Despite Gunbuster(1988) factoring into my theory I'll be focusing more on Diebuster (2004) and Tengen Toppa Gurren Lagann (2007), since Diebuster introduced heaps of new elements (such as Topless energy and the Buster Legion) into the Buster canon continuity, while the original Gunbuster's premise and 'science' was relatively simple--a robot built by humans to destroy a massive alien threat. Diebuster expanded on and overhauled Gunbuster's universe.
In Diebuster there are humans called Topless, who are in a sense 'without limit,' their spiritual powers (stemming from a pure and innocent self-confidence found more commonly in children) allowing them to produce limitless amounts of energy which in turn power their giant robots, the Buster Machines. This is almost identical to the Spiral Power of TTGL, where humanity (and any sentient beings capable of evolution) produce a magical Spiral Power that allows them to, you got it, power their giant robots, the Lagann.
In Diebuster and Gunbuster, the 'good-guy' robots have all been built either directly or indirectly, by humans, and have been shown to be extremely durable. Buster Machine #7 lies in hibernation for so long a comet forms around itself, and has lived for at least thousands (if not tens of thousands, as speculated by one character) of years, and Buster Machine Dix-Neuf, the oldest Buster Machine still in service, has been around for at least 500. The Spiral King Genome built and then maintained his own personal Gurren Lagann (the Lazengann) over the course of his entire reign, during which he forced all of humanity underground and established a completely new social hierarchy based around Beastmen. While Gurren was a Beastmen mech and supposedly relatively new Lagann was found buried underground in a state of dormancy similar to Buster #7's. One significant aspect of this longevity is that the robots are able to live long past the confines of human memory. Buster Machine 7 was completely forgotten by humanity (the single digit Busters being the stuff of legend by the time the story takes place), and throughout TTGL enemies and allies alike remark on the curiousity of a tiny cranial robot existing.
In Diebuster the emerging Topless factions of humanity cause the ancient defense systems of ages past, designed to defend against space monsters with similar abilities, to go haywire and turn on humanity. In TTGL a distant collective of beings, the Anti-Spiral, discovered that Spiral power's very limitlessness threatened the fundamental structure of the universe, and thus vowed to destroy all beings who possessed it in order to prevent a catastrophic buildup of Spiral, the so-called Spiral Nemesis.
In TTGL humanity grew dependent on Spiral energy, drawing the Anti-Spiral to them. In all-out war the Anti-Spiral prevailed, utterly annihilating humanity, but the human Lordgenome saw the futility of battle and betrayed his allies, forcing humanity underground to hide their Spiral signature and populating Earth with artificially produced Spiral-devoid Beastmen. One VERY IMPORTANT thing to consider when looking at this event is that EVEN BEFORE Lordgenome's betrayal of humanity, quasi-humanoid robots designed to defend against alien threats already existed (as shown in several OVAs released by Gainax), these robots supposedly being powered by Spiral energy. This means that Spiral-powered robots were one of humanity's prevalent defense mechanisms before humanity's collapse at the hands of LG and the Anti-Spiral. Keep that in mind.
In Diebuster, Gunbuster, and TTGL, a fundamental feature of the series' robots is their ability to combine, almost always in pairs. In Gunbuster there were Buster Machines 1 & 2, which combined to form the titular Gunbuster, and in Gurren Lagann the lanky Gurren and the cranial Lagann mesh to create the titular Gurren Lagann. Also, in both Diebuster and TTGL, the final-final form of the hero robots is an amalgam of millions of individuals combined to form a massive human-like figure dwarfing galaxies (in the shape of Nono is Diebuster and the shape of Kamina in TTGL).
