# Research claims and their status

## Mathematics offered in the manuscript

| Result | Exact scope | Relationship to prior work |
|---|---|---|
| Matching-support formula | Finite poset, infinity-category with finite limits, any fixed family of diagrams; arbitrary full starting level when left Kan extensions exist | Elementary consequence of standard Reedy matching and Kan composition. No new foundational cofinality theorem is claimed. |
| Upper-interval formula | Constant generation from an initial one-object level, all full subposets, sets through k-types and spaces | Ordinary endpoint directly meets Dey–Lesnick and Brüstle–Desrochers–Leblanc; higher proof uses standard matching machinery. |
| Boolean profile | Exact least subsets and wedge ranks, with every old comparison unchanged | Proposed explicit calculation/application; sphere skeleton topology is standard. |
| Compact growing profile | m distinct levels on 2m+2 objects; first Set-to-groupoid increase requires six objects under the initial-singleton assumptions | Uses classical McCord sphere models; six-object lower bound is proved directly. |
| Nonconstant example | Nine-object poset, two-object generating antichain, genuinely nonconstant generated diagrams | Shows the growing effect survives beyond constant output when generation still commutes with the coefficient inclusion. |
| Shrinking profile | C={x}⋆J⋆{t}, D=J; full C exactly when τ≤k BJ is contractible, otherwise C minus x | Distinctive explicit counterexample to unconditional monotonicity. Generation changes. The proof uses the canonical cocone, a space of contractions, and strictness of the empty space. Priority remains unestablished. |
| Infinite boundary | C=N, D={0}; resolvers exactly unbounded subsets containing 0, no minimal one | Elementary exact boundary: the finite support formula cannot simply be applied to arbitrary infinite chains. |
| Reader comparison | Same finite positions encoded by branching ancestry or compatible succession | Both profiles invariant; least subsets radically different. These are stipulated editorial encodings, not a derivation of Hegelian dependence. |

## Philosophical contribution

The paper argues that a formal preservation verdict must answer for both its represented relations and its coefficient semantics. Agreement on all old comparisons does not secure a new leastness claim. Conversely, coefficient invariance does not validate an interpretation of mediation. When generation also changes, even the direction of least-resolver change needs proof.

The paper does not claim to discover Hegel's distinction between a valid demonstration and adequate mediation. It credits his text and contemporary scholarship, particularly Hentrup. Its contribution is a set of exact calculations and a self-applied interpretive audit. It does not derive Hegel's sequence of categories.

## Remaining priority question

A focused primary-source search found direct ordinary antecedents, standard higher machinery, and the finite-sphere constructions. It did not establish first publication of the exact coefficient-profile and reversal formulations. A specialist could recognize an equivalent formulation in another literature. The article is therefore a proposed research contribution with proved statements, not a certified priority claim.