diff 56bc728..f6bf760 mutually_unbiased_bases_dim6_v2.txt
diff --git a/mutually_unbiased_bases_dim6_v2.txt b/mutually_unbiased_bases_dim6_v2.txt
index b63973e..e41a359 100644
--- a/mutually_unbiased_bases_dim6_v2.txt
+++ b/mutually_unbiased_bases_dim6_v2.txt
@@ -104,10 +104,10 @@ residual at a proposed center.
medskipnoindentIntroduction.
% =====================================================================
-Two orthonormal bases mathcal BcalB and mathcal CcalC of C^d are
+Two orthonormal bases B and C of C^d are
mutually unbiased (MUB) if
[
- |< b,c>|^2=frac 1d qquadfor every binmathcal B,calB, cinmathcal CcalC .
+ |< b,c>|^2=frac 1d for every binB, cinC .
]
Here < u,v>=sum_ju_jbarv_j.
For dge 2 let N(d) be the maximum size of a pairwise mutually
@@ -337,10 +337,10 @@ and U_0=I_6.
medskipnoindentA Lie-algebra consequence.
-An orthonormal basis mathcal BcalB of C^6 determines its Cartan
-subalgebra mathfrak h_frakh_{mathcal B}calB of traceless matrices that are
+An orthonormal basis B of C^6 determines its Cartan
+subalgebra mathfrak h_frakh_{B} of traceless matrices that are
diagonal in that basis. It is spanned by the matrices bb^*-I/6 for
-binmathcal B.calB. The Killing form on sl_6(C) is
+binB. The Killing form on sl_6(C) is
K(X,Y)=12tr(XY); two subalgebras are
Killing-orthogonal when this form vanishes on every pair of
elements. For unit vectors b,c,
@@ -360,7 +360,7 @@ the certified arithmetic, using the complete execution recorded above
and certified in Theorem~thm:main, the maximum
number of pairwise Killing-orthogonal Cartan subalgebras of
sl_6(C) that are all closed under the standard adjoint
-Xmapsto X^*=overlineoverline X^X^{,T} is three.
+X|-> X^*=overlineoverline X^X^{,T} is three.
Proof.
@@ -622,8 +622,8 @@ If 1+sum_{j=0}^4 e(u_j+Delta_j)=0 and |Delta_j|le w/2 then
In particular for every vinR^2,
- |vcdot c|le(5pi w^2)/(4)|v|
- +frac w2sum_{j=0}^4|vcdot J_j|.
+ |v· c|le(5pi w^2)/(4)|v|
+ +frac w2sum_{j=0}^4|v· J_j|.
@@ -655,15 +655,15 @@ In grid units h=NDelta, write
Let t be the position of a shared entry in either list. Every feasible
displacement satisfies, for every vinR^2,
- |Nvcdot c+(vcdot J_t)h_t|
- le E|v|+frac 12sum_{jne t}|vcdot J_j|.
+ |Nv· c+(v· J_t)h_t|
+ le E|v|+frac 12sum_{jne t}|v· J_j|.
-If a=vcdot J_tne 0 and the right side is r, this is the interval
+If a=v· J_tne 0 and the right side is r, this is the interval
constraint
- -(Nvcdot c)/(a)-frac r{|a|}
+ -(Nv· c)/(a)-frac r{|a|}
le h_tle
- -(Nvcdot c)/(a)+frac r{|a|}.
+ -(Nv· c)/(a)+frac r{|a|}.
The intervals obtained for the shared entry from X and Y must
intersect.
@@ -729,19 +729,19 @@ blocks r_X,r_Y of r_* satisfy |r_X|,|r_Y|le E.
[Support bounds for center reuse]
Let u' be a previously retained midpoint with the same selected
-position s, and let k=N(u-u')inmathbb Z^9.bbZ^9. Every feasible
+position s, and let k=N(u-u')inZ^9. Every feasible
displacement in the current bin pair satisfies
N^2|q-u'|^2=|k+h|^2
- le|k|^2+frac 94+2kcdot h .
+ le|k|^2+frac 94+2k· h .
Two upper bounds for the last term are
- 2kcdot h&lesum_{j=0}^8|k_j|,
+ 2k· h&lesum_{j=0}^8|k_j|,
- 2kcdot h&le
+ 2k· h&le
sum_{j=0}^8|k_j-(J_*^Tv)_j|
- +2vcdot p_*+2E(|v_X|+|v_Y|),
+ +2v· p_*+2E(|v_X|+|v_Y|),
vinR^4,
@@ -753,8 +753,8 @@ Expand the square in~(seed:distance-bound) and use
|h|^2le 9/4. The coordinate bounds give~(seed:box-support).
For the other bound write
[
- 2kcdot h
- =2(k-J_*^Tv)cdot h+2vcdot p_*+2vcdot r_* .
+ 2k· h
+ =2(k-J_*^Tv)· h+2v· p_*+2v· r_* .
]
The first term is at most sum_j|k_j-(J_*^Tv)_j| by the same
coordinate bound. Cauchy--Schwarz on the two residual blocks bounds
@@ -1197,7 +1197,7 @@ multiply the center entry by i times the corresponding component of
s; at fixed entries the variation is zero. Define the linear map
- mathcal J(s,calJ(s,theta)={}&
+ J(s,theta)={}&
(-C_1E_0^*-C_0E_1^*-DB_1^*)G_B^*
&+G_C(-E_0^*B_1-C_1^*D)P_K
+sqrt6,K_CdU[theta]K_B^*.
@@ -1212,10 +1212,10 @@ an exact completion at every seed center.
Every feasible completion associated to the cell satisfies
- |Z-mathcal J(s,calJ(s,theta)|_Fle E_r,
+ |Z-J(s,theta)|_Fle E_r,
E_r=sqrt{e_{rm off}^2+
- (frac{√(6cdot 42)}2,h^2+p)^2},
+ (frac{√(6· 42)}2,h^2+p)^2},
where
@@ -1269,7 +1269,7 @@ and adding the two product bounds gives~(completion:e-off).
In the third space the exact remainder is
K_C[Y-sqrt6,U-sqrt6,dU[theta]]K_B^*. Insert sqrt6,V and
apply the polar and unitary Taylor bounds: its norm is at most
-p+√(6cdot 42),h^2/2. This space is orthogonal to the first
+p+√(6· 42),h^2/2. This space is orthogonal to the first
two, so Pythagoras
proves~(completion:linear-remainder-bound).
@@ -1294,7 +1294,7 @@ and the c_k coefficient is
For the angular coefficients, if the entries of U are flattened in
row-major order, use
[
- W_{ij,l}=sqrt6,(K_C)_{i,lfloor l/2rfloor}
+ W_{ij,l}=sqrt6,(K_C)_{i,floor( l/2)}
overlineoverline{(K_B)_(K_B)_{j,lbmod 2}}, 0le l<4,
]
and contract with each angular derivative of U. These are the fixed
@@ -1306,7 +1306,7 @@ medskipnoindentNecessary flatness and separating inequalities.
% ---------------------------------------------------------------
We have bounded the total block displacement Z=D'-D by z and its
-remainder after the linear comparison mathcal J(s,calJ(s,theta) by E_r.
+remainder after the linear comparison J(s,theta) by E_r.
The next step uses these bounds inside the necessary equations
|D'_{ij}|=1.
@@ -1341,7 +1341,7 @@ following sixteen real constants and linear forms:
c_{ij}=frac{|D_{ij}|^2-1}{2},
J_{ij}(s,theta)
- =Re(overlineoverline{D_D_{ij}},mathcal J_calJ_{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
@@ -1353,7 +1353,7 @@ a flat part y_DinR^{16}, and write g=J_{rm all}^*y=(g_s,g_theta)
inR^9timesR^4. Every feasible completion of the cell satisfies
- ycdot c_{rm all}le{}&
+ y· c_{rm 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|).
@@ -1364,7 +1364,7 @@ coordinates arranged as a 4times 4 matrix.
Proof.
-Write R=Z-mathcal J(s,calJ(s,theta), so |R|_Fle E_r. Expanding the
+Write R=Z-J(s,theta), so |R|_Fle E_r. Expanding the
exact flatness equation gives
[
0=frac{|D'_{ij}|^2-1}{2}
@@ -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 ycdot c_{rm all} to the right. The linear contribution is
+than y· c_{rm 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
@@ -1578,12 +1578,12 @@ is used only to establish the Root assertions and is not serialized.
