Notation by work
COS Method
All notation · Glossary · Source coverage
626 symbol spellings in 1443 mathematical passages across 24 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 626 entries are available below.
*Notation table · 9 passages
Search aliases: *
Notation table · Read passage · D_glossary.tex, line 37
Saddlepoint tilt · An admissible exponential tilt satisfying κ'(θ_*)=q. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.
Exact mathematical passage
\kappa'(\theta_*)=q
+Notation table · 120 passages
Search aliases: +
Notation table · Read passage · D_glossary.tex, line 65
[p_-,p_+] · A probability enclosure. Acceptance at relative tolerance ε requires p_->0 and p_+/p_- -1≤ε.
Exact mathematical passage
[p_-,p_+]
=Notation table · 411 passages
Search aliases: =
Notation table · Read passage · D_glossary.tex, line 14
CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).
Exact mathematical passage
F(q)=\Pr(Q\le q)
ANotation table · 133 passages
Search aliases: A
Notation table · Read passage · D_glossary.tex, line 26
Gram representation · The operator AA^top or its dual A^top A. Their positive eigenvalues agree with multiplicity.
Exact mathematical passage
AA^\top
A^\topNotation table · 46 passages
Search aliases: A^top top
Notation table · Read passage · D_glossary.tex, line 26
Gram representation · The operator AA^top or its dual A^top A. Their positive eigenvalues agree with multiplicity.
Exact mathematical passage
AA^\top
A_bNotation table · 11 passages
Search aliases: A_b
Notation table · Read passage · D_glossary.tex, line 52
A, A_b · A real factor and its resident column blocks.
Exact mathematical passage
A_b
bNotation table · 125 passages
Search aliases: b
Notation table · Read passage · D_glossary.tex, line 52
A, A_b · A real factor and its resident column blocks.
Exact mathematical passage
A_b
CNotation table · 18 passages
Search aliases: C
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm COS}
dNotation table · 46 passages
Search aliases: d
Notation table · Read passage · D_glossary.tex, line 66
[d_-,d_+] · A base-ten logarithmic probability enclosure.
Exact mathematical passage
[d_-,d_+]
d_+Notation table · 11 passages
Search aliases: d_+
Notation table · Read passage · D_glossary.tex, line 66
[d_-,d_+] · A base-ten logarithmic probability enclosure.
Exact mathematical passage
[d_-,d_+]
d_-Notation table · 11 passages
Search aliases: d_-
Notation table · Read passage · D_glossary.tex, line 66
[d_-,d_+] · A base-ten logarithmic probability enclosure.
Exact mathematical passage
[d_-,d_+]
\mathbb ENotation table · 4 passages
Search aliases: E mathbb
Notation table · Read passage · frontmatter.tex, line 47
κ(s) · The cumulant generating function, log E e^sQ.
Exact mathematical passage
\log\mathbb E e^{sQ}
eNotation table · 80 passages
Search aliases: e
Notation table · Read passage · frontmatter.tex, line 47
κ(s) · The cumulant generating function, log E e^sQ.
Exact mathematical passage
\log\mathbb E e^{sQ}
e^{sQ}Notation table · 2 passages
Search aliases: e^sQ
Notation table · Read passage · frontmatter.tex, line 47
κ(s) · The cumulant generating function, log E e^sQ.
Exact mathematical passage
\log\mathbb E e^{sQ}
FNotation table · 58 passages
Search aliases: F
Notation table · Read passage · D_glossary.tex, line 14
CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).
Exact mathematical passage
F(q)=\Pr(Q\le q)
HNotation table · 101 passages
Search aliases: H
Notation table · Read passage · D_glossary.tex, line 53
H, K · Symmetric operators, with sample Gram K=AA^top and feature Gram H=A^top A in the streaming construction.
Exact mathematical passage
H
hNotation table · 53 passages
Search aliases: h
Notation table · Read passage · D_glossary.tex, line 61
P_h · Orthogonal structural phase projectors.
Exact mathematical passage
P_h
iNotation table · 57 passages
Search aliases: i
Notation table · Read passage · D_glossary.tex, line 32
Noncentrality · The squared mean displacement in a standardized Gaussian direction. The symbol δ_i in a noncentral chi-square component differs from a centered packet deviation.
Exact mathematical passage
\delta_i
JNotation table · 27 passages
Search aliases: J
Notation table · Read passage · frontmatter.tex, line 50
u_k, J_k · Frequency kπ/b and its exponentially weighted payoff integral.
Exact mathematical passage
J_k
jNotation table · 74 passages
Search aliases: j
Notation table · Read passage · D_glossary.tex, line 21
Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.
Exact mathematical passage
(\sum_j\lambda_j)^2/\sum_j\lambda_j^2
J_kNotation table · 18 passages
Search aliases: J_k
Notation table · Read passage · frontmatter.tex, line 50
u_k, J_k · Frequency kπ/b and its exponentially weighted payoff integral.
Exact mathematical passage
J_k
KNotation table · 32 passages
Search aliases: K
Notation table · Read passage · D_glossary.tex, line 53
H, K · Symmetric operators, with sample Gram K=AA^top and feature Gram H=A^top A in the streaming construction.
Exact mathematical passage
K
kNotation table · 122 passages
Search aliases: k
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm pkt}
\logNotation table · 41 passages
Search aliases: log
Notation table · Read passage · frontmatter.tex, line 47
κ(s) · The cumulant generating function, log E e^sQ.
Exact mathematical passage
\log\mathbb E e^{sQ}
MNotation table · 72 passages
Search aliases: M
Notation table · Read passage · D_glossary.tex, line 54
Σ, M, μ · Covariance, quadratic weight, and Gaussian mean.
Exact mathematical passage
M
mNotation table · 39 passages
Search aliases: m
Notation table · Read passage · D_glossary.tex, line 62
M_P,m · The mth centered power sum of a spectral packet P.
Exact mathematical passage
M_{P,m}
M_{P,m}Notation table · 6 passages
Search aliases: M_P,m
Notation table · Read passage · D_glossary.tex, line 62
M_P,m · The mth centered power sum of a spectral packet P.
Exact mathematical passage
M_{P,m}
NNotation table · 68 passages
Search aliases: N
Notation table · Read passage · D_glossary.tex, line 60
N · Retained COS mode count. The implementation argument is K.
Exact mathematical passage
N
nNotation table · 45 passages
Search aliases: n
Notation table · Read passage · frontmatter.tex, line 41
midrule A · A factor. In the streaming construction, A∈ R^n× m.
Exact mathematical passage
A\in\mathbb R^{n\times m}
ONotation table · 12 passages
Search aliases: O
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm COS}
PNotation table · 94 passages
Search aliases: P
Notation table · Read passage · D_glossary.tex, line 61
P_h · Orthogonal structural phase projectors.
Exact mathematical passage
P_h
pNotation table · 82 passages
Search aliases: p
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm pkt}
p_+Notation table · 10 passages
Search aliases: p_+
Notation table · Read passage · D_glossary.tex, line 65
[p_-,p_+] · A probability enclosure. Acceptance at relative tolerance ε requires p_->0 and p_+/p_- -1≤ε.
Exact mathematical passage
[p_-,p_+]
p_-Notation table · 12 passages
Search aliases: p_-
Notation table · Read passage · D_glossary.tex, line 65
[p_-,p_+] · A probability enclosure. Acceptance at relative tolerance ε requires p_->0 and p_+/p_- -1≤ε.
Exact mathematical passage
[p_-,p_+]
P_hNotation table · 29 passages
Search aliases: P_h
Notation table · Read passage · D_glossary.tex, line 61
P_h · Orthogonal structural phase projectors.
Exact mathematical passage
P_h
\PrNotation table · 5 passages
Search aliases: Pr
Notation table · Read passage · D_glossary.tex, line 14
CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).
Exact mathematical passage
F(q)=\Pr(Q\le q)
QNotation table · 107 passages
Search aliases: Q
Notation table · Read passage · D_glossary.tex, line 14
CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).
Exact mathematical passage
F(q)=\Pr(Q\le q)
qNotation table · 106 passages
Search aliases: q
Notation table · Read passage · D_glossary.tex, line 14
CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).
Exact mathematical passage
F(q)=\Pr(Q\le q)
\mathbb RNotation table · 14 passages
Search aliases: R mathbb
Notation table · Read passage · frontmatter.tex, line 41
midrule A · A factor. In the streaming construction, A∈ R^n× m.
Exact mathematical passage
A\in\mathbb R^{n\times m}
\mathbb R^{n\times m}Notation table · 4 passages
Search aliases: R^n× m mathbb times
Notation table · Read passage · frontmatter.tex, line 41
midrule A · A factor. In the streaming construction, A∈ R^n× m.
Exact mathematical passage
A\in\mathbb R^{n\times m}
\rmNotation table · 26 passages
Search aliases: rm
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm pkt}
SNotation table · 43 passages
Search aliases: S
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm COS}
sNotation table · 96 passages
Search aliases: s
Notation table · Read passage · D_glossary.tex, line 63
ρ_P(s) · The packet's complex-grid convergence radius parameter.
Exact mathematical passage
\rho_P(s)
\succeqNotation table · 8 passages
Search aliases: succeq
Notation table · Read passage · frontmatter.tex, line 53
Msucceq0 · Positive semidefiniteness.
Exact mathematical passage
M\succeq0
tNotation table · 65 passages
Search aliases: t
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm pkt}
\topNotation table · 114 passages
Search aliases: top
Notation table · Read passage · D_glossary.tex, line 26
Gram representation · The operator AA^top or its dual A^top A. Their positive eigenvalues agree with multiplicity.
Exact mathematical passage
AA^\top
uNotation table · 45 passages
Search aliases: u
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm sup}
u_kNotation table · 20 passages
Search aliases: u_k
Notation table · Read passage · frontmatter.tex, line 50
u_k, J_k · Frequency kπ/b and its exponentially weighted payoff integral.
Exact mathematical passage
u_k
\cdotNotation table · 4 passages
Search aliases: · cdot
Notation table · Read passage · frontmatter.tex, line 51
· _2 · Euclidean vector norm or induced matrix norm, as appropriate.
Exact mathematical passage
\|\cdot\|_2
\timesNotation table · 139 passages
Search aliases: × times
Notation table · Read passage · frontmatter.tex, line 41
midrule A · A factor. In the streaming construction, A∈ R^n× m.
Exact mathematical passage
A\in\mathbb R^{n\times m}
\deltaNotation table · 17 passages
Search aliases: δ delta
Notation table · Read passage · D_glossary.tex, line 32
Noncentrality · The squared mean displacement in a standardized Gaussian direction. The symbol δ_i in a noncentral chi-square component differs from a centered packet deviation.
Exact mathematical passage
\delta_i
\delta_iNotation table · 3 passages
Search aliases: δ_i delta
Notation table · Read passage · D_glossary.tex, line 32
Noncentrality · The squared mean displacement in a standardized Gaussian direction. The symbol δ_i in a noncentral chi-square component differs from a centered packet deviation.
Exact mathematical passage
\delta_i
\varepsilonNotation table · 22 passages
Search aliases: ε varepsilon
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm pkt}
\varepsilon_{\rm COS}Notation table · 1 passage
Search aliases: ε_rm COS varepsilon rm
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm COS}
\varepsilon_{\rm pkt}Notation table · 1 passage
Search aliases: ε_rm pkt varepsilon rm
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm pkt}
\varepsilon_{\rm sup}Notation table · 1 passage
Search aliases: ε_rm sup varepsilon rm
Notation table · Read passage · D_glossary.tex, line 64
ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.
Exact mathematical passage
\varepsilon_{\rm sup}
\etaNotation table · 18 passages
Search aliases: η eta
Notation table · Read passage · D_glossary.tex, line 58
θ, η · Admissible exponential tilts, with η>θ when used in a Chernoff support bound.
Exact mathematical passage
\eta
\thetaNotation table · 107 passages
Search aliases: θ theta
Notation table · Read passage · D_glossary.tex, line 37
Saddlepoint tilt · An admissible exponential tilt satisfying κ'(θ_*)=q. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.
Exact mathematical passage
\kappa'(\theta_*)=q
\theta_*Notation table · 9 passages
Search aliases: θ_* theta
Notation table · Read passage · D_glossary.tex, line 37
Saddlepoint tilt · An admissible exponential tilt satisfying κ'(θ_*)=q. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.
Exact mathematical passage
\kappa'(\theta_*)=q
\kappaNotation table · 49 passages
Search aliases: κ kappa
Notation table · Read passage · D_glossary.tex, line 37
Saddlepoint tilt · An admissible exponential tilt satisfying κ'(θ_*)=q. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.
Exact mathematical passage
\kappa'(\theta_*)=q
\kappa'Notation table · 11 passages
Search aliases: κ' kappa
Notation table · Read passage · D_glossary.tex, line 37
Saddlepoint tilt · An admissible exponential tilt satisfying κ'(θ_*)=q. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.
Exact mathematical passage
\kappa'(\theta_*)=q
\lambdaNotation table · 74 passages
Search aliases: λ lambda
Notation table · Read passage · D_glossary.tex, line 21
Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.
Exact mathematical passage
(\sum_j\lambda_j)^2/\sum_j\lambda_j^2
\lambda_jNotation table · 37 passages
Search aliases: λ_j lambda
Notation table · Read passage · D_glossary.tex, line 21
Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.
Exact mathematical passage
(\sum_j\lambda_j)^2/\sum_j\lambda_j^2
\lambda_j^2Notation table · 3 passages
Search aliases: λ_j^2 lambda
Notation table · Read passage · D_glossary.tex, line 21
Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.
Exact mathematical passage
(\sum_j\lambda_j)^2/\sum_j\lambda_j^2
\muNotation table · 25 passages
Search aliases: μ mu
Notation table · Read passage · D_glossary.tex, line 54
Σ, M, μ · Covariance, quadratic weight, and Gaussian mean.
Exact mathematical passage
\mu
\piNotation table · 19 passages
Search aliases: π pi
Notation table · Read passage · frontmatter.tex, line 50
u_k, J_k · Frequency kπ/b and its exponentially weighted payoff integral.
Exact mathematical passage
k\pi/b
\rhoNotation table · 55 passages
Search aliases: ρ rho
Notation table · Read passage · D_glossary.tex, line 63
ρ_P(s) · The packet's complex-grid convergence radius parameter.
Exact mathematical passage
\rho_P(s)
\rho_PNotation table · 11 passages
Search aliases: ρ_P rho
Notation table · Read passage · D_glossary.tex, line 63
ρ_P(s) · The packet's complex-grid convergence radius parameter.
Exact mathematical passage
\rho_P(s)
\SigmaNotation table · 14 passages
Search aliases: Σ Sigma
Notation table · Read passage · D_glossary.tex, line 54
Σ, M, μ · Covariance, quadratic weight, and Gaussian mean.
Exact mathematical passage
\Sigma
\inNotation table · 56 passages
Search aliases: ∈ in
Notation table · Read passage · frontmatter.tex, line 41
midrule A · A factor. In the streaming construction, A∈ R^n× m.
Exact mathematical passage
A\in\mathbb R^{n\times m}
\sumNotation table · 97 passages
Search aliases: ∑ sum
Notation table · Read passage · D_glossary.tex, line 21
Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.
Exact mathematical passage
(\sum_j\lambda_j)^2/\sum_j\lambda_j^2
\sum_jNotation table · 13 passages
Search aliases: ∑_j sum
Notation table · Read passage · D_glossary.tex, line 21
Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.
Exact mathematical passage
(\sum_j\lambda_j)^2/\sum_j\lambda_j^2
\leNotation table · 103 passages
Search aliases: ≤ le
Notation table · Read passage · D_glossary.tex, line 14
CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).
Exact mathematical passage
F(q)=\Pr(Q\le q)
\geNotation table · 68 passages
Search aliases: ≥ ge
Notation table · Read passage · frontmatter.tex, line 46
q · The threshold in P(Q≥ q).
Exact mathematical passage
P(Q\ge q)
A^{\mathsf T}Introduction passage · 6 passages
Search aliases: A^ T mathsf
Introduction passage · Read passage · 11_evidence.tex, line 482
The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(AA^ T). Three independent n=400, m=1200 genotype sets yielded numerical rank 397. Four upper-tail queries per spectrum were generated by choosing sadd…
Exact mathematical passage
\lambda=\tfrac12\operatorname{eig}(AA^{\mathsf T}).
BIntroduction passage · 57 passages
Search aliases: B
Introduction passage · Read passage · 12_synthesis.tex, line 46
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\mathcal P_h = J_{h1}\sqcup\cdots\sqcup J_{hB_h}.
\mathcal BIntroduction passage · 37 passages
Search aliases: B mathcal
Introduction passage · Read passage · 03_phases.tex, line 321
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
\mathcal B
\mathcal B_{\mathcal O}Introduction passage · 5 passages
Search aliases: B_ O mathcal mathcal
Introduction passage · Read passage · 03_phases.tex, line 329
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
\mathcal B_{\mathcal O} =\mathcal B-\sum_{e\in\mathcal O}\Delta_e.
\mathcal B_{gh}Introduction passage · 1 passage
Search aliases: B_gh mathcal
Introduction passage · Read passage · 03_phases.tex, line 323
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
(\Delta_e)_{hg}=\mathcal B_{hg},\qquad (\Delta_e)_{gh}=\mathcal B_{gh},
B_hIntroduction passage · 1 passage
Search aliases: B_h
Introduction passage · Read passage · 12_synthesis.tex, line 46
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\mathcal P_h = J_{h1}\sqcup\cdots\sqcup J_{hB_h}.
\mathcal B_{hg}Introduction passage · 5 passages
Search aliases: B_hg mathcal
Introduction passage · Read passage · 03_phases.tex, line 323
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
(\Delta_e)_{hg}=\mathcal B_{hg},\qquad (\Delta_e)_{gh}=\mathcal B_{gh},
cIntroduction passage · 41 passages
Search aliases: c
Introduction passage · Read passage · 03_phases.tex, line 522
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic. Its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
c_e\ge0
c_eIntroduction passage · 2 passages
Search aliases: c_e
Introduction passage · Read passage · 03_phases.tex, line 522
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic. Its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
c_e\ge0
\cosIntroduction passage · 4 passages
Search aliases: cos
Introduction passage · Read passage · 07_inversion.tex, line 18
Because Q≥0, K-COS uses the interval [0,b]. Throughout the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-θ+i u_k)(b-q)θ-i u_k ]. The prime on a cosine sum denotes half weight for k=0.
Exact mathematical passage
J_k(q,b,\theta) =\int_q^b e^{-\theta(x-q)}\cos(u_kx)\,dx. \label{eq:J-def}
\EIntroduction passage · 6 passages
Search aliases: E operatorname mathbb
Introduction passage · Read passage · 06_packets.tex, line 31
Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose …
Exact mathematical passage
\kappa(s)=\log \E e^{sQ} =-\frac12\sum_{j=1}^{r}\log(1-2s\lambda_j). \label{eq:cgf}
\mathcal EIntroduction passage · 5 passages
Search aliases: E mathcal
Introduction passage · Read passage · 03_phases.tex, line 320
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
\mathcal E
e^{(-\theta+\ii u_k)(b-q)}Introduction passage · 1 passage
Search aliases: e^(-θ+ii u_k)(b-q) e^(-θ+i u_k)(b-q) theta ii theta mathrm
Introduction passage · Read passage · 07_inversion.tex, line 24
the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-θ+i u_k)(b-q)θ-i u_k ]. The prime on a cosine sum denotes half weight for k=0.
Exact mathematical passage
J_k =\Re\left[ e^{\ii u_k q} \frac{1-e^{(-\theta+\ii u_k)(b-q)}}{\theta-\ii u_k} \right]. \label{eq:J-formula}
e^{-\theta(x-q)}Introduction passage · 6 passages
Search aliases: e^-θ(x-q) theta
Introduction passage · Read passage · 07_inversion.tex, line 18
Because Q≥0, K-COS uses the interval [0,b]. Throughout the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-θ+i u_k)(b-q)θ-i u_k ]. The prime on a cosine sum denotes half weight for k=0.
Exact mathematical passage
J_k(q,b,\theta) =\int_q^b e^{-\theta(x-q)}\cos(u_kx)\,dx. \label{eq:J-def}
e^{\ii u_k q}Introduction passage · 1 passage
Search aliases: e^ii u_k q e^i u_k q ii mathrm
Introduction passage · Read passage · 07_inversion.tex, line 24
the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-θ+i u_k)(b-q)θ-i u_k ]. The prime on a cosine sum denotes half weight for k=0.
Exact mathematical passage
J_k =\Re\left[ e^{\ii u_k q} \frac{1-e^{(-\theta+\ii u_k)(b-q)}}{\theta-\ii u_k} \right]. \label{eq:J-formula}
\operatorname{eig}Introduction passage · 1 passage
Search aliases: eig operatorname
Introduction passage · Read passage · 11_evidence.tex, line 482
The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(AA^ T). Three independent n=400, m=1200 genotype sets yielded numerical rank 397. Four upper-tail queries per spectrum were generated by choosing sadd…
Exact mathematical passage
\lambda=\tfrac12\operatorname{eig}(AA^{\mathsf T}).
gIntroduction passage · 32 passages
Search aliases: g
Introduction passage · Read passage · 03_phases.tex, line 322
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
e=\{h,g\}
\iiIntroduction passage · 7 passages
Search aliases: ii i mathrm
Introduction passage · Read passage · 07_inversion.tex, line 24
the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-θ+i u_k)(b-q)θ-i u_k ]. The prime on a cosine sum denotes half weight for k=0.
Exact mathematical passage
J_k =\Re\left[ e^{\ii u_k q} \frac{1-e^{(-\theta+\ii u_k)(b-q)}}{\theta-\ii u_k} \right]. \label{eq:J-formula}
J_{h1}Introduction passage · 1 passage
Search aliases: J_h1
Introduction passage · Read passage · 12_synthesis.tex, line 46
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\mathcal P_h = J_{h1}\sqcup\cdots\sqcup J_{hB_h}.
J_{hB_h}Introduction passage · 1 passage
Search aliases: J_hB_h
Introduction passage · Read passage · 12_synthesis.tex, line 46
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\mathcal P_h = J_{h1}\sqcup\cdots\sqcup J_{hB_h}.
\mathrm{MAF}Introduction passage · 1 passage
Search aliases: MAF mathrm
Introduction passage · Read passage · 11_evidence.tex, line 481
The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(AA^ T). Three independent n=400, m=1200 genotype sets yielded numerical rank 397. Four upper-tail queries per spectrum were generated by choosing sadd…
Exact mathematical passage
w_j=(1-\mathrm{MAF}_j)^{24}
\mathrm{MAF}_jIntroduction passage · 1 passage
Search aliases: MAF_j mathrm
Introduction passage · Read passage · 11_evidence.tex, line 481
The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(AA^ T). Three independent n=400, m=1200 genotype sets yielded numerical rank 397. Four upper-tail queries per spectrum were generated by choosing sadd…
Exact mathematical passage
w_j=(1-\mathrm{MAF}_j)^{24}
\maxIntroduction passage · 26 passages
Search aliases: max
Introduction passage · Read passage · 03_phases.tex, line 525
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic. Its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\max_jIntroduction passage · 1 passage
Search aliases: max_j max
Introduction passage · Read passage · 06_packets.tex, line 25
Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose …
Exact mathematical passage
\Re s < \frac{1}{2\lambda_{\max}}, \qquad \lambda_{\max}=\max_j\lambda_j,
\max_{x_e\in\{0,1\}}Introduction passage · 1 passage
Search aliases: max_x_e∈{0,1} max in
Introduction passage · Read passage · 03_phases.tex, line 525
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic. Its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\mathcal OIntroduction passage · 16 passages
Search aliases: O mathcal
Introduction passage · Read passage · 03_phases.tex, line 327
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
\mathcal O\subseteq\mathcal E
\mathcal PIntroduction passage · 5 passages
Search aliases: P mathcal
Introduction passage · Read passage · 12_synthesis.tex, line 42
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
\mathcal P_1Introduction passage · 1 passage
Search aliases: P_1 mathcal
Introduction passage · Read passage · 12_synthesis.tex, line 42
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
P_gIntroduction passage · 22 passages
Search aliases: P_g
Introduction passage · Read passage · 12_synthesis.tex, line 77
Storage refinement changes only the order and grouping of finite sums. By partition refinement invariance, the streamed products reconstruct the same A, A^top, and therefore the same H=A^top A. Since each P_h is fixed by the structural phase partition, every block P_hHP_g is unchanged. The phase graph is defined only by these blocks, so its path structure and light cones are unchanged. Finally, an exact matrix-free Krylov method depends only on repeated applications of the same operator to the s…
Exact mathematical passage
P_hHP_g
\mathcal P_HIntroduction passage · 1 passage
Search aliases: P_H mathcal
Introduction passage · Read passage · 12_synthesis.tex, line 42
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
\mathcal P_hIntroduction passage · 1 passage
Search aliases: P_h mathcal
Introduction passage · Read passage · 12_synthesis.tex, line 46
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\mathcal P_h = J_{h1}\sqcup\cdots\sqcup J_{hB_h}.
rIntroduction passage · 86 passages
Search aliases: r
Introduction passage · Read passage · 06_packets.tex, line 18
Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose …
Exact mathematical passage
Q=\sum_{j=1}^{r}\lambda_j Z_j^2, \qquad \lambda_j>0,\quad r\ge1. \label{eq:Q}
\ReIntroduction passage · 11 passages
Search aliases: Re
Introduction passage · Read passage · 06_packets.tex, line 25
Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose …
Exact mathematical passage
\Re s < \frac{1}{2\lambda_{\max}}, \qquad \lambda_{\max}=\max_j\lambda_j,
\sqcupIntroduction passage · 3 passages
Search aliases: sqcup
Introduction passage · Read passage · 12_synthesis.tex, line 42
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
\mathsf TIntroduction passage · 16 passages
Search aliases: T mathsf
Introduction passage · Read passage · 11_evidence.tex, line 482
The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(AA^ T). Three independent n=400, m=1200 genotype sets yielded numerical rank 397. Four upper-tail queries per spectrum were generated by choosing sadd…
Exact mathematical passage
\lambda=\tfrac12\operatorname{eig}(AA^{\mathsf T}).
wIntroduction passage · 9 passages
Search aliases: w
Introduction passage · Read passage · 11_evidence.tex, line 481
The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(AA^ T). Three independent n=400, m=1200 genotype sets yielded numerical rank 397. Four upper-tail queries per spectrum were generated by choosing sadd…
Exact mathematical passage
w_j=(1-\mathrm{MAF}_j)^{24}
w_jIntroduction passage · 3 passages
Search aliases: w_j
Introduction passage · Read passage · 11_evidence.tex, line 481
The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(AA^ T). Three independent n=400, m=1200 genotype sets yielded numerical rank 397. Four upper-tail queries per spectrum were generated by choosing sadd…
Exact mathematical passage
w_j=(1-\mathrm{MAF}_j)^{24}
\widetilde GIntroduction passage · 2 passages
Search aliases: widetilde G widetilde
Introduction passage · Read passage · 11_evidence.tex, line 335
where widetilde G_b denotes the centered and scaled block. The full weighted factor is never materialized.
Exact mathematical passage
\widetilde G_b
\widetilde G_bIntroduction passage · 2 passages
Search aliases: widetilde G_b widetilde
Introduction passage · Read passage · 11_evidence.tex, line 335
where widetilde G_b denotes the centered and scaled block. The full weighted factor is never materialized.
Exact mathematical passage
\widetilde G_b
xIntroduction passage · 34 passages
Search aliases: x
Introduction passage · Read passage · 03_phases.tex, line 525
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic. Its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
x_eIntroduction passage · 1 passage
Search aliases: x_e
Introduction passage · Read passage · 03_phases.tex, line 525
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic. Its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
ZIntroduction passage · 27 passages
Search aliases: Z
Introduction passage · Read passage · 06_packets.tex, line 18
Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose …
Exact mathematical passage
Q=\sum_{j=1}^{r}\lambda_j Z_j^2, \qquad \lambda_j>0,\quad r\ge1. \label{eq:Q}
Z_jIntroduction passage · 2 passages
Search aliases: Z_j
Introduction passage · Read passage · 06_packets.tex, line 23
Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose …
Exact mathematical passage
Z_j
Z_j^2Introduction passage · 6 passages
Search aliases: Z_j^2
Introduction passage · Read passage · 06_packets.tex, line 18
Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose …
Exact mathematical passage
Q=\sum_{j=1}^{r}\lambda_j Z_j^2, \qquad \lambda_j>0,\quad r\ge1. \label{eq:Q}
\betaIntroduction passage · 23 passages
Search aliases: β beta
Introduction passage · Read passage · 03_phases.tex, line 525
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic. Its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\beta_e^{(k)}Introduction passage · 6 passages
Search aliases: β_e^(k) beta
Introduction passage · Read passage · 03_phases.tex, line 525
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic. Its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\DeltaIntroduction passage · 9 passages
Search aliases: Δ Delta
Introduction passage · Read passage · 03_phases.tex, line 322
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
\Delta_e
\Delta_eIntroduction passage · 6 passages
Search aliases: Δ_e Delta
Introduction passage · Read passage · 03_phases.tex, line 322
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
\Delta_e
\kappa^{(m)}Introduction passage · 1 passage
Search aliases: κ^(m) kappa
Introduction passage · Read passage · 06_packets.tex, line 41
log(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose argument can wrap by 2π. For real θ<1/(2λ_max), κ^(m)(θ) =2^m-1(m-1)!∑_j=1^r fracλ_j^m(1-2θλ_j)^m, m≥1. At θ=0, this recovers the standard cumulant identity cum_m(Q)=2^m-1(m-1)!∑_jλ_j^m.
Exact mathematical passage
\kappa^{(m)}(\theta) =2^{m-1}(m-1)!\sum_{j=1}^{r} \frac{\lambda_j^m}{(1-2\theta\lambda_j)^m}, \qquad m\ge1. \label{eq:derivatives}
\lambda_j^mIntroduction passage · 2 passages
Search aliases: λ_j^m lambda
Introduction passage · Read passage · 06_packets.tex, line 41
log(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose argument can wrap by 2π. For real θ<1/(2λ_max), κ^(m)(θ) =2^m-1(m-1)!∑_j=1^r fracλ_j^m(1-2θλ_j)^m, m≥1. At θ=0, this recovers the standard cumulant identity cum_m(Q)=2^m-1(m-1)!∑_jλ_j^m.
Exact mathematical passage
\kappa^{(m)}(\theta) =2^{m-1}(m-1)!\sum_{j=1}^{r} \frac{\lambda_j^m}{(1-2\theta\lambda_j)^m}, \qquad m\ge1. \label{eq:derivatives}
\lambda_{\max}Introduction passage · 12 passages
Search aliases: λ_max lambda max
Introduction passage · Read passage · 06_packets.tex, line 25
Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose …
Exact mathematical passage
\Re s < \frac{1}{2\lambda_{\max}}, \qquad \lambda_{\max}=\max_j\lambda_j,
\tauIntroduction passage · 23 passages
Search aliases: τ tau
Introduction passage · Read passage · 03_phases.tex, line 525
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic. Its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\ldotsIntroduction passage · 13 passages
Search aliases: … ldots
Introduction passage · Read passage · 07_inversion.tex, line 13
Because Q≥0, K-COS uses the interval [0,b]. Throughout the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-θ+i u_k)(b-q)θ-i u_k ]. The prime on a cosine sum denotes half weight for k=0.
Exact mathematical passage
u_k=\frac{k\pi}{b}, \qquad k=0,1,\ldots,
\sum_eIntroduction passage · 1 passage
Search aliases: ∑_e sum
Introduction passage · Read passage · 03_phases.tex, line 525
Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk greedy heuristic. Its output remains certified because it never exceeds the additive risk budget. A fractional-knapsack relaxation supplies an upper …
Exact mathematical passage
\boxed{ \max_{x_e\in\{0,1\}} \sum_e c_ex_e \quad\text{subject to}\quad \sum_e\beta_e^{(k)}x_e\le\tau. }
\sum_{e\in\mathcal O}Introduction passage · 7 passages
Search aliases: ∑_e∈ O sum in mathcal
Introduction passage · Read passage · 03_phases.tex, line 329
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
\mathcal B_{\mathcal O} =\mathcal B-\sum_{e\in\mathcal O}\Delta_e.
\sum_{j=1}^{r}Introduction passage · 6 passages
Search aliases: ∑_j=1^r sum
Introduction passage · Read passage · 06_packets.tex, line 18
Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose …
Exact mathematical passage
Q=\sum_{j=1}^{r}\lambda_j Z_j^2, \qquad \lambda_j>0,\quad r\ge1. \label{eq:Q}
\intIntroduction passage · 9 passages
Search aliases: ∫ int
Introduction passage · Read passage · 07_inversion.tex, line 18
Because Q≥0, K-COS uses the interval [0,b]. Throughout the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-θ+i u_k)(b-q)θ-i u_k ]. The prime on a cosine sum denotes half weight for k=0.
