COS Method

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1 \chapter{Glossary and notation}\label{app:glossary}
2 \begingroup
3 \small
4 \renewcommand{\arraystretch}{1.08}
5 
6 
7 \begin{longtable}{@{}P{.25\linewidth}P{.70\linewidth}@{}}
8 \toprule
9 Term & Meaning\\
10 \midrule
11 \endhead
12 Analytic ledger & A bound on packet, support, and mode approximation errors in the real or complex arithmetic model. It does not, by itself, enclose numerical rounding.\\
13 Binary64 & The IEEE double-precision binary format. A finite stored value represents an exact dyadic rational, although earlier preprocessing may have rounded the intended input.\\
14 CDF & Cumulative distribution function, $F(q)=\Pr(Q\le q)$.\\
15 CGF & Cumulant generating function. In the complex transform domain, the analytic logarithm normalized at zero is used.\\
16 Central form & A quadratic form of a Gaussian vector with zero mean.\\
17 Certificate & A justified bound with a stated object and scope. Structural, analytic, numerical, and formal certificates are not interchangeable.\\
18 Centered packet & A nonempty spectral group represented by its center, centered moments, and a remainder bound valid on a specified complex grid.\\
19 COS & Fourier-cosine evaluation of a functional using characteristic-function coefficients.\\
20 Directed enclosure & An outer interval whose finite arithmetic is rounded outward and which is enlarged by proved analytic remainders.\\
21 Effective rank & The participation ratio $(\sum_j\lambda_j)^2/\sum_j\lambda_j^2$ for a nonzero positive spectrum. It need not be an integer.\\
22 Exact completion & Recovery of unresolved factor directions when a residual-rank argument permits the remaining range to be represented explicitly.\\
23 Exact input & The mathematical value encoded by the supplied literal or stored binary entry, not an unspecified quantity before rounding.\\
24 Fail-closed & A numerical policy that does not label a result certified unless the stated acceptance inequality is proved.\\
25 Folding & Reflection of a positive-support measure into a cosine period. It explains the alias contribution from mass outside the selected window.\\
26 Gram representation & The operator $AA^\top$ or its dual $A^\top A$. Their positive eigenvalues agree with multiplicity.\\
27 Krylov query & A vector or block-vector expression generated by repeated operator actions, often summarized as a polynomial in the operator.\\
28 LD & Linkage disequilibrium, the dependence among alleles at distinct loci represented by the synthetic genotype models.\\
29 Light cone & The phases reachable from a specified initial support in a bounded number of exact structural steps.\\
30 MAF & Minor-allele frequency, used to define a variant's weight in the synthetic kernel examples.\\
31 MiB & Mebibyte, equal to $2^{20}$ bytes. Memory tables report modeled payload, not total device allocation.\\
32 Noncentrality & The squared mean displacement in a standardized Gaussian direction. The symbol $\delta_i$ in a noncentral chi-square component differs from a centered packet deviation.\\
33 Phase & A structural subspace specified by an orthogonal projector. It need not coincide with a physical storage block.\\
34 Post-verification & Certification of a candidate numerical output using independently enclosed residuals rather than trust in the algorithm that proposed it.\\
35 Range core & A small factor or operator representation on a captured subspace. Exact spectral equivalence requires that the relevant range be captured.\\
36 Residual & The component not represented by a chosen subspace or approximation. Its trace, square-trace, norm, and rank carry different information.\\
37 Saddlepoint tilt & An admissible exponential tilt satisfying $\kappa'(\theta_*)=q$. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.\\
38 SKAT & Sequence kernel association test, used here as an application of Gaussian quadratic-form evaluation.\\
39 Spectral passport & The query-dependent collection of spikes, packet centers, moments, radii, and COS parameters needed for repeated evaluation.\\
40 Storage refinement & Splitting a factor block into disjoint smaller blocks while retaining all columns and hence the same Gram operator.\\
41 Survival probability & The upper-tail probability $\Pr(Q\ge q)$. For the nondegenerate continuous laws considered here, strict and non-strict thresholds agree.\\
42 \bottomrule
43 \end{longtable}
44 
45 \clearpage\section{Principal symbols}
46 \begin{longtable}{@{}P{.25\linewidth}P{.70\linewidth}@{}}
47 \toprule
48 Symbol & Convention\\
49 \midrule
50 \endhead
51 $Q$ & A quadratic random variable. In the candidate-eigensystem theorem, the local symbol $Q$ instead denotes a candidate basis matrix.\\
52 $A$, $A_b$ & A real factor and its resident column blocks.\\
53 $H$, $K$ & Symmetric operators, with sample Gram $K=AA^\top$ and feature Gram $H=A^\top A$ in the streaming construction.\\
54 $\Sigma$, $M$, $\mu$ & Covariance, quadratic weight, and Gaussian mean.\\
55 $\lambda_j$ & Spectral coefficients. Positivity is imposed whenever a positive-support theorem requires it.\\
56 $q$ & The tail threshold.\\
57 $\kappa$ & Cumulant generating function. The implementation name \texttt{K(s)} is mapped in Appendix~\ref{app:reproduction}.\\
58 $\theta$, $\eta$ & Admissible exponential tilts, with $\eta>\theta$ when used in a Chernoff support bound.\\
59 $b$ & Upper endpoint of the positive COS support window. The implementation calls this \texttt{U}.\\
60 $N$ & Retained COS mode count. The implementation argument is \texttt{K}.\\
61 $P_h$ & Orthogonal structural phase projectors.\\
62 $M_{P,m}$ & The $m$th centered power sum of a spectral packet $P$.\\
63 $\rho_P(s)$ & The packet's complex-grid convergence radius parameter.\\
64 $\varepsilon_{\rm pkt}$, $\varepsilon_{\rm sup}$, $\varepsilon_{\rm COS}$ & Packet, support/folding, and omitted-mode contributions to the analytic budget.\\
65 $[p_-,p_+]$ & A probability enclosure. Acceptance at relative tolerance $\varepsilon$ requires $p_->0$ and $p_+/p_- -1\le\varepsilon$.\\
66 $[d_-,d_+]$ & A base-ten logarithmic probability enclosure.\\
67 \bottomrule
68 \end{longtable}
69 
70 
71 \endgroup

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