---
id: F-002
title: The decreasing pattern is not always first missing
tier: census
polarity: negative
depends_on: [F-001]
supersedes: []
verifier: export PATH="$HOME/.cargo/bin:$PATH"; cargo run --release --manifest-path check_answer/Cargo.toml -- verified_math/F-002_decreasing-not-first/candidate.txt
date: 2026-07-11
---

## Statement

Under the strict-permutation interpretation pinned in F-001,

\[
F(826)=3,
\]

but the length-four block starting at `m=822` is strictly decreasing:

\[
\phi(823)=822 > \phi(824)=408 > \phi(825)=400 > \phi(826)=348.
\]

Thus the decreasing length-`F(826)+1` pattern has already appeared even though
not all patterns of that length have appeared.  Consequently
`DecreasingIsFirstMissingQuestion` is false: the displayed decreasing pattern
is **not** always the first pattern to fail/last pattern to appear.

## Proof / verification

The witness file is `candidate.txt`, containing `826 3 822`.  The independent
Rust gate from F-001 computes every totient through 826 and exhaustively
enumerates all strict rank patterns.  It finds all 6 length-three patterns (so
length three succeeds), but only 15 of 24 length-four patterns (so length four
fails).  Prefix extension ensures no later successful length can be skipped,
therefore `F(826)=3`.  It separately checks all adjacent inequalities in the
length-four block at `m=822`.

`checker-output.txt` records the accepting run.  The exact verifier command
recomputes the census from the witness and must print `PASS` with exit code 0.
