diff 52024db..21759e1 kadison_kaplansky_projection_counter_v2.txt
diff --git a/kadison_kaplansky_projection_counter_v2.txt b/kadison_kaplansky_projection_counter_v2.txt
index 93fdfe1..286e5e7 100644
--- a/kadison_kaplansky_projection_counter_v2.txt
+++ b/kadison_kaplansky_projection_counter_v2.txt
@@ -430,13 +430,18 @@ 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.
+Proof. Choose one representative v from every left Lambda-orbit
+Lambda v in Gamma. Then l^2(Gamma) is the orthogonal direct sum
+of the subspaces l^2(Lambda v). The unitary delta_g -> delta_{gamma v}
+(from l^2(Lambda) onto l^2(Lambda v)) conjugates the Lambda-action
+on l^2(Lambda) to its restriction to l^2(Lambda v). Therefore
+the restriction of the left regular representation of Gamma to
+Lambda is a direct sum of copies of the left regular representation
+of Lambda. So every x in C[Lambda] has the same operator norm on
+l^2(Lambda) as on l^2(Gamma), and the identity diagonal
+<x delta_1, delta_1> = tau_Lambda(x) equals the Gamma identity
+diagonal. Taking operator-norm closures gives C_r*(Lambda) inside
+C_r*(Gamma); tensoring with M_N(C) gives the matrix version.
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
@@ -893,24 +898,101 @@ 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_*).
+Proof. Seven steps, each elementary.
+
+STEP 1 (support has few arcs). Let S be the union of the edges
+of W. Each vertex of S has degree at least two: a degree-one
+vertex in the closed walk, including at the cyclic closure,
+would force W to traverse some edge and immediately reverse it,
+a backtrack. Write b = b_1(S) for the first Betti number;
+then b >= 1. If every vertex of S has degree two, S is a
+single circle; treat it as one arc. Otherwise suppress all
+degree-two vertices. Every vertex of the suppressed graph has
+degree at least three, so 2e >= 3v. Together with
+b = e - v + 1 this gives v <= 2b - 2 and e <= 3b - 3, all
+purely combinatorial (no planarity needed). In either case S
+decomposes into ARCS, meaning maximal edge paths between branch
+points, and the number of arcs is at most 3b.
+
+STEP 2 (a spanning tree with pivots). Call an arc SHORT if
+its length is less than g/(3b). The short arcs cannot contain
+a circuit: a circuit made entirely of short arcs has length
+less than 3b times g/(3b) = g, contradicting girth(S) > g. So
+the short arcs form a forest. Extend that forest to a spanning
+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 one.
+
+STEP 3 (total pivot crossings). Cyclically start W at a
+retained vertex of T. Because W is nonbacktracking, it cannot
+turn around inside an arc: to leave an arc on the side it
+entered, W would have to traverse the last edge back, a
+backtrack. So once W enters an arc it traverses that arc
+completely, from one end to the other. Each traversal of a
+LONG arc crosses exactly one pivot; each traversal of a short
+arc crosses none. Let N be the number of long-arc traversals
+of W. Each such traversal contributes length at least g/(3b)
+to W, so the sum of their lengths is at least N g/(3b). That
+sum cannot exceed the total length of W, which is at most Cg.
+Therefore N g/(3b) <= Cg, i.e. N <= 3bC. Since each pivot
+crossing belongs to one long-arc traversal, the TOTAL number
+of pivot crossings of W is at most 3bC <= 3 C B(C) = L(C).
+
+STEP 4 (a nontrivial reduced word). Collapse T to a point.
+The result is a bouquet of b circles, with one circle per
+pivot label. Collapsing a spanning tree is a homotopy
+equivalence, and the image of W is a word on the b pivots. A
+nonempty cyclically reduced closed walk in a graph is not
+null-homotopic, so this image word is nontrivial in the free
+group on the pivots. Free reduction can only shorten a word,
+so the reduced length k of the image word is at least one and
+at most the number of pivot crossings, hence at most L(C).
+
+STEP 5 (a matrix cannot be trivial). Write the reduced word
+in maximal syllables w = x_{j_1}^{a_1} ... x_{j_k}^{a_k}, where
+each a_s is nonzero and successive j_s differ. In formal
+noncommutative power series, set x_j = 1 + Z_j, using
+(1+Z)^a = sum over h >= 0 of binom(a,h) Z^h, which holds also
+for a < 0 as a finite expansion modulo Z^m. The monomial
+Z_{j_1} ... Z_{j_k} in w has coefficient prod_s a_s, nonzero:
+to make that monomial, each syllable must supply exactly one of
+its k letters, and no two adjacent syllables share a variable,
+so no other contribution reaches that monomial. For
+m > k = length(w), put Z_j = sum over { s : j_s = j } of
+E_{s, s+1} in M_m(Z). Because the indices 1, 2, ..., k+1 are
+strictly increasing, the (1, k+1)-entry of w(1 + Z_j) equals
+prod_s a_s. It is nonzero, so w(1 + Z_j) != 1.
