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The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E1070.

Unit-distance independence: E1070

This account read Dúcz--Varga Section 4 and the extension proofs of Matolcsi--Ruzsa--Varga--Zsámboki Section 3, Theorems 1--2. The first theorem identifies the supremum over finite planar graphs of geometric fractional chromatic number with that of ordinary fractional chromatic number, using Følner averaging in the planar isometry group. Solvable group amenability is an external premise here. The second identifies the latter supremum with the reciprocal of the infimum independence ratio, using integer boxes in the rational span of a finite point set. This is specific geometric content, not arbitrary abstract graph blow-up.

Consequently the claimed rational certificate chi_gf(G_29)>4.0007 would imply the existence of a finite planar unit-distance graph with independence ratio below one quarter. It does not say that G_29 itself has that ratio. The rational dual vector, congruence inputs and independent-set list needed for that certificate have not been inspected. The implication is therefore conditional on a successful certificate check, not an adoption of the claimed inequality.

Once a finite m-vertex graph with independence number a and rho=a/m<1/4 is available, the remaining all-large-n extension is elementary. Write n=qm+r with 0<=r<m. Place q translated copies far enough apart that all inter-copy distances exceed one; place r more isolated points similarly. This is an exact Euclidean realization, not a graph-theoretic replacement of vertices by coincident twins. Its independence number is qa+r, hence

$$f(n)\leq \rho n+(1-\rho)r \leq \rho n+(1-\rho)(m-1).$$

In particular f(n)<n/4 whenever n>(1-rho)(m-1)/(1/4-rho). More generally placing optimal finite sets far apart proves f(n+m)<=f(n)+f(m), so f(n)/n has a limit equal to its infimum. Its value is still not determined by one counterexample.

The next mathematical obligation is to obtain and replay the exact certificate, or supply another finite graph with the strict ratio. The all-large-size extension alone does not supply that graph.

Source interfaces

These source locators identify the uses in this account. Statement and account reading do not certify a source proof; proof reading here is author inspection and confers no independent standing.

• [[../library/discrete_geometry/ducz_2026_unit_distance_graph_plane_independence_ratio/_index|ducz 2026 unit distance graph plane independence ratio]]: Sections 3-4, definition of chi_gf, Lemma 1 certificate description, Theorem 1 (statement reading).
• [[../library/discrete_geometry/matolcsi_2025_fractional_chromatic_number_plane_is_at/_index|matolcsi 2025 fractional chromatic number plane is at]]: Section 3, Theorems 1-2 and proofs (proof reading).