diff --git a/mutually_unbiased_bases_dim6_v2.txt b/mutually_unbiased_bases_dim6_v2.txt
index e41a359..dc37947 100644
--- a/mutually_unbiased_bases_dim6_v2.txt
+++ b/mutually_unbiased_bases_dim6_v2.txt
@@ -97,7 +97,7 @@ representatives, are harmless: duplication weakens an exclusion but
 never omits a solution. ``Small ball'' always means the phase matrices
 within a stated Frobenius norm. A numerical center is a proposal and
 need not be an exact Hadamard matrix. Every conclusion comes from a
-necessary-condition inequality verified on a whole ball, not from the
+necessary-condition ineuality verified on a whole ball, not from the
 residual at a proposed center.
 
 % =====================================================================
@@ -166,7 +166,7 @@ is due to Bandyopadhyay, Boykin, Roychowdhury and Vatan (Algorithmica
 
 {smallhttps://www.sciencedirect.com/science/article/pii/0003491689903229}
 
-{smallhttps://arxiv.org/pdf/quant-ph/0103162}
+{small./pdf/unknown-nd-quant-ph0103162-small.pdf}
 
 Dimension six is the first case that is neither a prime power nor a
 small even dimension settled by other constructions. A tensor-product
@@ -178,11 +178,11 @@ reappears in his doctoral thesis, and it is Conjecture~1 of the
 Klappenecker--R"otteler paper. A survey of the composite-dimension
 setting is due to McNulty and Weigert.
 
-{smallhttps://arxiv.org/pdf/quant-ph/0309120}
+{small./pdf/unknown-nd-quant-ph0309120-small.pdf}
 
-{smallhttps://www.gerhardzauner.at/documents/gzdiplthd.pdf}
+{small./pdf/unknown-nd-small.pdf}
 
-{smallhttps://arxiv.org/pdf/2410.23997}
+{small./pdf/unknown-nd-2410.23997-small.pdf}
 
 Four attacks ran alongside. Grassl proved that the standard basis
 together with any member of the two-parameter Fourier family cannot
@@ -199,13 +199,13 @@ Raynal, Lu and Englert sharpened the numerical side by maximising the
 average distance between four bases and finding the maximum strictly
 below unity, with a two-parameter family of three bases attaining it.
 
-{smallhttps://arxiv.org/pdf/quant-ph/0406175}
+{small./pdf/unknown-nd-quant-ph0406175-small.pdf}
 
-{smallhttps://arxiv.org/pdf/math.OA/0609076}
+{small./pdf/unknown-nd-small.pdf}
 
-{smallhttps://arxiv.org/pdf/0901.4051}
+{small./pdf/unknown-nd-0901.4051-small.pdf}
 
-{smallhttps://arxiv.org/pdf/1103.1025}
+{small./pdf/1103.1025.pdf}
 
 The closest antecedent is the certified discretization of Jaming,
 Matolcsi, M'ora, Sz"ollH osi and Weiner. They proved that the
@@ -218,9 +218,9 @@ programme in Sections~2--4 of their survey. Those four devices carry
 the finite search throughout: what is added is exhaustive coverage of
 arbitrary second bases, with every error bounded.
 
-{smallhttps://arxiv.org/pdf/0902.0882}
+{small./pdf/unknown-nd-0902.0882-small.pdf}
 
-{smallhttps://arxiv.org/pdf/1201.0640}
+{small./pdf/unknown-nd-1201.0640-small.pdf}
 
 Two recent lines of work are complementary rather than overlapping.
 Matolcsi, Matszangosz, Varga and Weiner study triplets of MUBs under
@@ -231,9 +231,9 @@ finite cover of all second bases with the local exclusion, executed
 under a certified arithmetic contract, yields an unconditional upper
 bound.
 
-{smallhttps://arxiv.org/pdf/2503.14752}
+{small./pdf/unknown-nd-2503.14752-small.pdf}
 
-{smallhttps://arxiv.org/pdf/2608.18053}
+{small./pdf/unknown-nd-2608.18053-small.pdf}
 
 medskipnoindentThe strategy at a glance.
 
@@ -338,7 +338,7 @@ and U_0=I_6.
 medskipnoindentA Lie-algebra consequence.
 
 An orthonormal basis B of C^6 determines its Cartan
-subalgebra mathfrak h_frakh_{B} of traceless matrices that are
+subalgebra h_{B} of traceless matrices that are
 diagonal in that basis. It is spanned by the matrices bb^*-I/6 for
 binB. The Killing form on sl_6(C) is
 K(X,Y)=12tr(XY); two subalgebras are
@@ -371,7 +371,7 @@ gives the upper bound under its stated contract, and
 Lemma~lem:three-mub-six gives attainment.
 
 
-{smallhttps://arxiv.org/pdf/quant-ph/0506089}
+{small./pdf/unknown-nd-quant-ph0506089-small.pdf}
 
 % =====================================================================
 medskipnoindentEquivalences and an exhaustive seed cover.
@@ -386,9 +386,9 @@ programme is set out in Sections 2--3 of the later survey by Jaming,
 Matolcsi and M'ora. What is new is the specific symmetry reduction and
 shared-coordinate estimate that prove completeness of this exact cover.
 
-{smallhttps://arxiv.org/pdf/0902.0882}
+{small./pdf/unknown-nd-0902.0882-small.pdf}
 
-{smallhttps://arxiv.org/pdf/1201.0640}
+{small./pdf/unknown-nd-1201.0640-small.pdf}
 
 Throughout, phases are in cycles and e(t)=exp(2pi i t)
 entrywise. Matrix indices run 0,...,5, entries are stored in
@@ -511,7 +511,7 @@ Dephase the first row and column. Ordering the remaining columns along
 the complementary arc gives g_x=1-x_5 and the sorted list
 in~(seed:x-arrangement).
 
-If the final orientation inequality fails, interchange the two selected
+If the final orientation ineuality fails, interchange the two selected
 rows, use the old x_5 column as the starting column, and reverse the
 ordering. The new list is
 [
@@ -519,7 +519,7 @@ ordering. The new list is
 ]
 It has the same distinguished largest gap, and
 x'_1=x_5-x_4 whereas x'_5-x'_4=x_1. So this orientation satisfies
-the required inequality whenever the other fails. The construction
+the required ineuality whenever the other fails. The construction
 uses a chosen largest gap and is unchanged by ties or repeated points.
 
 Select a noninitial column and place it at position two. Permuting the
@@ -535,7 +535,7 @@ g_yge g_x.
 
 Set N=112 and w=1/N. Use closed bins
 
- I_b=[b/N,(b+1)/N], u_b=(b+tfrac12)/N, 0le b<N .
+ I_b=[b/N,(b+1)/N], u_b=(b+1/2)/N, 0le b<N .
 
