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# Argument selection for T625
This run uses the coordinated canonical skill in
`agents/runs/20261030T115517Z-build-skills/manifest.json`. The archived local
and live copies were read and are byte-identical, both SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The quantity is the volume G_F(n,k) of the Grassmann manifold of k-dimensional
subspaces of F^n. Its metric is ds² = Re tr(dP²)/2 on the orthogonal projector.
It is one geometric quantity; the field changes the space, not the quantity.
Real subspaces are unoriented. There is no variable metric scale or probability
normalization. Parameter order is (field, n, k), as in family issue 249 and T624.
* **field**, Symbolic, discrete_index: exactly real and complex, displayed as R
and C. These are the two matrix geometries sourced by EAS and PWTH. Both are
included with the same projector convention. Quaternionic geometry would need
an additional metric audit and is outside this family's present scope.
* **n**, Z, discrete_index: the ambient vector-space dimension, intrinsically
n >= 2. The stored bound is 2 <= n <= 16. This is a noncanonical practical
reference cutoff, independent of notation, for total geometric masses in
subspace integration and code/packing normalizations. It gives broad coverage
of small matrix sizes while keeping the table readily readable and permitting
every frame-fibre comparison with T624. Matching T624 is an additional check,
not the mathematical reason the family is useful. No application is asserted
to need each of these dimensions or to privilege 16.
* **k**, Z, discrete_index: the subspace dimension, intrinsically 1 <= k < n.
Every representative 1 <= k <= floor(n/2) is stored for each selected n.
Orthogonal complementation P -> I-P is an isometry for the specified metric.
Thus the representatives cover every positive-dimensional shape up to this
symmetry, including projective spaces, intermediate ranks and balanced
subspaces before increasing the ambient bound. Point cases k=0,n are omitted
consistently with the positive-dimensional domain; they are not numerical
approximations to 1. Complementary dimensions remain part of the definition.
This is a constrained discrete geometric family, with 64 shapes per field and
128 entries. It is not a Cartesian mesh of unrelated continuous arguments.
There are no hidden continuous or exploratory reference-input groups. The
finite dimension bound is a reference choice; that does not change the three
parameters' discrete geometric roles. Famous projective cases alone would
miss the main subspace-volume family, so they have not displaced the middle
ranks. `generate.py` has no imported selection dependencies, and its default
enumeration agrees with the complete-note and Programs section.
Checked sources: Edelman, Arias and Smith, Sections 2.3.2 and 2.4.1, equations
(2.28)–(2.31), (2.39); Pitaval, Wei, Tirkkonen and Hollanti,
arXiv:1607.03132v4, Section II-D, Theorem 1 equation (36), Appendix B-C
(91)–(92); Życzkowski and Sommers, quant-ph/0302197v3, Appendix A,
(A23)–(A28). These establish the family, scale and relationships, not the
noncanonical cutoff 16.
A separate critique actor must independently assess this selection and its
numerics. This document is the builder's rationale, not independent acceptance.
Fingerprint the live generate.py; also retain this rationale and check files
as evidence. No critique sidecar is created by the build.