A four-point failure of (J+, boxdot)-cubical excision

Summary

Excision is a basic homology principle: when a space is covered in the right way, removing the same part from a subspace and the whole space should not change their relative homology. This anonymous, unrefereed candidate gives a four-point failure for one precisely named cubical theory of closure spaces.

The source relative group is the integers. In the target, the identity square fills its generator, so the target group is zero. The inclusion therefore induces the map

which is not injective. The claim is deliberately narrow: integral normalized cubical homology built from the directed interval $J_+$ and the inductive product $\boxdot$.

Summary for specialists

Let $X=J_+\boxdot J_+$ and set

The singleton closures give $i(A)=\{11\}$ and $i(B)=\{00,10,01\}$, so $A,B$ form an interior cover. In the ordered relative one-basis $(e_x,e_y)$ and zero-basis $([00])$,

Thus $H_1(B,L)\cong\mathbb Z$, generated by $\eta=e_x-e_y$. A vertex-potential cocycle takes value $2$ on $\eta$ and annihilates every source two-boundary. In $(X,A)$ the identity square has relative boundary $\eta$, and the target $d_2$ contains the primitive column $(1,-1)^T$. Hence

Technical account: why the four points suffice

The inductive square has smallest neighbourhoods

The intersection $L=\{10,01\}$ is discrete. The only nondegenerate relative one-cubes are the two coordinate edges from $00$. Dimensions zero through two are enough because $H_1=\ker d_1/\operatorname{im}d_2$; no higher chain group enters the calculation.

The machine-readable certificate names all four source and eight target degree-two cubes and binds each one to its integer boundary column. This makes the enumeration order inspectable rather than implicit in code.

Evidence, assurance and limitations

The immutable package contains the canonical seven-page PDF and LaTeX, accessible Markdown, complete JSON result, Python and JavaScript encodings, 15 tests, six semantic mutations, source and novelty audits, internal review records, licences, a 53-entry manifest, and two hash-bound gate receipts over a 56-file payload.

Normal and optimized Python are byte-identical at the stable-result layer. JavaScript matches the complete stable schema. The mutations deliberately damage the cover, cycle, product convention, interval direction, degeneracy normalization and face signs; all are rejected. A fresh replay compares the complete source path-and-hash inventory before and after, so ignored Python bytecode cannot hide a mutation.

These checks establish public availability, integrity and producer replay. They do not establish unaffiliated rerun or reimplementation, proof-assistant formalization, external specialist review, editorial peer review, historical priority or research impact.

Relationship to earlier work

Bubenik and Milićević defined six cubical theories using three intervals and two products. They proved categorical-product excision and left the inductive-product small-chain cases open. Their Example 5.9 uses this same four-point square with a four-set cover. Here, $A$ is their member $D$, while $B$ is the union of the three lower members; this coarsening exposes the actual two-set relative map.

Milićević's 2025 excision theorem uses ordinary topological simplices, so it concerns a different singular theory. Jamil, Staecker and Ali study symmetric digital adjacency and graph maps, not the asymmetric $J_+$ closure. Related work remains listed as in preparation, so the release makes no first or priority claim.

Who should care, and why

Why the problem matters

Excision is one of the Eilenberg–Steenrod axioms. A minimal counterexample does more than show that a standard proof fails: it identifies the precise chain map where the axiom breaks, and it sharply separates the affected theory from nearby cubical, digital and simplicial constructions.

How to inspect or reproduce the checks

Use tag v0.2.0-candidate or the version DOI, not moving main. For the lightweight mathematics:

python3 verify_counterexample.py
python3 -O verify_counterexample.py
node verify_counterexample.mjs
PYTHONDONTWRITEBYTECODE=1 python3 -m unittest discover -s tests -v

For the complete manifest, inventory, concordance and PDF replay:

bash run_all.sh

Expected markers include PASS_PRODUCER_CONCORDANCE, PASS_PACKAGE, and PASS_ALL. Successful execution confirms the encoded finite predicates. It does not establish novelty, independence, or peer review.

The most valuable next projects

1. Determine excision for the remaining inductive theories based on $J_1$ and the ordinary interval $I$.
2. Seek an analogous failure—or a positive theorem—under symmetric graph closure restrictions.
3. Reconstruct the complete chain calculation in a materially separate stack.
4. Formalize the four-point closure, normalized cubical chains and relative map in a proof assistant.
5. Obtain an unaffiliated specialist source and priority audit.

What is in the evidence package

The all-files ZIP includes the DOI-bearing PDF and source, aligned Markdown, explicit claims and matrices, both exact encodings, tests and mutations, source/citation/novelty records, role-separated internal review, the supplied review response, licences, runtime declaration, receipts and complete manifest. The version DOI is the citation target; any correction should be a versioned successor rather than a silent edit.

@misc{fourpointinductivecubicalexcision2026,
  title        = {A four-point failure of (J+, boxdot)-cubical excision},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22171896},
  url          = {https://doi.org/10.5281/zenodo.22171896},
  version      = {0.2.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/four-point-inductive-cubical-excision/}
}