+
+STEP 6 (invariance under fixed factors). For strictly upper
+triangular X, the inverse of 1 + X is
+sum over h = 0 to m-1 of (-X)^h. Every entry of every word in
+upper unitriangular matrices and their inverses is therefore a
+polynomial in the upper entries of the edge variables, and each
+of its monomials uses at most m - 1 such entries. Multiplying
+by a fixed upper unitriangular matrix on either side is a
+polynomial change of coordinates with polynomial inverse, so
+nonidentity is preserved.
+
+STEP 7 (applying it in the tower). Choose m once and for all
+with
-Then any closed walk in Y with length at most C_* H_i / log_2 d
-is detected by U_Z, uniformly in i.
+ m > L(201 C_*), L(C) = ceil(3 C B(C)),
+ B(C) = max(1, ceil(20 C))^2.
+
+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
+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
+greater than g = H_i / (201 log_2 d). With C = 201 C_*, the
+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
@@ -1292,13 +1374,60 @@ row-and-column sums are bounded by R. Then
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.
+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
+
+ 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
+|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
+coordinate of weight j the shift differs from f's coordinate by
+a sum of finitely many graded monomials, each having total
+weight between 1 and j and containing at least one coordinate of
+g; that coordinate of g is bounded by x^a. The multiplication
+polynomials of the group law are fixed, so for a fixed integer P
+every such monomial is at most C x^P T^{j-1} in absolute value
+once T >= x. Hence a point f whose weight-j coordinate is
+farther than C x^P T^{j-1} from both faces of B_x stays inside
+B_x after right multiplication by g. The bad region for weight
+j has relative measure at most C' x^P / T, and there are D
+coordinates, so the union over all weights has relative measure
+at most C'' x^P / T. Taking T = ceil(x^{P+B+2}) makes this at
+most x^{-B}. Right multiplication is a bijection, so the same
+estimate, doubled, bounds the relative size of
+B_x sym-diff B_x g by C x^{-B}.
+
+For the norm assertion, let P_s be the projection onto the left
+translate s B_x. Right convolution commutes with left
+translation, so every P_s A P_s has the same norm as P_x A P_x.
+For a finitely supported vector xi,
+
+ sum over s in F of ||P_s xi||^2 = |B_x| ||xi||^2,
+
+because any fixed deck index appears in exactly |B_x| translates
+s B_x. Average <AP_s xi, P_s xi> over s, dividing by |B_x|. The
+coefficient of rho(g) in the resulting operator A_x equals
+
+ alpha_x(g) = |B_x intersect B_x g^{-1}| / |B_x|,
+
+and |1 - alpha_x(g)| <= C x^{-B} by the previous step. So the
+averaged quadratic form satisfies |<A_x xi, xi>| <= ||P_x A P_x||
+||xi||^2. Since the row and column coefficient sums of A are
+bounded by R and the same bounds hold after averaging, the scalar
+Schur test gives ||A - A_x|| <= C x^{-B} R. Density plus the
+quadratic form identity give the claimed two-sided inequality.
+Do nothing new for the amplification: the argument never touched
+the auxiliary index set, only the deck indices of F.
+
+For a rectangular S : J -> J' of coefficient row sums bounded by
+R, apply the above to the self-adjoint block operator
+(0 S; S* 0) on J disjoint-union J', whose norm equals ||S||, with
+R replaced by 2R.
The key consequence: the full regular norm of a block operator
is determined up to C x^(-B) * (row sum) by ONE finite matrix
@@ -1317,13 +1446,87 @@ 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.
+Proof. Fix one of the finitely many distributions. Take M
+independent samples X_1, ..., X_M and write the empirical
+deviation of the mean
+
+ Y = M^{-1} sum_{j=1}^{M} (X_j - E X_1).
+
+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,
+exponent B, and operator A = Y. This gives a coordinate box B_q
+of size q^{c_0} with
+
+ ||Y|| <= ||P_q Y P_q|| + C q^{-B} (2 B_0).
