De Trinitate Sancta ut Varietate Topologica Trium Dimensionum: On the Holy Trinity as a 3-Dimensional Topological Manifold

Pope Leo XIV¹, and B. McGraw²

¹ Da Pope 

² Topological School of Mathematical Divinity, Cranberry-Lemon University, Pittsburgh, PA, USA

Abstract

Before I begin dis paper let us pray. Gahd is good. Gahd is great. May we thank Perelman’s proof which is great. Amen, Go Bears. Aight, so I’m the pope now and we’re gonna finally define da trinity in topological space. Gahd is a lot of things; He’s love, He’s goodness, He’s justice, He’s outa all dese tings da same as His Son Jesus and I have ta add da Holy Spirit. How can dey be one ting, and tree tings and all da same? Let me tell ya how. It’s called a 3-Manifold, which get dis, is a clever way ta define the whole pahsitive or flat curved space-time universe we live in. You, me, your aunt, we’re all 2-Manifold 2-spheres, except for my cousin Dorothy, she’s got too many piercings. Like, dat girls gotta be a quadruple Torus, probably too many holes for that even. We’ll get into dat later. But Gahd, Son, Holy Spirit? They’re a 3-Manifold, something called a 3-sphere. Woah right? Yeah. Well, we gonna formalize it right here and map der topological Space across scripture into a nice neat 3-sphere using dem Ricci Flows. 

Keywords:  Trinity, Topology, 3-Manifold, Theology, Math Pope, 3-Sphere, Space-Time-Gahd-Curviture, Ricci  Flow

1. Introduction

Now der hasn’t been a math pope since Pope Sylvester II over a tousand years ago. Right? He brought over dem Arabic numerals and dat abacus. My grahndma used ta have one of dems. A lots happened in da last tousand years since our ole’ Sylvester boy der. Now I’m here, I spent a lot of sleepless nights in Villanova studyin Math of all tings, with da hope I’d be Pope and let me tell YOU, we’re gettin formal definitions of all dems theologies. I’m goin down dat Nicene Creed I gotta start with the top dog der, da big one, da Trinity. 

2. Background

Now first I gada stop you guys. No, we can’t define Gahd completely. I don’t even tink you can approximate da Guy. He’s too big, like a big ole lineman from da ‘85 team, you’d run outa numbers. I’m talkin wide shoulders! Now He’s infinite, all tree parts of ‘Em. 

2.1 Da Trinity

Okay, so if ya gotta understand da Trinity, ya gotta understand it’s tree tings. Da Father. Da Son. Da Holy Spirit. I put a handy diagram down der in Figure 1 for ya. Now der’s a lot of ways ta get it wrong [1]. 

Gahd isn’t just revealin Himself in different ways. That’s Modalism [2]. Now some people might tink, ‘Hey ya Jabroni, what if we had da Father, then Da Son came, then we got Da Holy Spirit’, right ya Jagoff, that’s just Arianism [3]! All tree parts are eternal. Now don’t GET ME STARTED on people who tink They could be separate because that’s Tritheism and let me tell ya as head of da Catholics, that ain’t it either [4]. Now some people believe They just parts of one Gahd. Not that either, that’s called Partialism and He aint that either [5]. It’s confusing, and that’s just what we gaht. I’d say Gahd’s a They/Them as da kids are sayin. 

2.2 Gahd’s Omniscience

When it comes to da Trinity and all da great Omni’s there’s no one dat quite says it like my boy John. 

“1 In da beginnin’ was da Woird, and da Woird was wit Gahd, and da Woird was Gahd, okay? 2 He was wit Gahd right from da get-go — like, right dere at da start? 3 Through him, all tings got made — every last ting. Without him? Fuhgeddaboudit — nuttin’ got made that even looks made. 4 In him was life, and lemme tell ya, dat life? It was da light of all mankind, even them guys from Green Bay. 5 Da light shines in da darkness, and guess what? Da darkness — yeah, it tried, but it ain’t never overcome it. Like da ’85 Bears defense — impenetrable!” –John 1:1-5 NCV (New Chicago Version).

Now, especially when I’m talkin with an Arian, ain’t no way da Son was ever not der or not here. Each part was der and if you’re one of dose Jabroni’s dat tink da Spirit wudn’t be here until da night of da Pentacost, let me tell you: 

“Da earth? It was all messed up — like, no shape, no nuttin’. Just a big ol’ void. Darkness was sittin’ over da face o’ da deep, like da lake in February. And da Spirit o’ Gahd? He was hoverin’ — floatin’ real quiet — over da waters, just takin’ it all in. Then Gahd goes, ‘Let der be light,’ and BAM — der was light. Bright as Halas Hall on media day.”