 They cover [0,1] including both endpoints. Sorted exact phases admit
 nondecreasing bin indices, so write the five indices for
@@ -596,9 +596,9 @@ w. Since x_5in[l/N,(l+1)/N] we have
 For sorted midpoint bin indices q_0,...,q_4 the gap used by
 mk is
 [
- maxBigl{(q_0+tfrac12)/(N),
+ maxBigl{(q_0+1/2)/(N),
   (q_1-q_0)/(N),...,(q_4-q_3)/(N),
- 1-(q_4+tfrac12)/(N)Bigr}.
+ 1-(q_4+1/2)/(N)Bigr}.
 ]
 For a Y tuple it sorts a copy solely for this computation, preserving
 the stored shared-first ordering.
@@ -636,19 +636,19 @@ with integral remainder gives
 ]
 Sum the five remainders and use the vanishing-sum hypothesis to
 obtain~(seed:taylor-residual). Dot with v, apply the triangle
-inequality and |Delta_j|le w/2 to get~(seed:sum-support).
+ineuality and |Delta_j|le w/2 to get~(seed:sum-support).
 
 
 The function sumok tests~(seed:sum-support) in the
 residual direction and in directions perpendicular to the five
-unit-exponential columns. Since the inequality holds for every real
+unit-exponential columns. Since the ineuality holds for every real
 direction, their numerical construction need not reproduce any
 prescribed exact direction. One-sided margins for evaluating the
 residuals and supports are established in the certified arithmetic.
 
 In grid units h=NDelta, write
 
- E=(5pi)/(4N), |Jh+Nc|le E, |h_j|letfrac12 .
+ E=(5pi)/(4N), |Jh+Nc|le E, |h_j|le1/2 .
 
 
 [Shared-coordinate intervals]
@@ -672,7 +672,7 @@ intersect.
 Proof. 
 Separate the shared term in~(seed:grid-error), take its dot
 product with v, and bound each remaining coordinate by 1/2. Solving
-the resulting absolute-value inequality gives~(seed:clip-interval)
+the resulting absolute-value ineuality gives~(seed:clip-interval)
 for either sign of a. The shared midpoint is identical in both lists,
 so the same exact grid displacement satisfies both sets of necessary
 intervals.
@@ -689,11 +689,11 @@ retained.
 
 The enumeration in seeds::build now has an unambiguous
 meaning. It visits every weakly increasing five-bin X tuple and
-applies~(seed:orientation-filter), the first inequality
-in~(seed:gap-inequalities), and sumok. Separately, for
+applies~(seed:orientation-filter), the first ineuality
+in~(seed:gap-ineualities), and sumok. Separately, for
 every possible first bin, it visits all weakly increasing four-bin Y
 tails and applies sumok. The program covout matches
-the shared bin, applies the second gap inequality, and intersects the
+the shared bin, applies the second gap ineuality, and intersects the
 shared displacement intervals. So every discarded exact solution would
 violate one of these necessary conditions. Numerical gap comparisons
 include outward slack (the certified arithmetic).
@@ -831,7 +831,7 @@ centers remain and are included in the output.
 
 The common selected-position permutation preserves all these distances.
 the certified arithmetic bounds the joint error of the nine binary
-midpoint phases by less than 10^{-14}, so the triangle inequality
+midpoint phases by less than 10^{-14}, so the triangle ineuality
 permits the enlarged radius 0.0320001. The centers need not themselves
 satisfy the exact Hadamard equations; their balls cover the necessary
 seed coordinates of all exact solutions.
@@ -847,7 +847,7 @@ Hadamard matrix" to a covering of the 2times 2 unitary group U(2).
 The reference frames are fixed at the seed center; the center itself is
 not assumed to be an exact Hadamard matrix. All statements are first
 proved in exact real and complex arithmetic; the certified arithmetic
-replaces each inequality by an outward-computed version with a certified
+replaces each ineuality by an outward-computed version with a certified
 margin.
 
 % ---------------------------------------------------------------
@@ -890,15 +890,15 @@ chord bound |e^{ix}-1|le |x| gives
 
 [Fixed frames]
 Let pinC^4 have unit-modulus entries, S=sum_j p_j, and
-M=({bf 1} p). The squared singular values of M are 4pm|S|. If
+M=({ 1} p). The squared singular values of M are 4pm|S|. If
 |S|<4 then
 
  M(M^*M)^{-1}
- =frac 1{16-|S|^2}(4{bf 1}-poverlineoverline SS,,;4p-{bf 1}S).
+ =frac 1{16-|S|^2}(4{ 1}-poverlineoverline SS,,;4p-{ 1}S).
 
 Under the same condition there is an orthonormal 4times 2 matrix K
 with range,K=ker M^*; it can be constructed by completing
-{bf 1}/2 and the normalized residual of p with projected coordinate
+{ 1}/2 and the normalized residual of p with projected coordinate
 vectors having nonzero residuals.
 
 
@@ -925,7 +925,7 @@ and write
 Here K_B^*K_B=K_C^*K_C=I_2. Put
 m_B=sigma_{min}(B_0), m_C=sigma_{min}(C_0), and
 m=min(m_B,m_C)>0, so that
-|G_B|_{rm op}=1/m_B, |G_C|_{rm op}=1/m_C. Fix the orthogonal
+|G_B|_{op}=1/m_B, |G_C|_{op}=1/m_C. Fix the orthogonal
 projections
 
  P_K=K_BK_B^*, P_B=I-P_K=G_BB_0,
@@ -933,12 +933,12 @@ projections
 
 Also
 
- A_0=|E_0|_{rm op}=√(2+|1+a|),
+ A_0=|E_0|_{op}=√(2+|1+a|),
  A_*=min(2,A_0+R_s).
 
-Both A_0 and |E'|_{rm op} are at most A_*: indeed A_0le 2,
-every entry of E' has modulus one so |E'|_{rm op}le |E'|_F=2,
-and |E'|_{rm op}le A_0+e_1le A_0+R_s.
+Both A_0 and |E'|_{op} are at most A_*: indeed A_0le 2,
+every entry of E' has modulus one so |E'|_{op}le |E'|_F=2,
+and |E'|_{op}le A_0+e_1le A_0+R_s.
 
