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Metadata 
bibkey: cfmp2026cycledegree9
authors: trureturing contributors
year: 2026
title: Pure three-cycle packets with degree-eight low edges and degree-nine high edges
doi: null
url: https://github.com/the-omega-institute/trureturing/pull/11934
claim: A paper-first strict hyper-ideal realization theorem for role-homogeneous pure three-cycle degree-(8,9) packets, together with finite-index Coxeter-sector non-vacuity.
license: citation-only
triage: anchor
strata_touched: []

The theorem and exact rational endpoint audit are in docs/develop/theory/CFMP_CYCLE_DEGREE9.md and verify_cycle9_box.py.

The shared cosh-length box is [73/50,2] on low edges and [53/40,347/200] on high edges. Zhao’s six-variable cosine monotonicity reduces all lower/upper faces to four endpoint tuples. Exact squared cosines are on the strict sides of 1/2 for degree eight and of cos(2*pi/9)^2 for degree nine; the latter is bracketed by the cubic 8q^3-6q+1=0.

Uemura, arXiv:2609.34108v1, Theorem 3.2 supplies the abstract invariant-box criterion; the packet-specific endpoint arithmetic below is separate. Zhao’s Lemma 3.4 supplies monotonicity. Luo–Yang, arXiv:1404.5365v2, Section 4.3, Corollary 4.12 and Section 5 formula (5.1), supply the co-volume differential; the displayed compact-box minimum argument yields an interior zero-curvature point. The analytic theorem assumes a valid finite connected orientable ideal triangulation with the packet roles; the Coxeter-sector construction below proves existence at some finite count, without an explicit minimal pairing or an N=24 claim. It does not resolve arbitrary minimum-eight or minimum-six CFMP, and no Lean formalization is asserted.

Primary references:

  • Costantino–Frigerio–Martelli–Petronio, Conjecture 0.8: https://arxiv.org/abs/math/0402339
  • Zhao, Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact 3-Manifolds with Boundary, arXiv:2601.15174: https://arxiv.org/html/2601.15174v2
  • Luo–Yang, Volume and rigidity of hyperbolic polyhedral 3-manifolds: https://arxiv.org/html/1404.5365v2#S4.SS3
  • Uemura, abstract invariant-box criterion, Theorem 3.2: https://arxiv.org/html/2609.34108v1

Non-vacuity provenance

Construction: https://github.com/the-omega-institute/trureturing/pull/12065

A finite non-vacuity witness is now supplied by a six-sector Coxeter quotient, not by an N=24 pairing. Start with the unique symmetric hyper-ideal tetrahedron with low angle pi/4 and high angle 2*pi/9. Its S3 sector is the [4,9,3] doubly-truncated Coxeter orthoscheme Q. The finite K=<c,d> sector group has order 6; KQ is exactly one target tetrahedron: A facets orbit as 1 base hexagon, B facets as 3 side hexagons, U truncations as 3 peripheral triangles, and V truncations as 1 center triangle. In the Coxeter development, <a,b> has 8 sectors around a generic low edge with trivial K stabilizer, while <b,c> has 18 sectors around a high edge and K stabilizer of order 2, giving target degrees 8 and 9. A torsion-free finite-index subgroup of the orientation-preserving parabolic subgroup gives a compact orientable quotient with geodesic boundary; collapsing boundary components yields a topological ideal triangulation. The local exact audit is verify_cycle9_coxeter_local.py; it checks the [4,9,3] relations, maximal finite parabolics, K intersections, and all angle sums over F_17/rational arithmetic.

Primary sources for this non-vacuity addition: Luo–Yang, arXiv:1404.5365v2, Proposition 4.1 and Proposition 4.4; Marshall, J. Austral. Math. Soc. 64 (1998), pp. 58–60 and 69–70; Felikson–Tumarkin, arXiv:math/0604248v3, introduction.