# Provenance of the argument's ingredients

Prepared 2026-10-10. This is a source-provenance note, separate from correctness reviews and the independent human assessment. Only primary papers, their publisher metadata/abstracts, and an author's publication listing support the conclusions below. AI-generated commentary, search-result summaries from aggregators, and absence of search hits do not establish provenance or novelty.

## What is explicit, and what remains unresolved

The facet/zonotope formula, affine covariance, and the simplex benchmark have explicit antecedents. The unrestricted simplex maximum was already the target of Brannen's 1996 conjecture, and the cited 2026 Feng–Hu–Liu–Xu paper already states counterexamples in every dimension at least nine. Henk's publisher abstract confirms an earlier exponential lower bound for centrally symmetric bodies. The exact product/tensorization mechanism in Henk has **not** been checked, because the full text was inaccessible; a power involving floor(n/3) is suggestive of repeated blocks but does not establish that mechanism.

The exact witness T_10 × T_10 and its application to the unrestricted simplex benchmark are the selected manuscript's claimed alternate construction. This investigation does not certify that this precise witness, the general product identity, or its block repetition is new. A literature search with inaccessible full texts cannot support that conclusion.

## Primary-source ledger

### Brannen, 1996 — the question itself

[Publisher abstract and bibliographic record](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S002557930001175X); [Cambridge publisher record](https://www.cambridge.org/core/journals/mathematika/article/abs/volumes-of-projection-bodies/0669377F656521BFBF76A22D5C62C3C1).

The primary abstract explicitly proposes the simplex as maximizer among all convex bodies, and a largest symmetric subset of a simplex in the symmetric class. Wiley records first publication 1 December 1996, Mathematika 43, 255–264. Cambridge's 26 February 2010 date is online publication of the older article, not the conjecture's origin. Full PDF not obtained: Wiley returned access failure and Cambridge supplied its abstract page. No page-specific claim about Brannen's product formulas is made.

### Ludwig, 2002 — exact covariance and facet representation

[Author-hosted primary PDF](https://dmg.tuwien.ac.at/ludwig/projection.pdf); [publisher DOI](https://doi.org/10.1016/S0001-8708(02)00021-X).

Checked PDF pages 1–2 and 5–6. Equation (1), p. 2, gives precisely Π(φK)=|det φ|φ^{-t}ΠK and translation invariance, attributing the result to Petty [25]. Page 5 gives brightness as half the sum of absolute scalar products with scaled facet normals, and Eq. (6) the centered-segment representation. Thus selected Lemmas 2 and 4 rederive established identities. Pages 5–6 discuss simplex projection-body facet normals but were not used to establish the exact normalized simplex constant. The paper's research theorem characterizes the projection operator by valuation and covariance properties; that theorem is not an input to the selected proof. The primary publisher identifies Advances in Mathematics 172 (2002), 158–168. Petty's original proceedings text was not retrieved, so historical priority before Ludwig is source-attributed rather than directly inspected.

### Saroglou, arXiv 2014 / journal 2015 — formulas and earlier product-shaped bodies

[Versioned primary PDF](https://arxiv.org/pdf/1409.4347v2); [version dates](https://arxiv.org/abs/1409.4347v2); [journal DOI](https://doi.org/10.1016/j.aam.2015.03.004).

Checked the introduction and §3, PDF pp. 1–3 and 7–8. Page 1 has the surface-area integral with factor 1/2; p. 2 notes the polytopal zonotope case and affine invariance. §3 Eqs. (7)–(9) records segment support functions, the determinant-sum zonotope volume, and projection-body volume from facet vectors. These are direct precedents for the selected proof's geometric toolkit. Page 3 describes Weil bodies as Cartesian products of symmetric one- or two-dimensional bodies in a second-projection-body fixed-point/class-reduction problem. This is **not** a statement of the selected tensorization identity or a product of ten-dimensional simplices. The arXiv record dates v1 to 15 September 2014 and v2 to 16 September 2014; journal publication is 2015. No first-inventor claim follows from these secondary attributions within a primary research paper.

### Henk, 2025 — exponential dimensional growth, mechanism unresolved

[Primary publisher abstract and dates](https://link.springer.com/article/10.1007/s10473-025-0107-9); [author publication listing](https://page.math.tu-berlin.de/~henk/publications.html).

