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# Argument selection for T624
This table holds one quantity, the volume of an ordered orthonormal frame space
in its ambient Frobenius metric. The scalar field specifies the frame space;
it does not switch the quantity being measured. Parameter order is (field,n,k).
- **field (discrete_index):** `real` and `complex` select the two matrix
geometries established in Edelman–Arias–Smith (Sections 2.2–2.4) and
Pitaval–Wei–Tirkkonen–Hollanti (Section II-C and Theorem 1). Their displays
are R and C. Quaternionic frames need a separately checked metric convention
and are outside this table's declared domain.
- **n (discrete_index):** the positive integer ambient dimension is part of
the frame-space definition, not a continuous variable sampled on integers.
We retain every n from 1 through 16. Sixteen is a noncanonical practical
reference bound, independent of numeral notation. It gives compact coverage
of small matrix normalization constants in frame integration and packing,
while keeping the reference and its verification easy to inspect. The cited
sources establish these uses and conventions; they do not demand every n
through this particular cutoff.
- **k (discrete_index):** the number of ordered orthonormal columns satisfies
1 <= k <= n. Every admissible k is retained at every included n. This covers
vectors, all intermediate rectangular frames and full orthogonal or unitary
frames before deepening any one slice. Complementing a subspace does not
identify its ordered frames, so no k <-> n-k reduction is applied here.
There are 2*(1+...+16)=272 entries. The first case in each field has n=k=1;
within each field n and then k increase. The zero-column singleton is outside
the positive-column domain. The full real group O(n) includes both components.
There is no adjustable metric scale, probability, or additional hidden input.
These are constrained discrete geometric indices, with a practical cutoff,
not an exploratory mesh of continuous arguments. The finite selection is a
reference choice; neither the cutoff nor every member is claimed to be famous.
The default generator enumeration and the completeness note state precisely
this same selection. `generate.py` contains all code controlling the selection
and has no external generator dependencies. `check.py` and `check_values.py`
only test it. This build is not the independent argument-selection review:
the separate critique must inspect these choices, compute numerical checks,
and create its own fingerprint-bound sidecar using its own reviewer_run.
Checked primary sources:
- EAS: https://www.cis.upenn.edu/~cis5150/Stiefel-Grassmann-manifolds-Edelman.pdf,
Sections 2.2–2.4, equations (2.22) and (2.39).
- PWTH: https://arxiv.org/html/1607.03132v4, Section II-C, equations (7)–(10),
Theorem 1, equations (34)–(35), and Appendix B.
- ZS: https://arxiv.org/pdf/quant-ph/0302197v3, Appendix A, (A23)–(A27).
The governing skill archive is
`agents/runs/20261030T111206Z-build-skills/manifest.json`.
Both archived versions have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The attached `skill-evidence.json` preserves the manifest and source paths.