[Quadratic and difference remainders]
-There is a real 30-vector mathcal R_P(x)calR_P(x) with
-F(P+x)=F(P)+L_Px+mathcal R_P(x).calR_P(x). Put
+There is a real 30-vector R_P(x) with
+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
-|(mathcal R_P(x))_calR_P(x))_{ij}|&lepi a_{ij}(x),
-|(mathcal R_P(x)-calR_P(x)-mathcal R_P(v))_calR_P(v))_{ij}|
+|(R_P(x))_{ij}|&lepi a_{ij}(x),
+|(R_P(x)-calR_P(x)-R_P(v))_{ij}|
&lepisqrt{a_{ij}(x-v)}
(sqrt{a_{ij}(x)}+sqrt{a_{ij}(v)}).
@@ -1596,9 +1596,9 @@ the two edge coordinates of M^*y. If
then for horizontal x,v,
- |Mmathcal R_P(x)calR_P(x)|&lepi b|x|^2,
+ |MR_P(x)|&lepi b|x|^2,
- |M(mathcal R_P(x)-calR_P(x)-mathcal R_P(v))calR_P(v))|
+ |M(R_P(x)-calR_P(x)-R_P(v))|
&lepi b,|x-v|(|x|+|v|).
@@ -1618,7 +1618,7 @@ Apply to each scalar row difference and sum over g. The unit factors
e(P_{ig}-P_{jg}) do not affect modulus. Cauchy--Schwarz in g
gives~(phase:edge-difference).
-Pairing Mmathcal R_P(x)calR_P(x) with a real unit y and using the
+Pairing MR_P(x) with a real unit y and using the
two-coordinate Cauchy--Schwarz inequality on each edge
proves~(phase:matrix-remainder). For the difference, apply
separately
@@ -1637,10 +1637,10 @@ over y; the same constant controls both remainders.
[Two computable remainder constants]
Two sufficient choices in~(phase:weighted-constant) are
-b=sqrt 8,|M| and b=sqrt 6,|mathcal ScalS|, where the
-30times 216 real matrix mathcal ScalS has row
+b=sqrt 8,|M| and b=sqrt 6,|S|, where the
+30times 216 real matrix S has row
[
- mathcal S_calS_{a,:}
+ S_{a,:}
=M_{:,a}^* ⊗ ({bf e}_i-{bf e}_j)^*
]
when a is either of the two real-coordinate indices of edge i>j.
@@ -1654,23 +1654,23 @@ u_i=|x_i|^2. Expansion gives
sum_{i>j}a_{ij}(x)&=6s^2,nonumber
sum_{i>j}a_{ij}(x)^2
- &=6sum_i u_i^2+s^4+2sum_{i,j}(x_icdot x_j)^2 .
+ &=6sum_i u_i^2+s^4+2sum_{i,j}(x_i· x_j)^2 .
-The cross terms use sum_{jne i}x_icdot x_j=-u_i. Also
-u_ile 5(s^2-u_i) and sum_{i,j}(x_icdot x_j)^2le s^4. Thus
+The cross terms use sum_{jne i}x_i· x_j=-u_i. Also
+u_ile 5(s^2-u_i) and sum_{i,j}(x_i· x_j)^2le s^4. Thus
sum_i u_i^2le (5/6)s^4, and the second sum
in~(phase:row-distance-identities) is at most 8s^4.
Cauchy--Schwarz on the edges, with sum w_{ij}(y)^2=|M^*y|^2,
-proves the first choice of b. For the second, apply mathcal ScalS to
+proves the first choice of b. For the second, apply S to
y ⊗ x_{:,g} and sum squared norms over g: for unit y this
-gives sum w_{ij}(y)^2a_{ij}(x)le|mathcal ScalS|^2s^2. Another
+gives sum w_{ij}(y)^2a_{ij}(x)le|S|^2s^2. Another
Cauchy--Schwarz, now using sum a_{ij}=6s^2, proves the claim. The row
-Gram of mathcal ScalS is computable without forming its 216 columns:
+Gram of S is computable without forming its 216 columns:
- (mathcal ScalSmathcal S^*)_calS^*)_{ab}
- =(({bf e}_i-{bf e}_j)cdot({bf e}_{i'}-{bf e}_{j'}))
- (M_{:,a}cdot M_{:,b}),
+ (S S^*)_calS^*)_{ab}
+ =(({bf e}_i-{bf e}_j)·({bf e}_{i'}-{bf e}_{j'}))
+ (M_{:,a}· M_{:,b}),
with the edge labels repeated for real and imaginary parts. This is the
sign-factor formula in buildH::patch.
@@ -1694,7 +1694,7 @@ Proof.
Let delta be horizontal with F(P+delta)=0 and put eta=T^*delta.
Write E=Pi-TT^*-ML_PPi. Taylor's formula at the exact solution gives
[
- (I-TT^*)delta=Edelta-MF(P)-Mmathcal R_P(calR_P(delta).
+ (I-TT^*)delta=Edelta-MF(P)-MR_P(
]
The certificate and Lemma~phase:remainder
prove~(phase:root-normal). For the squared inequality (no
@@ -1708,7 +1708,7 @@ this holds for arbitrary delta. For horizontal v, put
h'=delta-v and subtract Taylor formulas at delta and v. Since
h' is horizontal and F(P+delta)=0,
[
- (I-TT^*)h'=Eh'-MF(P+v)-M(mathcal R_P(calR_P(delta)-mathcal R_P(v)).calR_P(v)).
+ (I-TT^*)h'=Eh'-MF(P+v)-M(R_P(-R_P(v)).
]
Apply~(phase:matrix-difference). This
proves~(phase:root-difference), including the residual term at the
@@ -1742,7 +1742,7 @@ causes failure.
Certified absolute row sums of the column Grams of H_V,L_V bound their
squared operator norms. The bound mnorm also includes the
difference between the accumulated stored M and the exact product
-H_VL_V^*. The required additions and the 2.1cdot 10^{-11}
+H_VL_V^*. The required additions and the 2.1· 10^{-11}
Frobenius bound on this difference are proved in the certified
arithmetic. Thus mnorm bounds the actual stored map, and the
first choice in Lemma~phase:remainder-constants
@@ -1752,8 +1752,8 @@ For the optional tensor improvement the row Gram
(phase:tensor-gram) is formed, and a real change of basis U is
proposed with a checked upper beta for |U^*U-I|. If
beta<.1 then |U^{-1}|^2le 1/(1-beta). A certified absolute row
-sum bound r for U^*(mathcal ScalSmathcal S^*)UcalS^*)U gives
-|mathcal ScalS|^2le r/(1-beta). This uses congruence by an
+sum bound r for U^*(S S^*)UcalS^*)U gives
+|S|^2le r/(1-beta). This uses congruence by an
invertible matrix, not assumed orthogonal diagonalization. The second
choice in Lemma~phase:remainder-constants, with the one-sided
margins certified alongside it, gives the alternative C. Both
@@ -1789,7 +1789,7 @@ P and its transported center V=Phi P,
Hence a Root transported by Phi remains a valid Root certificate with
the same constants; the source writes such a transport through
-Root::transport. This is the map PmapstoPhi P used below.
+Root::transport. This is the map P|->Phi P used below.
After transport, replace P by its dephasing D_0(P). Dephasing is a
row and column rephasing; by Lemma~seed:equivalences this preserves
@@ -1839,9 +1839,9 @@ 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-mathcal J(s,calJ(s,theta)|_Fle E_r as
-in . Let mathcal I_s=(7,8,9,10,11,13,19,25,31)calI_s=(7,8,9,10,11,13,19,25,31)
-denote the nine seed indices, and mathcal I_DcalI_D the sixteen lower-right
+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)
+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.
@@ -1901,15 +1901,15 @@ route, not an exclusion of the cell.
Let P_0 be the stored Candidate phase matrix. On a successful
preparation path there is a correction ε, in cycles, with
- |ε|le 2cdot 10^{-6},
+ |ε|le 2· 10^{-6},
such that, up to entrywise integer choices, the actual feasible phase
matrix has the form
P_0+ε+frac u{2pi}.
