bibkey: cfmp2026fourcyclecurvature
authors: trureturing contributors
year: 2026
title: Sources for CFMP four-cycle barriers and cover-uniform curvature estimates
doi: null
url: https://raw.githubusercontent.com/the-omega-institute/trureturing/8009ec61bf9a08f36a96f37f0f666bc4e0b45da3/docs/develop/theory/CFMP_GEOMETRIC_REALIZATION.md
claim: The cited analytic and co-volume inputs support the restricted four-cycle incidence theorem and its universal real curvature estimate without establishing general minimum-eight realization.
license: citation-only
triage: anchor
strata_touched: []

Source map for adjacent low-valence clusters and curvature observations

This is a source map for Sections 16–20 of the existing docs/develop/theory/CFMP_GEOMETRIC_REALIZATION.md, not a claim that a new paper with this title has been published.

Verified locator

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

Primary mathematical inputs

Xinrong Zhao, Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact 3-Manifolds with Boundary, arXiv:2601.15174v2, revised February 5, 2026. https://arxiv.org/html/2601.15174v2

Theorem 1.1 treats general minimum valence nine. Lemma 3.4 gives the adjacent coordinate monotonicity used on the cosh-length cube; Lemma 2.2 supplies the actual six-variable formula; Proposition 2.4 states the genuine length-domain criterion. Section 2.2 records the co-volume gradient. These are existing inputs, not results newly proved by the four-cycle incidence theorem. The new restricted class has valence-eight edges and therefore is not covered by Theorem 1.1 merely on the basis of its minimum valence.

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 https://arxiv.org/pdf/2009.03731

Proposition 2.2 and the co-volume discussion supply smooth local convexity and the positive-definite length Hessian. The existence-dependent flow convergence result is not used to assume existence of the required metric. The finite-dimensional spectral and strong-monotonicity consequences in Section 18 are classical deductions from the actual geometric Hessian.

Feng Luo and Tian Yang, Volume and rigidity of hyperbolic polyhedral 3-manifolds, arXiv:1404.5365. https://arxiv.org/abs/1404.5365 https://arxiv.org/pdf/1404.5365

The co-volume argument following equation (4.2) gives a positive-definite Hessian in true hyper-ideal length space. This is the geometric input used by the compact-box spectral argument below.

Exact restricted result and its boundary

Each tetrahedron has four degree-eight local edges forming a four-cycle and two opposite edges of degree at least twelve. Shared global lengths and local multiplicities are retained. The same box uses critical floor 5/4, critical cap 2, and high cap 8/5 in cosh-length. Its decisive local cosines are 293/400, 1577/2236, and 37/43, with positive rational squared margins. Both sides of the box are proved inward before applying the actual co-volume minimum argument. No zero-curvature solution is assumed.

A fully specified six-tetrahedron manifold has degrees (12,8,8,8) and one genus-three boundary. Its finite cyclic covers give unbounded examples. The conditions allow each low edge two adjacent low edges, which the earlier low-edge matching condition excluded. They do not allow arbitrary low-edge clusters, all-eight tetrahedra, or unrestricted degrees nine through eleven. No general minimum-eight or full CFMP resolution is asserted.

Relation to the repository’s observer and diagonal material

The actual D5/S0/Diagonal/Lawvere/QualitativeEscape.lean at dev c57bf72f087dae3bb5f5cc886c850649d7cb5593 concerns a fixed-point-free twist and a listing’s diagonal escaping its range. It does not prove a curvature estimate, geometric compactness, or an identity about opposite tetrahedral edges. Its hypothesis and carrier cannot be omitted when interpreting it.

FORMAL_OBSERVER_COMPLETION_REFLECTION.md explicitly distinguishes exact observability from quantitative conditioning and retains complete mixed responses. This informs the use of the full Hessian, rather than just its diagonal entries. The uniformly controlled local Hessians give mu D <= -DK <= M D on the specified common compact true-geometry region. No numerical eigenvalue sample is treated as a proof of the uniform mu.