Exact mathematical passage
J_k(q,b,\theta) =\int_q^b e^{-\theta(x-q)}\cos(u_kx)\,dx. \label{eq:J-def}
\int_q^bIntroduction passage · 3 passages
Search aliases: ∫_q^b int
Introduction passage · Read passage · 07_inversion.tex, line 18
Because Q≥0, K-COS uses the interval [0,b]. Throughout the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-θ+i u_k)(b-q)θ-i u_k ]. The prime on a cosine sum denotes half weight for k=0.
Exact mathematical passage
J_k(q,b,\theta) =\int_q^b e^{-\theta(x-q)}\cos(u_kx)\,dx. \label{eq:J-def}
\subseteqIntroduction passage · 5 passages
Search aliases: ⊆ subseteq
Introduction passage · Read passage · 03_phases.tex, line 327
Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
\mathcal O\subseteq\mathcal E
\cdotsIntroduction passage · 9 passages
Search aliases: ⋯ cdots
Introduction passage · Read passage · 12_synthesis.tex, line 42
Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.
Exact mathematical passage
\{1,\ldots,m\}=\mathcal P_1\sqcup\cdots\sqcup\mathcal P_H
\mathbf1Use in source · 3 passages
Search aliases: 1 mathbf
Use in source · Read passage · 06_packets.tex, line 88
For any θ>0 in the transform domain, P(Q≥ q) =e^κ(θ)-θ q E_θ [e^-θ(Q-q) 1_{Q≥ q}]. This classical exponential-tilting identity lets K-COS evaluate the upper tail without subtracting a cumulative distribution function (CDF) value close to one.
Exact mathematical passage
\Pp(Q\ge q) =e^{\kappa(\theta)-\theta q} \E_{\theta}\!\left[e^{-\theta(Q-q)}\mathbf 1_{\{Q\ge q\}}\right]. \label{eq:tilt-tail}
\mathbf1_{[q,b]}Use in source · 1 passage
Search aliases: 1_[q,b] mathbf
Use in source · Read passage · 07_inversion.tex, line 108
[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted chi-square sum has an absolutely continuous law. Its folded measure therefore has no atom at the jump of h. The even periodic extension of h has boun…
Exact mathematical passage
h(x)=e^{-\theta(x-q)}\mathbf1_{[q,b]}(x)
\mathbf1_{\{Q\ge q\}}Use in source · 2 passages
Search aliases: 1_{Q≥ q} mathbf ge
Use in source · Read passage · 06_packets.tex, line 88
For any θ>0 in the transform domain, P(Q≥ q) =e^κ(θ)-θ q E_θ [e^-θ(Q-q) 1_{Q≥ q}]. This classical exponential-tilting identity lets K-COS evaluate the upper tail without subtracting a cumulative distribution function (CDF) value close to one.
Exact mathematical passage
\Pp(Q\ge q) =e^{\kappa(\theta)-\theta q} \E_{\theta}\!\left[e^{-\theta(Q-q)}\mathbf 1_{\{Q\ge q\}}\right]. \label{eq:tilt-tail}
aUse in source · 23 passages
Search aliases: a
Use in source · Read passage · 01_gaussian.tex, line 293
The numerical interface implements this reduction directly. In an isotropic control, Σ=I, M=aI, a>0, the law collapses to the exact scaled noncentral chi-square distribution Q/a∼χ^2_d( μ ^2).
Exact mathematical passage
\Sigma=I,\qquad M=aI,\qquad a>0,
A'_{ik}Use in source · 1 passage
Search aliases: A'_ik
Use in source · Read passage · B_lean.tex, line 86
The reindexing theorem declgramEntry_reindex handles a bijection e:V≃ V. It moves the block to e(s) and transforms the factor to A'_ik=A_i,e^-1(k). The resulting Gram entry is unchanged. Permuting only the coordinates without moving their factor columns falls outside this statement.
Exact mathematical passage
A'_{ik}=A_{i,e^{-1}(k)}
A^TUse in source · 1 passage
Search aliases: A^T
Use in source · Read passage · 05_spectrum.tex, line 684
Let A∈ R^n× m be supplied as an IEEE binary64 array. Each stored entry is treated as its exact dyadic rational value. The target operator is therefore the exact Gram matrix H=AA^T defined by those binary64 values, not an unspecified nearby real matrix.
Exact mathematical passage
H=AA^T
A_1Use in source · 6 passages
Search aliases: A_1
Use in source · Read passage · 01_gaussian.tex, line 52
[Representation equivalence] The same central quadratic-form law may be supplied by any of the following: item a nonnegative spectrum {λ_j}. item a positive semidefinite matrix H. item a factor A with H=AA^top or H=A^top A. item a replayable streamed factor A=[A_1 ⋯ A_B]. The nonzero spectra of AA^top and A^top A are identical, and AA^top=∑_bA_bA_b^top.
Exact mathematical passage
A=[A_1\ \cdots\ A_B].
A_1^\topUse in source · 2 passages
Search aliases: A_1^top top
Use in source · Read passage · 02_streaming.tex, line 103
Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.
Exact mathematical passage
[A_1\ \cdots\ A_B] \begin{bmatrix} A_1^\top\\ \vdots\\ A_B^\top \end{bmatrix} = \sum_bA_bA_b^\top.
A_2Use in source · 3 passages
Search aliases: A_2
Use in source · Read passage · 02_streaming.tex, line 422
The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0
Exact mathematical passage
A=[A_1\ A_2\ A_3],\qquad A_i\in\mathbb R^{4\times2}.
A_2^\topUse in source · 1 passage
Search aliases: A_2^top top
Use in source · Read passage · 02_streaming.tex, line 426
The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0 0 · 1 · 0 · 1 1 · 0 · 1 · 0 0 · 1 · 0 · 1, A_2A_2^top=A_3A_3^top=frac12 1 · 0 · -1 · 0 0 · 1 · 0 · -1 -1 · 0 · 1 · 0 0 · -1 · 0 · 1. The sum is K. Merging the last two blocks preserves it. Discarding A_3 instead changes the o…
Exact mathematical passage
A_1A_1^\top=\frac12 \begin{pmatrix}1&0&1&0\\0&1&0&1\\1&0&1&0\\0&1&0&1\end{pmatrix},\qquad A_2A_2^\top=A_3A_3^\top=\frac12 \begin{pmatrix}1&0&-1&0\\0&1&0&-1\\-1&0&1&0\\0&-1&0&1\end{pmatrix}.
A_3Use in source · 4 passages
Search aliases: A_3
Use in source · Read passage · 02_streaming.tex, line 422
The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0
Exact mathematical passage
A=[A_1\ A_2\ A_3],\qquad A_i\in\mathbb R^{4\times2}.
A_3^\topUse in source · 1 passage
Search aliases: A_3^top top
Use in source · Read passage · 02_streaming.tex, line 426
The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0 0 · 1 · 0 · 1 1 · 0 · 1 · 0 0 · 1 · 0 · 1, A_2A_2^top=A_3A_3^top=frac12 1 · 0 · -1 · 0 0 · 1 · 0 · -1 -1 · 0 · 1 · 0 0 · -1 · 0 · 1. The sum is K. Merging the last two blocks preserves it. Discarding A_3 instead changes the o…
Exact mathematical passage
A_1A_1^\top=\frac12 \begin{pmatrix}1&0&1&0\\0&1&0&1\\1&0&1&0\\0&1&0&1\end{pmatrix},\qquad A_2A_2^\top=A_3A_3^\top=\frac12 \begin{pmatrix}1&0&-1&0\\0&1&0&-1\\-1&0&1&0\\0&-1&0&1\end{pmatrix}.
A_BUse in source · 4 passages
Search aliases: A_B
Use in source · Read passage · 01_gaussian.tex, line 52
[Representation equivalence] The same central quadratic-form law may be supplied by any of the following: item a nonnegative spectrum {λ_j}. item a positive semidefinite matrix H. item a factor A with H=AA^top or H=A^top A. item a replayable streamed factor A=[A_1 ⋯ A_B]. The nonzero spectra of AA^top and A^top A are identical, and AA^top=∑_bA_bA_b^top.
Exact mathematical passage
A=[A_1\ \cdots\ A_B].
A_{b,I_r}Use in source · 2 passages
Search aliases: A_b,I_r
Use in source · Read passage · 02_streaming.tex, line 284
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ A_{b,I_r} = H_{b,I_r}-Q_{X,I_r}R_b. }
A_{b,I_s}^\topUse in source · 1 passage
Search aliases: A_b,I_s^top top
Use in source · Read passage · 02_streaming.tex, line 292
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ K_{I_r,I_s} = \sum_b A_{b,I_r}A_{b,I_s}^\top. }
A_b^{\mathsf T}Use in source · 3 passages
Search aliases: A_b^ T mathsf
Use in source · Read passage · 02_streaming.tex, line 127
[htbp] centering includegraphics[width=linewidth]figures_production/figure_02_exact_block_kernel_assembly.pdf captionEach resident column block contributes a complete sample-space Gram summand A_bA_b^ T. Adding the summands recovers AA^ T exactly. Storage partitioning changes residence and evaluation order, not the represented operator.
Exact mathematical passage
A_bA_b^{\mathsf T}
A_B^\topUse in source · 1 passage
Search aliases: A_B^top top
Use in source · Read passage · 02_streaming.tex, line 103
Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.
Exact mathematical passage
[A_1\ \cdots\ A_B] \begin{bmatrix} A_1^\top\\ \vdots\\ A_B^\top \end{bmatrix} = \sum_bA_bA_b^\top.
A_b^\topUse in source · 5 passages
Search aliases: A_b^top top
Use in source · Read passage · 01_gaussian.tex, line 57
The same central quadratic-form law may be supplied by any of the following: item a nonnegative spectrum {λ_j}. item a positive semidefinite matrix H. item a factor A with H=AA^top or H=A^top A. item a replayable streamed factor A=[A_1 ⋯ A_B]. The nonzero spectra of AA^top and A^top A are identical, and AA^top=∑_bA_bA_b^top.
Exact mathematical passage
AA^\top=\sum_bA_bA_b^\top.
a_dUse in source · 2 passages
Search aliases: a_d
Use in source · Read passage · 03_phases.tex, line 416
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
p(t)=\sum_{d=0}^{D}a_dt^d.
A_FUse in source · 2 passages
Search aliases: A_F
Use in source · Read passage · 12_synthesis.tex, line 118
starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the full variant space. Thus P_F=J_FJ_F^top and A_F=AJ_F. The compressed operator and its ambient embedding satisfy H_F=J_F^top HJ_F=A_F^top A_F, P_FHP_F=J_FH_FJ_F^top. T…
Exact mathematical passage
A_F=AJ_F
A_F^\topUse in source · 1 passage
Search aliases: A_F^top top
Use in source · Read passage · 12_synthesis.tex, line 119
^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the full variant space. Thus P_F=J_FJ_F^top and A_F=AJ_F. The compressed operator and its ambient embedding satisfy H_F=J_F^top HJ_F=A_F^top A_F, P_FHP_F=J_FH_FJ_F^top. These identities distinguish the compressed matrix from the larger …
Exact mathematical passage
H_F=J_F^\top HJ_F=A_F^\top A_F, \qquad P_FHP_F=J_FH_FJ_F^\top.
A_{F_k}Use in source · 1 passage
Search aliases: A_F_k
Use in source · Read passage · 12_synthesis.tex, line 107
ghts, and structural phase metadata fixed, and assume all factor-action blocks required for the retained phase operator are available. At depth k, let F_k be the graph k-neighborhood of F_0. Then no storage block lying wholly in a phase outside F_k can affect the k-step Krylov vector. Such a block need not be loaded when evaluating the requested power through the restricted factor A_F_k. Storage blocks mixing active and inactive phases require a separate refinement before this omission is valid.
Exact mathematical passage
A_{F_k}
A_iUse in source · 1 passage
Search aliases: A_i
Use in source · Read passage · 02_streaming.tex, line 422
The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0
Exact mathematical passage
A=[A_1\ A_2\ A_3],\qquad A_i\in\mathbb R^{4\times2}.
A_{i,e^{-1}(k)}Use in source · 1 passage
Search aliases: A_i,e^-1(k)
Use in source · Read passage · B_lean.tex, line 86
The reindexing theorem declgramEntry_reindex handles a bijection e:V≃ V. It moves the block to e(s) and transforms the factor to A'_ik=A_i,e^-1(k). The resulting Gram entry is unchanged. Permuting only the coordinates without moving their factor columns falls outside this statement.
Exact mathematical passage
A'_{ik}=A_{i,e^{-1}(k)}
A_{i1}Use in source · 1 passage
Search aliases: A_i1
Use in source · Read passage · B_lean.tex, line 139
If the two blocks overlap in one column, their sum repeats that column's outer product. For s=t={1} and A_i1=3, the union has contribution 9 while the sum of the two blocks is 18. This is why declgramEntry_union requires disjointness.
Exact mathematical passage
A_{i1}=3
A_{ik}Use in source · 3 passages
Search aliases: A_ik
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
A_JUse in source · 1 passage
Search aliases: A_J
Use in source · Read passage · 02_streaming.tex, line 323
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
A_jUse in source · 1 passage
Search aliases: A_j
Use in source · Read passage · 02_streaming.tex, line 435
For the declared vector v=(1,2,-1,0)^top, the action can be evaluated without forming any block Gram matrix: ∑_j=1^3 A_j(A_j^top v)=Kv=(2,3,-2,-1)^top. The symbolic check confirms both storage partitions. Every column remains represented. Dropping an operator coupling instead changes the operator and requires the error analysis developed next.
Exact mathematical passage
\sum_{j=1}^3 A_j(A_j^\top v)=Kv=(2,3,-2,-1)^\top.
a_jUse in source · 10 passages
Search aliases: a_j
Use in source · Read passage · 03_phases.tex, line 225
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
p(t)=\sum_{j=0}^d a_jt^j.
A_{J'}Use in source · 1 passage
Search aliases: A_J'
Use in source · Read passage · 02_streaming.tex, line 323
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
A_{J'}^\topUse in source · 1 passage
Search aliases: A_J'^top top
Use in source · Read passage · 02_streaming.tex, line 323
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
a_j^{-1/2}Use in source · 1 passage
Search aliases: a_j^-1/2
Use in source · Read passage · 09_intervals.tex, line 130
Since 1+(a_jt)^2≥(a_jt)^2, we have |ϕ_θ(t)| ≤ C_θ t^-r/2, C_θ=∏_j a_j^-1/2.
Exact mathematical passage
|\phi_\theta(t)| \le C_\theta t^{-r/2}, \qquad C_\theta=\prod_j a_j^{-1/2}.
A_J^\topUse in source · 1 passage
Search aliases: A_J^top top
Use in source · Read passage · 02_streaming.tex, line 323
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
A_j^\topUse in source · 1 passage
Search aliases: A_j^top top
Use in source · Read passage · 02_streaming.tex, line 435
For the declared vector v=(1,2,-1,0)^top, the action can be evaluated without forming any block Gram matrix: ∑_j=1^3 A_j(A_j^top v)=Kv=(2,3,-2,-1)^top. The symbolic check confirms both storage partitions. Every column remains represented. Dropping an operator coupling instead changes the operator and requires the error analysis developed next.
Exact mathematical passage
\sum_{j=1}^3 A_j(A_j^\top v)=Kv=(2,3,-2,-1)^\top.
A_{jk}Use in source · 2 passages
Search aliases: A_jk
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
\mathrm{an}Use in source · 4 passages
Search aliases: an mathrm
Use in source · Read passage · 07_inversion.tex, line 290
Let p= P(Q≥ q), and let widehat p be the finite K-COS approximation formed with [label=(roman*)] item an upper interval endpoint b>q, item an integer N≥1 of cosine modes, item a packet approximation satisfying [source reference] on the finite grid. Under the positive-spectrum and parameter hypotheses of Proposition [source reference], |p-widehat p| ≤ ε_an :=ε_pkt+ε_sup+ε_COS. If ε_an<|widehat p|, then frac|p-widehat p||p| ≤ fracε_an |widehat p|-ε_an.
Exact mathematical passage
|p-\widehat p| \le \eps_{\mathrm{an}} :=\eps_{\mathrm{pkt}}+\eps_{\mathrm{sup}}+\eps_{\mathrm{COS}}. \label{eq:combined}
\arctanUse in source · 1 passage
Search aliases: arctan
Use in source · Read passage · 11_evidence.tex, line 121
The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponential tilting, the COS expansion, spectral packets, or the K-COS error controller.
Exact mathematical passage
\Theta(u) = \frac12\sum_j h_j\arctan(\lambda_j u) - \frac12qu, \qquad R(u) = \prod_j \left(1+\lambda_j^2u^2\right)^{h_j/4}.
B^{\mathsf T}Use in source · 1 passage
Search aliases: B^ T mathsf
Use in source · Read passage · A_atlas.tex, line 116
A=UB+F, U^ TF=0, A^ TA=B^ TB+F^ TF. t=trR= F _F^2, v=tr(R^2), ρ_R= R _2. Let U contain d orthonormal sample-space directions. Project the factor into its captured range coordinates B=U^ TA and residual F=(I-UU^ T)A. Orthogonality eliminates the cross terms in the feature-space Gram matrix.
Exact mathematical passage
A=UB+F,\quad U^{\mathsf T}F=0,\quad A^{\mathsf T}A=B^{\mathsf T}B+F^{\mathsf T}F.
\mathcal B^jUse in source · 4 passages
Search aliases: B^j mathcal
Use in source · Read passage · 03_phases.tex, line 229
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
\boxed{ \rho\bigl((p(H)-p(H_0))X\bigr) \leq \sum_{j=1}^d |a_j|(\mathcal B^j-\mathcal B_0^j)\rho(X). }
\mathcal B^kUse in source · 15 passages
Search aliases: B^k mathcal
Use in source · Read passage · 03_phases.tex, line 100
[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).
Exact mathematical passage
\boxed{ \rho(H^kX)\leq \mathcal B^k\rho(X). }
\mathcal B^{k-1-j}Use in source · 4 passages
Search aliases: B^k-1-j mathcal
Use in source · Read passage · 03_phases.tex, line 166
Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
\rho\bigl((H^k-H_0^k)X\bigr) \leq \sum_{j=0}^{k-1} \mathcal B^{k-1-j}\mathcal R\mathcal B_0^j\rho(X).
B^\topUse in source · 8 passages
Search aliases: B^top top
Use in source · Read passage · 04_range.tex, line 142
[Exact range core] The nonzero eigenvalues of K=AA^top are exactly the nonzero eigenvalues of BB^top, with multiplicity. If range(Q_r)=range(A), then BB^top is positive definite on the core space and its entire spectrum is the nonzero spectrum of K.
Exact mathematical passage
BB^\top,
\mathcal B_{\mathcal O}^jUse in source · 2 passages
Search aliases: B_ O^j mathcal mathcal
Use in source · Read passage · 03_phases.tex, line 389
The first inequality in the second display is Theorem~[source reference]. For the transfer matrices, the exact noncommutative telescoping identity is B^k- B_ O^k = ∑_j=0^k-1 B^k-1-j ( B- B_ O) B_ O^j. All matrices and ρ_0 are nonnegative, and 0≤ B_ O≤ B entrywise. Hence B_ O^jρ_0 ≤ B^jρ_0. Substituting B- B_ O = ∑_e∈ OΔ_e and distributing the sum gives the componentwise majorant. The final scalar bound follows from the Euclidean triangle inequality.
Exact mathematical passage
\mathcal B^k-\mathcal B_{\mathcal O}^k = \sum_{j=0}^{k-1} \mathcal B^{k-1-j} (\mathcal B-\mathcal B_{\mathcal O}) \mathcal B_{\mathcal O}^j.
\mathcal B_{\mathcal O}^kUse in source · 5 passages
Search aliases: B_ O^k mathcal mathcal
Use in source · Read passage · 03_phases.tex, line 363
[Edge-additive leakage majorant] For every omitted set O⊆ E, 0≤ ( B^k- B_ O^k)ρ_0 ≤ ∑_e∈ Oq_e^(k) componentwise. Consequently, lVert(H^k-H_ O^k)XrVert_F ≤ lVert ( B^k- B_ O^k)ρ_0 rVert_2 ≤ ∑_e∈ Oβ_e^(k).
Exact mathematical passage
\boxed{ 0\le (\mathcal B^k-\mathcal B_{\mathcal O}^k)\rho_0 \le \sum_{e\in\mathcal O}q_e^{(k)} }
\mathcal B_0Use in source · 4 passages
Search aliases: B_0 mathcal
Use in source · Read passage · 03_phases.tex, line 125
Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0, 0, · otherwise, R= B- B_0. Symmetric pair retention preserves self-adjointness of H_0, although the following error theorem does not require positive semidefiniteness of H_0.
Exact mathematical passage
(\mathcal B_0)_{hg}= \begin{cases} \mathcal B_{hg},&(h,g)\in\mathcal E_0,\\ 0,&\text{otherwise}, \end{cases} \qquad \mathcal R=\mathcal B-\mathcal B_0.
\mathcal B_0^jUse in source · 3 passages
Search aliases: B_0^j mathcal
Use in source · Read passage · 03_phases.tex, line 166
Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
\rho\bigl((H^k-H_0^k)X\bigr) \leq \sum_{j=0}^{k-1} \mathcal B^{k-1-j}\mathcal R\mathcal B_0^j\rho(X).
\mathcal B_0^kUse in source · 8 passages
Search aliases: B_0^k mathcal
Use in source · Read passage · 03_phases.tex, line 140
[Thresholded-power certificate] For every k≥1, ρ((H^k-H_0^k)X) ≤ ( B^k- B_0^k)ρ(X). In particular, lVert(H^k-H_0^k)XrVert_F ≤ lVert( B^k- B_0^k)ρ(X)rVert_2.
Exact mathematical passage
\boxed{ \rho\bigl((H^k-H_0^k)X\bigr) \leq (\mathcal B^k-\mathcal B_0^k)\rho(X). }
B_kUse in source · 3 passages
Search aliases: B_k
Use in source · Read passage · 07_inversion.tex, line 51
Let widehatκ(s_k) be the packet approximation at s_k=θ+i u_k, and suppose |widehatκ(s_k)-κ(s_k)|≤ B_k. For disjoint packets, B_k can be chosen as the sum of the packet bounds [source reference].
Exact mathematical passage
|\widehat\kappa(s_k)-\kappa(s_k)|\le B_k. \label{eq:Bk}
\bar GUse in source · 1 passage
Search aliases: bar G bar
Use in source · Read passage · 11_evidence.tex, line 325
G_b · ⟶ G_b-bar G_b ⟶ widetilde G_b, widetilde G_b · ⟶ M_Xwidetilde G_b ⟶ A_b=M_Xwidetilde G_bW_b^1/2 ⟶ A_bA_b^top,
Exact mathematical passage
\begin{aligned} G_b&\longrightarrow G_b-\bar G_b \longrightarrow \widetilde G_b,\\ \widetilde G_b&\longrightarrow M_X\widetilde G_b \longrightarrow A_b=M_X\widetilde G_bW_b^{1/2} \longrightarrow A_bA_b^\top, \end{aligned}
\bar G_bUse in source · 1 passage
Search aliases: bar G_b bar
Use in source · Read passage · 11_evidence.tex, line 325
G_b · ⟶ G_b-bar G_b ⟶ widetilde G_b, widetilde G_b · ⟶ M_Xwidetilde G_b ⟶ A_b=M_Xwidetilde G_bW_b^1/2 ⟶ A_bA_b^top,
Exact mathematical passage
\begin{aligned} G_b&\longrightarrow G_b-\bar G_b \longrightarrow \widetilde G_b,\\ \widetilde G_b&\longrightarrow M_X\widetilde G_b \longrightarrow A_b=M_X\widetilde G_bW_b^{1/2} \longrightarrow A_bA_b^\top, \end{aligned}
\CUse in source · 1 passage
Search aliases: C mathbb
Use in source · Read passage · 06_packets.tex, line 115
[Centered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P,p(s)| ≤ fracn_P2 fracρ_P(s)^p+1(p+1)(1-ρ_P(s)).
Exact mathematical passage
s\in\C
\mathcal CUse in source · 8 passages
Search aliases: C mathcal
Use in source · Read passage · 05_spectrum.tex, line 30
Let E=H-H_0 be the self-adjoint error introduced by phase pruning. Using the same orthogonal phase projectors P_h, define C_hg = P_hEP_g _2.
Exact mathematical passage
\mathcal C_{hg} = \|P_hEP_g\|_2.
\mathcal C_{\mathcal O}Use in source · 1 passage
Search aliases: C_ O mathcal mathcal
Use in source · Read passage · 05_spectrum.tex, line 172
An execution schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.
Exact mathematical passage
\boxed{ \sum_{e\in\mathcal O}\beta_e^{(k)} \le \tau_K, \qquad \|\mathcal C_{\mathcal O}\|_2 \le \tau_\lambda. }
c_BUse in source · 1 passage
Search aliases: c_B
Use in source · Read passage · 03_phases.tex, line 597
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
\boxed{ T_{\rm pipe} = t_1 + \sum_{b=1}^{B-1}\max\{c_b,t_{b+1}\} + c_B. }
c_bUse in source · 5 passages
Search aliases: c_b
Use in source · Read passage · 03_phases.tex, line 588
The mathematical streaming identities do not require transfers and GEMMs to be serial. Consider B storage blocks. Let t_b be the transfer time for block b and c_b its device compute time.
Exact mathematical passage
c_b
\mathcal C_{hg}Use in source · 1 passage
Search aliases: C_hg mathcal
Use in source · Read passage · 05_spectrum.tex, line 30
Let E=H-H_0 be the self-adjoint error introduced by phase pruning. Using the same orthogonal phase projectors P_h, define C_hg = P_hEP_g _2.
Exact mathematical passage
\mathcal C_{hg} = \|P_hEP_g\|_2.
c_iUse in source · 3 passages
Search aliases: c_i
Use in source · Read passage · 01_gaussian.tex, line 206
Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.
Exact mathematical passage
\lambda_iY_i^2+2c_iY_i = \lambda_i \left( Y_i+\frac{c_i}{\lambda_i} \right)^2 - \frac{c_i^2}{\lambda_i}.
c_i^2Use in source · 2 passages
Search aliases: c_i^2
Use in source · Read passage · 01_gaussian.tex, line 206
Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.
Exact mathematical passage
\lambda_iY_i^2+2c_iY_i = \lambda_i \left( Y_i+\frac{c_i}{\lambda_i} \right)^2 - \frac{c_i^2}{\lambda_i}.
c_kUse in source · 3 passages
Search aliases: c_k
Use in source · Read passage · B_lean.tex, line 146
The separate definition weightedSquares(s,c,z)=∑_k∈ sc_k z_k^2 models a supplied diagonal quadratic form. Its coefficients need not be eigenvalues obtained by a proved diagonalization. It has three theorems: item declweightedSquares_mono: if l_k≤ u_k for every k∈ s, then ∑ l_kz_k^2≤∑ u_kz_k^2. item declweightedSquares_nonneg: if c_k≥0 for every k∈ s, then ∑ c_kz_k^2≥0. item declweightedSquares_tail_imp: coefficientwise order gives the pointwise implication q≤∑ l_kz_k^2⇒ q≤∑ u_kz_k^2. The first a…
Exact mathematical passage
\operatorname{weightedSquares}(s,c,z)=\sum_{k\in s}c_k z_k^2
\mathcal C_{\rm complete}Use in source · 2 passages
Search aliases: C_rm complete mathcal rm
Use in source · Read passage · 05_spectrum.tex, line 606
D . Consequently, the nonzero singular values of A are exactly the nonzero singular values, with multiplicity, of the stacked completion core C_rm complete = B D .
Exact mathematical passage
\mathcal C_{\rm complete} = \begin{bmatrix} B\\D \end{bmatrix}.
\mathcal C_{\rm complete}^\topUse in source · 1 passage
Search aliases: C_rm complete^top mathcal rm top
Use in source · Read passage · 05_spectrum.tex, line 618
The decomposition A=UB+F and the identity F=WD give the factorization. The matrix [U W] has orthonormal columns, so A^top A= C_rm complete^top C_rm complete. The nonzero singular values therefore agree with multiplicity. If the combined basis contains directions outside range(A), its row Gram may include additional zeros. Its entire spectrum is the nonzero spectrum of AA^top only when the combined basis spans exactly range(A).
Exact mathematical passage
A^\top A=\mathcal C_{\rm complete}^\top\mathcal C_{\rm complete}
C_\thetaUse in source · 5 passages
Search aliases: C_θ theta
Use in source · Read passage · 09_intervals.tex, line 130
Since 1+(a_jt)^2≥(a_jt)^2, we have |ϕ_θ(t)| ≤ C_θ t^-r/2, C_θ=∏_j a_j^-1/2.
Exact mathematical passage
|\phi_\theta(t)| \le C_\theta t^{-r/2}, \qquad C_\theta=\prod_j a_j^{-1/2}.
\mathrm{COS}Use in source · 2 passages
Search aliases: COS mathrm
Use in source · Read passage · 07_inversion.tex, line 240
[Omitted COS-mode certificate] Let N≥1 and b>q≥0. Set u_N=Nπ/b and β_N=β(u_N). If β_N>0, then the contribution of all exact tilted-COS modes k≥ N to the weighted tail functional is bounded by ε_COS ≤ e^κ(θ)-θ q frac4|ϕ_θ(u_N)|π (frac1N+frac1β_N).
Exact mathematical passage
\eps_{\mathrm{COS}} \le e^{\kappa(\theta)-\theta q} \frac{4|\phi_{\theta}(u_N)|}{\pi} \left(\frac1N+\frac1{\beta_N}\right). \label{eq:eps-cos}
\operatorname{cum}Use in source · 1 passage
Search aliases: cum operatorname
Use in source · Read passage · 06_packets.tex, line 49
ts logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose argument can wrap by 2π. For real θ<1/(2λ_max), κ^(m)(θ) =2^m-1(m-1)!∑_j=1^r fracλ_j^m(1-2θλ_j)^m, m≥1. At θ=0, this recovers the standard cumulant identity cum_m(Q)=2^m-1(m-1)!∑_jλ_j^m.
Exact mathematical passage
\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\sum_j\lambda_j^m.
\operatorname{cum}_mUse in source · 1 passage
Search aliases: cum_m operatorname
Use in source · Read passage · 06_packets.tex, line 49
ts logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose argument can wrap by 2π. For real θ<1/(2λ_max), κ^(m)(θ) =2^m-1(m-1)!∑_j=1^r fracλ_j^m(1-2θλ_j)^m, m≥1. At θ=0, this recovers the standard cumulant identity cum_m(Q)=2^m-1(m-1)!∑_jλ_j^m.
Exact mathematical passage
\operatorname{cum}_m(Q)=2^{m-1}(m-1)!\sum_j\lambda_j^m.
DUse in source · 14 passages
Search aliases: D
Use in source · Read passage · 03_phases.tex, line 416
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
p(t)=\sum_{d=0}^{D}a_dt^d.
\mathcal DUse in source · 1 passage
Search aliases: D mathcal
Use in source · Read passage · 12_synthesis.tex, line 11
The solver is organized around four independent decomposition axes: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. This display describes the pipeline schematically, not four endomorphisms proved to commute pairwise. The precise commutation theorem below is for exact storage refinement and a fixed structural phase partition. Full range capture or residual completion preserves the nonzero spectrum exactly. A truncated packet representation instead preserves the requested tail within i…
Exact mathematical passage
\boxed{ \mathcal D = \mathcal D_{\rm storage} \circ \mathcal D_{\rm phase} \circ \mathcal D_{\rm rank} \circ \mathcal D_{\rm spectrum}. }
D^{1/2}Use in source · 3 passages
Search aliases: D^1/2
Use in source · Read passage · 05_spectrum.tex, line 738
Set S=Q^TQ. The first hypothesis gives the Loewner inequalities (1-η)Ipreceq Spreceq(1+η)I. The eigenvalues of B=(QD^1/2)(QD^1/2)^T are the same as those of D^1/2SD^1/2. Hence (1-η)D preceq D^1/2SD^1/2 preceq (1+η)D, and eigenvalue monotonicity yields (1-η)d_i ≤ λ_i(B) ≤ (1+η)d_i. Weyl's perturbation inequality and H-B _2≤ρ then give |λ_i(H)-λ_i(B)|≤ρ. Combining the two bounds and using Hsucceq0 proves the claim.
Exact mathematical passage
B=(QD^{1/2})(QD^{1/2})^T
d_1Use in source · 1 passage
Search aliases: d_1
Use in source · Read passage · 05_spectrum.tex, line 691
A high-precision symmetric eigensolver is used only to generate candidate quantities Q∈ R^n× n, D=diag(d_1,…,d_n), d_1≥⋯≥ d_n≥0. No correctness claim depends on trusting this eigensolver.
Exact mathematical passage
Q\in\mathbb R^{n\times n}, \qquad D=\operatorname{diag}(d_1,\ldots,d_n), \qquad d_1\ge\cdots\ge d_n\ge0.
d_iUse in source · 2 passages
Search aliases: d_i
Use in source · Read passage · 05_spectrum.tex, line 716
[Candidate eigensystem post-certification] Let B=QDQ^T. Suppose directed-rounding computation proves Q^TQ-I _2≤η<1 and H-B _2≤ρ. Then the ordered eigenvalues of the positive semidefinite matrix H satisfy max{0,(1-η)d_i-ρ} ≤ λ_i(H) ≤ (1+η)d_i+ρ for every i.
