# Argument selection for T547

This build uses the coordinated skill archived in
`agents/runs/20261027T114054Z-build-skills/manifest.json`.
Both `skill-local.md` and `skill-live.md` have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The manifest and verified hashes are also attached as `skill-evidence.json`.
The family conventions were read from numberdb-data issue 220 before building.
This is the builder's justification; the controller must obtain a separate
independent argument-selection critique before offering the draft.

## Quantity and full parameter domain

There is one free parameter, `g`, a graph isomorphism class. Its category is
`discrete_index`. The graph is finite, simple, connected, undirected and
unweighted, with at least two vertices. The value is algebraic connectivity,
the second-smallest eigenvalue of the combinatorial Laplacian D-A.
Fiedler (1973), Section 1, page 298, gives precisely this definition. In the
family's descending eigenvalue notation it is mu_(n-1), with mu_n=0.
There is no normalized-Laplacian, edge-weight, coupling, root-order or quantity
parameter. Vertex and edge counts and degrees are derived from the graph.

## Selection, coverage and practical boundary

Every isomorphism class at orders 2, 3, 4, 5 and 6 is included. McKay's
connected simple graph catalogues give respectively 1, 2, 6, 21 and 112
classes. The 142 entries are an exhaustive census of small connected
structures. It includes trees and cyclic graphs, bipartite and non-bipartite
graphs, and regular and irregular graphs. Algebraic connectivity detects weak
connections between parts of a graph; restricting the census to complete
graphs and stars would mostly return small integers and miss these varied
small bottlenecks. No application is claimed for each individual graph.

Six vertices is a practical, noncanonical cutoff independent of numerical
notation. It already covers the structural contrasts just listed; order seven
would add 853 classes. The choice favours structural breadth at small order
over extending only highly symmetric families. No extra large named graphs
are appended. The one-vertex graph is outside the quantity's second-eigenvalue
domain. No graph is removed for having a rational value or sharing a value
with another graph. No exploratory continuous reference grid occurs, so
`reference_sample` and mixed-role assessments do not apply.

The published neighbouring graph tables hold some larger censuses; their
order-2-through-6 subsets, rather than all their keys, are the consistency
control. Counts alone do not prove the objects agree: the checks compare the
canonicalized sets from McKay's actual catalogue files with the generator and
with T237, T238, T397, T400, T545 and T546.

## Identity and display

The identity is `G.canonical_label(algorithm='sage').graph6_string()`, computed
with SageMath 10.9 and checked against every neighbouring identifier selected.
Graph6 is an encoding, not an intrinsic canonicalization. Explicit
`Graph(key, format='graph6')` decoding keeps literal backslashes intact.
Entries are ordered by vertex count and then ASCII order of graph6 strings.
The graph is the only parameter; the number column is labelled a(G).
Eighteen rows name complete graphs, paths, cycles or complete bipartite graphs,
after deriving their identifiers from those graph constructions.

## Measurement and independent controls

The 142 entries comprise 59 exact integers and 83 irrational algebraic values.
Rationality is determined from the exact minimal polynomial, never from digits.
The irrational values use certified AA root isolation followed by MPFI
interval conversion at 413 bits. The largest measured interval width is less
than 1.891e-124. All entire enclosures fit inside their written 100-significant-
digit intervals. The longest written value is 102 characters; the JSON entries
block including comments is 12,923 bytes. The standard dry-run measurement of
working-precision objects reports 127 characters and 16.5 KB.

An independent graph6 decoder and permutation determinant reproduce every
stored T238 Laplacian polynomial. Exact rational Sturm counts certify that
each written interval contains the least positive root, with no smaller
positive root. Independent mpmath symmetric diagonalization at 150 decimal
digits agrees on every graph. Controls are the graph6 specification's DQc
example, K2 with spectrum {0,2}, an edgeless graph, a repeated-root Sturm
example, and deliberately wrong intervals that must be rejected.

All five complete graphs, five paths, four cycles and nine complete-bipartite
part-size pairs in the census satisfy their closed forms. Fiedler's Section
4.4, page 304, lists the complete, path, cycle and star cases; Section 3.11,
page 303, gives the bipartite relation. The small exception K_(1,1)=K2 is
explicitly handled: its connectivity is 2, while min(a,b) applies to K_(a,b)
when a+b is at least 3. This follows directly from the two-vertex Laplacian.

## Checked sources

* Fiedler, Algebraic connectivity of graphs (1973), Section 1 and Sections
  3.11 and 4.4, DOI 10.21136/CMJ.1973.101168. Scanned pages 298, 303 and 304
  were read. https://snap.stanford.edu/class/cs224w-readings/fiedler73connectivity.pdf
* McKay, connected simple graph catalogues, rows for orders 2 through 7;
  the five graph2c.g6 through graph6c.g6 data files were downloaded and hashed.
  https://users.cecs.anu.edu.au/~bdm/data/graphs.html
* McKay, graph6 specification, graph6 definition and worked DQc example.
  https://users.cecs.anu.edu.au/~bdm/data/formats.txt
* Live documents T237, T238, T397 and T400 for shared canonical graph names;
  T545 and T546 for the same finite family. Full snapshots and lookup evidence
  are preserved with the run artifacts. Each selected graph identity agrees.