So here are the biggest parallels between Gunbuster, Diebuster, and TTGL: powerful combining robots built by humanity in order to combat alien menaces, capable of withstanding the test of time and powered by humanity's innate will to attain greater and greater heights. It's my belief that these machines are one and the same, all part of a chain of evolution and improvement throughout human history. Of course, in order to accept this theory two leaps in logic have to be made:
-1) that Spiral and Topless energy are one and the same. Pretty simple assumption to make. They both center around humanity (and in TTGL's case, evolution-capables)'s will to survive and grow stronger, both revolve around the idea of being without limitations (these limitations often being referenced as literally being vertical --> see 'Topless''s very name and TTGL's famous quote "This is the drill that will pierce the heavens!"). One could argue that this makes no sense because TTGL asserts that Spiral energy has existed in humanity since its conception, while Diebuster's Topless only emerged long after the conclusion of Gunbuster. However, you could easily say that the connection was only made as advances in science and understanding of Spiral energy became more profound. That is to say, Spiral/Topless (S/T) energy may not manifest externally until a certain threeshold of 'ultimate evolution' is reached (supported by the fact that the human beings in TTGL never transform into some otherworldly creatures, while the mole-pet does evolve... into a humanoid!) upon which S/T energy can be used and manifested in producing energy and mass (e.g. TTGL's titular Giga Drill, which literally is pulled out of nowhere. Part of the Anti-Spiral's fear of Spiral is that it allows wielders to break the fundamental laws of physics, tearing at the universe's foundations. Topless can achieve feats such as instantaneous warping, teleportation and throwing of planets, et cetera). Thus we can say that the progression of events is thus -
A) In gunbuster, humanity is, while APPROACHING the evolutionary threshold, not yet there.
B) By the point of Diebuster's story, humanity has BEGUN to break that threshold, resulting in small portions of humanity gaining Topless powers, but losing them as they age.
C) By the time of the Anti-Spiral War we can assume that a majority of humanity has broken this barrier. Humanity's capability for external S/T manifestation had become more advanced, with the robots involved in the final crusade angainst the Anti-Spiral are piloted by adult-voiced humans, and various adults (e.g. Kamina, and like half the misc. pilots involved in the final assault on Granzeboma) pilot Lagann as well, and it is noted several times that ALL humans are capable of producing Spiral.
Following up on the idea that S/T science is further along in TTGL than in DB, we can say that in DB's time Topless energy was not as well understood, while in TTGL we can see that it influences evolution. Thus one could say that the reason S/T energy does not manifest prior is due to said S/T being used up in the evolutionary process. This would not have been grasped until TTGL, with DB's universe treating Toplessness as an abnormal condition.
-2) humanity's expansion across the galaxy? By the end of Diebuster, the alien menace has been supposedly defeated, and settlements on Mars and Pluto already exist. So then, why is humanity back on Earth? This one's pretty simple--no matter how many advancements man made between Diebuster and TTGL, they were all erased in the blink of an eye by the relentless advance of the Anti-Spiral armies. After his coup, Lordgenome probably also took extra measures to ensure all of humanity was encapsulated within Earth to prevent any leakage of Spiral signature. Thus, the regression of humanity back into being Earthbound.
With these handled, now I can get onto the major connections (not the obvious superficial ones).
One: GUNS -> DRILLS. The original Gunbuster robot was all about guns. Its signature attacks involved splitting its limbs into tens of hundreds of energy cannons, embedding them into its enemy, and firing point-blank to achieve maximum destructive power. This technique we also see in Diebuster, with Buster #7 using it on the STAGE 3 Fluctuating Gravity Well monster in Episode 6... but with little to no effect.
^Here we can see that the evolution of the original alien menace from Gunbuster 1 has brought it to a point where the original techniques of gunbuster just won't cut it anymore. Even with the boundless Topless energy backing it up. But perhaps there's a more optimal usage of S/T ENERGY????
EXACTLY. It's FUCKING DRILLS. The ultimate combined form of Buster 7 and the Buster legion is adorned with two drill-shaped shoulder pauldrons, and the final, ultimate technique used to finish off the S3 FGW, the INAZUMA DOUBLE KICKU, forms a spiralling, drill-shape when piercing the enemy's hull.
