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# Source review for T611
The archived local/live skill manifest is
`agents/runs/20261029T183732Z-build-skills/manifest.json`. Both SHA-256 values
are `28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The run checked the archive hashes before using the instructions.
## Definitions and normalization
Inoue, *Organizing volumes of right-angled hyperbolic polyhedra*,
https://arxiv.org/pdf/0809.2111, Section 2 fixes curvature -1; Section 4
constructs the pentagonal flowers, glues them to form a trivalent sphere,
and gives the volume in Theorem 4.1. Thus L_n is a polyhedron, with n base
sides, not a manifold formed from several copies. The arcsine used in the
code is pi/2 minus the paper's principal arccosine. Their arguments lie in
(0,1); there is no branch choice left open.
Repovš and Vesnin, *On a family of hyperbolic Brunnian links and their volumes*,
https://repovs.splet.arnes.si/files/2024/11/Repovs-Vesnin-corrected-version-25.2.2024.pdf,
Section 2, Figure 4 and equation (1), define A_n and quote Thurston's formula.
The reference is to one polyhedron; the paper separately describes multiples
of its volume in link complements. The quoted six-decimal numerical examples
are only coarse checks, not input data.
## Resolved printed sign discrepancy
Mednykh and Vesnin, *Löbell manifolds revised*,
https://ftp.gwdg.de/pub/EMIS/journals/SEMR/v4/p605-609.pdf,
page 606 explicitly states vol(manifold)=8 vol(polyhedron). However,
equation (2) on that page prints +Lambda(2 theta-pi/2), while Inoue's
Theorem 4.1 has a minus. The PDF was inspected visually as well as extracted;
this is not an OCR confusion. The definitions and normalizations agree.
Theorem 2 on page 608 independently derives
vol(polyhedron L_n) = n * integral_[pi/n,pi/3]
acosh(abs(cos(t)/cos(2*t))) dt.
This follows from the Schläfli formula applied to 2n truncated orthoschemes.
Numerical integration of this expression at 115 and 135 decimal digits agrees
with Inoue's formula for every n=5,...,30. At n=5 it gives
4.30620760073080865291983715984..., also matching the published dodecahedron
value (OEIS A321209, https://oeis.org/A321209). The printed plus sign instead
gives 2.22788471968557577993010875807.... Their difference is exactly the
change from +Lambda(2 theta-pi/2) to -Lambda(2 theta-pi/2), after dividing the
manifold formula by eight. The discrepancy is a sign error in that displayed
formula, not an alternative definition. The generator therefore uses Inoue's
formula; the independent checker uses the other paper's Theorem 2 integral.
## Controls and scope of the evidence
The integral checker first verifies integral_0^1 1 dx=1,
Lambda(pi/4)=Catalan/2, and the Schläfli integral from pi/4 to pi/3=Catalan/2.
It analytically removes the logarithmic singularities before quadrature.
Near pi/3 a tiny negative radicand from endpoint rounding is bounded by
100 times the *outer* working epsilon and replaced by zero. The independent
integrals agree across 115- and 135-digit runs to relative tolerance 1e-105.
They are cross-checks, not rigorous enclosures. Arb balls from the generator
provide the certified error claims.
The octahedron has volume 4 Catalan. Also A_4 and L_6 have equal volume;
for L_6 the sine of theta is 1/sqrt(3), and the numerical comparison uses the
independent integral. The face/vertex counts follow directly from the
combinatorics: A_n is four-valent with F=2n+2; L_n is trivalent with the same
F. Counting face-edge incidences gives E=4n or 6n and V=2n or 4n, satisfying
Euler's relation. All finite instances are checked in the supplementary run.
## Corpus and selection
The corpus and screening helpers were queried afresh. Numeric lookup for
3.6638623767088760602 matched manifold-volume tables T219/T225, which index
different objects. Lookups of the dodecahedral and A_4 volumes found no
same-family table. T85, T131, T175, T222, T223, T224 and the matching manifold
contexts were read. The family did not yet have a Löbell growth companion.
The selected 54 rows and both free parameters are justified separately in
argument-selection.md. No publication, offer or independent critique is
performed by this build session.