bibkey: cfmp2026criticaltransition
authors: trureturing contributors
year: 2026
title: Sources for critical six-valent transition stars and cover-stable realization
doi: null
url: https://raw.githubusercontent.com/the-omega-institute/trureturing/8009ec61bf9a08f36a96f37f0f666bc4e0b45da3/docs/develop/theory/CFMP_GEOMETRIC_REALIZATION.md
claim: The cited geometric inputs support the explicitly restricted critical-star construction and its continuous six-occurrence angle estimate without establishing unrestricted CFMP.
license: citation-only
triage: anchor
strata_touched: []

Source map for Sections 31-35 of CFMP_GEOMETRIC_REALIZATION

This note is a source map for the existing theory owner, not a claim of a published paper or an independently established priority result.

Verified locator

• URL: https://raw.githubusercontent.com/the-omega-institute/trureturing/8009ec61bf9a08f36a96f37f0f666bc4e0b45da3/docs/develop/theory/CFMP_GEOMETRIC_REALIZATION.md

Primary geometric inputs

Francesco Costantino, Roberto Frigerio, Bruno Martelli and Carlo Petronio, Triangulations of 3-manifolds, hyperbolic relative handlebodies, and Dehn filling, arXiv:math/0402339, Conjecture 0.8. https://arxiv.org/abs/math/0402339 The external question concerns geometric realization of the given triangulation. The present theory retains the strictly hyperideal case with all boundary components closed and of genus at least two.

Xinrong Zhao, Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact 3-Manifolds with Boundary, arXiv:2601.15174v2, February 5, 2026. https://arxiv.org/html/2601.15174v2 https://arxiv.org/abs/2601.15174 Lemma 2.2, Proposition 2.4 and Lemma 3.4 supply the actual six-variable cosine, genuine length domain and neighbour/opposite monotonicity. Theorem 1.1 is the general minimum-nine result.

Ke Feng, Huabin Ge and Bobo Hua, Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds, Geometry & Topology 26 (2022), 1349-1384, DOI 10.2140/gt.2022.26.1349. https://arxiv.org/abs/2009.03731

Feng Luo and Tian Yang, Volume and rigidity of hyperbolic polyhedral 3-manifolds, arXiv:1404.5365. https://arxiv.org/abs/1404.5365 These are the existing co-volume and true-geometry rigidity inputs already assigned in the theory. The new work does not reclassify their theorems as new results, or use an existence-dependent convergence result to assume existence of the required metric.

Exact new ordinary theorem and example

Low/high caps remain 2 and 5/4. Critical degree-six low edges have floor 4/3. Each of their occurrences has at least three high neighbours; at least four have four high neighbours and a critical opposite. No equality of different global edge lengths is assumed. Other low and high edges obey the existing seven/eight and eighteen-degree budgets. The critical bounds are 4/7 on the lower face, 43/99 at favourable upper occurrences, and 49/sqrt(6534) at the other upper occurrences. The strict mixed budget is 4 acos(43/99)+2 acos(49/sqrt(6534))>2pi, proved by the double-angle identity and a positive rational square gap 3896615/192119202.

A single face-pairing permutation in the existing eight-tetrahedron packet is replaced: (1,3)<->(1,0) uses 2310 instead of 1230. The resulting actual edge degrees are (6,17,25), with one genus-six vertex link. There are two M tetrahedra, five P4 tetrahedra and one C tetrahedron. The six-degree star has four favourable occurrences and two P4 endpoints adjoining seventeen-degree low edges. Complete edge classes and normal circles are written in the theory; independent finite checks also trace oriented-edge classes and all vertex-link fans. These finite checks do not prove the unrestricted conjecture.

The intrinsic opposite condition uses its degree class. It therefore persists through unbranched covers even if global edge identities split. The connected n-fold cyclic covers have 8n tetrahedra and Euler characteristic -5n. The fixed maximum degree 25 allows a common lower floor and hence use of the previously credited local Hessian compactness input. No explicit numerical Hessian eigenvalue bound is claimed.

Analytic scope

The arbitrary finite occurrence carrier has cardinality six; all five non-target real coordinates vary on continuous faces, and a finite favourable subset has at least four members. The conditions are actual coordinate caps and opposite floors, not cosine bounds or desired total angles. The derivative comparison, endpoint estimates, double-angle gap, and occurrence aggregation give one explicit positive two-sided margin.

This continuous six-occurrence estimate does not construct the manifold, establish the co-volume minimum, or prove the cover family. The full CFMP conjecture and unrestricted minimum-eight case remain outside the restricted hypotheses. The independent-opposite-one obstruction is retained without alteration.