diff f6bf760..6fbd3ac switch_chain_polynomial_mixing_v2.txt
diff --git a/switch_chain_polynomial_mixing_v2.txt b/switch_chain_polynomial_mixing_v2.txt
index 9aecaba..96bc2e4 100644
--- a/switch_chain_polynomial_mixing_v2.txt
+++ b/switch_chain_polynomial_mixing_v2.txt
@@ -42,7 +42,7 @@ Three simple objects recur throughout:
being an average over the pair-resampling fiber". H measures how
much a test function f on graphs sees the pair resampling.
-The main engine inequality H^2 >= H is proved in two pieces:
+The main engine ineuality H^2 >= H is proved in two pieces:
(a) triple of pairs inside a 3-vertex set (Sections 4-5)
(b) pairs that share no vertex (Sections 6-7)
Then Section 8 assembles the two pieces and shows the kernel of H is
@@ -214,7 +214,7 @@ For self-adjoint operators, the shorthand A >= B means
<f, (A - B) f> >= 0 for every real f.
Call this the quadratic-form order. It is only partial: A >= B and
-B >= A are both true iff A = B. The engine inequality H^2 >= H is
+B >= A are both true iff A = B. The engine ineuality H^2 >= H is
understood in this order.
--------------------------------------------------------------------------
@@ -299,7 +299,7 @@ Two useful facts.
* H kills constants. Each h_a does, so H 1 = 0, and
<1, H 1> = 0 confirms the constants sit in the kernel.
-The engine inequality is
+The engine ineuality is
(ENGINE) H^2 >= H.
@@ -307,7 +307,7 @@ In eigenvalue terms (writing H for a symmetric operator on the finite-
dimensional space of all real functions on Omega_d, orthogonal
decomposition relative to pi_d): on the 1-dimensional space of
constants, H acts by 0, so both H and H^2 act by 0 there and the
-inequality is trivially true. On the orthogonal complement (the
+ineuality is trivially true. On the orthogonal complement (the
"centered" functions), ENGINE says that every eigenvalue lambda of H
restricted to centered functions satisfies
@@ -329,7 +329,7 @@ sharing a vertex do NOT commute: each resamples the shared vertex's
neighborhood, and the two choices clash. Pairs with a and b disjoint
also do NOT commute: they both touch the four edges between the two
pairs (Carstens-Berger-Strona noticed this "dependence of undirected
-trades" in 2018). So the engine inequality is not a triviality that
+trades" in 2018). So the engine ineuality is not a triviality that
falls out of a commuting family of operators. Each family needs its
own estimate.
@@ -345,7 +345,7 @@ of the whole paper. A sloppier constant (say 3) would still give
polynomial mixing; a constant that GREW with n would not. So the tight
factor 2 is earned.
-The result is equivalent to a conditional-projection inequality proved
+The result is equivalent to a conditional-projection ineuality proved
by Fu-Qin-Wang (2026, Theorem 5.1) in the bipartite setting. Their
proof goes through a swap chain and a variance argument; the version
below uses "transpositions and spectral decomposition", which is more
@@ -357,7 +357,7 @@ group heat-flow method of Carlen-Lieb-Loss (2006). Only the count-coupling step
Fu-Qin-Wang attribution.
--------------------------------------------------------------------------
-3.1 Two-row warm-up: what does the inequality say?
+3.1 Two-row warm-up: what does the ineuality say?
--------------------------------------------------------------------------
Before the technicalities, look at the simplest case. Fix a 2-row
@@ -381,7 +381,7 @@ have Var(X + Y) = Var(X) + Var(Y). Here X and Y are NOT independent:
the choice of U_1 and U_2 is complementary. Sometimes the negative
cross term can be as bad as -Var(X) - Var(Y), in which case the
variance of X + Y can be ZERO even when X, Y vary independently --
-think of X = -Y. The whole content of the 3-row inequality is that
+think of X = -Y. The whole content of the 3-row ineuality is that
when there are THREE rows instead of two, you still have enough
freedom in the third coordinate that the variance cannot collapse:
@@ -436,7 +436,7 @@ each g_{i,lambda} is again a function of row i.
