The Envelope Forcing and the Phase Dependence: Two Arithmetic Results on Daniel's Day-Counts in the Hebrew Calendar

This note records two arithmetic properties of the Hebrew calendar and their verification. It predicts nothing: it asserts no application of Daniel's numbers to any year, names no future event, and is not to be read, at any later time, as having anticipated any outcome. Result one, the envelope forcing: with the terminus fixed at 21 Tishri (Hoshana Rabbah) — a day inside the 8–24 Tishri corridor window — the reverse-counted 1,335-day midpoint lands on days 14 through 16 of the corridor month in every year of the tested range, and in the Adar-band years on the Purim date-signature of the first Adar (principal case: Hoshana Rabbah 5794 minus 1,335 days = February 17, 2030 = 14 Adar I 5790, exactly), produced by arithmetic that contains no Purim input at any point. Result two, the phase dependence: while the corridor property survives all 57 aviv-plausible intercalation patterns, the selection of particular years does not — under each of the nineteen possible phases of the Metonic cycle, every year is selected in exactly nine of the nineteen phases, the union over the phases covers all years, and the year-lists of distinct phases can be entirely disjoint; any year-list derived from such a selection therefore carries the phase of its cycle as a premise. Both results were computed on an independently written engine and have been independently reproduced, exactly, on separately written engines. Includes a standalone verification script (Python standard library only) reproducing both results from scratch. The author's own dated application is deposited separately (DOI 10.5281/zenodo.21426633) and earns nothing from this note. Corrections, if ever needed, will be issued only as new versions of this record; the original remains on file unaltered.