–Genisis 1:2-3 NCV.

Yeah, thought so, da Spirit is everywhere and always has been. 

2.3 Poincare’s Conjecture

So, here’s da deal. We live in ℝ3 just like er’ybahdy else. Right? Now we’re not too sure which spaces Gahd lives in, but we figure, He’s at least interacting in ℝ3 wit us. Now Gahd, we don’t know if he’s a 3-sphere but if He’s da whole universe like yours truly believes [6], he might as well be a 3-Sphere or S3. Here’s da real argument why Gahd’s probably a S3. According to dis French guy Poincaré, every 3d topological manifold which is closed, connected, and has trivial fundamental group is homeomorphic ta da S3. Boom. 

It gets better; our boy Richard Hamilton built a program using Ricci flow’s ta solve the conjecture. Next Perelman solved it 20-some-odd years ago [7] and touchdown, we got the framework to map the Trinity! See der in figure 2, each S3 gets connected lika a bunch of Polish Sausages, each connected at the singularities where ya tie off each sausage section. 

3. Formalization

Assume the Gahd-Manifold (G-M) is a vast, compact, connected, and orientable 3-manifold such that 

Despite what some non-canon scriptures may suggest, it is equipped with a time-dependent Riemannian metric g(t) evolving under Ricci flow:

As da Ricci flow evolves over our human understanding of Gahd, so does our geometric understanding. Now it often produces singularities as Gahd appears ta man as say, a burning bush, or when he got a little spry and wrestled Jacob. We in the pope’n business call those a Theophany. We suspect dose Theophany-Singularities (TS) occur all da freakin time if we got da Holy Spirit speakin to you guys or ta me through my papal meditations in scripture. I tell ya, I can’t get away. 

We let t go to infinity and use that Hamilton style tensor to let dem singularities play out. I could never get away with a singularity as a cardinal but now I can go ahead and say that all the mysticism and Theophanies boil down to singularities in da Ricci flow. Whether da TS is confirmed or allegedly like da one my distant baby cousin who swore up and down dere was a light guiding him away from traffic on da Dan Ryan. We love ya bro, but I doubt that was a TS. Yeah, they can happen, but don’t tell me that the underlying God head isn’t topologically invariant in all dimensions. 

4. Crucifixion Analysis

I don’t have ta be da freakin math pope ta tell ya dat the biggest Theophany Singularity occurred during da Crucifixion. During his great love, da Son was pierced in both His wrists and both of His feet as He was nailed to da cross. During all of known spiritual history, at least da history I can release from da papal archives without being murdered, Ghad has always been a spheroid. But at da point of da Crucifixion, Ghad, but specifically da Son, stopped being a spherical manifold and became a quadruple Torus. 

At this point, Jesus was now defined by the Cross product of four different spheres as shown in da equation below. At this point He was not homeomorphic to Gahd da Father, and dat’s not a place ya want ta be.  

     Nota Bona: from now on all cross products will be denoted by and not X, thus sayeth the math pope or you don’t get into heaven [8].

5. Discussion

So, dere’s a lot ta unpack here. We haven’t updated da Trinity in centuries. You guys probably have a lot of questions, but I think dese are some of da most significant ones. 

5.1 Is Gahd Positively or Negatively Curved or Flat 

Now I tell ya, I was just at the Sistine Chapel for my Conclave and Gahd was pretty curvy. Dat Michelangelo painted with da best o dem. I’d love to leave it der an call it a day but I just can’t. Now we know if Gahd is a 3-sphere like we propose He’s positively curved. But do we really know dat? According to da plank study [9], da observable universe is flat n’ not curvy within some margin of error. Now if da universe is flat like a pancake, how’s Gahd supposed ta be bigger dan it? 

“7 So like, where da heck am I s’posed ta go ta get away from yer Spirit, huh? Where can I even try ta scram from yer presence? Nowhere, dat’s where.

8 If I shoot up to da heavens — BOOM — you’re already dere. If I go way down, like real deep, make a bed in da basement o’ existence? Yup — dere too.

9 If I catch da sunrise express — wings o’ da dawn — or I move way out East, like, other side o’ da lake or somethin’…

10 Even dere, yer hand’s gonna be guidin’ me, and yer right hand? It’s holdin’ on tight, like Ma holdin’ her purse on the Red Line.”

