D Glossary and notation
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| Term | Meaning |
|---|---|
| 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. |
| 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. |
| CDF | Cumulative distribution function, \(F(q)=\Pr(Q\le q)\). |
| CGF | Cumulant generating function. In the complex transform domain, the analytic logarithm normalized at zero is used. |
| Central form | A quadratic form of a Gaussian vector with zero mean. |
| Certificate | A justified bound with a stated object and scope. Structural, analytic, numerical, and formal certificates are not interchangeable. |
| Centered packet | A nonempty spectral group represented by its center, centered moments, and a remainder bound valid on a specified complex grid. |
| COS | Fourier-cosine evaluation of a functional using characteristic-function coefficients. |
| Directed enclosure | An outer interval whose finite arithmetic is rounded outward and which is enlarged by proved analytic remainders. |
| 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. |
| Exact completion | Recovery of unresolved factor directions when a residual-rank argument permits the remaining range to be represented explicitly. |
| Exact input | The mathematical value encoded by the supplied literal or stored binary entry, not an unspecified quantity before rounding. |
| Fail-closed | A numerical policy that does not label a result certified unless the stated acceptance inequality is proved. |
| Folding | Reflection of a positive-support measure into a cosine period. It explains the alias contribution from mass outside the selected window. |
| Gram representation | The operator \(AA^\top\) or its dual \(A^\top A\). Their positive eigenvalues agree with multiplicity. |
| Krylov query | A vector or block-vector expression generated by repeated operator actions, often summarized as a polynomial in the operator. |
| LD | Linkage disequilibrium, the dependence among alleles at distinct loci represented by the synthetic genotype models. |
| Light cone | The phases reachable from a specified initial support in a bounded number of exact structural steps. |
| MAF | Minor-allele frequency, used to define a variant's weight in the synthetic kernel examples. |
| MiB | Mebibyte, equal to \(2^{20}\) bytes. Memory tables report modeled payload, not total device allocation. |
| 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. |
| Phase | A structural subspace specified by an orthogonal projector. It need not coincide with a physical storage block. |
| Post-verification | Certification of a candidate numerical output using independently enclosed residuals rather than trust in the algorithm that proposed it. |
| Range core | A small factor or operator representation on a captured subspace. Exact spectral equivalence requires that the relevant range be captured. |
| Residual | The component not represented by a chosen subspace or approximation. Its trace, square-trace, norm, and rank carry different information. |
| 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. |
| SKAT | Sequence kernel association test, used here as an application of Gaussian quadratic-form evaluation. |
| Spectral passport | The query-dependent collection of spikes, packet centers, moments, radii, and COS parameters needed for repeated evaluation. |
| Storage refinement | Splitting a factor block into disjoint smaller blocks while retaining all columns and hence the same Gram operator. |
| Survival probability | The upper-tail probability \(\Pr(Q\ge q)\). For the nondegenerate continuous laws considered here, strict and non-strict thresholds agree. |
D.1 Principal symbols
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| Symbol | Convention |
|---|---|
| \(Q\) | A quadratic random variable. In the candidate-eigensystem theorem, the local symbol \(Q\) instead denotes a candidate basis matrix. |
| \(A\), \(A_b\) | A real factor and its resident column blocks. |
| \(H\), \(K\) | Symmetric operators, with sample Gram \(K=AA^\top\) and feature Gram \(H=A^\top A\) in the streaming construction. |
| \(\Sigma\), \(M\), \(\mu\) | Covariance, quadratic weight, and Gaussian mean. |
| \(\lambda_j\) | Spectral coefficients. Positivity is imposed whenever a positive-support theorem requires it. |
| \(q\) | The tail threshold. |
| \(\kappa\) | Cumulant generating function. The implementation name K(s) is mapped in Appendix C. |
| \(\theta\), \(\eta\) | Admissible exponential tilts, with \(\eta>\theta\) when used in a Chernoff support bound. |
| \(b\) | Upper endpoint of the positive COS support window. The implementation calls this U. |
| \(N\) | Retained COS mode count. The implementation argument is K. |
| \(P_h\) | Orthogonal structural phase projectors. |
| \(M_{P,m}\) | The \(m\)th centered power sum of a spectral packet \(P\). |
| \(\rho_P(s)\) | The packet's complex-grid convergence radius parameter. |
| \(\varepsilon_{\rm pkt}\), \(\varepsilon_{\rm sup}\), \(\varepsilon_{\rm COS}\) | Packet, support/folding, and omitted-mode contributions to the analytic budget. |
| \([p_-,p_+]\) | A probability enclosure. Acceptance at relative tolerance \(\varepsilon\) requires \(p_->0\) and \(p_+/p_- -1\le\varepsilon\). |
| \([d_-,d_+]\) | A base-ten logarithmic probability enclosure. |