Thus we can assume that the ultimate form of Topless energy is Spiral energy (perhaps indicative of some sort of spiral wave nature of the energy itself), and that the optimal form of usage of said energy is in literally a spiralling motion, best materialised in drills. Thus the next generation of S/T robots uses drills instead.. Certain iterations of the Gurren Lagann even have the same drill-shaped shoulder pauldrons as BN7!
Even the Spiral Nemesis, the deadly catastrophe brought about by overuse of Spiral predicted by the Anti-Spiral, has a presence in DB. The splitting of the Fluctuating Gravity Well, and in turn its host black hole, creates what one character calls a "naked singularity," something which "should not be possible in this universe." The power of Spiral is so great that it does what cannot, and should not be done, sometimes creating dangerous reality-warping situations. It's not hard to move from this occurrence to perhaps a predictably much more dangerous, impossible to contain event occurring as humanity's capacity for S/T grows.
So here is my new timeline, which integrates all three series into one big continuity:
1- Humanity evolves.
2- the intergalactic menace of the 'space monsters,' later called the 'fluctuating gravity wells,' emerges.
3- To combat the threat, humanity creates an Absolute Defense Line system designed to protect against the aliens.
4- The first test of the Absolute Defense Line. The experimental weapon Gunbuster, piloted by Noriko Takay and Kazuki Amano, repels the invaders.
5- Later on, Gunbuster once again departs, holding the supermassive ship the Black Hole Bomb, to go and destroy the entire home solar system of the aliens.
(here, the timeline splits due to space time dilation).
5a - time passes. The Buster machines continue to be produced. Topless humans begin to emerge, produced by humanity's approach to the 'S/T threshold.' Topless retain powers until a point of 'expiration,' at which they lose everything.
5b - in the far reaches of the solar system Gunbuster (the original Buster Machine) deploys the Black Hole Bomb and obliterates the alien's home system before beginning the long journey home.
6a - Buster Machines (starting at least with or after 2 and with or before 7) begin to be built solely to accommodate Topless, using their limitless energy to surpass new heights. However, we can see that some Busters were built with non-Topless in mind, as shown by Buster 9's secret cockpit, which appears to draw out innate, hidden S/T energy from humans even after expiration. These Buster Machines are sentient, and can communicate with Topless (and apparently non-Topless, but non-Topless cannot understand the robots). The Absolute Defense Line system turns on the Topless humans, whose evolution has progressed to the point where they, like the supposedly fully evolved 'space monsters,' are able to externally manifest S/T energy.
6b - same as 5b.
7a - Buster 7 and Buster 9 destroy the FGW Stage 3, causing a naked singularity due to their combined Spiral energy as well as the spiralling nature of their attack. This may possibly be one of the events that alert the Anti-Spiral, off in their distant corner of the universe, to the dangers of Spiral power.
7b - same as 5b and 6b.
8 - Buster Machines 1 and 2 return to Earth. The alien menace both at home (FGW S3) and abroad (home star system of the space monsters) now destroyed, humanity is free to develop its tech, etc.
8b - The Anti-Spiral purges itself of Spiral, becomes a hive mind and declares war on Spiral life-forms.
9 - humanity has grown further, to the point where Spiral is ubiquitous amongst Humans, regardless of age, et cetera. Robots powered by Spiral, supposedly sentient in a manner similar to the Busters, are still used by humanity to defend against attacks.
10 - The Anti-Spiral detect humanity's Spiral and attack. LordGenome betrays his race and drives humanity underground, hoping to hide the remainder of humanity's Spiral signatures and save them from the Anti-Spiral.
11- the events of TTGL. (One thing I forgot to note is that despite the extreme lengths of interstellar travel endured during the second half of TTGL, space time dilation does not take effect--e.g. the people on Earth can see Granzeboma and TTGL fighting from trillions of lightyears away. Perhaps S/T can also warp light, space and time subconsciously, for reasons unknown)
Sorry if this is all over the place, I've been writing this ofr like an hour lol. If you have any questions or objections, feel free to tell me itt or thru vm/pm/irc/etc, I'd love to talk to u abt it.