The Rarest Item in TF2

Source information and caveats below!
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Music:
Stardust Speedway Zone Act 1 - Sonic Mania [OST]
[Blue Archive] Mitsukiyo - Mischievous Step
[Blue Archive] Mitsukiyo - Lemonade Diary
Pink Guy - Fried Noodles [DVNZXL Remix]

Sources and Caveats
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Certain types of items were omitted from consideration, specifically glitched items, promotional items, and gifts from Valve. Quantifying the statistical probability of occurrence for these items is not possible, for obvious reasons (i.e. a gift is a gift).

Australium Drop Spreadsheet:
https://docs.google.com/spreadsheets/...

Article on TF2's unusual drop rate:
http://tf2finance.com/rate/

Item Counts, Item Grade Abundance, Stat-Clock Abundance:
TF2tools.net

TF2 Skin Sticker and Pattern Information:
https://steamcommunity.com/sharedfile...

CSGO's case odds:
https://www.pcgamesn.com/counter-stri...
https://www.csgo.com.cn/news/gamebroa...

By law, Chinese companies must publicly display the results of all lottery results (including case openings):
https://www.csgo.com.cn/hd/1707/lotte...

Given the abysmal state of item servers, previous versions of the TF2tools webpages were accessed via the Wayback machine:
https://web.archive.org/web/201606041...

Item grade abundances change with the gradual increase in the number of trade-ups performed. As time goes on, lower grade items are exchanged for higher grade items (hence elite grades going from 0.8% to 1% over the course of about a year, reflecting item consumption and generation, not a change in case odds).

Wear abundance changes are negligible over the lifespan of TF2 skins, unimpacted by trade-ups given the differing nature from CSGO trade-ups.

Note again that the odds shown in the Steam guide are not official, and appear to be largely derived from guesswork or first-order estimates.