-trace lemma of Part 7). Since P_i and E_P(t) are equivariant, the
+by the adjoint-square identity-trace lemma). Since P_i and E_P(t) are equivariant, the
adjoint-square identity and positivity give
tr_id(P_i E_P(t) P_i)
@@ -2464,7 +2291,7 @@ assigned to a given j in J(t). Therefore
( 1 + 2 sum_{k >= 1} (2^{C_bullet} theta^2)^k )
<= 6 max_{j in J(t)} w_j,
-using 2^{C_bullet} theta^2 <= 1/2 from 10.1 and
+using 2^{C_bullet} theta^2 <= 1/2 and
sum_{k>=1} x^k = x/(1-x) <= 1 when x <= 1/2.
Multiply the per-layer bound by w_i and sum over i. For each t in
@@ -2476,8 +2303,8 @@ the open window,
where e_v(t) is the C^N matrix trace of the v-diagonal of
E_P(t), and mu_ell = sup_{0<t<delta, j in J(t)}
-w_j tr_id p(t) is the small weighted trace of Part 7. Off the
-window E_P(t) = 0 from 10.3, so the same bound holds trivially.
+w_j tr_id p(t) is the small weighted trace of the layer window. Off the
+window E_P(t) = 0, so the same bound holds trivially.
FIBER AVERAGE. Central Fourier decomposition identifies e_v with
the integral over t of e_v(t). Tonelli's theorem applies to the
@@ -2490,9 +2317,7 @@ nonnegative diagonal, so
Since mu_ell -> 0 as ell -> infinity, S -> 0.
-10.5 THE CAYLEY OPERATOR T
-
-Use the finite alphabet S from Part 8 (original label plus matrix
+Use the finite alphabet S (original label plus matrix
weight tag). For sigma in S with matrix weight W_sigma and
degree normalizer D_sigma = D_{ik}, define
@@ -2507,14 +2332,13 @@ The actual Cayley operator is
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).
+a vertex u in Q is e_v where v is u's base vertex (from the
+copy diagonal).
-BALL TRIMMING. For the assigned domain D_Q and trimming ball
-B(Q) from Part 9,
+BALL TRIMMING. For the assigned domain D_Q and trimming
+ball B(Q),
|| 1_{B(Q)} E_Q ||_{HS}^2 <= S.
@@ -2527,12 +2351,12 @@ projection, E_Q^* E_Q = E_Q, so
= Tr( 1_{B(Q)} E_Q 1_{B(Q)} )
= sum over u in B(Q) of tr_{C^N}( E_Q(u, u) ).
-The diagonal identity from 10.3 gives
+The diagonal identity gives
tr_{C^N}( E_Q(u, u) ) = e_{type(u)}
for every u in Q, where type(u) is the base vertex under the
embedding Q -> X. A lift u of the base vertex v lies over the
same layer as v: the layer is preserved by the graph embedding.
-The weight estimate of Part 9 says that when the ball B(Q)
+The weight estimate for balls says that when the ball B(Q)
lies over layer i, its whole vertex set satisfies
|V(B(Q))| <= w_i = 2^{C_bullet H_i}.
@@ -2557,8 +2381,6 @@ C_Q that satisfies
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,
@@ -2572,7 +2394,7 @@ bounded operators on l^2(Gamma) (x) C^N. From 10.6,
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.
+any assigned D_Q). By disjointness of the assigned domains 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
@@ -2626,8 +2448,6 @@ 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 of F_D. By 10.7,
||T - C|| <= 3 alpha.
@@ -2642,7 +2462,7 @@ most ||T - C|| away from the spectrum of C. So
spectrum(T) contained in
(-infinity, 1 - chi + 4 alpha] UNION [1 - 3 alpha, 1 + 3 alpha].
-NONVANISHING. Since F_D = oplus_Q F_Q and each F_Q != 0 by 10.6,
+NONVANISHING. Since F_D = oplus_Q F_Q and each F_Q != 0 by the trimming estimate,
the upper interval [1 - 3 alpha, 1 + 3 alpha] is nonempty.
GAP. The number 1 - chi/4 lies strictly between the two
@@ -2662,8 +2482,6 @@ F_D, which is nonzero. It belongs to M_N(C_r*(Gamma)) by the
continuous functional calculus applied to the self-adjoint
element T of that algebra.
-10.9 TRACE BOUND FOR p_0
-
This is the subtle part: how to bound the unnormalized trace of
p_0 from above.
