A semantic degradation contract and state lower bound for EPUB knowl fallbacks

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

EPUB is a packaged electronic-publication format. A knowl is an expandable content fragment used to reveal supplementary information in a web presentation. Support for such an interaction depends on the reading environment as well as the packaged document.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The source's categorical answer is a historical design opinion. It is not a current verified claim about every EPUB reader or publication workflow.

Catalogue screening limitation: PreTeXt software design, documentation, or publication feature request; no mathematical problem is specified by the supplied passage.

The source snapshot labels this record partially solved; this is catalogue metadata rather than independent verification of a mathematical result or the current completion of a project.

Statement recorded in the supplied source.

Is ePub good enough? No, because of knowls. [S2]

Source status.

The supplied catalogue labels this source record partially solved; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness.

Short English statement

Assess whether an EPUB presentation meets the intended needs involving expandable content.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20002180 (AIM-INFRASTRUCTURE-0080). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 AIM Workshop Problem Lists, source linked by catalogue record AIM-INFRASTRUCTURE-0080. https://github.com/PreTeXtBook/community/wiki/Moderated-Problem-Session.

Preparation and review.

The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 20002180 and code AIM-INFRASTRUCTURE-0080 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Literature and context check

The primary community-session source was consulted. The record is a broad software-design request, and no mathematical theorem follows from its editorial catalogue title. The finite model below is explicitly auxiliary; it does not certify an actual implementation or current feature status. The EPUB 3.3 specification was consulted for current format context; it discusses scripting. The source's brief historical objection is therefore not treated as a theorem that every EPUB format or reader forbids interactivity. No actual reader-support claim is made.

Research attempt: The state requirement for independently toggled knowls

Model $n\ge1$ independently toggled panels. A state is the subset $S\subseteq\{1,\ldots,n\}$ of currently open panels; the observable output identifies exactly that subset. Toggling panel $i$ replaces $S$ by its symmetric difference with $\{i\}$. Starting from the empty subset, any subset is reachable by toggling each of its elements once. There are therefore $2^n$ reachable, observably distinct states.

Any deterministic emulator preserving these observations requires at least $2^n$ distinguishable internal states. Otherwise two distinct reachable subsets $S,T$ would correspond to the same emulator state by the pigeonhole principle. The emulator's current observable output is a function of its state and would be equal for those two histories, contradicting the required different visible subsets. If state is stored in $b$ binary bits with no other varying memory, it has at most $2^b$ possible values, so $2^b\ge2^n$ and $b\ge n$. An $n$-bit indicator vector achieves the bound: flip bit $i$ on action $i$ and display exactly the set of one bits.

A static fallback that opens every panel preserves the union of panel contents but not this interaction behavior. For $n=1$ it already merges the closed and open states, losing one observable distinction. A collection of static pages could represent all $2^n$ subsets, but then it pays for that many separate state representations; navigation semantics and link availability still need implementation. The lower bound is about the selected state model, not a claim that an EPUB file must literally contain exponentially many pages or that a scripting runtime cannot encode states compactly.

Verification and adversarial audit

The construction and counterexample are checked in the written finite model. No actual PreTeXt build, browser, EPUB reader, or publishing service was tested. All runtime and identity assumptions used by the proof are stated explicitly.

Audit corrections.

Independent visible panel states are the explicit observable. The bound is distinguishable states or binary memory, not file size, and current EPUB capabilities are not inferred from a workshop remark.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

This is a restricted route investigation. The full feature requires implementation, supported-version decisions, conformance tests, and user-facing evaluation that were not performed here. The original agenda remains unresolved, and no novelty or completed software feature is claimed.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://github.com/PreTeXtBook/community/wiki/Moderated-Problem-Session (consulted 2026-10-11).
• https://www.w3.org/TR/epub-33/ (consulted 2026-10-11).

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration and repository-format review; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.