 The numerical construction of B_0,C_0 fixes particular coordinate
 choices for K_B,K_C. Acceptance checks on rank and residual norms
@@ -962,7 +962,7 @@ For a feasible completion let Y=K_C^*D'K_B. Then
  6I-Y^*Y&=K_B^*B'^*B'K_B+K_B^*D'^*P_CD'K_Bsucceq 0,
 
  tr(6I-Y^*Y)
- &le(b_1')^2+frac{(|E'|_{rm op}b_1'+sqrt6,c_1')^2}{m_C^2}le q,
+ &le(b_1')^2+frac{(|E'|_{op}b_1'+sqrt6,c_1')^2}{m_C^2}le q,
 
 where
 
@@ -983,7 +983,7 @@ by K_B and subtracting Y^*Y=K_B^*D'^*(I-P_C)D'K_B proves
 most (b_1')^2. Off-diagonal column orthogonality gives
 C'^*D'=-E'^*B', hence
 C_0^*D'K_B=-E'^*Delta B K_B-Delta C^*D'K_B. Now P_C=G_CC_0^* and
-|D'|_{rm op}lesqrt6, giving the first inequality
+|D'|_{op}lesqrt6, giving the first ineuality
 in~(completion:deficit-bound). Cauchy--Schwarz with
 (completion:seed-norm) gives the second.
 
@@ -992,15 +992,15 @@ nonnegative with sum at most q<6, so Y has positive singular values
 √(6-ell_j) and its polar factor is unitary. Moreover
 [
  sqrt6-√(6-ell_j)
- =frac{ell_j}{sqrt6+√(6-ell_j)}
- lefrac{ell_j}{sqrt6+√(6-q)}.
+ =ell_j/sqrt6+√(6-ell_j)
+ leell_j/sqrt6+√(6-q).
 ]
 Taking the Euclidean norm of these two differences and using
 √(ell_1^2+ell_2^2)le ell_1+ell_2le q
 proves~(completion:polar-bound).
 
 
-The inequality q<6 is checked using outward bounds; in the specified
+The ineuality q<6 is checked using outward bounds; in the specified
 parameter range the stronger q<.32 follows from the accepted frame
 checks and the seed radius. Hence every
 feasible completion of an accepted seed ball is assigned at least one
@@ -1019,7 +1019,7 @@ Every Vin U(2) has a representation
  -e(-gamma)sin(2pi t)&e(-beta)cos(2pi t)
  ,
 
-with 0lealphaletfrac12, 0le tletfrac14,
+with 0lealphale1/2, 0le tle1/4,
 0lebeta,gammale 1. For any integer N_u divisible by four, the
 products of closed intervals of width 1/N_u in these ranges cover the
 parameter domain, with bin counts N_u/2,N_u/4,N_u,N_u.
@@ -1096,9 +1096,9 @@ medskipnoindentComparison with a cell center.
 
 For the fixed seed and unitary cell define
 
- D=D_{rm span}+D_{rm ker},
- D_{rm span}=-C_0E_0^*G_B^*,
- D_{rm ker}=sqrt6,K_CUK_B^*, Z=D'-D .
+ D=D_{span}+D_{ker},
+ D_{span}=-C_0E_0^*G_B^*,
+ D_{ker}=sqrt6,K_CUK_B^*, Z=D'-D .
 
 The symbol D throughout denotes this comparison block, which need not
 be entrywise flat; it is not asserted to be the lower-right block of any
@@ -1129,14 +1129,14 @@ The right projections P_B,P_K are complementary and orthogonal, and
 within the right P_K-part the left projections P_C,I-P_C are
 complementary and orthogonal; this proves the decomposition and the
 orthogonality. Row orthogonality gives D'B'^*=-C'E'^*. Since
-P_B=B_0^*G_B^*, D_{rm ker}P_B=0, and D_{rm span}P_B=D_{rm span},
+P_B=B_0^*G_B^*, D_{ker}P_B=0, and D_{span}P_B=D_{span},
 subtracting the comparison block produces~(completion:piece-B).
 The identity C'^*D'=-E'^*B', together with B_0P_K=0 and
 P_CDP_K=0, gives~(completion:piece-C). The definitions of Y,
 K_B, K_C give~(completion:piece-K).
 
 Put r_{EB}=√(e_1^2+(b_1')^2). The four lower rows of H' are
-orthogonal and have norm sqrt6, so |[C' D']|_{rm op}=sqrt6.
+orthogonal and have norm sqrt6, so |[C' D']|_{op}=sqrt6.
 Therefore
 
  |ZP_B|_F&lefrac{sqrt6,r_{EB}+A_0c_1'}{m_B},
@@ -1162,17 +1162,17 @@ Pythagoras for the three pieces gives~(completion:z-bound).
 [A row-block operator bound]
 For every cell midpoint Uin U(2),
 
- |[C_0 D]|_{rm op}le
- L:=max(sqrt6, |C_0|_{rm op}√(1+(A_0/m_B)^2)).
+ |[C_0 D]|_{op}le
+ L:=max(sqrt6, |C_0|_{op}√(1+(A_0/m_B)^2)).
 
-Also |DP_K|_{rm op}=sqrt6 and
+Also |DP_K|_{op}=sqrt6 and
 
- |D|_{rm op}lemax(sqrt6, frac{|C_0|_{rm op}A_0}{m_B}).
+ |D|_{op}lemax(sqrt6, frac{|C_0|_{op}A_0}{m_B}).
 
 
 
 Proof. 
-The two terms D_{rm span},D_{rm ker} have orthogonal right domains
+The two terms D_{span},D_{ker} have orthogonal right domains
 and orthogonal ranges, so the cross terms in DD^* vanish. Using
 G_B^*G_B=(B_0B_0^*)^{-1} gives
 
@@ -1180,11 +1180,11 @@ G_B^*G_B=(B_0B_0^*)^{-1} gives
  =C_0[I+E_0^*(B_0B_0^*)^{-1}E_0]C_0^*+6K_CK_C^* .
 
 The first summand has range in ranC_0 and operator norm
-at most |C_0|_{rm op}^2(1+A_0^2/m_B^2); the second is six times the
+at most |C_0|_{op}^2(1+A_0^2/m_B^2); the second is six times the
 projection onto the orthogonal complement. The total norm is the larger
 of these two, giving~(completion:L-definition). Omitting the
 C_0C_0^* contribution gives~(completion:D-operator-bound).
-Finally DP_K=D_{rm ker}, whose two nonzero singular values are
+Finally DP_K=D_{ker}, whose two nonzero singular values are
 sqrt6.
 
 
@@ -1214,12 +1214,12 @@ Every feasible completion associated to the cell satisfies
 
  |Z-J(s,theta)|_Fle E_r,
  
- E_r=sqrt{e_{rm off}^2+
+ E_r=sqrt{e_{off}^2+
        (frac{√(6· 42)}2,h^2+p)^2},
 
 where
 
- e_{rm off}=
+ e_{off}=
  (R_s^2(L+A_0+1))/(2m)+(zR_s)/(m).
 