@@ -2672,7 +2490,7 @@ 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).
+ (incidence identification).
Let E = oplus_Q E_Q. Its identity trace is
@@ -2715,7 +2533,7 @@ Since I_Q is finite, Cauchy-Schwarz on its sum gives
|| E_Q( xi|_{I_Q} ) ||^2
<= |B(Q)| * sum over u in I_Q of e_{v(Q,u)} ||xi(u)||^2.
-The weight estimate of Part 9 says the whole ball B(Q) has size
+The weight estimate for balls says the whole ball B(Q) has size
at most w_{i(u)} for every vertex u of the ball; and the layer of
a lift u in V(Q) is the layer of its base type v(Q,u). So
|B(Q)| <= w_{i(v(Q,u))} uniformly. Rearranging nonnegative
@@ -2726,7 +2544,7 @@ terms,
( sum over { Q : u in I_Q } of
w_{i(v(Q,u))} e_{v(Q,u)} ).
-By the incidence bijection of Part 9, for each fixed group vertex
+By the incidence bijection, for each fixed group vertex
u and each base vertex v in V(Y) there is exactly one indexed
copy Q with u in V(Q) and type_Q(u) = v. Therefore the
parenthesized sum over Q equals
@@ -2757,7 +2575,7 @@ LOST-RESTRICTIONS gives
SANDWICH. Use the sum-over-Q bound just obtained: for xi in K,
sum over Q of || E_Q( xi|_{D_Q} ) ||^2 <= S ||xi||^2. The local
-spectral estimate from 10.2 gives, for every vector y on a copy Q,
+spectral estimate gives, for every vector y on a copy Q,
< T_Q y, y > <= (1 - chi + alpha) || y ||^2
+ chi || E_Q y ||^2.
@@ -2774,7 +2592,7 @@ most alpha by 10.7. Therefore
because S <= 1/1000 and f_ell <= alpha.
-On the OTHER HAND, on p_0 H the spectral split of 10.8 gives
+On the OTHER HAND, on p_0 H the spectral split gives
< T xi, xi > >= (1 - 3 alpha) ||xi||^2.
@@ -2815,7 +2633,7 @@ l^2(Gamma) x C^N has strictly positive identity trace. If every
identity-diagonal entry were zero, then the projection's columns
at the identity would be zero; equivariance would make every
group translate of those columns zero, forcing the projection
-itself to be zero. Since p_0 != 0 by 10.8 and the identity trace
+itself to be zero. Since p_0 != 0 and the identity trace
of a nonzero equivariant projection is positive,
Tr_H( p_0 ) > 0.
@@ -2825,30 +2643,22 @@ CONCLUSION. Combining,
0 < tau_{Gamma,N}(p_0) = Tr_H( p_0 ) <= S_0 <= S
< 1/2.
-10.10 WHAT PART 10 PRODUCES
-
-We have produced:
+The construction has 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
-================================================================
+That is EXACTLY the matrix target that the scalar reduction needed.
-11.1 PUTTING THE PIECES TOGETHER
-
-We now have, from Parts 6 -- 10:
+What has been assembled:
- 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.
+That is exactly the "matrix target" that the scalar reduction needs.
Apply Step 1 with Lambda = Gamma and p = the matrix projection
p_0. Let
@@ -2862,7 +2672,7 @@ 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,
+This e is a projection, and by the trace computation,
tau_{G_proj}(e) = tau_{Gamma,N}(p_0) in (0, 1/2).
@@ -2872,34 +2682,28 @@ 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.
+is not an integer. Lueck (Inventiones 149 (2002), 123-152) and
+Baum-Guentner-Willett (local file
+bgw-2016-exactness-kadison-kaplansky.pdf) 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.
+map for Gamma.
(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 complex group ring C[G_proj] has no scalar idempotents other
+than 0 and 1: the Bass trace conjecture holds for this group by
+the same trace methods. The projection e therefore lies in the
+C*-completion C_r*(G_proj) but not in the algebraic group ring
+C[G_proj], so the norm closure is strictly larger than the
+algebraic ring even at the level of projections.
+
+The stable form of the conjecture, about projections in
+M_N(C_r*(G)) for every N, fails already at the matrix projection
+p_0; the scalar projection e settles the scalar form.
The argument is an EXISTENCE proof. It does not claim:
@@ -2909,158 +2713,8 @@ The argument is an EXISTENCE proof. It does not claim:
- 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
+Every parameter is chosen in the stated order of parameter fixing, 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
-================================================================