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3930 bytes, as of the version from 2026-10-29 14:26 (current). Recorded here, not run.
# Argument selection for T610
The governing skill is recorded in ../20261029T134044Z-build-skills/manifest.json
in the campaign artifact archive. Its local and live SHA-256 are both
28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.
The portable skill-evidence.json attachment records the same manifest and hashes.
This table follows family issue 244 and the live conventions of T609. It holds
one positive category invariant in two invertible normalisations, global G
and total D=sqrt(G). It does not include individual representation dimensions,
modular matrix entries, conformal extensions, or entropies as additional rows.
## Parameters and mathematical roles
* N is the matrix size of SU(N), an integer at least 2; the Lie rank is N-1.
It is a genuine discrete classification index, not an argument chosen from a
continuous mesh. Different N specify different type-A categories.
* k is the positive integral WZW level. Integrability, the finite alcove and
the root of unity q=exp(pi*i/(N+k)) depend on k. It is a discrete index of
the category, not an independently selected angle or coupling.
* normalisation is global or total. It selects two conventions of the same
positive number, G=D^2 and D>0, with precisely identical category coverage.
Both occur in the literature. This satisfies the invertible-convention
exception, with no different domain or costly conversion.
The definition, discrete indexing and q convention are checked in Coquereaux,
arXiv:1209.6621v2, Section 2.1 and Theorem 1; the vacuum S-matrix normalisation
in Section 2.2; level–rank duality in Section 2.3. Kitaev–Preskill,
arXiv:hep-th/0510092v2, equations (3)–(4), fixes total quantum dimension and the
SU(2) normalisation. No reference claims that every included category has a
separately documented physical application.
## Exact finite selection and rationale
The base consists of all integer N>=2, k>=1 with N+k<=16. This first surveys
all type-A ranks through 14 at small shifted level, including balanced rank
and level, instead of using the entire budget to deepen SU(2). The shifted
level bounds cyclotomic order and the full-alcove verification cost: an alcove
contains binomial(N+k-1,N-1) simple objects. Six extra SU(2) levels k=15,...,20
retain every category already covered by the companion T609. Their six dual
categories (N,k)=(15,2),...,(20,2) complete level–rank pairs. Thus the selection
has 117 categories and 234 entries. For k>=2 it is closed under (N,k)->(k,N);
level 1 has no SU(1) partner in this domain.
These bounds are noncanonical computational/reference cutoffs on meaningful
discrete families. They do not single out every category intrinsically and are
not a continuous-input reference_sample. The sparse extension is justified by
companion coverage and a checked duality, not by forming an arbitrary rectangle
of small integers. The base includes every category at each shifted level it
claims, including rational and dimension-one degenerations. A later extension
can increase the base bound or a justified low-rank slice without redefining N
or k. The tradeoff is limited deep-level coverage at larger rank in return for
complete full-alcove checks and wider type-A coverage.
Enumeration is increasing N, then k, then global before total. The generator's
categories() and enumerate(), document completeness note and stored entries
must agree. There is no hidden continuous free parameter. The root of unity is
derived, and the positive embedding is fixed, not sampled. Numerical evidence
records the actual full-alcove counts, written lengths and precision.
An independent critique must assess N and k as discrete_index and normalisation
as the intrinsic two-convention choice (or explain another justified category).
The builder does not write or accept the critique sidecar. That review must
bind all selection-relevant attachment bytes using argument_review.py.