Second, apply J to the sum g_1 + g_2 + g_3. For a single transposition
tau, at most two of the three g_i change (because a transposition
swaps exactly 2 labels, so it moves labels into and out of exactly 2 of
-the 3 U-blocks -- same for W-blocks). The elementary inequality
+the 3 U-blocks -- same for W-blocks). The elementary ineuality
(x + y + z)^2 <= 3 (x^2 + y^2 + z^2)
@@ -460,7 +460,7 @@ observation (transposition of a U-label swaps in/out of exactly 2
U-blocks) and the transitivity of the permutation action are the only
places specific facts about U, W are used. Empty blocks and singleton
label sets cause no trouble: a one-state space has no nonzero centered
-functions, so both sides of the inequality are zero there.
+functions, so both sides of the ineuality are zero there.
--------------------------------------------------------------------------
3.3 The random-block-count case
@@ -551,7 +551,7 @@ centered functions):
So lambda <= 2. That is Lemma 3.2.
--------------------------------------------------------------------------
-3.4 Combining: the three-row variance inequality
+3.4 Combining: the three-row variance ineuality
--------------------------------------------------------------------------
Now put the two pieces together. Fix row sums r_1, r_2, r_3 and column
@@ -560,7 +560,7 @@ be a uniform random binary matrix with those prescribed margins, and
let S_i denote the i-th row of M. For arbitrary real functions f_1,
f_2, f_3:
- LEMMA 3.3 (three-row variance inequality).
+ LEMMA 3.3 (three-row variance ineuality).
Var(f_1(S_1) + f_2(S_2) + f_3(S_3))
<= 2 ( Var(f_1(S_1)) + Var(f_2(S_2)) + Var(f_3(S_3)) ).
@@ -806,7 +806,7 @@ Similarly, h_{ {i,k} } is "conditional on row j", h_{ {i,j} } is
where sum is over the three rows.
--------------------------------------------------------------------------
-4.4 The variance inequality
+4.4 The variance ineuality
--------------------------------------------------------------------------
Let f be a function on the cell, centered in the sense E[f] = 0
@@ -888,15 +888,15 @@ That is Proposition 4.1. QED.
There is one last technical point. Proposition 4.1 is proved cell by
cell (each cell has its own uniform measure and its own model as a 3-
-row matrix). Does the cellwise inequality on H_T^2 >= H_T imply the
-global inequality on H_T^2 >= H_T?
+row matrix). Does the cellwise ineuality on H_T^2 >= H_T imply the
+global ineuality on H_T^2 >= H_T?
Yes, because H_T PRESERVES CELLS. Each cell is a "block diagonal"
subspace for H_T: the operator h_a maps cell-constant functions to
cell-constant functions and does not mix cells. So H_T = sum over
cells of "H_T restricted to cell". The quadratic form <f, H_T^2 f> is
a sum over cells of the cellwise <f_cell, H_T^2|_{cell} f_cell>. Same
-for <f, H_T f>. The cellwise inequality implies the global one.
+for <f, H_T f>. The cellwise ineuality implies the global one.
(There is one harmless corner case: a cell might be a SINGLE graph.
That is, its conditional measure puts all mass on one point. Then the
@@ -904,7 +904,7 @@ cell's function space is 1-dimensional, f is constant on the cell, and
H_T acts by 0, both on H_T and H_T^2. Trivially H_T^2 >= H_T.)
So Proposition 4.1 holds on Omega_d for every triple T. This finishes
-the first half of the engine inequality.
+the first half of the engine ineuality.
End of Section 4.
@@ -1252,14 +1252,14 @@ length x^2 (1 - rho^2). Add:
- (y^2 + x^2)(1 - rho^2) = - ||h_a f_uv||^2 - ||h_b f_uv||^2.