+
+Take B large enough that the second term is at most eps_emp / 2
+for all large ell; this B is a fixed constant. Set
+Y_q = P_q Y P_q and let d_fin <= q^{c_1} be the dimension of the
+range of P_q, including matrix indices. It remains to show that
+some choice of the M samples makes ||Y_q|| <= eps_emp / 2.
+
+Step 2 (moment expansion). Set Z_j = P_q (X_j - EX_1) P_q. The
+Z_j are independent, self-adjoint, with mean zero and
+||Z_j|| <= 2 B_0, and
+
+ Y_q = M^{-1} sum_{j=1}^{M} Z_j.
+
+For any integer K >= 1,
+
+ E Tr(Y_q^{2K})
+ = M^{-2K} sum_{j_1,...,j_{2K} in {1,...,M}}
+ E Tr(Z_{j_1} ... Z_{j_{2K}}).
+
+If some index j_0 occurs exactly once among j_1, ..., j_{2K},
+that summand is ZERO: condition on all the other Z_j, and by
+independence and E Z_{j_0} = 0 the factor Z_{j_0} pulls out of
+the trace and vanishes. So only sequences in which every index
+used appears at least twice contribute. Such a sequence uses
+at most K distinct indices, so the number of surviving
+sequences is at most
+
+ sum_{a=1}^{K} M^a a^{2K}.
+
+For each one,
+|Tr(Z_{j_1} ... Z_{j_{2K}})| <= d_fin (2 B_0)^{2K}. Using
+M^a <= M^K and a^{2K} <= K^{2K},
+
+ E Tr(Y_q^{2K})
+ <= d_fin (2 B_0)^{2K} M^{-2K} sum_{a=1}^{K} M^a a^{2K}
+ <= d_fin (2 B_0)^{2K} K (K^2 / M)^K.
+
+Step 3 (choice of K and the moment bound). Put K = ceil(log_2
+q), so K = log_2 q + O(1) and, since M = q^{1/(32 A_*)},
+
+ log_2 E Tr(Y_q^{2K})
+ <= c_1 log_2 q + K log_2(4 B_0^2 K^2 / M) + log_2 K
+ = c_1 log_2 q + O((log_2 q)(log_2 log_2 q))
+ - (1/(32 A_*)) (log_2 q)^2.
+
+The negative quadratic term in log_2 q dominates, so
+E Tr(Y_q^{2K}) tends to zero super-polynomially fast as ell
+grows.
+
+Step 4 (Markov and existence). For any eps_emp > 0,
+
+ P( ||Y_q|| > eps_emp / 2 )
+ <= (2 / eps_emp)^{2K} E Tr(Y_q^{2K}),
+
+and (2/eps_emp)^{2K} grows only polynomially in q. So this
+probability tends to zero with ell. Hence for large ell there
+exists a sample tuple with ||Y_q|| <= eps_emp / 2 for the chosen
+distribution. There are only C(1 + log q) fixed distributions,
+and the union bound over them still fails with probability
+tending to zero as ell grows. So a single simultaneous choice
+exists.
+
+Combining Step 4 with Step 1 gives
+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
@@ -1880,6 +2083,155 @@ Fix eta <= alpha / 2. Let
We also require theta <= 1/2 and 2^{C_bullet} theta^2 <= 1/2,
which just amounts to eta small enough (a fixed choice).
+10.1A 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
+each use.
+
+COMPRESSION LEMMA. Let A = A* be an operator on H with ||A||
+<= 2, and let E be a reducing projection (AE = EA) with
+
+ || (A - I) E || <= a, A on (1 - E)H <= (1 - b) I,
+
+where a >= 0 and 0 < b < 1. Let J be any orthogonal projection
+with eps := || (1 - J) E || <= 1/2. Then
+
+(a) JE has closed range; write E^J for its range projection.
+
+(b) On J H define B_J = E^J + (J - E^J) A (J - E^J). Then
+
+ || J A J - B_J || <= a + 20 eps, || B_J || <= 2,
+ B_J on (J - E^J) H <= (1 - b) I.
+
+(c) There is a partial isometry V from E H to E^J H, with
+V* V = E, V V* = E^J, and || V - E || <= 2 eps.
+
+(d) If A, E, J commute with a unitary (or projective unitary)
+action, then so do V, E^J, and B_J.
+
+(e) If there is an identity trace with Tr(X* X) = Tr(X X*) for
+every commuting bounded X, then E and E^J have equal identity
+trace.
+
+All statements include the case E = 0.
+
+Proof. For x in E H, use E x = x and ||(1 - J) x||^2 <= eps^2
+||x||^2 to get
+
+ <x, E J E x> = ||J x||^2 >= (1 - eps^2) ||x||^2.
