A pyramid with a Ramsey base is Ramsey

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lemma_2_3: Proves the finite-witness compactness lemma relative to the exact Rado selection principle.

theorem_1_2: Gives the complete color-induction proof that adding a point outside a Ramsey base's affine hull preserves the Ramsey property.

theorem_2_1: States the classical Cartesian product theorem used in Moore's pyramid proof.

theorem_2_2: States the Frankl–Rödl theorem that finite affinely independent sets are Ramsey.

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Kenneth Moore, A pyramid with a Ramsey base is Ramsey, arXiv:2608.09649v1, submitted 10 August 2026 at 14:26:02 UTC (versioned record, canonical five-page PDF). The retained PDF is this exact first version. Acquisition identity, source reading and external-input records are in source_snapshot.json. The arXiv record (https://arxiv.org/abs/2608.09649, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

A finite Euclidean configuration $T$ is Ramsey if, for every positive integer $r$, all $r$-colorings of some $\mathbb R^N$ contain a monochromatic congruent copy of $T$. Moore proves that if a finite set $B$ is Ramsey and $z$ lies outside $\operatorname{aff}(B)$, then $B\cup\{z\}$ is Ramsey. The projection of $z$ onto the affine hull may lie anywhere, and its nonzero perpendicular height may be arbitrarily small.

The complete Theorem 1.2 proof inducts on $r$. A finite $(r-1)$-color witness is realized as a set of apices, each completing a fiber in a Ramsey product. A monochromatic copy of that product either extends to a pyramid in the same color or leaves all apices in the other $r-1$ colors. The auxiliary simplex is affinely independent because of added orthogonal coordinates.

The source's essential chain is recorded with these scopes:

• Lemma 2.3 has a complete finite-configuration compactness proof relative to the exact external Rado selection principle. Moore states the lemma and cites the standard hypergraph compactness argument.
• Theorem 2.1 is the exact classical product theorem from Euclidean Ramsey theorems I, Theorem 20, printed p. 357. Its proof remains external.
• Theorem 2.2 is the exact Frankl–Rödl simplex input, obtained from their 1990 Theorem 5.1. Its deep proof remains external.
• Theorem 1.2 contains every deduction of Moore's proof, including the isometry extension and both color cases.

All five source pages, including the figure and references, were read visually. The finite-witness proof and expanded linear-algebra details are identified compilation additions. The same-paper proof chain is fully reconstructed relative to the named external inputs. The external simplex, product and selection theorems themselves are not reproved here.

The paper answers Conjecture 8 of Ivan, Leader and Walters, Generalised prisms and Euclidean Ramsey theory, arXiv:2606.13472v1, p. 9. This is a construction result relevant to #174; it does not settle the general classification of finite Ramsey sets. The competing spherical and subtransitive classifications are recorded as conjectures in Moore's introduction, not proved by this paper.

Mirabi's preprint was submitted on 12 August 2026 and gives a materially different proof of the same main conclusion. Mirabi reports private circulation of the argument in June 2026 and independence from Moore; that chronology is an author's account, not an independently established priority determination.

As of the primary-source search, this record identifies an arXiv preprint, with no journal acceptance or formal proof certification located in that bounded search. Pálvölgyi's arXiv:2608.10865v2, p. 24 cites Moore and Mirabi for this closure result. That citation is evidence of uptake, not an independent proof check. Moore's p. 5 acknowledges using ChatGPT 5.6 during brainstorming; that acknowledgment is provenance, not validation of the mathematics. No Lean build or formal-code audit is claimed here.

Bears on. #174.