-Now use the elementary inequality 2 xy rho >= -(x^2 + y^2) rho >=
+Now use the elementary ineuality 2 xy rho >= -(x^2 + y^2) rho >=
-(x^2 + y^2) (both valid because 0 < rho < 1 and 2|xy| <= x^2 + y^2):
2 <h_a f_uv, h_b f_uv> = 2 xy rho (1 - rho^2)
>= -(x^2 + y^2)(1 - rho^2)
= - ||h_a f_uv||^2 - ||h_b f_uv||^2.
-That is the plane inequality.
+That is the plane ineuality.
--------------------------------------------------------------------------
5.6 The error lines belong to the slice projections
@@ -1325,7 +1325,7 @@ w = v - E_a v = v + rho u (since E_a v = -rho u):
Similarly ||R_{b,a} f||^2 >= ||h_b f_uv||^2 = x^2 (1 - rho^2).
-Combine with the plane inequality:
+Combine with the plane ineuality:
2 <h_a f_uv, h_b f_uv>
>= -(x^2 + y^2)(1 - rho^2)
@@ -1363,12 +1363,12 @@ there and both sides are zero. Otherwise:
same independence argument, same conclusion.
* Case s = 2 with center present and both probabilities positive:
- preceding calculation, giving exactly the stated inequality.
+ preceding calculation, giving exactly the stated ineuality.
Sum over cells (all involved operators preserve cells), done. End.
All that remains to prove (Proposition 5.1 to be summed in Section 7)
-is controlling the two error terms via a slice inequality, which is
+is controlling the two error terms via a slice ineuality, which is
the content of Section 6.
--------------------------------------------------------------------------
@@ -1403,7 +1403,7 @@ using the convention Q_{ij} = 0 if z_{ij} = 0. On X_{N,q},
P( x_i = x_j ) = p = ( q(q-1) + r(r-1) ) / ( N(N-1) ), r := N - q.
-LEMMA 6.1 (slice inequality).
+LEMMA 6.1 (slice ineuality).
S := sum_{i<j} Q_{ij} <= G.
@@ -1494,7 +1494,7 @@ plug into ||Bc||^2 = mu_1 ||c||^2 and solve:
mu_1 = ( 2q - N )^2 / ( 2 (N - 2) p ) <= N / 2.
-The inequality uses q, r >= 1 and q + r = N:
+The ineuality uses q, r >= 1 and q + r = N:
N(N-2) p - (2q - N)^2 = ( 2N / (N-1) )( qr - (N-1) ) >= 0.
@@ -1590,7 +1590,7 @@ are preserved by all these operators) gives the corollary.
SECTION 7. ASSEMBLY: H^2 >= H AND THE KERNEL IS CONSTANTS
--------------------------------------------------------------------------
-We assemble the local estimates into the global engine inequality, and
+We assemble the local estimates into the global engine ineuality, and
verify that H's kernel is exactly the constants (which upgrades "gap on
centered functions" into a non-degenerate gap on the switch chain).
@@ -1674,10 +1674,10 @@ By Corollary 6.2, the bracket is >= 0 for every a. Hence <f, Df> >= 0.
End.
--------------------------------------------------------------------------
-7.3 The engine inequality
+7.3 The engine ineuality
--------------------------------------------------------------------------
-PROPOSITION 7.2 (engine inequality). H^2 >= H.
+PROPOSITION 7.2 (engine ineuality). H^2 >= H.
Proof. From the regrouping identity:
@@ -1874,7 +1874,7 @@ Therefore the gap of P_d is at least
Also: the holding probability 1/2 guarantees nonnegative eigenvalues.
Formally, P_d = (I + Q)/2 where Q is the reversible Markov kernel
consisting of the proposal step including rejected proposals. Jensen's
-inequality and stationarity show that Q is a contraction on L^2(pi_d),
+ineuality and stationarity show that Q is a contraction on L^2(pi_d),
so its eigenvalues lie in [-1, 1]; hence every eigenvalue of P_d on the
centered subspace lies in [0, 1 - gamma_0].