 
@@ -1261,7 +1261,7 @@ output pieces componentwise by
  frac 1mL&A_0 A_0&sqrt6
  x y.
 ]
-For the second component use |DP_K|_{rm op}=sqrt6. Since
+For the second component use |DP_K|_{op}=sqrt6. Since
 Lgesqrt6 the matrix has operator norm at most L+A_0. The combined
 seed Taylor contribution is therefore at most R_s^2(L+A_0)/(2m),
 and adding the two product bounds gives~(completion:e-off).
@@ -1287,9 +1287,9 @@ the b_k coefficient is
 ]
 and the c_k coefficient is
 [
- -mathbf 1_bf1_{i=k},i c_k
+ -1_bf1_{i=k},i c_k
        (overlineoverline{(G_B)_(G_B)_{j0}}+overlineoverline{a(G_B)_a(G_B)_{j1}})
- -(G_C)_{i1}i c_kbar(D_{rm ker})_{kj}.
+ -(G_C)_{i1}i c_kbar(D_{ker})_{kj}.
 ]
 For the angular coefficients, if the entries of U are flattened in
 row-major order, use
@@ -1302,7 +1302,7 @@ and moving parts of block::setup, makeD, and
 makeJ.
 
 % ---------------------------------------------------------------
-medskipnoindentNecessary flatness and separating inequalities.
+medskipnoindentNecessary flatness and separating ineualities.
 % ---------------------------------------------------------------
 
 We have bounded the total block displacement Z=D'-D by z and its
@@ -1318,7 +1318,7 @@ A feasible completion associated to a cell must satisfy
 
 
 Proof. 
-Every D'_{ij} has modulus one, and the reverse triangle inequality
+Every D'_{ij} has modulus one, and the reverse triangle ineuality
 gives ||D_{ij}|-1|le |D'_{ij}-D_{ij}|. Sum squares
 and use~(completion:z-bound).
 
@@ -1334,7 +1334,7 @@ by
 The corresponding exact row and column sums vanish, so Taylor's formula
 gives c_X+J_Xs+rho_X=0 and c_Y+J_Ys+rho_Y=0 with
 
- |rho_X|letfrac12 R_s^2, |rho_Y|letfrac12 R_s^2.
+ |rho_X|le1/2 R_s^2, |rho_Y|le1/2 R_s^2.
 
 Take real and imaginary parts of these two equations and append the
 following sixteen real constants and linear forms:
@@ -1343,17 +1343,17 @@ following sixteen real constants and linear forms:
  J_{ij}(s,theta)
  =Re(overlineoverline{D_D_{ij}},J_{ij}(s,theta)).
 
-Denote the resulting real constant vector by c_{rm all}inR^{20} and
-the real linear matrix by J_{rm all}inR^{20times 13}, with the
+Denote the resulting real constant vector by c_{all}inR^{20} and
+the real linear matrix by J_{all}inR^{20times 13}, with the
 first nine columns corresponding to s and the last four to theta.
 
 [Directional exclusion]
 Let yinR^{20} be arbitrary, split as two pairs y_X,y_YinR^2 and
-a flat part y_DinR^{16}, and write g=J_{rm all}^*y=(g_s,g_theta)
+a flat part y_DinR^{16}, and write g=J_{all}^*y=(g_s,g_theta)
 inR^9timesR^4. Every feasible completion of the cell satisfies
 
  
- y· c_{rm all}le{}&
+ y· c_{all}le{}&
  R_s|g_s|+h|g_theta|_1+E_r|Dcirc y_D|_F
  &+frac 12 z^2max(0,max_{i,j}(-y_{D,ij}))
  +frac 12 R_s^2(|y_X|+|y_Y|).
@@ -1374,7 +1374,7 @@ exact flatness equation gives
 ]
 Weight by y_D, and similarly weight the two complex seed equations
 by y_X,y_Y through their real coordinate pairs. Move every term other
-than y· c_{rm all} to the right. The linear contribution is
+than y· c_{all} to the right. The linear contribution is
 bounded by R_s|g_s|+h|g_theta|_1 using the seed ball and angle
 box; real Cauchy--Schwarz bounds the flat remainder contribution by
 E_r|Dcirc y_D|_F; the seed remainders contribute the last term
@@ -1388,19 +1388,19 @@ to the right its contribution is
                sum_{i,j}|Z_{ij}|^2 .
 ]
 Use |Z|_Fle z. This proves the remaining term and the stated
-inequality.
+ineuality.
 
 
 [Dual certificates]
 A cell contains no feasible completion if~(completion:norm-test)
 fails, or if some real direction y violates
 (completion:directional-test). The choice of y may be heuristic;
-only its final inequality must be certified. Positive normalization
+only its final ineuality must be certified. Positive normalization
 of y preserves the certificate.
 
 
 Proof. 
-Both inequalities are necessary for every feasible completion in the
+Both ineualities are necessary for every feasible completion in the
 cell. All terms of~(completion:directional-test) are positively
 homogeneous in y, so a strict violation is a separating, or dual,
 certificate, independent of how the direction was found.
@@ -1409,7 +1409,7 @@ certificate, independent of how the direction was found.
 The implementation stacks exactly these twenty equations in
 eq. The routine trial::pass proposes directions by
 regularized linear solves and weight updates; it excludes only when
-trial::certificate verifies the final inequality. That
+trial::certificate verifies the final ineuality. That
 routine checks finiteness, divides by the largest absolute coordinate,
 and treats the resulting stored coefficients as the direction. It uses
 the negative part of the flat weights, as
@@ -1473,7 +1473,7 @@ successful processing of all prescribed seed indices, including all
 shards, with no failed seed or unresolved cell. Retaining some valid
 phase balls from an unsuccessful seed does not certify the unprocessed
 remainder; these conditions are part of the computational obligation.
-They do not follow from the local inequalities alone.
+They do not follow from the local ineualities alone.
 
 % =====================================================================
 medskipnoindentCertified phase balls and their atlas.
@@ -1583,7 +1583,7 @@ F(P+x)=F(P)+L_Px+R_P(x). Put
 a_{ij}(x)=|x_i-x_j|^2 (where x_i is row i of x). Then
 
 |(R_P(x))_{ij}|&lepi a_{ij}(x), 
-|(R_P(x)-calR_P(x)-R_P(v))_{ij}|
+|(R_P(x)-R_P(x)-R_P(v))_{ij}|
 &lepisqrt{a_{ij}(x-v)}
       (sqrt{a_{ij}(x)}+sqrt{a_{ij}(v)}).
 
@@ -1598,7 +1598,7 @@ then for horizontal x,v,
 
  |MR_P(x)|&lepi b|x|^2,
  
- |M(R_P(x)-calR_P(x)-R_P(v))|
+ |M(R_P(x)-R_P(x)-R_P(v))|
  &lepi b,|x-v|(|x|+|v|).
  
