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# Argument selection for T528
The sole free parameter is the integer graph excess k = edges minus vertices,
with mathematical domain k >= -1 for connected graphs. It is a discrete_index:
k+1 is the number of independent cycles, so successive indices distinguish
successive sparse graph enumeration problems, not samples of a continuous
function. All graphs are finite, simple, undirected, and labelled on {1,...,n}.
The limit is n tending to infinity at fixed k. No embedding, weights, probability,
continuous coupling, or Brownian scaling parameter is varied.
Janson (2007), Section 4, equation (20), fixes the normalization
C(n,n+k) ~ rho_k n^(n+(3k-1)/2). Equations (21) and (34) give its relation
to auxiliary recurrence and generating-function coefficients. Those auxiliary
coefficients are not additional stored quantities. Section 5, equation (36),
identifies rho_k as the (k+1)st standard Brownian excursion area moment divided
by (k+1)!, providing a second interpretation of the same quantity.
The default enumeration is all k=0,...,11 in ascending order: connected graphs
with one through twelve independent cycles. This is a short initial reference
for the fixed-excess expansion, with no gaps. Eleven is an editorial depth
limit, not a singularity or canonical mathematical cutoff. The family proposal
uses this depth; its mathematical justification is the discrete cycle-count
hierarchy, not similarity to another table or capacity to compute more rows.
The exact tree value rho_{-1}=1 is given as a formula and used as a control;
its uninformative numeric row is omitted. The completeness note describes
exactly this choice and the parameter declaration keeps k>=-1 unrestricted
above. No quantiles or statistical critical values are involved.
Source: S. Janson, *Brownian excursion area, Wright's constants in graph
enumeration, and other Brownian areas*, Probability Surveys 4 (2007), 80–145,
https://arxiv.org/abs/0704.2289 (v3), equations (20), (21), (34)–(36).
Instructions: agents/runs/20261026T210447Z-build-skills/manifest.json;
the attached skill-manifest.json is an exact copy. Both archived skill versions
and the fetched live version have SHA-256
b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18.
This build justification is not an independent argument-selection acceptance.
The controller must obtain that in the separate critique stage.