-Here the raw angular matrix u is s at mathcal I_s,calI_s, is d at
-mathcal I_D,calI_D, and is zero elsewhere, so in particular
+Here the raw angular matrix u is s at I_s, is d at
+I_D, and is zero elsewhere, so in particular
|u|lesqrt{R_s^2+arc^2}.
@@ -1917,7 +1917,7 @@ Let X:R^{13}toR^{36} give s at the seed positions and, at the
lower-right positions, give
(X(s,theta))_j
- =Im(D_j^circbarmathcal J(s,calJ(s,theta)_j).
+ =Im(D_j^circbarJ(s,theta)_j).
Then
@@ -1925,7 +1925,7 @@ Then
|e_D|le E_r,
|e_{sin}|_1lemathtt{arc}^3/6,
-where both errors are supported on mathcal I_D.calI_D.
+where both errors are supported on I_D.
Proof.
@@ -1934,13 +1934,13 @@ something other than the seed displacement. The exact flatness equation
|D'_{ij}|=1 forces D'_j=m_je(d_j) up to a scalar. From
Lemma~completion:linear-remainder,
[
- D'_j=D_j+mathcal J(s,calJ(s,theta)_j+R_j, |R|le E_r,
+ D'_j=D_j+J(s,theta)_j+R_j, |R|le E_r,
]
and Lemma~phase:arc-bounds bounds |d_j|lemathtt{arc}. Divide
D'_j by m_j: the real part is cos d_j, the imaginary part is
sin d_j. Thus
[
- Im(D_j^circbarmathcal J(s,calJ(s,theta)_j)
+ Im(D_j^circbarJ(s,theta)_j)
=m_j^{-1}Im(D_j^circbar(D'_j-D_j))
+O(E_r)= d_j+O(E_r)+O(arc^3).
]
@@ -1951,13 +1951,13 @@ further: the seed coordinates in u are exactly s, and the identity
|u|^2=|s|^2+|d|^2 gives~(phase:raw-displacement).
The stored center P_0 differs from an exact solution by
ε+u/(2pi) up to integer entries. Dephasing an exact phase matrix
-changes P_0 by less than 2cdot 10^{-6} cycles per entry, as proved
+changes P_0 by less than 2· 10^{-6} cycles per entry, as proved
in the certified arithmetic.
[A direct displacement bound]
-Either of the following is an upper bound for Deltacdot u:
+Either of the following is an upper bound for Delta· u:
S_1={}&R_s|(X^*Delta)_s|
+h|(X^*Delta)_theta|_1
@@ -1965,7 +1965,7 @@ Either of the following is an upper bound for Deltacdot u:
+|Delta_D|_∞arc^3/6,
S_2={}&R_s|Delta_s|
+weighted
- (sum_{jinmathcal I_D}calI_DDelta_j^2/m_j)^{1/2}.
+ (sum_{jinI_D}/m_j)^{1/2}.
With S_Delta=min(S_1,S_2), a valid upper bound is
@@ -1982,7 +1982,7 @@ Proof.
Equation~(phase:raw-support), the seed ball, and the angle box
give~(phase:support-linear). Direct weighted Cauchy--Schwarz
using~(phase:arc-weighted) gives~(phase:support-weighted).
-Since Delta is horizontal, DeltacdotPi u=Deltacdot u.
+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.
@@ -2016,7 +2016,7 @@ For each real test vector z, the support is at most
r|A^*z|+q|B^*z|, which by the inequality
(x+y)^2le(1+beta)x^2+(1+1/beta)y^2 is at
most (z^*Gz)^{1/2}. Diagonalize G: the vectors satisfying
-|zcdot x|le(z^*Gz)^{1/2} for every z are exactly G^{1/2} times
+|z· x|le(z^*Gz)^{1/2} for every z are exactly G^{1/2} times
the unit ball, including when G is singular.
@@ -2028,7 +2028,7 @@ the seed and e_D contributions to T^*u lies in an ellipsoid whenever
+(1+1/.45)E_r^2T_D^*T_D.
The remaining sine error has norm at most
-(max_{jinmathcal I_D}calI_D|T_{j,:}|)arc^3/6.
+(max_{jinI_D}|T_{j,:}|)arc^3/6.
There is also a bound using the raw seed and weighted angle balls with a
single offset U_0=2pi T^*Delta:
@@ -2065,19 +2065,19 @@ gamma U^*A^{-1}U. The triangle bound is immediate; convexity of norm
gives the last claim.
-Let mathcal E(G;calE(G;{U_sigma}) denote any certified upper bound from
+Let E(G;{U_sigma}) denote any certified upper bound from
Lemma~phase:ellipsoid-norm. Two valid tangent bounds are
|T^*delta|
le{}&
- frac{mathcal E(G_calE(G_{rm lin};{U_sigma})+
- max_{jinmathcal I_D}calI_D|T_{j,:}|,arc^3/6+
+ frac{E(G_{rm 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{mathcal E(G_calE(G_{rm dir};{U_0})+
+ frac{E(G_{rm dir};{U_0})+
horsqrt{R_s^2+arc^2}}{2pi}
+sqrt{1+ort},|ε|.
@@ -2088,9 +2088,9 @@ 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|<1.5cdot 10^{-5},
+|widehatwidehat A-T^*XA-T^*X|<1.5· 10^{-5},
|(s,theta)|<.226, so this substitution costs less than
-3.4cdot 10^{-6} radians. For both covariance choices a diagonal
+3.4· 10^{-6} radians. For both covariance choices a diagonal
10^{-7}I is added. The inverses in the second choice are certified
upper invmag values. proves PSD
domination of the exact covariance expressions in the stored-coefficient
@@ -2142,9 +2142,9 @@ 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
-mathcal S(P,r_P)calS(P,r_P)cupmathcal S(Q,r_Q)calS(Q,r_Q) is contained in
+S(P,r_P)calS(P,r_P)cupS(Q,r_Q)calS(Q,r_Q) is contained in
[
- mathcal S(P,calS(P, max{r_P, d_{PQ}+r_Q+allowance}).
+ S(P, max{r_P, d_{PQ}+r_Q+allowance}).
]
@@ -2153,7 +2153,7 @@ 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
-mathcal ScalS in Definition~refine:solution-set is closed under this
+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.
@@ -2182,7 +2182,7 @@ projects it, and evaluates its norm. On success it adds 10^{-7} to the
computed norm, exceeding the dephasing and projection errors certified
by the arithmetic bounds above. Atlas::add grows the selected
radius to at least the incoming radius plus this distance, with another
-10^{-8} added and a separate 2cdot 10^{-8}-margin comparison to the
+10^{-8} added and a separate 2· 10^{-8}-margin comparison to the
cap. It remains valid when an alternative map found by variants
supplied the distance; only existence of that allowed map is needed.
@@ -2220,7 +2220,7 @@ bin the four ordered tail bins number at most
binom{115}{4}=6{,}913{,}340. The former bound is below the signed
32-bit limit. Pair traversals use range loops and 64-bit counters. All
bin differences lie between -111 and 111; their nine-coordinate
-square sum is at most 9cdot 111^2, so the integer distance and bucket
+square sum is at most 9· 111^2, so the integer distance and bucket
arithmetic fit the stated types. Stored five-coordinate differences fit
the signed eight-bit type used by the cover.
@@ -2244,13 +2244,13 @@ and no regularity assumption on the Hadamard locus is made. The tangent
coordinates used below organize an exhaustive search; the normal
estimates from control the remaining
directions. A phase matrix is identified with a vector in R^{36};
-stars mean real transpose; mathcal V=imcalV=imPi is the
+stars mean real transpose; V=im is the
horizontal subspace.
For a certified center P and radius Rge 0 put
- mathcal S(P,R)=calS(P,R)=
- bigl{[e(P+delta)]:deltainmathcal V,calV, |delta|le R,
+ S(P,R)=calS(P,R)=
+ bigl{[e(P+delta)]:deltainV, |delta|le R,
F(P+delta)=0bigr},
where brackets denote the equivalences of
@@ -2311,7 +2311,7 @@ are row and column rephasings. Consequently P+d_q and Q differ only
by integer phases and rephasings.
