Regretting A Fact

He regretted (the fact) that he had lied.

He	regretted	(the fact) [[that he had lied]]
Senser	Process: mental: emotive	Phenomenon: metaphenomenon

Halliday & Matthiessen (2004: 471):

A fact clause serving as Head in a nominal group without a fact noun can be related to the first class of fact noun — that of 'cases', since such a fact clause is always agnate with an expanded nominal group with fact as Head.  Whether the nominal group has a fact noun as Head or not, the fact clause is embedded.  Because there is no projecting process involved, to which it could be paratactically or hypotactically related, a fact can appear only in embedded form: either as Qualifier to a 'fact' noun, or as a nominalisation on its own.

the	fact	[[that he had lied]]
Deictic	Thing	Qualifier

vs

[[that he had lied]]

Qualifier/Head

Hot Lover Boy

Did he tell you his real name or that he was called Hot Lover Boy?

Did

he

tell

you

his real name

or

(did)

(he)

(tell)

(you)

that

he

was called

Hot Lover Boy

1

+ 2

a

“ b

Process:

Sayer

verbal

Receiver

Verbiage

Process:

Sayer

verbal

Receiver

Value/ Identified

Process: relational

Token/ Identifier

This analysis recovers the information that is understood by both speaker and addressee, 

but which is so Given in such discourse, it can be left unsaid.

Forbearers

At these moments	America	has carried on	not simply because of the skill or vision of those in high office	but	because	We the People	have remained	faithful to the ideals of our forbearers*, and true to our founding documents
a					x b cause: reason
Location	Actor	Process: material	Cause: reason			Carrier	Process: relational	Attribute
Adjunct	Subject	Finite/Predicator	Adjunct			Subject	Finite/Predicator	Complement
Theme: marked	Rheme			Theme			Rheme

Reasoning:

The 'but' relates to 'not simply' and anticipates a second prepositional phrase related by paratactic extension in a phrase complex realising a circumstance of Cause: reason.*  However, the writer has instead opted for an enhancing dependent clause of Cause: reason, thereby creating a logical inconsistency.

The hypotactic relation between the clauses can be made more apparent by removing the preceding circumstance of Cause: reason (which also removes the need for 'but'):

At these moments	America	has carried on	because	We the People	have remained	faithful to the ideals of our forbearers, and true to our founding documents
a			x b cause: reason
Location	Actor	Process: material		Carrier	Process: relational	Attribute
Adjunct	Subject	Finite/Predicator		Subject	Finite/Predicator	Complement
Theme: marked	Rheme		Theme		Rheme

* An extending paratactic prepositional phrase complex such as:

not simply because of the skill or vision of those in high office,

but because of the fact that We the People have remained faithful to the ideals of our forbearers, and true to our founding documents 

not simply	because of	the skill or vision [of those in high office]	but	because of	the fact [[that We the People have remained faithful to the ideals of our forbearers, and true to our founding documents]]
1			+ 2
	Process	Range		Process	Range

∞

* forbear

verb: politely or patiently restrain an impulse to do something; refrain.

"he modestly forbears to include his own work"

synonyms: refrain, abstain, desist, keep, restrain oneself, stop oneself, hold back,withhold

Given that the writer screwed up on the lexis, it's hardly surprising they screwed up on the grammar as well, by trying to set up a logico-semantic relation between a prepositional phrase and a clause.

The theory accounts for an instance like this the same way it accounts for instances like he done real good.

On the mismatch between potential and instance, see here.