================================================================
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
================================================================