Exact mathematical passage
\boxed{ \max\{0,(1-\eta)d_i-\rho\} \le \lambda_i(H) \le (1+\eta)d_i+\rho }
d_nUse in source · 1 passage
Search aliases: d_n
Use in source · Read passage · 05_spectrum.tex, line 691
A high-precision symmetric eigensolver is used only to generate candidate quantities Q∈ R^n× n, D=diag(d_1,…,d_n), d_1≥⋯≥ d_n≥0. No correctness claim depends on trusting this eigensolver.
Exact mathematical passage
Q\in\mathbb R^{n\times n}, \qquad D=\operatorname{diag}(d_1,\ldots,d_n), \qquad d_1\ge\cdots\ge d_n\ge0.
\mathcal D_{\rm phase}Use in source · 1 passage
Search aliases: D_rm phase mathcal rm
Use in source · Read passage · 12_synthesis.tex, line 11
The solver is organized around four independent decomposition axes: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. This display describes the pipeline schematically, not four endomorphisms proved to commute pairwise. The precise commutation theorem below is for exact storage refinement and a fixed structural phase partition. Full range capture or residual completion preserves the nonzero spectrum exactly. A truncated packet representation instead preserves the requested tail within i…
Exact mathematical passage
\boxed{ \mathcal D = \mathcal D_{\rm storage} \circ \mathcal D_{\rm phase} \circ \mathcal D_{\rm rank} \circ \mathcal D_{\rm spectrum}. }
\mathcal D_{\rm rank}Use in source · 1 passage
Search aliases: D_rm rank mathcal rm
Use in source · Read passage · 12_synthesis.tex, line 11
The solver is organized around four independent decomposition axes: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. This display describes the pipeline schematically, not four endomorphisms proved to commute pairwise. The precise commutation theorem below is for exact storage refinement and a fixed structural phase partition. Full range capture or residual completion preserves the nonzero spectrum exactly. A truncated packet representation instead preserves the requested tail within i…
Exact mathematical passage
\boxed{ \mathcal D = \mathcal D_{\rm storage} \circ \mathcal D_{\rm phase} \circ \mathcal D_{\rm rank} \circ \mathcal D_{\rm spectrum}. }
\mathcal D_{\rm spectrum}Use in source · 1 passage
Search aliases: D_rm spectrum mathcal rm
Use in source · Read passage · 12_synthesis.tex, line 11
The solver is organized around four independent decomposition axes: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. This display describes the pipeline schematically, not four endomorphisms proved to commute pairwise. The precise commutation theorem below is for exact storage refinement and a fixed structural phase partition. Full range capture or residual completion preserves the nonzero spectrum exactly. A truncated packet representation instead preserves the requested tail within i…
Exact mathematical passage
\boxed{ \mathcal D = \mathcal D_{\rm storage} \circ \mathcal D_{\rm phase} \circ \mathcal D_{\rm rank} \circ \mathcal D_{\rm spectrum}. }
\mathcal D_{\rm storage}Use in source · 1 passage
Search aliases: D_rm storage mathcal rm
Use in source · Read passage · 12_synthesis.tex, line 11
The solver is organized around four independent decomposition axes: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. This display describes the pipeline schematically, not four endomorphisms proved to commute pairwise. The precise commutation theorem below is for exact storage refinement and a fixed structural phase partition. Full range capture or residual completion preserves the nonzero spectrum exactly. A truncated packet representation instead preserves the requested tail within i…
Exact mathematical passage
\boxed{ \mathcal D = \mathcal D_{\rm storage} \circ \mathcal D_{\rm phase} \circ \mathcal D_{\rm rank} \circ \mathcal D_{\rm spectrum}. }
\detUse in source · 1 passage
Search aliases: det
Use in source · Read passage · 05_spectrum.tex, line 388
[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, κ_H(θ) =-frac12logdet(I-2θ H) satisfies κ_H(θ) ≤ κ_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.
Exact mathematical passage
\kappa_H(\theta) =-\frac12\log\det(I-2\theta H)
\operatorname{diag}Use in source · 3 passages
Search aliases: diag operatorname
Use in source · Read passage · 01_gaussian.tex, line 353
1 · -1 · -1 · 1 satisfies V^top K V=diag(1,1,2,2). For Z∼ N(0,I_4), Q=Z^top KZ overset d= Y+2W, Y,W independent with law χ^2_2. The spectrum is therefore (1,1,2,2) and E Q=6. The six-dimensional feature Gram A^top A has these same four nonzero eigenvalues and two additional zeros.
Exact mathematical passage
V^\top K V=\operatorname{diag}(1,1,2,2)
\operatorname{Disjoint}Use in source · 1 passage
Search aliases: Disjoint operatorname
Use in source · Read passage · B_lean.tex, line 55
The principal refinement theorem is declgramEntry_union. Its hypotheses are finite sets s,t⊆ V and Disjoint(s,t). It proves, for all sample coordinates i,j, gramEntry(s∪ t,A,i,j) =gramEntry(s,A,i,j)+gramEntry(t,A,i,j). Disjointness prevents repeated columns from being counted twice. It is a mathematical condition, not a performance assumption. The theorem applies to empty blocks and zero factors. It does not require positive weights, since A is already supplied as a real factor.
Exact mathematical passage
\operatorname{Disjoint}(s,t)
\operatorname{dist}Use in source · 1 passage
Search aliases: dist operatorname
Use in source · Read passage · A_atlas.tex, line 85
∑_hP_h=I, g→ hiff P_hHP_g≠0. dist(g,h)>k Longrightarrow P_hH^kP_g=0. Choose a complete orthogonal family of phase projectors on feature space and form H=A^ TA. An edge from phase g to phase h exists when the transfer block P_hHP_g is nonzero. Each operator application follows one graph edge. An iterate of depth k cannot reach a phase farther than k edges away.
Exact mathematical passage
\operatorname{dist}(g,h)>k\quad\Longrightarrow\quad P_hH^kP_g=0.
\downarrowUse in source · 1 passage
Search aliases: downarrow
Use in source · Read passage · 05_spectrum.tex, line 646
Residual information supplies a stopping rule for approximate iteration and a handoff to exact completion: c streamed / pruned Krylov [2mm] downarrow [-1mm] residual moments (t,v,ρ) [2mm] downarrow [-1mm] continue Krylov, · residual range still large, exact streamed residual completion, · completion rank affordable.
Exact mathematical passage
\boxed{ \begin{array}{c} \text{streamed / pruned Krylov}\\[2mm] \downarrow\\[-1mm] \text{residual moments }(t,v,\rho)\\[2mm] \downarrow\\[-1mm] \begin{cases} \text{continue Krylov}, & \text{residual range still large},\\ \text{exact streamed residual completion}, & \text{completion rank affordable}. \end{cases} \end{array} }
EUse in source · 20 passages
Search aliases: E
Use in source · Read passage · 01_gaussian.tex, line 136
For the right-tail path, the cumulant generating function is κ(s) = -frac12∑_jlog(1-2sλ_j), with real domain max_λ_j<0frac12λ_j < s < min_λ_j>0frac12λ_j. Here a missing negative or positive spectral set gives a domain endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain whenever a positive spectral component is present. The negative-only case with at least one nonzero coefficient has a positive saddlepoint only for E[Q…
Exact mathematical passage
q>E[Q]=\sum_j\lambda_j
e^{(-\theta+iu_k)(b-q)}Use in source · 1 passage
Search aliases: e^(-θ+iu_k)(b-q) theta
Use in source · Read passage · A_atlas.tex, line 197
J_k=∫_q^be^-θ(x-q)cos(u_kx) dx=Re [e^iu_kqfrac1-e^(-θ+iu_k)(b-q)θ-iu_k]. S_N=frac2b∑_k=0^N-1^prime J_k Re ϕ_θ(u_k), ϕ_θ(t)=e^κ(θ+it)-κ(θ). On a positive-support interval [0,b], the cosine frequencies are u_k=kπ/b. The payoff is zero below the threshold and decays exponentially above it. Integrating each cosine mode against that payoff gives an explicit coefficient.
Exact mathematical passage
J_k=\int_q^be^{-\theta(x-q)}\cos(u_kx)\,dx=\Re\!\left[e^{iu_kq}\frac{1-e^{(-\theta+iu_k)(b-q)}}{\theta-iu_k}\right].
e^{-(x-y)/4}Use in source · 1 passage
Search aliases: e^-(x-y)/4
Use in source · Read passage · 01_gaussian.tex, line 361
Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×10^-5. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor A and as…
Exact mathematical passage
f_Q(x)=\int_0^x\frac12e^{-y/2}\frac14e^{-(x-y)/4}\,dy =\frac12\bigl(e^{-x/4}-e^{-x/2}\bigr).
e^{-1}Use in source · 1 passage
Search aliases: e^-1
Use in source · Read passage · B_lean.tex, line 86
The reindexing theorem declgramEntry_reindex handles a bijection e:V≃ V. It moves the block to e(s) and transforms the factor to A'_ik=A_i,e^-1(k). The resulting Gram entry is unchanged. Permuting only the coordinates without moving their factor columns falls outside this statement.
Exact mathematical passage
A'_{ik}=A_{i,e^{-1}(k)}
e^{-10}Use in source · 1 passage
Search aliases: e^-10
Use in source · Read passage · worked_enclosure_result.tex, line 5
approximately 0.0246131, which fails the requested tolerance 0.02. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below 0.013731. The displayed, coarser interval also meets the requested tolerance 0.02. An independently evaluated interval for the exact expression 2e^-10-e^-20 lies entirely inside the COS enclosure.
Exact mathematical passage
2e^{-10}-e^{-20}
e^{-20}Use in source · 1 passage
Search aliases: e^-20
Use in source · Read passage · worked_enclosure_result.tex, line 5
approximately 0.0246131, which fails the requested tolerance 0.02. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below 0.013731. The displayed, coarser interval also meets the requested tolerance 0.02. An independently evaluated interval for the exact expression 2e^-10-e^-20 lies entirely inside the COS enclosure.
Exact mathematical passage
2e^{-10}-e^{-20}
e^{-q/0.14}Use in source · 1 passage
Search aliases: e^-q/0.14
Use in source · Read passage · 11_evidence.tex, line 234
Retained low-rank limitation. For independent Y_1,Y_2∼χ^2_2, the quadratic form Q=0.07(Y_1-Y_2) is Laplace distributed with scale 0.14, so for q≥0, P(Q≥ q)=frac12e^-q/0.14. The computed values agree with this exact law through 10^-200 to roughly 10^-9 decades or better. However, the implemented analytic COS-series certificate rejects these low-rank mixed cases because the tilted characteristic function decays only as |t|^-2. The implementation reports this as a certificate failure rather than si…
Exact mathematical passage
P(Q\ge q)=\frac12e^{-q/0.14}.
e^{-q/2}Use in source · 1 passage
Search aliases: e^-q/2
Use in source · Read passage · 01_gaussian.tex, line 366
Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×10^-5. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor A and as…
Exact mathematical passage
P(Q\ge q)=2e^{-q/4}-e^{-q/2},\qquad q\ge0. \label{eq:worked-exact-tail}
e^{-q/4}Use in source · 1 passage
Search aliases: e^-q/4
Use in source · Read passage · 01_gaussian.tex, line 366
Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×10^-5. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor A and as…
Exact mathematical passage
P(Q\ge q)=2e^{-q/4}-e^{-q/2},\qquad q\ge0. \label{eq:worked-exact-tail}
e^{-x}Use in source · 1 passage
Search aliases: e^-x
Use in source · Read passage · 09_intervals.tex, line 225
For the control Q=0.02χ^2_80, the survival probability has the finite exact formula P(Q≥ q) = e^-x ∑_j=0^39 fracx^jj!, x=fracq0.04.
Exact mathematical passage
P(Q\ge q) = e^{-x} \sum_{j=0}^{39} \frac{x^j}{j!}, \qquad x=\frac{q}{0.04}.
e^{-x/2}Use in source · 2 passages
Search aliases: e^-x/2
Use in source · Read passage · 01_gaussian.tex, line 360
Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×10^-5. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The n
Exact mathematical passage
\tfrac12e^{-x/2}
e^{-x/4}Use in source · 2 passages
Search aliases: e^-x/4
Use in source · Read passage · 01_gaussian.tex, line 360
Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×10^-5. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with th
Exact mathematical passage
\tfrac14e^{-x/4}
e^{-y/2}Use in source · 1 passage
Search aliases: e^-y/2
Use in source · Read passage · 01_gaussian.tex, line 361
Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×10^-5. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor A and as…
Exact mathematical passage
f_Q(x)=\int_0^x\frac12e^{-y/2}\frac14e^{-(x-y)/4}\,dy =\frac12\bigl(e^{-x/4}-e^{-x/2}\bigr).
e^{-z}Use in source · 1 passage
Search aliases: e^-z
Use in source · Read passage · A_atlas.tex, line 261
z=q/0.04, P(Q≥ q)=e^-z∑_k=0^39fracz^kk!. log_10P(Q≥ q)=frac-z+log (∑_k=0^39z^k/k!)log(10). For an even-degree chi-square variable, the upper tail has a finite positive-sum formula. The control Q=0.02χ^2_80 is a gamma law of integer shape forty and scale 0.04. Its survival probability can therefore be evaluated independently of cosine inversion.
Exact mathematical passage
z=q/0.04,\qquad P(Q\ge q)=e^{-z}\sum_{k=0}^{39}\frac{z^k}{k!}.
e^{-\theta(b-q)}Use in source · 5 passages
Search aliases: e^-θ(b-q) theta
Use in source · Read passage · 07_inversion.tex, line 141
nd] Let the finite positive spectrum have rank r≥1, and let b>q≥0, b>0, and 0<θ<1/(2λ_max). Let τ_θ(b)= P_θ(Q>b). Then the combined error from (i) replacing the tilted density on [0,b] by its full-coefficient cosine periodization and (ii) omitting the original integral over (b,∞) is bounded by ε_sup ≤ e^κ(θ)-θ q τ_θ(b) (1+e^-θ(b-q)).
Exact mathematical passage
\eps_{\mathrm{sup}} \le e^{\kappa(\theta)-\theta q}\, \tau_{\theta}(b) \left(1+e^{-\theta(b-q)}\right). \label{eq:eps-support}
e^{-\theta(Q-q)}Use in source · 3 passages
Search aliases: e^-θ(Q-q) theta
Use in source · Read passage · 06_packets.tex, line 88
For any θ>0 in the transform domain, P(Q≥ q) =e^κ(θ)-θ q E_θ [e^-θ(Q-q) 1_{Q≥ q}]. This classical exponential-tilting identity lets K-COS evaluate the upper tail without subtracting a cumulative distribution function (CDF) value close to one.
Exact mathematical passage
\Pp(Q\ge q) =e^{\kappa(\theta)-\theta q} \E_{\theta}\!\left[e^{-\theta(Q-q)}\mathbf 1_{\{Q\ge q\}}\right]. \label{eq:tilt-tail}
e^{B_k}Use in source · 1 passage
Search aliases: e^B_k
Use in source · Read passage · 07_inversion.tex, line 66
[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).
Exact mathematical passage
\eps_{\mathrm{pkt}} = \frac{2}{b} e^{\kappa(\theta)-\theta q} \sum_{k=0}^{N-1}{}' |J_k|\bigl(e^{B_k}-1\bigr). \label{eq:eps-pkt}
e^{iu_kq}Use in source · 1 passage
Search aliases: e^iu_kq
Use in source · Read passage · A_atlas.tex, line 197
J_k=∫_q^be^-θ(x-q)cos(u_kx) dx=Re [e^iu_kqfrac1-e^(-θ+iu_k)(b-q)θ-iu_k]. S_N=frac2b∑_k=0^N-1^prime J_k Re ϕ_θ(u_k), ϕ_θ(t)=e^κ(θ+it)-κ(θ). On a positive-support interval [0,b], the cosine frequencies are u_k=kπ/b. The payoff is zero below the threshold and decays exponentially above it. Integrating each cosine mode against that payoff gives an explicit coefficient.
Exact mathematical passage
J_k=\int_q^be^{-\theta(x-q)}\cos(u_kx)\,dx=\Re\!\left[e^{iu_kq}\frac{1-e^{(-\theta+iu_k)(b-q)}}{\theta-iu_k}\right].
e^\topUse in source · 1 passage
Search aliases: e^top top
Use in source · Read passage · 11_evidence.tex, line 371
item directly through the factor, Q=frac12 A^top e _2^2. item through the sample kernel, Q=frac12 e^top Ke. item through the eigencoordinates of K.
Exact mathematical passage
Q=\frac12 e^\top Ke.
e^{\widehat\kappa(s_k)-\theta q}Use in source · 1 passage
Search aliases: e^widehatκ(s_k)-θ q widehat kappa theta
Use in source · Read passage · 07_inversion.tex, line 79
Write Δ_k=widehatκ(s_k)-κ(s_k). Then |e^widehatκ(s_k)-θ q-e^κ(s_k)-θ q| ≤ |e^κ(s_k)-θ q| (e^|Δ_k|-1). Moreover Reκ(θ+i u_k)-κ(θ) =log|ϕ_θ(u_k)|≤0, so |e^κ(s_k)-θ q| ≤ e^κ(θ)-θ q. Substitute this into the finite cosine sum and apply the triangle inequality.
Exact mathematical passage
|e^{\widehat\kappa(s_k)-\theta q}-e^{\kappa(s_k)-\theta q}| \le |e^{\kappa(s_k)-\theta q}|\,(e^{|\Delta_k|}-1).
e^{|\Delta_k|}Use in source · 1 passage
Search aliases: e^|Δ_k| Delta
Use in source · Read passage · 07_inversion.tex, line 79
Write Δ_k=widehatκ(s_k)-κ(s_k). Then |e^widehatκ(s_k)-θ q-e^κ(s_k)-θ q| ≤ |e^κ(s_k)-θ q| (e^|Δ_k|-1). Moreover Reκ(θ+i u_k)-κ(θ) =log|ϕ_θ(u_k)|≤0, so |e^κ(s_k)-θ q| ≤ e^κ(θ)-θ q. Substitute this into the finite cosine sum and apply the triangle inequality.
Exact mathematical passage
|e^{\widehat\kappa(s_k)-\theta q}-e^{\kappa(s_k)-\theta q}| \le |e^{\kappa(s_k)-\theta q}|\,(e^{|\Delta_k|}-1).
e^{\varepsilon_k}Use in source · 2 passages
Search aliases: e^ε_k varepsilon
Use in source · Read passage · 08_adaptive.tex, line 69
After separating the likelihood-ratio prefactor, the packet contribution contains sums of the form frac2b ∑_k |J_k| ( e^ε_k-1 ). Direct evaluation can overflow even when the candidate is destined to be rejected by the adaptive planner. The implementation instead computes log( frac2b ∑_k |J_k| ( e^ε_k-1 ) ) by log-sum-exp arithmetic, and compares this quantity with the logarithm of the requested tolerance before converting a final accepted bound back to float64.
Exact mathematical passage
\frac{2}{b} \sum_k |J_k| \left( e^{\varepsilon_k}-1 \right).
e^{\theta x-\kappa(\theta)}Use in source · 1 passage
Search aliases: e^θ x-κ(θ) theta kappa theta
Use in source · Read passage · 06_packets.tex, line 75
Define the exponentially tilted law by f_θ(x)=e^θ x-κ(θ)f(x). Its characteristic function is ϕ_θ(t) =exp{κ(θ+i t)-κ(θ)}, and E_θQ=κ'(θ). Therefore the choice [source reference] moves the query q to the mean of the tilted law.
Exact mathematical passage
f_{\theta}(x)=e^{\theta x-\kappa(\theta)}f(x). \label{eq:tilt-density}
e^{\kappa(s_k)-\theta q}Use in source · 2 passages
Search aliases: e^κ(s_k)-θ q kappa theta
Use in source · Read passage · 07_inversion.tex, line 79
Write Δ_k=widehatκ(s_k)-κ(s_k). Then |e^widehatκ(s_k)-θ q-e^κ(s_k)-θ q| ≤ |e^κ(s_k)-θ q| (e^|Δ_k|-1). Moreover Reκ(θ+i u_k)-κ(θ) =log|ϕ_θ(u_k)|≤0, so |e^κ(s_k)-θ q| ≤ e^κ(θ)-θ q. Substitute this into the finite cosine sum and apply the triangle inequality.
Exact mathematical passage
|e^{\widehat\kappa(s_k)-\theta q}-e^{\kappa(s_k)-\theta q}| \le |e^{\kappa(s_k)-\theta q}|\,(e^{|\Delta_k|}-1).
e^{\kappa(\theta)-\theta q}Use in source · 8 passages
Search aliases: e^κ(θ)-θ q kappa theta theta
Use in source · Read passage · 06_packets.tex, line 88
For any θ>0 in the transform domain, P(Q≥ q) =e^κ(θ)-θ q E_θ [e^-θ(Q-q) 1_{Q≥ q}]. This classical exponential-tilting identity lets K-COS evaluate the upper tail without subtracting a cumulative distribution function (CDF) value close to one.
Exact mathematical passage
\Pp(Q\ge q) =e^{\kappa(\theta)-\theta q} \E_{\theta}\!\left[e^{-\theta(Q-q)}\mathbf 1_{\{Q\ge q\}}\right]. \label{eq:tilt-tail}
e^{\kappa(\theta+\ii u_k)-\theta q}Use in source · 1 passage
Search aliases: e^κ(θ+ii u_k)-θ q e^κ(θ+i u_k)-θ q kappa theta ii theta kappa theta mathrm theta
Use in source · Read passage · 07_inversion.tex, line 35
If the exact tilted characteristic function is used, the N-mode finite-window expression is T_N,b =frac2b∑_k=0^N-1' J_k Re[e^κ(θ+i u_k)-θ q]. The cancellation of κ(θ) in [source reference] is useful numerically: the likelihood-ratio prefactor from [source reference] and the normalization in [source reference] do not need to be formed separately inside each mode.
Exact mathematical passage
T_{N,b} =\frac{2}{b}\sum_{k=0}^{N-1}{}' J_k\, \Re\left[e^{\kappa(\theta+\ii u_k)-\theta q}\right]. \label{eq:tail-cos}
e^{\kappa(\theta+it)-\kappa(\theta)}Use in source · 1 passage
Search aliases: e^κ(θ+it)-κ(θ) kappa theta kappa theta
Use in source · Read passage · A_atlas.tex, line 198
J_k=∫_q^be^-θ(x-q)cos(u_kx) dx=Re [e^iu_kqfrac1-e^(-θ+iu_k)(b-q)θ-iu_k]. S_N=frac2b∑_k=0^N-1^prime J_k Re ϕ_θ(u_k), ϕ_θ(t)=e^κ(θ+it)-κ(θ). On a positive-support interval [0,b], the cosine frequencies are u_k=kπ/b. The payoff is zero below the threshold and decays exponentially above it. Integrating each cosine mode against that payoff gives an explicit coefficient.
Exact mathematical passage
S_N=\frac2b\sum_{k=0}^{N-1}{}^{\prime}\, J_k\,\Re\,\phi_\theta(u_k),\qquad\phi_\theta(t)=e^{\kappa(\theta+it)-\kappa(\theta)}.
\mathcal E_0Use in source · 2 passages
Search aliases: E_0 mathcal
Use in source · Read passage · 03_phases.tex, line 121
Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,
Exact mathematical passage
H_0=\sum_{(h,g)\in\mathcal E_0}P_hHP_g.
E_{\rm modes}Use in source · 1 passage
Search aliases: E_rm modes rm
Use in source · Read passage · A_atlas.tex, line 229
|p-widehat p|≤ E_rm packet+E_rm support+E_rm modes (analytic packet path). S_θ(q)∈ S_N+[-E,E], E=E_rm support+E_rm series (exact-spectrum interval path). Packet approximation changes the retained-grid transform. Support truncation and cosine periodization change the finite-window functional. Mode truncation removes infinitely many terms. These three mathematical approximation errors have different refinement controls and must be accounted for separately.
Exact mathematical passage
|p-\widehat p|\le E_{\rm packet}+E_{\rm support}+E_{\rm modes}\quad\text{(analytic packet path)}.
E_{\rm packet}Use in source · 1 passage
Search aliases: E_rm packet rm
Use in source · Read passage · A_atlas.tex, line 229
|p-widehat p|≤ E_rm packet+E_rm support+E_rm modes (analytic packet path). S_θ(q)∈ S_N+[-E,E], E=E_rm support+E_rm series (exact-spectrum interval path). Packet approximation changes the retained-grid transform. Support truncation and cosine periodization change the finite-window functional. Mode truncation removes infinitely many terms. These three mathematical approximation errors have different refinement controls and must be accounted for separately.
Exact mathematical passage
|p-\widehat p|\le E_{\rm packet}+E_{\rm support}+E_{\rm modes}\quad\text{(analytic packet path)}.
E_{\rm series}Use in source · 4 passages
Search aliases: E_rm series rm
Use in source · Read passage · 09_intervals.tex, line 156
Let ν=frac r2>1, N≥1. For a finite transform retaining modes 0,…,N-1, E_rm series ≤ frac2C_θbθ (fracbπ)^ν [ N^-ν + fracN^1-νν-1 ].
Exact mathematical passage
\boxed{ E_{\rm series} \le \frac{2C_\theta}{b\theta} \left(\frac{b}{\pi}\right)^\nu \left[ N^{-\nu} + \frac{N^{1-\nu}}{\nu-1} \right]. }
E_{\rm support}Use in source · 4 passages
Search aliases: E_rm support rm
Use in source · Read passage · 09_intervals.tex, line 97
Therefore E_rm support ≤ P_θ(Q>b) ( 1+e^-θ(b-q) ).
Exact mathematical passage
\boxed{ E_{\rm support} \le P_\theta(Q>b) \left( 1+e^{-\theta(b-q)} \right). }
E_\thetaUse in source · 1 passage
Search aliases: E_θ theta
Use in source · Read passage · 09_intervals.tex, line 53
Choose any 0<θ<frac12λ_max. Exponential tilting gives P(Q≥ q) = e^κ(θ)-θ q S_θ(q), where S_θ(q) = E_θ [ e^-θ(Q-q) 1_{Q≥ q} ].
Exact mathematical passage
S_\theta(q) = E_\theta\!\left[ e^{-\theta(Q-q)} 1_{\{Q\ge q\}} \right].
\E_{\theta}Use in source · 2 passages
Search aliases: E_θ E theta operatorname mathbb theta
Use in source · Read passage · 06_packets.tex, line 85
Define the exponentially tilted law by f_θ(x)=e^θ x-κ(θ)f(x). Its characteristic function is ϕ_θ(t) =exp{κ(θ+i t)-κ(θ)}, and E_θQ=κ'(θ). Therefore the choice [source reference] moves the query q to the mean of the tilted law.
Exact mathematical passage
\E_{\theta}Q=\kappa'(\theta)
\mathbb E_\thetaUse in source · 1 passage
Search aliases: E_θ mathbb theta
Use in source · Read passage · A_atlas.tex, line 149
κ(s)=-frac12∑_jlog(1-2sλ_j), 0<θ<frac12λ_max. P(Q≥ q)=e^κ(θ)-θ q E_θ[e^-θ(Q-q)1_{Q≥ q}]. For a positive spectrum, the cumulant generating function is a sum of logarithms. A real tilt inside its domain defines a normalized reweighted law. The original upper tail becomes an exponential prefactor times a bounded weighted expectation under that law.
Exact mathematical passage
P(Q\ge q)=e^{\kappa(\theta)-\theta q}\,\mathbb E_\theta[e^{-\theta(Q-q)}\mathbf1_{\{Q\ge q\}}].
\operatorname{Enclosed}Use in source · 1 passage
Search aliases: Enclosed operatorname
Use in source · Read passage · B_lean.tex, line 172
The interval module uses two elementary predicates: Enclosed(x,L,U) · :Longleftrightarrow L≤ x≤ U, ErrorBound(x,widehat x,r) · :Longleftrightarrow |x-widehat x|≤ r. The equivalence declerrorBound_iff_enclosed proves |x-widehat x|≤ r Longleftrightarrow widehat x-r≤ x≤widehat x+r. There is no extra nonnegativity hypothesis: if this predicate holds, then the radius is necessarily nonnegative. The definition accepts arbitrary real inputs, and the theorem states the exact logical equivalence.
Exact mathematical passage
\operatorname{Enclosed}(x,L,U)&\;:\Longleftrightarrow\;L\le x\le U,\\ \operatorname{ErrorBound}(x,\widehat x,r)&\;:\Longleftrightarrow\; |x-\widehat x|\le r.
\epsUse in source · 13 passages
Search aliases: eps ε varepsilon
Use in source · Read passage · 07_inversion.tex, line 60
[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).
Exact mathematical passage
|\widehat T_{N,b}-T_{N,b}| \le \eps_{\mathrm{pkt}}, \label{eq:packet-tail-bound}
\eps_{\mathrm{an}}Use in source · 4 passages
Search aliases: eps_an ε_an eps mathrm varepsilon mathrm
Use in source · Read passage · 07_inversion.tex, line 290
Let p= P(Q≥ q), and let widehat p be the finite K-COS approximation formed with [label=(roman*)] item an upper interval endpoint b>q, item an integer N≥1 of cosine modes, item a packet approximation satisfying [source reference] on the finite grid. Under the positive-spectrum and parameter hypotheses of Proposition [source reference], |p-widehat p| ≤ ε_an :=ε_pkt+ε_sup+ε_COS. If ε_an<|widehat p|, then frac|p-widehat p||p| ≤ fracε_an |widehat p|-ε_an.
Exact mathematical passage
|p-\widehat p| \le \eps_{\mathrm{an}} :=\eps_{\mathrm{pkt}}+\eps_{\mathrm{sup}}+\eps_{\mathrm{COS}}. \label{eq:combined}
\eps_{\mathrm{COS}}Use in source · 2 passages
Search aliases: eps_COS ε_COS eps mathrm varepsilon mathrm
Use in source · Read passage · 07_inversion.tex, line 240
[Omitted COS-mode certificate] Let N≥1 and b>q≥0. Set u_N=Nπ/b and β_N=β(u_N). If β_N>0, then the contribution of all exact tilted-COS modes k≥ N to the weighted tail functional is bounded by ε_COS ≤ e^κ(θ)-θ q frac4|ϕ_θ(u_N)|π (frac1N+frac1β_N).
Exact mathematical passage
\eps_{\mathrm{COS}} \le e^{\kappa(\theta)-\theta q} \frac{4|\phi_{\theta}(u_N)|}{\pi} \left(\frac1N+\frac1{\beta_N}\right). \label{eq:eps-cos}
\eps_{\mathrm{pkt}}Use in source · 4 passages
Search aliases: eps_pkt ε_pkt eps mathrm varepsilon mathrm
Use in source · Read passage · 07_inversion.tex, line 60
[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).
Exact mathematical passage
|\widehat T_{N,b}-T_{N,b}| \le \eps_{\mathrm{pkt}}, \label{eq:packet-tail-bound}
\eps_{\mathrm{sup}}Use in source · 2 passages
Search aliases: eps_sup ε_sup eps mathrm varepsilon mathrm
Use in source · Read passage · 07_inversion.tex, line 141
nd] Let the finite positive spectrum have rank r≥1, and let b>q≥0, b>0, and 0<θ<1/(2λ_max). Let τ_θ(b)= P_θ(Q>b). Then the combined error from (i) replacing the tilted density on [0,b] by its full-coefficient cosine periodization and (ii) omitting the original integral over (b,∞) is bounded by ε_sup ≤ e^κ(θ)-θ q τ_θ(b) (1+e^-θ(b-q)).
Exact mathematical passage
\eps_{\mathrm{sup}} \le e^{\kappa(\theta)-\theta q}\, \tau_{\theta}(b) \left(1+e^{-\theta(b-q)}\right). \label{eq:eps-support}
\operatorname{ErrorBound}Use in source · 1 passage
Search aliases: ErrorBound operatorname
Use in source · Read passage · B_lean.tex, line 172
The interval module uses two elementary predicates: Enclosed(x,L,U) · :Longleftrightarrow L≤ x≤ U, ErrorBound(x,widehat x,r) · :Longleftrightarrow |x-widehat x|≤ r. The equivalence declerrorBound_iff_enclosed proves |x-widehat x|≤ r Longleftrightarrow widehat x-r≤ x≤widehat x+r. There is no extra nonnegativity hypothesis: if this predicate holds, then the radius is necessarily nonnegative. The definition accepts arbitrary real inputs, and the theorem states the exact logical equivalence.
Exact mathematical passage
\operatorname{Enclosed}(x,L,U)&\;:\Longleftrightarrow\;L\le x\le U,\\ \operatorname{ErrorBound}(x,\widehat x,r)&\;:\Longleftrightarrow\; |x-\widehat x|\le r.
\expUse in source · 4 passages
Search aliases: exp
Use in source · Read passage · 05_spectrum.tex, line 450
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehatκ_H(θ)-θ q}, where widehatκ_H is the preceding MGF upper bound.
