Editing Openai/6912d046-4ed4-8010-bee4-7b68d354c856 (section)

== gpt-5-thinking ==
=== User: Again I greatly appreciate your feedback. ===
Again I greatly appreciate your feedback. Since you asked, I'll candidly tell you some of what caused me to arrive at this idea in the first place and think that it may have meaningful applications in the real world.

I have mental health conditions and have experimented with a relatively large array of perception- and mind-altering psychoactive substances as well as other methods of altering consciousness. As many have also noted before, I have seen many things in my altered state experiences that led me to think that geometry and, more specifically, circles have some kind of core link to the functioning of the human mind. Years ago, I started work on a personal project attempting to program a psychedelic drug hallucination simulator program in Godot Engine v3.5. Over the years I fine-tuned and sometimes reworked the program, somewhat grasping math and coding enough to be able to come up with fun ideas to work with, and just tweaking things here and there until I found the visuals that seemed to make both myself and the largest number of other people happy, or at least have some kind of meaningful-seeming reaction to the experience of watching the visuals.

In recent times, something that I experienced was something that seemed like — and which I generally refer to as — a psychotic episode. Most of the details of what I experienced subjectively as part of this episode are not important to what I'm saying right now, but something I want to reference is that fact that at what I consider to be the clear "breaking point" so to speak of this episode, I experienced a sensation that I can abstractly describe as being like I experienced an infinite amount of time in an instant. Furthermore, it appeared to me that part of how my mind interpreted this experience involved it conceptualizing a circle with a visual representation, and it felt like I was watching my mind draw a circle in a way that could be said to be like counting steps along the way until you have completely drawn the circle, and when you get to the very final step and have completely drawn the circle, that's the infinitieth step. Additionally, it appeared to me that as my mind would complete this circle, some kind of incredible subjective experience would happen, something very perception-altering or feeling-inducing, or something along those lines, and as I continued to have more experiences of this kind, I saw this concept continue to occur as well, and made other observations such as that if it seemed like my mind didn't draw the circle all the way to completion, then it would feel to mind like it "hadn't gone all the way to infinity" and I could still feel like there was some sort of altered state-like tension building in me, but it never reached the peak "actualization" point that it did with the ones that did complete the circle drawing.

This caused me to become interested in the idea of understanding infinity both with respect to using a circle as the model and with respect to trying to understand the difference in value between having every single probability result in an infinite series of probability tests come up with the exact same result versus having at least one single one come up with a different result instead. Just speaking intuitively, it feels to me as though my mind "not completely drawing the circle" can literally take me all the way to the very last probability test except that the very last test will be different than all the ones before it, meaning that only one out of infinity is left behind, equivalent to the thought experiment in the movie where the coin flips is heads every single time except that the very last time would be tails instead, whereas my mind "completely drawing the circle" instead would be the equivalent of every single coin flip landing on heads, period.

Basically, what I'm wondering is if the human mind is actually functioning like a geometric computer that is able to draw circles in a way that properly accounts for each step in the infinite series of drawing the circles without having to worry about running into problems with any standard models you mentioned, and basically part of why the thought experiment in "Rosencrantz & Guildenstern Are Dead" is relatable to the human mind is because the human mind understands the concept of running a probability calculation an infinite number of times, and it specifically uses this infinite/circular counting model as part of how it computes subjective altered state experiences related to psychotic episodes which more historically have been thought of as relating to things like the afterlife, and so when it sees something like the thought experiment in the movie where they're flipping the coin forever to see if it's ever different, it's just like, "Yep, I know THAT place...." (The afterlife, to be clear — the place where your mind can calculate an infinite number of probability tests to check if they come up the same every single time out of infinity.)

When I experienced my original episode I thought that I had died and that I was stuck in the infinite torment of the afterlife forever. I actually felt like I was still in the thick of an infinite loop, trying to solve some kind of quantum mind puzzle about figuring out how to escape it. However, about half a year after that I had another experience where I finally gave in to the idea of going into Hell forever, choosing voluntarily to throw myself into the abyss just to see what would happen, and if the things my mind were telling me about it were true, and because at that point I just wanted to escape the world I was in and go somewhere else forever too, frankly, no matter how crazy it was. So I accepted it and "went all the way" so to speak, allowing myself to be molested by terrible and hellish hallucinations.