Section 19 constructs genuine cone metrics on connected covers for which mean squared curvature tends to zero while a single moving edge has a fixed nonzero defect. These metrics are not a single Ricci-flow trajectory. This is a norm and changing-index-set phenomenon, not an instance of Lawvere’s listing theorem. It is consistent with the unnormalized weighted residual bound and the cover-uniform contraction of the explicitly specified flow dot(l)=D^(-1)K(l) inside the common region.

Verification and remaining problem

Exact checks cover the three local substitutions, the three rational sign margins, and the face-pairing/link/incidence conditions of the displayed manifold. Sixteen actual global-box corners and one curvature root were checked numerically; the root residual was about 4.44e-15. The displayed finite-difference Hessian eigenvalues are diagnostics only. The all-cover statements follow from the written proofs, not an enumeration of covers.

The cited sources do not state the precise four-cycle incidence theorem. No exhaustive priority claim follows from this source map. The co-volume, compactness and matrix tools are classical and credited above.

The general CFMP obligation remains to handle all permitted local low-valence patterns and their shared variables, or to exclude flat tetrahedra in the global maximizing angle structure. Small average error on larger objects cannot replace genuine nondegenerate compactness and an undiluted residual criterion on a fixed object.

Universal real envelopes and global incidence

The analytic comparison retains six independent real coordinates, the original numerator, both radicands, and real square roots. On the whole cube [1,2], differentiating the quotient reduces neighbour monotonicity to the nonnegative interval sign of Q=xow+xv+yo+yvw+z(1-w^2); the other coordinate comparisons follow from the tetrahedral symmetries. Denominator positivity is proved on the same cube.

For a finite incidence map T x Fin(6) -> E, one global real vector is read through six vertex-induced frames and occurrences are counted with multiplicity. Under the stated four-cycle colouring and low/high fibre cardinalities, the construction gives a nonempty box, strict cosine-domain inclusion, and one positive eta on every lower and upper curvature face. No angle bound, small-enough parameter, curvature sign, co-volume function or solution is an extra hypothesis. The fixed eta is independent of incidence size; the chosen high-edge floor tends to one as that size grows, so no uniform Hessian lower bound across all these boxes follows.

Exact derivative identities, tetrahedral symmetries, endpoint substitutions, and rational squared angle margins support the universal calculation. Ninety- six face corners and 1000 interior samples are finite diagnostics only and do not prove the universal inequality.

For the separate face-colour observation, each face gluing permutes three edges while preserving their global labels. Equal opposite colours identify every face count with the number of low opposite pairs; reindexing across one gluing and finite-path induction make this number constant on each connected component.

This rules out a proposed direct mixture of all-high, one-low-opposite-pair, low-four-cycle and all-low tetrahedra in a connected complex made exclusively of those balanced types. It does not rule out the earlier matching theorem’s allowed single noncritical low occurrence, which is unbalanced. A three-edge path has face signatures (2,1,1,2) and is a concrete local transition type; no geometric realization of arbitrary complexes using it is asserted.

Theory Sections 21-24 include the corresponding proofs and a prescribed-cone-curvature extension. The latter minimizes the existing co-volume plus a linear term for sup-norm target curvature less than eta. At nonzero curvature, the edge ends induce cone points on the truncated boundary as well; totally geodesic boundary is asserted only away from these endpoints. The perturbation corollary requires a pointwise uniform error bound and does not assert convergence.

Finite diagnostics validate six vertex frames, all 64 Boolean edge-colour patterns, 6144 paired-pattern/face-permutation cases, the original genuine six-tetrahedron packet, nine high-precision floor cases, 816 actual global boundary states, and 100 independent angle-relabel comparisons. The full unbounded proofs are separate from these finite checks and retain the remaining manifold and co-volume obligations stated above.