The publisher abstract states normalized projection-body volume 2^n for n-dimensional zonotopes with n+1 generators, and an optimal symmetric-body upper constant at least 2^n(9/8)^{floor(n/3)}. This explicitly precedes the selected manuscript's interest in exponential dimensional growth. Publisher dates: received 16 August 2024, revised 16 September 2024, published 24 January 2025, pp. 96–103. Its PDF endpoint redirects to subscription preview; the author's relevant entry contains no PDF link. **Unresolved:** exact proof construction, product identity, block repetition and any numbered equation supporting those mechanisms. The author listing's pagination differs from publisher metadata; publisher pagination is used. No inference from the bound alone is promoted to explicit product-method provenance.

### Feng–Hu–Liu–Xu, 2026 — earlier general counterexamples using a different construction

[Primary PDF](https://archive.ymsc.tsinghua.edu.cn/pacm_download/743/12781--2026.8.26.pdf).

Checked p. 2 Eq. (1.3) and Theorem 1.3, and §4 pp. 13–21, especially Proposition 4.1/Eq. (4.3), Proposition 4.5, Eq. (4.19) and Proposition 4.6. Eq. (1.3) explicitly gives the same simplex constant (n+1)n^n/n!. Theorem 1.3 states counterexamples in every n≥9. Proposition 4.1 uses projected cube facet-area vectors, Minkowski existence, and a simplex-section estimate. Proposition 4.5 obtains a dimension-nine strict inequality via first variation. Eq. (4.19) and Proposition 4.6 raise dimension by zero-coordinate extension and a pyramid-volume identity. Those inspected mechanisms differ from the selected direct Cartesian product. This inspection establishes provenance of ideas and stated scope, not a correctness audit of the prior proofs. The selected bibliography identifies a 2026 preprint; the URL's apparent August date is not independent proof of its first public availability.

## Map to the selected manuscript

Line numbers refer to pinned expanded source SHA-256 `628bd94e51987a4ab5879c572a01837154b2ace52c1be9fe9d672178951541df`.

| Selected ingredient | Native source | Provenance classification |
|---|---|---|
| Facet brightness and segment representation | Lemma 2, lines 153–191 | Explicit earlier identities: Ludwig Eq. (6); Saroglou p. 1 and §3. |
| Affine covariance and invariant normalization | Lemma 4, lines 245–303 | Explicit earlier identity: Ludwig p. 2 Eq. (1), attributed there to Petty. |
| Simplex value c_d | Proposition 5, lines 311–356 | Benchmark already explicit in Feng p. 2 Eq. (1.3). The local cube-extrusion derivation is supplied by the selected manuscript; its historical novelty was not determined. |
| Cartesian product multiplicativity | Proposition 3, lines 193–238 | Locally derived here from classical facet data. Exact earlier occurrence not established by this bounded search. Weil-body products are related context, not equivalent statements. |
| T_10 × T_10 witness | Theorem 1, lines 90–100 and 388–409 | Selected alternate witness; earlier general disproof explicitly acknowledged at lines 80–88. Exact-witness novelty unresolved. |
| Repeated ten-dimensional blocks and residue factors | Corollary 6, lines 423–484 | Local mechanism is explicit. Henk supplies a prior exponential bound; his exact block/product mechanism remains uninspected. Feng's inspected dimension lift is pyramidal. |

## How to describe the amount of harder work

The source-grounded framing is: an elementary alternate construction assembled from classical projection-body identities, with an explicit product witness and a quantitative consequence. Most named prerequisites are classical and reproved locally. The exact witness's provenance remains open, rather than established by a failed search.

As an **inference about proof machinery**, the Feng construction uses substantially more apparatus in the inspected passages: Minkowski existence, section/projection estimates and first variation, followed by lifting. It is reasonable to call that construction more involved than the selected facet/product/extrusion route. It is not reasonable to imply that the selected manuscript is the first disproof, that its classical identities required newly solving a hard research problem, or that proof length measures the difficulty of discovering the idea. This note makes no correctness or panel verdict.

## Saved evidence and limits

`download_manifest.json` records URLs, resolved endpoints, hashes for obtained PDFs, and access failures. Ludwig and Saroglou PDFs/texts are stored here; Feng's already pinned downloaded PDF/text is in `../sources/literature/` with its manifest. Brannen/Henk original full proofs remain inaccessible. Their primary publisher abstracts support only the statements identified above. Search hits from nonprimary and AI-assisted commentary were excluded. No correspondence was sent, no subscription was purchased, no model review was requested, and no historical novelty claim was inferred from missing results.