Preface
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A Gaussian quadratic-form probability is often presented as an inversion problem. In a large calculation, the spectrum required by that inversion must first be obtained. Data may not fit in memory. A structural approximation may discard couplings. A reduced basis may leave part of the range unresolved. Each choice affects what can ultimately be said about the probability.
This book connects those choices. It begins with the exact Gaussian reduction to a weighted sum of independent squares and ends with an interval for the requested tail probability. Between them lie streamed factors, phase blocks, range cores, spectral intervals, and local transform expansions. Changing representation reduces the work only if the mathematical object is preserved or the resulting error is controlled.
K-COS is the transform method at the end of this construction. Exponential tilting moves an upper-tail query toward the center of a reweighted law. Centered spectral packets reduce repeated evaluations of the cumulant generating function. Fourier-cosine inversion evaluates the weighted tail functional. Packet approximation, finite support, and omitted modes contribute separate errors. Their separate bounds make refinement possible.
The distinction between exact representation and controlled approximation runs throughout the argument. So does the distinction between a real-arithmetic theorem and a computed interval with outward rounding. These are the interfaces that determine which parts of the method can be composed.
The four parts follow that dependency: Gaussian laws, structured operators and spectral recovery, packet and inversion analysis, then numerical enclosure and acceptance. One four-dimensional example accompanies the development from factor to independently checkable probability. Further visual studies, formal correspondence, reproducible computation, notation, and references are collected at the back.
The intended reader knows finite-dimensional linear algebra, Gaussian probability, and basic numerical analysis. Written proofs carry the mathematical argument. The accompanying numerical and formal sources provide distinct, explicitly delimited forms of supporting evidence.