The Crypt Cycle as a Source Counter: Crypt Fission is Set by a Cell Division Quota, Not by Crypt Size

Crypt fission replicates colonic crypts. It sets crypt number during infancy, continues slowly throughadult life, and is elevated in familial adenomatous polyposis (FAP) and in adenoma, where it is theroute by which one altered crypt becomes many. What triggers an individual fission is unsettled.The available accounts are spatial: a crypt divides once its size, or its stem cell population, crossesa threshold. Each asks the crypt to measure a magnitude about itself, and no apparatus for thatmeasurement is known.We ask whether fission requires spatial self-measurement at all, and test the alternative that it isscheduled in division-time. The hypothesis is an ordinal quota: the exemption that makes a niche sitea source is itself finite, good for S retained-daughter divisions, after which the source set doubles andthe crypt partitions. This adds one rule to a published open-loop renewal model, in the single placethat model leaves unbounded, and asks the crypt to count rather than to measure.Four results follow. First, the ratio of fission rate to renewal rate is fixed whenever the renewal andsource-cycle timescales scale together, which establishes that fission is paced by division rather thanby elapsed time, though not that it is counted, since a per-division hazard gives the same constant.Second, the source count must cycle between K and 2K across the quota. Third, three measurementsseparate a quota from a per-division probability, two of them without imaging a lesion. Fourth, anycounter must survive some seven hundred occupant replacements per quota, which excludes factorspartitioned at division by a saturation argument, the APC/β-catenin loop, which is closed feedbackand oscillates rather than counts, and APC promoter methylation, which is absent from normal colon.Crypt size is then a correlate of fission and not its cause. The significance is that an ordinalmechanism removes the sensing requirement that every spatial account incurs, and makes the elevatedFAP index legible as a prevalence rather than a rate. We identify no molecule that counts, and give thespecification any candidate must meet.