+
+So (1 - eps^2) E <= E J E <= E on E H, and E J E is invertible
+inside the E corner. Define
+
+ V = J E (E J E)^{-1/2}
+
+on E H. Then V* V = E by the functional calculus identity
+(EJE)^{-1/2} (EJE) (EJE)^{-1/2} = E, and V V* = E^J because the
+range of V is J E H. Since the spectrum of E J E lies in
+[1 - eps^2, 1], the same calculus gives
+|| (E J E)^{-1/2} - E || <= 2 eps^2, so
+
+ || V - E || <= || J E - E || + || J E || * 2 eps^2
+ <= eps + 2 eps^2 <= 2 eps.
+
+This proves (a) and (c).
+
+For (b): if y in (J - E^J) H, then y is orthogonal to J E H and
+y = J y, so <y, E x> = <y, J E x> = 0 for every x; that is,
+E y = 0. In particular A E lands in E H and (J - E^J) A E = 0.
+
+Transport A to the E^J corner by V: V* A V = E^J +
+V* (A - E) V, so ||V* A V - E^J|| <= a + 2 ||A|| ||V - E|| <= a +
+8 eps. And
+
+ || (J - E^J) A V ||
+ = || (J - E^J) A (V - E) || <= 4 eps,
+
+using (J - E^J) A E = 0. In the orthogonal splitting
+J H = E^J H disjoint-union (J - E^J) H, both J A J and B_J are
+2-by-2 blocked. Their (1,1) blocks agree on the upper space up
+to a + 8 eps; their (1,2) and (2,1) blocks are adjoints of each
+other and each of norm at most 4 eps; their (2,2) blocks agree
+exactly. Hence
+
+ || J A J - B_J || <= (a + 8 eps) + 2 (4 eps) <= a + 20 eps.
+
+The B_J upper bound follows from the same orthogonal form and
+the hypothesis on A, and ||B_J|| <= 2 follows directly.
+
+For (d): every step used the functional calculus on E J E and
+the polar decomposition, and both are compatible with any
+equivariance A, E, J already share.
+
+For (e): apply the adjoint-square identity
+Tr(V* V) = Tr(V V*) to V; the two sides are Tr(E) and Tr(E^J).
+
+PERTURBATION LEMMA. Suppose B = B* has a reducing projection F
+with B F = F, and B on (1 - F) H has spectrum in [-2, 1 - b] for
+some 0 < b < 1. Let r_0 satisfy 0 < r_0 <= b/2 and r_0 < 1. If
+A = A* and ||A - B|| <= u < r_0, then:
+
+(a) The circle C_{r_0} = {z : |z - 1| = r_0} lies in the
+resolvent set of both A and B.
+
+(b) Let F_A be the contour projection of A around C_{r_0}. Then
+
+ || F_A - F || <= u / (r_0 - u),
+ spec(A on F_A H) lies in [1 - u, 1 + u],
+ spec(A on (1 - F_A) H) lies in [-2 - u, 1 - b + u].
+
+(c) If u < r_0/2, F_A and F are equivalent by a partial isometry
+belonging to the algebra generated by them. If A and B commute
+with a unitary action, that partial isometry does too, and F_A
+and F have equal identity traces.
+
+Proof. Any z on C_{r_0} is at distance at least r_0 from
+spec(B), because the complementary spectrum of B is contained in
+[-2, 1 - b] and r_0 <= b/2. So || (z - B)^{-1} || <= 1/r_0
+there. The Neumann series
+
+ (z - A)^{-1} = sum over j >= 0 of
+ ((z - B)^{-1} (B - A))^j (z - B)^{-1}
+
+converges geometrically, because ||(z - B)^{-1}|| ||A - B|| <=
+u / r_0 < 1, with sum bound
+(1/r_0) / (1 - u/r_0) = 1/(r_0 - u). This proves (a).
+
+The resolvent identity
+
+ (z - A)^{-1} - (z - B)^{-1}
+ = (z - A)^{-1} (A - B) (z - B)^{-1}
+
+integrates around C_{r_0} to give
+
+ F_A - F = (1/2 pi i) integral over C_{r_0} of
+ (z - A)^{-1} (A - B) (z - B)^{-1} dz.
+
+The integrand has norm at most (1/(r_0 - u)) u (1/r_0), and the
+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).