Fix a parent radius R and an exact solution displacement
-deltainmathcal VcalV with |delta|le R. Set
+deltainV with |delta|le R. Set
eta=T^*delta,
h'=delta-d_q, d=|h'|
@@ -2457,7 +2457,7 @@ alignment~(refine:alignment).
|eta-eta_q|le t and widehatwidehat D(t)D(t)le a
⇒
- [e(P+delta)]inmathcal S(Q,a)calS(Q,a)
+ [e(P+delta)]inS(Q,a)calS(Q,a)
quadfor every parent solution delta.
@@ -2465,7 +2465,7 @@ 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
make 25 midpoint trials, each with m=(ell+u)/2: if
-widehatwidehat D(m)<a-2D(m)<a-2cdot 10^{-7}, replace ell by m; otherwise
+widehatwidehat D(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}),
@@ -2477,7 +2477,7 @@ Every usable returned threshold certifies
[
|eta-eta_q|lerho
⇒
- [e(P+delta)]inmathcal S(Q,a)calS(Q,a)
+ [e(P+delta)]inS(Q,a)calS(Q,a)
]
for every parent solution delta. The search need not find the largest
possible threshold; in particular its correctness does not require
@@ -2487,11 +2487,11 @@ monotonicity of the numerical function widehatwidehat D.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-2cdot 10^{-7}, and
+trial widehatwidehat D(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
an upper bound for |h'| at this solution, so
-|h'|<a-2cdot 10^{-7}<a. Apply~(refine:eval-semantics). No
+|h'|<a-2· 10^{-7}<a. Apply~(refine:eval-semantics). No
comparison between widehatwidehat DD at distinct arguments is needed.
@@ -2545,16 +2545,16 @@ become a blocker.
Suppose Stage B runs on a finite list of records with valid Root
certificates, under the arithmetic assumptions and error bounds
-described above. On a completed run, let mathcal AcalA be the
-final atlas of children and mathcal BcalB the list of input parents
+described above. On a completed run, let A be the
+final atlas of children and B the list of input parents
whose refinement failed. Then
- ∪_{(P,R),input}mathcal S(P,R)calS(P,R)
+ ∪_{(P,R),input}S(P,R)calS(P,R)
&⊆
- ∪_{(Q,r)inmathcal A}calAmathcal S(Q,r)calS(Q,r)
- &qquadcup
- ∪_{(P,R)inmathcal B}calBmathcal S(P,R).calS(P,R).
+ ∪_{(Q,r)inA} S(Q,r)calS(Q,r)
+ & cup
+ ∪_{(P,R)inB} S(P,R).calS(P,R).
In particular, a completed run with no bad parents replaces the input
@@ -2585,7 +2585,7 @@ only after every base box has either been rejected as outside, marked
by a valid whole-box relation, or solved as above. Hence success for
the current parent proves coverage of its entire sphere, permitting its
addition to the blocker list. If the parent fails, it is not added and
-is retained in mathcal B.calB. Any children produced while attempting it
+is retained in B. Any children produced while attempting it
remain valid stored spheres and are reused without assuming the failed
parent has been covered.
@@ -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,
- mathcal S(Q_i,r_i)⊆calS(Q_i,r_i)⊆mathcal S(Q_i,calS(Q_i,tau),
+ S(Q_i,r_i)⊆calS(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
@@ -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
- |ycdot(c+Jx-Beta)|lemathcal E(y;X,I),calE(y;X,I),
+ |y·(c+Jx-Beta)|leE(y;X,I),calE(y;X,I),
where
- mathcal E(y;X,I):=calE(y;X,I):={}&nN_y+piBigl[
+ E(y;X,I):=calE(y;X,I):={}&nN_y+piBigl[
|y|_∞(KX+r)^2
+min{G_yX^2, K|Scirc y|IX}notag
&hspace{28mm}
@@ -2765,9 +2765,9 @@ where
For a bin, this gives the necessary support inequality
- |ycdot c|
+ |y· c|
le asum_{j>0}|(J^*y)_j|+r_H|B^*y|
- +mathcal E(y;X_M,I_M).calE(y;X_M,I_M).
+ +E(y;X_M,I_M).calE(y;X_M,I_M).
@@ -2839,7 +2839,7 @@ are four bin layers.
For vinR^6 and sge 0, define the closed vector ball
- mathcal B_calB_{vec}(v,s)=
+ B_{vec}(v,s)=
Bigl{(lambda e(v+x))/(sqrt6):
|lambda|=1, xinR^6, sum_jx_j=0, |x|le sBigr}.
@@ -2847,12 +2847,12 @@ For vinR^6 and sge 0, define the closed vector ball
[Certified clustering]
Suppose each retained bin midpoint u is assigned to a center v and
-an integer vector minmathbb Z^6.bbZ^6. Let
+an integer vector minZ^6. Let
[
h=u-v-m-frac16(sum_j(u_j-v_j-m_j)){bf 1}.
]
If the assigned radius is at least |h|+X_M, then
-mathcal B_calB_{vec}(v,s) covers the whole bin up to global
+B_{vec}(v,s) covers the whole bin up to global
phase. The final assignment in cluster has this property.
@@ -2895,7 +2895,7 @@ assume that the selected wrapped lift minimizes torus distance.
[Distinct labels]
-Two vectors in the same ball mathcal B_calB_{vec}(v,s) with
+Two vectors in the same ball B_{vec}(v,s) with
s<.1 are neither orthogonal nor mutually unbiased. Hence the twelve
vectors in two further mutually unbiased orthonormal bases must occupy
twelve distinct retained balls.
@@ -2949,43 +2949,43 @@ actual representatives with displacements x_u,x_v, set
The actual unscaled inner product is, up to conjugation and global
phase, S'=sum_jQ_je(z_j). Since |S'-S|le 2pisqrt6,r_e,
-necessary direct tests for orthogonality (mathcal O)calO) and
-unbiasedness (mathcal U)calU) are
+necessary direct tests for orthogonality (O) and
+unbiasedness (U) are
- (|S|)/(2pisqrt6)le r_e (mathcal O),calO),
- (||S|/sqrt6-1|)/(2pi)le r_e (mathcal U).calU).
+ (|S|)/(2pisqrt6)le r_e (O),
+ (||S|/sqrt6-1|)/(2pi)le r_e (U).
- (D_{mathcal O}calO)_{:,j}=
+ (D_{O}{:,j}=
-ImQ_j ReQ_j,
- (D_{mathcal U}calU)_j=
- frac{ReS,(D_{mathcal O}calO)_{1j}
- +ImS,(D_{mathcal O}calO)_{2j}}6.
+ (D_{U}
+ frac{ReS,(D_{O}{1j}
+ +ImS,(D_{O}{2j}}6.
[Edge Taylor estimates]
For any actual pair from two distinct closed vector balls, with the
data of~(exclude:edge-data), and for
-S_{mathcal O}calO=ReS,
-S_{mathcal U}calU=(|S|^2-6)/(4pi),
+S_{O},
+S_{U}|S|^2-6)/(4pi),
- |Re(S'-S)-(D_{mathcal O}calOz)_1|
- &le E_{mathcal O}calO:=4pisqrt6,r_e^2,
- |(|S'|^2-6)/(4pi)-S_{mathcal U}calU-D_{mathcal U}calUz|
- &le E_{mathcal U}calU:=
+ |Re(S'-S)-(D_{O}|
+ &le E_{O}:=4pisqrt6,r_e^2,
+ |(|S'|^2-6)/(4pi)-S_{U}-D_{U}|
+ &le E_{U}:=
(2pimax(|S|,sqrt6))/(sqrt6),r_e^2 +4pi r_e^2.
Therefore the necessary support conditions
- |S_{mathcal O}calO|
- &le r_e|D_{mathcal O}calO|_F+E_{mathcal O}calO,
+ |S_{O}|
+ &le r_e|D_{O}|_F+E_{O},
- |S_{mathcal U}calU|
- &le r_e|D_{mathcal U}calU|+E_{mathcal U}calU
+ |S_{U}|
+ &le r_e|D_{U}|+E_{U}
hold for every actual pair.
@@ -2995,13 +2995,13 @@ Proof.
