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# Argument selection for T607
This build follows the matching archived local and live canonical skill in
`agents/runs/20261029T103821Z-build-skills/manifest.json`. Both SHA-256 digests
are `28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The family proposal is numberdb-data#242; the proposal was reassessed against
that skill. This document is the builder's justification, not an independent
critique or an acceptance sidecar.
The quantity is the representation-dimension Dirichlet series at positive even
arguments. Algebra type names what the number is of. The exponent is an argument
of that same function. There is no second quantity, group quotient, root-wise
exponent, or adjustable coupling concealed in either parameter.
## Algebra: discrete_index
The infinite domain is complex simple Lie algebras up to isomorphism, indexed
by Cartan type. We select every such algebra of rank at most two: A1, A2, B2,
and G2. C2 is an alias of B2, not a fifth entry. A1 x A1 is semisimple but not
simple, so it is excluded. MIT 18.745 Lecture 21, section 21.3, Theorem 21.10
lists the rank-two reduced systems; the reducible case is A1 x A1:
https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/mit18_745_f20_lec21.pdf
This breadth includes the rank-one baseline, simply laced A2, and both
non-simply-laced rank-two types. The practical rank bound admits exact,
independently checkable formulas for every included type, before increasing
the depth for just A1 or A2. Rank three brings a third highest-weight index
and materially more complicated exact formulas. It is a future extension,
not a missing point inside the stated coverage.
## m: arithmetic_structure
The infinite domain is positive integers m, with zeta argument s=2m.
These arguments form the Witten volume special-value family:
Z_g(2m) is a rational multiple of pi^(2mN), where N is the number of positive
roots. This is a family-specific theorem, not a generic integer mesh on a
continuous function. Its moduli-volume motivation is described in Zhao,
*Witten volume formulas for semi-simple Lie algebras*, introduction and (2.1),
https://arxiv.org/abs/1001.3630, and Komori-Matsumoto-Tsumura IV, introduction,
(1.2)-(1.4), https://arxiv.org/abs/0907.0972.
For each algebra we select the full prefix m=1,2,3,4. The finite cutoff is a
practical low-order reference choice, not a theorem that singles out four.
The first four B2 and G2 coefficients have independent published exact
examples (Zhao 2009, section 4, page 8; KMT IV, Example 2.2). Including all
four moments permits comparisons across all four types and does not prune
the family to a single famous constant. Deeper moments approach 1 quickly;
the first four retain a useful low-order reference without extending only the
easier types for the sake of entry count. There is no quota or minimum count.
The rationale is identical for all four algebra groups. No group is an
exploratory continuous reference sample, so reference_sample and mixed would
misclassify this particular selection. Integer m is genuinely the even
special-value index. No numeral base selects the arguments.
## Normalization and enumeration
The sum includes the trivial representation exactly once and counts distinct
irreducibles with equal dimension separately. Compact group names refer to
the simply connected forms. B2 therefore means Spin(5)=Sp(2), not SO(5).
The rank-two dimension denominators are 2, 6 and 120. The independent sources
use root-system sums without the corresponding factors 2^s, 6^s and 120^s;
checks explicitly restore those factors.
The generator enumerates A1, A2, B2, G2 in that order, and m=1 through 4 within
each type. There are exactly 16 entries. The parameter declarations describe
the infinite domains; Data properties states these finite bounds and reasons.
`generate.py` has no local imports or data-file dependencies. `checks.py`
contains independent reference coefficients but does not define enumeration.
The critic should fingerprint the current `generate.py` attachment and inspect
this document, the checks, all formulas and the completeness note afresh.