Title: Determinate Preservation: Hegelian Sublation and the Homotopy of Resolving Levels
Authors: Petri Mikael Autio
Suggested primary category: math.HO
Suggested secondary category: math.CT
Comments: 29 pages; source and reproducibility materials included.
Affiliation: not supplied

Abstract:
Lawvere's categorical rendering of Hegelian sublation makes preservation
precise through least resolving levels. We study two choices on which
such leastness depends: the relations represented and the coefficient
semantics governing generation and reconstruction.
For diagrams over a finite poset, a matching-map argument gives the least
full reconstruction subset for any specified diagram family, and hence
for arbitrary starting levels. For constant generation this becomes an
upper-interval criterion. We compute coefficient-relative resolution
profiles from sets through truncated homotopy types. A Boolean family
has explicit wedge-of-sphere obstructions and arbitrarily many distinct
least levels, although every earlier resolution comparison remains
unchanged. A compact family realizes $m$ distinct levels on $2m+2$
objects; for an initial one-object level, the first set-to-groupoid jump
requires exactly six. The jump
also survives nonconstant generation. Without preservation of generation,
a second family reverses the inclusion: richer coefficients require fewer
positions. A further comparison, grounded in
an interactive reader of the Science of Logic, orders the same
positions by ancestry or succession. Both profiles are invariant under
coefficient enrichment, yet their least subsets differ sharply.
Together these calculations show why old-test agreement does not secure
later leastness, and why coefficient stability does not establish an
interpretation. Reading these results alongside Hegel's accounts of
retention, mediation, and method, we defend determinate preservation:
a reconstruction must make its asserted mode of retention and its
interpretive warrant answerable together. The resulting comparison with
nLab and contemporary formalizations separates mathematical necessity
within a representation from justification of the relations and tests
that produce it.
