First Failure of Coherent Realization in a Steel Array

Definitions and mathematical statement

Let $\langle N_\xi:\xi\leq\Theta\rangle$ be the source's Steel array, with $e$-robustness as the criterion for adding extenders. The construction of the array, the parameter $e$, ambient iterability/foundational assumptions and the precise robustness condition are not supplied.

For each index, the failure in question is witnessed by a countable premouse/structure $P$, an elementary map $\sigma_0:P\to N_\xi$, and a countable normal iteration $I$ of $P$ for which the source's coherent realization does not exist. To make the first-index quantifier explicit, put \[ F=\{\xi\leq\Theta:\exists(P,\sigma_0,I) \text{ with the stated countability, elementarity and normality,} \text{ admitting no coherent realization}\}. \] The first question asks to identify $N_{\xi_*}$ with $\xi_*=\min F$, if $F$ is nonempty. It must include the absence of such failure at every earlier array index, not merely an example of one failure. Existence of a first failure within the displayed array is not independently assumed if the omitted source construction does not establish it.

The second question asks whether this $N_{\xi_*}$ has an iteration containing an alternating chain. This is an iteration of the first failing array model itself, not automatically the same countable iteration $I$ of its countable witness $P$. No extra countability or strategy requirement is imposed on this second iteration without the source definition.

"Steel array", "$e$-robust", "coherent realization" and "alternating chain" are technical source terms. In particular coherent realization requires the intended compatibility through iteration/extender data, not just an embedding of a final model; its diagrams, allowed branches and well-foundedness conditions are absent. Normal iteration may also refer to a tree-based convention needing explicit branch/strategy data. These definitions cannot be reconstructed from the names alone, so no specific first ordinal/model or alternating-chain theorem is invented.

The supplied literature triage checked 21 August 2026 reports no primary paper located in its targeted search resolving this specialized first-failure/alternating-chain question. That bounded search finding is not an independent proof of openness or impossibility.

Short English statement

Identify the first model of the specified $e$-robust Steel array with a countable coherent-realization failure, and ask whether that model itself admits an alternating-chain iteration. The array construction and technical realization/chain definitions are missing, so no particular index or model is claimed.

Sources

• Local UnsolvedMath registry, permanent ID 30000471, exact source code catalogue entry; First Failure of Coherent Realization in a Steel Array. Proposed year 2006, status open. The corrected_verified statement restores the array, $e$-robustness criterion, countable realization data and both questions; its claimed report inspection is source metadata.
• Supplied report citation: Feinstrukturtheorie und Innere Modelle (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/20, pp. 1215--1244, https://doi.org/10.4171/owr/2006/20. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21, citing only "original Oberwolfach report", https://doi.org/10.4171/owr/2006/20. It reports no primary resolution found in the targeted search. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims.