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3039 bytes, as of the version from 2026-10-29 02:59 (current). Recorded here, not run.
# Argument selection for T598
The two free parameters are discrete indices. No probability, correlation,
length or continuous argument is selected independently.
* **d (simplex dimension): discrete_index.** A regular simplex exists uniquely
up to similarity in every integer dimension d >= 2. Its internal angles are
invariants of that similarity class. The finite range 2 through 12 covers
planar and spatial cases and a modest higher-dimensional sequence.
* **k (face dimension): discrete_index.** For a fixed d, the complete range
0 <= k < d exhausts the geometrically distinct proper-face orbits. Symmetry
makes all faces of a given dimension equivalent; labelled faces are not
repeated. Including every face type gives complete profiles, covering
vertices, edges, intermediate faces and facets before adding depth to only
the vertex sequence.
The 77 pairs are ordered by increasing d and then increasing k. The cutoff 12
is noncanonical: it balances reference breadth with certification, cancellation
control and independent high-precision checking. It is not the domain of the
infinite family. The generator, table completeness note and program use this
same range. The checks measure runtime, numerical retention and entry length.
There is no exploratory reference grid, arbitrary angular mesh or continuous
input disguised as an index. Exact facets and the equilateral-triangle vertex
belong to the same geometric profiles and are retained as rational values.
Dimensions zero and one and the whole-simplex face add only universal boundary
cases and are omitted from the stored selection.
Let m=d-k. Orthogonal projection off the face directions gives tangent-cone
rays u_j=e_j-(e_0+...+e_k)/(k+1). Their Gram matrix is I_m+J_m/(k+1), whose
inverse is I_m-J_m/(d+1). Thus the Gaussian covariance parameter r=-1/(d+1)
and correlation -1/d are forced by geometry, not free inputs. The full sphere
has measure 1. Uniform edge scaling does not alter the angles. Internal and
external angles are different named quantities; the latter are held in T597.
Checked sources: Kabluchko and Zaporozhets, *Absorption probabilities for
Gaussian polytopes and regular spherical simplices*, Advances in Applied
Probability 52 (2020), 588–616, DOI 10.1017/apr.2020.7, arXiv:1704.04968v3.
Proposition 1.2(b) gives the internal angles with vertex-count indexing;
Section 2.5 proves the tangent-cone description. Equations (1.4), (1.5), (1.19)
define the covariance and integral. Source n=d+1 and face index k_source=k+1.
The archived arXiv PDF has this numbering; the generated HTML renumbers the
same result as Proposition 6. The sources establish the family and convention,
not an intrinsic reason for the finite cutoff 12.
The governing instructions are the matching archived local and live skills in
`agents/runs/20261029T024405Z-build-skills/manifest.json`. Both SHA-256 digests:
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
This build justification is not an independent critique or an acceptance.