You get 1 admin while an Administrator is in your pool, when you promote said admin you lose 1 and gain 5, so in effect it only goes up by 4, but you gain 5 due to the -1.

I am not saying something is not amiss, just not sure what it is tbh.

When an Administrator is in the leader pool it gives this:-

<UnitGlobalStats>
<UnitType>Administrator</UnitType>
<Stat>
<EffectType>AdministrationAwarded</EffectType>
<Target>
<TargetType>Faction</TargetType>
</Target>
<BonusType>Flat</BonusType>
<Value>1</Value>
</Stat>
</UnitGlobalStats>

You also gain 4 from the hireing of an Administrator as this code shows:-

<UnitSpecialization>
<InternalName>Unit_Administrator</InternalName>
<DisplayName>Unit_Administrator_Name</DisplayName>
<Description>Unit_Administrator_Dec</Description>
<GlobalType>Administrator</GlobalType>
<GlobalCapAmount>10</GlobalCapAmount>
<Icon>Citizen_Administrator.png</Icon>
<Image>Crusade_Administrator.png</Image>
<Background>Citizen_BG_Admiral.png</Background>
<SelectSFX>UI/CITIZENS/ADMINISTRATOR_SELECT</SelectSFX>
<ImprovementType>AllImprovements</ImprovementType>
<CanTravel>false</CanTravel>
<Triggers>
<OnEvent>OnUnitGenerated</OnEvent>
<PerformAction>
<Action>AwardSpecificResource</Action>
<ValueParam>4</ValueParam>
<StringParam>Administration</StringParam>
</PerformAction>
</Triggers>
</UnitSpecialization>

When you then promote said Admin, you lose 1 Admin due to removing the Administrator from the Global Pool, then gain 5 more from the Promotion seen in this bit of code:-

<UnitSpecialization>
<InternalName>Promotion_Minister</InternalName>
<DisplayName>Promotion_Minister_Name</DisplayName>
<Description>Promotion_Minister_Dec</Description>
<GlobalType>Minister</GlobalType>
<Icon>GC3_Build_Starbases_Icon.png</Icon>
<Image>Crusade_Administrator.png</Image>
<Background>Citizen_BG_Admiral.png</Background>
<SelectSFX>UI/CITIZENS/ADMINISTRATOR_SELECT</SelectSFX>
<ImprovementType>AllImprovements</ImprovementType>
<CanTravel>false</CanTravel>
<Cost>
<EffectType>PromethionCost</EffectType>
<Target>
<TargetType>Faction</TargetType>
</Target>
<BonusType>OneTime</BonusType>
<Value>10</Value>
</Cost>
<Cost>
<EffectType>EpimetheusPollenCost</EffectType>
<Target>
<TargetType>Faction</TargetType>
</Target>
<BonusType>OneTime</BonusType>
<Value>1</Value>
</Cost>
<Triggers>
<OnEvent>OnUnitGenerated</OnEvent>
<PerformAction>
<Action>AwardSpecificResource</Action>
<ValueParam>5</ValueParam>
<StringParam>Administration</StringParam>
</PerformAction>
<DescriptionText>Promotion_Minister_ActionTriggerDec_0</DescriptionText>
</Triggers>
<Triggers>
<OnEvent>OnUnitGenerated</OnEvent>
<PerformAction>
<Action>KillUnitSelf</Action>
</PerformAction>
<DescriptionText>Promotion_Minister_ActionTriggerDec_1</DescriptionText>
</Triggers>
<Prerequ>
<UnitTypeFrom>Administrator</UnitTypeFrom>
</Prerequ>
</UnitSpecialization>

In total you gain 9 Admins per Promoted Administrator. if you pay attention to the tooltips you should in effect gain a total of 10, 5 from the Hireing :-

<String>Administrators provide 5 points for administration which in turn are consumed by Survey Ships, Colonies, and Starbases.</String>

And another 5 from the Promotion:-

<String>[ICON=Stat_Admin_Icon] Provides 5 more administrative resources. </String>

So yeah, either the Tooltip is wrong, or we are not getting what we should

Well, however you put it, when you assign an admin to a survey ship and it gets destroyed, the admin is gone - I've never heard of an admin citizen getting lost in this manner. He (or she) is clearly not the same kind of administrator, and the game also uses subtly different names. Therefore the tool tip should clearly state how many freely usable admins you get, and that is 5+4 at the moment - or fix it to provide 5+5 freely usable admins, if that was the original intention.

It's not sensible to advertise a number of 'items' only to silently subtract one afterwards on the premise that the 'container' that comes with the package accounts for one item but at the same time cannot be used as such. Yes I know, that's how advertisement often works in the real world, but that doesn't mean it's how we want it in a game.