9 Directed probability enclosures
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An analytic remainder estimate controls the error of a mathematical approximation. A numerical enclosure must also control the arithmetic used to evaluate it. This chapter joins directed arithmetic to explicit remainder bounds, yielding an interval for an exactly supplied positive spectrum under the stated arithmetic trust assumptions.
The notation follows the analytic development: \(\kappa\) is the cumulant generating function, \(b\) the support endpoint, \(N\) the retained mode count, and \(J_k\) the payoff coefficient. The implementation-name correspondence is given in Appendix C.
9.1 Interval target
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For a positive central quadratic form
the output is not merely a point estimate. It is
Hence the corresponding multiplicative uncertainty satisfies
The finite transform and exponential prefactor are evaluated by directed-rounding interval arithmetic. The resulting interval is enlarged by two independently proved analytic remainders: a support/periodization alias bound and an omitted-COS-mode bound.
9.2 No exact saddlepoint is required
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Choose any
Exponential tilting gives
where
The interval proof does not require \(\theta\) to be the exact saddlepoint. A high-precision saddlepoint is used only as an efficient design choice. Once serialized as an exact decimal input, it is simply a valid tilt inside the MGF domain.
The same is true of the support endpoint \(b\) and the auxiliary Chernoff parameter \(\eta\). Correctness requires only
9.3 Rigorous support and alias control
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Under the tilted measure,
The full-characteristic-function cosine expansion periodizes the tilted density on \([0,b]\). Because the original positive quadratic form is supported on \([0,\infty)\), the total positive-tail mass aliased back into one fundamental interval is at most \(P_\theta(Q>b)\).
The actual payoff omitted above \(b\) is at most
Therefore
9.4 Rigorous infinite-mode control
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Define
The normalized tilted characteristic function satisfies
Since
we have
For the payoff coefficient
the elementary bound
holds.
Let
For a finite transform retaining modes \(0,\ldots,N-1\),
This implemented interval backend therefore requires \(r>2\). Lower-rank positive forms can use a different tail bound and are not claimed by this theorem.
9.5 Final enclosure theorem
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Let \(\boldsymbol S_N\) be the directed-rounding interval containing the finite \(N\)-mode tilted COS sum, and set
Then
If the lower endpoint of this interval is positive, directed rounding gives
All finite arithmetic in this expression is enclosed by the interval backend.
9.6 Completing the running calculation
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The factor, spectrum, and tilt of Sections 1.5–6.3 now determine a complete positive-spectrum calculation at \(q=40\). The exact diagonalization supplies the coefficients \((1,1,2,2)\). The implementation receives those coefficients as exact decimal literals, together with a serialized decimal approximation to the admissible tilt. Correctness does not require that decimal to equal the radical saddlepoint exactly.
With \(b=512\), \(N=16384\), auxiliary tilt \(\eta=0.245\), and 60 interval decimal digits, the supplied backend returns a logarithmic interval. Rounding its lower endpoint downward and its upper endpoint upward gives the readable enclosure
At \(N=8192\), the directed ratio upper bound is 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.
This same fixed enclosure does not meet tolerance \(0.001\). The acceptance routine rejects that tighter request. This is an intended distinction between an accurate point estimate and an established enclosure width. The rank-four example has slower Fourier decay than the rank-eighty control developed later, so its mode count should not be interpreted as a general cost prediction.
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The certificate in this example concerns the tail evaluation for the exact spectrum. Its connection to the matrix is the explicit algebraic diagonalization above. It does not test a black-box eigensolver. Chapter 5 gives the separate post-verification theorem required for a general computed eigensystem.
9.7 Independent exact interval control through \(10^{-5000}\)
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For the control
the survival probability has the finite exact formula
This finite expression was evaluated independently by directed-rounding interval arithmetic, without using K-COS. The K-COS enclosure and this exact control enclosure are compared at thresholds targeting
With \(N=128\) and 90 interval decimal digits, the resulting K-COS widths are listed below. Widths and ratios are computed from the full outward endpoints before rounding the displayed values. At all three stored inputs, the independent exact-law interval is fully contained in the K-COS interval.
| target | width in \(\log_{10}p\) | multiplicative ratio minus one |
|---|---|---|
| \(10^{-100}\) | \(8.93\times10^{-13}\) | \(2.06\times10^{-12}\) |
| \(10^{-1000}\) | \(7.01\times10^{-12}\) | \(1.62\times10^{-11}\) |
| \(10^{-5000}\) | \(3.34\times10^{-11}\) | \(7.70\times10^{-11}\) |
Thus the \(10^{-5000}\) result is not merely a point estimate agreeing with a reference. For the exact supplied spectrum, it comes with a directed-rounding logarithmic enclosure whose probability ratio satisfies
The displayed upper bound is rounded upward from the directed ratio bound.
9.8 Arithmetic width versus analytic width
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At the \(10^{-5000}\) control, the interval precision was swept from 40 to 120 decimal digits. The total certified width remained essentially
decades.
Over the same sweep, the arithmetic-only width contracted from approximately
to
decades.
This separation is important. Increasing numerical precision cannot improve an analytic truncation theorem. Conversely, once interval arithmetic is far below the analytic error budget, numerical rounding is no longer the limiting uncertainty.
9.9 Guarantee boundary
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The directed enclosure has the following scope:
Guarantee boundary
For an exact supplied positive decimal spectrum of rank greater than two, the backend returns a directed-rounding logarithmic probability enclosure enlarged by proved analytic support/alias and omitted-mode bounds.
The result is conditional on correctness of the interval-arithmetic library implementation. The implementation uses mpmath.iv, whose interval layer applies downward and upward endpoint rounding.
A completely end-to-end guarantee beginning from an arbitrary floating-point matrix requires one more layer: either a rigorously enclosed matrix-to-spectrum computation, or an exact/residual completion step whose linear algebra is also performed with verified interval or ball arithmetic.
The supplied interval backend evaluates the exact spectrum directly. Packet centers, moments, and their propagated errors are not evaluated with directed arithmetic in this implementation. The packetized path therefore retains its separate analytic guarantee.