"separately for TP & Spouse as $3,875 [6 mos.* $7,750/12]"

Where did $3,875 come from?

"in 'b' above of $3,875 for TP and $1,292 for spouse."

For tax year 2023, Family coverage for HSA gives a contribution limit of $7,750. You don't split that as if that is equal into individual limits. The combined or shared limit, no matter which HSA accounts get contributed to, is the family limit. For instance, one might have the coverage through an employer and that employer provides towards contribution, so it can all go into the one HSA account.

An exception is the catch up provision, since it is age-related, has to go to the HSA of the person attaining the age, even if they covered under their spouse's employer family plan.

For instance, a person with a family plan gets $7,750/12 for the month's limit, but they share that with the spouse. It can be split however they want to, as HSA accounts stand individually but the person making the contribution is only eligible because they are covered appropriately.

An adult child (up to age 26) is allowed to be on the family plan. If they cannot be claimed as a dependent, they are entitled to their own Family Limit HSA contribution for that year, of $7,750, in addition to the parents having to share/split their contribution.

Example:

Paul is 55 and Mary is 50 and her employer offers the family medical plan qualifying coverage and an HSA amount and Junior is 25 and employed.

Paul puts $1,000 into his HSA.

Junior puts $7,750 into his HSA.

Mary's employer puts $1,500 into her HSA.

Mary puts $7,750 minus $1,500 = $6,250 into her HSA.