Theorem 2: Atomicity & Countability
Exactly one unit posting per tick; concurrency at one tick is impossible
Statement
Atomic tick. There exists an indivisible unit step τ₀ such that per tick exactly one elementary posting of size δ occurs. Two distinct postings at the same tick are impossible under double‑entry and conservation.
In plain English
Time, at the ledger level, clicks like a metronome. Each click, one thing happens. Not two. If two “happened at once,” either you’ve secretly changed the coin size (now it’s 2δ) or you’ve created a bookkeeping glitch that breaks conservation. So events are countable— like frames in a film—and the universe has a smallest beat, τ₀.
- Why inevitable: two‑at‑once merges into a bigger coin (illegal) or leaves pages unbalanced (illegal).
- What it buys: clean cause→effect ordering and the discrete rhythm used in T7’s 8‑beat cycle.
Formal setting
- Ledger (T1): a linearly ordered abelian group with generator δ > 0 and page maps ϕ, ψ respecting double‑entry.
- Tick index: a total order (ℤ or ℕ) labeling posting events; finiteness excludes dense orderings for effective updates.
- Closed chain conservation: sums over closed finite chains telescope to 0.
Key lemmas (outline)
- Granularity lemma: If two postings share the same tick, they can be merged to a single δ‑sized posting only by violating the positive cone or by redefining δ (forbidden by T1’s δ‑nonrescalability).
- Conservation lemma: Two independent postings at one tick generically create an unmatched credit/debit, producing a nonzero loop on a finite closed chain, contradicting conservation.
- Order lemma: If ticks were dense, there would be no minimal separation between postings, contradicting T1’s Archimedean order induced by δ and finiteness.
Proof sketch
Assume by contradiction two distinct postings occur at the same tick t. Either (i) they combine to an effective posting of size 2δ, rescaling δ against T1, or (ii) they target disjoint pages, generating a non‑telescoping surplus/deficit around the nearest closed chain, violating conservation. Therefore exactly one δ‑posting occurs per tick. Countability follows from the total order on ticks with a minimal separation τ₀.
Edge cases
- Parallelism: Apparent concurrency resolves to a total posting order at ledger granularity; micro‑ordering below τ₀ is undefined/irrelevant.
- Composite operations: Multi‑edge updates decompose into a tick‑ordered list of δ‑postings.
Implications
- T7 minimal period: The 3‑cube requires at least |V| = 8 ticks; atomicity fixes the step size.
- T5 integer split‑count: Per‑tick splits are integer‑valued; minimizing total cost over k ≥ 1 selects k = 1 and φ.
- Discrete continuity (T3): The continuity equation arises from sums over tick‑ordered chains.