time_emergence_certificate
plain-language theorem explainer
Time emerges as discrete tick counts on ledger snapshots, organized into minimal 8-tick epochs with the arrow of time given by strict defect reduction under recognition steps, which admit no reverse. Researchers in discrete spacetime or emergent-time models would cite this for the 2^3 period and one-way recognition. The proof is a term-mode triple that directly supplies the epoch equality, discreteness fact, and the irreversibility lemma applied to any strict defect decrease.
Claim. Let epoch length be the duration of one ledger update cycle. Then epoch length equals 8, which equals $2^3$, and for every recognition step where output defect is strictly less than input defect there exists no reverse recognition step with matching input-output defects while advancing the tick index.
background
The TimeEmergence module formalizes time as the tick counter on successive ledger snapshots rather than an external continuum, with epochs as the minimal complete update cycles. A RecognitionStep is the structure carrying an input LedgerSnapshot, output snapshot, exact tick advance of +1, and non-increasing defect, where defect is the J-cost functional. The module draws the eight_tick value from DimensionForcing for D=3. Upstream results include epoch_length_eq proving the length is 8 by reflexivity and recognition_irreversible establishing that strict defect reduction precludes any reversing step.
proof idea
The proof is a term-mode construction of the three-way conjunction. It supplies epoch_length_eq for the first conjunct, time_is_discrete for the equality to 2^3, and a lambda that applies recognition_irreversible to any step satisfying the strict defect inequality.
why it matters
This is the certificate for F-004 and F-006, confirming time as tick count and the arrow of time via defect decrease. It directly realizes the eight-tick octave (T7) in the forcing chain once D=3 is fixed. The result closes the minimal-cycle and irreversibility requirements for the Recognition Composition Law and later ledger arguments.
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