This guide to resolve any issues users get when they try to change nature or PID of a Pokémon from Gen 3 or Gen 4.
DO NOT USE ANY POKéMON WITH MODIFIED PID ONLINE!
Foreword
The reason why people get various PID-Mismatch or Invalid Encounter type of messages, is because they did not modify the Pokémon correctly.
In Generations 3 and 4, due to an in-depth understanding of how the Pseudo-Random Number Generator works (as well as game mechanics), the Personality Value (PID) is tied to various other values related to the Pokémon generation procedure in-game, such as Individual Values (IVs), Gender, Ability slot, Nature, and Encounter Slot. (Encounter slot refers to the programmed data in a table, that hosts a list of level and species that can be encountered in a map)
TL;DR: Due to how RNG is called, PID, IVs, Nature, Gender, Ability, Encounter slot are related and can be derived.
Basic Preparation
1. Pokefinder will be the tool we use to find correct information (PID/IVs etc)
2. Time and patience. I cannot stress this enough.
3. Know whether the specific Pokémon is Stationary/Gift or Grass/Cave/Surfing/Fishing/Rock Smash wild encounters.
4. Begin with 'priority of information' in mind. Which is more important: PID or IVs? That will be your starting point.
This will be basis for choosing Search via PID or Search via IVs below.
Note: Do know that there are scenarios where the PID or IVs you want would be impossible. Learn to adapt to such scenarios.
5. If you're trying to make a shiny, and TID/SID isn't all that important to you, ignore shininess for now.
(There's a Shinification portion further down. Run through the regular steps first)
Search via PID - Stationary/Gift
Before we begin:
If Listed Below, Method 1:
For DPP:
Starter gift, Wild Roamers, Fossil Gifts, NPC Egg gifts, Platinum's Porygon Gift, Gift Manaphy Egg, In-game gifts
For HGSS:
Starters gift, Wild Roamers, Fossil Gifts, NPC Egg gifts, Sinjoh Ruins Dragon, Gift Manaphy Egg, In-game gifts (including Game Corner prizes)
Else:
DPPT uses Method J, and HGSS uses Method K.
Note: HGSS Lake of Rage Red Gyarados is via Chained Shiny Method.
Let's say for this example, we're trying to fix Darkrai caught in Platinum.
1. Pokefinder > Gen 3 Tools > PID to IV (yes, this isn't a typo)
2. Paste PID into the textbox, and Generate.
3. While keeping PID to IV window open, go back to main window Gen 4 - Stationary
4. Select Searcher, select Method 1, paste in the IVs from a Method 1 spread from the PID to IV window,
set the delay and frame to as per the image below, and Search.
We are reconfirming that the spread can be found for DPPT HGSS.
Note: If the Pokémon isn't Stationary Encounter (eg: Gift, Roaming), Lead has to be None.
5. Once we confirm the PID/IV spread indeed exists (in example above, 2nd line), copy the IVs to the Pokémon.
Search via PID - Wild
Before we begin: Wild Encounters in DPPT uses Method J, and HGSS uses Method K.
Example: I have a PID, but do not know what IVs it should have
1. Launch Pokefinder > Gen 3 Tools > PID to IV (not a typo)
2. Paste PID into the textbox, and Generate. 
3. While keeping PID to IV window open, go back to main window Gen 4 - Wild
4. Select intended game as profile. If you don't have a profile, set up a simple one.
Since this example is based on Gible from Platinum, that'll be my chosen game.
5. Select Searcher, select Method (DPPT uses Method J, and HGSS uses Method K),
Input Encounter type and Location (refer to PKHeX), Input Time [of day], Input Frame & Delay as per below,
and Input IVs (refer to PID to IV), then hit Search.
We are reconfirming that the spread can be found for DPPT HGSS.
Note: Time is important, as some Routes and encounters vary based on Time.
Note: Make sure you select correct encounter types too:
Also: Cave encounters uses the same procedure as Grass encounters.
6. Check if the PID generated and the Met Level given is the same as the one on your Pokémon (or the Level you decided on)
Met level only applies if on Generation 4 save. If it's been transferred up, Met level changes.
Two things you can do if the Level differs:
a. Modify the Met Level on your Pokémon
b. Cycle through Location and/or Time (if it's in a cave system where Encounter Rate changes at every level)
If you go through (b), and you still cannot get the desired PID, it's likely this PID/IV/level combo is not possible to encounter for this Pokémon at the selected location. Go back to Step 2, select a different one.
7. If PID and level matches, simply modify your Pokémon with the correct stats generated, such as IVs.
Search via IVs - Stationary/Gift
Before we begin:
If Listed Below, Method 1:
For DPP:
Starter gift, Wild Roamers, Fossil Gifts, NPC Egg gifts, Platinum's Porygon Gift, Gift Manaphy Egg
For HGSS:
Starters gift, Wild Roamers, Fossil Gifts, NPC Egg gifts, Sinjoh Ruins Dragon, Gift Manaphy Egg
Else:
DPPT uses Method J, and HGSS uses Method K.
Note: HGSS Lake of Rage Red Gyarados is via Chained Shiny Method.
Example: I want to modify a Darkrai captured in Platinum to be Flawless/Near Flawless IVs, but I do not know what the PID should have.
[For this example, we are using Method 1, even though Darkrai should be Method J; old images]
1. Open Pokefinder > Gen 4 - Stationary
2. Select Searcher, Method 1, input wanted IVs, set frame and delay as per image below, Search.
Note: If the Pokémon isn't Stationary Encounter (eg: Gift, Roaming), Lead has to be None.
3. Copy PID and Nature over to the Pokémon
Search via IVs - Wild
Before we begin: Wild Encounters in DPPT uses Method J, and HGSS uses Method K.
Example: I have a set of IVs I want, but do not know what PID it should be
1. Pokefinder > Gen 4 - Wild
2. Select intended game as profile. If you don't have a profile, set up a simple one.
Since this example is based on Luxray (evolved Shiny) in Platinum, that'll be my chosen game.
3. Select Searcher, select Method (DPPT uses Method J, and HGSS uses Method K),
Input Encounter type and Location (refer to PKHeX), Input Time [of day], Input Frame & Delay as per below,
and Input IVs, then hit Search.
You may want to allow some range for IVs, cause not every encounter and location allows for Flawless IVs.
4. Pick the PID and details you like, input it into PKHeX.
Shinification
There's 2 ways to make a Pokémon shiny:
a. When Search via IVs (for shiny PID), input the intended TID/SID and select Shiny Only
This method isn't perfect, as not every PID/IV/Nature combo would result in a shiny Pokémon for the particular chosen TID/SID combo
b. After obtaining wanted PID/IV/Nature combo, use the PID to find shiny TID/SID combo.
Using this Darkrai as an example:
B1. Pokefinder > Gen 4 - IDs
B2. Paste the PID in, set Delay, and Search. After a few seconds, hit Cancel, and the results will appear.
If you want a specific TID, click on Search for TID, and input wanted TID.
B3. Now, copy and paste the found TID and SID into the Pokémon
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