Write e(z_j)-1=2pi iz_j+ε_j with
|ε_j|le2pi^2z_j^2 by the integral Taylor remainder.
Multiply by Q_j=a_j+ib_j and separate real and imaginary parts:
-Re(Q_j(e(z_j)-1))approx-2pi b_j z_j=(D_{mathcal O}calOz)_j
+Re(Q_j(e(z_j)-1))approx-2pi b_j z_j=(D_{O}
and Im(Q_j(e(z_j)-1))approx2pi a_jz_j up to a
residual of modulus at most 4pi^2 r_e^2 per coordinate. Summing over
j and using |z|le r_e proves~(exclude:orth-residual).
For unbiasedness, expand (|S'|^2-6)/(4pi) at S. The constant is
-S_{mathcal U}calU, the linear term is D_{mathcal U}calUz (the division by
+S_{U}, the linear term is D_{U} (the division by
4pi normalizes the derivative exactly), and the quadratic remainder
is at most (2pi|S|/sqrt6)r_e^2+4pi^2r_e^2; substituting the
residual of the orthogonality calculation
@@ -3017,7 +3017,7 @@ medskipnoindentThe joint test with one shared tangent variable.
For each ball t=0,1, form c^t,J^t,B^t,S^t from
(exclude:single-sums) at its actual center. Write its radius as s_t and put
- mathcal E_t^calE_t^{ball}(y):=
+ E_t^{ball}(y):=
n(sum_{i,j}y_i^2(J^t_{ij})^2)^{1/2}
+piBigl[
|y|_∞(Ks_t+r)^2
@@ -3031,13 +3031,13 @@ second one here), with X=s_t and I=s_t; they are valid on the whole
closed Euclidean vector ball because the bound was derived for the
worst case over that ball.
-For an edge type Ein{mathcal O,calO,mathcal UcalU} write ell=2 or
+For an edge type Ein{O,U} write ell=2 or
1 for the dimension of D_E, and let q_EinR^ell be the
constant part of the linearized edge equation:
- q_{mathcal O}calO := (ReS, ImS) inR^2,
+ q_{O} := (ReS, ImS) inR^2,
- q_{mathcal U}calU := (|S|^2-6)/(4pi) inR^1,
+ q_{U} := (|S|^2-6)/(4pi) inR^1,
with D_E and E_E as in~(exclude:edge-derivatives)
and~(exclude:orth-residual)--(exclude:unbiased-residual).
@@ -3047,14 +3047,14 @@ and~(exclude:orth-residual)--(exclude:unbiased-residual).
Suppose one vector from each of the two balls is unbiased to the
coordinate basis and to the same second basis, whose phase matrix
is e(P+delta)/sqrt6, and the two vectors have relation
-Ein{mathcal O,calO,mathcal UcalU}. For every y_0,y_1inR^6 and
+Ein{O,U}. For every y_0,y_1inR^6 and
y_einR^ell,
- &|y_0cdot c^0+y_1cdot c^1+y_ecdot q_E|notag
+ &|y_0· c^0+y_1· c^1+y_e· q_E|notag
&quadle
sum_{t=0}^1Bigl[
s_tbigl|(J^t)^*y_t+(-1)^tD_E^*y_ebigr|
- +mathcal E_t^calE_t^{ball}(y_t)Bigr]notag
+ +E_t^{ball}(y_t)Bigr]notag
&+r_Hbigl|(B^0)^*y_0+(B^1)^*y_1bigr|
+E_E|y_e|.
@@ -3064,7 +3064,7 @@ y_einR^ell,
Proof.
For each single vector the linearized equation is
c^t+J^tx_t-B^teta with weighted residual bounded by
-mathcal E_t^calE_t^{ball}(y_t). The edge equation is
+E_t^{ball}(y_t). The edge equation is
q_E+D_E(x_0-x_1) with residual norm at most E_E.
Divide the edge equation by y_e, expand each x_t-dependent part
@@ -3106,8 +3106,8 @@ medskipnoindentExact graph exclusion.
% ---------------------------------------------------------------
Build one vertex for each retained ball. Join two distinct vertices by
-an mathcal OcalO edge iff their orthogonality type has not been excluded
-by the direct and joint tests above; similarly for mathcal UcalU edges
+an O edge iff their orthogonality type has not been excluded
+by the direct and joint tests above; similarly for U edges
and unbiasedness. Both graphs are undirected and have no self-loops.
Every genuine pair retains its corresponding edge by the preceding
lemmas. Extra edges are harmless: retained pairwise possibilities need
@@ -3117,8 +3117,8 @@ 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 mathcal O-cliques,calO-cliques,
- Atimes B⊆ E(mathcal U).calU).
+ |A|=|B|=6, A,B are O-cliques,calO-cliques,
+ Atimes B⊆ E(U).
The following finite search tests this necessary configuration.
@@ -3135,7 +3135,7 @@ Proof.
First consider clique on a candidate set P and target size
m. For m=0 it succeeds, and for m=1 it tests nonemptiness. For
mge2 it repeatedly selects the least remaining vertex i, removes
-it from P, and searches Pcap N_{mathcal O}calO(i) for an
+it from P, and searches Pcap N_{O} for an
(m-1)-clique. For m=2 that search is just nonemptiness. Every
clique has a unique least vertex, so this enumerates every possibility.
Terminating when fewer than m candidates remain is also exact.
@@ -3152,7 +3152,7 @@ clique, and it is performed only when the remaining first-clique target
is six. Recursion then runs clique on the first-pool target
six, and for each resulting candidate A, on the second-pool target
six. When clique succeeds, its exact enumeration is used to
-test Atimes A'⊆ E(mathcal U)calU) by checking every cross pair
+test Atimes A'⊆ E(U) by checking every cross pair
in order.
The search cap and any other finite work bound turn into failure, not
@@ -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 mathcal O-calO- and mathcal U-graphscalU-graphs on the retained balls,
+ the O-calO- 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
@@ -3185,10 +3185,10 @@ Suppose two such bases existed. Each of their twelve vectors is
covered by some retained ball (Lemma~exclude:cluster-cover).
Lemma~exclude:distinct-labels puts them in twelve distinct balls.
Within each basis the six vectors are pairwise orthogonal, and every
-intra-pair survived its mathcal OcalO direct and joint tests because the
-tests are necessary conditions; thus they are joined by mathcal OcalO
+intra-pair survived its O direct and joint tests because the
+tests are necessary conditions; thus they are joined by O
edges. Across the two bases every pair is unbiased, and similarly
-they survive the mathcal UcalU tests, so the cross-pairs are mathcal
+they survive the U tests, so the cross-pairs are mathcal
UcalU edges. This produces a configuration~(exclude:graph-configuration),
contradicting the third hypothesis.
@@ -3210,20 +3210,20 @@ indicated multiplication. The stored tangent and normal bounds are
widehatwidehat n=n_0+n=n_0+iota r+Gamma r^2+10^{-9}.
The upper squared-magnitude threshold in~(exclude:direct-tests) is
-increased by 2cdot10^{-7}, and its lower threshold, when tau<1, is
+increased by 2·10^{-7}, and its lower threshold, when tau<1, is
decreased by that amount. The combined squared-distance threshold, the
right side of~(exclude:scalar-linear), and the right side
-of~(exclude:bin-support) each receive a 2cdot 10^{-7}
+of~(exclude:bin-support) each receive a 2· 10^{-7}
comparison allowance.
For a pair, the radius sum s_u+s_v is increased by 10^{-10}. The
distance tests~(exclude:edge-distance) permit a further
-2cdot 10^{-7} on that radius. Both edge residual bounds
-E_{mathcal O}calO and E_{mathcal U}calU are increased by 2cdot 10^{-8};
+2· 10^{-7} on that radius. Both edge residual bounds
+E_{O} and E_{U} are increased by 2· 10^{-8};
the direct comparisons~(exclude:orth-support)
-and~(exclude:unbiased-support) receive another 2cdot 10^{-7}.
+and~(exclude:unbiased-support) receive another 2· 10^{-7}.
The right side of the joint
-comparison~(exclude:joint-inequality) receives 3cdot 10^{-7}.
+comparison~(exclude:joint-inequality) receives 3· 10^{-7}.