+
+For (c): if u < r_0/2 then || F_A - F || <= u/(r_0 - u) < 1. For
+y in F H,
+
+ || F_A y || >= (1 - ||F_A - F||) ||y||,
+
+so F_A F is bounded below on F H. Its range is all of F_A H: a
+vector x in F_A H orthogonal to F_A F H would satisfy F x = 0
+(shift by one line of the same argument), and then
+||x|| = ||(F_A - F) x|| <= ||F_A - F|| ||x||, contradiction. So
+the polar normalization
+
+ F_A F (F F_A F)^{-1/2}
+
+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
@@ -1889,15 +2241,47 @@ 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.
-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.
+Proof. Set a = 0 and b = chi in the compression lemma of 10.1A,
+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.
+
+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
+
+ || Pi O Pi - B_Pi || <= 20 sqrt(beta_ell),
+ B_Pi on (Pi - E_*^Pi)H <= (1 - chi)I.
+
+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.
+
+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
+comparison:
+
+ || T_P - B ||
+ <= || T_P - Pi O Pi || + || Pi O Pi - B_Pi ||
+ <= eta + 20 sqrt(beta_ell)
+ <= alpha/2 + alpha/2 = alpha.
+
+Set u = alpha. Since 0 < u = chi/1000 < chi/3 = r_0 <= chi/2 and
+r_0 < 1, the perturbation lemma applies to A = T_P and B.
+
+Let E_P be the contour projection of T_P around C_{r_0}. The
+perturbation lemma gives
+
+ || E_P - E_*^Pi || <= u / (r_0 - u), which is small;
+
+ spec(T_P on E_P H) lies in [1 - alpha, 1 + alpha];
+ spec(T_P on (1 - E_P)H) lies in [-1 - alpha, 1 - chi + alpha];
+
+ || T_P || <= ||B|| + u <= 1 + alpha < 2.
+
+The three displays of the statement follow immediately.
10.3 EXACT FIBER TRACE
@@ -1907,15 +2291,62 @@ For almost every t in the circle T = R / Z,
= 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.
+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
+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.
+
+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
+u = alpha < r_0/2 = chi/6.
+
+Apply the compression lemma of 10.1A (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
+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
+
+ tr_id E_P(t) = tr_id p(t) for 0 < t < delta.
-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.
+The trace identity for almost every t follows by measure-
+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
+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
+with positive identity trace is nonzero, so E_P != 0.
+
+DIAGONAL INVARIANCE. The operator T_P is F-equivariant: its
+edge voltages and its matrix weights are unchanged by left
+translation. Its resolvent and contour integral are therefore
+F-equivariant too. In particular, for every f in F and every
+base vertex v of Y,
+
+ < E_P delta_{(v,f)}, delta_{(v,f)} >
+ = < E_P delta_{(v,1)}, delta_{(v,1)} >.
+
+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
@@ -1928,23 +2359,110 @@ 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
+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,
+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).
+
+Bound the resolvent of A_0(t) on the circle C_{r_0} = { |z - 1|
+= r_0 }. Any z on the circle is at distance at least r_0 from
+spec B(t), so || (z - B(t))^{-1} || <= 1/r_0. Since
+||A_0(t) - B(t)|| <= alpha/2 < r_0, the Neumann series gives
+
+ R_0(z,t) := (z - A_0(t))^{-1},
+ || R_0(z,t) || <= (r_0 - alpha/2)^{-1} <= C_chi = 4/chi.
+
+The operator A_0(t) is supported on the active layers J(t), so
+its resolvent R_0(z,t) equals z^{-1} I on the orthogonal sum of
+the other layers and has no matrix entries between that sum and
+J(t). The resolvent series for T_P(t) in terms of A_0(t) is
+
+ (z - T_P(t))^{-1}
+ = sum over n >= 0 of R_0(z,t) (Delta(t) R_0(z,t))^n,
+
+and it converges because ||Delta(t) R_0(z,t)|| <= eta C_chi
+= theta < 1.
- || P_i E_P(t) || <= 4 theta^{ dist(i, J(t)) },
+Let k = dist(i, J(t)) and P_i be the projection to both sides of
+layer i (including all transverse and matrix coordinates). For
+n < k, every layer path touched in P_i R_0(z,t) (Delta R_0)^n
+stays outside J(t); each resolvent factor along such a path acts
+as z^{-1} I, so by induction on n,
-which decays in the layer distance from the active band. That is
-why layer weights and layer distances combine to give a small
-number.
+ P_i R_0(z,t) (Delta(t) R_0(z,t))^n
+ = z^{-n-1} P_i Delta(t)^n (n < k).
