back to table · edit · history · where entries came from · files · download
5565 bytes, as of the version from 2026-11-10 01:14 (current). Recorded here, not run.
# Argument selection for T701
The governing skill is recorded in `skill-manifest.json`, copied from
`agents/runs/20261110T005551Z-build-skills/manifest.json` in this run's source
snapshot. The local and live copies both have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
This build justification is not an independent critique acceptance.
## Quantity and marking
Only the real elliptic dilogarithm companion J is stored. Brunault, equation
(14), defines its logarithmic average and Bernoulli correction; Samart's
introduction gives the same definition. D and J are distinct scalar components
of an elliptic regulator. T700 stores D, not J. All logarithms are natural.
Set b=0 for discriminants divisible by four and b=1 otherwise, and use the
principal marked lattice with tau=(-b+i sqrt(|Delta|))/2 and q=exp(2 pi i tau).
The square root is positive. Odd discriminants give negative real q.
## Delta: arithmetic_structure
Take -3,-4,-7,-8,-11,-12,-16,-19,-27,-28,-43,-67,-163, in increasing absolute
value. These are exactly the imaginary quadratic orders of class number one,
equivalently the orders with rational singular modulus. The Sage CM manual's
`cm_orders(1)` and the live rational singular moduli table T83 support this
identification. Nonmaximal orders are included: Delta is an order discriminant,
not necessarily a field discriminant. This complete arithmetic slice includes
rectangular and rhombic lattices and the extra-automorphism cases. The marking
is never silently changed by a modular transformation. Nonprincipal classes
and higher class numbers are outside this initial table coverage.
## N: discrete_index
N is the exact torsion order, not an independently sampled real input. We cover
3 <= N <= 6. Orders 3,4,6 occur in the weight-two torsion regulator setting of
Guillera and Rogers, Theorem 4. Those formulas concern D, so they establish
family context, not a separate application for every J value. The square-lattice
rotation relates J at i/4 to D at 1/4 and supplies a direct Mahler-integral
control. Order 5 contributes the next prime group, while 6 is the first group
with mixed prime order. This bounded arithmetic reference tests the whole
finite group structure and permits recognition beyond named examples. Six is
a noncanonical practical bound independent of numeral notation, not a claimed
intrinsic threshold. We prefer all thirteen CM orders and all directions at
these small orders to a deep single-axis sequence. No publication is claimed
for every combination. Orders one and two give zero by oddness and are omitted.
## r and s: joint arithmetic_structure
The point is u=(r tau+s)/N. Enumerate 0 <= r,s < N with gcd(N,r,s)=1, which
is exactly the finite group locus of order N. Retain the lexicographically
smaller pair between (r,s) and (-r,-s) modulo N. This representative convention
is noncanonical and notation-independent. The signed value is preserved;
the omitted inverse has its negative. These two coordinates are constrained
finite-group coordinates, not independently sampled continuous arguments.
For real q the function is unchanged by conjugating x=exp(2 pi i u), and is
negated by x -> 1/x. Thus it vanishes whenever the conjugate of x represents
the inverse point. In this marking that occurs precisely on the locus specified
by both 2r=0 (mod N) and br=0 (mod N). Remove that locus algebraically, never
by a numerical threshold. Other coincidences are retained. The additional zero at Delta=-3, N=3,
(r,s)=(1,2) is retained exactly: multiplication by tau fixes this point
modulo the lattice, while reindexing the Eisenstein-Kronecker expression
multiplies the regulator by 1/tau. Since tau is not 1, the regulator is zero. The gcd, inversion
and zero conditions must be reviewed jointly. The result is 362 rows: 26 on
each of five rectangular orders and 29 on each of eight rhombic orders.
## Extent and consistency
The enumeration, completeness note, definition of the marked arguments and
attached generator describe the same selection. No probability parameters,
decimal meshes, independently varying continuous couplings, second quantities,
or extension of T700 are present. The cutoff and representative choices are
explicit finite reference choices inside an arithmetic/discrete construction;
there is no additional `reference_sample` input family requiring a separate
reference policy. The critic should decide that classification independently.
## Sources checked
- F. Brunault, On Zagier's conjecture for base changes of elliptic curves,
section 2, equation (14) and Proposition 6: logarithmic definition, correction,
marking, and negative Eisenstein-Kronecker sign.
https://perso.ens-lyon.fr/francois.brunault/recherche/LEF.pdf
- D. Samart, The elliptic trilogarithm and Mahler measures of K3 surfaces,
arXiv:1309.7730v2, section 1: same J-series; its displayed positive lattice
sign disagrees with Brunault and must not govern the computation.
- J. Guillera and M. Rogers, Mahler measure and the WZ algorithm,
arXiv:1006.1654v2, Theorem 4: weight-two torsion context and Mahler control.
- SageMath CM documentation, `cm_orders(1)`:
https://doc.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/cm.html
- Live T700 and the family coverage map in numberdb-data issue 311, read during
this run, for shared parameter order and marking. T83 and T114 were read for
the arithmetic slice and B3 convention. Other live tables in the family map
were inspected for scope and do not store this named quantity.