 
@@ -1619,7 +1619,7 @@ e(P_{ig}-P_{jg}) do not affect modulus. Cauchy--Schwarz in g
 gives~(phase:edge-difference).
 
 Pairing MR_P(x) with a real unit y and using the
-two-coordinate Cauchy--Schwarz inequality on each edge
+two-coordinate Cauchy--Schwarz ineuality on each edge
 proves~(phase:matrix-remainder). For the difference, apply
 separately
 [
@@ -1630,7 +1630,7 @@ separately
  (sum_{i>j}w_{ij}a_{ij}(x))^{1/2}
  le b|x-v||x|,
 ]
-and the same inequality with v in the last factor. Take the supremum
+and the same ineuality with v in the last factor. Take the supremum
 over y; the same constant controls both remainders.
 
 
@@ -1641,7 +1641,7 @@ b=sqrt 8,|M| and b=sqrt 6,|S|, where the
 30times 216 real matrix S has row
 [
  S_{a,:}
-   =M_{:,a}^* ⊗ ({bf e}_i-{bf e}_j)^*
+   =M_{:,a}^* ⊗ ({ e}_i-{ e}_j)^*
 ]
 when a is either of the two real-coordinate indices of edge i>j.
 Consequently pi times the smaller of any certified upper bounds for
@@ -1669,7 +1669,7 @@ Cauchy--Schwarz, now using sum a_{ij}=6s^2, proves the claim. The row
 Gram of S is computable without forming its 216 columns:
 
  (S S^*)_calS^*)_{ab}
-  =(({bf e}_i-{bf e}_j)·({bf e}_{i'}-{bf e}_{j'}))
+  =(({ e}_i-{ e}_j)·({ e}_{i'}-{ e}_{j'}))
            (M_{:,a}· M_{:,b}),
 
 with the edge labels repeated for real and imaginary parts. This is the
@@ -1697,7 +1697,7 @@ Write E=Pi-TT^*-ML_PPi. Taylor's formula at the exact solution gives
  (I-TT^*)delta=Edelta-MF(P)-MR_P(
 ]
 The certificate and Lemma~phase:remainder
-prove~(phase:root-normal). For the squared inequality (no
+prove~(phase:root-normal). For the squared ineuality (no
 projector assumption on TT^*),
 [
  |delta-Teta|^2
@@ -1840,7 +1840,7 @@ medskipnoindentDisplacement bounds for a retained cell.
 Fix a retained cell. Its seed displacement satisfies |s|le R_s, its
 angular box satisfies |theta_a|le h, 1le ale 4, and
 Z=D'-D has |Z|_Fle z and |Z-J(s,theta)|_Fle E_r as
-in . Let I_s=(7,8,9,10,11,13,19,25,31)calI_s=(7,8,9,10,11,13,19,25,31)
+in . Let I_s=(7,8,9,10,11,13,19,25,31)I_s=(7,8,9,10,11,13,19,25,31)
 denote the nine seed indices, and I_D the sixteen lower-right
 indices (i,j) with 2le i,jle 5. The first row and column have zero
 raw displacement.
@@ -1972,7 +1972,7 @@ With S_Delta=min(S_1,S_2), a valid upper bound is
 
  |delta|le
  (|Delta|^2+
-       frac{R_s^2+arc^2}{(2pi)^2}
+       R_s^2+arc^2/(2pi)^2
        +(S_Delta)/(pi))^{1/2}
        +|ε|.
 
@@ -1984,7 +1984,7 @@ give~(phase:support-linear). Direct weighted Cauchy--Schwarz
 using~(phase:arc-weighted) gives~(phase:support-weighted).
 Since Delta is horizontal, Delta·Pi u=Delta· u.
 Expand the squared norm of Delta+Pi u/(2pi), contract by Pi, and
-add |Piε|le|ε| by the triangle inequality.
+add |Piε|le|ε| by the triangle ineuality.
 
 
 % ---------------------------------------------------------------
@@ -2013,7 +2013,7 @@ beta>0, the sum As+Bv belongs to the ellipsoid
 
 Proof. 
 For each real test vector z, the support is at most
-r|A^*z|+q|B^*z|, which by the inequality
+r|A^*z|+q|B^*z|, which by the ineuality
 (x+y)^2le(1+beta)x^2+(1+1/beta)y^2 is at
 most (z^*Gz)^{1/2}. Diagonalize G: the vectors satisfying
 |z· x|le(z^*Gz)^{1/2} for every z are exactly G^{1/2} times
@@ -2023,7 +2023,7 @@ the unit ball, including when G is singular.
 By Lemma~phase:ellipsoid-containment with beta=.45, the sum of
 the seed and e_D contributions to T^*u lies in an ellipsoid whenever
 
- G_{rm lin}succeq
+ G_{lin}succeq
     1.45R_s^2(T^*X_s)(T^*X_s)^*
        +(1+1/.45)E_r^2T_D^*T_D.
 
@@ -2032,7 +2032,7 @@ The remaining sine error has norm at most
 There is also a bound using the raw seed and weighted angle balls with a
 single offset U_0=2pi T^*Delta:
 
- G_{rm dir}succeq
+ G_{dir}succeq
     1.8R_s^2T_s^*T_s+
        (1+1/.8)weighted^2
                      T_D^*diag(1/m_j)T_D.
@@ -2070,14 +2070,14 @@ Lemma~phase:ellipsoid-norm. Two valid tangent bounds are
 
  |T^*delta|
  le{}&
- frac{E(G_{rm lin};{U_sigma})+
+ frac{E(G_{lin};{U_sigma})+
       max_{jinI_D}|T_{j,:}|,arc^3/6+
       horsqrt{R_s^2+arc^2}}{2pi}
        +sqrt{1+ort},|ε|,
  
  |T^*delta|
  le{}&
- frac{E(G_{rm dir};{U_0})+
+ frac{E(G_{dir};{U_0})+
       horsqrt{R_s^2+arc^2}}{2pi}
        +sqrt{1+ort},|ε|.
  