Exact mathematical passage
\boxed{ \Pr(Q\ge q) \le \inf_{0<2\theta(c+\rho)<1} \exp\{\widehat\kappa_H(\theta)-\theta q\}, }
\operatorname{expm1}Use in source · 5 passages
Search aliases: expm1 operatorname
Use in source · Read passage · 10_accuracy.tex, line 79
The directed-rounding backend encloses the finite transform and exponential prefactor and then enlarges that enclosure by the proved support/alias and infinite-mode error bounds. Therefore the true tail probability lies inside the returned interval. The controller encloses expm1(log(10)(d_+-d_-)) by directed rounding and sets certified=True only when the upper endpoint of that ratio enclosure is no larger than the lower endpoint of the exact requested tolerance enclosure. High precision alone is…
Exact mathematical passage
\operatorname{expm1}(\log(10)(d_+-d_-))
fUse in source · 6 passages
Search aliases: f
Use in source · Read passage · 01_gaussian.tex, line 361
Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×10^-5. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor A and as…
Exact mathematical passage
f_Q(x)=\int_0^x\frac12e^{-y/2}\frac14e^{-(x-y)/4}\,dy =\frac12\bigl(e^{-x/4}-e^{-x/2}\bigr).
F^{\mathsf T}Use in source · 2 passages
Search aliases: F^ T mathsf
Use in source · Read passage · A_atlas.tex, line 116
A=UB+F, U^ TF=0, A^ TA=B^ TB+F^ TF. t=trR= F _F^2, v=tr(R^2), ρ_R= R _2. Let U contain d orthonormal sample-space directions. Project the factor into its captured range coordinates B=U^ TA and residual F=(I-UU^ T)A. Orthogonality eliminates the cross terms in the feature-space Gram matrix.
Exact mathematical passage
A=UB+F,\quad U^{\mathsf T}F=0,\quad A^{\mathsf T}A=B^{\mathsf T}B+F^{\mathsf T}F.
F^2Use in source · 2 passages
Search aliases: F^2
Use in source · Read passage · 05_spectrum.tex, line 358
This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form. The second records how concentrated the residual spectrum is. The third is its largest coefficient.
Exact mathematical passage
\boxed{ t=\|F\|_F^2, \qquad v=\|FF^\top\|_F^2, \qquad \rho=\|F\|_2^2. }
F^\topUse in source · 3 passages
Search aliases: F^top top
Use in source · Read passage · 05_spectrum.tex, line 328
[Exact core--residual decomposition] A^top A = B^top B+F^top F. In particular the unresolved spectral term R=F^top F is positive semidefinite.
Exact mathematical passage
\boxed{ A^\top A = B^\top B+F^\top F. }
F_0Use in source · 5 passages
Search aliases: F_0
Use in source · Read passage · 12_synthesis.tex, line 101
[Exact frontier-to-I/O reduction] Suppose a validated exact phase graph of the fixed operator H=A^top A is already supplied and the initial Krylov block is supported on a phase set F_0. Keep covariate projections, weights, and structural phase metadata fixed, and assume all factor-action blocks required for the retained phase operator are available. At depth k, let F_k be the graph k-neighborhood of F_0. Then no storage block lying wholly in a phase outside F_k can affect the k-step Krylov vecto…
Exact mathematical passage
F_0
F_bUse in source · 2 passages
Search aliases: F_b
Use in source · Read passage · 07_inversion.tex, line 105
[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted chi-square sum has an absolutely continuous law. Its folded measure therefore has no atom at the jump of h. The even periodic extension of h has boun…
Exact mathematical passage
F_b:[0,\infty)\to[0,b]
F_{b\#}Use in source · 2 passages
Search aliases: F_b#
Use in source · Read passage · 07_inversion.tex, line 109
[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted chi-square sum has an absolutely continuous law. Its folded measure therefore has no atom at the jump of h. The even periodic extension of h has boun…
Exact mathematical passage
\int h\,d(F_{b\#}\nu_\theta) =\frac2b\sum_{k=0}^{\infty}{}' J_k\,\Re\phi_\theta(k\pi/b).
F_kUse in source · 6 passages
Search aliases: F_k
Use in source · Read passage · 12_synthesis.tex, line 103
xact frontier-to-I/O reduction] Suppose a validated exact phase graph of the fixed operator H=A^top A is already supplied and the initial Krylov block is supported on a phase set F_0. Keep covariate projections, weights, and structural phase metadata fixed, and assume all factor-action blocks required for the retained phase operator are available. At depth k, let F_k be the graph k-neighborhood of F_0. Then no storage block lying wholly in a phase outside F_k can affect the k-step Krylov vector.…
Exact mathematical passage
F_k
f_QUse in source · 1 passage
Search aliases: f_Q
Use in source · Read passage · 01_gaussian.tex, line 361
Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×10^-5. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor A and as…
Exact mathematical passage
f_Q(x)=\int_0^x\frac12e^{-y/2}\frac14e^{-(x-y)/4}\,dy =\frac12\bigl(e^{-x/4}-e^{-x/2}\bigr).
f_{\theta}Use in source · 1 passage
Search aliases: f_θ theta
Use in source · Read passage · 06_packets.tex, line 75
Define the exponentially tilted law by f_θ(x)=e^θ x-κ(θ)f(x). Its characteristic function is ϕ_θ(t) =exp{κ(θ+i t)-κ(θ)}, and E_θQ=κ'(θ). Therefore the choice [source reference] moves the query q to the mean of the tilted law.
Exact mathematical passage
f_{\theta}(x)=e^{\theta x-\kappa(\theta)}f(x). \label{eq:tilt-density}
\operatorname{factorEnergy}Use in source · 1 passage
Search aliases: factorEnergy operatorname
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
GUse in source · 19 passages
Search aliases: G
Use in source · Read passage · 02_streaming.tex, line 15
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
G\in\mathbb R^{n\times m}, \qquad W=\operatorname{diag}(w_1,\ldots,w_m), \quad w_j\ge 0,
G_{[:,J_b]}Use in source · 1 passage
Search aliases: G_[:,J_b]
Use in source · Read passage · 02_streaming.tex, line 71
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
G_bUse in source · 9 passages
Search aliases: G_b
Use in source · Read passage · 02_streaming.tex, line 71
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
G_b^\topUse in source · 6 passages
Search aliases: G_b^top top
Use in source · Read passage · 02_streaming.tex, line 84
[Additive block Gram identity] K = AA^top = ∑_b=1^B A_bA_b^top = M_X ( ∑_b=1^B G_bW_bG_b^top ) M_X.
Exact mathematical passage
\boxed{ K = AA^\top = \sum_{b=1}^B A_bA_b^\top = M_X \left( \sum_{b=1}^B G_bW_bG_b^\top \right) M_X. }
\operatorname{gramEntry}Use in source · 2 passages
Search aliases: gramEntry operatorname
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
H^kUse in source · 16 passages
Search aliases: H^k
Use in source · Read passage · 03_phases.tex, line 26
[Polyphase light cone] If the graph distance from phase g to phase h exceeds k, then P_hH^kP_g=0.
Exact mathematical passage
\boxed{ P_hH^kP_g=0. }
H^{k-1-j}Use in source · 1 passage
Search aliases: H^k-1-j
Use in source · Read passage · 03_phases.tex, line 159
Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
H^k-H_0^k = \sum_{j=0}^{k-1}H^{k-1-j}EH_0^j.
H_{\mathcal O}Use in source · 2 passages
Search aliases: H_ O mathcal
Use in source · Read passage · 03_phases.tex, line 334
. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
H_{\mathcal O} =H-\sum_{\{h,g\}\in\mathcal O} \bigl(P_hHP_g+P_gHP_h\bigr).
H_{\mathcal O}^kUse in source · 1 passage
Search aliases: H_ O^k mathcal
Use in source · Read passage · 03_phases.tex, line 372
[Edge-additive leakage majorant] For every omitted set O⊆ E, 0≤ ( B^k- B_ O^k)ρ_0 ≤ ∑_e∈ Oq_e^(k) componentwise. Consequently, lVert(H^k-H_ O^k)XrVert_F ≤ lVert ( B^k- B_ O^k)ρ_0 rVert_2 ≤ ∑_e∈ Oβ_e^(k).
Exact mathematical passage
\boxed{ \lVert(H^k-H_{\mathcal O}^k)X\rVert_F \le \left\lVert (\mathcal B^k-\mathcal B_{\mathcal O}^k)\rho_0 \right\rVert_2 \le \sum_{e\in\mathcal O}\beta_e^{(k)}. }
H_0Use in source · 13 passages
Search aliases: H_0
Use in source · Read passage · 03_phases.tex, line 121
Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,
Exact mathematical passage
H_0=\sum_{(h,g)\in\mathcal E_0}P_hHP_g.
H_0^jUse in source · 1 passage
Search aliases: H_0^j
Use in source · Read passage · 03_phases.tex, line 159
Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
H^k-H_0^k = \sum_{j=0}^{k-1}H^{k-1-j}EH_0^j.
H_0^kUse in source · 7 passages
Search aliases: H_0^k
Use in source · Read passage · 03_phases.tex, line 140
[Thresholded-power certificate] For every k≥1, ρ((H^k-H_0^k)X) ≤ ( B^k- B_0^k)ρ(X). In particular, lVert(H^k-H_0^k)XrVert_F ≤ lVert( B^k- B_0^k)ρ(X)rVert_2.
Exact mathematical passage
\boxed{ \rho\bigl((H^k-H_0^k)X\bigr) \leq (\mathcal B^k-\mathcal B_0^k)\rho(X). }
H_bUse in source · 4 passages
Search aliases: H_b
Use in source · Read passage · 02_streaming.tex, line 264
A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.
Exact mathematical passage
H_b=G_bW_b^{1/2}
H_{b,I_r}Use in source · 2 passages
Search aliases: H_b,I_r
Use in source · Read passage · 02_streaming.tex, line 275
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ R_b = \sum_r Q_{X,I_r}^\top H_{b,I_r}, }
H_FUse in source · 1 passage
Search aliases: H_F
Use in source · Read passage · 12_synthesis.tex, line 119
^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the full variant space. Thus P_F=J_FJ_F^top and A_F=AJ_F. The compressed operator and its ambient embedding satisfy H_F=J_F^top HJ_F=A_F^top A_F, P_FHP_F=J_FH_FJ_F^top. These identities distinguish the compressed matrix from the larger …
Exact mathematical passage
H_F=J_F^\top HJ_F=A_F^\top A_F, \qquad P_FHP_F=J_FH_FJ_F^\top.
H_F^kUse in source · 1 passage
Search aliases: H_F^k
Use in source · Read passage · 12_synthesis.tex, line 125
rdinate inclusion from the retained coordinates F=F_k into the full variant space. Thus P_F=J_FJ_F^top and A_F=AJ_F. The compressed operator and its ambient embedding satisfy H_F=J_F^top HJ_F=A_F^top A_F, P_FHP_F=J_FH_FJ_F^top. These identities distinguish the compressed matrix from the larger ambient matrix. Since every path of length at most k from F_0 stays in F, H^kX=J_F H_F^k J_F^top X. This includes the zero output outside F. Blocks wholly outside the retained phase set may therefore be om…
Exact mathematical passage
H^kX=J_F H_F^k J_F^\top X.
h_jUse in source · 3 passages
Search aliases: h_j
Use in source · Read passage · 11_evidence.tex, line 120
The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponential tilting, the COS expansion, spectral packets, or the K-COS error controller.
Exact mathematical passage
h_j
IUse in source · 41 passages
Search aliases: I
Use in source · Read passage · 01_gaussian.tex, line 19
Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.
Exact mathematical passage
Z\sim N(0,I)
\mathrm iUse in source · 1 passage
Search aliases: i mathrm
Use in source · Read passage · 06_packets.tex, line 208
A packet with a nonzero remainder. To see the approximation itself, take a separate declared packet {1.96,1.98,2.02,2.04}, μ=2, and evaluate its contribution at s=0.1+0.4 i. The deviations are -0.04,-0.02,0.02,0.04. In particular, M_1=M_3=0, M_2=0.004, M_4=0.00000544. The factor α=2s/(1-4s) gives ρ=0.04|α|<1, so Theorem~[source reference] applies. Retaining successive even moment orders gives the values below. Both columns are rounded point evaluations. The theorem, not these rounded comparisons…
Exact mathematical passage
s=0.1+0.4\,\mathrm i
I_1Use in source · 1 passage
Search aliases: I_1
Use in source · Read passage · 02_streaming.tex, line 267
A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.
Exact mathematical passage
I_1,\ldots,I_R
I_2Use in source · 1 passage
Search aliases: I_2
Use in source · Read passage · 04_range.tex, line 198
Why the nonzero-spectrum qualification matters. Take instead A=1 · 0 0 · 0, Q_r=I_2, B=Q_r^top A=A. The captured subspace contains the range of A, but is larger than that range. Both Gram matrices have spectrum {1,0}. Their nonzero spectrum is {1}. Thus the nonzero spectrum of AA^top cannot be identified with the entire spectrum of BB^top. Choosing the compact basis Q_r=(1,0)^top instead gives B=(1,0) and the one-dimensional core Gram (1). The extra zero records an unused basis direction, not an…
Exact mathematical passage
A=\begin{pmatrix}1&0\\0&0\end{pmatrix},\qquad Q_r=I_2,\qquad B=Q_r^\top A=A.
I_4Use in source · 1 passage
Search aliases: I_4
Use in source · Read passage · 01_gaussian.tex, line 353
1 · -1 · -1 · 1 satisfies V^top K V=diag(1,1,2,2). For Z∼ N(0,I_4), Q=Z^top KZ overset d= Y+2W, Y,W independent with law χ^2_2. The spectrum is therefore (1,1,2,2) and E Q=6. The six-dimensional feature Gram A^top A has these same four nonzero eigenvalues and two additional zeros.
Exact mathematical passage
Z\sim N(0,I_4)
I_RUse in source · 1 passage
Search aliases: I_R
Use in source · Read passage · 02_streaming.tex, line 267
A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.
Exact mathematical passage
I_1,\ldots,I_R
I_rUse in source · 3 passages
Search aliases: I_r
Use in source · Read passage · 02_streaming.tex, line 275
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ R_b = \sum_r Q_{X,I_r}^\top H_{b,I_r}, }
I_sUse in source · 1 passage
Search aliases: I_s
Use in source · Read passage · 02_streaming.tex, line 292
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ K_{I_r,I_s} = \sum_b A_{b,I_r}A_{b,I_s}^\top. }
\iffUse in source · 1 passage
Search aliases: iff
Use in source · Read passage · A_atlas.tex, line 84
∑_hP_h=I, g→ hiff P_hHP_g≠0. dist(g,h)>k Longrightarrow P_hH^kP_g=0. Choose a complete orthogonal family of phase projectors on feature space and form H=A^ TA. An edge from phase g to phase h exists when the transfer block P_hHP_g is nonzero. Each operator application follows one graph edge. An iterate of depth k cannot reach a phase farther than k edges away.
Exact mathematical passage
\sum_hP_h=I,\qquad g\to h\iff P_hHP_g\ne0.
\infUse in source · 1 passage
Search aliases: inf
Use in source · Read passage · 05_spectrum.tex, line 450
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehatκ_H(θ)-θ q}, where widehatκ_H is the preceding MGF upper bound.
Exact mathematical passage
\boxed{ \Pr(Q\ge q) \le \inf_{0<2\theta(c+\rho)<1} \exp\{\widehat\kappa_H(\theta)-\theta q\}, }
\inf_{0<2\theta(c+\rho)<1}Use in source · 1 passage
Search aliases: inf_0<2θ(c+ρ)<1 inf theta rho
Use in source · Read passage · 05_spectrum.tex, line 450
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehatκ_H(θ)-θ q}, where widehatκ_H is the preceding MGF upper bound.
Exact mathematical passage
\boxed{ \Pr(Q\ge q) \le \inf_{0<2\theta(c+\rho)<1} \exp\{\widehat\kappa_H(\theta)-\theta q\}, }
\mathcal JUse in source · 4 passages
Search aliases: J mathcal
Use in source · Read passage · 02_streaming.tex, line 321
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\mathcal J
J'Use in source · 1 passage
Search aliases: J'
Use in source · Read passage · 02_streaming.tex, line 323
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
\mathcal J'Use in source · 2 passages
Search aliases: J' mathcal
Use in source · Read passage · 02_streaming.tex, line 322
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\mathcal J'
J_1Use in source · 1 passage
Search aliases: J_1
Use in source · Read passage · 02_streaming.tex, line 65
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
\{1,\ldots,m\} = J_1\sqcup\cdots\sqcup J_B.
J_BUse in source · 1 passage
Search aliases: J_B
Use in source · Read passage · 02_streaming.tex, line 65
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
\{1,\ldots,m\} = J_1\sqcup\cdots\sqcup J_B.
J_bUse in source · 1 passage
Search aliases: J_b
Use in source · Read passage · 02_streaming.tex, line 71
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
J_FUse in source · 5 passages
Search aliases: J_F
Use in source · Read passage · 12_synthesis.tex, line 116
For every phase h∉ F_k and every starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the full variant space. Thus P_F=J_FJ_F^top and A_F=AJ_F. The compressed operator and its ambient embedding satisfy H_F=J_F^top HJ_F=A_F^…
Exact mathematical passage
J_F
J_F^\topUse in source · 3 passages
Search aliases: J_F^top top
Use in source · Read passage · 12_synthesis.tex, line 117
∉ F_k and every starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the full variant space. Thus P_F=J_FJ_F^top and A_F=AJ_F. The compressed operator and its ambient embedding satisfy H_F=J_F^top HJ_F=A_F^top A_F, P_FHP_F=…
Exact mathematical passage
P_F=J_FJ_F^\top
k^{-1}Use in source · 1 passage
Search aliases: k^-1
Use in source · Read passage · 07_inversion.tex, line 124
has no atom at the jump of h. The even periodic extension of h has bounded Fej er means, which converge at every continuity point. Dominated convergence identifies the integral of those means with ∫ h d(F_b#ν_θ). Their cosine coefficients are the Ces aro weighted coefficients 2J_k/b, and folding preserves the cosine moments Reϕ_θ(kπ/b). Finally J_k=O(k^-1) and |ϕ_θ(kπ/b)|=O(k^-r/2), as follows directly from the product formula below. Thus the coefficient products are O(k^-1-r/2). Their absolute …
Exact mathematical passage
J_k=O(k^{-1})
k^{-1-r/2}Use in source · 1 passage
Search aliases: k^-1-r/2
Use in source · Read passage · 07_inversion.tex, line 127
ted convergence identifies the integral of those means with ∫ h d(F_b#ν_θ). Their cosine coefficients are the Ces aro weighted coefficients 2J_k/b, and folding preserves the cosine moments Reϕ_θ(kπ/b). Finally J_k=O(k^-1) and |ϕ_θ(kπ/b)|=O(k^-r/2), as follows directly from the product formula below. Thus the coefficient products are O(k^-1-r/2). Their absolute sum converges. The Ces aro sums and ordinary sum have the same limit, proving the displayed identity. This argument justifies the weighte…
Exact mathematical passage
O(k^{-1-r/2})
k^{-r/2}Use in source · 1 passage
Search aliases: k^-r/2
Use in source · Read passage · 07_inversion.tex, line 125
p of h. The even periodic extension of h has bounded Fej er means, which converge at every continuity point. Dominated convergence identifies the integral of those means with ∫ h d(F_b#ν_θ). Their cosine coefficients are the Ces aro weighted coefficients 2J_k/b, and folding preserves the cosine moments Reϕ_θ(kπ/b). Finally J_k=O(k^-1) and |ϕ_θ(kπ/b)|=O(k^-r/2), as follows directly from the product formula below. Thus the coefficient products are O(k^-1-r/2). Their absolute sum converges. The Ces…
Exact mathematical passage
|\phi_\theta(k\pi/b)|=O(k^{-r/2})
k^{-\beta_N-1}Use in source · 2 passages
Search aliases: k^-β_N-1 beta
Use in source · Read passage · 07_inversion.tex, line 264
Using the cosine coefficient magnitude 2|ϕ_θ(u_k)|/b and [source reference], the omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1+fracN^-β_Nβ_N, which yields [source reference].
Exact mathematical passage
\sum_{k=N}^{\infty}\frac{|\phi_{\theta}(u_k)|}{k} \le |\phi_{\theta}(u_N)|N^{\beta_N} \sum_{k=N}^{\infty}k^{-\beta_N-1}.
K_{I_r,I_s}Use in source · 1 passage
Search aliases: K_I_r,I_s
Use in source · Read passage · 02_streaming.tex, line 292
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ K_{I_r,I_s} = \sum_b A_{b,I_r}A_{b,I_s}^\top. }
K_{ij}Use in source · 1 passage
Search aliases: K_ij
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
K_sUse in source · 3 passages
Search aliases: K_s
Use in source · Read passage · B_lean.tex, line 75
The same refinement has formal consequences for the operator action and the quadratic statistic: (K_s∪ tv)_i · =(K_s v)_i+(K_t v)_i, v^top K_s∪ tv · =v^top K_s v+v^top K_t v. These are declmatrixAction_gram_union and declquadraticForm_gram_union. The independent declstreamedAction_union proves the corresponding streamed accumulation identity directly.
Exact mathematical passage
(K_{s\cup t}v)_i&=(K_s v)_i+(K_t v)_i,\\ v^\top K_{s\cup t}v&=v^\top K_s v+v^\top K_t v.
K_{s\cup t}Use in source · 1 passage
Search aliases: K_s∪ t cup
Use in source · Read passage · B_lean.tex, line 75
The same refinement has formal consequences for the operator action and the quadratic statistic: (K_s∪ tv)_i · =(K_s v)_i+(K_t v)_i, v^top K_s∪ tv · =v^top K_s v+v^top K_t v. These are declmatrixAction_gram_union and declquadraticForm_gram_union. The independent declstreamedAction_union proves the corresponding streamed accumulation identity directly.
Exact mathematical passage
(K_{s\cup t}v)_i&=(K_s v)_i+(K_t v)_i,\\ v^\top K_{s\cup t}v&=v^\top K_s v+v^\top K_t v.
K_tUse in source · 1 passage
Search aliases: K_t
Use in source · Read passage · B_lean.tex, line 75
The same refinement has formal consequences for the operator action and the quadratic statistic: (K_s∪ tv)_i · =(K_s v)_i+(K_t v)_i, v^top K_s∪ tv · =v^top K_s v+v^top K_t v. These are declmatrixAction_gram_union and declquadraticForm_gram_union. The independent declstreamedAction_union proves the corresponding streamed accumulation identity directly.
Exact mathematical passage
(K_{s\cup t}v)_i&=(K_s v)_i+(K_t v)_i,\\ v^\top K_{s\cup t}v&=v^\top K_s v+v^\top K_t v.
LUse in source · 11 passages
Search aliases: L
Use in source · Read passage · 08_adaptive.tex, line 11
The packet planner accepts a finite positive spectrum, an upper-tail query q>∑_jλ_j, and a requested relative tolerance τ>0. Its initial support parameter satisfies L>0, and its mode count is a positive integer.
Exact mathematical passage
L>0
lUse in source · 9 passages
Search aliases: l
Use in source · Read passage · 02_streaming.tex, line 377
For fixed n,p,s,c, fracM_rm streamM_rm full = O ( fraccm + fracsm + fracpm ), so device memory remains bounded as total variant count grows.
Exact mathematical passage
\boxed{ \frac{M_{\rm stream}}{M_{\rm full}} = O\!\left( \frac{c}{m} + \frac{s}{m} + \frac{p}{m} \right), }
L^\topUse in source · 1 passage
Search aliases: L^top top
Use in source · Read passage · 11_evidence.tex, line 579
Hierarchical refinement gave relative discrepancy 3.17×10^-16. For H=L^top L with block-bidiagonal L, the smallest eigenvalue was approximately 2.86 and the largest norm outside the exact light cone was zero. The QR-core singular-spectrum discrepancy was 5.89×10^-16 in relative ℓ_2 norm. These are finite controls, not universal floating-point error bounds.
Exact mathematical passage
H=L^\top L
l_kUse in source · 3 passages
Search aliases: l_k
Use in source · Read passage · B_lean.tex, line 152
The separate definition weightedSquares(s,c,z)=∑_k∈ sc_k z_k^2 models a supplied diagonal quadratic form. Its coefficients need not be eigenvalues obtained by a proved diagonalization. It has three theorems: item declweightedSquares_mono: if l_k≤ u_k for every k∈ s, then ∑ l_kz_k^2≤∑ u_kz_k^2. item declweightedSquares_nonneg: if c_k≥0 for every k∈ s, then ∑ c_kz_k^2≥0. item declweightedSquares_tail_imp: coefficientwise order gives the pointwise implication q≤∑ l_kz_k^2⇒ q≤∑ u_kz_k^2. The first a…
Exact mathematical passage
l_k\le u_k
\lceilUse in source · 2 passages
Search aliases: lceil
Use in source · Read passage · 05_spectrum.tex, line 539
[Residual-rank bracket] lceil max{ fractρ, fract^2v } rceil ≤ rankF ≤ n-r. The zero-residual case is interpreted separately as rank zero.
Exact mathematical passage
\boxed{ \left\lceil \max\left\{ \frac{t}{\rho}, \frac{t^2}{v} \right\} \right\rceil \le \operatorname{rank}F \le n-r. }
\log_{10}Use in source · 18 passages
Search aliases: log_10 log
Use in source · Read passage · 05_spectrum.tex, line 192
[htbp] centering includegraphics[width=linewidth]figures_production/figure_14_dual_budget_tail_bridge.pdf captionTail brackets under a fixed query budget and varying operator budgets. The dotted reference is at log_10p=-8. A sufficiently loose operator budget admits a strong coupling and widens the bracket even though the query-local constraint still passes.
Exact mathematical passage
\log_{10}p=-8
\LongleftrightarrowUse in source · 3 passages
Search aliases: Longleftrightarrow
Use in source · Read passage · 03_phases.tex, line 18
Let the phase projectors form an orthogonal decomposition of variant space: P_h^top=P_h=P_h^2, P_hP_g=0 (h≠ g), ∑_hP_h=I. Thus P_h projects onto variant phase h. Again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.
Exact mathematical passage
g\to h \quad\Longleftrightarrow\quad P_hHP_g\ne0.
\LongrightarrowUse in source · 2 passages
Search aliases: Longrightarrow
Use in source · Read passage · A_atlas.tex, line 85
∑_hP_h=I, g→ hiff P_hHP_g≠0. dist(g,h)>k Longrightarrow P_hH^kP_g=0. Choose a complete orthogonal family of phase projectors on feature space and form H=A^ TA. An edge from phase g to phase h exists when the transfer block P_hHP_g is nonzero. Each operator application follows one graph edge. An iterate of depth k cannot reach a phase farther than k edges away.
Exact mathematical passage
\operatorname{dist}(g,h)>k\quad\Longrightarrow\quad P_hH^kP_g=0.
\lVertUse in source · 15 passages
Search aliases: lVert
Use in source · Read passage · 03_phases.tex, line 81
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
\rho(X)_g=\lVert P_gX\rVert_F,
M^{1/2}Use in source · 5 passages
Search aliases: M^1/2
Use in source · Read passage · 01_gaussian.tex, line 245
xact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈range(M^1/2S). In particular, Σsucc0 is sufficient. For singular covariance the shift is the square…
Exact mathematical passage
T=M^{1/2}S
M^{k-1}Use in source · 1 passage
Search aliases: M^k-1
Use in source · Read passage · 03_phases.tex, line 211
[Scalar threshold law] For any induced matrix norm, let M=max{lVert BrVert,lVert B_0rVert}. Then lVert B^k- B_0^krVert ≤ kM^k-1lVert RrVert. Hence weak discarded coupling produces a first-order error bound proportional to the discarded transfer norm and to the Krylov depth.
Exact mathematical passage
\boxed{ \lVert\mathcal B^k-\mathcal B_0^k\rVert \leq kM^{k-1}\lVert\mathcal R\rVert. }
M^\topUse in source · 2 passages
Search aliases: M^top top
Use in source · Read passage · 01_gaussian.tex, line 79
The positive case is not the natural boundary of the central theory. Let X∼ N(0,Σ), M=M^top, where M may be indefinite, and again write S=Σ^1/2. The symmetric operator H=SMS has a real signed spectrum.
Exact mathematical passage
X\sim N(0,\Sigma), \qquad M=M^\top,
M_1Use in source · 1 passage
Search aliases: M_1
Use in source · Read passage · 06_packets.tex, line 209
A packet with a nonzero remainder. To see the approximation itself, take a separate declared packet {1.96,1.98,2.02,2.04}, μ=2, and evaluate its contribution at s=0.1+0.4 i. The deviations are -0.04,-0.02,0.02,0.04. In particular, M_1=M_3=0, M_2=0.004, M_4=0.00000544. The factor α=2s/(1-4s) gives ρ=0.04|α|<1, so Theorem~[source reference] applies. Retaining successive even moment orders gives the values below. Both columns are rounded point evaluations. The theorem, not these rounded comparisons…
Exact mathematical passage
M_1=M_3=0,\quad M_2=0.004,\quad M_4=0.00000544.
M_2Use in source · 1 passage
Search aliases: M_2
Use in source · Read passage · 06_packets.tex, line 209
A packet with a nonzero remainder. To see the approximation itself, take a separate declared packet {1.96,1.98,2.02,2.04}, μ=2, and evaluate its contribution at s=0.1+0.4 i. The deviations are -0.04,-0.02,0.02,0.04. In particular, M_1=M_3=0, M_2=0.004, M_4=0.00000544. The factor α=2s/(1-4s) gives ρ=0.04|α|<1, so Theorem~[source reference] applies. Retaining successive even moment orders gives the values below. Both columns are rounded point evaluations. The theorem, not these rounded comparisons…
Exact mathematical passage
M_1=M_3=0,\quad M_2=0.004,\quad M_4=0.00000544.
M_3Use in source · 1 passage
Search aliases: M_3
Use in source · Read passage · 06_packets.tex, line 209
A packet with a nonzero remainder. To see the approximation itself, take a separate declared packet {1.96,1.98,2.02,2.04}, μ=2, and evaluate its contribution at s=0.1+0.4 i. The deviations are -0.04,-0.02,0.02,0.04. In particular, M_1=M_3=0, M_2=0.004, M_4=0.00000544. The factor α=2s/(1-4s) gives ρ=0.04|α|<1, so Theorem~[source reference] applies. Retaining successive even moment orders gives the values below. Both columns are rounded point evaluations. The theorem, not these rounded comparisons…
Exact mathematical passage
M_1=M_3=0,\quad M_2=0.004,\quad M_4=0.00000544.
M_4Use in source · 1 passage
Search aliases: M_4
Use in source · Read passage · 06_packets.tex, line 209
A packet with a nonzero remainder. To see the approximation itself, take a separate declared packet {1.96,1.98,2.02,2.04}, μ=2, and evaluate its contribution at s=0.1+0.4 i. The deviations are -0.04,-0.02,0.02,0.04. In particular, M_1=M_3=0, M_2=0.004, M_4=0.00000544. The factor α=2s/(1-4s) gives ρ=0.04|α|<1, so Theorem~[source reference] applies. Retaining successive even moment orders gives the values below. Both columns are rounded point evaluations. The theorem, not these rounded comparisons…
Exact mathematical passage
M_1=M_3=0,\quad M_2=0.004,\quad M_4=0.00000544.
M_{P,0}Use in source · 1 passage
Search aliases: M_P,0
Use in source · Read passage · 06_packets.tex, line 111
Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.
Exact mathematical passage
M_{P,0}=n_P
M_{P,1}Use in source · 2 passages
Search aliases: M_P,1
Use in source · Read passage · 06_packets.tex, line 111
Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.
Exact mathematical passage
M_{P,1}=0
M_{\rm block}Use in source · 2 passages
Search aliases: M_rm block rm
Use in source · Read passage · 03_phases.tex, line 621
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm block}
M_{\rm double}Use in source · 1 passage
Search aliases: M_rm double rm
Use in source · Read passage · 03_phases.tex, line 623
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm single}=M_{\rm fixed}+M_{\rm block}, \qquad \boxed{ M_{\rm double}=M_{\rm fixed}+2M_{\rm block}. }
M_{\rm fixed}Use in source · 2 passages
Search aliases: M_rm fixed rm
Use in source · Read passage · 03_phases.tex, line 622
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm fixed}
M_{\rm full}Use in source · 1 passage
Search aliases: M_rm full rm
Use in source · Read passage · 02_streaming.tex, line 377
For fixed n,p,s,c, fracM_rm streamM_rm full = O ( fraccm + fracsm + fracpm ), so device memory remains bounded as total variant count grows.
Exact mathematical passage
\boxed{ \frac{M_{\rm stream}}{M_{\rm full}} = O\!\left( \frac{c}{m} + \frac{s}{m} + \frac{p}{m} \right), }
M_{\rm single}Use in source · 1 passage
Search aliases: M_rm single rm
Use in source · Read passage · 03_phases.tex, line 623
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm single}=M_{\rm fixed}+M_{\rm block}, \qquad \boxed{ M_{\rm double}=M_{\rm fixed}+2M_{\rm block}. }
M_{\rm stream}Use in source · 1 passage
Search aliases: M_rm stream rm
Use in source · Read passage · 02_streaming.tex, line 377
For fixed n,p,s,c, fracM_rm streamM_rm full = O ( fraccm + fracsm + fracpm ), so device memory remains bounded as total variant count grows.