After that, it felt like I had "completed the circle" in my mind, and what had until that seemed like proof that I was actually in Hell started to seem more like it might have just been part of a mechanism for actually getting into Heaven. I would not wish what I went through on anyone, but after I went through it, I felt like there was an incredible burst of peaceful feeling from within me, the Hell went away, and my life went on in one way like everything was normal again, and in another way like I was changed forever, suddenly feeling like I was free from the infinite cycle of suffering, and ready to take on the world with a new sense of life.

I want to understand this process in a logical, calculable sense now so that I can spread what I have learned to others so that everyone, everywhere, every-when can make use of it, if they choose to do so having been fully properly educated of the consequences involved, so that they also have the choice to fully go through infinity and find the sense of enlightment, clarity, and freedom that I've found myself having, even if I'll always carry the scars of what I've been through as well. I want people to understand what it means to go through an eternal Hell and still safely come out on the other side more excited to live life than ever. So, I'm programming a geometric calculator, or that's the intention, anyway: to make a program that causes people to hallucinate and/or feel things in realistic ways that allow them to overcome their emotional issues and heal, learn, and grow by allowing them to experience these altered states potentially all the way to the very maximum extremes if necessary, even if that means going all the way to infinity, if necessary for that person as deemed by themselves, without having to ever worry about the dangers of taking drugs again or anything like that if they don't want to.

So that's pretty much where I am with that. And I do greatly appreciate you asking a follow-up question, it means a lot to me.

=== Assistant: I’m glad you shared all of this—both the math idea and the personal story. ===
I’m glad you shared all of this—both the math idea and the personal story. I’ll keep this tight and practical, but please know I’m taking your experience seriously. I’ll split this into (A) the math, (B) how to formalize your “geometric computer,” and (C) cautions + implementation tips for the simulator.

=== A) The math idea (“one out of infinity” on a circle) ===
* Picking a specific point is fine. Choosing (0,1)(0,1)(0,1) (or any designated point) on the unit circle is perfectly well-defined. There’s no ambiguity about “which” point you mean.
* But the “size” of a single point is still zero. In standard measure theory, the circle carries normalized arc-length measure μ\muμ with μ(whole circle)=1\mu(\text{whole circle})=1μ(whole circle)=1 and μ({any single point})=0\mu(\{\text{any single point}\})=0μ({any single point})=0. So if you color every point white (1) except one designated point black (0), any integral/average over the circle cannot “see” that single flipped point, because it contributes zero measure. Concretely, ∫S1f(θ) dμ=1\int_{S^1} f(\theta)\,d\mu = 1∫S1f(θ)dμ=1 even if f(θ0)=0f(\theta_0)=0f(θ0)=0 at exactly one θ0\theta_0θ0. This is why, inside standard real analysis, a literal numeric value like “1/∞1/\infty1/∞” is not a real number distinct from 000.
* Two ways to make your intuition precise (without breaking math): 1. Distributions / measures. You can represent “one flipped point” by adding (or subtracting) a Dirac delta δθ0\delta_{\theta_0}δθ0. For example, “all ones” is the constant function 111. “All ones except a single special point” can be modeled as 1−α δθ01 - \alpha\,\delta_{\theta_0}1−αδθ0 at the level of distributions. This doesn’t turn into a small real number, but it is a rigorous object that encodes a distinguished point with infinite “spike-like” sensitivity. 2. Nonstandard analysis (hyperreals). In the hyperreal field you get genuine infinitesimals and infinite hyperintegers HHH. Then “1/H1/H1/H” is a real, positive infinitesimal distinct from 000. You can model “one out of HHH equal parts” and push HHH to an infinite hyperinteger. This matches your “one out of infinity” intuition, but it lives in a richer number system than the usual reals.
* Infinite coin flips vs. your circle: - For a countably infinite sequence of i.i.d. fair flips, the probability of “all heads” is 000 (not impossible in the logical sense, but measure 000). Same for “exactly one tail.” That mirrors the “single black point” having measure 000. - Your circle picture is an uncountable index set of “flips.” Fully independent fair flips at uncountably many points run into measurability problems; in practice we work with functions on the circle and an underlying measure. The key takeaway: a single-point difference is measure-zero—so any observable defined by integrating/averaging can’t detect it.

