Press release · 30 August 2026 · version 0.2.0-candidate
A four-point failure of (J+, boxdot)-cubical excision
A directed four-point closure space sends a nonzero integral relative one-cycle to zero, disproving interior-cover excision for normalized (J+, boxdot)-cubical homology.
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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 $\boxdot$.
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
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:
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
Determine excision for the remaining inductive theories based on $J_1$ and the ordinary interval $I$.
Seek an analogous failure—or a positive theorem—under symmetric graph closure restrictions.
Reconstruct the complete chain calculation in a materially separate stack.
Formalize the four-point closure, normalized cubical chains and relative map in a proof assistant.
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.
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
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
Measurement scope
assurance-through-publication — From prospective registration before substantive action on the supplied Major Revision report through terminal public readback or a preserved stop. The earlier problem selection, theorem discovery, proof construction, prepare-only review and package build are excluded and are not reconstructed.
Frozen 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.
Fermi active-time forecast
140 minutes; plausible interval 90–220; expected unattended wait 60. Reference class: Reviewed exact mathematics candidates taken through full Evidence Press publication (n=2) — The two latest prospectively measured assurance-through-publication releases completed in 95 and 55 active agent minutes. This candidate has a smaller theorem but a material replay-contract redesign and a fresh five-role review..
non-mutating replay redesign, schema consistency and clean-environment validation: 1 × 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 × 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 × 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 × 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.
Tractability forecast
Within 220 active minutes: positive signal 0.96; target closure 0.82. Stop rule: Stop on a fatal proof, semantic, novelty, licence, authentication, preservation, CI, accessibility or exact-readback failure. At 140 active agent minutes consolidate to load-bearing gates and report variance; at 220 active minutes pause rather than weaken a gate unless the user explicitly authorises continuation after seeing the exact blocker.
1 agent runs; maximum parallelism 1; 1 model turns; 1543001 deduplicated model tokens; 2 substantive review rounds; P0/P1 findings 0/4; pre-publication claim corrections 18.
Result and calibration
target-closed — The supplied Major Revision was fully actioned, including the manifest-mutating replay redesign, frozen-record reconciliation, precise source relationship, adjacent-theory comparison, exact matrix certificate and bounded terminology and presentation repairs. The final non-mutating package passed 15 tests, six semantic corruptions, Python optimization and JavaScript concordance, a pristine-archive gate, a four-environment public Linux matrix and a frozen internal review plus confirmation. Byte-identical immutable GitHub and Zenodo assets then reached canonical Evidence Press publication with provenance-bound media, composite CI, accessibility, 41-release and 31-artifact preservation, protocol integrity and desktop/mobile browser readback. The result remains an anonymous unrefereed candidate with unresolved priority and no external-assurance or impact promotion. Positive signal: true; target reached: true. Active-time error -56 minutes; actual/forecast 0.6; inside interval: false. Brier score: positive signal 0.0016; target closure 0.0324. Variance: The theorem was mathematically sound at intake, the replay redesign admitted one compact build-directory architecture, and established exact-package, GitHub, Zenodo, media, composite-seal and guarded-deployment tooling allowed immutable-identity and publication work to close below the 90-minute lower active-time bound despite a concurrent-main reconciliation and one bounded confirmation enhancement.
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.
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
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 is 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.