Exact mathematical passage
\boxed{ \frac{M_{\rm stream}}{M_{\rm full}} = O\!\left( \frac{c}{m} + \frac{s}{m} + \frac{p}{m} \right), }
M_XUse in source · 25 passages
Search aliases: M_X
Use in source · Read passage · 02_streaming.tex, line 23
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
M_X=I-Q_XQ_X^\top, \qquad A=M_XGW^{1/2}, \qquad K=AA^\top.
M_X^{\mathsf T}Use in source · 1 passage
Search aliases: M_X^ T mathsf
Use in source · Read passage · A_atlas.tex, line 68
Q_X^ TQ_X=I, M_X=I-Q_XQ_X^ T, M_X^2=M_X=M_X^ T. A=M_XGW^1/2, K=AA^ T. For a covariate range basis Q_X with orthonormal columns, the residualizer M_X=I-Q_XQ_X^ T removes the component in the covariate span. It is symmetric and idempotent. Applying it to the weighted feature factor restricts the kernel to variation left after the null-model covariates have been removed.
Exact mathematical passage
Q_X^{\mathsf T}Q_X=I,\quad M_X=I-Q_XQ_X^{\mathsf T},\quad M_X^2=M_X=M_X^{\mathsf T}.
M_X^2Use in source · 2 passages
Search aliases: M_X^2
Use in source · Read passage · 02_streaming.tex, line 115
A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.
Exact mathematical passage
M_X^\top=M_X=M_X^2
M_X^\topUse in source · 1 passage
Search aliases: M_X^top top
Use in source · Read passage · 02_streaming.tex, line 115
A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.
Exact mathematical passage
M_X^\top=M_X=M_X^2
\operatorname{matrixAction}Use in source · 1 passage
Search aliases: matrixAction operatorname
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
\max_{j\in P}Use in source · 2 passages
Search aliases: max_j∈ P max in
Use in source · Read passage · 06_packets.tex, line 116
[Centered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P,p(s)| ≤ fracn_P2 fracρ_P(s)^p+1(p+1)(1-ρ_P(s)).
Exact mathematical passage
\alpha_P(s)=\frac{2s}{1-2s\mu_P}, \qquad \rho_P(s)=|\alpha_P(s)|\max_{j\in P}|\delta_j|. \label{eq:alpha-rho}
\max_{\lambda_j<0}Use in source · 1 passage
Search aliases: max_λ_j<0 max lambda
Use in source · Read passage · 01_gaussian.tex, line 127
For the right-tail path, the cumulant generating function is κ(s) = -frac12∑_jlog(1-2sλ_j), with real domain max_λ_j<0frac12λ_j < s < min_λ_j>0frac12λ_j. Here a missing negative or positive spectral set gives a domain endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain whenever a positive spectral component is present. The negative-only case with at least one nonzero coefficient has a positive saddlepoint only for E[Q…
Exact mathematical passage
\boxed{ \max_{\lambda_j<0}\frac{1}{2\lambda_j} < s < \min_{\lambda_j>0}\frac{1}{2\lambda_j}. }
\minUse in source · 2 passages
Search aliases: min
Use in source · Read passage · 01_gaussian.tex, line 127
For the right-tail path, the cumulant generating function is κ(s) = -frac12∑_jlog(1-2sλ_j), with real domain max_λ_j<0frac12λ_j < s < min_λ_j>0frac12λ_j. Here a missing negative or positive spectral set gives a domain endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain whenever a positive spectral component is present. The negative-only case with at least one nonzero coefficient has a positive saddlepoint only for E[Q…
Exact mathematical passage
\boxed{ \max_{\lambda_j<0}\frac{1}{2\lambda_j} < s < \min_{\lambda_j>0}\frac{1}{2\lambda_j}. }
\min_{\lambda_j>0}Use in source · 1 passage
Search aliases: min_λ_j>0 min lambda
Use in source · Read passage · 01_gaussian.tex, line 127
For the right-tail path, the cumulant generating function is κ(s) = -frac12∑_jlog(1-2sλ_j), with real domain max_λ_j<0frac12λ_j < s < min_λ_j>0frac12λ_j. Here a missing negative or positive spectral set gives a domain endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain whenever a positive spectral component is present. The negative-only case with at least one nonzero coefficient has a positive saddlepoint only for E[Q…
Exact mathematical passage
\boxed{ \max_{\lambda_j<0}\frac{1}{2\lambda_j} < s < \min_{\lambda_j>0}\frac{1}{2\lambda_j}. }
N^{-\beta_N}Use in source · 1 passage
Search aliases: N^-β_N beta
Use in source · Read passage · 07_inversion.tex, line 271
omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1+fracN^-β_Nβ_N, which yields [source reference].
Exact mathematical passage
\sum_{k=N}^{\infty}k^{-\beta_N-1} \le N^{-\beta_N-1}+\int_N^{\infty}x^{-\beta_N-1}\,dx =N^{-\beta_N-1}+\frac{N^{-\beta_N}}{\beta_N},
N^{-\beta_N-1}Use in source · 1 passage
Search aliases: N^-β_N-1 beta
Use in source · Read passage · 07_inversion.tex, line 271
omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1+fracN^-β_Nβ_N, which yields [source reference].
Exact mathematical passage
\sum_{k=N}^{\infty}k^{-\beta_N-1} \le N^{-\beta_N-1}+\int_N^{\infty}x^{-\beta_N-1}\,dx =N^{-\beta_N-1}+\frac{N^{-\beta_N}}{\beta_N},
N^{-\nu}Use in source · 2 passages
Search aliases: N^-ν nu
Use in source · Read passage · 09_intervals.tex, line 156
Let ν=frac r2>1, N≥1. For a finite transform retaining modes 0,…,N-1, E_rm series ≤ frac2C_θbθ (fracbπ)^ν [ N^-ν + fracN^1-νν-1 ].
Exact mathematical passage
\boxed{ E_{\rm series} \le \frac{2C_\theta}{b\theta} \left(\frac{b}{\pi}\right)^\nu \left[ N^{-\nu} + \frac{N^{1-\nu}}{\nu-1} \right]. }
N^{1-\nu}Use in source · 2 passages
Search aliases: N^1-ν nu
Use in source · Read passage · 09_intervals.tex, line 156
Let ν=frac r2>1, N≥1. For a finite transform retaining modes 0,…,N-1, E_rm series ≤ frac2C_θbθ (fracbπ)^ν [ N^-ν + fracN^1-νν-1 ].
Exact mathematical passage
\boxed{ E_{\rm series} \le \frac{2C_\theta}{b\theta} \left(\frac{b}{\pi}\right)^\nu \left[ N^{-\nu} + \frac{N^{1-\nu}}{\nu-1} \right]. }
N^{\beta_N}Use in source · 1 passage
Search aliases: N^β_N beta
Use in source · Read passage · 07_inversion.tex, line 264
Using the cosine coefficient magnitude 2|ϕ_θ(u_k)|/b and [source reference], the omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1+fracN^-β_Nβ_N, which yields [source reference].
Exact mathematical passage
\sum_{k=N}^{\infty}\frac{|\phi_{\theta}(u_k)|}{k} \le |\phi_{\theta}(u_N)|N^{\beta_N} \sum_{k=N}^{\infty}k^{-\beta_N-1}.
n_PUse in source · 9 passages
Search aliases: n_P
Use in source · Read passage · 06_packets.tex, line 99
Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.
Exact mathematical passage
n_P\ge1
oUse in source · 9 passages
Search aliases: o
Use in source · Read passage · 03_phases.tex, line 621
If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page pinning, and library workspaces can reduce realized overlap.
Exact mathematical passage
M_{\rm block}
\oversetUse in source · 6 passages
Search aliases: overset
Use in source · Read passage · 01_gaussian.tex, line 30
Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.
Exact mathematical passage
\boxed{ Q\overset d= \sum_{j=1}^r\lambda_jZ_j^2. }
P_1Use in source · 1 passage
Search aliases: P_1
Use in source · Read passage · 03_phases.tex, line 79
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
P_1,\ldots,P_q
P_FUse in source · 2 passages
Search aliases: P_F
Use in source · Read passage · 12_synthesis.tex, line 117
∉ F_k and every starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the full variant space. Thus P_F=J_FJ_F^top and A_F=AJ_F. The compressed operator and its ambient embedding satisfy H_F=J_F^top HJ_F=A_F^top A_F, P_FHP_F=…
Exact mathematical passage
P_F=J_FJ_F^\top
P_h^2Use in source · 1 passage
Search aliases: P_h^2
Use in source · Read passage · 03_phases.tex, line 10
Let the phase projectors form an orthogonal decomposition of variant space: P_h^top=P_h=P_h^2, P_hP_g=0 (h≠ g), ∑_hP_h=I. Thus P_h projects onto variant phase h. Again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.
Exact mathematical passage
P_h^\top=P_h=P_h^2,\qquad P_hP_g=0\ (h\ne g),\qquad \sum_hP_h=I.
P_h^\topUse in source · 1 passage
Search aliases: P_h^top top
Use in source · Read passage · 03_phases.tex, line 10
Let the phase projectors form an orthogonal decomposition of variant space: P_h^top=P_h=P_h^2, P_hP_g=0 (h≠ g), ∑_hP_h=I. Thus P_h projects onto variant phase h. Again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.
Exact mathematical passage
P_h^\top=P_h=P_h^2,\qquad P_hP_g=0\ (h\ne g),\qquad \sum_hP_h=I.
P_jUse in source · 1 passage
Search aliases: P_j
Use in source · Read passage · 03_phases.tex, line 35
Insert the projector resolution ∑_jP_j=I between successive factors of H. Every nonzero term in P_hH^kP_g corresponds to a phase path of length k. If no such path exists, every term vanishes.
Exact mathematical passage
\sum_jP_j=I
P_qUse in source · 1 passage
Search aliases: P_q
Use in source · Read passage · 03_phases.tex, line 79
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
P_1,\ldots,P_q
P_TUse in source · 4 passages
Search aliases: P_T
Use in source · Read passage · 01_gaussian.tex, line 246
:final-2-3 For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈range(M^1/2S). In particular, Σsucc0 is sufficient. For singular covariance the shift is the squared deterministic …
Exact mathematical passage
P_T
P_\thetaUse in source · 4 passages
Search aliases: P_θ theta
Use in source · Read passage · 09_intervals.tex, line 78
Under the tilted measure, P_θ(Q>b) ≤ exp{ κ(η)-κ(θ)-(η-θ)b }.
Exact mathematical passage
P_\theta(Q>b) \le \exp\left\{ \kappa(\eta)-\kappa(\theta)-(\eta-\theta)b \right\}.
\mathcal P_{\Lambda,q,\varepsilon}Use in source · 1 passage
Search aliases: P_Λ,q,ε mathcal Lambda varepsilon
Use in source · Read passage · A_atlas.tex, line 182
s_k=θ+ifrackπb, ρ_P(s_k)<1 on every retained frequency. (Λ,q,ε)⟼ P_Λ,q,ε. A distance between eigenvalues is insufficient to decide whether a packet is useful. The expansion must be controlled at the actual complex arguments s_k=θ+i kπ/b selected by the tail query. The tilt, support endpoint, and retained mode count all change that grid.
Exact mathematical passage
(\Lambda,q,\varepsilon)\longmapsto\mathcal P_{\Lambda,q,\varepsilon}.
\mathcal P_{\Lambda,q,\tau}Use in source · 2 passages
Search aliases: P_Λ,q,τ mathcal Lambda tau
Use in source · Read passage · 08_adaptive.tex, line 29
The resulting object is query dependent. Schematically, (Λ,q,τ) ⟼ P_Λ,q,τ ⟼ (widehat p,certificate), where P_Λ,q,τ stores exact spikes, packet centers, centered moments, packet radii, and the selected COS parameters.
Exact mathematical passage
(\Lambda,q,\tau) \longmapsto \mathcal P_{\Lambda,q,\tau} \longmapsto (\widehat p,\text{certificate}), \label{eq:compiler-map}
\mathrm{pkt}Use in source · 4 passages
Search aliases: pkt mathrm
Use in source · Read passage · 07_inversion.tex, line 60
[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).
Exact mathematical passage
|\widehat T_{N,b}-T_{N,b}| \le \eps_{\mathrm{pkt}}, \label{eq:packet-tail-bound}
\PpUse in source · 3 passages
Search aliases: Pp P mathbb
Use in source · Read passage · 06_packets.tex, line 88
For any θ>0 in the transform domain, P(Q≥ q) =e^κ(θ)-θ q E_θ [e^-θ(Q-q) 1_{Q≥ q}]. This classical exponential-tilting identity lets K-COS evaluate the upper tail without subtracting a cumulative distribution function (CDF) value close to one.
Exact mathematical passage
\Pp(Q\ge q) =e^{\kappa(\theta)-\theta q} \E_{\theta}\!\left[e^{-\theta(Q-q)}\mathbf 1_{\{Q\ge q\}}\right]. \label{eq:tilt-tail}
\Pp_{\theta}Use in source · 1 passage
Search aliases: Pp_θ P_θ Pp theta mathbb theta
Use in source · Read passage · 07_inversion.tex, line 137
[Support and periodization bound] Let the finite positive spectrum have rank r≥1, and let b>q≥0, b>0, and 0<θ<1/(2λ_max). Let τ_θ(b)= P_θ(Q>b). Then the combined error from (i) replacing the tilted density on [0,b] by its full-coefficient cosine periodization and (ii) omitting the original integral over (b,∞) is bounded by ε_sup ≤ e^κ(θ)-θ q τ_θ(b) (1+e^-θ(b-q)).
Exact mathematical passage
\tau_{\theta}(b)=\Pp_{\theta}(Q>b).
\preceqUse in source · 2 passages
Search aliases: preceq
Use in source · Read passage · 05_spectrum.tex, line 734
Set S=Q^TQ. The first hypothesis gives the Loewner inequalities (1-η)Ipreceq Spreceq(1+η)I. The eigenvalues of B=(QD^1/2)(QD^1/2)^T are the same as those of D^1/2SD^1/2. Hence (1-η)D preceq D^1/2SD^1/2 preceq (1+η)D, and eigenvalue monotonicity yields (1-η)d_i ≤ λ_i(B) ≤ (1+η)d_i. Weyl's perturbation inequality and H-B _2≤ρ then give |λ_i(H)-λ_i(B)|≤ρ. Combining the two bounds and using Hsucceq0 proves the claim.
Exact mathematical passage
(1-\eta)I\preceq S\preceq(1+\eta)I.
\primeUse in source · 1 passage
Search aliases: prime
Use in source · Read passage · A_atlas.tex, line 198
J_k=∫_q^be^-θ(x-q)cos(u_kx) dx=Re [e^iu_kqfrac1-e^(-θ+iu_k)(b-q)θ-iu_k]. S_N=frac2b∑_k=0^N-1^prime J_k Re ϕ_θ(u_k), ϕ_θ(t)=e^κ(θ+it)-κ(θ). On a positive-support interval [0,b], the cosine frequencies are u_k=kπ/b. The payoff is zero below the threshold and decays exponentially above it. Integrating each cosine mode against that payoff gives an explicit coefficient.
Exact mathematical passage
S_N=\frac2b\sum_{k=0}^{N-1}{}^{\prime}\, J_k\,\Re\,\phi_\theta(u_k),\qquad\phi_\theta(t)=e^{\kappa(\theta+it)-\kappa(\theta)}.
Q^TUse in source · 4 passages
Search aliases: Q^T
Use in source · Read passage · 05_spectrum.tex, line 704
[Candidate eigensystem post-certification] Let B=QDQ^T. Suppose directed-rounding computation proves Q^TQ-I _2≤η<1 and H-B _2≤ρ. Then the ordered eigenvalues of the positive semidefinite matrix H satisfy max{0,(1-η)d_i-ρ} ≤ λ_i(H) ≤ (1+η)d_i+ρ for every i.
Exact mathematical passage
B=QDQ^T.
Q^\topUse in source · 2 passages
Search aliases: Q^top top
Use in source · Read passage · 04_range.tex, line 63
Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥, Q_⊥^top Q_⊥=I, Q^top Q_⊥=0. Here the columns of Q_⊥ form an orthonormal basis of the nonzero residual range. Zero residual directions are omitted.
Exact mathematical passage
A=QR, \qquad Q^\top Q=I,
Q_+Use in source · 3 passages
Search aliases: Q_+
Use in source · Read passage · 05_spectrum.tex, line 123
For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.
Exact mathematical passage
Q_-=\sum_i\ell_iZ_i^2, \qquad Q=\sum_i\lambda_iZ_i^2, \qquad Q_+=\sum_i u_iZ_i^2.
Q_-Use in source · 3 passages
Search aliases: Q_-
Use in source · Read passage · 05_spectrum.tex, line 123
For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.
Exact mathematical passage
Q_-=\sum_i\ell_iZ_i^2, \qquad Q=\sum_i\lambda_iZ_i^2, \qquad Q_+=\sum_i u_iZ_i^2.
q_0Use in source · 5 passages
Search aliases: q_0
Use in source · Read passage · 01_gaussian.tex, line 219
[Exact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈range(M^1/2S). In particular, Σsucc0 is sufficient. For singular covariance the shift is the squa…
Exact mathematical passage
\boxed{ Q \overset d= q_0 + \sum_i \lambda_i \chi^2_1(\delta_i), }
q_e^{(d)}Use in source · 1 passage
Search aliases: q_e^(d)
Use in source · Read passage · 03_phases.tex, line 420
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
q_e^{(p)} = \sum_{d=1}^{D}|a_d|q_e^{(d)}, \qquad \beta_e^{(p)}=\lVert q_e^{(p)}\rVert_2.
q_e^{(k)}Use in source · 3 passages
Search aliases: q_e^(k)
Use in source · Read passage · 03_phases.tex, line 343
For depth k≥1 and starting phase profile ρ_0=ρ(X), define the edge risk vector q_e^(k) = ∑_j=0^k-1 B^k-1-jΔ_e B^jρ_0 and its scalar risk β_e^(k)=lVert q_e^(k)rVert_2. Unlike the raw block norm B_hg, this quantity depends on the start profile and on Krylov depth.
Exact mathematical passage
\boxed{ q_e^{(k)} = \sum_{j=0}^{k-1} \mathcal B^{k-1-j}\Delta_e\mathcal B^j\rho_0 }
q_e^{(p)}Use in source · 1 passage
Search aliases: q_e^(p)
Use in source · Read passage · 03_phases.tex, line 420
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
q_e^{(p)} = \sum_{d=1}^{D}|a_d|q_e^{(d)}, \qquad \beta_e^{(p)}=\lVert q_e^{(p)}\rVert_2.
Q_rUse in source · 11 passages
Search aliases: Q_r
Use in source · Read passage · 04_range.tex, line 125
Suppose an orthonormal basis Q_r captures the factor range: range(A)⊆range(Q_r). Then A=Q_rB, B=Q_r^top A.
Exact mathematical passage
Q_r
Q_r^\topUse in source · 4 passages
Search aliases: Q_r^top top
Use in source · Read passage · 04_range.tex, line 130
Suppose an orthonormal basis Q_r captures the factor range: range(A)⊆range(Q_r). Then A=Q_rB, B=Q_r^top A.
Exact mathematical passage
A=Q_rB, \qquad B=Q_r^\top A.
Q_XUse in source · 6 passages
Search aliases: Q_X
Use in source · Read passage · 02_streaming.tex, line 21
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
Q_X\in\mathbb R^{n\times p}
Q_{X,I_r}Use in source · 1 passage
Search aliases: Q_X,I_r
Use in source · Read passage · 02_streaming.tex, line 284
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ A_{b,I_r} = H_{b,I_r}-Q_{X,I_r}R_b. }
Q_{X,I_r}^\topUse in source · 1 passage
Search aliases: Q_X,I_r^top top
Use in source · Read passage · 02_streaming.tex, line 275
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ R_b = \sum_r Q_{X,I_r}^\top H_{b,I_r}, }
Q_X^{\mathsf T}Use in source · 2 passages
Search aliases: Q_X^ T mathsf
Use in source · Read passage · A_atlas.tex, line 68
Q_X^ TQ_X=I, M_X=I-Q_XQ_X^ T, M_X^2=M_X=M_X^ T. A=M_XGW^1/2, K=AA^ T. For a covariate range basis Q_X with orthonormal columns, the residualizer M_X=I-Q_XQ_X^ T removes the component in the covariate span. It is symmetric and idempotent. Applying it to the weighted feature factor restricts the kernel to variation left after the null-model covariates have been removed.
Exact mathematical passage
Q_X^{\mathsf T}Q_X=I,\quad M_X=I-Q_XQ_X^{\mathsf T},\quad M_X^2=M_X=M_X^{\mathsf T}.
Q_X^\topUse in source · 4 passages
Search aliases: Q_X^top top
Use in source · Read passage · 02_streaming.tex, line 23
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
M_X=I-Q_XQ_X^\top, \qquad A=M_XGW^{1/2}, \qquad K=AA^\top.
Q_\perpUse in source · 6 passages
Search aliases: Q_⊥ perp
Use in source · Read passage · 04_range.tex, line 69
Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥, Q_⊥^top Q_⊥=I, Q^top Q_⊥=0. Here the columns of Q_⊥ form an orthonormal basis of the nonzero residual range. Zero residual directions are omitted.
Exact mathematical passage
C=Q^\top U, \qquad F=U-QC, \qquad F=Q_\perp R_\perp, \qquad Q_\perp^\top Q_\perp=I, \qquad Q^\top Q_\perp=0.
Q_\perp^\topUse in source · 1 passage
Search aliases: Q_⊥^top perp top
Use in source · Read passage · 04_range.tex, line 69
Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥, Q_⊥^top Q_⊥=I, Q^top Q_⊥=0. Here the columns of Q_⊥ form an orthonormal basis of the nonzero residual range. Zero residual directions are omitted.
Exact mathematical passage
C=Q^\top U, \qquad F=U-QC, \qquad F=Q_\perp R_\perp, \qquad Q_\perp^\top Q_\perp=I, \qquad Q^\top Q_\perp=0.
\operatorname{quadraticForm}Use in source · 1 passage
Search aliases: quadraticForm operatorname
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
RUse in source · 35 passages
Search aliases: R
Use in source · Read passage · 02_streaming.tex, line 267
A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.
Exact mathematical passage
I_1,\ldots,I_R
\RUse in source · 3 passages
Search aliases: R mathbb
Use in source · Read passage · B_lean.tex, line 25
The factor has a finite sample index type I and an arbitrary variant index type V. A variant block is a finite set s⊆ V. The factor entries are exact real numbers, A:I× V→ R. The Lean type is declRow -> Variant -> Real. The sample coordinates carry a declFintype instance. The variants carry a declDecidableEq instance to support finite-set unions, singleton blocks, and reindexing.
Exact mathematical passage
A:I\times V\to\R
\mathcal RUse in source · 4 passages
Search aliases: R mathcal
Use in source · Read passage · 03_phases.tex, line 125
Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0, 0, · otherwise, R= B- B_0. Symmetric pair retention preserves self-adjointness of H_0, although the following error theorem does not require positive semidefiniteness of H_0.
Exact mathematical passage
(\mathcal B_0)_{hg}= \begin{cases} \mathcal B_{hg},&(h,g)\in\mathcal E_0,\\ 0,&\text{otherwise}, \end{cases} \qquad \mathcal R=\mathcal B-\mathcal B_0.
R^2Use in source · 2 passages
Search aliases: R^2
Use in source · Read passage · 05_spectrum.tex, line 350
This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form. The second records how concentrated the residual spectrum is. The third is its largest coefficient.
Exact mathematical passage
t=\operatorname{tr}R, \qquad v=\operatorname{tr}(R^2), \qquad \rho=\|R\|_2.
r^2Use in source · 2 passages
Search aliases: r^2
Use in source · Read passage · 11_evidence.tex, line 290
[htbp] centering captionSynthetic rare-variant simulation scenarios. Adjacent r^2 is computed within the simulated LD blocks after variant standardization. lrrrrr scenario · n · m · rank · median adjacent r^2 · effective rank
Exact mathematical passage
r^2
\mathbb R^{4\times2}Use in source · 1 passage
Search aliases: R^4×2 mathbb times
Use in source · Read passage · 02_streaming.tex, line 422
The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0
Exact mathematical passage
A=[A_1\ A_2\ A_3],\qquad A_i\in\mathbb R^{4\times2}.
\mathbb R^{m\times s}Use in source · 2 passages
Search aliases: R^m× s mathbb times
Use in source · Read passage · 02_streaming.tex, line 184
Let V∈ R^m× s be partitioned conformally: V= V_1
Exact mathematical passage
V\in\mathbb R^{m\times s}
\mathbb R^{n\times c_b}Use in source · 1 passage
Search aliases: R^n× c_b mathbb times
Use in source · Read passage · A_atlas.tex, line 37
A=[A_1 ⋯ A_B], AA^ T=∑_b=1^B A_bA_b^ T. KV=∑_b A_b(A_b^ TV), A_b∈ R^n× c_b, V∈ R^n× s. Partition a factor by columns and keep every block. Horizontal concatenation turns the sample-space Gram matrix into a sum of block Gram matrices. Changing the block widths or the order in which blocks are read preserves the represented operator.
Exact mathematical passage
KV=\sum_b A_b(A_b^{\mathsf T}V),\qquad A_b\in\mathbb R^{n\times c_b},\quad V\in\mathbb R^{n\times s}.
\mathbb R^{n\times n}Use in source · 1 passage
Search aliases: R^n× n mathbb times
Use in source · Read passage · 05_spectrum.tex, line 691
A high-precision symmetric eigensolver is used only to generate candidate quantities Q∈ R^n× n, D=diag(d_1,…,d_n), d_1≥⋯≥ d_n≥0. No correctness claim depends on trusting this eigensolver.
Exact mathematical passage
Q\in\mathbb R^{n\times n}, \qquad D=\operatorname{diag}(d_1,\ldots,d_n), \qquad d_1\ge\cdots\ge d_n\ge0.
\mathbb R^{n\times p}Use in source · 1 passage
Search aliases: R^n× p mathbb times
Use in source · Read passage · 02_streaming.tex, line 21
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
Q_X\in\mathbb R^{n\times p}
\mathbb R^{n\times r}Use in source · 1 passage
Search aliases: R^n× r mathbb times
Use in source · Read passage · 05_spectrum.tex, line 314
Let U∈ R^n× r have orthonormal columns and define B=U^top A, F=(I-UU^top)A. Then A=UB+F, U^top F=0.
Exact mathematical passage
U\in\mathbb R^{n\times r}
\mathbb R^{n\times s}Use in source · 4 passages
Search aliases: R^n× s mathbb times
Use in source · Read passage · 02_streaming.tex, line 135
[One-pass sample-space streaming action] For every block vector V∈ R^n× s, KV = M_X ∑_b=1^B G_bW_bG_b^top M_XV.
Exact mathematical passage
V\in\mathbb R^{n\times s}
r_1Use in source · 1 passage
Search aliases: r_1
Use in source · Read passage · 05_spectrum.tex, line 529
The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.
Exact mathematical passage
r_1,\ldots,r_s
R_bUse in source · 4 passages
Search aliases: R_b
Use in source · Read passage · 02_streaming.tex, line 271
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
R_b=Q_X^\top H_b.
r_gUse in source · 1 passage
Search aliases: r_g
Use in source · Read passage · 05_spectrum.tex, line 48
For any vector x, let r_g= P_gx _2. Then P_hEx _2 ≤ ∑_g P_hEP_g _2 P_gx _2 = ( C r)_h. Orthogonality of the phase decomposition gives Ex _2^2 = ∑_h P_hEx _2^2 ≤ C r _2^2 ≤ C _2^2 r _2^2 = C _2^2 x _2^2. Taking the supremum over unit vectors proves the claim.
Exact mathematical passage
r_g=\|P_gx\|_2.
R_hUse in source · 3 passages
Search aliases: R_h
Use in source · Read passage · 04_range.tex, line 10
Let R be a newly generated Krylov residual and R_h=P_hR.
Exact mathematical passage
R_h=P_hR.
r_hUse in source · 3 passages
Search aliases: r_h
Use in source · Read passage · 04_range.tex, line 16
[Minimal exact phase width] Let r_h=rank(R_h). Every phase-supported basis spanning range(R_h) has at least r_h vectors, and a rank-revealing QR or compact SVD attains exactly r_h.
Exact mathematical passage
r_h=\operatorname{rank}(R_h).
r_{hg}Use in source · 1 passage
Search aliases: r_hg
Use in source · Read passage · 04_range.tex, line 45
If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).
Exact mathematical passage
r_{hg}
r_iUse in source · 2 passages
Search aliases: r_i
Use in source · Read passage · 05_spectrum.tex, line 530
The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.
Exact mathematical passage
t=\sum_i r_i, \qquad v=\sum_i r_i^2, \qquad r_i\le\rho.
r_i^2Use in source · 2 passages
Search aliases: r_i^2
Use in source · Read passage · 05_spectrum.tex, line 530
The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.
Exact mathematical passage
t=\sum_i r_i, \qquad v=\sum_i r_i^2, \qquad r_i\le\rho.
R_{P,p}Use in source · 4 passages
Search aliases: R_P,p
Use in source · Read passage · 06_packets.tex, line 123
tered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P,p(s)| ≤ fracn_P2 fracρ_P(s)^p+1(p+1)(1-ρ_P(s)).
Exact mathematical passage
\kappa_P(s) &=-\frac{n_P}{2}\log(1-2s\mu_P) +\frac12\sum_{m=1}^{p} \frac{\alpha_P(s)^m}{m}M_{P,m} +R_{P,p}(s), \label{eq:packet-expansion}
r_{\rm eff}Use in source · 1 passage
Search aliases: r_rm eff rm
Use in source · Read passage · 11_evidence.tex, line 305
Effective rank here is the participation ratio r_rm eff=frac(∑_jλ_j)^2∑_jλ_j^2. It measures spectral concentration, not an integer count of nonzero directions. The corresponding spectra therefore span a substantial range of concentration: stronger LD reduces effective rank even when the nominal quadratic-form rank remains close to the sample residual dimension.
Exact mathematical passage
r_{\rm eff}=\frac{(\sum_j\lambda_j)^2}{\sum_j\lambda_j^2}.
r_sUse in source · 1 passage
Search aliases: r_s
Use in source · Read passage · 05_spectrum.tex, line 529
The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.
Exact mathematical passage
r_1,\ldots,r_s
R_\perpUse in source · 3 passages
Search aliases: R_⊥ perp
Use in source · Read passage · 04_range.tex, line 69
Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥, Q_⊥^top Q_⊥=I, Q^top Q_⊥=0. Here the columns of Q_⊥ form an orthonormal basis of the nonzero residual range. Zero residual directions are omitted.
Exact mathematical passage
C=Q^\top U, \qquad F=U-QC, \qquad F=Q_\perp R_\perp, \qquad Q_\perp^\top Q_\perp=I, \qquad Q^\top Q_\perp=0.
\operatorname{range}Use in source · 14 passages
Search aliases: range operatorname
Use in source · Read passage · 01_gaussian.tex, line 246
s λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈range(M^1/2S). In particular, Σsucc0 is sufficient. For singular covariance the shift is the squared deterministic component perpendicular to the random …
Exact mathematical passage
\operatorname{range}(T)
\operatorname{rank}Use in source · 4 passages
Search aliases: rank operatorname
Use in source · Read passage · 04_range.tex, line 16
[Minimal exact phase width] Let r_h=rank(R_h). Every phase-supported basis spanning range(R_h) has at least r_h vectors, and a rank-revealing QR or compact SVD attains exactly r_h.
Exact mathematical passage
r_h=\operatorname{rank}(R_h).
\rceilUse in source · 2 passages
Search aliases: rceil
Use in source · Read passage · 05_spectrum.tex, line 539
[Residual-rank bracket] lceil max{ fractρ, fract^2v } rceil ≤ rankF ≤ n-r. The zero-residual case is interpreted separately as rank zero.
Exact mathematical passage
\boxed{ \left\lceil \max\left\{ \frac{t}{\rho}, \frac{t^2}{v} \right\} \right\rceil \le \operatorname{rank}F \le n-r. }
\rVertUse in source · 15 passages
Search aliases: rVert
Use in source · Read passage · 03_phases.tex, line 81
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
\rho(X)_g=\lVert P_gX\rVert_F,
\rVert_2Use in source · 9 passages
Search aliases: rVert_2 rVert
Use in source · Read passage · 03_phases.tex, line 85
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
\mathcal B_{hg}=\lVert P_hHP_g\rVert_2.
\rVert_FUse in source · 6 passages
Search aliases: rVert_F rVert
Use in source · Read passage · 03_phases.tex, line 81
Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.
Exact mathematical passage
\rho(X)_g=\lVert P_gX\rVert_F,
\rVert_F^2Use in source · 1 passage
Search aliases: rVert_F^2 rVert
Use in source · Read passage · 03_phases.tex, line 180
ing Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
\lVert Y\rVert_F^2=\sum_h\lVert P_hY\rVert_F^2,
\boldsymbol SUse in source · 4 passages
Search aliases: S boldsymbol
Use in source · Read passage · 09_intervals.tex, line 175
Let S_N be the directed-rounding interval containing the finite N-mode tilted COS sum, and set E=E_rm support+E_rm series. Then S_θ(q) ∈ S_N+[-E,E].
