Two constants in the numerical estimates of inter-universal Teichmüller theory IV: 12/(ℓ(ℓ+1)) and 12/ℓ² (A fifth report)

A fifth report in the series begun by 10.5281/zenodo.22537342. The constants χ = 12/(ℓ(ℓ+1)) and τ = 12/ℓ² occur at different steps of the numerical estimates surrounding inter-universal Teichmüller theory IV. The first is the exact ratio of the stage-one coefficient 1/(2ℓ) to the mean of j²/(2ℓ) over j = 1, …, (ℓ−1)/2, and appears in Scholze–Stix's approximate calculation. The second is used in the upper-bound rearrangement in the proof of [IUT IV, Theorem 1.10]. Both are located in the original texts and their difference is computed with exact rational arithmetic. What is compared here are two scalar expressions under a common set of inputs, stated as a hypothesis. This is not a comparison of the two sources' full estimates, whose positive terms, divisor supports and approximation conventions differ and are kept separate throughout. The same rearrangement and the same slack term are already in print in Joshi, arXiv:2403.10430v2, p. 70; no priority is claimed for either. These calculations do not determine a possible-image hull, and they adjudicate neither inter-universal Teichmüller theory nor the Scholze–Stix objection. The deposit contains the paper, an exact-arithmetic check program with its frozen output, and a quotation checker with a manifest that pins each cited printing by SHA-256. Every quoted passage is located in the printing the manifest names; locating a candidate is not a verdict of verbatim identity, and each candidate was also read on the source page. Large language models were used in drafting, computation and source checking. The checks are the author's own; no external review took place.