The Least Element

A lemma that was false as written. The premise it needed was the first line of the picture.

Claim. The hourglass tipped once. The first grain is the first record. Grains pass one per click. So every initial segment of the record history is finite. The order is total and well-founded. The least element exists and is unique. AP44 Proposition 1 stands on the premise. The premise is supplied by dated note.

Intuition. Discreteness alone is not enough. One grain per click over a past with no beginning is the integers. The integers have no first. What does the work is that the sand started falling.

Givens. AP44 §2 Proposition 1. AP20 (R as a monoid; the undeniable premise). AP45 (one grain per event). The author’s sequence of the tip. Three reviewers’ counterexample.

Theorem — the tip. Before the tip there is no record (S). The first grain is the first record (B). Grains pass the neck one at a time, one per click (C). So every initial segment of the history is finite. Discreteness with a beginning is well-founded. Discreteness alone is not. The order is total. R orders records by sequential composition in a monoid. One grain after another. The universe is processed one grain at a time, whichever site’s neck the grain passes (The Keys; §1; AP52 §2). Two records are always one before the other. That order is the record’s. No site reads it (AP52 §6). A total, strict, well-founded order has a unique least element. For any configuration it is its constitution record.

Grade. The tip: Ruled (“yes — said it”). Well-foundedness: Derived. The premise is the corpus’s own. Not a repair.

Connections. The premise is §1’s first line. It is AP44’s own application of the proposition to the arena.

The integers have no first grain. The universe does.