Exact mathematical passage
\boldsymbol S_N
S'Use in source · 5 passages
Search aliases: S'
Use in source · Read passage · 12_synthesis.tex, line 60
[Storage-phase commutation theorem] Let A=M_XGW^1/2, H=A^top A, and let S and S' be any two exact storage refinements that respect the same structural phase partition. Then: [label=(roman*)] item the exact operator H is identical under S and S'. item every phase coupling P_hHP_g is identical under S and S'. item the exact phase graph and every k-step polyphase light cone are identical under S and S'. item any exact matrix-free Krylov method applied through streamed products produces the same Kry…
Exact mathematical passage
S'
S^{\mathsf T}Use in source · 1 passage
Search aliases: S^ T mathsf
Use in source · Read passage · A_atlas.tex, line 88
The finite control is a nine-phase chain with H=2I+S+S^ T and a unit start in phase five. The printed entries are exact integers in H^kx, and the shaded support expands by one phase at each step. The script checks every zero outside the cone.
Exact mathematical passage
H=2I+S+S^{\mathsf T}
s_kUse in source · 10 passages
Search aliases: s_k
Use in source · Read passage · 07_inversion.tex, line 46
Let widehatκ(s_k) be the packet approximation at s_k=θ+i u_k, and suppose |widehatκ(s_k)-κ(s_k)|≤ B_k. For disjoint packets, B_k can be chosen as the sum of the packet bounds [source reference].
Exact mathematical passage
\widehat\kappa(s_k)
S_NUse in source · 1 passage
Search aliases: S_N
Use in source · Read passage · A_atlas.tex, line 198
J_k=∫_q^be^-θ(x-q)cos(u_kx) dx=Re [e^iu_kqfrac1-e^(-θ+iu_k)(b-q)θ-iu_k]. S_N=frac2b∑_k=0^N-1^prime J_k Re ϕ_θ(u_k), ϕ_θ(t)=e^κ(θ+it)-κ(θ). On a positive-support interval [0,b], the cosine frequencies are u_k=kπ/b. The payoff is zero below the threshold and decays exponentially above it. Integrating each cosine mode against that payoff gives an explicit coefficient.
Exact mathematical passage
S_N=\frac2b\sum_{k=0}^{N-1}{}^{\prime}\, J_k\,\Re\,\phi_\theta(u_k),\qquad\phi_\theta(t)=e^{\kappa(\theta+it)-\kappa(\theta)}.
\boldsymbol S_NUse in source · 4 passages
Search aliases: S_N boldsymbol
Use in source · Read passage · 09_intervals.tex, line 175
Let S_N be the directed-rounding interval containing the finite N-mode tilted COS sum, and set E=E_rm support+E_rm series. Then S_θ(q) ∈ S_N+[-E,E].
Exact mathematical passage
\boldsymbol S_N
S_\thetaUse in source · 4 passages
Search aliases: S_θ theta
Use in source · Read passage · 09_intervals.tex, line 46
Choose any 0<θ<frac12λ_max. Exponential tilting gives P(Q≥ q) = e^κ(θ)-θ q S_θ(q), where S_θ(q) = E_θ [ e^-θ(Q-q) 1_{Q≥ q} ].
Exact mathematical passage
P(Q\ge q) = e^{\kappa(\theta)-\theta q} S_\theta(q),
\sinUse in source · 1 passage
Search aliases: sin
Use in source · Read passage · 11_evidence.tex, line 110
The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponential tilting, the COS expansion, spectral packets, or the K-COS error controller.
Exact mathematical passage
P(Q\ge q) = \frac12 + \frac1\pi \int_0^\infty \frac{\sin\Theta(u)}{u\,R(u)}\,du,
\operatorname{spec}Use in source · 1 passage
Search aliases: spec operatorname
Use in source · Read passage · A_atlas.tex, line 21
X=Σ^1/2Z, Z∼ N(0,I), H=Σ^1/2MΣ^1/2. X^ TMXoverset d=∑_j=1^rλ_jZ_j^2, λ_j∈spec(H)∖{0}. For a central Gaussian vector with covariance Σsucceq0 and symmetric M, the quadratic form becomes a weighted sum of independent standard-normal squares after whitening and orthogonal diagonalization. The operator may originate in a kernel, a signal model, or another symmetric construction. Once its spectrum has been correctly obtained, the tail layer sees those coefficients.
Exact mathematical passage
X^{\mathsf T}MX\overset d=\sum_{j=1}^{r}\lambda_jZ_j^2,\qquad \lambda_j\in\operatorname{spec}(H)\setminus\{0\}.
\operatorname{streamedAction}Use in source · 1 passage
Search aliases: streamedAction operatorname
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
\succUse in source · 1 passage
Search aliases: succ
Use in source · Read passage · 01_gaussian.tex, line 252
2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈range(M^1/2S). In particular, Σsucc0 is sufficient. For singular covariance the shift is the squared deterministic component perpendicular to the random range in these weighted coordinates.
Exact mathematical passage
\Sigma\succ0
\mathrm{sup}Use in source · 2 passages
Search aliases: sup mathrm
Use in source · Read passage · 07_inversion.tex, line 141
nd] Let the finite positive spectrum have rank r≥1, and let b>q≥0, b>0, and 0<θ<1/(2λ_max). Let τ_θ(b)= P_θ(Q>b). Then the combined error from (i) replacing the tilted density on [0,b] by its full-coefficient cosine periodization and (ii) omitting the original integral over (b,∞) is bounded by ε_sup ≤ e^κ(θ)-θ q τ_θ(b) (1+e^-θ(b-q)).
Exact mathematical passage
\eps_{\mathrm{sup}} \le e^{\kappa(\theta)-\theta q}\, \tau_{\theta}(b) \left(1+e^{-\theta(b-q)}\right). \label{eq:eps-support}
TUse in source · 28 passages
Search aliases: T
Use in source · Read passage · 01_gaussian.tex, line 245
xact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈range(M^1/2S). In particular, Σsucc0 is sufficient. For singular covariance the shift is the square…
Exact mathematical passage
T=M^{1/2}S
t^{-r/2}Use in source · 3 passages
Search aliases: t^-r/2
Use in source · Read passage · 09_intervals.tex, line 130
Since 1+(a_jt)^2≥(a_jt)^2, we have |ϕ_θ(t)| ≤ C_θ t^-r/2, C_θ=∏_j a_j^-1/2.
Exact mathematical passage
|\phi_\theta(t)| \le C_\theta t^{-r/2}, \qquad C_\theta=\prod_j a_j^{-1/2}.
T^{-\nu}Use in source · 1 passage
Search aliases: T^-ν nu
Use in source · Read passage · 11_evidence.tex, line 139
The four spectra include balanced, skewed, dominant-spike, and higher-rank cases. Each reference integral was evaluated at two truncation limits. The omitted integral after the frequency cutoff T>0 was bounded using frac1π ∫_T^∞fracduuR(u) ≤ fracCπνT^-ν, ν=frac12∑_jh_j, C=∏_jλ_j^-h_j/2.
Exact mathematical passage
\frac1\pi \int_T^\infty\frac{du}{uR(u)} \le \frac{C}{\pi\nu}T^{-\nu}, \qquad \nu=\frac12\sum_jh_j, \qquad C=\prod_j\lambda_j^{-h_j/2}.
T^2Use in source · 1 passage
Search aliases: T^2
Use in source · Read passage · 05_spectrum.tex, line 427
Factor the determinant through the known core: I-2θ(C+R) = (I-2θ C)^1/2 (I-2θ T) (I-2θ C)^1/2, where T=(I-2θ C)^-1/2 R (I-2θ C)^-1/2. Since (I-2θ C)^-1/2 _2^2=α, we have trT≤α t, tr(T^2)≤α^2v, T _2≤αρ. For every eigenvalue x of T, -frac12log(1-2θ x) ≤ θ x + fracθ^2x^2 1-2θαρ. The scalar inequality follows from the power series of -log(1-u) and a geometric bound on all terms of degree at least two. Summing over the eigenvalues of T proves the result.
Exact mathematical passage
\operatorname{tr}T\le\alpha t, \qquad \operatorname{tr}(T^2)\le\alpha^2v, \qquad \|T\|_2\le\alpha\rho.
t^2Use in source · 5 passages
Search aliases: t^2
Use in source · Read passage · 02_streaming.tex, line 311
With sample tile height t and resident variant width c, the dominant working storage, including a resident covariate tile, is O(tc+pc+tp+t^2). This count excludes the backing input store, completed output blocks, allocator overhead, and library workspaces.
Exact mathematical passage
O(tc+pc+tp+t^2).
t^dUse in source · 1 passage
Search aliases: t^d
Use in source · Read passage · 03_phases.tex, line 416
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
p(t)=\sum_{d=0}^{D}a_dt^d.
t^jUse in source · 1 passage
Search aliases: t^j
Use in source · Read passage · 03_phases.tex, line 225
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
p(t)=\sum_{j=0}^d a_jt^j.
T^\topUse in source · 1 passage
Search aliases: T^top top
Use in source · Read passage · 01_gaussian.tex, line 282
xpand (μ+SZ)^top M(μ+SZ) = Z^top SMSZ + 2μ^top MSZ + μ^top Mμ and rotate by V. Completing the square gives the displayed noncentral components. For a null vector of SMS, M^1/2Sv=0, hence v^top SMμ = (M^1/2Sv)^top M^1/2μ = 0, so there is no unresolved stochastic linear term in a zero-eigenvalue direction. For the positive eigenvectors v_i of T^top T=SMS, the vectors Tv_i/sqrtλ_i form an orthonormal basis of range(T). Their inner products with b are c_i/sqrtλ_i. Hence the subtracted sum equals P_T…
Exact mathematical passage
T^\top T=SMS
t_0Use in source · 5 passages
Search aliases: t_0
Use in source · Read passage · 07_inversion.tex, line 211
[Power envelope from one frequency] For t≥ t_0>0, |ϕ_θ(t)| ≤ |ϕ_θ(t_0)| (fractt_0)^-β(t_0).
Exact mathematical passage
t\ge t_0>0
t_1Use in source · 1 passage
Search aliases: t_1
Use in source · Read passage · 03_phases.tex, line 597
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
\boxed{ T_{\rm pipe} = t_1 + \sum_{b=1}^{B-1}\max\{c_b,t_{b+1}\} + c_B. }
T_bUse in source · 1 passage
Search aliases: T_b
Use in source · Read passage · 02_streaming.tex, line 161
[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.
Exact mathematical passage
T_b=G_b^\top U, \qquad Y\leftarrow Y+G_b(W_bT_b).
t_bUse in source · 3 passages
Search aliases: t_b
Use in source · Read passage · 03_phases.tex, line 587
The mathematical streaming identities do not require transfers and GEMMs to be serial. Consider B storage blocks. Let t_b be the transfer time for block b and c_b its device compute time.
Exact mathematical passage
t_b
t_{b+1}Use in source · 1 passage
Search aliases: t_b+1
Use in source · Read passage · 03_phases.tex, line 597
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
\boxed{ T_{\rm pipe} = t_1 + \sum_{b=1}^{B-1}\max\{c_b,t_{b+1}\} + c_B. }
T_{N,b}Use in source · 2 passages
Search aliases: T_N,b
Use in source · Read passage · 07_inversion.tex, line 35
If the exact tilted characteristic function is used, the N-mode finite-window expression is T_N,b =frac2b∑_k=0^N-1' J_k Re[e^κ(θ+i u_k)-θ q]. The cancellation of κ(θ) in [source reference] is useful numerically: the likelihood-ratio prefactor from [source reference] and the normalization in [source reference] do not need to be formed separately inside each mode.
Exact mathematical passage
T_{N,b} =\frac{2}{b}\sum_{k=0}^{N-1}{}' J_k\, \Re\left[e^{\kappa(\theta+\ii u_k)-\theta q}\right]. \label{eq:tail-cos}
T_{\rm pipe}Use in source · 2 passages
Search aliases: T_rm pipe rm
Use in source · Read passage · 03_phases.tex, line 597
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
\boxed{ T_{\rm pipe} = t_1 + \sum_{b=1}^{B-1}\max\{c_b,t_{b+1}\} + c_B. }
T_{\rm serial}Use in source · 1 passage
Search aliases: T_rm serial rm
Use in source · Read passage · 03_phases.tex, line 593
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
T_{\rm serial}=\sum_{b=1}^{B}(t_b+c_b),
\operatorname{tr}Use in source · 3 passages
Search aliases: tr operatorname
Use in source · Read passage · 05_spectrum.tex, line 350
This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form. The second records how concentrated the residual spectrum is. The third is its largest coefficient.
Exact mathematical passage
t=\operatorname{tr}R, \qquad v=\operatorname{tr}(R^2), \qquad \rho=\|R\|_2.
UUse in source · 42 passages
Search aliases: U
Use in source · Read passage · 02_streaming.tex, line 158
[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.
Exact mathematical passage
U=M_XV
U^{\mathsf T}Use in source · 4 passages
Search aliases: U^ T mathsf
Use in source · Read passage · A_atlas.tex, line 116
A=UB+F, U^ TF=0, A^ TA=B^ TB+F^ TF. t=trR= F _F^2, v=tr(R^2), ρ_R= R _2. Let U contain d orthonormal sample-space directions. Project the factor into its captured range coordinates B=U^ TA and residual F=(I-UU^ T)A. Orthogonality eliminates the cross terms in the feature-space Gram matrix.
Exact mathematical passage
A=UB+F,\quad U^{\mathsf T}F=0,\quad A^{\mathsf T}A=B^{\mathsf T}B+F^{\mathsf T}F.
u^2Use in source · 1 passage
Search aliases: u^2
Use in source · Read passage · 11_evidence.tex, line 121
The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponential tilting, the COS expansion, spectral packets, or the K-COS error controller.
Exact mathematical passage
\Theta(u) = \frac12\sum_j h_j\arctan(\lambda_j u) - \frac12qu, \qquad R(u) = \prod_j \left(1+\lambda_j^2u^2\right)^{h_j/4}.
U^\topUse in source · 6 passages
Search aliases: U^top top
Use in source · Read passage · 05_spectrum.tex, line 220
There is a second route that does not use the pruned operator as the final spectral approximation. Let U be any orthonormal sample-space basis, possibly generated cheaply from a pruned or approximate Krylov process, and define B=U^top A, F=(I-UU^top)A, δ= F _2.
Exact mathematical passage
B=U^\top A, \qquad F=(I-UU^\top)A, \qquad \delta=\|F\|_2.
U_{hg}Use in source · 2 passages
Search aliases: U_hg
Use in source · Read passage · 04_range.tex, line 40
If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).
Exact mathematical passage
P_hHP_g = U_{hg}V_{hg}^\top
u_iUse in source · 6 passages
Search aliases: u_i
Use in source · Read passage · 05_spectrum.tex, line 102
[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.
Exact mathematical passage
\boxed{ \ell_i = \max\{0,\mu_i-\varepsilon\} \le \lambda_i \le u_i = \max\{0,\mu_i+\varepsilon\}. }
u_NUse in source · 5 passages
Search aliases: u_N
Use in source · Read passage · 07_inversion.tex, line 238
[Omitted COS-mode certificate] Let N≥1 and b>q≥0. Set u_N=Nπ/b and β_N=β(u_N). If β_N>0, then the contribution of all exact tilted-COS modes k≥ N to the weighted tail functional is bounded by ε_COS ≤ e^κ(θ)-θ q frac4|ϕ_θ(u_N)|π (frac1N+frac1β_N).
Exact mathematical passage
u_N=N\pi/b
\uparrowUse in source · 1 passage
Search aliases: uparrow
Use in source · Read passage · 06_packets.tex, line 71
By [source reference], κ'(θ)=∑_jfracλ_j1-2θλ_j is continuous and strictly increasing on the stated interval. It equals E Q at θ=0 and diverges as θuparrow1/(2λ_max). The intermediate value theorem and strict monotonicity give existence and uniqueness.
Exact mathematical passage
\theta\uparrow1/(2\lambda_{\max})
VUse in source · 31 passages
Search aliases: V
Use in source · Read passage · 01_gaussian.tex, line 104
Write X=SZ with Z∼ N(0,I) and diagonalize SMS=VΛ V^top. Rotational invariance of the standard normal gives Z^top VΛ V^top Z overset d= ∑_jλ_jZ_j^2. No positivity assumption on M is needed for this central reduction.
Exact mathematical passage
SMS=V\Lambda V^\top.
vUse in source · 30 passages
Search aliases: v
Use in source · Read passage · 01_gaussian.tex, line 65
If AA^top u=λ u with λ>0, then A^top u≠0 and A^top A(A^top u)=λ A^top u. Conversely, an eigenvector v of A^top A with positive eigenvalue maps to the eigenvector Av of AA^top. These maps restrict to inverse isomorphisms of the corresponding eigenspaces after scaling, so multiplicities agree. Horizontal concatenation gives AA^top=∑_b A_bA_b^top by direct matrix multiplication. Orthogonal diagonalization then identifies the associated Gaussian quadratic-form laws.
Exact mathematical passage
v
V^\topUse in source · 5 passages
Search aliases: V^top top
Use in source · Read passage · 01_gaussian.tex, line 104
Write X=SZ with Z∼ N(0,I) and diagonalize SMS=VΛ V^top. Rotational invariance of the standard normal gives Z^top VΛ V^top Z overset d= ∑_jλ_jZ_j^2. No positivity assumption on M is needed for this central reduction.
Exact mathematical passage
SMS=V\Lambda V^\top.
v^\topUse in source · 6 passages
Search aliases: v^top top
Use in source · Read passage · 01_gaussian.tex, line 274
Expand (μ+SZ)^top M(μ+SZ) = Z^top SMSZ + 2μ^top MSZ + μ^top Mμ and rotate by V. Completing the square gives the displayed noncentral components. For a null vector of SMS, M^1/2Sv=0, hence v^top SMμ = (M^1/2Sv)^top M^1/2μ = 0, so there is no unresolved stochastic linear term in a zero-eigenvalue direction. For the positive eigenvectors v_i of T^top T=SMS, the vectors Tv_i/sqrtλ_i form an orthonormal basis of range(T). Their inner products with b are c_i/sqrtλ_i. Hence the subtracted sum equals P_…
Exact mathematical passage
v^\top SM\mu = (M^{1/2}Sv)^\top M^{1/2}\mu = 0,
V_1Use in source · 1 passage
Search aliases: V_1
Use in source · Read passage · 02_streaming.tex, line 185
Let V∈ R^m× s be partitioned conformally: V= V_1 vdots V_B .
Exact mathematical passage
V= \begin{bmatrix} V_1\\ \vdots\\ V_B \end{bmatrix}.
V_BUse in source · 1 passage
Search aliases: V_B
Use in source · Read passage · 02_streaming.tex, line 185
Let V∈ R^m× s be partitioned conformally: V= V_1 vdots V_B .
Exact mathematical passage
V= \begin{bmatrix} V_1\\ \vdots\\ V_B \end{bmatrix}.
V_bUse in source · 1 passage
Search aliases: V_b
Use in source · Read passage · 02_streaming.tex, line 195
[Streamed AV] AV = M_X ∑_b=1^B G_bW_b^1/2V_b.
Exact mathematical passage
\boxed{ AV = M_X \sum_{b=1}^B G_bW_b^{1/2}V_b. }
V_{hg}^\topUse in source · 2 passages
Search aliases: V_hg^top top
Use in source · Read passage · 04_range.tex, line 40
If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).
Exact mathematical passage
P_hHP_g = U_{hg}V_{hg}^\top
v_iUse in source · 4 passages
Search aliases: v_i
Use in source · Read passage · 01_gaussian.tex, line 282
proof Expand (μ+SZ)^top M(μ+SZ) = Z^top SMSZ + 2μ^top MSZ + μ^top Mμ and rotate by V. Completing the square gives the displayed noncentral components. For a null vector of SMS, M^1/2Sv=0, hence v^top SMμ = (M^1/2Sv)^top M^1/2μ = 0, so there is no unresolved stochastic linear term in a zero-eigenvalue direction. For the positive eigenvectors v_i of T^top T=SMS, the vectors Tv_i/sqrtλ_i form an orthonormal basis of range(T). Their inner products with b are c_i/sqrtλ_i. Hence the subtracted sum equ…
Exact mathematical passage
v_i
v_jUse in source · 2 passages
Search aliases: v_j
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
\vdotsUse in source · 2 passages
Search aliases: vdots
Use in source · Read passage · 02_streaming.tex, line 103
Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.
Exact mathematical passage
[A_1\ \cdots\ A_B] \begin{bmatrix} A_1^\top\\ \vdots\\ A_B^\top \end{bmatrix} = \sum_bA_bA_b^\top.
WUse in source · 31 passages
Search aliases: W
Use in source · Read passage · 01_gaussian.tex, line 354
1 · -1 · -1 · 1 satisfies V^top K V=diag(1,1,2,2). For Z∼ N(0,I_4), Q=Z^top KZ overset d= Y+2W, Y,W independent with law χ^2_2. The spectrum is therefore (1,1,2,2) and E Q=6. The six-dimensional feature Gram A^top A has these same four nonzero eigenvalues and two additional zeros.
Exact mathematical passage
Q=Z^\top KZ\ \overset d=\ Y+2W, \qquad Y,W\text{ independent with law }\chi^2_2.
W^{1/2}Use in source · 6 passages
Search aliases: W^1/2
Use in source · Read passage · 02_streaming.tex, line 23
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
M_X=I-Q_XQ_X^\top, \qquad A=M_XGW^{1/2}, \qquad K=AA^\top.
w_1Use in source · 1 passage
Search aliases: w_1
Use in source · Read passage · 02_streaming.tex, line 15
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
G\in\mathbb R^{n\times m}, \qquad W=\operatorname{diag}(w_1,\ldots,w_m), \quad w_j\ge 0,
W_bUse in source · 5 passages
Search aliases: W_b
Use in source · Read passage · 02_streaming.tex, line 71
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
W_b^{1/2}Use in source · 7 passages
Search aliases: W_b^1/2
Use in source · Read passage · 02_streaming.tex, line 71
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
W_{J_b,J_b}Use in source · 1 passage
Search aliases: W_J_b,J_b
Use in source · Read passage · 02_streaming.tex, line 71
Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].
Exact mathematical passage
G_b=G_{[:,J_b]}, \qquad W_b=W_{J_b,J_b}, \qquad A_b=M_XG_bW_b^{1/2}.
w_mUse in source · 1 passage
Search aliases: w_m
Use in source · Read passage · 02_streaming.tex, line 15
Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.
Exact mathematical passage
G\in\mathbb R^{n\times m}, \qquad W=\operatorname{diag}(w_1,\ldots,w_m), \quad w_j\ge 0,
\operatorname{weightedSquares}Use in source · 1 passage
Search aliases: weightedSquares operatorname
Use in source · Read passage · B_lean.tex, line 146
The separate definition weightedSquares(s,c,z)=∑_k∈ sc_k z_k^2 models a supplied diagonal quadratic form. Its coefficients need not be eigenvalues obtained by a proved diagonalization. It has three theorems: item declweightedSquares_mono: if l_k≤ u_k for every k∈ s, then ∑ l_kz_k^2≤∑ u_kz_k^2. item declweightedSquares_nonneg: if c_k≥0 for every k∈ s, then ∑ c_kz_k^2≥0. item declweightedSquares_tail_imp: coefficientwise order gives the pointwise implication q≤∑ l_kz_k^2⇒ q≤∑ u_kz_k^2. The first a…
Exact mathematical passage
\operatorname{weightedSquares}(s,c,z)=\sum_{k\in s}c_k z_k^2
\widehat pUse in source · 7 passages
Search aliases: widehat p widehat
Use in source · Read passage · 07_inversion.tex, line 283
[K-COS analytic error certificate] Let p= P(Q≥ q), and let widehat p be the finite K-COS approximation formed with [label=(roman*)] item an upper interval endpoint b>q, item an integer N≥1 of cosine modes, item a packet approximation satisfying [source reference] on the finite grid. Under the positive-spectrum and parameter hypotheses of Proposition [source reference], |p-widehat p| ≤ ε_an :=ε_pkt+ε_sup+ε_COS. If ε_an<|widehat p|, then frac|p-widehat p||p| ≤ fracε_an |widehat p|-ε_an. end
Exact mathematical passage
\widehat p
\widehat TUse in source · 2 passages
Search aliases: widehat T widehat
Use in source · Read passage · 07_inversion.tex, line 59
[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).
Exact mathematical passage
\widehat T_{N,b}
\widehat T_{N,b}Use in source · 2 passages
Search aliases: widehat T_N,b widehat
Use in source · Read passage · 07_inversion.tex, line 59
[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).
Exact mathematical passage
\widehat T_{N,b}
\widehat xUse in source · 6 passages
Search aliases: widehat x widehat
Use in source · Read passage · B_lean.tex, line 172
The interval module uses two elementary predicates: Enclosed(x,L,U) · :Longleftrightarrow L≤ x≤ U, ErrorBound(x,widehat x,r) · :Longleftrightarrow |x-widehat x|≤ r. The equivalence declerrorBound_iff_enclosed proves |x-widehat x|≤ r Longleftrightarrow widehat x-r≤ x≤widehat x+r. There is no extra nonnegativity hypothesis: if this predicate holds, then the radius is necessarily nonnegative. The definition accepts arbitrary real inputs, and the theorem states the exact logical equivalence.
Exact mathematical passage
\operatorname{Enclosed}(x,L,U)&\;:\Longleftrightarrow\;L\le x\le U,\\ \operatorname{ErrorBound}(x,\widehat x,r)&\;:\Longleftrightarrow\; |x-\widehat x|\le r.
\widehat\kappaUse in source · 7 passages
Search aliases: widehatκ widehat kappa
Use in source · Read passage · 05_spectrum.tex, line 450
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehatκ_H(θ)-θ q}, where widehatκ_H is the preceding MGF upper bound.
Exact mathematical passage
\boxed{ \Pr(Q\ge q) \le \inf_{0<2\theta(c+\rho)<1} \exp\{\widehat\kappa_H(\theta)-\theta q\}, }
\widehat\kappa_HUse in source · 2 passages
Search aliases: widehatκ_H widehat kappa
Use in source · Read passage · 05_spectrum.tex, line 450
[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehatκ_H(θ)-θ q}, where widehatκ_H is the preceding MGF upper bound.
Exact mathematical passage
\boxed{ \Pr(Q\ge q) \le \inf_{0<2\theta(c+\rho)<1} \exp\{\widehat\kappa_H(\theta)-\theta q\}, }
\widetilde\sigmaUse in source · 3 passages
Search aliases: widetildeσ widetilde sigma
Use in source · Read passage · 05_spectrum.tex, line 231
[Range-core singular-value certificate] For A∈ R^n× m, list the singular values of A and B in decreasing order and pad both lists with zeros to length n. Write these lists as σ_i(A) and widetildeσ_i(B), respectively. They are the square roots of the n eigenvalues of AA^top and (UB)(UB)^top. Then, for 1≤ i≤ n, | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\widetilde\sigma_i(B)
\widetilde\sigma_iUse in source · 3 passages
Search aliases: widetildeσ_i widetilde sigma
Use in source · Read passage · 05_spectrum.tex, line 231
[Range-core singular-value certificate] For A∈ R^n× m, list the singular values of A and B in decreasing order and pad both lists with zeros to length n. Write these lists as σ_i(A) and widetildeσ_i(B), respectively. They are the square roots of the n eigenvalues of AA^top and (UB)(UB)^top. Then, for 1≤ i≤ n, | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\widetilde\sigma_i(B)
XUse in source · 66 passages
Search aliases: X
Use in source · Read passage · 01_gaussian.tex, line 14
Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.
Exact mathematical passage
X\sim N(0,\Sigma), \qquad M\succeq0,
X^{\mathsf T}Use in source · 1 passage
Search aliases: X^ T mathsf
Use in source · Read passage · A_atlas.tex, line 21
X=Σ^1/2Z, Z∼ N(0,I), H=Σ^1/2MΣ^1/2. X^ TMXoverset d=∑_j=1^rλ_jZ_j^2, λ_j∈spec(H)∖{0}. For a central Gaussian vector with covariance Σsucceq0 and symmetric M, the quadratic form becomes a weighted sum of independent standard-normal squares after whitening and orthogonal diagonalization. The operator may originate in a kernel, a signal model, or another symmetric construction. Once its spectrum has been correctly obtained, the tail layer sees those coefficients.
Exact mathematical passage
X^{\mathsf T}MX\overset d=\sum_{j=1}^{r}\lambda_jZ_j^2,\qquad \lambda_j\in\operatorname{spec}(H)\setminus\{0\}.
x^{-\beta_N-1}Use in source · 1 passage
Search aliases: x^-β_N-1 beta
Use in source · Read passage · 07_inversion.tex, line 271
omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1+fracN^-β_Nβ_N, which yields [source reference].
Exact mathematical passage
\sum_{k=N}^{\infty}k^{-\beta_N-1} \le N^{-\beta_N-1}+\int_N^{\infty}x^{-\beta_N-1}\,dx =N^{-\beta_N-1}+\frac{N^{-\beta_N}}{\beta_N},
x^2Use in source · 1 passage
Search aliases: x^2
Use in source · Read passage · 05_spectrum.tex, line 435
ginproof Factor the determinant through the known core: I-2θ(C+R) = (I-2θ C)^1/2 (I-2θ T) (I-2θ C)^1/2, where T=(I-2θ C)^-1/2 R (I-2θ C)^-1/2. Since (I-2θ C)^-1/2 _2^2=α, we have trT≤α t, tr(T^2)≤α^2v, T _2≤αρ. For every eigenvalue x of T, -frac12log(1-2θ x) ≤ θ x + fracθ^2x^2 1-2θαρ. The scalar inequality follows from the power series of -log(1-u) and a geometric bound on all terms of degree at least two. Summing over the eigenvalues of T proves the result.
Exact mathematical passage
-\frac12\log(1-2\theta x) \le \theta x + \frac{\theta^2x^2} {1-2\theta\alpha\rho}.
x^jUse in source · 1 passage
Search aliases: x^j
Use in source · Read passage · 09_intervals.tex, line 225
For the control Q=0.02χ^2_80, the survival probability has the finite exact formula P(Q≥ q) = e^-x ∑_j=0^39 fracx^jj!, x=fracq0.04.
Exact mathematical passage
P(Q\ge q) = e^{-x} \sum_{j=0}^{39} \frac{x^j}{j!}, \qquad x=\frac{q}{0.04}.
X^\topUse in source · 2 passages
Search aliases: X^top top
Use in source · Read passage · 01_gaussian.tex, line 20
Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.
Exact mathematical passage
Q=X^\top MX = Z^\top(SMS)Z.
YUse in source · 11 passages
Search aliases: Y
Use in source · Read passage · 01_gaussian.tex, line 206
Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.
Exact mathematical passage
\lambda_iY_i^2+2c_iY_i = \lambda_i \left( Y_i+\frac{c_i}{\lambda_i} \right)^2 - \frac{c_i^2}{\lambda_i}.
yUse in source · 1 passage
Search aliases: y
Use in source · Read passage · 01_gaussian.tex, line 361
Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×10^-5. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor A and as…
Exact mathematical passage
f_Q(x)=\int_0^x\frac12e^{-y/2}\frac14e^{-(x-y)/4}\,dy =\frac12\bigl(e^{-x/4}-e^{-x/2}\bigr).
Y_1Use in source · 2 passages
Search aliases: Y_1
Use in source · Read passage · 11_evidence.tex, line 229
Retained low-rank limitation. For independent Y_1,Y_2∼χ^2_2, the quadratic form Q=0.07(Y_1-Y_2) is Laplace distributed with scale 0.14, so for q≥0, P(Q≥ q)=frac12e^-q/0.14. The computed values agree with this exact law through 10^-200 to roughly 10^-9 decades or better. However, the implemented analytic COS-series certificate rejects these low-rank mixed cases because the tilted characteristic function decays only as |t|^-2. The implementation reports this as a certificate failure rather than si…
Exact mathematical passage
Y_1,Y_2\sim\chi^2_2
Y_2Use in source · 2 passages
Search aliases: Y_2
Use in source · Read passage · 11_evidence.tex, line 229
Retained low-rank limitation. For independent Y_1,Y_2∼χ^2_2, the quadratic form Q=0.07(Y_1-Y_2) is Laplace distributed with scale 0.14, so for q≥0, P(Q≥ q)=frac12e^-q/0.14. The computed values agree with this exact law through 10^-200 to roughly 10^-9 decades or better. However, the implemented analytic COS-series certificate rejects these low-rank mixed cases because the tilted characteristic function decays only as |t|^-2. The implementation reports this as a certificate failure rather than si…
Exact mathematical passage
Y_1,Y_2\sim\chi^2_2
Y_iUse in source · 1 passage
Search aliases: Y_i
Use in source · Read passage · 01_gaussian.tex, line 206
Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.
Exact mathematical passage
\lambda_iY_i^2+2c_iY_i = \lambda_i \left( Y_i+\frac{c_i}{\lambda_i} \right)^2 - \frac{c_i^2}{\lambda_i}.
