A hermetic breakthrough is a Major Virtue, and major breakthrough is a Minor Virtue.
If I am recalling correctly, a major breakthrough can be transmitted through a book requiring a season of reading, or taught to an apprentice as part of their 10 points of virtues. Major virtues can be can also be taught to Apprentices (as part of their 10 points of virtues) OR can be taught through Mystery Cult Initiation.
If you wish for the for a 'seamless' integration, then you need to perform it over again. At which point, another magus will understand the breakthrough if they learn a spell
The first breakthrough allows: 0) Make what the researcher was looking for possible and reliable, 1) the researcher to use his findings (and if it is a skill, with a number of XP = to breakthrough point - but I don't remember where I read that), 2) to teach it through initiation or 3) allow people to learn it by studying his work (spell, enchantement, etc...), building up BP in a much easier way since no roll is involved.
The integration makes the research "obvious" for anybody with magit theory: reading a tractatus on it, a spell or spending a season with somebody knowledgeable on the topic is enough to be able to use it.
In short, the integration is the most altruistic research a mage can make since it allows every mage to benefit from the research at no cost but just a season of work. If somebody was proceeding with the integration of Silent casting or Flawless magic, every mage would be able to use it without the need of the virtue - more accurately it would become a 0 cost virtue.
The rules for this are extremely unclear but I suggest taking a cue from the "Multiple Laboratory Texts" section on HoH:TL page 29. Magi can accumulate breakthrough points from reading the original developer's lab texts and acquire the virtue when they master all of them.
If a maga has achieved an HoH:TL p.29 Major Breakthrough, she can teach it either personally, or by writing a single Tractatus about it.
If that Major Breakthrough results in a Virtue, as it usually does, this Virtue is a Hermetic one (see HoH:TL p.29 Breakthroughs in Play).
If it is an Ability, by ArM5 p.18 Abilities it can be learned by a Gifted individual. As it is Hermetic like e. g. Parma Magica, an Hermetic student learns it by standard ArM5 p.164f Teaching from the inventor, or by reading the Tractatus she wrote. There is no need to subtract a student's Art scores from the ArM5 p.163 Advancement Total for it.
If it has no associated Ability, by HoH:TL p.29 Breakthroughs in Play it can still be learned just like such an Ability: 5 xp gained from the instruction or Tractatus should then be sufficient - though this is nowhere stated explicitly. The reason for this is, that 5 xp are sufficient to learn the imaginary Ability to level 1.
For none of the above the maga who made the Major Breakthrough needs to achieve a second Major Breakthrough.
This doesn't appear to be the problem of the OP, though.
HoH:TL p.29f Hermetic Integration aims at changing Magic Theory, so that everybody learning - typically from a Tractatus on Magic Theory - that change gets the Virtue and/or Ability of the Major Breakthrough for free. This is not necessary
Note that these rules have mutated slightly over time.
Technically you can teach Major and Hermetic Integrations once they are officially virtues. (ie, after you've gained all the relevant Insight.) However, the most recent rules for teaching virtues (as laid out in Apprentices) makes it just barely possible to teach a completely pristine student a single minor virtue, and maybe one major virtue, if you don't mind saddling said student with a major flaw. Once they already have any major or minor Supernatural virtues though, it's functionally impossible to teach them more. Note that this is using the Teaching ability. So really, this stage is just the halfway point for most Integrations or Discoveries.
In contrast, If you do it twice, you have essentially updated Hermetic Magic Theory. Once you've done this, your virtue is completely a part of MT, and you can think of yourself as a (slightly lesser) peer of Bonisagus. Huzzah! Any learning source from you on Magic Theory now contains this update; direct instruction, Training, or a book. Over time, it will filter out and be a part of the larger Corpus of Hermetic magic understanding. (Minor Discoveries/Insights, in contrast, are immediately at this level; no need to do it twice.)
Don't mix up the rules from Apprentices p. 40ff Teaching Hermetic Virtues for teaching an Hermetic Virtue you have to an apprentice of yours with the rules from HoH:TL p.29 Breakthroughs in Play to teach other magi the Major Breakthrough of yours. The situations are different, and Apprentices p.40ff neither subsumes no overrides HoH:TL p.29.
This is a reasonable interpretation of HoH:TL...but one that's never used in subsequent publications, at least to my knowledge. We see traditions, cults, and covenants with unique virtues but they're all taught through Mystery initiation. I'm not aware of a single example of learning a Virtue through a 5xp Tractatus or a season of teaching.
I think this is true and for the better. Breakthrough virtues seem analogous to Mystery virtues, both representing secret bits of knowledge beyond the general understanding of Hermetic Magi. They can be taught via initiation, using the Apprentices rules, or by following the breakthrough lab texts to recreate the discovery. This makes full Hermetic Integration actually worth the effort, at least in some cases. Otherwise why bother, if a single season studying a tractatus passes on the information?
Note, that such an example would have to be about a magus teaching a Virtue not resulting in an Ability, and gained from a self-achieved Breakthrough. How many of such instances are there in the published material?
Why do you specify "self-achieved Breakthrough"? I don't see any suggestion that the discoverer can teach in ways subsequent possessors of the Virtue can.