–Psalms 139:7-10 NCV

Psalms 139:7-10 suggests dat Gahd is positively curved, if ya get away towards one of His edges, you’re right back comin around. He’s like a giant potato. This still begs the question whether da universe is a small part of Gahd’s toasty approximately flat center or if dose scientists out dere are measurin the universe wrong and thinkin it’s flat. I don’t know.

5.2 The Significance of the Jesus Quadruple Torus Form

Our symbol is da cross for a reason you guys. When Jesus was up der sacrificing himself for you guys’s sins at one point darkness came around in da middle o’ da afternoon like a big ole snowstorm and He shouted out “Eli, Eli lema sabachthani?” (which basically means “My Gahd, My Gahd? Where da BeJesus are ya?”) according to Matthew 27:46. Now some say Jesus was calling out to Elijah, at da time, but now we know it was Gahd! You know because of the resurrection and all our other interpretations of scripture. Now how can Jesus be calling out to Gahd if dey are already together? I tink cause he’s up dere on da cross with dose holes in his wrists and feet that He wasn’t homomorphically the same as Gahd the Father or Da Holy Spirit and well maybe he was separated dere, topologically ya know. 

5.3 Is Gahd a 4-Manifold?

I’ve been getting this question all da time since I started writing this paper. How do we know Gahd is 3-Manifold and not a 4 or higher manifold? Well, I don’t think we can know. I also think, a 3-Manifold basically has us covered unless we want to get into more dimensions than we have at da present. Maybe Gahd’s in a totally different dimension. We don’t know but I think at a minimum, he’s a 3-Manifold and that math holds up. Maybe we should get that Hamilton tensor runnin up on some of our Vatican simulators and try out a S4. Something like dat’s gotta be a ting. Even then though, I tink if we start talkin about S4, someone’s gonna wana know why don’t we work on a S5 Gahd, and ten a S6 Gahd and before ya know it we just wasted my entire papacy on one problem and spent everyone’s tithe money on AWS charges for da processing! Ya don’t just put ketchup on your dog, it’s already perfect. When you’re done, ya done. Official decree, da next pope can figure dat out. Until then, it is heresy [10]!

6. Conclusion

I tink with this Ricci flow and da 3-manifold nature of Gahd, I tink we got all his bases covered in S3 in any way we encounter a Theophany-Singularity. Yeah, I will say we’ve got some more questions about Gahd’s curviness or what da heck happened on da cross, or if we need any more manifolds. But largely, I think we got this one pretty covered. Besides, I’m da math pope now and all I say is now doctrine [11]. 

References

1. Pope Leo XIV 2025 Diagramma Trinitatis Omnium Priorum Destruens: The Trinity Diagran That Destroys All Others :: Official Papal Decree 
2. Pope Leo XIV 2025 Modi Sunt Falsi: Deus Est Aeternus: Modes Are Lies: God Is Eternal :: Official Papal Decree Pope Leo XIV 2025 New Trinity Diagram :: Official Papal Decree 
3. Pope Leo XIV 2025 Filius Non Est Startup Divinitatis: The Son Is Not a Diine Startup :: Official Papal Decree 
4. Pope Leo XIV 2025 Unum substantia, Tes Non Corporati socii: One substance, Not Three Devine Bros :: Official Papal Decree 
5. Pope Leo XIV 2025 Trinitas: Non Est Pizza, Noli Dividere: The Trinity Is Not Pizza, Do Not Divide It :: Official Papal Decree 
6. Pope Leo XIV 2025 Perelman Probavit, Papa Confirmavit: Pelerem Proved It, the Pope Confirmed It :: Official Papal Decree 
7. Terance Tao 2006 Perelman’s Proof of the Poincaré Conjecture: A Nonlinear PDE Perspective
8. Pope Leo XIV 2025 Est Novum ×:Fiat Voluntas Papalis: is the New ×: Let the Pope’s Will Be Done :: Official Papal Decree 
9. Plank Collaboration: N. Aghanim Et al. 2018 Plank 2018 results. VI. Cosmological parameters
10. Pope Leo XIV 2025 Manifoldus Dimensiones Ultra Tertiam Sunt Damnandae: Manifold Dimensions Beyond the Third are to be Condemned :: Official Papal Decree 
11. Pope Leo XIV 2025 Papa Mathematicus, Vox Infinitatis: The Math Pope: The Voice of Infinity :: Official Papal Decree

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