Y_i^2Use in source · 1 passage
Search aliases: Y_i^2
Use in source · Read passage · 01_gaussian.tex, line 206
Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.
Exact mathematical passage
\lambda_iY_i^2+2c_iY_i = \lambda_i \left( Y_i+\frac{c_i}{\lambda_i} \right)^2 - \frac{c_i^2}{\lambda_i}.
zUse in source · 12 passages
Search aliases: z
Use in source · Read passage · 06_packets.tex, line 158
s positive real part for each 1-2sλ_j and for 1-2sμ_P. The disk condition gives positive real part for 1-α_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing over j∈ P gives [source reference]. The remainder satisfies |∑_m≥ p+1fracz^mm| ≤ ∑_m≥ p+1frac|z|^mp+1 =frac|z|^p+1(p+1)(1-|z|). Using |z|≤ρ_P(s) and sum…
Exact mathematical passage
-\log(1-z)=\sum_{m\ge1}\frac{z^m}{m}.
z^kUse in source · 2 passages
Search aliases: z^k
Use in source · Read passage · A_atlas.tex, line 261
z=q/0.04, P(Q≥ q)=e^-z∑_k=0^39fracz^kk!. log_10P(Q≥ q)=frac-z+log (∑_k=0^39z^k/k!)log(10). For an even-degree chi-square variable, the upper tail has a finite positive-sum formula. The control Q=0.02χ^2_80 is a gamma law of integer shape forty and scale 0.04. Its survival probability can therefore be evaluated independently of cosine inversion.
Exact mathematical passage
z=q/0.04,\qquad P(Q\ge q)=e^{-z}\sum_{k=0}^{39}\frac{z^k}{k!}.
z^mUse in source · 2 passages
Search aliases: z^m
Use in source · Read passage · 06_packets.tex, line 158
s positive real part for each 1-2sλ_j and for 1-2sμ_P. The disk condition gives positive real part for 1-α_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing over j∈ P gives [source reference]. The remainder satisfies |∑_m≥ p+1fracz^mm| ≤ ∑_m≥ p+1frac|z|^mp+1 =frac|z|^p+1(p+1)(1-|z|). Using |z|≤ρ_P(s) and sum…
Exact mathematical passage
-\log(1-z)=\sum_{m\ge1}\frac{z^m}{m}.
Z^\topUse in source · 7 passages
Search aliases: Z^top top
Use in source · Read passage · 01_gaussian.tex, line 20
Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.
Exact mathematical passage
Q=X^\top MX = Z^\top(SMS)Z.
z^\topUse in source · 1 passage
Search aliases: z^top top
Use in source · Read passage · 11_evidence.tex, line 361
For each scenario, 100 Gaussian null residual draws were evaluated in three ways. Under the standardized Gaussian null, write e=M_Xz with z∼ N(0,I). Because M_XA=A, the statistic used here is Q=frac12 z^top Kz, so its nonzero mixture coefficients are frac12λ_j(K). The factor frac12 belongs to this case-study normalization, not to the general quadratic-form definition.
Exact mathematical passage
Q=\frac12 z^\top Kz
Z_1^2Use in source · 1 passage
Search aliases: Z_1^2
Use in source · Read passage · 04_range.tex, line 193
0 · 2 · 2. The large operator has eigenvalues 4,2,0. Its two-dimensional core has exactly the nonzero eigenvalues 4,2. The Gaussian quadratic form therefore has the law 4Z_1^2+2Z_2^2, whether it is computed from the full factor, the full Gram matrix, or the core.
Exact mathematical passage
4Z_1^2+2Z_2^2
Z_2^2Use in source · 1 passage
Search aliases: Z_2^2
Use in source · Read passage · 04_range.tex, line 193
0 · 2 · 2. The large operator has eigenvalues 4,2,0. Its two-dimensional core has exactly the nonzero eigenvalues 4,2. The Gaussian quadratic form therefore has the law 4Z_1^2+2Z_2^2, whether it is computed from the full factor, the full Gram matrix, or the core.
Exact mathematical passage
4Z_1^2+2Z_2^2
Z_iUse in source · 2 passages
Search aliases: Z_i
Use in source · Read passage · 05_spectrum.tex, line 122
For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.
Exact mathematical passage
Z_i
Z_i^2Use in source · 3 passages
Search aliases: Z_i^2
Use in source · Read passage · 05_spectrum.tex, line 123
For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.
Exact mathematical passage
Q_-=\sum_i\ell_iZ_i^2, \qquad Q=\sum_i\lambda_iZ_i^2, \qquad Q_+=\sum_i u_iZ_i^2.
z_k^2Use in source · 4 passages
Search aliases: z_k^2
Use in source · Read passage · B_lean.tex, line 146
The separate definition weightedSquares(s,c,z)=∑_k∈ sc_k z_k^2 models a supplied diagonal quadratic form. Its coefficients need not be eigenvalues obtained by a proved diagonalization. It has three theorems: item declweightedSquares_mono: if l_k≤ u_k for every k∈ s, then ∑ l_kz_k^2≤∑ u_kz_k^2. item declweightedSquares_nonneg: if c_k≥0 for every k∈ s, then ∑ c_kz_k^2≥0. item declweightedSquares_tail_imp: coefficientwise order gives the pointwise implication q≤∑ l_kz_k^2⇒ q≤∑ u_kz_k^2. The first a…
Exact mathematical passage
\operatorname{weightedSquares}(s,c,z)=\sum_{k\in s}c_k z_k^2
|Use in source · 53 passages
Search aliases: |
Use in source · Read passage · 01_gaussian.tex, line 161
The packet representation is sign aware: positive and negative eigenvalues are clustered separately, and each same-sign packet stores a center μ and centered moments. The local expansion remains -frac12log(1-2s(μ+δ)) = -frac12log(1-2sμ) + frac12∑_n≥1 fracα(s)^nδ^nn, α(s)=frac2s1-2sμ, with the same geometric remainder condition |α(s)|max|δ|<1.
Exact mathematical passage
|\alpha(s)|\max|\delta|<1.
|^{-2}Use in source · 1 passage
Search aliases: |^-2
Use in source · Read passage · 11_evidence.tex, line 240
^2_2, the quadratic form Q=0.07(Y_1-Y_2) is Laplace distributed with scale 0.14, so for q≥0, P(Q≥ q)=frac12e^-q/0.14. The computed values agree with this exact law through 10^-200 to roughly 10^-9 decades or better. However, the implemented analytic COS-series certificate rejects these low-rank mixed cases because the tilted characteristic function decays only as |t|^-2. The implementation reports this as a certificate failure rather than silently accepting the numerical value.
Exact mathematical passage
|t|^{-2}
|^{-r/2}Use in source · 1 passage
Search aliases: |^-r/2
Use in source · Read passage · 01_gaussian.tex, line 178
A useful distinction appears in the Fourier tail. Under the saddlepoint tilt, |ϕ_θ(t)| = ∏_j [ 1+ ( frac2λ_j t1-2θλ_j )^2 ]^-1/4. If r coefficients are nonzero, then asymptotically |ϕ_θ(t)|=O(|t|^-r/2). Thus certification difficulty depends strongly on spectral rank, not only on how small the requested probability is.
Exact mathematical passage
|\phi_\theta(t)|=O(|t|^{-r/2}).
|^mUse in source · 1 passage
Search aliases: |^m
Use in source · Read passage · 06_packets.tex, line 162
alpha_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing over j∈ P gives [source reference]. The remainder satisfies |∑_m≥ p+1fracz^mm| ≤ ∑_m≥ p+1frac|z|^mp+1 =frac|z|^p+1(p+1)(1-|z|). Using |z|≤ρ_P(s) and summing n_P terms yields [source reference].
Exact mathematical passage
\left|\sum_{m\ge p+1}\frac{z^m}{m}\right| \le \sum_{m\ge p+1}\frac{|z|^m}{p+1} =\frac{|z|^{p+1}}{(p+1)(1-|z|)}.
|^{p+1}Use in source · 1 passage
Search aliases: |^p+1
Use in source · Read passage · 06_packets.tex, line 162
alpha_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing over j∈ P gives [source reference]. The remainder satisfies |∑_m≥ p+1fracz^mm| ≤ ∑_m≥ p+1frac|z|^mp+1 =frac|z|^p+1(p+1)(1-|z|). Using |z|≤ρ_P(s) and summing n_P terms yields [source reference].
Exact mathematical passage
\left|\sum_{m\ge p+1}\frac{z^m}{m}\right| \le \sum_{m\ge p+1}\frac{|z|^m}{p+1} =\frac{|z|^{p+1}}{(p+1)(1-|z|)}.
\alphaUse in source · 17 passages
Search aliases: α alpha
Use in source · Read passage · 01_gaussian.tex, line 150
The packet representation is sign aware: positive and negative eigenvalues are clustered separately, and each same-sign packet stores a center μ and centered moments. The local expansion remains -frac12log(1-2s(μ+δ)) = -frac12log(1-2sμ) + frac12∑_n≥1 fracα(s)^nδ^nn, α(s)=frac2s1-2sμ, with the same geometric remainder condition |α(s)|max|δ|<1.
Exact mathematical passage
-\frac12\log(1-2s(\mu+\delta)) = -\frac12\log(1-2s\mu) + \frac12\sum_{n\ge1} \frac{\alpha(s)^n\delta^n}{n}, \qquad \alpha(s)=\frac{2s}{1-2s\mu},
\alpha^2Use in source · 2 passages
Search aliases: α^2 alpha
Use in source · Read passage · 05_spectrum.tex, line 393
[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, κ_H(θ) =-frac12logdet(I-2θ H) satisfies κ_H(θ) ≤ κ_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.
Exact mathematical passage
\boxed{ \kappa_H(\theta) \le \kappa_C(\theta) + \theta\alpha t + \frac{\theta^2\alpha^2v} {1-2\theta\alpha\rho}. }
\alpha_PUse in source · 8 passages
Search aliases: α_P alpha
Use in source · Read passage · 06_packets.tex, line 116
[Centered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P,p(s)| ≤ fracn_P2 fracρ_P(s)^p+1(p+1)(1-ρ_P(s)).
Exact mathematical passage
\alpha_P(s)=\frac{2s}{1-2s\mu_P}, \qquad \rho_P(s)=|\alpha_P(s)|\max_{j\in P}|\delta_j|. \label{eq:alpha-rho}
\beta_{(1)}Use in source · 1 passage
Search aliases: β_(1) beta
Use in source · Read passage · 03_phases.tex, line 464
Suppose each retained phase pair has the same communication or scheduling cost, and fix a budget τ≥0. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).
Exact mathematical passage
\beta_{(1)}\le\beta_{(2)}\le\cdots\le\beta_{(N)}.
\beta_{(2)}Use in source · 1 passage
Search aliases: β_(2) beta
Use in source · Read passage · 03_phases.tex, line 464
Suppose each retained phase pair has the same communication or scheduling cost, and fix a budget τ≥0. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).
Exact mathematical passage
\beta_{(1)}\le\beta_{(2)}\le\cdots\le\beta_{(N)}.
\beta_{(i)}Use in source · 2 passages
Search aliases: β_(i) beta
Use in source · Read passage · 03_phases.tex, line 471
[Cardinality-optimal certified pruning] Let r∈{0,…,N} be the largest integer satisfying ∑_i=1^rβ_(i)≤τ. Then omitting the r least-risk phase pairs is feasible, and no feasible omitted set can contain more than r pairs under the scalar additive certificate.
Exact mathematical passage
\sum_{i=1}^{r}\beta_{(i)}\le\tau.
\beta_{(N)}Use in source · 1 passage
Search aliases: β_(N) beta
Use in source · Read passage · 03_phases.tex, line 464
Suppose each retained phase pair has the same communication or scheduling cost, and fix a budget τ≥0. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).
Exact mathematical passage
\beta_{(1)}\le\beta_{(2)}\le\cdots\le\beta_{(N)}.
\beta_e^{(p)}Use in source · 2 passages
Search aliases: β_e^(p) beta
Use in source · Read passage · 03_phases.tex, line 420
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
q_e^{(p)} = \sum_{d=1}^{D}|a_d|q_e^{(d)}, \qquad \beta_e^{(p)}=\lVert q_e^{(p)}\rVert_2.
\beta_NUse in source · 6 passages
Search aliases: β_N beta
Use in source · Read passage · 07_inversion.tex, line 239
[Omitted COS-mode certificate] Let N≥1 and b>q≥0. Set u_N=Nπ/b and β_N=β(u_N). If β_N>0, then the contribution of all exact tilted-COS modes k≥ N to the weighted tail functional is bounded by ε_COS ≤ e^κ(θ)-θ q frac4|ϕ_θ(u_N)|π (frac1N+frac1β_N).
Exact mathematical passage
\beta_N=\beta(u_N)
\delta^nUse in source · 1 passage
Search aliases: δ^n delta
Use in source · Read passage · 01_gaussian.tex, line 150
The packet representation is sign aware: positive and negative eigenvalues are clustered separately, and each same-sign packet stores a center μ and centered moments. The local expansion remains -frac12log(1-2s(μ+δ)) = -frac12log(1-2sμ) + frac12∑_n≥1 fracα(s)^nδ^nn, α(s)=frac2s1-2sμ, with the same geometric remainder condition |α(s)|max|δ|<1.
Exact mathematical passage
-\frac12\log(1-2s(\mu+\delta)) = -\frac12\log(1-2s\mu) + \frac12\sum_{n\ge1} \frac{\alpha(s)^n\delta^n}{n}, \qquad \alpha(s)=\frac{2s}{1-2s\mu},
\delta_jUse in source · 7 passages
Search aliases: δ_j delta
Use in source · Read passage · 06_packets.tex, line 100
Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.
Exact mathematical passage
\mu_P=\frac{1}{n_P}\sum_{j\in P}\lambda_j, \qquad \delta_j=\lambda_j-\mu_P,
\delta_j^mUse in source · 1 passage
Search aliases: δ_j^m delta
Use in source · Read passage · 06_packets.tex, line 106
Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.
Exact mathematical passage
M_{P,m}=\sum_{j\in P}\delta_j^m, \qquad m\ge0. \label{eq:centeredmoments}
\Delta_kUse in source · 2 passages
Search aliases: Δ_k Delta
Use in source · Read passage · 07_inversion.tex, line 78
Write Δ_k=widehatκ(s_k)-κ(s_k). Then |e^widehatκ(s_k)-θ q-e^κ(s_k)-θ q| ≤ |e^κ(s_k)-θ q| (e^|Δ_k|-1). Moreover Reκ(θ+i u_k)-κ(θ) =log|ϕ_θ(u_k)|≤0, so |e^κ(s_k)-θ q| ≤ e^κ(θ)-θ q. Substitute this into the finite cosine sum and apply the triangle inequality.
Exact mathematical passage
\Delta_k=\widehat\kappa(s_k)-\kappa(s_k)
\varepsilon_kUse in source · 2 passages
Search aliases: ε_k varepsilon
Use in source · Read passage · 08_adaptive.tex, line 69
After separating the likelihood-ratio prefactor, the packet contribution contains sums of the form frac2b ∑_k |J_k| ( e^ε_k-1 ). Direct evaluation can overflow even when the candidate is destined to be rejected by the adaptive planner. The implementation instead computes log( frac2b ∑_k |J_k| ( e^ε_k-1 ) ) by log-sum-exp arithmetic, and compares this quantity with the logarithm of the requested tolerance before converting a final accepted bound back to float64.
Exact mathematical passage
\frac{2}{b} \sum_k |J_k| \left( e^{\varepsilon_k}-1 \right).
\varepsilon_{\rm req}Use in source · 5 passages
Search aliases: ε_rm req varepsilon rm
Use in source · Read passage · 10_accuracy.tex, line 14
For an exact positive spectrum and threshold q, the solver accepts a requested relative multiplicative accuracy ε_rm req>0 and returns a certified interval [p_-,p_+] only when fracp_+p_--1 ≤ ε_rm req.
Exact mathematical passage
\varepsilon_{\rm req}>0
\varepsilon_{\rm requested}Use in source · 1 passage
Search aliases: ε_rm requested varepsilon rm
Use in source · Read passage · 12_synthesis.tex, line 288
Accuracy-driven positive tail solver · certified / fail-closed · A result is accepted only after p_+/p_- - 1≤ε_rm requested is proved.
Exact mathematical passage
p_+/p_- - 1\le\varepsilon_{\rm requested}
\ThetaUse in source · 2 passages
Search aliases: Θ Theta
Use in source · Read passage · 11_evidence.tex, line 110
The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponential tilting, the COS expansion, spectral packets, or the K-COS error controller.
Exact mathematical passage
P(Q\ge q) = \frac12 + \frac1\pi \int_0^\infty \frac{\sin\Theta(u)}{u\,R(u)}\,du,
\theta^2Use in source · 3 passages
Search aliases: θ^2 theta
Use in source · Read passage · 05_spectrum.tex, line 393
[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, κ_H(θ) =-frac12logdet(I-2θ H) satisfies κ_H(θ) ≤ κ_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.
Exact mathematical passage
\boxed{ \kappa_H(\theta) \le \kappa_C(\theta) + \theta\alpha t + \frac{\theta^2\alpha^2v} {1-2\theta\alpha\rho}. }
\kappa''Use in source · 1 passage
Search aliases: κ'' kappa
Use in source · Read passage · 08_adaptive.tex, line 16
[leftmargin=2em] item Solve κ'(θ_*)=q. item Initialize an upper interval b=q+Lsqrtκ''(θ_*)+sqrtκ^(4)(θ_*) and a mode count N. item Retain the largest spectral spikes exactly and place the remaining sorted eigenvalues into coarse packets. item Evaluate ρ_P(θ_*+i u_k) and the packet remainder bound on the actual COS grid. Split packets until the propagated packet contribution is within its budget. item Evaluate the support/alias certificate from [source reference]--[source reference]. item Evaluat…
Exact mathematical passage
b=q+L\sqrt{\kappa''(\theta_*)+\sqrt{\kappa^{(4)}(\theta_*)}}
\kappa^{(4)}Use in source · 1 passage
Search aliases: κ^(4) kappa
Use in source · Read passage · 08_adaptive.tex, line 16
[leftmargin=2em] item Solve κ'(θ_*)=q. item Initialize an upper interval b=q+Lsqrtκ''(θ_*)+sqrtκ^(4)(θ_*) and a mode count N. item Retain the largest spectral spikes exactly and place the remaining sorted eigenvalues into coarse packets. item Evaluate ρ_P(θ_*+i u_k) and the packet remainder bound on the actual COS grid. Split packets until the propagated packet contribution is within its budget. item Evaluate the support/alias certificate from [source reference]--[source reference]. item Evaluat…
Exact mathematical passage
b=q+L\sqrt{\kappa''(\theta_*)+\sqrt{\kappa^{(4)}(\theta_*)}}
\kappa_CUse in source · 1 passage
Search aliases: κ_C kappa
Use in source · Read passage · 05_spectrum.tex, line 393
[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, κ_H(θ) =-frac12logdet(I-2θ H) satisfies κ_H(θ) ≤ κ_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.
Exact mathematical passage
\boxed{ \kappa_H(\theta) \le \kappa_C(\theta) + \theta\alpha t + \frac{\theta^2\alpha^2v} {1-2\theta\alpha\rho}. }
\kappa_HUse in source · 2 passages
Search aliases: κ_H kappa
Use in source · Read passage · 05_spectrum.tex, line 388
[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, κ_H(θ) =-frac12logdet(I-2θ H) satisfies κ_H(θ) ≤ κ_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.
Exact mathematical passage
\kappa_H(\theta) =-\frac12\log\det(I-2\theta H)
\kappa_PUse in source · 2 passages
Search aliases: κ_P kappa
Use in source · Read passage · 06_packets.tex, line 123
tered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P,p(s)| ≤ fracn_P2 fracρ_P(s)^p+1(p+1)(1-ρ_P(s)).
Exact mathematical passage
\kappa_P(s) &=-\frac{n_P}{2}\log(1-2s\mu_P) +\frac12\sum_{m=1}^{p} \frac{\alpha_P(s)^m}{m}M_{P,m} +R_{P,p}(s), \label{eq:packet-expansion}
\LambdaUse in source · 6 passages
Search aliases: Λ Lambda
Use in source · Read passage · 01_gaussian.tex, line 104
Write X=SZ with Z∼ N(0,I) and diagonalize SMS=VΛ V^top. Rotational invariance of the standard normal gives Z^top VΛ V^top Z overset d= ∑_jλ_jZ_j^2. No positivity assumption on M is needed for this central reduction.
Exact mathematical passage
SMS=V\Lambda V^\top.
\lambda_1Use in source · 2 passages
Search aliases: λ_1 lambda
Use in source · Read passage · 01_gaussian.tex, line 29
Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.
Exact mathematical passage
\lambda_1,\ldots,\lambda_r
\lambda_iUse in source · 18 passages
Search aliases: λ_i lambda
Use in source · Read passage · 01_gaussian.tex, line 206
Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.
Exact mathematical passage
\lambda_iY_i^2+2c_iY_i = \lambda_i \left( Y_i+\frac{c_i}{\lambda_i} \right)^2 - \frac{c_i^2}{\lambda_i}.
\lambda_j^{-h_j/2}Use in source · 1 passage
Search aliases: λ_j^-h_j/2 lambda
Use in source · Read passage · 11_evidence.tex, line 139
The four spectra include balanced, skewed, dominant-spike, and higher-rank cases. Each reference integral was evaluated at two truncation limits. The omitted integral after the frequency cutoff T>0 was bounded using frac1π ∫_T^∞fracduuR(u) ≤ fracCπνT^-ν, ν=frac12∑_jh_j, C=∏_jλ_j^-h_j/2.
Exact mathematical passage
\frac1\pi \int_T^\infty\frac{du}{uR(u)} \le \frac{C}{\pi\nu}T^{-\nu}, \qquad \nu=\frac12\sum_jh_j, \qquad C=\prod_j\lambda_j^{-h_j/2}.
\lambda_rUse in source · 2 passages
Search aliases: λ_r lambda
Use in source · Read passage · 01_gaussian.tex, line 29
Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.
Exact mathematical passage
\lambda_1,\ldots,\lambda_r
\mu^\topUse in source · 2 passages
Search aliases: μ^top mu top
Use in source · Read passage · 01_gaussian.tex, line 236
[Exact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈range(M^1/2S). In particular, Σsucc0 is sufficient. For singular covariance the shift is the squa…
Exact mathematical passage
q_0 = \mu^\top M\mu - \sum_i\frac{c_i^2}{\lambda_i} \ge0.
\mu_iUse in source · 3 passages
Search aliases: μ_i mu
Use in source · Read passage · 05_spectrum.tex, line 86
Assume Hsucceq0 and let the eigenvalues of H and H_0, in decreasing order, be λ_i and μ_i. If H-H_0 _2≤ε, Weyl's inequality gives |λ_i-μ_i|≤ε.
Exact mathematical passage
\lambda_i \qquad\text{and}\qquad \mu_i.
\mu_PUse in source · 9 passages
Search aliases: μ_P mu
Use in source · Read passage · 06_packets.tex, line 100
Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.
Exact mathematical passage
\mu_P=\frac{1}{n_P}\sum_{j\in P}\lambda_j, \qquad \delta_j=\lambda_j-\mu_P,
\nuUse in source · 7 passages
Search aliases: ν nu
Use in source · Read passage · 07_inversion.tex, line 104
[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted chi-square sum has an absolutely continuous law. Its folded measure therefore has no atom at the jump of h. The even periodic extension of h has boun…
Exact mathematical passage
\nu_\theta
\nu_\thetaUse in source · 3 passages
Search aliases: ν_θ nu theta
Use in source · Read passage · 07_inversion.tex, line 104
[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted chi-square sum has an absolutely continuous law. Its folded measure therefore has no atom at the jump of h. The even periodic extension of h has boun…
Exact mathematical passage
\nu_\theta
\rho_0Use in source · 9 passages
Search aliases: ρ_0 rho
Use in source · Read passage · 03_phases.tex, line 341
For depth k≥1 and starting phase profile ρ_0=ρ(X), define the edge risk vector q_e^(k) = ∑_j=0^k-1 B^k-1-jΔ_e B^jρ_0 and its scalar risk β_e^(k)=lVert q_e^(k)rVert_2. Unlike the raw block norm B_hg, this quantity depends on the start profile and on Krylov depth.
Exact mathematical passage
\rho_0=\rho(X)
\rho_RUse in source · 2 passages
Search aliases: ρ_R rho
Use in source · Read passage · A_atlas.tex, line 117
A=UB+F, U^ TF=0, A^ TA=B^ TB+F^ TF. t=trR= F _F^2, v=tr(R^2), ρ_R= R _2. Let U contain d orthonormal sample-space directions. Project the factor into its captured range coordinates B=U^ TA and residual F=(I-UU^ T)A. Orthogonality eliminates the cross terms in the feature-space Gram matrix.
Exact mathematical passage
t=\operatorname{tr}R=\|F\|_F^2,\quad v=\operatorname{tr}(R^2),\quad \rho_R=\|R\|_2.
\sigmaUse in source · 2 passages
Search aliases: σ sigma
Use in source · Read passage · 05_spectrum.tex, line 231
[Range-core singular-value certificate] For A∈ R^n× m, list the singular values of A and B in decreasing order and pad both lists with zeros to length n. Write these lists as σ_i(A) and widetildeσ_i(B), respectively. They are the square roots of the n eigenvalues of AA^top and (UB)(UB)^top. Then, for 1≤ i≤ n, | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\sigma_i(A)
\Sigma^{1/2}Use in source · 4 passages
Search aliases: Σ^1/2 Sigma
Use in source · Read passage · 01_gaussian.tex, line 19
Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.
Exact mathematical passage
S=\Sigma^{1/2}
\sigma_iUse in source · 2 passages
Search aliases: σ_i sigma
Use in source · Read passage · 05_spectrum.tex, line 231
[Range-core singular-value certificate] For A∈ R^n× m, list the singular values of A and B in decreasing order and pad both lists with zeros to length n. Write these lists as σ_i(A) and widetildeσ_i(B), respectively. They are the square roots of the n eigenvalues of AA^top and (UB)(UB)^top. Then, for 1≤ i≤ n, | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.
Exact mathematical passage
\sigma_i(A)
\tau_KUse in source · 1 passage
Search aliases: τ_K tau
Use in source · Read passage · 05_spectrum.tex, line 172
An execution schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.
Exact mathematical passage
\boxed{ \sum_{e\in\mathcal O}\beta_e^{(k)} \le \tau_K, \qquad \|\mathcal C_{\mathcal O}\|_2 \le \tau_\lambda. }
\tau_{\theta}Use in source · 6 passages
Search aliases: τ_θ tau theta
Use in source · Read passage · 07_inversion.tex, line 137
[Support and periodization bound] Let the finite positive spectrum have rank r≥1, and let b>q≥0, b>0, and 0<θ<1/(2λ_max). Let τ_θ(b)= P_θ(Q>b). Then the combined error from (i) replacing the tilted density on [0,b] by its full-coefficient cosine periodization and (ii) omitting the original integral over (b,∞) is bounded by ε_sup ≤ e^κ(θ)-θ q τ_θ(b) (1+e^-θ(b-q)).
Exact mathematical passage
\tau_{\theta}(b)=\Pp_{\theta}(Q>b).
\tau_\lambdaUse in source · 1 passage
Search aliases: τ_λ tau lambda
Use in source · Read passage · 05_spectrum.tex, line 172
An execution schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.
Exact mathematical passage
\boxed{ \sum_{e\in\mathcal O}\beta_e^{(k)} \le \tau_K, \qquad \|\mathcal C_{\mathcal O}\|_2 \le \tau_\lambda. }
\phiUse in source · 21 passages
Search aliases: ϕ phi
Use in source · Read passage · 01_gaussian.tex, line 166
A useful distinction appears in the Fourier tail. Under the saddlepoint tilt, |ϕ_θ(t)| = ∏_j [ 1+ ( frac2λ_j t1-2θλ_j )^2 ]^-1/4. If r coefficients are nonzero, then asymptotically |ϕ_θ(t)|=O(|t|^-r/2). Thus certification difficulty depends strongly on spectral rank, not only on how small the requested probability is.
Exact mathematical passage
\left|\phi_\theta(t)\right| = \prod_j \left[ 1+ \left( \frac{2\lambda_j t}{1-2\theta\lambda_j} \right)^2 \right]^{-1/4}.
\phi_\thetaUse in source · 9 passages
Search aliases: ϕ_θ phi theta
Use in source · Read passage · 01_gaussian.tex, line 166
A useful distinction appears in the Fourier tail. Under the saddlepoint tilt, |ϕ_θ(t)| = ∏_j [ 1+ ( frac2λ_j t1-2θλ_j )^2 ]^-1/4. If r coefficients are nonzero, then asymptotically |ϕ_θ(t)|=O(|t|^-r/2). Thus certification difficulty depends strongly on spectral rank, not only on how small the requested probability is.
Exact mathematical passage
\left|\phi_\theta(t)\right| = \prod_j \left[ 1+ \left( \frac{2\lambda_j t}{1-2\theta\lambda_j} \right)^2 \right]^{-1/4}.
\phi_{\theta}Use in source · 12 passages
Search aliases: ϕ_θ phi theta
Use in source · Read passage · 06_packets.tex, line 80
Define the exponentially tilted law by f_θ(x)=e^θ x-κ(θ)f(x). Its characteristic function is ϕ_θ(t) =exp{κ(θ+i t)-κ(θ)}, and E_θQ=κ'(θ). Therefore the choice [source reference] moves the query q to the mean of the tilted law.
Exact mathematical passage
\phi_{\theta}(t) =\exp\{\kappa(\theta+\ii t)-\kappa(\theta)\}, \label{eq:tilt-cf}
\chiUse in source · 17 passages
Search aliases: χ chi
Use in source · Read passage · 01_gaussian.tex, line 219
[Exact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈range(M^1/2S). In particular, Σsucc0 is sufficient. For singular covariance the shift is the squa…
Exact mathematical passage
\boxed{ Q \overset d= q_0 + \sum_i \lambda_i \chi^2_1(\delta_i), }
\chi^2Use in source · 4 passages
Search aliases: χ^2 chi
Use in source · Read passage · 02_streaming.tex, line 51
Storage blocking controls residence. Structural phases encode linkage-disequilibrium (LD) windows, annotations, frequency bands, or other coordinate groups. Krylov methods compress the operator to the rank actually requested. Spectral packets compress the final generalized-χ^2 transform.
Exact mathematical passage
\chi^2
\chi^2_1Use in source · 1 passage
Search aliases: χ^2_1 chi
Use in source · Read passage · 01_gaussian.tex, line 219
[Exact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈range(M^1/2S). In particular, Σsucc0 is sufficient. For singular covariance the shift is the squa…
Exact mathematical passage
\boxed{ Q \overset d= q_0 + \sum_i \lambda_i \chi^2_1(\delta_i), }
\chi^2_2Use in source · 2 passages
Search aliases: χ^2_2 chi
Use in source · Read passage · 01_gaussian.tex, line 354
1 · -1 · -1 · 1 satisfies V^top K V=diag(1,1,2,2). For Z∼ N(0,I_4), Q=Z^top KZ overset d= Y+2W, Y,W independent with law χ^2_2. The spectrum is therefore (1,1,2,2) and E Q=6. The six-dimensional feature Gram A^top A has these same four nonzero eigenvalues and two additional zeros.
Exact mathematical passage
Q=Z^\top KZ\ \overset d=\ Y+2W, \qquad Y,W\text{ independent with law }\chi^2_2.
\chi^2_{80}Use in source · 9 passages
Search aliases: χ^2_80 chi
Use in source · Read passage · 09_intervals.tex, line 221
For the control Q=0.02χ^2_80, the survival probability has the finite exact formula P(Q≥ q) = e^-x ∑_j=0^39 fracx^jj!, x=fracq0.04.
Exact mathematical passage
Q=0.02\chi^2_{80},
\chi^2_dUse in source · 1 passage
Search aliases: χ^2_d chi
Use in source · Read passage · 01_gaussian.tex, line 297
The numerical interface implements this reduction directly. In an isotropic control, Σ=I, M=aI, a>0, the law collapses to the exact scaled noncentral chi-square distribution Q/a∼χ^2_d( μ ^2).
Exact mathematical passage
Q/a\sim\chi^2_d(\|\mu\|^2).
\ellUse in source · 7 passages
Search aliases: ℓ ell
Use in source · Read passage · 05_spectrum.tex, line 102
[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.
Exact mathematical passage
\boxed{ \ell_i = \max\{0,\mu_i-\varepsilon\} \le \lambda_i \le u_i = \max\{0,\mu_i+\varepsilon\}. }
\ell_2Use in source · 1 passage
Search aliases: ℓ_2 ell
Use in source · Read passage · 11_evidence.tex, line 582
Hierarchical refinement gave relative discrepancy 3.17×10^-16. For H=L^top L with block-bidiagonal L, the smallest eigenvalue was approximately 2.86 and the largest norm outside the exact light cone was zero. The QR-core singular-spectrum discrepancy was 5.89×10^-16 in relative ℓ_2 norm. These are finite controls, not universal floating-point error bounds.