@@ -2088,7 +2088,7 @@ off is the computed 2pi T^*Delta; AB is the
 computed T^*X; U holds the 16 corners. The covariance uses
 the stored AB coefficients and charges their ideal-coefficient
 error separately outside the ellipsoid.  gives
-|widehatwidehat A-T^*XA-T^*X|<1.5· 10^{-5},
+|Ahat-T^*XA-T^*X|<1.5· 10^{-5},
 |(s,theta)|<.226, so this substitution costs less than
 3.4· 10^{-6} radians. For both covariance choices a diagonal
 10^{-7}I is added. The inverses in the second choice are certified
@@ -2142,7 +2142,7 @@ Suppose two Root records (P,r_P), (Q,r_Q) satisfy
  |delta_P|le R_P, |delta_Q|le R_Q, F(P+delta_P)=0,
 ]
 for a certified upper distance d_{PQ}. Then the union
-S(P,r_P)calS(P,r_P)cupS(Q,r_Q)calS(Q,r_Q) is contained in
+S(P,r_P)S(P,r_P)cupS(Q,r_Q)S(Q,r_Q) is contained in
 [
  S(P, max{r_P, d_{PQ}+r_Q+allowance}).
 ]
@@ -2152,7 +2152,7 @@ Proof.
 Take Q+delta_Q with F(Q+delta_Q)=0 and |delta_Q|le r_Q. The
 alignment d_q=Pi(P-Q-N) chosen by wrap gives an equivalent
 solution of the first center, with displacement of norm at most
-d_{PQ}+r_Q by the triangle inequality. The phase-ball solution set
+d_{PQ}+r_Q by the triangle ineuality. The phase-ball solution set
 S in Definition~refine:solution-set is closed under this
 equivalence, so it contains the aligned representative. Inflate to
 r_Pvee d_{PQ}+r_Q and add the outward allowance.
@@ -2249,7 +2249,7 @@ horizontal subspace.
 
 For a certified center P and radius Rge 0 put
 
- S(P,R)=calS(P,R)=
+ S(P,R)=S(P,R)=
  bigl{[e(P+delta)]:deltainV, |delta|le R,
        F(P+delta)=0bigr},
 
@@ -2291,7 +2291,7 @@ horizontal.
 
 
 Proof. 
-The Gram bound gives the first inequality. The eigenvalues of TT^*
+The Gram bound gives the first ineuality. The eigenvalues of TT^*
 lie in [0,1+o_P], so those of I-TT^* lie in [-o_P,1]; its norm is
 at most max(1,o_P)le 1+o_P. Finally,
 |Ax|^2=|x|^2-2|zeta|^2+|Tzeta|^2
@@ -2354,10 +2354,10 @@ tangent bounds apply to h':
 
 
 Proof. 
-The triangle inequality gives dle R+|d_q|. The identity
+The triangle ineuality gives dle R+|d_q|. The identity
 |x|^2le(1+o_P)|T^*x|^2+|(I-TT^*)x|^2 from
 Lemma~refine:approximate-projection, applied to h', gives the
-second inequality in~(refine:parent-distance).
+second ineuality in~(refine:parent-distance).
 
 For~(refine:normal-three): nu=A(delta-d_q), so
 |nu|le|Adelta|+|Ad_q|le n_P(R)+n_q=nu_0 by the parent Root
@@ -2387,7 +2387,7 @@ sigma|nu|. This makes explicit why no extra hor
 correction is needed at this step: the horizontal projection appears in
 S_h once, and both T and S_h^*T are controlled by the Gram bounds.
 
-For~(refine:child-distance), apply the squared inequality at the
+For~(refine:child-distance), apply the squared ineuality at the
 child Root to the horizontal exact-solution displacement h' of Q:
 d^2=|h'|^2le n_Q(d)^2+(1+o_Q)|S^*h'|^2.
 
@@ -2445,7 +2445,7 @@ medskipnoindentThe rho search for usable tangent thresholds.
 Fix a certified parent Root P,T and a reference center Q,S (an atlas
 child or an earlier blocker) with certified constants
 o_Q,alpha_Q,gamma_Q,mu_Q,ε_Q. For every tangent threshold
-tge 0, let widehatwidehat D(t)D(t) denote a certified upper bound for
+tge 0, let Dhat(t)D(t) denote a certified upper bound for
 |h'|=|delta-d_q| over every parent solution delta with
 |eta-eta_q|le t. Such a bound is obtained by evaluating
 (refine:iteration-step) at
@@ -2455,17 +2455,17 @@ upper. The source stores it back through the
 alignment~(refine:alignment).
 
 
- |eta-eta_q|le t  and  widehatwidehat D(t)D(t)le a
+ |eta-eta_q|le t  and  Dhat(t)D(t)le a
   ⇒
- [e(P+delta)]inS(Q,a)calS(Q,a)
+ [e(P+delta)]inS(Q,a)S(Q,a)
  quadfor every parent solution delta.
 
 
 For a queried sphere radius 0le ale .126, the source's init
 searches for a usable tangent distance rho as follows. Set ell=0,
-u=1.01a. If widehatwidehat D(0)>a-10^D(0)>a-10^{-7}, report no relation. Otherwise
+u=1.01a. If Dhat(0)>a-10^D(0)>a-10^{-7}, report no relation. Otherwise
 make 25 midpoint trials, each with m=(ell+u)/2: if
-widehatwidehat D(m)<a-2D(m)<a-2· 10^{-7}, replace ell by m; otherwise
+Dhat(m)<a-2D(m)<a-2· 10^{-7}, replace ell by m; otherwise
 replace u by m. Finally set
 
  rho=max(0,ell-10^{-8}),
@@ -2477,28 +2477,28 @@ Every usable returned threshold certifies
 [
  |eta-eta_q|lerho
  ⇒
- [e(P+delta)]inS(Q,a)calS(Q,a)
+ [e(P+delta)]inS(Q,a)S(Q,a)
 ]
 for every parent solution delta. The search need not find the largest
 possible threshold; in particular its correctness does not require
-monotonicity of the numerical function widehatwidehat D.D.
+monotonicity of the numerical function Dhat.D.
 
 
 Proof. 
 A positive returned rho>10^{-7} cannot arise from the initial value
 ell=0. Thus the final stored ell was assigned by a successful
-trial widehatwidehat D(D(ell)<a-2· 10^{-7}, and
+trial Dhat(D(ell)<a-2· 10^{-7}, and
 rho=ell-10^{-8}<ell. Take any parent solution delta with
-|eta-eta_q|lerholeell. By construction widehatwidehat D(D(ell) is
+|eta-eta_q|lerholeell. By construction Dhat(D(ell) is
 an upper bound for |h'| at this solution, so
 |h'|<a-2· 10^{-7}<a. Apply~(refine:eval-semantics). No
-comparison between widehatwidehat DD at distinct arguments is needed.
+comparison between DDhat at distinct arguments is needed.
 
