---
id: F-006
title: Exact first cutoff for all length-four strict patterns
tier: census
polarity: positive
depends_on: [F-001]
supersedes: []
verifier: export PATH="$HOME/.cargo/bin:$PATH"; cargo run --release --manifest-path check_answer/Cargo.toml --bin pattern_threshold -- verified_math/F-006_exact-length4-threshold/candidate.txt
date: 2026-07-11
---

## Statement

The least cutoff `N` by which every one of the 24 strict ordering patterns of
four consecutive totient values has appeared is

\[
N=185{,}910{,}725.
\]

The unique pattern newly appearing at this endpoint has rank vector
`[2,3,1,0]` (rank zero is smallest).  Equivalently, if the four values are
`a,b,c,d`, its inequalities are `d<c<a<b`.

## Proof / verification

`candidate.txt` contains `4 185910725`.  The independent
`pattern_threshold` checker computes every totient through that cutoff with
integer arithmetic and scans every valid length-four block.  It finds fewer
than 24 patterns through `N-1`, then exactly 24 through `N`, with `[2,3,1,0]`
new at the final endpoint.  Thus both existence by `N` and nonexistence of at
least one pattern before `N` are exhaustive, not sampled.

`checker-output.txt` records the accepting run.  The checker's separately
written sieve is unit-tested against the gcd definition through 1000.
