The quick answer is, you're on the right lines.
Larks' I & II was the emerging of my personal voice as a writer / composer (although "composer" claims too much). My "guitar voice" was already reasonably well developed, but composing wasn't my prime direction until Ian McDonald left Crimson in December 1969. ITWOP, "Lizard" & "Islands" might be seen as the development & working through of learnt & acquired materials before my personal voice began to speak.
During the middle of 1971 I began working with recognisably Larks' material, running lines which appear on LTIA I, at the new flat in Holland's Park that I had just moved into. This was a vocabulary that I didn't feel worked for that particular version of Crimson, which looked more towards America than Europe.
My way of working through this "new vocabulary" was primarily instinctive & intuitive: I didn't "know" what I was doing, or playing. Rather, I began and kept going following where the material was leading. Discovery rather than invention, perhaps.
As the material became more formalised (during mid-1972 for the Jamie-David-John-Billy & Bob band) LTIA I was conceived as the beginning of a KC show, and LTIA II as the end. LTIA I as a whole contains more ideas & input from all the team than LTIA II, which follows more closely my own overall "vision" (this doesn't aim to detract from the contributions of the other members).
If you were to ask my aim, briefly, it was to access the energy & power of Hendrix (the Afro-American tradition) but to expand the vocabulary to access what was available in the European tradition; notably, via Bartok (the string quartets) & the early Stravinsky of ROS & Firebird. The question I posed myself might be put like this: "What would Hendrix sound like playing The ROS or a Bartok string quartet?" If an older man might look back at this and be struck by that young man's arrogance, well, an ignorance of limitations sometimes allows the young of any age to achieve impossible things!
I didn't, and don't, have the technical qualifications or capacity to "know" what was involved. But I did read music and spent my unemployment in London with Persichetti's "20th. Century Harmony" and Steven's "Bartok" while practising guitar. We might recall that the young Stravinsky of The ROS didn't "know" what he was doing either: for him it was more an instinctive & intuitive process. And Bartok himself described his own compositional process as instinctive & intuitive.
If musical material does emerge instinctively & intuitively, we might postulate that anyone who adopts this approach might be accessing the same "pool" of "information". Formally, this leads to the arithmetic approach (tonal harmony) and the geometric approach (Bartok "Axis" & the Golden Section). The first has the characteristic of stability, the second of dynamism. The first is crystalline, the second vital. This applies vertically (pitch, and therefore melody & harmony) & horizontally (rhythm). Forgive me for telling you how to suck eggs - you know this already.
When I discovered Lendvai c. 1986 it rationalised what I had discovered in practice over the preceding 15 years (although Lendvai appears to remain persona non grata with Western musicologists). The octotonic (double symmetrical) scale seemed to me, as a rock guitarist, very obvious: the scale was both major & minor, or straight & "blue". To include both forms of the third as equally legitimate in a scale of 8 notes wasn't a very great conceptual leap.
My approach during 1971-73 while developing material for Crim was to play the themes while consulting the body: how did the stresses, accents and lines relate to the fundamental pulse? This is not something which the mind can answer. So, I spent many hours testing ideas against my feet in the kitchen of Thornhill Cottage, Holt, near Wimborne, Dorset.
LTIA I began appearing in mid-1971, and LTIA II probably during early-mid 1972. When I first threw out the main riff/motif to John & Bill during the earliest jamming (Command Studios) they didn't pick up on it. The second time I threw it out (Richmond Athletic Club) they did - and how! The 10/8 theme I believe may have emerged during that playing in Richmond. The violin material from LTIA I is primarily David's. My own Englishness is never far away ("Book Of Saturday" and "Pie Jesu" for example) but in Larks' my Englishness drifts towards the continent.
Crim composing, or "fixing materials", is an ongoing process: the material often appears while playing, and when it is well-formed ("written") may well continue to develop during performance. I consider all material from all my writing life as fair game & available for transformation, transmutation & developing variation today where several decades of material is somehow telescoped into a present ongoing & maturing "now". So it didn't matter to me that motifs from 1972/3 ("Larks'" & "Fracture") presented themselves for consideration during TCOL.
If any writing is "true" its formal properties will also be sound. If Larks' is "true" then inevitably its form will appear to be "based on cosmological principles. i.e.Golden Section in terms of structure and, possibly, pitches". If the work is of a quality, its form will be inevitable rather than intentional. (This may be both involutionary and evolutionary).
I'm not sure that the composer "knows" what is happening, or can fully understand what is involved in an authentic piece of work: their understanding is as limited as anyone else's when it comes to developing a relationship with their externalised work. For example, my Mother knew me well, but there were parts of me which developed when I grew up and left home that she would only have been able to know and interpret in terms of her own knowing. So, even for my Mother, aspects of her son escaped her understanding. And more so with my Father.
Back from a day playing with my Sister.
We called our cousins in Ross-On-Wye this morning for news of Auntie Evie, our Mother's sister. Auntie has been in a nursing home for a couple of years after she kept falling down. Her memory left her some years ago, although the place it left her in was by all accounts a happy place. Two weeks ago Auntie had a stroke. We learnt that she died this morning at 10.30 UK local time, or 04.30 Nashville. Cousin Jean was with her when she flew away, as Cousin Jean was also with Nans, the Mother of Edie & Evelyn, when she flew away.
We grew up very close to Edie's sister's family just outside Ross, visiting each year when young for a week's holiday. Edith & Evelyn were exceptionally tight sisters & their easy dissolution into gurgles of laughter were as incomprehensible to an outsider as they were infectious.
Evelyn Harris, nee Green, was 88 years of age. Her Mother died at 95, her Sister at 77.