Problem 2.5 — Knowing that d(φ)≥ 1, can we decide whether it is ≥ 2?

Definitions and mathematical statement

Let $\varphi$ range over the source's intended objects and let $d(\varphi)$ be its invariant. Neither the objects, their effective representation, nor the definition of $d$ is supplied. Problem 2.5 asks two successive decision questions:

1. Given $\varphi$ with the promise $d(\varphi)\geq1$, can one decide whether $d(\varphi)\geq2$?
2. Given $\varphi$ with the promise $d(\varphi)\geq2$, can one decide whether $d(\varphi)\geq3$?

In an algorithmic interpretation, for $r=1,2$ put $A_r=\{\varphi:d(\varphi)\geq r\}$. Each request is for an algorithm halting on every valid promised input in $A_r$ and correctly deciding membership in $A_{r+1}$. This does not assert computability of the full invariant or ask for a procedure for arbitrary thresholds. The phrase "can we decide" may also require confirmation as an effective algorithmic request rather than an informal criterion.

The two threshold promises must be retained. Without the omitted definition and input model, no particular distance, filtration, genus restriction or decision algorithm can be substituted, and these are only conditional formulations of the supplied questions.

Short English statement

Under the promise that the omitted invariant is at least $1$, decide if it is at least $2$; under the promise that it is at least $2$, decide if it is at least $3$. Its definition and input model are missing.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000162$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 2.5, PDF page 156: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; repeated local numbering disambiguated by page among 329 headings.
• No definition of $d$, effective encoding, individual attribution or additional primary citation is supplied. The registry literature summary is truncated.