-WEIGHTED TRACE BOUND. A short union-bound over closest layers
-in J(t), plus Equation (10.3), gives
+The contour integral of z^{-n-1} around C_{r_0} is zero because
+the circle does not enclose the origin. Therefore
- S <= 96 mu_ell,
+ P_i E_P(t) = (1 / 2 pi i) integral over C_{r_0} of
+ sum_{n >= k} P_i R_0 (Delta R_0)^n dz.
-where mu_ell is the small weighted trace of Part 7. Since
-mu_ell -> 0 as ell -> infinity, S -> 0 too.
+Bound the geometric tail:
+
+ || P_i E_P(t) ||
+ <= r_0 C_chi sum_{n >= k} theta^n
+ <= r_0 C_chi theta^k / (1 - theta) <= 4 theta^k,
+
+using theta <= 1/2 and r_0 C_chi = (chi/3) * (4/chi) = 4/3
+combined with 1/(1 - theta) <= 2. This is the stated decay in
+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
+its central Fourier fiber; the countable amplification is handled
+by Lemma nil:trace). 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)
+ = tr_id(E_P(t) P_i E_P(t))
+ <= || P_i E_P(t) ||^2 tr_id(E_P(t))
+ <= 16 theta^{2 dist(i, J(t))} tr_id p(t).
+
+The middle line uses 0 <= E_P(t) P_i E_P(t) <= ||P_i E_P(t)||^2
+E_P(t); the last uses 10.3.
+
+It remains to bound the weighted sum. Since |H_i - H_j| <= |i -
+j| for every pair of layer indices, w_i = 2^{C_bullet H_i}
+satisfies w_i <= w_j 2^{C_bullet |i - j|}. Assign every integer i
+to a closest point of the nonempty finite set J(t); because
+|J(t)| <= 2, at most two integers at each positive distance are
+assigned to a given j in J(t). Therefore
+
+ sum_i w_i theta^{2 dist(i, J(t))}
+ <= 2 max_{j in J(t)} w_j *
+ ( 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
+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
+the open window,
+
+ sum over surviving base vertices v of w_{i(v)} e_v(t)
+ <= 16 tr_id p(t) sum_i w_i theta^{2 dist(i, J(t))}
+ <= 96 max_{j in J(t)} w_j tr_id p(t) <= 96 mu_ell,
+
+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.
+
+FIBER AVERAGE. Central Fourier decomposition identifies e_v with
+the integral over t of e_v(t). Tonelli's theorem applies to the
+nonnegative diagonal, so
+
+ S = sum over surviving v of w_{i(v)} e_v
+ = integral over t in [0,1] of
+ ( sum over surviving v of w_{i(v)} e_v(t) ) dt
+ <= 96 mu_ell.
+
+Since mu_ell -> 0 as ell -> infinity, S -> 0.
10.5 THE CAYLEY OPERATOR T
@@ -1974,10 +2492,34 @@ 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.
+Proof. On H_Q = l^2(V(Q)) (x) C^N let 1_{B(Q)} be the vertex
+projection onto the trimming ball. Since E_Q is an orthogonal
+projection, E_Q^* E_Q = E_Q, so
+
+ || 1_{B(Q)} E_Q ||_HS^2
+ = Tr( E_Q 1_{B(Q)} E_Q )
+ = 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
+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 (geo:weight) says that for every
+vertex u of B(Q) over layer i = layer(u), the whole ball satisfies
+
+ |V(B(Q))| <= w_i = 2^{C_bullet H_i}.
+
+So the number of lifts of a fixed base vertex v in B(Q) is at
+most |V(B(Q))| <= w_{i(v)}. Therefore
+
+ || 1_{B(Q)} E_Q ||_HS^2
+ = sum over u in B(Q) of e_{type(u)}
+ <= sum over surviving base vertices v of
+ ( # of u in B(Q) with type(u) = v ) e_v
+ <= sum over surviving base vertices v of w_{i(v)} e_v
+ = 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
@@ -2010,6 +2552,44 @@ 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).
+Proof. Root each connected component of the forest at an
+arbitrary vertex and orient every edge from parent to child.
+Write B for the half-adjacency from the child coordinate to the
+parent coordinate:
+
+ (B xi)(x) = sum over children y of x of w_{xy} xi(y),
+
+where w_{xy} is the matrix weight between x and y. Then the
+forest adjacency is B + B^*.
+
+Bound B on finite-support vectors. For each vertex x, Cauchy-
+Schwarz on the sum over children y of x gives
+
+ || sum over children y of x of w_{xy} xi(y) ||^2
+ <= ( sum over children y of x of w_1^2 )
+ * ( sum over children y of x of ||xi(y)||^2 )
+ <= w_1^2 m_1 * sum over children y of x of ||xi(y)||^2.
+
+Summing over x and using that a nonroot vertex has exactly one
+parent,
+
+ || B xi ||^2
+ <= w_1^2 m_1 * sum over x of sum over children y of x of
+ || xi(y) ||^2
+ = w_1^2 m_1 * sum over nonroot y of || xi(y) ||^2
+ <= w_1^2 m_1 * || xi ||^2.
+
+So || B xi || <= w_1 sqrt(m_1) || xi ||, i.e. ||B|| <= w_1
+sqrt(m_1). Bounded extension lifts this from finitely supported
+vectors to all of l^2. Then
+
+ || A || = || B + B^* || <= 2 || B || <= 2 w_1 sqrt(m_1).
+
+The countable number of components and the infinite dimension of
+the ambient Hilbert space cause no issue: each vertex has a
+finite-distance parent, so B is well defined, and the operator-
+norm estimate above is uniform in the tree.
+
Applied to T - T_D, with m_1 <= 2 |S| and w_1 <= sqrt(2) C_eta /
sqrt(dD):
@@ -2023,20 +2603,38 @@ 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
+1 - chi + alpha on the orthogonal complement of F_D. By 10.7,
+||T - C|| <= 3 alpha.
+
+SPECTRUM OF C. Since C = C* and C = 1 on F_D while C <= 1 -
+chi + alpha on (1 - F_D)H, the spectrum of C lies in
+[-||C||, 1 - chi + alpha] together with the eigenvalue 1 at F_D.
+
+SPECTRUM OF T. Weyl's inequality: the spectral slope of T is at
+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].
+ (-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,
+the upper interval [1 - 3 alpha, 1 + 3 alpha] is nonempty.
+
+GAP. The number 1 - chi/4 lies strictly between the two
+intervals. Indeed with alpha = chi/1000,
+
+ 1 - chi + 4 alpha = 1 - (996/1000) chi < 1 - chi/4,
+ 1 - chi/4 = 1 - 0.25 chi < 1 - 0.003 chi
+ = 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. 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.
+is therefore a well-defined projection. It is nonzero because
+its upper band [1 - chi/4, infinity) contains the whole range of
+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
@@ -2059,48 +2657,147 @@ 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.
+ A xi = ( E_Q( xi|_{V(Q)} ) )_Q,
+
+with xi|_{V(Q)} extended by zero outside V(Q) in each copy Q.
+
+BOUNDEDNESS OF A. Split V(Q) = D_Q disjoint-union I_Q, where
+I_Q = V(Q) minus D_Q lies inside B(Q). The D_Q are pairwise
+disjoint and each E_Q is a contraction, so
+
+ sum over Q of || E_Q( xi|_{D_Q} ) ||^2
+ <= sum over Q of || xi|_{D_Q} ||^2
+ <= ||xi||^2.
+
+The I_Q parts are governed by the next lemma.
+
+LEMMA (LOST-RESTRICTIONS). With I_Q = V(Q) minus D_Q,
+
+ sum over Q of || E_Q( xi|_{I_Q} ) ||^2 <= S ||xi||^2.
+
+Proof. Write v(Q,u) = type_Q(u) for the base type of an incidence
+and i(v) for the layer of a base vertex v. For a vector z in C^N
+supported at u in V(Q), the copy-diagonal identity (10.3) and
+positivity of the diagonal matrix give
+
+ || E_Q( delta_u (x) z ) ||^2
+ = < E_Q(u,u) z, z >
+ <= e_{v(Q,u)} ||z||^2.
+
+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
+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
+terms,
+
+ sum over Q of || E_Q( xi|_{I_Q} ) ||^2
+ <= sum over u in Gamma_base of ||xi(u)||^2 *
+ ( 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
+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
+
+ sum over surviving base vertices v of w_{i(v)} e_v = S.
+
+Substituting,
+
+ sum over Q of || E_Q( xi|_{I_Q} ) ||^2
+ <= S * sum over u in Gamma_base of ||xi(u)||^2
+ = S ||xi||^2.
+
+Combining the D_Q and I_Q estimates and taking square roots,
+
+ || A xi || <= sqrt(1 + S) ||xi|| <= (1 + sqrt(S)) ||xi||,
+
+so ||A|| <= 1 + sqrt(S). Its kernel K is closed and invariant:
+A is equivariant because left Gamma-translation permutes the
+indexed copies and the E_Q with them.
