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3415 bytes, as of the version from 2026-10-29 19:37 (current). Recorded here, not run.
# Argument selection for T612
The sole parameter is `n` of type `Z`, the number of sides of either base of
one compact right-angled Löbell polyhedron L_n. Its domain is all integers
n >= 5. Its role is **discrete_index**: it changes an actual combinatorial
object, not a freely sampled real argument. The face-reflection generating
set is fixed by the object. There are no selected angles, shape parameters,
roots, alternate word metrics, or quantity selectors.
The enumeration includes every n = 5,...,30 in increasing order, 26 entries.
Inoue, Section 4, defines two pentagonal flowers glued along their boundary
to give a trivalent sphere and proves existence of their right-angled
hyperbolic realizations. The index is the same as L_n in the companion T611.
The cutoff 30 is a practical, noncanonical choice: it covers all initial
small and moderate combinatorial sizes, from the dodecahedron through 62
faces and 120 vertices, permitting comparison with the same compact objects
in T611. It is independent of numeral notation. It is not an intrinsic upper
bound or an assertion of documented use of every individual member.
Breadth was considered before depth: this proposal concerns only the full
face-reflection groups of compact Löbell polyhedra. Its one combinatorial
family is covered without gaps or famous-only pruning. Ideal antiprisms,
other compact right-angled polyhedra, deformed angles, and orientation-
preserving subgroups are different proposed objects or metrics, not missing
slices of this parameter. The antiprism case was separately assessed and
rejected in the family proposal because its rates are the elementary odd
integers; it is not silently mixed into this table. Volumes remain in T611.
The family has a direct published growth-series formula: Kellerhals and
Kolpakov, Section 4.2, equation (4.3). Substitution of the face count gives
one quadratic irrational for each L_n. Small and moderate polyhedra thus
supply a plausible identification family beyond the single named
polyhedron L_5. The inexpensive computation could extend further, but a
larger size target alone would not justify a longer listing. The current
extent makes a compact reference and aligns with T611 without copying a
continuous mesh from an unrelated function.
Sources checked:
- Taiyo Inoue, *Organizing volumes of right-angled hyperbolic polyhedra*,
Section 4, https://arxiv.org/abs/0809.2111.
- Ruth Kellerhals and Alexander Kolpakov, *The minimal growth rate of
cocompact Coxeter groups in hyperbolic 3-space*, Section 2.2 for the
minimum-length convention, equation (2.5) for the finite-subgroup formula,
and Section 4.2, equation (4.3), for compact right-angled polyhedra.
https://homeweb.unifr.ch/kellerha/pub/CJM-KellKolp-J0844-FINAL.pdf.
- Live T611: `family=lobell`, n = 5,...,30; its definition and combinatorial
conventions were read, along with the issue #245 family-coverage map.
The generator's default enumeration, Programs snippet and complete-note all
state this same range. The complete-note marks the infinite family incomplete.
This build-stage justification requires a separate independent critique; it
is not that critique's acceptance or fingerprint sidecar.
Policy evidence: agents/runs/20261029T192200Z-build-skills/manifest.json.
The archived local and live canonical skills both have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.