# T456 build notes

This draft stores the lower-tail quantiles of the limiting, scaled two-sided
Kolmogorov-Smirnov statistic. It follows family #197's 14 exact rational
probabilities, ascending order, symbol kappa_p, and 100-digit proven rigour.
It does not hold finite-sample or one-sided statistic quantiles.

Numerical production: exact-rational bisection, Arb alternating-series CDF,
and a first-omitted-term remainder. The 461-bit field retained at least 110
decimal digits across all 14 balls. Each output decimal has 100 significant
digits and denotes the interval one unit of its last place on either side.

Independent checking: MPFI arithmetic evaluates the positive transformed
series with a geometric remainder, and proves the CDF has opposite signs
relative to p at every written interval's endpoints. All 14 also agree with
mpmath theta-function inversion at 130 decimal working precision and SciPy
1.17.1 at machine precision. The controls include the nonzero exact identity
F(sqrt(pi/2)) = Gamma(1/4)/(2*pi)^(3/4), and rejection of each interval shifted
by 100 last-place units. Every control enclosure has diameter below 1e-135.

The gamma control follows from the theta-elliptic relations in DLMF 20.9,
the identity theta3^4 = theta2^4 + theta4^4 and the modular transformation
at tau=i in DLMF 20.7, and the lemniscatic elliptic integral value.
It is a control of the checking machinery, not an extra table entry.

Sources read: SciPy kstwobign reference and tutorial, and Paul van Mulbregt,
Computing the Cumulative Distribution Function and Quantiles of the limit
of the Two-sided Kolmogorov-Smirnov Statistic, arXiv:1803.00426, equations
(3) and (16). The second supplies the transformed CDF for checking.

Conventions resolved: for this two-sided KS statistic a test at level alpha
uses kappa_(1-alpha), without an alpha/2 adjustment. Programs give mpmath
and SciPy examples; there is no invented base-R qkolmogorov call.

Corpus: no matching family was found by screening, focused text searches,
or numerical lookups near 0.8275735551899077, 1.3580986393225505 and
1.6276236115189502. The existing probability theory tag is used. The empty
statistics tag is not used; T452, T453, T454, T455 and T456 would justify it
as a subject heading if a person decides to activate it.

Publication dependency: T454 and T455 are readable to this account but their
unauthenticated pages return 404. The draft links them by their valid stable
T-number identifiers. The controller/reviewer should address those draft
links before publication. No studentized-range table was found; no address
has been invented for it. No request provenance appears in mathematical prose.

This build does not publish, offer for review, or conduct the separate critique.