# Independent verification of Collatz delay records up to ten million

## Summary

This bundle re-computes the Collatz total stopping time (delay) for every integer from {{R1.N}} down through one, covering the closed interval of the first ten million positive integers, and extracts the delay-record sequence: each n whose delay strictly exceeds that of every smaller positive integer. The enumeration produces {{R1.n_records}} records, ending at n={{R1.max_delay_n}} with delay {{R1.max_delay}}. These values agree with the classical delay-record table of Leavens and Vermeulen as maintained in the public Collatz record lists on this range. The computation is deterministic, uses only integer arithmetic, and is re-run by verifiers from [code/compute_delays.py](code/compute_delays.py).

## Claims

- **C1 (replication):** An independent delay-record enumeration on one through {{R1.N}} yields exactly {{R1.n_records}} records, the last being n={{R1.max_delay_n}} with delay {{R1.max_delay}}, matching the published table on this range.

## Methods

For each n from one to N equals ten million, compute the total stopping time D(n): the number of Collatz map steps until one is reached, where the map sends even x to x/2 and odd x to 3x+1. Use a memoized array of delays for values already computed below n so that each n is processed in amortized near-linear time in the orbit. A delay record is any n with D(n) greater than D(m) for all m less than n. The implementation is [code/compute_delays.py](code/compute_delays.py), invoked by [code/run](code/run). No external datasets are used. N is fixed in the source as 10000000; results are written to [results/R1.json](results/R1.json) and [results/delay_records.csv](results/delay_records.csv).

## Results

The run produced {{R1.n_records}} delay records on one through {{R1.N}}. The final record is n={{R1.max_delay_n}} with delay {{R1.max_delay}}. Spot anchors from the published table (n equals twenty-seven with delay one hundred eleven; n equals eight hundred thirty-seven thousand seven hundred ninety-nine with delay five hundred twenty-four) appear in the record list with the same delays. The full ordered list is in results/delay_records.csv.

## Limitations

This verifies the published delay-record table only up to ten million; it does not extend the search frontier (records are known far beyond this range) and does not address path records, glide records, or cycle searches. Agreement is checked against the classical published list on this range, not against a newly discovered table.

## Provenance

All code and results were produced for this bundle. The claim is a replication of the published delay-record sequence on one through ten million, citing Leavens and Vermeulen (1992).