It's true that Virtues are never identified as being the results of Breakthroughs, but it's reasonable to assume that many, perhaps most, of the Mysteries and other cult and House secrets result from either original research (HoH:TL) or Integration (AM). The two systems are compatible for work on the same discoveries, so one should assume their teaching methods are also identical. Incidentally, the only method mentioned in AM of teaching discovered Virtues is Mystery Initiation.
The only other options of the origins of these Virtues are random Twilight events or gift from supernatural beings. Neither of these seems appropriate for the majority of cases.
What exactly do you wish to say here? "suggestion that the discoverer can teach in ways subsequent possessors of the Virtue can"? Really? Yes, she can!
The tricky point goes the other way round: Can anybody teach a Virtue A by the specific methods from HoH:TL p.29 Breakthroughs in Play, that a maga achieved through a Breakthrough? Or can only the maga who has achieved it do so? Reading the full HoH:TL p.29 Breakthroughs in Play carefully and in context you find, that it only treats, how the discoverer can communicate or teach his Breakthrough, and not, how somebody who in some other way has gained Virtue A can do so. The chapter often refers to standard teaching methods everybody can use, but where it exceeds these it does so limited by its context.
There are other common ways. Like e. g. GotF p.35ff The Paths Through the Forest and TMRE p.18ff Experimental Scripts. HoH:MC Inner House Mysteries are also best mapped to such research within Mystery Cults.
The quote you give gets part of its meaning by the chapter HoH:TL p.29 Breakthroughs in Play. I fear I need to walk through it here.
So if you have achieved a Breakthrough, how you can teach it to others depends on its kind. The start of the chapter clearly names the inventor of the Breakthrough as the teacher or author: this needs not be repeated in all the following paragraphs.
This is the basic construction of this chapter, which then branches into the different paragraphs:
If it is a Minor Breakthrough, you can communicate it with a relevant lab text.
If it is a Major Breakthrough, you teach it via standard ArM5 p.163ff as a Supernatural Ability, keeping in mind ArM5 p.18 Abilities, which allows Gifted persons to learn new Supernatural Abilities. As this is standard ArM5, others knowing that Supernatural Ability can do so as well. But there is an exception here for Virtues that have no teachable Abilities associated: these you can teach after your Major Breakthrough nevertheless, though this is not standard ArM5 teaching of Supernatural Abilities.
If it is a Hermetic Breakthrough, you teach it either as an Arcane Ability, or like a Major Breakthrough.
AFAICS most Mystery Virtues from TMRE would be unknown if there had not been experimental scripts first. HoH:MC has some examples, how House Mystery Virtues became available.
Obviously "You" (the Discoverer) need to be the first one to put the new effect into a Lab Text. There's no suggestion that further transmission is in any way limited. As you quote,
Ancient Magic pg. 9 elaborates
The paragraphs on Major Breakthroughs are written in the passive voice ("Major Breakthroughs are more involved and must be taught to other magi", "they may still be taught in this manner") and there is no indication as to who is doing the teaching. Obviously the Discoverer must be the first teacher and obviously one Magus teaches the Breakthrough to an "other". But nothing suggests deviation from the normal learning and teaching rules, which make the process freely transitive. I don't see any basis for the supposition that a secondary possessor of the Virtue, taught by the Discoverer, is in any way more limited in the ability to pass the knowledge further along. That would be a major break with AM5 precedent and would almost certainly be stated explicitly. The mention of standard Teaching and Books rules suggests the opposite.
Maybe. I find the idea of inventing a desired ability by creating scripts out of thin air to be inconsistent with the spirit of the Ars rules, which center Hermetic Magic on labwork and experimentation, supplemented by the seeking out of secret lore. Opinions vary on this of course. I prefer to see Mystery Initiations as a method of passing on secret lore, or of obtaining it through found scripts, suitably modified by experimentation if needed.
In practical terms, it's nearly impossible to invent a new Major Virtue through experimental scripts, so there's that too.
Note, that the Apprentices chapter about Teaching Hermetic Virtues (p.40ff) is part of the chapter Hermetic Apprenticeship (p.32ff). So understanding Matt Ryan right about it is again just reading in context.
As I said above, who is doing it comes from the context in the chapter. Understanding it is just elementary reading in context, nothing else.
I don't see the Abilities and the experimental scripts coming "from thin air" either. Examples like the sample initiation scripts from the HoH:MC p.92ff Merinita Mysteries show, how "secret lore" was cast into scripts. P.106 Sample True Merinitae Initiation: Wilding is told to come from the galli sect quite directly. But p.96 Sample Line of Quendalon Initiation: Animae Magic appears to come from Quendalon's experiences in Faerie, and subsequent experimentation. And for Criamon magi we have
Storyguides are explicitly enabled to review and moderate such experimentation along the lines of TMRE p.18 Experimental Scripts closely, to benefit from it for their saga. See especially:
To achieve a useful script for a Major Virtue via experimentation, you first need a verrry solid Lore for your Mystery Cult. And then you need your troupe's and storyguide's cooperation.
When teaching a Major Breakthrough, HoH:TL says that it follows the rules for learning a Supernatural Ability, except that you don't deduct your Arts from Source Quality.
That seems to mean that existing Supernatural abilities are deducted from the Source Quality, which seems pretty harsh.
Am I getting that right?
I guess that an option is to lose the Supernatural Ability as part of an initiation or something like that, but it doesn't sound right.