# Argument selection for T597

This is the external solid angle at one face of a regular simplex. It is
not a collection of unrelated Gaussian probabilities. The full sphere has
measure 1. The simplex is conv(e_0,...,e_d) in its d-dimensional affine hull;
its edge length is sqrt(2), which does not affect angles. A k-face has k+1
vertices. The normal cone is measured in its own (d-k)-dimensional span.

## Each free parameter

- **d, discrete_index.** The integer d >= 2 selects the dimension of a regular
  simplex. These are the regular simplices in Euclidean dimensions, a discrete
  geometric family. The finite extent 2 <= d <= 12 is a noncanonical resource
  and reference-scope choice, not a change to the infinite domain.
- **k, discrete_index.** For each d, every integer 0 <= k < d selects one face
  orbit under simplex symmetry. We include all these orbits. This covers
  vertices, edges, facets, and every intermediate face type rather than
  extending just the vertex sequence or selecting famous examples.

The derived dimension m=d-k and covariance parameter r=1/(k+1) are not free
inputs. The covariance is I_m+r J_m and its off-diagonal correlation is
r/(1+r)=1/(k+2). It is forced by the normal cone: its generating vectors have
Gram matrix I_m-J_m/(d+1), whose inverse is I_m+J_m/(k+1). The generator
constructs integer dimensions exactly and never substitutes rounded
correlations. No continuous sampling, rational reference grid, or arbitrary
coupling is present, so neither reference_sample nor mixed applies here.

## Breadth, extent, and controls

The default enumeration is every (d,k) with 2 <= d <= 12, 0 <= k < d, ordered
by increasing d and then k. There are 77 entries. All face types are included
before extending dimension further. This gives complete face profiles for
small and moderate dimensions with a short, readable value in every row.
The cutoff allows full validated quadrature, independent computations at
125 and 155 decimal digits, and geometric/algebraic checks in one run.
Actual timing, retained precision and entry lengths are recorded in
numerical-evidence.md. The cutoff is not a storage target or a claim that
higher-dimensional angles lack interest.

Dimensions zero and one and the whole-simplex face are omitted because they
add only universal boundary cases. The exact facet values 1/2 and exact
vertex values 1/(d+1) remain: they belong to each complete face profile and
supply normalization controls. There is no famous-only pruning or arbitrary
minimum count. The table has one named quantity; internal angles are a
separate table, not a value of a quantity selector.

## Checked sources

1. Z. Kabluchko and D. Zaporozhets, *Absorption probabilities for Gaussian
   polytopes and regular spherical simplices*, Advances in Applied Probability
   52 (2020), 588–616, DOI 10.1017/apr.2020.7. Published Proposition 1.2(a)
   specifies the normal angle at a (k-1)-face of a simplex with n vertices.
   Substitution n=d+1 and k_source=k+1 gives g_(d-k)(1/(k+1)). Equations
   (1.4), (1.7), Proposition 1.4 and Section 2.2 fix the covariance,
   integration and polar-normal conventions. The full published text was read.
2. The same authors, *Expected volumes of Gaussian polytopes, external angles,
   and multiple order statistics*, Trans. Amer. Math. Soc. 372 (2019),
   1709–1733, DOI 10.1090/tran/7708, arXiv:1706.08092. Equation (5), Section
   2.1 and equations (16), (20) corroborate the normalization and explain how
   external angles enter intrinsic volumes. The preprint was read. This
   supports the geometric family, not a claim of a published use of every
   individual dimension through 12.

## Policy and independent review

The controlling archived manifest is
`agents/runs/20261029T020934Z-build-skills/manifest.json`.
Both archived versions, `skill-local.md` and `skill-live.md`, have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
Their bytes were checked against that manifest. `skill-evidence.json` records
both reads. This document justifies the build; it does not constitute the
independent argument-selection review. The separate critique actor must
check enumeration against stored entries and completeness, independently
check numerical claims, and fingerprint generate.py and all reviewed source
dependencies. checks.py imports generate.py; generate.py has no local
computational dependencies. No critique sidecar has been written by this run.