--- title: "A four-point failure of (J+, boxdot)-cubical excision" date: 2026-08-30 version: "0.2.0-candidate" doi: 10.5281/zenodo.22171896 pdf: https://github.com/ipitchford/four-point-inductive-cubical-excision/releases/download/v0.2.0-candidate/four-point-inductive-cubical-excision-v0.2.0-candidate.pdf repository: https://github.com/ipitchford/four-point-inductive-cubical-excision archive: https://zenodo.org/records/22171896 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A four-point failure of (J+, boxdot)-cubical excision ## Summary Excision is a basic homology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 $$ \mathbb Z \longrightarrow 0, $$ 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 $$ A=\{10,01,11\},\qquad B=\{00,10,01\},\qquad L=A\cap B. $$ 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 $([00])$, $$ d_1=[-1\ -1],\qquad d_2^{(B,L)}=0_{2\times4}. $$ 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 $$ H_1^{(J_+,\boxdot)}(B,L)\cong\mathbb Z,\qquad H_1^{(J_+,\boxdot)}(X,A)=0. $$ ## Technical account: why the four points suffice The inductive square has smallest neighbourhoods | Point | Smallest neighbourhood | |---|---| | $00$ | $\{00\}$ | | $10$ | $\{00,10\}$ | | $01$ | $\{00,01\}$ | | $11$ | $\{10,01,11\}$ | 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 | Audience | Potential use | Required caution | |---|---|---| | Closure-space topologists | A minimal obstruction for one open inductive cubical branch | Do not generalize to all six theories | | Computational reviewers | A tiny exact matrix fixture with hostile convention mutations | Reimplement independently rather than importing producer code | | Formalizers | Four points, labeled bases and a short integer proof | Continuity and normalization conventions still need formalization | | Research agents | A worked example of turning a failed proof method into an actual relative-map test | Internal replay is not external validation | | Interested readers | A concrete example of a nonzero cycle becoming a square boundary | Candidate publication is not field consensus | ## 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: ```sh 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: ```sh 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. ## Open directions for follow-up research - Determine excision for the inductive theories based on J1 and the ordinary interval I. - Determine whether an analogous failure exists under symmetric graph-closure restrictions. - Independently reconstruct the four-point chain complexes in a materially separate stack. - Formalize the finite closure, continuity, normalization and relative-homology calculation in a proof assistant. - Obtain an unaffiliated specialist source and priority assessment, including the related work listed in preparation. ## Research process, metrics and reusable methods This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:four-point-inductive-cubical-excision - Attempt and metric receipts: ep-attempt:four-point-inductive-cubical-excision-assurance-publication: published / positive; scope assurance-through-publication; target Action every required review item, redesign replay so a clean cross-environment run is successful and non-mutating, obtain frozen internal editorial closure, publish synchronized immutable GitHub and Zenodo assets, and complete both guarded Evidence Press release cycles without exceeding the exact (J_+, inductive-product, integral H_1) claim boundary.; active forecast 140 minutes (90-220); Fermi components non-mutating replay redesign, schema consistency and clean-environment validation: 1 x 30/45/70 minutes low/central/high (One compact Python and JavaScript package, but the manifest, stable and volatile receipts, PDF build and clean-tree test must be separated coherently.); manuscript, source relationship, literature and frozen editorial closure: 1 x 20/30/50 minutes low/central/high (One six-page note, one supplied Major Revision, four primary related-work records, one five-role round and at most one confirmation.); public research repository, CI, immutable release and Zenodo identity: 1 x 15/25/40 minutes low/central/high (One dependency-light anonymous candidate with standard licences, one tagged CI run and three principal archive assets.); Evidence Press page, media, two composite seals, deployments and readback: 1 x 25/40/60 minutes low/central/high (One new reader-first release using established art, audio, thumbnail, A/B then C/D protocol seals and zero-cost guarded deployment tooling.); positive-signal/closure probabilities 0.96/0.82 within 220 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 84/unknown/2/8/0/6; cycles positive/negative/inconclusive 0/0/0; falsification gates 6; architectures tested/rejected 1/0; result target-closed; target reached true; forecast error -56 minutes; ratio 0.6; inside interval false; positive-signal/target-closure Brier scores 0.0016/0.0324; missing telemetry activeHumanMinutes: No instrument captured human direction or review time at the prospective attempt boundary.; uncachedInputTokens: The runtime does not expose an uncached-input counter.. Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json - Intended aims: science - Artifact roles: research-output, evidence-assessment, method-demonstration, communication - Decision object: counterexample — An exact four-point interior cover whose integral normalized (J+, boxdot)-cubical relative-homology map is Z to 0. Scope: The directed interval J+ with the inductive box product, normalized cubical chains, integral coefficients and degree one; not other intervals, products, graph restrictions, coefficient systems, novelty or external validation. - Reusable methods: Counterexample- and proxy-first analysis (counterexample-proxy-first); Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: discovery, assurance, publication - Semantic bridge: explicit — The package derives the four-point closure from the cited J+ inductive-product definitions, states the exact relative excision map, and labels every basis column needed for H1. Remaining risks: The live AIM statement does not select one of the six cubical theories.; Related in-preparation work is unavailable for collision testing.; All executable and internal-review evidence remains producer coordinated.. - Human judgement gates: Keep the result limited to integral normalized (J+, boxdot)-cubical homology.; Check the written relative-chain proof rather than treating replay as a proof oracle.; Keep producer concordance separate from independent reproduction.; Keep bounded novelty search separate from first or priority language. - Next assurance action: Obtain an authenticated unaffiliated reconstruction and closure-space specialist assessment, then investigate the remaining inductive interval theories. - Claim ceiling: An anonymous unrefereed four-point counterexample for integral normalized (J+, boxdot)-cubical homology with public immutable assets and producer-side replay; not historical priority, unaffiliated validation, formal verification, external specialist review, or editorial peer review. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Faster or more reliable resolution and assurance of topology open problems in AI-assisted finite counterexample reconstruction, proof compression, source audit, review repair and guarded candidate publication; design none; comparator No matched conventional workflow was registered.; estimand No effect on discovery time, human effort, compute, correction rate, proof quality, assurance time, uptake, citation or field outcomes was estimated.; no real-world effect evidence asserted - Parent handoffs: depends-on-claim https://arxiv.org/abs/2112.13421; inherited claim: Bubenik and Milićević define the interval-and-product cubical theories, prove categorical-product excision and pose the inductive-product excision cases as open.; inherited ceiling: The source fixes the definitions and question but does not validate this candidate's computation, proof, novelty or priority. ## Verification status Anonymous, unrefereed algebraic-topology candidate at internal PASS_WITH_NOTES. The finite theorem has a direct chain-level proof and same-producer exact replay. The candidate coarsens Bubenik and Milićević Example 5.9 but makes no first or priority claim because related work remains listed in preparation. The result is only for integral normalized (J+, boxdot)-cubical homology. ## References 1. Bubenik, P., & Milićević, N. (2021). Eilenberg–Steenrod homology and cohomology theories for Čech's closure spaces. arXiv:2112.13421v1. 2. Bubenik, P., & Milićević, N. (2024). Homotopy, homology, and persistent homology using closure spaces. Journal of Applied and Computational Topology 8, 579–641. 3. Milićević, N. (2025). Singular homology of roots of unity. Topology and its Applications 366, 109291. 4. Jamil, S. S., Staecker, P. C., & Ali, D. (2022). Computability of digital cubical singular homology of c1-digital images. arXiv:2205.07457. 5. Bubenik, P. (2025). Relative cell complexes in closure spaces. Canadian Mathematical Bulletin 68, 1338–1346.