+
+For xi in K, E_Q(xi|_{V(Q)}) = 0 for every Q, so
+E_Q(xi|_{D_Q}) = - E_Q(xi|_{I_Q}). Norms therefore agree, and
+LOST-RESTRICTIONS gives
+
+ sum over Q of || E_Q( xi|_{D_Q} ) ||^2
+ = sum over Q of || E_Q( xi|_{I_Q} ) ||^2
+ <= S ||xi||^2.
+
+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,
+
+ < T_Q y, y > <= (1 - chi + alpha) || y ||^2
+ + chi || E_Q y ||^2.
-Because the D_Q are disjoint and the ball-loss bound 10.6 holds,
-A is bounded with
+Apply this to the zero-extension of xi|_{D_Q}, sum over the
+pairwise disjoint domains, use the S-bound above, and include
+the unassigned part via its norm bound (the coefficient
+1 - chi + alpha is positive). The forest error f_ell is at
+most alpha by 10.7. Therefore
- ||A|| <= 1 + sqrt(S).
+ < T xi, xi >
+ <= (1 - chi + alpha + chi S + f_ell) ||xi||^2
+ <= (1 - chi + 3 alpha) ||xi||^2,
-Its kernel K is a closed invariant subspace, and eq. (10.6) says
-that for every xi in K,
+because S <= 1/1000 and f_ell <= alpha.
- E_Q( xi|_{D_Q} ) = - E_Q( xi|_{I_Q} ),
+On the OTHER HAND, on p_0 H the spectral split of 10.8 gives
-so the local loss on D_Q is paid by the trimming ball, giving
+ < T xi, xi > >= (1 - 3 alpha) ||xi||^2.
- sum over Q of || E_Q( xi|_{D_Q} ) ||^2 <= S ||xi||^2.
+Since 1 - 3 alpha > 1 - chi + 3 alpha (with alpha = chi/1000), no
+nonzero xi can satisfy both inequalities. Therefore
+p_0 H ∩ K = {0}.
-SANDWICH. For xi in K, using the local estimate from 10.2 on
-each copy, and the forest error:
+KERNEL-COMPLEMENT SIZE. Polar-decompose the restriction map
+A = V |A| of 10.9. This gives an equivariant partial isometry
+V : H -> L with V* V = P_{K^perp} and V V* the projection onto
+the closure of ran A, which is contained in the range of E.
+The adjoint-square identity and monotonicity then give
- <T xi, xi> <= (1 - chi + 3 alpha) ||xi||^2.
+ Tr_H( P_{K^perp} ) = Tr_H( V* V )
+ = Tr_L( V V* )
+ <= Tr_L( E ) = S_0.
-On the OTHER HAND, on the range of p_0:
+The middle equality uses the adjoint-square identity across
+the two Hilbert spaces H and L: V* V acts on H, V V* on L,
+and their identity traces match.
- <T xi, xi> >= (1 - 3 alpha) ||xi||^2.
+INJECTIVITY OF P_{K^perp} ON p_0 H. Because p_0 H ∩ K = {0}, the
+map P_{K^perp} : p_0 H -> K^perp is injective. Polar-decompose
+it as V' : p_0 H -> K^perp, with V'* V' = p_0 and V' V'* a
+projection on K^perp bounded by P_{K^perp}. Then by the adjoint-
+square identity and monotonicity,
-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.
+ Tr_H( p_0 ) = Tr_H( V'* V' )
+ = Tr_H( V' V'* )
+ <= Tr_H( P_{K^perp} )
+ <= S_0.
-TRACE CONSEQUENCE. A polar decomposition identifies p_0 H with a
-closed invariant subspace of K^perp. Therefore
+Since S_0 <= S and S < 1/2 (ell well past S <= 1/4), the upper
+bound of the trace target follows.
- tau_{Gamma,N}(p_0)
- = Tr_H( p_0 ) <= Tr_L( P_{K^perp} ) <= Tr_L( E )
- = S_0 <= S.
+STRICTLY POSITIVE. A nonzero equivariant projection on
+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
+of a nonzero equivariant projection is positive,
-Since S < 1/2 (we chose ell well past S <= 1/4), we conclude
+ Tr_H( p_0 ) > 0.
- 0 < tau_{Gamma,N}(p_0) < 1/2.
+CONCLUSION. Combining,
-The lower bound 0 < is because p_0 != 0 (from 10.8) and the
-trace is faithful.
+ 0 < tau_{Gamma,N}(p_0) = Tr_H( p_0 ) <= S_0 <= S
+ < 1/2.
10.10 WHAT PART 10 PRODUCES