A branch-profile audit of the CBH participant note

Definitions and mathematical statement

Participant contribution, not a research question. A.1 is David Asperó's first-person statement that he hoped to learn about the Cofinal Branches Hypothesis and Woodin's reported refutation, in a workshop where much of Woodin's work was described as unpublished. The supplied source verification identifies Chapter A as "Participant Contributions". The publication-status comment is historical to the 2004 document, not a present-day bibliographic claim. There is no quantified conjecture or requested construction in this entry.

Auxiliary branch-profile explanation from the aid. Fix an iteration tree $\mathcal T$ in an explicitly specified class for which the relevant direct-limit models are defined. Put \[ c(\mathcal T)=\#\{b:b\text{ is a cofinal branch of }\mathcal T\}, \qquad w(\mathcal T)=\#\{b:b\text{ is cofinal and its direct limit is well founded}\}. \] Then $0\le w(\mathcal T)\le c(\mathcal T)$. In the aid's local convention the CBH existence condition is $w(\mathcal T)\ge1$, while the UBH uniqueness condition is $w(\mathcal T)\le1$. Consequently:

• A CBH counterexample has $w=0$ and locally satisfies the UBH inequality.
• A UBH counterexample has $w\ge2$ and locally satisfies the CBH inequality.
• One tree cannot witness both failures. Global failures over a common tree class therefore require different trees.

These are elementary consequences of the fixed definitions, not an unconditional existence theorem about iteration trees. The full global hypotheses must specify the tree class, model, extenders, normality and well-foundedness conventions; none is supplied by the participant sentence.

The aid reports two distinct published Neeman--Steel witnesses under assumptions it labels (A1)--(A2): Theorem 2.1 with profile $(c,w)=(2,2)$ and Theorem 3.4 with profile $(c,w)=(1,0)$. The assumptions and publication are not identified in this record. These profiles are therefore preserved as attributed conditional reports needing source recovery, not instantiated theorems or evidence that the same tree refutes both hypotheses. They are also not silently identified with every detail of Woodin's historically reported refutation.

Short English statement

This is Asperó's 2004 statement of workshop interests, not an open problem. The aid's auxiliary branch counts distinguish failure of existence from failure of uniqueness: the former has no well-founded branch and the latter at least two. Its two named conditional examples require missing theorem hypotheses and references.

Sources

• Local UnsolvedMath registry, permanent ID 20002373, source code catalogue entry; canonical aim-logic-notes.json, notes[148].
• AIM Recent Advances in Core Model Theory, version 29 November 2004, Chapter A "Participant Contributions", A.1 David Aspero, p. 3: https://aimath.org/WWN/coremodel/coremodel.pdf. These document details are supplied verification, not a new visit.
• The aid names Neeman--Steel, Theorems 2.1 and 3.4, and assumptions (A1)--(A2), without title, date, publication entry or the assumption text. Woodin's work is mentioned in the participant's historical remarks.
• Registry research attempt 1, dated 2026-08-14: context only, status at run not a problem, rigor "proved", novelty confidence low. The explicitly machine-generated, non-peer-reviewed aid requires independent verification before citation.