Cluster assignment radii include the 10^{-9} allowance described in
Lemma~exclude:cluster-cover.
@@ -3572,12 +3572,12 @@ permissible wherever the proof subsequently uses that integer lift. No
accuracy premise is imposed on atan2: Candidate checks the unit
exponential of the phase it proposes. Decimal literals are either used
as exact stored parameters, or allowed a relative translation error of
-at most 5cdot 10^{-16} when compared with a displayed decimal.
+at most 5· 10^{-16} when compared with a displayed decimal.
Put
- u=2cdot 10^{-16}, tau=10^{-300},
- gamma_m=2.1cdot 10^{-16}(m+1).
+ u=2· 10^{-16}, tau=10^{-300},
+ gamma_m=2.1· 10^{-16}(m+1).
For a correctly rounded basic operation with finite exact result z at
the scales below,
@@ -3604,7 +3604,7 @@ instead of two increases that constant by one more. For the
square-sum/square-root norms used here,
|fl(|a|)-|a||
- le 2.2cdot 10^{-16}(m+2)|a|+10^{-140}.
+ le 2.2· 10^{-16}(m+2)|a|+10^{-140}.
If the computed components approximate other components, their
Euclidean difference is added to this bound.
@@ -3615,8 +3615,8 @@ Iterating |z_k|le (1+u)|z_{k-1}|+u+btau for a length-m dot product
with the standard addition/rounding analysis
(Lemma 3.1 of Higham's 2002 book) gives an interpolation of
the form (1+u)^msum_{i=1}^m|a_ib_i| plus a O(mtau) term. The bound
-gamma_m=2.1cdot 10^{-16}(m+1) is a conservative choice: for
-mle 2000 one has (1+u)^m-1le mule 2mcdot 10^{-16}, which is
+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
@@ -3632,14 +3632,14 @@ medskipnoindentPi and exponential evaluations.
The constant pi is stored as the correctly rounded binary64 literal.
Its absolute error is at most 10^{-16} and its relative error is at
most u. The constants sqrt 6 and its reciprocal are formed by the
-indicated square-root and division; 2cdot 10^{-15} is an ample
+indicated square-root and division; 2· 10^{-15} is an ample
absolute error bound for each.
The source evaluates cosine through degree 18 and sine through degree
19. In the jth nonconstant term, the error in -y^2 is reused j
times, and there are j further multiplications and j divisions.
The small integer denominators are exact. Its relative error is
-therefore at most (1+u)^{3j}-1<5.5cdot 10^{-15} for jle 9. For
+therefore at most (1+u)^{3j}-1<5.5· 10^{-15} for jle 9. For
additive underflow errors, compare with a recurrence using the same
multiplicative rounding factors but zero additive errors. Its terms
have magnitude below two, and each recurrence contracts the previous
@@ -3655,8 +3655,8 @@ below .4; the odd-term sum is below 1, its part after y below
first is at most .786^2/12, and for the odd terms after y it is at
most .786^2/20. Their geometric majorants give the claimed absolute
sums without evaluating a transcendental function. The first omitted
-cosine and sine terms are below 3.33cdot 10^{-21} and
-1.25cdot 10^{-22}, respectively; both are below 10^{-16} even in
+cosine and sine terms are below 3.33· 10^{-21} and
+1.25· 10^{-22}, respectively; both are below 10^{-16} even in
the worst scale considered here. Thus
|fl(e^{iy}) - e^{iy}| le 10^{-14}
@@ -3681,11 +3681,11 @@ medskipnoindentDephasing and projection.
% ---------------------------------------------------------------
For |p_j|le 4, the three operations in unwrapped dephasing
-p_{ij}-p_{i0}-p_{0j}+p_{00} have entry error below 8cdot 10^{-15};
+p_{ij}-p_{i0}-p_{0j}+p_{00} have entry error below 8· 10^{-15};
their intermediate magnitudes are at most 8, 12, and 16.
Subtracting the integer selected by nearbyint gives error
below 10^{-14} against the exact dephase minus that same integer. A
-wrapped difference of two such arrays has error below 3cdot 10^{-14}
+wrapped difference of two such arrays has error below 3· 10^{-14}
per entry against the ideal difference with the combined selected
integers. The one-sided Candidate version, whose other array is already
a stored dephased phase, has the same bound. These comparisons do not
@@ -3693,8 +3693,8 @@ assume identical rounding choices for an independently formed ideal
phase.
For a stored wrapped array, |z_j|le 1/2. Its row and column sums
-have error below 4cdot 10^{-15}, and its length-36 grand sum below
-1.4cdot 10^{-13}. Dividing by 6 or 36 and combining the means
+have error below 4· 10^{-15}, and its length-36 grand sum below
+1.4· 10^{-13}. Dividing by 6 or 36 and combining the means
gives projection-operation error below 10^{-14} per component.
Projection contracts the prior input error. Consequently the computed
projected difference has Frobenius error below 10^{-11}. All distance
@@ -3703,7 +3703,7 @@ against 10^{-7}.
Thus Atlas::find's returned √(s)+10^{-7} is an upper
distance for the selected ideal projected lift. A merge uses radius
-r+distance+10^{-8} and checks a cap with 2cdot 10^{-8}
+r+distance+10^{-8} and checks a cap with 2· 10^{-8}
slack. Its sum-rounding error is below 10^{-14}. The same arithmetic
applies to assign and to the local sphere-radius additions.
Searching only some buckets or transforms may miss a merge; it cannot
@@ -3721,11 +3721,11 @@ including the comparison threshold, by 10^{-14}. Their slack is
10^{-8}. The orientation filter is integral.
For a list of five varying phases, the computed residual c in
-sumok has component error below 2cdot 10^{-13}. Indeed its
+sumok has component error below 2· 10^{-13}. Indeed its
five exponential errors, after division by 2pi, total less than
-4.8cdot 10^{-14} in complex modulus, and the divisions, additions,
-and constant error cost less than 3cdot 10^{-15}. The sharper
-componentwise bound 1.1cdot 10^{-13} will also be useful. Each test
+4.8· 10^{-14} in complex modulus, and the divisions, additions,
+and constant error cost less than 3· 10^{-15}. The sharper
+componentwise bound 1.1· 10^{-13} will also be useful. Each test
direction is regarded as an exact stored pair; its norm is less than
1.1. The five column perturbations, the residual perturbation, and
ordinary arithmetic in chk cost together less than 10^{-11}.
@@ -3736,19 +3736,19 @@ feasible list.
In supdual, all proposed entries are checked by
|v_a|le 10^4, with a negated comparison that also rejects NaN. They
are then exact stored test coefficients. The four right-hand-side
-errors against -nc are below 3cdot 10^{-11} each. The signed term
+errors against -nc are below 3· 10^{-11} each. The signed term
consequently has input error at most
[
- 2sum_{a=1}^4 |v_a|,3cdot 10^{-11}le 2.4cdot 10^{-6}<2.5cdot 10^{-6}.
+ 2sum_{a=1}^4 |v_a|,3· 10^{-11}le 2.4· 10^{-6}<2.5· 10^{-6}.
]
Its absolute term sum is below 10^7, so its arithmetic error is below
10^{-7}. Each of the nine row-dot residuals has error below
-4cdot 10^{-9}: its four input contributions are at most
-4cdot 10^4ε_{exp}, and its short dot costs less than
+4· 10^{-9}: its four input contributions are at most
+4· 10^4ε_{exp}, and its short dot costs less than
10^{-10}. Each pair norm has error below 10^{-10} by
(round:norm). The remainder coefficient uses 1.251 instead of
1.25, which leaves positive slack after its own rounding. The final
-absolute scale is below 2cdot 10^7. Summing these contributions shows
+absolute scale is below 2· 10^7. Summing these contributions shows
that the ideal support expression is at most the computed proposal plus
10^{-4}. The actual addition is 0.01. If no usable proposal improves
the initial 10^{10} bound, that initial bound already exceeds the
@@ -3758,12 +3758,12 @@ Integer differences lie in [-111,111], including those stored in
int8_t; all their squared/absolute sums are exact and small.
The comparison radius is
[
- RN2=fl((fl(.032cdot112))^2)
+ RN2=fl((fl(.032·112))^2)
-.0001
]
with the source's indicated operation ordering. The total possible
error before the subtraction is below 10^{-12}, so this is smaller
-than the ideal (.032cdot112)^2. This reasoning requires the
+than the ideal (.032·112)^2. This reasoning requires the
pow(x,2) calls in covout to be lowered to binary64
multiplication, as checked for the fresh build documented in
. It is not a claim about an arbitrary library
@@ -3786,37 +3786,37 @@ code. We never need the selected coordinate to maximize the ideal
projection score.
For frame, the four-term sum S has error below
-2.5cdot 10^{-13}; its squared modulus has error below
-2.2cdot 10^{-12}, and its modulus below 2.6cdot 10^{-13} by
+2.5· 10^{-13}; its squared modulus has error below
+2.2· 10^{-12}, and its modulus below 2.6· 10^{-13} by
(round:norm). If the stored m=√(4-|S|) passes m>1.25, the
ideal m>1.24. Using |sqrt a-sqrt b|=|a-b|/(sqrt a+sqrt b) at
-these lower bounds gives errors below 3cdot 10^{-13} in m and
+these lower bounds gives errors below 3· 10^{-13} in m and
top=√(4+|S|). In particular 16-|S|^2>6.1; its error is
-below 2.3cdot 10^{-12}, so its reciprocal has error below
-7cdot 10^{-14}. The two pseudoinverse numerators have modulus at most
-eight and error below 5.1cdot 10^{-13}. Division/scaling then gives
+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,
the second inequality following from the lower singular bound.
The first constant basis column is exact. The next column has numerator
-error below 1.3cdot 10^{-13} and denominator
-√(4-|S|^2/4)>1.5 with error below 2cdot 10^{-13}. Its ideal
+error below 1.3· 10^{-13} and denominator
+√(4-|S|^2/4)>1.5 with error below 2· 10^{-13}. Its ideal
entries have modulus at most one, so the quotient error is below
-4cdot 10^{-13}. For each of the remaining two columns, use the actual
+4· 10^{-13}. For each of the remaining two columns, use the actual
selected coordinate projection q. In the first such projection the
-earlier two products give entry error below 9cdot 10^{-13}, and its
-norm error is below 2cdot 10^{-12}. The accepted computed norm is
+earlier two products give entry error below 9· 10^{-13}, and its
+norm error is below 2· 10^{-12}. The accepted computed norm is
greater than .449999, so the ideal projection is nonzero and
-normalization gives entry error below 7cdot 10^{-12}. In the last
+normalization gives entry error below 7· 10^{-12}. In the last
projection the extra column-product error gives entry error below
-1.6cdot 10^{-11} and norm error below 3.5cdot 10^{-11}. The same
-denominator check gives entry error below 1.2cdot 10^{-10}. This
+1.6· 10^{-11} and norm error below 3.5· 10^{-11}. The same
+denominator check gives entry error below 1.2· 10^{-10}. This
argument inductively constructs the ideal orthonormal complements with
the selected coordinates. The two-product kernel projector has entry
-error below 3cdot 10^{-10}; multiplying two complement entries and
-sqrt6 gives a W entry error below 8cdot 10^{-10}.
+error below 3· 10^{-10}; multiplying two complement entries and
+sqrt6 gives a W entry error below 8· 10^{-10}.
The following table records the rest of the propagation, with the source
array names. Its errors include ordinary operation rounding.
@@ -3828,17 +3828,17 @@ small
Quantity & Absolute error per entry & Ideal scale used
base & 10^{-11} & two factors of modulus at most 2 times G
-fixed & 2cdot 10^{-9} & projector times a factor at most 1.62
-bm, cm & 2cdot 10^{-12} & unit entry times G
-uc.U, angular derivatives & 2cdot 10^{-13} & products of unit phases/trig entries
-K, D & 4cdot 10^{-9} & four W products and base
-mag & 4cdot 10^{-8} & |D|<4.6
-sqrt{mag} & 5cdot 10^{-9} & norm Lipschitz
+fixed & 2· 10^{-9} & projector times a factor at most 1.62
+bm, cm & 2· 10^{-12} & unit entry times G
+uc.U, angular derivatives & 2· 10^{-13} & products of unit phases/trig entries
+K, D & 4· 10^{-9} & four W products and base
+mag & 4· 10^{-8} & |D|<4.6
+sqrt{mag} & 5· 10^{-9} & norm Lipschitz
jac & 10^{-8} & ideal modulus at most 6
-flat real J & 1cdot 10^{-7} & dot of D and jac
-flat c & 5cdot 10^{-8} & (|D|^2-1)/2
-pre-equation c & 4cdot 10^{-13} & short unit-exponential sums
-pre-equation coefficients & 6cdot 10^{-14} & unit exponentials
+flat real J & 1· 10^{-7} & dot of D and jac
+flat c & 5· 10^{-8} & (|D|^2-1)/2
+pre-equation c & 4· 10^{-13} & short unit-exponential sums
+pre-equation coefficients & 6· 10^{-14} & unit exponentials
@@ -3847,17 +3847,17 @@ medskipnoindentThe horizontal-projector identity..
An entry of the exact double-centering projector is the product of two
centering-matrix entries. The computed PI differs by less than
10^{-15} per entry. In the matrix L_P, each row has only 12 nonzero
-entries, with error below 1.3cdot 10^{-13} from the edge products.
-Forming LP gives error below 5cdot 10^{-12} per entry
+entries, with error below 1.3· 10^{-13} from the edge products.
+Forming LP gives error below 5· 10^{-12} per entry
against L_PPi; the ideal entry magnitude is at most 8.4. In the
residual Pi-TT^*-ML_PPi each computed entry then has error below
10^{-9}. The propagated LP error is at most
-30cdot 2cdot 5cdot 10^{-12}=3cdot 10^{-10}; the remaining PI,
+30· 2· 5· 10^{-12}=3· 10^{-10}; the remaining PI,
T-product, and short accumulation errors fit in the remainder of the
budget, with absolute sum below 560. The 1296 entry errors have
-Frobenius total below 3.6cdot 10^{-8}. Norm accumulation is below
+Frobenius total below 3.6· 10^{-8}. Norm accumulation is below
10^{-8} even with residual norm 20161. Thus the actual
-2cdot 10^{-6} addition in id suffices.
+2· 10^{-6} addition in id suffices.
% ---------------------------------------------------------------
@@ -3878,18 +3878,18 @@ For the auxiliary maps M: the source forms M=H_VL_V^* with at most
25 columns of H_V and L_V, each checked to magnitude at most 2.
Absolute row sums of the column Grams of H_V and L_V give upper
bounds on |H_V| and |L_V|; their product overestimates
-|M|. Adding the 2.1cdot 10^{-11} Frobenius bound on the
+|M|. Adding the 2.1· 10^{-11} Frobenius bound on the
difference H_VL_V^*-M (where M denotes the accumulated stored
product) gives a valid mnorm.
The two candidate choices for C are sqrt 8|M| and
-sqrt 6|mathcal ScalS|; the source takes the smaller of upper bounds
+sqrt 6|S|; the source takes the smaller of upper bounds
for these two quantities. Each is a valid pi b in
Lemma~phase:remainder by Lemma~phase:remainder-constants.
The root evaluation thresholds are val<10^{-18} for the
overall pass, ort<10^{-5}, hor<10^{-5},
-id<10^{-4} after the 2cdot 10^{-6} addition, and
-mnormcdotmathtt{froot}lemathtt{n0}.
+id<10^{-4} after the 2· 10^{-6} addition, and
+mnorm·mathtt{froot}lemathtt{n0}.
% ---------------------------------------------------------------
medskipnoindentCandidate conversion and ellipsoidal coverage.
@@ -3904,16 +3904,16 @@ magnitude test sqrt{mag_j}>.15 implies m_j>.14999998.
Normalizing D by its computed magnitude has unit-direction error
below
[
- frac{2cdot 4cdot 10^{-9}}{.14999998}
- +norm/division rounding<5.5cdot 10^{-8}.
+ frac{2· 4· 10^{-9}}{.14999998}
+ +norm/division rounding<5.5· 10^{-8}.
