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# Argument selection for Values of the Wilton function (T678)
The sole free parameter is the exact real argument x. Its role is
`arithmetic_structure` throughout: periodic points of the ordinary Gauss map
T(x)={1/x}, selected by primitive quadratic discriminant. This is the same
arithmetic family used in the Brjuno companion T677. It is not a reference mesh
of convenient radicals, and there are no mixed or exploratory groups.
The archived canonical skill manifest is
`agents/runs/20261102T123618Z-build-skills/manifest.json`.
Both `skill-local.md` and `skill-live.md` have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
Their bytes were compared before the build. The archive, not an older policy
quoted in a campaign note, governs this selection. This build justification
is not the required independent critique or its argument-review sidecar.
## Family, conventions, and purpose
Balazard and Martin, arXiv:1305.4395, §§2.1–2.2 and 3.1–3.2, define the
ordinary Gauss iterates, products beta, and alternating Wilton series. Their
Propositions 8–9 give the functional equation and finite iteration formula.
At periodic quadratic points this becomes a finite signed logarithmic sum
divided by 1-(-1)^ell P. Thus the map that defines the function also selects
the argument structure, rather than a generic library evaluation selecting it.
These values give exact-orbit benchmarks for the signed sum, including both
odd and even periods, all starting phases, and negative outputs.
By Galois's criterion, applied to 1/x, purely periodic regular continued
fractions x=[0;overline(a_1,...,a_ell)] are precisely the quadratic irrationals
0<x<1 with conjugate x'<-1. The checked source is M. Waldschmidt,
*Arithmétique modulaire, fractions continues, corps finis*, CIMPA–Mauritanie
lecture notes (2016), §3.6, Lemma 47, pp. 37–38:
https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/NouakchottCoursCIMPA2016.pdf
The PDF text was accessible through the web research tool. Direct download
from the build environment timed out; no local PDF is asserted as archived.
Use the primitive polynomial A*t^2+B*t-K with A>0, K>0, gcd(A,B,K)=1.
The positive root is (-B+sqrt(D))/(2A), D=B^2+4AK, and its stable identity is
quad_A_B_mK. Every row labels that exact argument, not its value or a rounded
approximation. The polynomial variable t is auxiliary. D is an order/polynomial
discriminant and is not required to be fundamental. Natural real logarithms
and the first summand log(1/x) fix the normalization and sign convention.
## Finite extent and breadth
Choose every such argument with positive nonsquare D<=100, A+B-K>0,
and A-B-K<0. The discriminant bound 100 is a noncanonical computational
boundary, independent of decimal spelling of the arguments. It keeps the
radicals and exact orbit metadata short while covering 40 discriminants and
45 cycles with least periods 1, 2, 3, 4, 5, 6, 7, and 9. This covers more than
fixed points: it includes 9 fixed points and 132 longer-period starting points,
including nonfundamental orders. Distinct phases are retained because W(x)
depends on the starting argument; no cyclic-shift, reversal, or field quotient
is taken. The 141-entry count is an outcome, not a target.
General quadratic arguments with a preperiod are a useful unfilled direction.
Attaching every short prefix to every orbit would increase combinatorial
breadth without an equally compact completeness statement. The present table
first finishes these full bounded-discriminant periodic cycles, as does T677.
Larger discriminants may be added later using the same identifiers. Rationals,
divergent inputs, arbitrary real grids, and other continued-fraction maps are
outside this computation. No different named function is stored as a selector.
## Exhaustiveness and checks
Since x'<-1 and x<1, B=-A(x+x')>0 and sqrt(D)=A(x-x')>A.
Also B<sqrt(D), since K>0. Hence A,B<=9 at D<=100, and
K=(D-B^2)/(4A)<=24. The generator tests this finite box, with harmless
extra rejected boundary values. Independent enumeration factors
(D-B^2)/4 by discriminant and recovers exactly the same triples.
Under a=floor(1/x), the transformed triple is
(K, 2aK-B, A+Ba-Ka^2). The floor is exact using
(B+isqrt(D))//(2K), since D is nonsquare. Every cycle stays in the selection;
its primitive matrix fixed-point polynomial also recovers the original triple.
The period counts by starting point are 1:9, 2:28, 3:21, 4:24, 5:10, 6:24,
7:7, and 9:18. Every partial quotient is at most 9.
The generator, the document's completeness note, the exact labels, and the
stored entry set are checked together. `checks.py` independently sums 800
terms through integer convergents and a separate complete-quotient recurrence,
with an explicit tail and rounding enclosures. Full numerical outcomes are in
`numerical-evidence.json` and `stored-evidence.json`. A separate critique must
assess this selection and bind its verdict to the current draft and attached
source bytes using the campaign fingerprint command.