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# Argument selection for T548
This is the builder's justification; the independent critique belongs to a
separate campaign stage. The governing archived skill manifest is
`agents/runs/20261027T121056Z-build-skills/manifest.json`. Both versions have
SHA-256 `e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The exact manifest and hashes are recorded in `skill-evidence.json`.
## Quantity and free parameter
The sole parameter `g` identifies an isomorphism class of finite simple,
connected, undirected, unweighted graphs with at least two vertices. Its type
is Symbolic and its category is `discrete_index`. It names what the energy is
of, not a choice among several quantities. The order n, edge count m,
adjacency matrix A, diagonal degree matrix D and mean degree 2m/n are all
derived from the graph. There are no independently selected weights,
couplings, root indices, temperatures or reference inputs.
The convention is Gutman and Zhou's Laplacian energy,
LE(G) = sum_i |mu_i - 2m/n|, with multiplicities, for L = D-A. It is not
normalized Laplacian energy or signless Laplacian energy. See their definition
in section 2, equation (9), and the regular-graph identity in section 3,
Lemma 1: https://doi.org/10.1016/j.laa.2005.09.008 . The primary paper was read
at https://aimath.org/WWN/matrixspectrum/gutman2006.pdf through the web reader;
a direct HTTP download redirected to a missing page, which is recorded in
`source-manifest.json` in the run artifacts.
## Exhaustive family, finite extent, and breadth
Include every connected isomorphism class on 2 through 6 vertices. This is an
exhaustive structural census, not a sample of real arguments. It includes
irregular degree patterns, trees and cyclic graphs, bipartite and
non-bipartite graphs, and regular graphs. In particular, irregular degrees
make the mean-centered Laplacian spectrum different from the adjacency
spectrum. Restricting the selection to complete graphs or other regular
families would largely repeat graph energy by a theorem, and would miss the
main distinction of this invariant. Members need no separate application.
No graph is removed for having a rational or repeated value.
Six is a practical, noncanonical cutoff, independent of the numeral system.
The counts are 1, 2, 6, 21 and 112, hence 142 total. Order seven would add
853 classes, mostly extending the same structural coverage at substantially
more rows. Breadth is obtained by completing all classes through six, rather
than extending only a few familiar named families to larger order. The
single-vertex graph is outside the family's common domain; it would have
energy zero. No larger named examples or extra combinations are appended.
The completeness note and default enumeration state exactly these bounds.
Counts and complete catalogue sets were checked against Brendan McKay's
connected graph catalogues, the Simple graphs section at
https://users.cecs.anu.edu.au/~bdm/data/graphs.html . The downloaded catalogue
records and their source hashes are in `census-reference.json`.
## Identifiers and order
The key is `G.canonical_label(algorithm='sage').graph6_string()`, computed
with SageMath 10.9. Graph6 is the encoding, not the canonicalization algorithm.
The encoding is specified at
https://users.cecs.anu.edu.au/~bdm/data/formats.txt . Decode explicitly with
`Graph(key, format='graph6')` and retain backslashes literally.
Entries are ordered by vertex count, then ASCII graph6 order. Tests compare
all 142 keys against the 2–6 vertex subsets of the live T237, T238, T397 and
T400 and the sibling T545, T546 and T547 snapshots. McKay's independent
catalogues provide a separate census control. Neither common keys nor a
pre-existing table's range alone is the rationale for this choice.
The values target 100 significant digits with certified enclosures, while
rational values are returned exactly. Readability, serialized size and
precision are measured in the attached numerical evidence. This precision
policy is not a selection criterion.