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# Argument selection for Values of the Brjuno function (T677)
This build follows the matching archived canonical skill versions recorded in
`agents/runs/20261102T115856Z-build-skills/manifest.json`. Both local and live
SHA-256 digests are
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The attached `skill-manifest.json` and `skill-evidence.json` preserve that record.
This is the builder's rationale, not the required independent critique.
## Meaning and role of the parameter
There is one free parameter, the exact real quadratic irrational argument
$x\in(0,1)$. The table holds one named quantity, the ordinary regular-continued-
fraction Brjuno function $B(x)$, using natural logarithms and first term
$\log(1/x)$. Buff–Chéritat, Definition 1, fixes this convention; neither the
continued-fraction denominator convergence-test series nor a nearest-integer
version is numerically interchangeable with it.
The selected parameter has role `arithmetic_structure` throughout. Periodic
points of the function's own Gauss map provide a family tied to its definition.
Galois's criterion, applied to $1/x$, identifies them exactly as quadratic
irrationals $0<x<1$ whose conjugate satisfies $x'<-1$. See Waldschmidt,
*Arithmétique modulaire, fractions continues, corps finis*, §3.6, Lemma 47,
pp. 37–38 (PDF pages 37–38). The initial zero in $[0;\overline{a_1,\ldots,a_\ell}]$
is not part of the repeated word. No decimal mesh, rational grid, or independent
coupling parameter is hidden in these arguments.
## Finite extent and breadth
Take every primitive integer polynomial $At^2+Bt-K$ with $A>0$, $K>0$,
$\gcd(A,B,K)=1$, $A+B-K>0$, and $A-B-K<0$, of positive nonsquare discriminant
$D=B^2+4AK\leq100$. Store its positive root $(-B+\sqrt D)/(2A)$.
These strict inequalities are exactly the required root conditions. The
primitive polynomial is unique, fixing the stable identity `quad_A_B_mK`.
The real parameter label displays this radical, not the function value.
The bound 100 is a noncanonical computational choice expressed through an
arithmetic invariant, independent of numeral notation. It keeps root labels
small while covering nonfundamental orders as well as fundamental field
discriminants, nine fixed points and cycles of lengths 2, 3, 4, 5, 6, 7 and 9.
The build reproduces 141 arguments, 40 discriminants and 45 cycles. The largest
partial quotient is 9. Increasing only the fixed-point index would have less
breadth; arbitrarily attaching prefixes to every cycle would add another
uncontrolled selection. General preperiodic quadratics are deferred, not
excluded from the parameter's mathematical domain.
Every starting phase is retained. Rotations, reversals, or field isomorphism
are not equivalences of function values. Full-cycle retention also makes the
functional equation testable throughout the selection. This is complete
coverage within a bounded arithmetic family, not an assertion that every
member has a separately documented application. It is not a mixed or
exploratory reference sample. Rationals and divergent arguments are omitted.
The Wilton companion is a separate named function and is not a selector here.
## Exhaustiveness and reproduction
The root conditions imply $B>0$ and $|A-K|<B$. Thus
$D=B^2+4AK>(A-K)^2+4AK=(A+K)^2$, so $A,K<\sqrt D$, and also $B<\sqrt D$.
The generator searches the safe box $1\leq A,B\leq\lfloor\sqrt{100}\rfloor$,
$1\leq K\leq25$, filtering all conditions. The independent checker instead
loops over discriminants and $B$, then factors $(D-B^2)/4=AK$. The two sets
agree exactly. Each polynomial is primitive and irreducible, so there are no
duplicate arguments.
The generator follows the exact polynomial substitution under the Gauss map.
The checker uses a separate complete-quotient recurrence in pairs $(h,k)$,
where $1/x_j=(\sqrt D+h)/k$, and independently detects each least period.
The two methods agree for every phase; every successor lies in the selection.
The period-length counts are 1:9, 2:28, 3:21, 4:24, 5:10, 6:24, 7:7, 9:18.
Entries are ordered by increasing discriminant, then $(A,B,K)$, so the
reciprocal golden-ratio input comes first. Written values have 101 characters,
including the decimal point; all carry 100 significant digits.
## Checked sources and related tables
- Buff and Chéritat, *The Brjuno function continuously estimates the size of
quadratic Siegel disks*, Annals of Mathematics 164 (2006), 265–312,
Definition 1; DOI 10.4007/annals.2006.164.265, arXiv math/0401044.
Author PDF: https://www.math.univ-toulouse.fr/~buff/Preprints/Continuity/Continuity.pdf
- Waldschmidt, §3.6, Lemma 47, Galois's pure-periodicity criterion:
https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/NouakchottCoursCIMPA2016.pdf
- T131 fixes the maximal-order regulator normalization; its $D=5$ entry is
$\log\varphi$ and supplies an external numerical control.
- T430 holds the Markov quadratic irrationals, including $\varphi$ and
$1+\sqrt2$, whose reciprocals are two selected fixed points.
- T174 studies the transfer operator of the same Gauss map, a different quantity.
The family issue #300 and its live coverage map were read before building.
All ten listed existing family neighbors were fetched and inspected. Searches
for Brjuno, Bruno, Yoccoz and Wilton, the screen helpers, and three numerical
lookups found no duplicate. The four proposed tags already exist. No Wilton
companion was public at build time, so no companion identifier was invented.
Request provenance belongs to the issue, not to mathematical table prose.