 
 For a previously completed parent used as a blocker, a is its fixed
 input radius and this lemma suffices. For an atlas child, the search
 uses the allowed cap a=tau; when boxes are assigned to that child,
-the code evaluates widehatwidehat DD again at an upper tangent distance for
+the code evaluates DDhat again at an upper tangent distance for
 those boxes and grows the stored radius to that certified distance, with
 an additional margin. It then checks both that the tangent argument is
 at most rho and that the new radius is at most tau; failing either
@@ -2550,11 +2550,11 @@ final atlas of children and B the list of input parents
 whose refinement failed. Then
 
  
- ∪_{(P,R),input}S(P,R)calS(P,R)
+ ∪_{(P,R),input}S(P,R)S(P,R)
  &⊆
- ∪_{(Q,r)inA} S(Q,r)calS(Q,r)
+ ∪_{(Q,r)inA} S(Q,r)S(Q,r)
  & cup
- ∪_{(P,R)inB} S(P,R).calS(P,R).
+ ∪_{(P,R)inB} S(P,R).S(P,R).
  
 
 In particular, a completed run with no bad parents replaces the input
@@ -2605,7 +2605,7 @@ tau rather than its current achieved radius. If a Root at index i
 is emitted before the run ends, every later state satisfies
 
  r_iletau,
- S(Q_i,r_i)⊆calS(Q_i,r_i)⊆S(Q_i,tau),
+ S(Q_i,r_i)⊆S(Q_i,r_i)⊆S(Q_i,tau),
 
 with exactly the same center, tangent columns, and certificate
 constants as in its batch. No future merge changes these data. At
@@ -2697,7 +2697,7 @@ torus, so this covers every boundary. Let u be a bin midpoint, with
 u_0=0, and let
 [
  a=frac 1{2M},  d_0=0, |d_j|le a (j>0), 
- x=d-frac 16(sum_j d_j){bf 1}.
+ x=d-frac 16(sum_j d_j){ 1}.
 ]
 Then the bin vector equals e(u+x)/sqrt6 up to global phase.
 
@@ -2751,11 +2751,11 @@ The scalar bound is the same with the first change at most
 Suppose |x|le X and |x|_∞le I. For every yinR^6,
 unbiasedness implies
 
- |y·(c+Jx-Beta)|leE(y;X,I),calE(y;X,I),
+ |y·(c+Jx-Beta)|leE(y;X,I),E(y;X,I),
 
 where
 
- E(y;X,I):=calE(y;X,I):={}&nN_y+piBigl[
+ E(y;X,I):=E(y;X,I):={}&nN_y+piBigl[
    |y|_∞(KX+r)^2
    +min{G_yX^2, K|Scirc y|IX}notag
  &hspace{28mm}
@@ -2763,11 +2763,11 @@ where
    +frac 1{sqrt6}|Scirc y|_∞ r^2Bigr].
  
 
-For a bin, this gives the necessary support inequality
+For a bin, this gives the necessary support ineuality
 
  |y· c|
  le asum_{j>0}|(J^*y)_j|+r_H|B^*y|
-       +E(y;X_M,I_M).calE(y;X_M,I_M).
+       +E(y;X_M,I_M).E(y;X_M,I_M).
 
 
 
@@ -2794,7 +2794,7 @@ The bin routine Layer::test applies~(exclude:direct-tests),
 then the scalar test~(exclude:scalar-linear)
 (see below), and finally~(exclude:bin-support) for proposed
 directions. Proposed directions come from an inexact linear solve; the
-proposal method is immaterial because the final inequality holds for
+proposal method is immaterial because the final ineuality holds for
 every real y and is checked at the stored test direction.
 
 There is a useful scalar test before the tangent decomposition. Set
@@ -2849,7 +2849,7 @@ For vinR^6 and sge 0, define the closed vector ball
 Suppose each retained bin midpoint u is assigned to a center v and
 an integer vector minZ^6. Let
 [
- h=u-v-m-frac16(sum_j(u_j-v_j-m_j)){bf 1}.
+ h=u-v-m-frac16(sum_j(u_j-v_j-m_j)){ 1}.
 ]
 If the assigned radius is at least |h|+X_M, then
 B_{vec}(v,s) covers the whole bin up to global
@@ -2925,7 +2925,7 @@ medskipnoindentBin geometry and edge tests.
 Fix a bin with midpoint uinR^6 (u_0=0) and halfwidth
 a=1/(2M). Any phase vector in the bin is u+d with d_0=0 and
 |d_j|le a for jge1. The horizontal displacement
-x=d-frac16(sum_jd_j){bf 1} satisfies sum_jx_j=0 and, with the
+x=d-frac16(sum_jd_j){ 1} satisfies sum_jx_j=0 and, with the
 constants X_M,L_M,I_M of ,
 
  |x|le X_M, |x|_1le L_M, |x|_∞le I_M.
@@ -2935,7 +2935,7 @@ constants X_M,L_M,I_M of ,
 Proof. 
 sum_jx_j=0 is immediate from the definition.
 Write d=barbar d+(d-d+(d-barbar d)d) with barbar d=(1/6)d=(1/6)sum_jd_j. Then
-|x|^2=|d-barbar dd{bf 1}|^2=|d|^2-6barbar d^2d^2le|d|^2le5a^2,
+|x|^2=|d-barbar dd{ 1}|^2=|d|^2-6barbar d^2d^2le|d|^2le5a^2,
 so X_Mgesqrt5,a suffices. Each |d_j-barbar d|d|le 2a, so
 L_Mge10a and I_Mge2a also suffice. 
 accounts for the 10^{-12} margin.
@@ -3006,7 +3006,7 @@ S_{U}, the linear term is D_{U} (the division by
 is at most (2pi|S|/sqrt6)r_e^2+4pi^2r_e^2; substituting the
 residual of the orthogonality calculation
 gives~(exclude:unbiased-residual). Cauchy--Schwarz on
-|z|le r_e gives the support inequalities.
+|z|le r_e gives the support ineualities.
 
 
 
@@ -3076,7 +3076,7 @@ support over |x_t|le s_t give the first two norm terms.
 The single-vector coefficients of the shared tangent eta are -B^0
 and -B^1, and combine to give the coefficient
 -(B^0)^*y_0-(B^1)^*y_1. Support this over |eta|le r_H. Bound
-the three residuals by the triangle inequality and Cauchy's inequality.
+the three residuals by the triangle ineuality and Cauchy's ineuality.
 
 
 This is exactly the comparison implemented by graph::joint.
@@ -3089,11 +3089,11 @@ array, in contrast, is shared and accumulates both -(B^t)^*y_t
 contributions before its norm is taken.
 
 As with the single-vector test, the proposed direction comes from an
-inexact linear solve; only its final inequality must be certified. The
-final inequality holds for every real y; the numeric choice of
+inexact linear solve; only its final ineuality must be certified. The
+final ineuality holds for every real y; the numeric choice of
 direction is not part of the certificate.
 