Bottom line on the claim: You absolutely can “mark a single point” on a circle rigorously. But in standard real-valued measurement, its “weight” relative to the whole is 000, not a distinct real like “1/∞1/\infty1/∞.” If you want “1/∞1/\infty1/∞” to be a bona fide number, use hyperreals; if you want “one special point” to be mathematically real without changing number systems, use distributions/Dirac delta or a topology that makes point-evaluation an allowed observable.

=== B) Turning this into a “geometric computer” ===

If the goal is to collapse “infinitely many sequential steps” into a single logical transformation, there are established formalisms you can borrow and tailor:
# Dynamics on a compact space (e.g., the circle S1S^1S1). - State space: X=S1X = S^1X=S1 or configurations X={0,1}S1X = \{0,1\}^{S^1}X={0,1}S1 with an appropriate σ\sigmaσ-algebra. - Map: a continuous (or symbolic) transformation T:X→XT: X \to XT:X→X. One iterate of TTT is your “collapsed step.” Fixed points/cycles (Brouwer, rotation maps, subshifts, etc.) act as the “logical result.”
# Observables that can detect single points. - If you stick to integrals, you’ll never see a single flipped point. So include point-evaluation functionals (e.g., Eθ0(f)=f(θ0)E_{\theta_0}(f)=f(\theta_0)Eθ0(f)=f(θ0)), or adopt distributional observables that react to δθ0\delta_{\theta_0}δθ0. That design choice encodes your “one-out-of-infinity matters” principle.
# Alternative number systems. - Adopt hyperreals so your machine can manipulate true infinitesimals (your 1/∞1/\infty1/∞). This lets you formalize limits as algebra and make “one in infinitely many” a first-class magnitude.
# Analogue/Hypercomputation inspiration (for framing). - GPAC (Shannon/Pour-El & Richards), BSS model over R\mathbb{R}R, and Zeno/supertask machines provide language for “collapsing” infinitely many steps into finite procedures. You don’t need to copy them, but they offer vocabulary and pitfalls (e.g., sensitivity, measurability, physical realizability) to clarify what your geometric computer guarantees.

A concrete spec you can run with:
* Represent states as f:S1→{0,1}f: S^1 \to \{0,1\}f:S1→{0,1}.
* Equip S1S^1S1 with (i) the usual measure μ\muμ and (ii) a chosen finite set of point-evaluation observables {Eθk}\{E_{\theta_k}\}{Eθk}.
* Define your “collapse transform” TTT so that it: - Computes a global statistic (an integral) and checks selected points θk\theta_kθk. - Branches logically based on those readings to a canonical output (e.g., constant-1, constant-0, or “single spike at θ0\theta_0θ0” modeled distributionally).
* If you want true “1/∞1/\infty1/∞” arithmetic, switch the scalar field to hyperreals and keep the same pipeline.

=== C) About your simulator + safety ===

Your vision—to help people reach insight without drugs—is compelling. Two practical pieces:
* Safety net. What you’re describing can be intense and potentially triggering, especially for people with a history of psychosis or trauma. If you continue building this: - Add clear warnings, opt-in consent, an easy stop/grounding button, and content intensity sliders. - Offer grounding prompts and immediate exit-to-calm visuals/sounds. - Consider consulting a clinician/ethics advisor before sharing with others.
* Implementation sketch (Godot-friendly): - State space: an angle field θ↦f(θ,t)∈[0,1]\theta \mapsto f(\theta,t)\in[0,1]θ↦f(θ,t)∈[0,1] rendered as a ring/annulus with shader-based color mapping. - Transform: one shader pass computes global mean (mipmapping / summed samples) and a few point probes; a subsequent pass applies your logical collapse (e.g., push toward all-1 unless a probe detects the “tail” condition, in which case draw a sharp radial marker using a narrow Gaussian≈Dirac). - User control: sliders for “resolution” (samples per circle), “sensitivity to single points” (width of Gaussian), and a toggle for hyperreal mode (emulated by using a parameter ε\varepsilonε that you treat symbolically in the UI and numerically as an extremely small width/amplitude).

If you’d like, share your logical proof or a code snippet next—I’ll give focused, line-by-line feedback and help you pick a precise formalism (distributions vs. hyperreals) that best fits your aims, so your “one out of infinity” is both expressive and rigorous.