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5641 bytes, as of the version from 2026-10-29 13:21 (current). Recorded here, not run.
# Argument selection for T609
This table contains one quantity, the positive quantum dimension of an individual
integrable SU(N) representation at positive integral level k. Parameters are N,
k, weight, in that order. All three are genuine discrete indices. No angle,
continuous coupling, probability, normalization switch, or quantity switch is
freely selected. Angles are derived from the representation and its level.
## Sources and conventions
The family conventions were read from numberdb-data issue 244 and are preserved
in family-244.txt in the run artifacts. Coquereaux, arXiv:1209.6621v2, Sections
2.1–2.2, defines symmetric q-integers, integrability, the quantum Weyl product,
and q=exp(pi*i/(k+dual Coxeter number)). For type A_(N-1) the dual Coxeter number
is N and integrability means sum(a_i)<=k. Coquereaux, arXiv:1003.2589, Section
2.2, relates quantum dimensions to characters and the vacuum column of the
modular S matrix. Long roots have squared length 2. N is the matrix size, not
the Lie rank. Dynkin labels follow the A-type chain, with the first fundamental
weight defining the N-dimensional classical representation.
## Every free parameter and the finite extent
* N (discrete_index): all matrix sizes 2, 3, 4, 5. This covers the first four
simple type-A ranks, including higher-rank characters beyond the SU(2)
trigonometric specialization. It gives breadth before spending effort on a
longer SU(2) list. Higher types and higher ranks are not claimed covered.
* k (discrete_index): every level 1..20 for N=2, 1..8 for N=3, 1..5 for N=4,
and 1..4 for N=5. Level is the integral affine integrability bound, not an
integer label for a continuous mesh. These finite cutoffs are noncanonical
reference budgets. Smaller levels expose the first nontrivial fusion
categories; the rank-dependent cutoffs balance the rapidly growing number
binomial(k+N-1,N-1) of weights in a complete alcove. Four higher-rank levels
are retained before taking rank one deeper. The purpose is identification
of dimensions from small-rank, low-level character and fusion calculations,
not a claim of a documented application for every row.
* weight (discrete_index): every tuple (a_1,...,a_(N-1)) of nonnegative integers
with sum at most k. The alcove is a finite, mathematically defined set of all
simple objects, rather than a Cartesian product of arbitrary label ranges.
There is no independent cutoff on an individual Dynkin coordinate. The
vacuum, invertible objects, dual pairs, and all equal-dimension objects are
retained with their own identities. This makes dimension-sum and symmetry
checks possible and prevents the fundamental representations from standing
in for the entire theory. Dimension-one rows are exact and do not supply
the justification for the nontrivial family by themselves.
Enumeration is by increasing N, then increasing k, then increasing sum of
Dynkin labels, with lexicographic order within each sum. The symbolic machine
key is the comma-separated tuple without spaces, displayed as the tuple in
parentheses. This is the same convention as the family. Tuple length is not a
polynomial-variable count. The resulting counts are 230, 164, 125, 125 by N,
for 644 rows in 37 complete alcoves. This count is a consequence of the chosen
coverage, not a target used to choose it. Serialized sizes and the smallest
certified precision are recorded in numerical-evidence.json. The completed measurement is 112,494 bytes
of flat YAML, at most 101 characters per written value, and at least 421 bits
of generator ball accuracy at 429 working bits. All 530 nonrational entries
carry 100 significant digits; 114 rational entries are exact integers.
No continuous-input reference_sample is present. Calling these genuine discrete
indices does not make their finite cutoffs canonical; those are explicitly a
practical reference choice. The selection is independent of numeral spelling.
The rationale does not rely on the absence of a table or a library evaluator.
## Corpus context and boundaries
Live screening found no WZW or fusion-dimension table. Existing T60, T61, T99,
T122, T256, T397 and T607 were read. T61 supplies the cosine factor in fundamental
SU(2) dimensions; T397 includes coincident path spectral radii through k=6.
Those overlaps do not cover arbitrary highest weights or higher ranks. T122
contains character polynomials only for small partitions, whereas this table
stores numerical specializations over complete integrable alcoves. No repeats
property is used: these are mathematical identities, not copied computations.
Global dimensions, total dimensions, modular matrices and 6j symbols remain
outside this table. No T-number is invented for the unbuilt global companion.
The new tag representation theory is proposed for T609, T122, T256 and T607,
but only the existing algebraic, groups and physics tags are applied here.
## Policy evidence and independent review
Manifest: agents/runs/20261029T130110Z-build-skills/manifest.json in the source
snapshot. Both archived skill-local.md and skill-live.md were read and hashed;
both have SHA-256 28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.
The copies match. skill-evidence.json records this manifest and the byte counts.
The current archived skill, not the older historical digest in campaign-gate.md,
governs this selection. This document is the builder's justification, not an
independent acceptance. A separate critique must evaluate all parameters,
perform numerical checks, and fingerprint generate.py and this attachment.
The builder neither offers nor publishes the draft.