Rick Neuheisel has a "good vibe" but it's the wrong case

In pursuing (and eventually landing) defensive specialist assistant coach DeWayne Walker on his UCLA football staff, recently hired head coach Rick Neuheisel stated he had "a good vibe" after talking with Walker. Unfortunately, coach Neuheisel's vibe did not get the pronoun case right. The sentence in question:

"The goal is for he and I to lock arms and do this together."

The compound subject "he and I" follows a preposition. Odd as it may sound, him and me is used as the objective case subject in its own clause.

The infinitive, "to lock" is of interest. Infinitives may be used as adverbs, and as such will express themselves as adverbs by showing: condition, degree (or comparison), manner, reason, purpose, or result. We are focusing on the use of the infinitive of purpose in Coach Neuheisel's sentence: to lock arms... How will "we" do this?" By locking arms. "How" is an adverb question, "to lock arms" answers the adverb question. Without the preposition "for," and with syntax rearranged, a correction variation would be:

He and I will do this together by locking arms, this is our goal.

The rule: The infinitive of purpose is frequently the object of the preposition for and has a regular objective-case subject. (Harper's English Grammar, John B. Opdycke). Thus, Coach Neuheisel should have said:

The goal is for him and me to lock arms and do this together.