-The inequalities just proved are necessary in exact real arithmetic. In
+The ineualities just proved are necessary in exact real arithmetic. In
 the implementation, each rejection is a strict violation of an
 outward-enlarged bound; equalities and unresolved comparisons retain
 the candidate.  certifies these comparisons,
@@ -3117,7 +3117,7 @@ By Lemma~exclude:cluster-cover and
 Lemma~exclude:distinct-labels, two additional mutually unbiased
 orthonormal bases would give disjoint vertex sets A,B satisfying
 
- |A|=|B|=6, A,B are O-cliques,calO-cliques,
+ |A|=|B|=6, A,B are O-cliques,O-cliques,
  Atimes B⊆ E(U).
  
 
@@ -3170,7 +3170,7 @@ Suppose
       (exclude:direct-tests), (exclude:scalar-linear), or
       (exclude:bin-support), or been accepted and covered by the
       assigned ball;
- the O-calO- and U-graphscalU-graphs on the retained balls,
+ the O-O- and U-graphscalU-graphs on the retained balls,
       after excluding any pair by the direct or joint tests conditional
       on eta, admit no
       configuration~(exclude:graph-configuration) as decided by a
@@ -3206,8 +3206,8 @@ For a bin the source forms its halfwidth with a 10^{-12} increase,
 then increases each of X_M,L_M,I_M by a further 10^{-12} after the
 indicated multiplication. The stored tangent and normal bounds are
 
- widehatwidehat r_H=√(1+r_H=√(1+omega),r+10^{-9},
- widehatwidehat n=n_0+n=n_0+iota r+Gamma r^2+10^{-9}.
+ rhat_H=√(1+r_H=√(1+omega),r+10^{-9},
+ nhat=n_0+n=n_0+iota r+Gamma r^2+10^{-9}.
 
 The upper squared-magnitude threshold in~(exclude:direct-tests) is
 increased by 2·10^{-7}, and its lower threshold, when tau<1, is
@@ -3223,7 +3223,7 @@ E_{O} and E_{U} are increased by 2· 10^{-8};
 the direct comparisons~(exclude:orth-support)
 and~(exclude:unbiased-support) receive another 2· 10^{-7}.
 The right side of the joint
-comparison~(exclude:joint-inequality) receives 3· 10^{-7}.
+comparison~(exclude:joint-ineuality) receives 3· 10^{-7}.
 Cluster assignment radii include the 10^{-9} allowance described in
 Lemma~exclude:cluster-cover.
 
@@ -3525,7 +3525,7 @@ medskipnoindentCertified arithmetic for the recorded execution.
 
 % =====================================================================
 
-The outward error estimates behind each inequality used above are fixed
+The outward error estimates behind each ineuality used above are fixed
 below. Every certified rejection is a strict violation
 of an outward bound computed under the primitive-operation model below.
 Nothing here assumes that a floating-point center is an exact solution,
@@ -3619,7 +3619,7 @@ gamma_m=2.1· 10^{-16}(m+1) is a conservative choice: for
 mle 2000 one has (1+u)^m-1le mule 2m· 10^{-16}, which is
 smaller than gamma_m. Direct substitution of the accumulated
 additive terms into the recurrence produces the tau bound. The
-norm inequality is the same argument applied to the square-sum and
+norm ineuality is the same argument applied to the square-sum and
 then to the correctly rounded square root, with m+2 factors and a
 final 10^{-140} additive allowance.
 
@@ -3796,9 +3796,9 @@ below 2.3· 10^{-12}, so its reciprocal has error below
 7· 10^{-14}. The two pseudoinverse numerators have modulus at most
 eight and error below 5.1· 10^{-13}. Division/scaling then gives
 
- |Delta G_{ij}|<10^{-12}, |G_{ij}^{rm ideal}|<.81,
+ |Delta G_{ij}|<10^{-12}, |G_{ij}^{ideal}|<.81,
 
-the second inequality following from the lower singular bound.
+the second ineuality following from the lower singular bound.
 
 The first constant basis column is exact. The next column has numerator
 error below 1.3· 10^{-13} and denominator
@@ -3867,7 +3867,7 @@ medskipnoindentRoot certificates and normal constants.
 
 The tangent operator has the certified bound
 
- |T|_{rm op}lesqrt{1+ort}<1.00001,
+ |T|_{op}lesqrt{1+ort}<1.00001,
 
 since T is a genuine Root. The evaluation of T^*T-I for the stored
 T (each |T_{ja}|le 1.01) yields ort<10^{-5}; the
@@ -3922,7 +3922,7 @@ times the Jacobian bound six, plus the 10^{-8} Jacobian error and
 short-dot rounding, is below 10^{-6} per entry. The other used X
 entries are exact. The mi subtraction of 10^{-6} therefore
 gives a positive magnitude lower bound greater than .148, and
-invmag_j=1/(widehatwidehat m_j-10^m_j-10^{-6})+10^{-9} is an upper bound
+invmag_j=1/(mhat_j-10^m_j-10^{-6})+10^{-9} is an upper bound
 for 1/m_j, below 6.8.
 
 The flat-sum error is below 10^{-6}, so the 10^{-5} addition in
@@ -3950,7 +3950,7 @@ and short-dot rounding. The error in the whole X array is at most
 16 affected positions. Multiplication by T^* and the less than
 2· 10^{-13} rounding of each dot give
 
- |AB-T^*X_{rm ideal}|<1.5· 10^{-5}.
+ |AB-T^*X_{ideal}|<1.5· 10^{-5}.
 
 The parameter-vector norm is below
 √(.203^2+4(pi/64)^2)<.226, so substitution of stored AB costs
@@ -4095,7 +4095,7 @@ coefficients. Define exact stored arrays
  
  a_y&=sum_i y_i c_i,& h_a&=sum_i y_iB_{ia},
  z_j&=sum_i y_iJ_{ij},&
- widetilde z_j&=sum_i y_iD^{rm re}_{ij},
+ widetilde z_j&=sum_i y_iD^{re}_{ij},
  N_J&=sum_{i,j}y_i^2J_{ij}^2,& N_S&=sum_i y_i^2|S_i|^2,
  M_S&=max_i y_i^2|S_i|^2,& m_y&=max_i|y_i|,
   M_z&=max_j(z_j^2+widetilde z_j^2).&
@@ -4107,8 +4107,8 @@ The support compared with |a_y| is the stored evaluation of
  &+piBigl[
  m_y(KX+r)^2+
  minbigl{√(M_z),X, K√(N_S),x_∞bigr}X
- +2frac{√(N_S)}{sqrt6}Xr
- +frac{√(M_S)}{sqrt6}r^2 Bigr].
+ +2√(N_S)/sqrt6Xr
+ +√(M_S)/sqrt6r^2 Bigr].
  
 
 This is the necessary support bound of the Exclusion stage.