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# Argument selection for T571
The table records one quantity: intrinsic Hilbert–Schmidt volume of trace-one,
positive semidefinite density matrices of fixed rank, with metric
`ds^2 = Tr(d rho^2)`. Motion of the support subspace is included. The field,
matrix size, and rank select different spaces whose volumes are taken, not
different quantities or different units.
The family issue #227 was read before research or computation. Its proposed
selection was reassessed under the coordinated canonical skill archived at
`agents/runs/20261028T042605Z-build-skills/manifest.json`. Both archived copies
were read and match SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
* `field` is a discrete index with the two identities `real` and `complex`.
These specify real symmetric and complex Hermitian density matrices,
respectively; beta is 1 and 2. Both geometries are treated in the primary
source, sections II–V and VII. Including both provides a comparison of the
two standard matrix spaces. Quaternionic spaces are outside the declared
domain; this is not an unspecified sampling of arbitrary scalar fields.
* `N` is the integer matrix size, with domain N >= 2. We include every size
from 2 through 16. Matrix dimension is an intrinsic discrete size of the
quantum-state problem. The finite cutoff is a noncanonical resource and
reference-coverage choice, independent of numeral notation. It includes
the elementary disk/ball controls, small multilevel systems, and moderate
dimensions, not only powers of two or individually named systems. Increasing
N first would deepen the same family; covering both fields and every rank
supplies the more useful breadth within this budget.
* `rank` is the actual positive integer rank r, with domain 1 <= r <= N.
Every admissible rank is included for each selected field and size. The
pure-state stratum, generic boundary, and interior are endpoints and slices
of one geometric volume family. Intermediate ranks give all boundary
depths, so none are removed for lacking an individual famous application.
In the source the rank-deficiency parameter is n; our nullity d=N-r is
that n. It must not be interpreted as our rank.
These are three discrete indices of a documented matrix-space family. There
is no selected spectrum, real coupling, angular mesh, quantile, or exploratory
continuous reference group. The triangular dependence 1 <= r <= N describes
the geometry's domain, not a product of unrelated simple inputs. The default
order is real then complex; within each field it is increasing N, then
increasing r, beginning with the elementary pure-state cases.
The total is 2*sum(N for N in range(2,17)) = 270 entries. `generate.py`
enumerates exactly this selection and exposes max_N for deliberate extensions.
The table's completeness note records both the finite bound and its rationale;
its parameter constraints retain the infinite domain. The dry run measured
52.4 KB of values and keys, with longest working-precision representation
140 characters, so the selection leaves ample room for future extension.
It was not chosen to reach a size limit.
Sources actually checked:
* K. Życzkowski and H.-J. Sommers, *Hilbert–Schmidt volume of the set of mixed
quantum states*, J. Phys. A 36 (2003), 10115–10130,
https://arxiv.org/pdf/quant-ph/0302197v3. Metric (3.1); eigenvalue integral
(4.3); complex rank-stratum formula (5.3); real counterpart (7.8); unit-trace
flag factors (3.10), (A27)–(A30). These establish the family and conventions,
not a canonical upper bound of 16.
* N. G. de Bruijn, *On some multiple integrals involving determinants*,
J. Indian Math. Soc. (N.S.) 19 (1955), 133–151,
https://pure.tue.nl/ws/files/1920642/597510.pdf, especially (1.1) and
(4.3)–(4.5), read from the scanned original. These support the independent
exact-moment checks; they do not determine the argument cutoff.
This is the builder's justification. The required independent critique and
its argument-review sidecar belong to a separate controller stage.