]
Candidate additionally checks a computed chord smaller than 10^{-7}
between this normalized value and the exponential of its proposed
phase. With (round:exp-budget), the true unit chord error is less
-than 1.56cdot 10^{-7}. If a circular phase distance in cycles is
+than 1.56· 10^{-7}. If a circular phase distance in cycles is
vin[0,1/2], its chord is 2sin(pi v)ge 4v. Across 16 entries
the phase-center correction therefore has norm below
-1.56cdot 10^{-7}, which is below the 2cdot 10^{-6} allowance used
+1.56· 10^{-7}, which is below the 2· 10^{-6} allowance used
in . The nine seed entries use exact stored
seed centers and an exact integer subtraction.
@@ -3946,22 +3946,22 @@ medskipnoindentTangent offsets and external errors..
The computed off approximates 2pi T^*Delta within 10^{-9}
in norm, by~(round:t-operator), the projected-difference error,
and short-dot rounding. The error in the whole X array is at most
-√(208)cdot 10^{-6} in Frobenius norm; there are 13 columns and
+√(208)· 10^{-6} in Frobenius norm; there are 13 columns and
16 affected positions. Multiplication by T^* and the less than
-2cdot 10^{-13} rounding of each dot give
+2· 10^{-13} rounding of each dot give
- |AB-T^*X_{rm ideal}|<1.5cdot 10^{-5}.
+ |AB-T^*X_{rm ideal}|<1.5· 10^{-5}.
The parameter-vector norm is below
√(.203^2+4(pi/64)^2)<.226, so substitution of stored AB costs
-less than 3.4cdot 10^{-6} radians. Formation of the corner offsets
+less than 3.4· 10^{-6} radians. Formation of the corner offsets
adds less than 10^{-11}, using coefficients of magnitude below 25.
The same bound applies to convex combinations of these corners.
For the second covariance, use stored weighted and
invmag, with rational beta=.8. Each D-weighted dot has
-absolute sum less than 16(1.00001)^2cdot 6.8<110. Its
-two-product-per-term accumulation has error below 5cdot 10^{-13} by
+absolute sum less than 16(1.00001)^2· 6.8<110. Its
+two-product-per-term accumulation has error below 5· 10^{-13} by
(round:dot) with one extra factor. The multiplying coefficient is
below 225. Coefficient translation and the subsequent arithmetic,
even at absolute intermediate scale 26000, give total entry error
@@ -3982,15 +3982,15 @@ For an exact stored symmetric ktimes k matrix G with kle 10 and
Proof.
-An absolute row sum has error below 3cdot 10^{-13}. The code adds
+An absolute row sum has error below 3· 10^{-13}. The code adds
10^{-10} to their maximum. For its other bound, put
m=max_{ij}|G_{ij}| and A=|G^2|_F. Because one diagonal sum of
squares contains a term m^2, Age m^2. Then
-|fl(sqrt{fro4})-A|<4cdot 10^{-13}A+10^{-139}
+|fl(sqrt{fro4})-A|<4· 10^{-13}A+10^{-139}
by dot-product and norm rounding. Taking sqrt A costs less than
-4cdot 10^{-13}sqrt A+10^{-69}, and since sqrt Ale 100 the total
+4· 10^{-13}sqrt A+10^{-69}, and since sqrt Ale 100 the total
possible downward error is below 10^{-10}. The code adds
-2cdot 10^{-7}. The exact √(|G^2|_F) bounds |G|. Both
+2· 10^{-7}. The exact √(|G^2|_F) bounds |G|. Both
returned candidates are upper bounds, so their minimum is also an upper
bound, and its value is nonnegative and at most 100.1.
@@ -4026,7 +4026,7 @@ medskipnoindentThe base grid and the outside test.
Use the stored positive width w_B as an exact grid parameter. The
allowed range is 10^{-4}le w_Ble 0.05. The stored grid radius
E=sqrt{1+ort_P},R+10^{-7} exceeds the true tangent radius
-by more than 0.999cdot 10^{-7}. For
+by more than 0.999· 10^{-7}. For
L=leftlceil E/w_B+0.500001rightrceil, the quotient scale is at
most 1261 and its error is below 10^{-12}. The 0.000001 excess
over the half-cell term is much larger, so the integer base range
@@ -4074,17 +4074,17 @@ regardless of literal translation; the Root residual gives
]
Storage and elementary arithmetic are bounded directly: the phase
-magnitude is below 4.0; the six-term S error is below 6cdot
+magnitude is below 4.0; the six-term S error is below 6·
10^{-14}; |,|S|/sqrt 6-4,|<10^{-13}. The arrays J and B
-have errors below 3cdot 10^{-11} and 4cdot 10^{-11}. The
+have errors below 3· 10^{-11} and 4· 10^{-11}. The
rounded constants a,X,I_M,L_M,1/sqrt6,(2pi)^2 each carry unit
relative error u, and the two required products and the sum carry
at most 3u relative. Each bin-face primitive check has constant
-2cdot 10^{-7} above
+2· 10^{-7} above
r_n|J|+(r_t/4), using |x_j|le a; its threshold is 10^{-6}.
-The combined bound uses the same 2cdot 10^{-7}. The nested
-squared-magnitude crossing band is 2cdot 10^{-7}; the squared
-increment is 7cdot 10^{-7}.
+The combined bound uses the same 2· 10^{-7}. The nested
+squared-magnitude crossing band is 2· 10^{-7}; the squared
+increment is 7· 10^{-7}.
medskipnoindentThe directional single-vector support..
The heuristic solve is accepted only with finite coordinates and a
@@ -4114,19 +4114,19 @@ The support compared with |a_y| is the stored evaluation of
This is the necessary support bound of the Exclusion stage.
Its coefficient errors against ideal arrays satisfy
[
- |Delta a_y|<7cdot 10^{-12},
- |Delta z_j|,|Deltawidetilde z_j|<7cdot 10^{-12},
- |Delta h_a|<4.3cdot 10^{-11}.
+ |Delta a_y|<7· 10^{-12},
+ |Delta z_j|,|Deltawidetilde z_j|<7· 10^{-12},
+ |Delta h_a|<4.3· 10^{-11}.
]
The first two use six input errors and a short dot; the last uses six
B errors and the same dot estimate. The two bracketed products cost
less than 10^{-14} each; the short absolute sums and the
normalization by m_yge 10^{-100} cost less than 10^{-15}.
Squares and square-root evaluation at the used scales cost less than
-10^{-15}. Three times either dot error is below 3cdot 10^{-11}.
+10^{-15}. Three times either dot error is below 3· 10^{-11}.
Thus the certified support contribution is within 10^{-11} of the
ideal expression inside pi, and the whole error against the ideal
-support is below 10^{-10}. The code adds 2cdot 10^{-7}.
+support is below 10^{-10}. The code adds 2· 10^{-7}.
% ---------------------------------------------------------------
medskipnoindentThe joint coefficient support.
@@ -4137,16 +4137,16 @@ The joint test is separated over the three tangent pieces and has at
most 6+6+2=14 rows. The proposed direction is checked finite and
normalized as before, so its stored entries have magnitude at most one.
-The constant pairing has error below 1.5cdot 10^{-11}. Each
-accumulated shared-Hadamard coefficient has error below 9cdot 10^{-11}:
-it receives at most twelve B input errors of 7cdot 10^{-12} plus
+The constant pairing has error below 1.5· 10^{-11}. Each
+accumulated shared-Hadamard coefficient has error below 9· 10^{-11}:
+it receives at most twelve B input errors of 7· 10^{-12} plus
short-dot rounding. Each vector-phase coefficient, including its edge
-contribution, has error below 9cdot 10^{-12}: six single-vector
+contribution, has error below 9· 10^{-12}: six single-vector
entries and at most two edge entries each contribute at most 12u.
Combining, the ideal support is at most the computed support plus
-3cdot 10^{-10}. The code adds 3cdot 10^{-7}.
+3· 10^{-10}. The code adds 3· 10^{-7}.
-The final comparison allowance of 3cdot 10^{-7} is monotone: NaN and
+The final comparison allowance of 3· 10^{-7} is monotone: NaN and
infinity are rejected by the direction checks, so only finite proposals
enter. Enlarging the proposed direction or its stored coefficients can
only raise the computed support, so the comparison remains a rigorous