diff --git a/exact_quantum_factoring_v2.txt b/exact_quantum_factoring_v2.txt
index 453a17f..80976c4 100644
--- a/exact_quantum_factoring_v2.txt
+++ b/exact_quantum_factoring_v2.txt
@@ -186,7 +186,7 @@ Definition 1. For an input integer N ge 2, set
     n = max{128, floor(log_2 N) + 1},   B = 2^n.
 
 Then N < B. The cutoff at 128 is a fixed constant; it has no effect
-on polynomial-time uniformity and simplifies every strict inequality
+on polynomial-time uniformity and simplifies every strict ineuality
 below by avoiding small-n edge cases. These n and B apply
 throughout.
 
@@ -509,7 +509,7 @@ cross-product gives
     |j/d - p/v| = |j v - p d| / (d v) >= 1/(d v) >= 1/((B-1) v)
                > 1/(v B) + 1/(2 Q).
 
-For the strict inequality, note that 2 Q > v B (B-1) because v < B and
+For the strict ineuality, note that 2 Q > v B (B-1) because v < B and
 Q = B^16. This contradicts (**) together with |k/Q - j/d| <= 1/(2Q).
 Thus j/d = p/v. It remains only to check 0 <= p < v < B and the exact
 rule k(d,j) = k; these checks accept precisely the valid labels. These
@@ -642,7 +642,7 @@ Since |A_t| <= 1/d + 1/Q and |I(u)| <= 1/d, we have
                      = |(A_t - I)(A_t + I)|  (up to real part)
                      <= (8/Q)(3/d) <= 24/Q <= 32/Q.
 
-This gives the first inequality in (ii). The upper bound |I(u)| <= 1/d
+This gives the first ineuality in (ii). The upper bound |I(u)| <= 1/d
 follows from |g| <= 1.
 
 For the lower bound on |I|, temporarily replace g in the integral by
@@ -654,7 +654,7 @@ magnitude
     >= 2/(pi d),
 
 where the quotient is interpreted as 1 at eta = 0, and the last
-inequality follows from concavity of sin on [0, pi/2]: sin(x)/x is
+ineuality follows from concavity of sin on [0, pi/2]: sin(x)/x is
 decreasing on [0, pi/2], so on |pi eta| <= pi/2 it is at least
 sin(pi/2)/(pi/2) = 2/pi.
 
@@ -1022,7 +1022,7 @@ requires a valid encoding.
 
 List transition. It generates K independent uniform b_m-bit integers.
 Entries outside [0, m) and nonunits are ignored. A list is GOOD if some
-usable entry a has unequal values of
+usable entry a has uneual values of
 
     v_2( ord_{p^e}(a) ),   p^e || m,
 
@@ -1034,7 +1034,7 @@ A canonical good list has first entry equal to the Chinese remainder
 residue that is 1 on the component corresponding to the smallest prime
 and -1 on all other components, followed by zeros. Its first entry has
 2-adic levels 0 on the smallest-prime component and 1 on every other,
-so unequal by inspection. No entry beyond the first is usable, so this
+so uneual by inspection. No entry beyond the first is usable, so this
 canonical list spawns one order transition.
 
 Order transition. For each usable entry a in a list, run K independent
@@ -1108,10 +1108,10 @@ Hence sum_t prod_i c_i(t) <= phi(m)/2. Define
 
     F = phi(m) - sum_t prod_i c_i(t)  >=  phi(m)/2.
 
-F is exactly the count of units with at least one unequal level pair.
+F is exactly the count of units with at least one uneual level pair.
 
 Good probability. A single uniform b_m-bit integer is a unit with
-unequal levels with probability F/2^{b_m}. Across K independent
+uneual levels with probability F/2^{b_m}. Across K independent
 draws,
 
     s = 1 - (1 - F/2^{b_m})^K.
@@ -1163,21 +1163,21 @@ lambda(r) = 2^{-ceil(log_2 r) - 10} with ceil(log_2 r) <= n. So
 phi(r) lambda(r) has denominator dividing 2^{n+10}. Then s has
 denominator dividing 2^{(n+10)K}, so s has at most (n+10)K = O(n^6) bits.
 
-Suppose the list is good (some usable entry has unequal levels) and
+Suppose the list is good (some usable entry has uneual levels) and
 every order transition on its usable entries is good (each returns r_a
 = ord_m(a) for its input a). The split procedure finds a
 proper nontrivial divisor of m.
 
-Choose an entry a with unequal levels. Let r = ord_m(a); this is the
+Choose an entry a with uneual levels. Let r = ord_m(a); this is the
 output of the good order transition on a. Write p_i^{e_i} || m for the
 distinct odd prime powers and r_i = ord_{p_i^{e_i}}(a). Then
 
     r = lcm_i r_i,   and so   v_2(r) = max_i v_2(r_i).
 
 Let M = { i : v_2(r_i) = v_2(r) } be the highest-level set. By
-definition M is nonempty. Since levels are unequal, some i has
+definition M is nonempty. Since levels are uneual, some i has
 v_2(r_i) < v_2(r), so the complement of M is also nonempty. Since the
-levels are unequal and every level is >= 0, the maximum v_2(r) is at
+levels are uneual and every level is >= 0, the maximum v_2(r) is at
 least 1; otherwise all levels would be 0 and equal. So r is even.
 
 Compute a^{r/2} modulo each p_i^{e_i}.
@@ -1303,7 +1303,7 @@ Denominators are bounded by 2^{nK} and 2^{(n+10)K} respectively, so
 they have polynomial bit length.
 
 (iii) All-good runs complete and generate D. Suppose every transition
-output is good. Then every list has an unequal-level entry, and every
+output is good. Then every list has an uneual-level entry, and every
 order transition for that entry returns r = ord_m(a). By 4.7, the
 resulting gcd is a proper nontrivial divisor of m. Recurse until all
 factors are prime (Lemma 20 identifies them). The final tree is
@@ -1626,12 +1626,12 @@ writing pi(R) for the count of primes <= R,
 
     log_2 C <= pi(R) log_2 R <= (log_2 P_R) log_2 R,
 
-where the second inequality uses log_2 P_R >= pi(R) (since each l >= 2
+where the second ineuality uses log_2 P_R >= pi(R) (since each l >= 2
 contributes log_2 l >= 1). Combining with the lower bound on C:
 
     log_2 P_R >= (R - log_2(R + 1)) / log_2 R > n^6.
 
-For the last inequality: 8 log_2 n + 1 < n for n >= 128 (the bound
+For the last ineuality: 8 log_2 n + 1 < n for n >= 128 (the bound
 holds at n = 128 and is monotonic in n), so a short computation gives
 (R - log_2(R+1)) / log_2 R > (n^8 - 1 - n) / n > n^6.