Exact mathematical passage
\ell_2
\ell_iUse in source · 6 passages
Search aliases: ℓ_i ell
Use in source · Read passage · 05_spectrum.tex, line 102
[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.
Exact mathematical passage
\boxed{ \ell_i = \max\{0,\mu_i-\varepsilon\} \le \lambda_i \le u_i = \max\{0,\mu_i+\varepsilon\}. }
\leftarrowUse in source · 1 passage
Search aliases: ← leftarrow
Use in source · Read passage · 02_streaming.tex, line 161
[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.
Exact mathematical passage
T_b=G_b^\top U, \qquad Y\leftarrow Y+G_b(W_bT_b).
\toUse in source · 11 passages
Search aliases: → to
Use in source · Read passage · 01_gaussian.tex, line 144
in endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain whenever a positive spectral component is present. The negative-only case with at least one nonzero coefficient has a positive saddlepoint only for E[Q]<q<0. Indeed, κ'(θ) increases from E[Q] to zero as θ→∞. For q≥0 its survival probability is zero, and no finite positive saddlepoint solves the query.
Exact mathematical passage
\theta\to\infty
\mapstoUse in source · 1 passage
Search aliases: ↦ mapsto
Use in source · Read passage · 04_range.tex, line 152
Since A=Q_rB and Q_r^top Q_r=I, one has AA^top=Q_r(BB^top)Q_r^top. The map x↦ Q_rx is an isometry from the core space onto range(Q_r). On that range the two operators are conjugate, and AA^top vanishes on its orthogonal complement. Hence their nonzero eigenvalues agree with multiplicity. If the captured range equals range(A), then B has full row rank, so BB^top is positive definite. Without that equality, the core may retain additional zero eigenvalues.
Exact mathematical passage
x\mapsto Q_rx
\RightarrowUse in source · 1 passage
Search aliases: ⇒ Rightarrow
Use in source · Read passage · B_lean.tex, line 157
ot be eigenvalues obtained by a proved diagonalization. It has three theorems: item declweightedSquares_mono: if l_k≤ u_k for every k∈ s, then ∑ l_kz_k^2≤∑ u_kz_k^2. item declweightedSquares_nonneg: if c_k≥0 for every k∈ s, then ∑ c_kz_k^2≥0. item declweightedSquares_tail_imp: coefficientwise order gives the pointwise implication q≤∑ l_kz_k^2⇒ q≤∑ u_kz_k^2. The first and third allow signed coefficients because squares remain nonnegative. The second has an explicit nonnegativity hypothesis.
Exact mathematical passage
q\le\sum l_kz_k^2\Rightarrow q\le\sum u_kz_k^2
\notinUse in source · 1 passage
Search aliases: ∉ notin
Use in source · Read passage · 12_synthesis.tex, line 112
For every phase h∉ F_k and every starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the full variant space. Thus P_F=J_FJ_F^top and A_F=AJ_F. The compressed operator and its ambient embedding satisfy H_F=J_F^top HJ_F=A_F^…
Exact mathematical passage
h\notin F_k
\prodUse in source · 7 passages
Search aliases: ∏ prod
Use in source · Read passage · 01_gaussian.tex, line 166
A useful distinction appears in the Fourier tail. Under the saddlepoint tilt, |ϕ_θ(t)| = ∏_j [ 1+ ( frac2λ_j t1-2θλ_j )^2 ]^-1/4. If r coefficients are nonzero, then asymptotically |ϕ_θ(t)|=O(|t|^-r/2). Thus certification difficulty depends strongly on spectral rank, not only on how small the requested probability is.
Exact mathematical passage
\left|\phi_\theta(t)\right| = \prod_j \left[ 1+ \left( \frac{2\lambda_j t}{1-2\theta\lambda_j} \right)^2 \right]^{-1/4}.
\prod_jUse in source · 5 passages
Search aliases: ∏_j prod
Use in source · Read passage · 01_gaussian.tex, line 166
A useful distinction appears in the Fourier tail. Under the saddlepoint tilt, |ϕ_θ(t)| = ∏_j [ 1+ ( frac2λ_j t1-2θλ_j )^2 ]^-1/4. If r coefficients are nonzero, then asymptotically |ϕ_θ(t)|=O(|t|^-r/2). Thus certification difficulty depends strongly on spectral rank, not only on how small the requested probability is.
Exact mathematical passage
\left|\phi_\theta(t)\right| = \prod_j \left[ 1+ \left( \frac{2\lambda_j t}{1-2\theta\lambda_j} \right)^2 \right]^{-1/4}.
\prod_{j=1}^rUse in source · 1 passage
Search aliases: ∏_j=1^r prod
Use in source · Read passage · 09_intervals.tex, line 116
Define a_j= frac2λ_j1-2θλ_j. The normalized tilted characteristic function satisfies |ϕ_θ(t)| = ∏_j=1^r ( 1+(a_jt)^2 )^-1/4.
Exact mathematical passage
|\phi_\theta(t)| = \prod_{j=1}^r \left( 1+(a_jt)^2 \right)^{-1/4}.
\prod_{j=1}^{r}Use in source · 1 passage
Search aliases: ∏_j=1^r prod
Use in source · Read passage · 07_inversion.tex, line 189
[Tilted characteristic-function modulus] Define a_j=frac2λ_j1-2θλ_j>0. Then |ϕ_θ(t)| =∏_j=1^r(1+(a_jt)^2)^-1/4. For t>0, define the logarithmic decay rate β(t) =-fracdlog|ϕ_θ(t)|dlog t =frac12∑_j=1^r frac(a_jt)^21+(a_jt)^2. Set β(0)=0 by continuous extension. Then β(t) is nondecreasing for t≥0.
Exact mathematical passage
|\phi_{\theta}(t)| =\prod_{j=1}^{r}\left(1+(a_jt)^2\right)^{-1/4}. \label{eq:phi-modulus}
\sum_{(h,g)\in\mathcal E_0}Use in source · 1 passage
Search aliases: ∑_(h,g)∈ E_0 sum in mathcal
Use in source · Read passage · 03_phases.tex, line 121
Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,
Exact mathematical passage
H_0=\sum_{(h,g)\in\mathcal E_0}P_hHP_g.
\sum_bUse in source · 5 passages
Search aliases: ∑_b sum
Use in source · Read passage · 01_gaussian.tex, line 57
The same central quadratic-form law may be supplied by any of the following: item a nonnegative spectrum {λ_j}. item a positive semidefinite matrix H. item a factor A with H=AA^top or H=A^top A. item a replayable streamed factor A=[A_1 ⋯ A_B]. The nonzero spectra of AA^top and A^top A are identical, and AA^top=∑_bA_bA_b^top.
Exact mathematical passage
AA^\top=\sum_bA_bA_b^\top.
\sum_{b=1}^BUse in source · 4 passages
Search aliases: ∑_b=1^B sum
Use in source · Read passage · 02_streaming.tex, line 84
[Additive block Gram identity] K = AA^top = ∑_b=1^B A_bA_b^top = M_X ( ∑_b=1^B G_bW_bG_b^top ) M_X.
Exact mathematical passage
\boxed{ K = AA^\top = \sum_{b=1}^B A_bA_b^\top = M_X \left( \sum_{b=1}^B G_bW_bG_b^\top \right) M_X. }
\sum_{b=1}^{B}Use in source · 1 passage
Search aliases: ∑_b=1^B sum
Use in source · Read passage · 03_phases.tex, line 593
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
T_{\rm serial}=\sum_{b=1}^{B}(t_b+c_b),
\sum_{b=1}^{B-1}Use in source · 1 passage
Search aliases: ∑_b=1^B-1 sum
Use in source · Read passage · 03_phases.tex, line 597
[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.
Exact mathematical passage
\boxed{ T_{\rm pipe} = t_1 + \sum_{b=1}^{B-1}\max\{c_b,t_{b+1}\} + c_B. }
\sum_{d=0}^{D}Use in source · 1 passage
Search aliases: ∑_d=0^D sum
Use in source · Read passage · 03_phases.tex, line 416
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
p(t)=\sum_{d=0}^{D}a_dt^d.
\sum_{d=1}^{D}Use in source · 1 passage
Search aliases: ∑_d=1^D sum
Use in source · Read passage · 03_phases.tex, line 420
[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.
Exact mathematical passage
q_e^{(p)} = \sum_{d=1}^{D}|a_d|q_e^{(d)}, \qquad \beta_e^{(p)}=\lVert q_e^{(p)}\rVert_2.
\sum_gUse in source · 3 passages
Search aliases: ∑_g sum
Use in source · Read passage · 03_phases.tex, line 108
Using ∑_gP_g=I and submultiplicativity, lVert P_hHXrVert_F · =lVert∑_gP_hHP_gXrVert_F
Exact mathematical passage
\sum_gP_g=I
\sum_hUse in source · 4 passages
Search aliases: ∑_h sum
Use in source · Read passage · 03_phases.tex, line 10
Let the phase projectors form an orthogonal decomposition of variant space: P_h^top=P_h=P_h^2, P_hP_g=0 (h≠ g), ∑_hP_h=I. Thus P_h projects onto variant phase h. Again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.
Exact mathematical passage
P_h^\top=P_h=P_h^2,\qquad P_hP_g=0\ (h\ne g),\qquad \sum_hP_h=I.
\sum_iUse in source · 6 passages
Search aliases: ∑_i sum
Use in source · Read passage · 01_gaussian.tex, line 219
[Exact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈range(M^1/2S). In particular, Σsucc0 is sufficient. For singular covariance the shift is the squa…
Exact mathematical passage
\boxed{ Q \overset d= q_0 + \sum_i \lambda_i \chi^2_1(\delta_i), }
\sum_{i=1}^{r}Use in source · 1 passage
Search aliases: ∑_i=1^r sum
Use in source · Read passage · 03_phases.tex, line 471
[Cardinality-optimal certified pruning] Let r∈{0,…,N} be the largest integer satisfying ∑_i=1^rβ_(i)≤τ. Then omitting the r least-risk phase pairs is feasible, and no feasible omitted set can contain more than r pairs under the scalar additive certificate.
Exact mathematical passage
\sum_{i=1}^{r}\beta_{(i)}\le\tau.
\sum_{i=1}^{r+1}Use in source · 1 passage
Search aliases: ∑_i=1^r+1 sum
Use in source · Read passage · 03_phases.tex, line 483
If r=N, every candidate pair can be omitted and optimality is immediate. Otherwise, among all subsets of cardinality r+1, the smallest possible sum of risks is ∑_i=1^r+1β_(i). By maximality of r, that sum exceeds τ. Hence no set of r+1 omitted pairs is feasible. The prefix of length r is feasible by construction.
Exact mathematical passage
\sum_{i=1}^{r+1}\beta_{(i)}.
\sum_{i=1}^sUse in source · 1 passage
Search aliases: ∑_i=1^s sum
Use in source · Read passage · 05_spectrum.tex, line 558
The inequality t≤ sρ gives s≥ t/ρ. Cauchy--Schwarz gives t^2 = (∑_i=1^s r_i)^2 ≤ s∑_i=1^s r_i^2 =sv, so s≥ t^2/v. Finally, every column of F lies in range(U)^⊥, whose dimension is n-r.
Exact mathematical passage
t^2 = \left(\sum_{i=1}^s r_i\right)^2 \le s\sum_{i=1}^s r_i^2 =sv,
\sum_{i\in I}Use in source · 2 passages
Search aliases: ∑_i∈ I sum in
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
\sum_{J'\in\mathcal J'}Use in source · 1 passage
Search aliases: ∑_J'∈ J' sum in mathcal
Use in source · Read passage · 02_streaming.tex, line 323
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
\sum_{j=0}^{39}Use in source · 1 passage
Search aliases: ∑_j=0^39 sum
Use in source · Read passage · 09_intervals.tex, line 225
For the control Q=0.02χ^2_80, the survival probability has the finite exact formula P(Q≥ q) = e^-x ∑_j=0^39 fracx^jj!, x=fracq0.04.
Exact mathematical passage
P(Q\ge q) = e^{-x} \sum_{j=0}^{39} \frac{x^j}{j!}, \qquad x=\frac{q}{0.04}.
\sum_{j=0}^dUse in source · 1 passage
Search aliases: ∑_j=0^d sum
Use in source · Read passage · 03_phases.tex, line 225
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
p(t)=\sum_{j=0}^d a_jt^j.
\sum_{j=0}^{k-1}Use in source · 5 passages
Search aliases: ∑_j=0^k-1 sum
Use in source · Read passage · 03_phases.tex, line 159
Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.
Exact mathematical passage
H^k-H_0^k = \sum_{j=0}^{k-1}H^{k-1-j}EH_0^j.
\sum_{j=1}^3Use in source · 1 passage
Search aliases: ∑_j=1^3 sum
Use in source · Read passage · 02_streaming.tex, line 435
For the declared vector v=(1,2,-1,0)^top, the action can be evaluated without forming any block Gram matrix: ∑_j=1^3 A_j(A_j^top v)=Kv=(2,3,-2,-1)^top. The symbolic check confirms both storage partitions. Every column remains represented. Dropping an operator coupling instead changes the operator and requires the error analysis developed next.
Exact mathematical passage
\sum_{j=1}^3 A_j(A_j^\top v)=Kv=(2,3,-2,-1)^\top.
\sum_{j=1}^dUse in source · 1 passage
Search aliases: ∑_j=1^d sum
Use in source · Read passage · 03_phases.tex, line 229
[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).
Exact mathematical passage
\boxed{ \rho\bigl((p(H)-p(H_0))X\bigr) \leq \sum_{j=1}^d |a_j|(\mathcal B^j-\mathcal B_0^j)\rho(X). }
\sum_{j=1}^rUse in source · 2 passages
Search aliases: ∑_j=1^r sum
Use in source · Read passage · 01_gaussian.tex, line 30
Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.
Exact mathematical passage
\boxed{ Q\overset d= \sum_{j=1}^r\lambda_jZ_j^2. }
\sum_{j\in I}Use in source · 2 passages
Search aliases: ∑_j∈ I sum in
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
\sum_{J\in\mathcal J}Use in source · 1 passage
Search aliases: ∑_J∈ J sum in mathcal
Use in source · Read passage · 02_streaming.tex, line 323
[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.
Exact mathematical passage
\boxed{ \sum_{J\in\mathcal J}A_JA_J^\top = \sum_{J'\in\mathcal J'}A_{J'}A_{J'}^\top = AA^\top. }
\sum_{j\in P}Use in source · 3 passages
Search aliases: ∑_j∈ P sum in
Use in source · Read passage · 06_packets.tex, line 100
Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.
Exact mathematical passage
\mu_P=\frac{1}{n_P}\sum_{j\in P}\lambda_j, \qquad \delta_j=\lambda_j-\mu_P,
\sum_kUse in source · 2 passages
Search aliases: ∑_k sum
Use in source · Read passage · 08_adaptive.tex, line 69
After separating the likelihood-ratio prefactor, the packet contribution contains sums of the form frac2b ∑_k |J_k| ( e^ε_k-1 ). Direct evaluation can overflow even when the candidate is destined to be rejected by the adaptive planner. The implementation instead computes log( frac2b ∑_k |J_k| ( e^ε_k-1 ) ) by log-sum-exp arithmetic, and compares this quantity with the logarithm of the requested tolerance before converting a final accepted bound back to float64.
Exact mathematical passage
\frac{2}{b} \sum_k |J_k| \left( e^{\varepsilon_k}-1 \right).
\sum_{k=0}^{39}Use in source · 2 passages
Search aliases: ∑_k=0^39 sum
Use in source · Read passage · A_atlas.tex, line 261
z=q/0.04, P(Q≥ q)=e^-z∑_k=0^39fracz^kk!. log_10P(Q≥ q)=frac-z+log (∑_k=0^39z^k/k!)log(10). For an even-degree chi-square variable, the upper tail has a finite positive-sum formula. The control Q=0.02χ^2_80 is a gamma law of integer shape forty and scale 0.04. Its survival probability can therefore be evaluated independently of cosine inversion.
Exact mathematical passage
z=q/0.04,\qquad P(Q\ge q)=e^{-z}\sum_{k=0}^{39}\frac{z^k}{k!}.
\sum_{k=0}^{N-1}Use in source · 3 passages
Search aliases: ∑_k=0^N-1 sum
Use in source · Read passage · 07_inversion.tex, line 35
If the exact tilted characteristic function is used, the N-mode finite-window expression is T_N,b =frac2b∑_k=0^N-1' J_k Re[e^κ(θ+i u_k)-θ q]. The cancellation of κ(θ) in [source reference] is useful numerically: the likelihood-ratio prefactor from [source reference] and the normalization in [source reference] do not need to be formed separately inside each mode.
Exact mathematical passage
T_{N,b} =\frac{2}{b}\sum_{k=0}^{N-1}{}' J_k\, \Re\left[e^{\kappa(\theta+\ii u_k)-\theta q}\right]. \label{eq:tail-cos}
\sum_{k=0}^{\infty}Use in source · 1 passage
Search aliases: ∑_k=0^∞ sum infty
Use in source · Read passage · 07_inversion.tex, line 109
[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted chi-square sum has an absolutely continuous law. Its folded measure therefore has no atom at the jump of h. The even periodic extension of h has boun…
Exact mathematical passage
\int h\,d(F_{b\#}\nu_\theta) =\frac2b\sum_{k=0}^{\infty}{}' J_k\,\Re\phi_\theta(k\pi/b).
\sum_{k=N}^{\infty}Use in source · 3 passages
Search aliases: ∑_k=N^∞ sum infty
Use in source · Read passage · 07_inversion.tex, line 252
Using the cosine coefficient magnitude 2|ϕ_θ(u_k)|/b and [source reference], the omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1+fracN^-β_Nβ_N, which yields [source reference].
Exact mathematical passage
e^{\kappa(\theta)-\theta q} \frac{4}{\pi} \sum_{k=N}^{\infty}\frac{|\phi_{\theta}(u_k)|}{k}.
\sum_{k\in s}Use in source · 4 passages
Search aliases: ∑_k∈ s sum in
Use in source · Read passage · B_lean.tex, line 40
There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The factor convention is the sample Gram AA^top, matching declstream_sample_gram. It is not the dual Gram A^top A. The case-study statistic Q=tfrac12 v^to…
Exact mathematical passage
\operatorname{gramEntry}(s,A,i,j)&=\sum_{k\in s}A_{ik}A_{jk},\\[6pt] \operatorname{matrixAction}(K,v,i)&=\sum_{j\in I}K_{ij}v_j,\\[6pt] \operatorname{streamedAction}(s,A,v,i)&=\sum_{k\in s}A_{ik}\sum_{j\in I}A_{jk}v_j,\\[6pt] \operatorname{quadraticForm}(K,v)&=\sum_{i\in I}v_i\operatorname{matrixAction}(K,v,i),\\[6pt] \operatorname{factorEnergy}(s,A,v)&=\sum_{k\in s}\left(\sum_{i\in I}A_{ik}v_i\right)^2.
\sum_{m=1}^{p}Use in source · 2 passages
Search aliases: ∑_m=1^p sum
Use in source · Read passage · 06_packets.tex, line 123
tered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P,p(s)| ≤ fracn_P2 fracρ_P(s)^p+1(p+1)(1-ρ_P(s)).
Exact mathematical passage
\kappa_P(s) &=-\frac{n_P}{2}\log(1-2s\mu_P) +\frac12\sum_{m=1}^{p} \frac{\alpha_P(s)^m}{m}M_{P,m} +R_{P,p}(s), \label{eq:packet-expansion}
\sum_{m\ge p+1}Use in source · 1 passage
Search aliases: ∑_m≥ p+1 sum ge
Use in source · Read passage · 06_packets.tex, line 162
alpha_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing over j∈ P gives [source reference]. The remainder satisfies |∑_m≥ p+1fracz^mm| ≤ ∑_m≥ p+1frac|z|^mp+1 =frac|z|^p+1(p+1)(1-|z|). Using |z|≤ρ_P(s) and summing n_P terms yields [source reference].
Exact mathematical passage
\left|\sum_{m\ge p+1}\frac{z^m}{m}\right| \le \sum_{m\ge p+1}\frac{|z|^m}{p+1} =\frac{|z|^{p+1}}{(p+1)(1-|z|)}.
\sum_{m\ge1}Use in source · 1 passage
Search aliases: ∑_m≥1 sum ge
Use in source · Read passage · 06_packets.tex, line 158
s positive real part for each 1-2sλ_j and for 1-2sμ_P. The disk condition gives positive real part for 1-α_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing over j∈ P gives [source reference]. The remainder satisfies |∑_m≥ p+1fracz^mm| ≤ ∑_m≥ p+1frac|z|^mp+1 =frac|z|^p+1(p+1)(1-|z|). Using |z|≤ρ_P(s) and sum…
Exact mathematical passage
-\log(1-z)=\sum_{m\ge1}\frac{z^m}{m}.
\sum_{n\ge1}Use in source · 1 passage
Search aliases: ∑_n≥1 sum ge
Use in source · Read passage · 01_gaussian.tex, line 150
The packet representation is sign aware: positive and negative eigenvalues are clustered separately, and each same-sign packet stores a center μ and centered moments. The local expansion remains -frac12log(1-2s(μ+δ)) = -frac12log(1-2sμ) + frac12∑_n≥1 fracα(s)^nδ^nn, α(s)=frac2s1-2sμ, with the same geometric remainder condition |α(s)|max|δ|<1.
Exact mathematical passage
-\frac12\log(1-2s(\mu+\delta)) = -\frac12\log(1-2s\mu) + \frac12\sum_{n\ge1} \frac{\alpha(s)^n\delta^n}{n}, \qquad \alpha(s)=\frac{2s}{1-2s\mu},
\sum_rUse in source · 1 passage
Search aliases: ∑_r sum
Use in source · Read passage · 02_streaming.tex, line 275
[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.
Exact mathematical passage
\boxed{ R_b = \sum_r Q_{X,I_r}^\top H_{b,I_r}, }
\sum_{\{h,g\}\in\mathcal O}Use in source · 1 passage
Search aliases: ∑_{h,g}∈ O sum in mathcal
Use in source · Read passage · 03_phases.tex, line 334
. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.
Exact mathematical passage
H_{\mathcal O} =H-\sum_{\{h,g\}\in\mathcal O} \bigl(P_hHP_g+P_gHP_h\bigr).
\setminusUse in source · 1 passage
Search aliases: ∖ setminus
Use in source · Read passage · A_atlas.tex, line 21
X=Σ^1/2Z, Z∼ N(0,I), H=Σ^1/2MΣ^1/2. X^ TMXoverset d=∑_j=1^rλ_jZ_j^2, λ_j∈spec(H)∖{0}. For a central Gaussian vector with covariance Σsucceq0 and symmetric M, the quadratic form becomes a weighted sum of independent standard-normal squares after whitening and orthogonal diagonalization. The operator may originate in a kernel, a signal model, or another symmetric construction. Once its spectrum has been correctly obtained, the tail layer sees those coefficients.
Exact mathematical passage
X^{\mathsf T}MX\overset d=\sum_{j=1}^{r}\lambda_jZ_j^2,\qquad \lambda_j\in\operatorname{spec}(H)\setminus\{0\}.
\circUse in source · 3 passages
Search aliases: ∘ circ
Use in source · Read passage · 01_gaussian.tex, line 309
The solver separates four logically different representations: storage blocks ∘ structural phases ∘ Krylov/range compression ∘ spectral tail packets. Storage blocks answer where data reside. Structural phases answer how the operator propagates. Krylov/range compression answers which subspace has actually been resolved. Spectral packets answer how the final transform is evaluated. These decompositions should not be conflated.
Exact mathematical passage
\boxed{ \text{storage blocks} \circ \text{structural phases} \circ \text{Krylov/range compression} \circ \text{spectral tail packets}. }
\inftyUse in source · 17 passages
Search aliases: ∞ infty
Use in source · Read passage · 01_gaussian.tex, line 135
For the right-tail path, the cumulant generating function is κ(s) = -frac12∑_jlog(1-2sλ_j), with real domain max_λ_j<0frac12λ_j < s < min_λ_j>0frac12λ_j. Here a missing negative or positive spectral set gives a domain endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain whenever a positive spectral component is present. The negative-only case with at least one nonzero coefficient has a positive saddlepoint only for E[Q…
Exact mathematical passage
-\infty
\cupUse in source · 2 passages
Search aliases: ∪ cup
Use in source · Read passage · B_lean.tex, line 57
The principal refinement theorem is declgramEntry_union. Its hypotheses are finite sets s,t⊆ V and Disjoint(s,t). It proves, for all sample coordinates i,j, gramEntry(s∪ t,A,i,j) =gramEntry(s,A,i,j)+gramEntry(t,A,i,j). Disjointness prevents repeated columns from being counted twice. It is a mathematical condition, not a performance assumption. The theorem applies to empty blocks and zero factors. It does not require positive weights, since A is already supplied as a real factor.
Exact mathematical passage
\operatorname{gramEntry}(s\cup t,A,i,j) =\operatorname{gramEntry}(s,A,i,j)+\operatorname{gramEntry}(t,A,i,j).
\int_0^xUse in source · 1 passage
Search aliases: ∫_0^x int
Use in source · Read passage · 01_gaussian.tex, line 361
Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×10^-5. This formula will be used as an independent control, not as a shortcut inside the COS calculation. The next part begins with the factor A and as…
Exact mathematical passage
f_Q(x)=\int_0^x\frac12e^{-y/2}\frac14e^{-(x-y)/4}\,dy =\frac12\bigl(e^{-x/4}-e^{-x/2}\bigr).
\int_0^\inftyUse in source · 1 passage
Search aliases: ∫_0^∞ int infty
Use in source · Read passage · 11_evidence.tex, line 110
The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponential tilting, the COS expansion, spectral packets, or the K-COS error controller.
Exact mathematical passage
P(Q\ge q) = \frac12 + \frac1\pi \int_0^\infty \frac{\sin\Theta(u)}{u\,R(u)}\,du,
\int_N^{\infty}Use in source · 1 passage
Search aliases: ∫_N^∞ int infty
Use in source · Read passage · 07_inversion.tex, line 271
omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1+fracN^-β_Nβ_N, which yields [source reference].
Exact mathematical passage
\sum_{k=N}^{\infty}k^{-\beta_N-1} \le N^{-\beta_N-1}+\int_N^{\infty}x^{-\beta_N-1}\,dx =N^{-\beta_N-1}+\frac{N^{-\beta_N}}{\beta_N},
\int_T^\inftyUse in source · 1 passage
Search aliases: ∫_T^∞ int infty
Use in source · Read passage · 11_evidence.tex, line 139
The four spectra include balanced, skewed, dominant-spike, and higher-rank cases. Each reference integral was evaluated at two truncation limits. The omitted integral after the frequency cutoff T>0 was bounded using frac1π ∫_T^∞fracduuR(u) ≤ fracCπνT^-ν, ν=frac12∑_jh_j, C=∏_jλ_j^-h_j/2.
Exact mathematical passage
\frac1\pi \int_T^\infty\frac{du}{uR(u)} \le \frac{C}{\pi\nu}T^{-\nu}, \qquad \nu=\frac12\sum_jh_j, \qquad C=\prod_j\lambda_j^{-h_j/2}.
\simUse in source · 11 passages
Search aliases: ∼ sim
Use in source · Read passage · 01_gaussian.tex, line 14
Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.
Exact mathematical passage
X\sim N(0,\Sigma), \qquad M\succeq0,
\simeqUse in source · 1 passage
Search aliases: ≃ simeq
Use in source · Read passage · B_lean.tex, line 85
The reindexing theorem declgramEntry_reindex handles a bijection e:V≃ V. It moves the block to e(s) and transforms the factor to A'_ik=A_i,e^-1(k). The resulting Gram entry is unchanged. Permuting only the coordinates without moving their factor columns falls outside this statement.
Exact mathematical passage
e:V\simeq V
\approxUse in source · 6 passages
Search aliases: ≈ approx
Use in source · Read passage · 06_packets.tex, line 186
The running example under an exponential tilt. For spectrum (1,1,2,2), the cumulant generator is κ(s)=-log(1-2s)-log(1-4s) on Re s<1/4. At q=40, the saddlepoint equation reduces to 160θ^2-112θ+17=0, with frac21-2θ+frac41-4θ=40, θ_* = frac14-sqrt2640≈0.2225245122. The other root lies outside the transform domain. The tilted mean is 40, while the likelihood-ratio identity restores the original tail probability.
Exact mathematical passage
\frac{2}{1-2\theta}+\frac{4}{1-4\theta}=40, \qquad \theta_* = \frac{14-\sqrt{26}}{40}\approx0.2225245122.
\neUse in source · 5 passages
Search aliases: ≠ ne
Use in source · Read passage · 01_gaussian.tex, line 63
If AA^top u=λ u with λ>0, then A^top u≠0 and A^top A(A^top u)=λ A^top u. Conversely, an eigenvector v of A^top A with positive eigenvalue maps to the eigenvector Av of AA^top. These maps restrict to inverse isomorphisms of the corresponding eigenspaces after scaling, so multiplicities agree. Horizontal concatenation gives AA^top=∑_b A_bA_b^top by direct matrix multiplication. Orthogonal diagonalization then identifies the associated Gaussian quadratic-form laws.
Exact mathematical passage
A^\top u\ne0
\leqUse in source · 10 passages
Search aliases: ≤ leq
Use in source · Read passage · 03_phases.tex, line 94
[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).
Exact mathematical passage
\boxed{ \rho(HX)\leq \mathcal B\rho(X) }
\geqUse in source · 2 passages
Search aliases: ≥ geq
Use in source · Read passage · 03_phases.tex, line 99
[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).
Exact mathematical passage
k\geq0
\subsetUse in source · 1 passage
Search aliases: ⊂ subset
Use in source · Read passage · 06_packets.tex, line 99
Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.
Exact mathematical passage
P\subset\{1,\ldots,r\}
\perpUse in source · 7 passages
Search aliases: ⊥ perp
Use in source · Read passage · 04_range.tex, line 69
Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥, Q_⊥^top Q_⊥=I, Q^top Q_⊥=0. Here the columns of Q_⊥ form an orthonormal basis of the nonzero residual range. Zero residual directions are omitted.
Exact mathematical passage
C=Q^\top U, \qquad F=U-QC, \qquad F=Q_\perp R_\perp, \qquad Q_\perp^\top Q_\perp=I, \qquad Q^\top Q_\perp=0.
\longrightarrowUse in source · 8 passages
Search aliases: ⟶ longrightarrow
Use in source · Read passage · 03_phases.tex, line 61
This is the key composition law: mathematical locality ⟶ active phases ⟶ active storage blocks.
Exact mathematical passage
\boxed{ \text{mathematical locality} \longrightarrow \text{active phases} \longrightarrow \text{active storage blocks}. }
\longmapstoUse in source · 2 passages
Search aliases: ⟼ longmapsto
Use in source · Read passage · 08_adaptive.tex, line 29
The resulting object is query dependent. Schematically, (Λ,q,τ) ⟼ P_Λ,q,τ ⟼ (widehat p,certificate), where P_Λ,q,τ stores exact spikes, packet centers, centered moments, packet radii, and the selected COS parameters.
Exact mathematical passage
(\Lambda,q,\tau) \longmapsto \mathcal P_{\Lambda,q,\tau} \longmapsto (\widehat p,\text{certificate}), \label{eq:compiler-map}
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- monographs/cos-method/README.md · 0 mathematical passages
- monographs/cos-method/book/K_COS_Monograph.tex · 0 mathematical passages
- monographs/cos-method/book/chapters/01_gaussian.tex · 115 mathematical passages
- monographs/cos-method/book/chapters/02_streaming.tex · 89 mathematical passages
- monographs/cos-method/book/chapters/03_phases.tex · 158 mathematical passages
- monographs/cos-method/book/chapters/04_range.tex · 54 mathematical passages
- monographs/cos-method/book/chapters/05_spectrum.tex · 180 mathematical passages
- monographs/cos-method/book/chapters/06_packets.tex · 80 mathematical passages
- monographs/cos-method/book/chapters/07_inversion.tex · 127 mathematical passages
- monographs/cos-method/book/chapters/08_adaptive.tex · 22 mathematical passages
- monographs/cos-method/book/chapters/09_intervals.tex · 64 mathematical passages
- monographs/cos-method/book/chapters/10_accuracy.tex · 48 mathematical passages
- monographs/cos-method/book/chapters/11_evidence.tex · 174 mathematical passages
- monographs/cos-method/book/chapters/12_synthesis.tex · 57 mathematical passages
- monographs/cos-method/book/chapters/A_atlas.tex · 110 mathematical passages
- monographs/cos-method/book/chapters/B_lean.tex · 60 mathematical passages
- monographs/cos-method/book/chapters/C_reproduction.tex · 5 mathematical passages
- monographs/cos-method/book/chapters/D_glossary.tex · 40 mathematical passages
- monographs/cos-method/book/frontmatter.tex · 28 mathematical passages
- monographs/cos-method/book/references.tex · 1 mathematical passage
- monographs/cos-method/book/replay_results.tex · 8 mathematical passages
- monographs/cos-method/book/worked_enclosure_result.tex · 11 mathematical passages
- monographs/cos-method/book/worked_packet_table.tex · 12 mathematical passages
- monographs/cos-method/lean/README.md · 0 mathematical passages
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